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\section{Introduction}
For the past two decades, additivity conjectures have been extensively
studied in quantum information theory e.g.
\cite{Bennett_EtAl, Pomeransky, AmosovEtAl, Osawa, Shor,
hayden_winter_2008}.
In this paper, we concentrate on the issue of additivity of classical
Holevo capacity of a quantum channel $\Phi$, denoted henceforth by
$C(\Phi)$.
The quantity $C(\Phi)$
is the number of classical bits of information per channel use that
can reliably be transmitted in the limit of infinitely many independent
uses of $\Phi$. Capacities of classical memoryless
channels are known to be additive, that is, the capacity of two
channels $\Phi$ and $\Psi$, used independently, is the sum of the
individual capacities. In other words,
$C(\Phi \otimes \Psi) = C(\Phi) + C(\Psi)$.
This additivity property leads to
a single letter characterization of the capacity of classical channels
viz. the capacity is nothing but the mutual information between the
input and channel output maximised over all possible input distributions
for one channel use
\cite{vajda_shannon_weaver_1950}.
For a long time, in analogy with the classical setting, it was generally
believed that the classical Holevo capacity of a quantum channel is
additive. In fact, this belief was proven to be true for several classes
of quantum channels e.g.
\cite{KingUnital, FujiwaraEtAl, KingDepolarising, ShorEntang,
KingEtAl}. Thus, it came as a major surprise to
the community
when Hastings, in a major breakthrough, showed that there are indeed
quantum channels with superadditive classical Holevo capacity
\cite{hastings_2009} i.e. there are quantum channels
$\Phi$, $\Psi$ such that $C(\Phi \otimes \Psi) > C(\Phi) + C(\Psi)$.
Hastings' proof proceeds by showing that a
Haar random unitary leads to such channels with high probability,
in the sense that the unitary, when viewed suitably, is the
Stinespring dilation of a quantum channel with superadditive classical
Holevo capacity.
The drawback of using Haar random unitaries is that they are inefficient
to implement. In fact, it takes at least
$\Omega(n^2 \log (1/\epsilon))$
random bits in order to pick an $n \times n$ Haar random unitary to
within a precision of $\epsilon$ in the $\ell_2$-distance
\cite{Vershynin}. Hence, it is of considerable interest to find an explicit
efficiently implementable unitary that gives rise to a quantum channel
with superadditive classical Holevo capacity.
In this paper, we take the first
step in this direction. We show that with high probability
a uniformly random $n \times n$ unitary from an
approximate $n^{2/3}$-design
leads to a quantum channel with
superadditive classical Holevo capacity. Though no efficient algorithms for
implementing approximate $n^{2/3}$-designs are known, nevertheless,
it is known that a uniformly random unitary from an exact
$n^{2/3}$-design can be sampled using
only $O(n^{2/3} \log n)$ random bits \cite[Theorem 3.3]{Kuperberg}.
Also, efficient constructions of approximate $(\log n)^{O(1)}$-designs
are known \cite{sen:zigzag, brandao2012local}.
Thus, our work can be viewed as
a partial derandomisation of Hastings' result, and a step towards the
quest of finding an explicit quantum channel with superadditive classical
Holevo capacity.
Hastings' proof was considerably simplified by Aubrun, Szarek and
Werner~\cite{aubrun_szarek_werner_2010_main} who showed that existence
of channels with subadditive minimum output von Neumann entropy follows
from a sharp Dvoretzky-like theorem which states that, under the
Haar measure, random subspaces of large dimension make a Lipschitz function
take almost constant value.
Dvoretzky's original theorem \cite{Dvoretzky}
stated that any centrally
symmetric convex body can be embedded with low distortion into a
section of a high dimensional unit $\ell_2$-sphere.
Milman \cite{Milman} extended Dvoretzky's theorem by proving that,
with high probability, Haar random subspaces of an appropriate dimension
make a Lipschitz function take almost constant value. Dvoretzky's theorem
becomes the special case of Milman's theorem where the Lipschitz
function happens to be norm induced by the centrally symmetric convex
body i.e. the norm under which the convex body becomes the unit ball.
Milman's work started a whole body of research sharpening the various
parameters of the extended Dvoretzky theorem
e.g. \cite{Schechtman, Gordon} etc. However, all these works
use Haar random subspaces. A Haar random subspace of $\mathbb{C}^n$ of
dimension $d$
can be obtained by applying a Haar random unitary to a fixed subspace
of dimension $d$ e.g. the subspace spanned by the first $d$ standard
basis vectors of $\mathbb{C}^n$. Our work is the first one to replace the
Haar random unitary in any Dvoretzky-type theorem by a uniformly
random unitary chosen from an
approximate $t$-design for a suitable value of $t$. In other words,
our main technical result is an Aubrun-Szarek-Werner style result for
approximate $t$-designs instead of Haar random unitaries.
As a corollary, we obtain the subadditivity of minimum output von
Neumann entropy for unitaries chosen from an approximate $n^{2/3}$-design.
As another corollary, we obtain the subadditivity
of minimum output R\'{e}nyi $p$-entropy for all $p > 1$ for
quantum channels arising from unitaries
chosen from an approximate unitary
$(n^{1.7} \log n)$-design.
Such a unitary can in fact be chosen from an exact
$(n^{1.7} \log n)$-design using only
$n^{1.7} (\log n)^2$ random
bits \cite{Kuperberg}, which is
much less than $\Omega(n^2)$ random bits required to choose a
Haar random unitary. Subadditivity
of minimum output R\'{e}nyi $p$-entropy for all $p > 1$
was originally proved for Haar random unitaries
by Hayden and Winter \cite{hayden_winter_2008}.
To prove our main technical result, we use a concentration of
measure result by
Low~\cite{low_2009} for approximate unitary $t$-designs,
combined with a stratified analysis of the variational
behaviour of Lipschitz functions on the unit sphere in high
dimension.
We need such a fine grained stratified analysis for the
following reason. Aubrun, Szarek and
Werner~\cite{aubrun_szarek_werner_2010_main} worked with the function
$f(M) := \lVert MM^\dag - (I/k) \rVert_2$, where the argument $M$
is a $k^3$-tuple rearranged to form a $k \times k^2$ matrix. They
found subspaces of dimension $k^2$ where $f$ took almost constant value.
For this, they had to do a two step analysis. The global Lipschitz
constant of $f$ was $2$ which, under naive Dvoretzky type arguments,
would only guarantee the existence of subpaces of dimension
$\frac{k^2}{\log k}$ where $f$ is almost constant. This does not
suffice to find a counter example to minimum output von Neumann
entropy. In order to shave off the $\log k$ term in the denominator,
they had to use several sophisticated arguments. One of them was the
observation that there is a high probablity subset $T$ of
$\mathbb{S}_{\mathbb{C}^{k^3}}$
on which the Lipschitz constant of $f$ was $k^{-1/2}$. They exploited
this by their two step analysis, where they separately analysed the
behaviour of $f$ on $T$ and on $T^c$, and managed to shave off
the $\log k$ term. For us, since we are working with designs, we
need the function to be a polynomial. Hence, instead of $f$, we have
to work with $f^2$. This seemingly trivial change introduces severe
technical difficulties. The main reason behind them is that the Lipschitz
constant of $f^2$ is about twice the Lipschitz constant for $f$
but the variation that we are looking to bound is around square of the
earlier variation! This contradiction lies at the heart of the technical
difficulty. In order to overcome this, we have to partition
$\mathbb{S}_{\mathbb{C}^{k^3}}$ into a number of sets
$\Omega_1, \Omega_2, \ldots, \Omega_{\log k}$,
called `layers',
with local Lipschitz constants for $f^2$ running as
$k^{-3/2}, 2^3 k^{-3/2}, 3^3 k^{-3/2}, \ldots, (\log k)^3 k^{-3/2}$.
We have to bound
the variation of $f^2$ individually on $\Omega_i$ as well as put
them together to bound the variation on large subspheres of
$\mathbb{S}_{\mathbb{C}^{k^3}}$. This leads to a challenging stratified
analysis, which
forms the main technical advance of this paper.
Another tool developed in this work which should find use in other
situations also, is a systematic way to approximate a monotonic
differentiable function and its derivative using moderate degree
polynomials. This tool is crucially used to prove strict subadditivity
of R\'{e}nyi $p$-entropy for any $p > 1$ for channels whose unitary
Stinespring dilation is chosen from an approximate design instead of
a Haar random unitary.
The power of our
stratified analysis shows up in the consequence that the dimension of
the subspace on which the Lipschitz function is almost constant depends
only on the smallest local Lipschitz constant, provided some mild
niceness conditions are satisfied. This gives larger dimensional subspaces
than a naive analysis which would depend on the global Lipschitz
constant. In fact, the stratified analysis
allows us to prove a sharper Dvoretzky-type theorem even for the
Haar measure. As a result, we can recover Aubrun, Szarek and Werner's
result for the function $f$ directly and elegantly
instead of applying their Dvoretzky-type result twice which is rather
messy. Another powerful consequence of our stratified analysis is that
with probability
exponentially close to one random, over Haar measure or $t$-design
measure, large subspaces
make the Lipschitz function almost constant. In contrast,
Aubrun, Szarek and Werner could only guarantee constant probability close
to one for the Haar measure, and they did not consider $t$-designs.
They also stated without providing details that the existence probability
could be made exponentially close to one using a deep Levy-type lemma for
unitary matrices. In contrast our stratified analysis uses only
the elementary Levy lemma for the unit sphere, yet it manages to prove
existence with probability exponentially close to one.
The rest of the paper is organised as follows. Section~\ref{sec:prelim}
contains notations, symbols
definitions and preliminary tools required for the
paper.
Section~\ref{sec:main} states and proves the main technical theorems viz.
the stratified analyses for Haar measure and approximate $t$-designs.
Section \ref{sec:vonNeumannentropy} describes the
application to subadditivity of minimum output von Neumann entropy.
Section~\ref{sec:Renyipentropy} describes the
application to subadditivity of minimum output R\'{e}nyi $p$-entropy
for $p > 1$. Section~\ref{sec:conclusion} concludes the paper and states
some open problems for future work.
\section{Preliminaries}
\label{sec:prelim}
All Hilbert spaces used in this paper are finite dimensional.
The $n$ dimensional space over complex numbers, $\mathbb{C}^n$, is endowed
with the standard inner product aka the dot product:
$\langle x, y \rangle := \sum_{i=1}^n x_i^* y_i$.
The unit radius sphere in $\mathbb{C}^n$ is denoted by $\mathbb{S}_{\mathbb{C}^n}$.
The symbol $\mathcal{M}_{k,d}$
denotes the Hilbert space of $k \times d$ linear operators over the
complex field under the Hilbert-Schmidt inner product
$\langle M, N \rangle := \mathrm{Tr}\, [M^\dag N]$,
and $\mathcal{M}_d := \mathcal{M}_{d,d}$.
Let $\mathcal{U}(n)$ denote the set of $n \times n$ unitary matrices with
complex entries.
For a composite Hilbert space
$\mathbb{C}^k \otimes \mathbb{C}^d$,
the notation $\mathrm{Tr}\,_{\mathbb{C}^d}[\cdot]$ denotes the operation of
taking partial trace i.e.
tracing out the mentioned subsystem $\mathbb{C}^d$. We use $\mathrm{Tr}\,[\cdot]$
to denote the trace of the underlying operator.
Fix standard bases for Hilbert spaces $A \cong \mathbb{C}^k$, $B \cong \mathbb{C}^d$.
Let $\ket{e_i}^A$, $\ket{e_i}^B$ denote standard basis vectors of
$A$, $B$ respectively. Any vector $x \in A \otimes B$ can be
written as $x = \sum_{i,j} \alpha_{ij} \ket{e_i}^A \otimes \ket{e_j}^B$.
We use $\mathrm{op}_{d \rightarrow k}(x)$ to denote the operator
$\sum_{i,j} \alpha_{ij} \ket{e_i}^A \otimes \bra{e_j}^B$
in $\mathcal{M}_{k, d}$. Conversely, given an operator
$M = \sum_{ij} m_{ij} \ket{e_i}^A \otimes \bra{e_j}^B$ in
$\mathcal{M}_{k,d}$, we let
$\mathrm{vec}(M) := \sum_{ij} m_{ij} \ket{e_i}^A \otimes \ket{e_j}^B$ denote
the vector in $\mathbb{C}^k \otimes \mathbb{C}^d$.
For Hermitian positive semidefinite operators $M$, we define
$M^\alpha$ for any $\alpha > 0$ to be the unique Hermitian
operator obtained by keeping the eigenbasis same and
taking the $\alpha$th power of the eigenvalues. We can define
$\log M$ similarly.
For $p > 1$, the notation
$\lVert M \rVert_p$ denotes the Schatten $p$-norm
of the matrix $M$, which is
nothing but the $\ell_p$-norm of the vector of its singular values.
Alternatively, $\lVert M \rVert_p = (\mathrm{Tr}\, [(M^\dagger M )^{p/2}])^{1/p}$.
Then $p=2$ gives the Hilbert Schmidt
norm aka the Frobenius norm which is nothing but
$\lVert M \rVert_2 = \lVert \mathrm{vec}(M) \rVert_2$.
Also, $p = \infty$ gives the operator norm aka spectral norm
which is nothing but
$
\lVert M \rVert_\infty =
\max_{v: \lVert v \rVert_2 = 1} \lVert M v \rVert_2.
$
Unless stated otherwise, the symbol $\rho$ denotes a quantum
state aka density matrix which is nothing but a Hermitian, positive
semidefinite matrix with unit trace. A rank one density matrix is called
a pure state. By the spectral theorem, any density matrix is a convex
combination of pure states.
The notation $\mathcal{D}(\mathbb{C}^d)$ denotes the convex
set of all $d \times d$ density matrices. We use $\ket{\cdot}$ to denote
a unit vector. By a slight abuse of notation, we shall often use a unit
vector $\ket{\psi}$ to denote a pure state $\ket{\psi}\bra{\psi}$.
A linear mapping $\Phi: \mathcal{M}_m \to \mathcal{M}_d $ is called
a superoperator. A superoperator is trace preserving if
$\mathrm{Tr}\, \Phi(M) = \mathrm{Tr}\, M$ for all $M \in \mathcal{M}_m$. It is said to be
positive if $\Phi(M)$ is positive semidefinite for all positive
semidefinite $M$. Furthermore, $\Phi$ is said to be completely positive
if $\Phi \otimes \mathbb{I}$ is a positive superoperator for identity
superoperators $\mathbb{I}$ of all dimensions.
Completely positive and trace preserving (CPTP) superoperators
are referred to as quantum channels.
Unless stated otherwise, $\Phi$, $\Psi$ are used to denote
quantum channels.
A compact convex set $\mathcal{S}$ in $\mathbb{C}^n$ is called a convex body.
The radius $r(\mathcal{S})$ of a convex body $\mathcal{S}$ is defined as
\[
r(\mathcal{S}) := \min_{x \in \mathcal{S}} \max_{y \in \mathcal{S}} \lVert x - y \rVert_2.
\]
Any point $x \in \mathcal{S}$ achieving the minimum above is said to be a
centre of $\mathcal{S}$.
The convex body $\mathcal{S}$ is said
to be centrally symmetric iff for every $x \in \mathbb{C}^n$,
$x \in \mathcal{S} \leftrightarrow -x \in \mathcal{S}$. The zero vector is a centre
of a centrally symmetric convex body. A centrally symmetric convex body
lying in $\mathbb{C}^n$
can be thought of as the unit sphere of a suitable notion of norm
in $\mathbb{C}^n$. Conversely for any norm in $\mathbb{C}^n$, the unit sphere under
the norm forms a centrally symmetric convex body.
\subsection{Entropies and norms}
\begin{definition}
The von Neumann entropy of a quantum state
$\rho$ is defined as
\[
S(\rho) := -\mathrm{Tr}\, [\rho \log \rho].
\]
For all $p > 1$, the R\'{e}nyi $p$-entropy of a quantum state
$\rho$ is defined as
\[
S_p(\rho) := \frac{1}{1-p} \log \mathrm{Tr}\, \rho^p =
-\frac{p}{p-1} \log \lVert \rho \rVert_p.
\]
\end{definition}
It turns out that
$
S(\rho) = \lim_{p \downarrow 1} S_p(\rho) =: S_1(\rho).
$
\noindent
Also, it can be shown that for $p \geq 1$, $S_p(\cdot)$
is concave in its argument.
\begin{definition}
For $p \geq 1$, the minimum output R\'{e}nyi $p$-entropy of
a quantum channel $\Phi$ is defined as :
\[
S_p^{\mathrm{min}}(\Phi) :=
\min_{\rho \in \mathcal{D}(\mathbb{C}^m)} S_p(\Phi(\rho))
\]
\end{definition}
By an easy concavity argument it can be seen that above
minimum is achieved on a pure state.
Equivalently, to obtain $S_p^{\mathrm{min}}(\Phi)$ for $p > 1$
we must maximise
$\lVert \Phi(\rho) \rVert_p$ for all input states $\rho$. This quantity is
also known as the
$1 \rightarrow p$ superoperator norm of superoperator
$\Phi: \mathcal{M}_m \rightarrow \mathcal{M}_d$:
\[
\lVert \Phi \rVert_{1 \rightarrow p}
:= \max_{M \in \mathcal{M}_m: \lVert M \rVert_1 = 1}
\lVert \Phi(M) \rVert_p.
\]
By an easy convexity argument it can be seen that the above
maximum is achieved on a pure state i.e.
\[
\lVert \Phi \rVert_{1 \rightarrow p} =
\max_{x \in \mathbb{C}^m: \lVert x \rVert_2 = 1} \lVert \ket{x}\bra{x} \rVert_{p}.
\]
Thus, the additivity conjecture for minimal output p-R\'{e}nyi $p$-entropy,
$p > 1$,
for quantum channels $\Phi$ and $\Psi$ is equivalent to
multiplicativity of $1 \rightarrow p$-norms of quantum channels viz.
$
\lVert \Phi \otimes \Psi \rVert_{1 \rightarrow p}
\stackrel{?}{=} \lVert \Phi \rVert_{1 \rightarrow p} \cdot
\lVert \Psi \rVert_{1 \rightarrow p}.
$
This equivalence will be used in Section~\ref{sec:Renyipentropy} to
give a counter example to additivity
conjecture for all $p > 1$ where the Stinespring dilation of the
quantum channel will be described
from a unitary chosen uniformly at random from an approximate $t$-design.
The equivalent result for Haar random unitaries was originally
proved by Hayden and Winter \cite{hayden_winter_2008}.
We heavily use the one-one correspondence between quantum channels and
subspaces of
composite Hilbert spaces, originally proved by Aubrun, Szarek and Werner
\cite{aubrun_szarek_werner_2010}, in this paper.
Let $\mathcal{W}$ be a subspace of
$\mathbb{C}^k \otimes \mathbb{C}^d$ of dimension $m$. Identify $\mathcal{W}$ with
$\mathbb{C}^m$ through an isometry
$V : \mathbb{C}^m \to \mathbb{C}^k \otimes \mathbb{C}^d$ whose range is
$\mathcal{W}$. Then, the corresponding quantum channel
$\Phi_\mathcal{W} : \mathcal{M}_m \to \mathcal{M}_k$ is defined by
$
\Phi_\mathcal{W}(\rho) := \mathrm{Tr}\,_{\mathbb{C}^d}(V \rho V^\dagger).
$
Using this equivalence and the fact that for $p > 1$ the
$1 \rightarrow p$-superoperator norm is achieved on pure input
states, we can write \cite{aubrun_szarek_werner_2010}
\begin{equation}
\label{eq:redefpnorm}
\lVert \Phi_\mathcal{W} \rVert_{1 \rightarrow p} =
\max_{x \in \mathcal{W}: \lVert x \rVert_2 = 1}
\lVert \mathrm{Tr}\,_{\mathbb{C}^d} \ket{x}\bra{x} \rVert_p =
\max_{x \in \mathcal{W}: \lVert x \rVert_2 = 1}
\lVert \mathrm{op}_{d \rightarrow k}(x) \rVert_{2p}^2.
\end{equation}
In an important paper, Shor \cite{Shor} proved that several
additivity conjectures
for quantum channels were in fact equivalent to the additivity
of minimum output von Neumann entropy of a quantum channel.
More specifically, Shor showed that if there is a quantum channel $\Phi$
whose minimum output von Neumann entropy is subadditive, then there
are quantum channels $\Psi_1$, $\Psi_2$ exhibiting superadditive
classical Holevo capacity viz.
$C(\Psi_1 \otimes \Psi_2) > C(\Psi_1) + C(\Psi_2)$. This equivalence
was used as a starting point by Hastings \cite{hastings_2009} in his
proof that there are
channels with superadditive classical Holevo capacity. Aubrun, Szarek and
Werner \cite{aubrun_szarek_werner_2010_main}, as well as this paper
also have the same starting point. For this, we need the following
fact.
\begin{fact}[\mbox{\cite[Lemma 2]{aubrun_szarek_werner_2010_main}}]
\label{fact:minoutputentropy}
Let a quantum
channel $\Phi_\mathcal{W}: \mathcal{M}_m \rightarrow \mathcal{M}_k$ be described by a subspace
$\mathcal{W} \leq \mathbb{C}^k \otimes \mathbb{C}^d$ of dimension $m$. Then,
\begin{eqnarray*}
S_{\mathrm{min}}(\Phi_\mathcal{W})
& = &
\log k -
k \cdot
\max_{\rho \in \mathcal{D}(\mathbb{C}^m)} \lVert \Phi(\rho) - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}
\rVert_2^2 \\
& = &
\log k -
k \cdot
\max_{x \in \mathcal{W}: \lVert x \rVert_2 = 1}
\lVert
(\mathrm{op}_{d \rightarrow k}(x)) (\mathrm{op}_{d \rightarrow k}(x))^\dagger
- \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}
\rVert_2^2.
\end{eqnarray*}
\end{fact}
We will need the following result proved by Hayden and
Winter \cite{hayden_winter_2008}
that upper bounds $S_p^{\mathrm{min}}(\Phi \otimes \bar{\Phi})$ where
$\bar{\Phi}$ denotes the CPTP superoperator obtained by taking complex
conjugate of the CPTP superoperator $\Phi$.
\begin{fact}
\label{fact:maxeigenvalue}
Let $V : \mathbb{C}^m \rightarrow \mathbb{C}^k \otimes \mathbb{C}^d$ be an
isometry describing the quantum channel
$\Phi: \rho \mapsto \mathrm{Tr}\,_{\mathbb{C}^d} [V \rho V^\dagger]$. Let
$\ket{\phi}$ denote the maximally entangled state in
$\mathbb{C}^m \otimes \mathbb{C}^m$. Suppose $m \leq d$.
Then $(\Phi \otimes \bar{\Phi})(\ket{\phi} \bra{\phi})$ has a
singular value
not less than $\frac{m}{kd}$.
Hence for all $p > 1$,
\[
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow p}
\geq
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow \infty} \geq
\frac{m}{kd}.
\]
Moreover,
\[
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi}) \leq
2 \log k - \frac{m}{kd} \log k +
O \left(\frac{m}{kd} \log \frac{d}{m} + \frac{1}{k}\right).
\]
\end{fact}
\subsection{Polynomial approximation of monotonic functions}
We will need the following facts about step functions and their analytic
and polynomial approximations when we prove our result on strict
subadditivity of minimum output R\'{e}nyi $p$-entropy for
channels chosen from approximate $t$-designs.
\begin{definition}
\label{def:stepfunction}
The {\em (Heaviside) step function} is a function
$\mathbb{R} \rightarrow [0, 1]$ defined as follows:
\[
s(x) :=
\begin{array}{l l}
0 & \mbox{for $x < 0$} \\
\frac{1}{2} & \mbox{for $x = 0$} \\
1 & \mbox{for $x > 0$}.
\end{array}
\]
\end{definition}
\begin{definition}
\label{def:errorfunction}
The {\em error function} is a function $\mathbb{R} \rightarrow (-1, 1)$ defined
as follows:
\[
\mathrm{erf}(x) :=
\frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt.
\]
\end{definition}
\noindent
The error function is a monotonically increasing function.
For positive $x$, $\mathrm{erf}(x)$ is nothing but the probability that the
normal distribution with mean $0$ and variance $1/2$ gives a point in
the interval $[-x, x]$. From the error function,
we get the so-called {\em sigmoid function}
$\Phi(x) := \frac{1}{2} + \frac{1}{2} \mathrm{erf}(x)$ which is nothing but
the cumulative distribution function of the above normal distribution.
The sigmoid function is a monotonically increasing function approximating
the step function in the following sense. Let $0 < \epsilon < 1$.
\begin{equation}
\label{eq:PhiVsS}
\Phi(x) ~
\begin{array}{l l l}
= & s(x) = \frac{1}{2} & \mbox{for $x=0$}, \\
> & s(x) = 0 & \mbox{for $x<0$}, \\
< & \frac{1}{2} & \mbox{for $x<0$}, \\
< & s(x) = 1 & \mbox{for $x>0$}, \\
> & \frac{1}{2} & \mbox{for $x>0$}, \\
> & s(x) - \epsilon = 1 - \epsilon &
\mbox{for
$x>\sqrt{\ln \epsilon^{-1}}$
}, \\
< & s(x) + \epsilon = \epsilon &
\mbox{for
$x<-\sqrt{\ln \epsilon^{-1}}$
},
\end{array}
~~~
\Phi'(x) ~
\begin{array}{l l l}
= & \frac{1}{\sqrt{\pi}} & \mbox{for $x=0$}, \\
< & \frac{1}{\sqrt{\pi}} & \mbox{for $x \neq 0$}, \\
> & 0 & \mbox{for all $x$}, \\
< & \epsilon &
\mbox{for
$|x| > \sqrt{\ln \epsilon^{-1}}$
}.
\end{array}
\end{equation}
The last two statements for $\Phi(x)$ above hold for
small $\epsilon$ and follow from
the bound
$
1 - \Phi(x) \leq
\frac{1}{2 x \sqrt{\pi}} e^{-x^2}.
$
The error function has the following rapidly converging Maclaurin
series:
\[
\mathrm{erf}(x) =
\frac{2}{\sqrt{\pi}}
\sum_{i=0}^\infty (-1)^i \frac{x^{2i+1}}{i! (2i+1)}.
\]
It is obtained by integrating termwise the Maclaurin series
$
e^{-x^2} =
\sum_{i=0}^\infty (-1)^i \frac{x^{2i}}{i!}.
$
Since both the above series are alternating series of positive and
negative terms,
truncating the Maclaurin expansion of $\Phi(x)$ at $i = n$ for
odd $n > x^2$ gives us
a polynomial $p_n(x)$ of degree $2n+1$ such that
\begin{equation}
\label{eq:PhiVsP}
p_n(x) ~
\begin{array}{l l l}
= & \Phi(x) = \frac{1}{2} & \mbox{for $x=0$}, \\
> & \Phi(x) & \mbox{for $-\sqrt{n} \leq x < 0$}, \\
< & \Phi(x) & \mbox{for $0 < x \leq \sqrt{n}$}, \\
> & \Phi(x) - \epsilon
& \mbox{for
$
0 \leq x \leq \frac{\epsilon^{\frac{1}{2n}} \sqrt{n}}{2}
$
},\\
< & \Phi(x) + \epsilon
& \mbox{for
$
-\frac{\epsilon^{\frac{1}{2n}} \sqrt{n}}{2} \leq x \leq 0
$
}.
\end{array}
\end{equation}
Moreover, the derivative $p'_n(x)$ is a polynomial of degree $2n$
satisfying
\begin{equation}
\label{eq:PhiprimeVsPprime}
p'_n(x) ~
\begin{array}{l l l}
= & \Phi'(x) = \frac{1}{\sqrt{\pi}} & \mbox{for $x=0$}, \\
\leq & \Phi'(x)
& \mbox{for
$
-\sqrt{n} \leq x \leq \sqrt{n}
$
}, \\
> & \Phi'(x) - \epsilon
& \mbox{for
$
-\frac{\epsilon^{\frac{1}{2n}} \sqrt{n}}{2}
\leq x \leq \frac{\epsilon^{\frac{1}{2n}} \sqrt{n}}{2}
$
}.
\end{array}
\end{equation}
For the last two claims in Equation~\ref{eq:PhiVsP} and the last claim in
Equation~\ref{eq:PhiprimeVsPprime},
we used Stirling's approximation
$
n^n e^{-n} < n!
$
which holds for all positive integers $n$.
We will also need to upper bound the sum of absolute values of the
coefficients of $p_n(x)$, denoted by $\alpha(p_n(x))$. For this we
observe that
$
\alpha(p_n(x)) =
|p_n(\sqrt{-1})| \leq
\frac{1}{2} + \frac{e}{\sqrt{\pi}}.
$
We can now conclude that for $m > 0$, $0 \leq q \leq A$,
\begin{equation}
\label{eq:alpha}
\begin{array}{rcl}
\alpha(p_n(m(x-q)))
& = &
\frac{1}{2} + \frac{1}{\sqrt{\pi}}
\alpha(
\sum_{i=0}^n (-1)^i
\frac{(m(x-q))^{2i+1}}{i! (2i + 1)}
)
\;\leq\;
\frac{1}{2} + \frac{1}{\sqrt{\pi}}
\alpha(
\sum_{i=0}^n
\frac{(m(x+q))^{2i+1}}{i! (2i + 1)}
) \\
& = &
\frac{1}{2} + \frac{1}{\sqrt{\pi}}
\sum_{i=0}^n
\frac{(m(1+q))^{2i+1}}{i! (2i + 1)}
\;\leq\;
\frac{1}{2} + \frac{1}{\sqrt{\pi}}
\sum_{i=0}^\infty
\frac{(m(1+A))^{2i+1}}{i!} \\
& = &
\frac{1}{2} + \frac{m(1+A) e^{(m(1+A))^2}}{\sqrt{\pi}}
\;\leq\;
e^{2 (m(1+A))^2}.
\end{array}
\end{equation}
Let $f: [0, A] \rightarrow \mathbb{R}$ be a continuous non-decreasing function.
The {\em global Lipschitz constant} of $f$ is defined by
\[
L :=
\sup_{x,y \in [0, A], x < y}
\frac{f(y) - f(x)}{y - x}.
\]
If $L$ is finite, then we say that $f$ is $L$-Lipschitz.
Let $\epsilon > 0$.
For an element $x \in [0, A]$, the {\em $\epsilon$-smoothed local
Lipschitz constant of $f$ at $x$} is defined by
\[
L^\epsilon_x :=
\sup_{x,y \in f^{-1}((f(x) - \epsilon, f(x) + \epsilon)), x < y}
\frac{f(y) - f(x)}{y - x}.
\]
It is obvious that $L_x^\epsilon \leq L$. If $f$ is differentiable,
then $f'(x) \leq L^\epsilon_x$.
We now give a general proposition showing how to approximate
a continuous non-decreasing Lipschitz function by a polynomial
of moderate degree.
\begin{proposition}
\label{prop:poly}
Let $f: [0,A] \rightarrow [0,1]$ be a continuous non-decreasing onto
function with global Lipschitz constant $L$.
Fix $0 < \epsilon < 1$.
Let $L_x^\epsilon$ denote the $\epsilon$-smoothed local Lipschitz
constant of $f$ at $x$.
Let $n$ be the minimum positive odd integer satisfying
$
m A \leq \frac{\epsilon^{\frac{1}{n}} \sqrt{n}}{2},
$
where
$
m := \frac{2 L}{\epsilon} \sqrt{\ln \epsilon^{-2}}.
$
Define
$
m_x := \frac{2 L_x^\epsilon}{\epsilon} \sqrt{\ln \epsilon^{-2}}.
$
Then there is a polynomial $p(x)$ of degree at most $2n + 1$
such that
\[
p(x) - 2 \epsilon \leq f(x) \leq p(x) + 3 \epsilon, ~~~
-m \epsilon^2 < p'(x) < \epsilon m_x + m \epsilon^2, ~~~
\forall x \in [0,A].
\]
Moreover the sum of absolute values of the coefficients of $p(x)$,
denoted by $\alpha(p(x))$, is at most
$e^{2 ((A+1)m)^2}$.
\end{proposition}
\begin{proof}
Subdivide the range $[0,1]$ into $t := \lceil 1/\epsilon \rceil$ many
closed subintervals each of length $\epsilon$ except possibly the last one
whose length $\epsilon'$ may be less than $\epsilon$.
Denote their inverse images
under $f$ by $I_1, I_2, \ldots, I_{t}$.
For $1 \leq i < t$, let $p_i$ be the single
point intersection of closed subintervals $I_i$ and $I_{i+1}$; define
$p_0 := 0$, $p_t := A$.
The subinterval $I_i$, $1 \leq i < t$ is of length at least
$
\frac{\epsilon}{2 L_{p_i}^{\epsilon/2}} +
\frac{\epsilon}{2 L_{p_{i-1}}^{\epsilon/2}},
$
$I_t$ is of length at least
$
\frac{\epsilon'}{2 L_{p_t}^{\epsilon/2}} +
\frac{\epsilon'}{2 L_{p_{t-1}}^{\epsilon/2}}.
$
Observe that $\max_i L_{p_i}^{\epsilon/2} \leq L$.
Define the function
\[
g_1(x) :=
\epsilon \sum_{i=1}^{t - 1} s(x - p_i).
\]
Then $g_1(x) \leq f(x) \leq g_1(x) + \epsilon$ for all $x \in [0,A]$.
Define
$
m_i := \frac{2 L_{p_i}^{\epsilon/2}}{\epsilon} \sqrt{\ln \epsilon^{-2}},
$
$1 \leq i \leq t$. Then $m \geq \max_i m_i$.
Approximate the step function $s(x - p_i)$ by the sigmoid function
$\Phi(m_i (x - p_i))$. By Equation~\ref{eq:PhiVsS},
\[
\Phi(m_i (x-p_i)) ~
\begin{array}{l l l}
= & s(x - p_i) = \frac{1}{2} & \mbox{for $x=p_i$}, \\
> & s(x - p_i) = 0 & \mbox{for $x<p_i$}, \\
< & \frac{1}{2} & \mbox{for $x<p_i$}, \\
< & s(x - p_i) = 1 & \mbox{for $x>p_i$}, \\
> & \frac{1}{2} & \mbox{for $x>p_i$}, \\
> & s(x - p_i) - \epsilon^2 = 1 - \epsilon^2
& \mbox{for $x> p_i +\frac{\epsilon}{2 L_{p_i}^{\epsilon/2}}$}, \\
< & s(x - p_i) + \epsilon^2 = \epsilon^2
& \mbox{for $x< p_i - \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}}$}.
\end{array}
\]
Define the function
\[
g_2(x) :=
\epsilon \sum_{i=1}^{t - 1} \Phi(m_i (x - p_i)).
\]
It is now easy to see that
$
g_2(x) - \epsilon \leq g_1(x) \leq g_2(x) + \epsilon
$
for all $x \in [0, A]$. Thus,
\[
g_2(x) - \epsilon \leq f(x) \leq g_2(x) + 2 \epsilon ~~~
\forall x \in [0,A].
\]
Also,
\[
0 < g'_2(x) < \epsilon m_i + m \epsilon^2, ~~~
\mbox{if\ }
x \in
[
p_i - \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}},
p_i + \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}}
]
\mbox{\ for some $i$},
\]
and
$
0 < g'_2(x) < m \epsilon^2
$
otherwise.
We now approximate the sigmoid function $\Phi(m_i (x-p_i))$ by
the polynomial $p_n(m_i (x - p_i))$ for
$m_i A \leq m A < \frac{\epsilon^{\frac{1}{n}} \sqrt{n}}{2}$, $n$ odd.
From Equations~\ref{eq:PhiVsP}, \ref{eq:PhiprimeVsPprime} we get
\[
p_n(m_i (x-p_i)) ~
\begin{array}{l l l}
= & \Phi(m_i (x-p_i)) = \frac{1}{2} & \mbox{for $x=p_i$}, \\
> & \Phi(m_i (x - p_i)) & \mbox{for $0 \leq x<p_i$}, \\
< & \Phi(m_i (x - p_i)) & \mbox{for $p_i < x \leq A$}, \\
> & \Phi(m_i (x - p_i)) - \epsilon^2 & \mbox{for $p_i \leq x \leq A$}, \\
< & \Phi(m_i (x - p_i)) + \epsilon^2 & \mbox{for $0 \leq x \leq p_i$},
\end{array}
\]
\[
p'_n(m_i (x-p_i)) ~
\begin{array}{l l l}
= & \Phi'(m_i (x-p_i)) = \frac{m_i}{\sqrt{\pi}} & \mbox{for $x=p_i$}, \\
\leq & \Phi'(m_i (x - p_i)) & \mbox{for $0 \leq x \leq A$}, \\
> & \Phi'(m_i (x - p_i)) - \epsilon^2 & \mbox{for $0 \leq x \leq A$}.
\end{array}
\]
Define the degree $2n+1$ polynomial
\[
p(x) :=
\epsilon \sum_{i=1}^{t - 1} p_n(m_i (x - p_i)).
\]
It is now easy to see that
\[
p(x) - \epsilon^2 \leq g_2(x) \leq p(x) + \epsilon^2,
~~~
p'(x) \leq g'_2(x) \leq p'(x) + m \epsilon^2,
\]
for all $x \in [0, A]$. Thus,
\[
p(x) - 2 \epsilon \leq f(x) \leq p(x) + 3 \epsilon ~~~
\forall x \in [0,A],
\]
and
\[
-m \epsilon^2 < p'(x) < \epsilon m_i + m \epsilon^2, ~~~
\mbox{if\ }
x \in
[
p_i - \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}},
p_i + \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}}
]
\mbox{\ for some $i$},
\]
and
$
-m \epsilon^2 < p'(x) < m \epsilon^2
$
otherwise. Now observe that if
$
x \in
[
p_i - \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}},
p_i + \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}}
],
$
$m_x \geq m_i$. Hence we can always say that
\[
-m \epsilon^2 < p'(x) < \epsilon m_x + m \epsilon^2 ~~~
\forall x \in [0, A].
\]
Finally by Equation~\ref{eq:alpha},
\[
\alpha(p(x)) \leq
\epsilon \sum_{i=1}^{t-1} \alpha(p_n(m_i (x-p_i))) \leq
\epsilon \sum_{i=1}^{t-1} e^{2((A+1)m_i)^2} \leq
e^{2((A+1)m)^2}.
\]
This completes the proof of the proposition.
\end{proof}
\paragraph{Remarks:}
\ \\
\noindent
1. \
Any continuous non-decreasing Lipschitz
function on a closed bounded interval can be converted into a function
of the above type by translating the domain and the range and scaling
the range.
\noindent
2. \
A similar proposition can be proved for approximating a monotonically
non-increasing Lipschitz function by a polynomial.
\subsection{Concentration results for Lipschitz functions}
We now state some basic definitions and facts from geometric functional
analysis that will be used in the proof of our main result.
\begin{definition}
A function $f:X \rightarrow \mathbb{C} $ defined over a metric
space $X$ is said to
be $L$-Lipschitz if
$\forall x, y \in X$ it satisfies the following inequality:
\[
\lvert f(x)-f(y) \rvert \leq L \cdot d(x,y).
\]
\end{definition}
\begin{definition}
Let $X$ be a compact metric space. An $\epsilon$-net $\mathcal{N}$ of
$X$ is a finite set of points such that for any point $x \in X$, there is
a point $x' \in \mathcal{N}$ such that $d(x,x') \leq \epsilon$.
\end{definition}
\noindent
Note that compactness guarantees that finite sized $\epsilon$-nets exist
for all $\epsilon > 0$.
We will need the following definition and fact from
\cite{aubrun_szarek_werner_2010_main}.
\begin{definition}
A function $f:X \rightarrow \mathbb{C} $ defined over a normed linear
space $X$ is said to be circled if
$f(e^{i\theta} x) = f(x)$ for all
$\theta \in \mathbb{R}$ and $x \in X$.
\end{definition}
\begin{fact}
\label{fact:extension}
Let $f: X \rightarrow \mathbb{R}$ be a function defined on a metric space $X$.
Suppose there exists a subset $Y \subseteq X$ such that $f$ restricted
to $Y$ is $L$-Lipschitz. Then there is a function
$\hat{f}: X \rightarrow \mathbb{R}$
that is $L$-Lipschitz on all of $X$ satisfying $\hat{f}(y) = f(y)$ for all
$y \in Y$. If $X$ is a normed linear space over real or complex numbers
and $f$ is circled
then the extension $\hat{f}$ is also circled.
\end{fact}
\begin{proof}
{\bf (Sketch)}
Define
$
\hat{f}(x) := \inf_{y \in Y} [f(y) + L d(x, y)].
$
\end{proof}
In this paper, we endow $\mathbb{C}^n$ with the $\ell_2$-metric and
$\mathbb{U}(n)$ with the Schatten $\ell_2$-metric aka Frobenius metric. The
following fact gives a reasonably tight upper bound on the
size of an $\epsilon$-net of $\mathbb{S}_{\mathbb{C}^n}$.
\begin{fact}[\mbox{\cite[Corollary~4.2.13]{Vershynin}}]
\label{fact:net}
Let $\epsilon > 0$. There exists an $\epsilon$-net of $\mathbb{S}_{\mathbb{C}^n}$ of
size less than
$
(\frac{3}{\epsilon})^{2n}.
$
\end{fact}
A fundamental result about concentration of Lipschitz functions defined
on the unit sphere or the unitary group, known as Levy's lemma, lies at
the heart of all
proofs of Dvoretzky-type theorems via the probabilistic method.
We now state the version of Levy's lemma that will be used in this
paper.
\begin{fact}[Levy's lemma, \mbox{\cite[Corollary~4.4.28]{AGZ}}]
\label{fact:levy}
Consider the Haar probability measure on $\mathbb{S}_{\mathbb{C}^n}$.
Let $f: \mathbb{S}_{\mathbb{C}^n} \rightarrow \mathbb{C}$ be an $L$-Lipshitz function.
Let $\mu := \mathbb{E}_x[f(x)]$ and $\lambda > 0$. Then
\[
\Pr_x(\lvert f(x) - \mu \rvert \geq \lambda) \leq
2 \exp(-\frac{n \lambda^2}{4 L^2}).
\]
\end{fact}
An elementary proof of the above fact, without explicitly calculated
constants, can be found in \cite[Theorem~5.1.4]{Vershynin}.
For our work, we need a measure concentration inequality like Levy's
lemma for difference of function values on two distinct arbitrary
points which is sensitive to the distance between those points.
Such an inequality is stated in the following fact.
\begin{fact}[\mbox{\cite[Lemma~9]{aubrun_szarek_werner_2010_main}}]
\label{fact:LevyLipschitz}
Let $f: \mathbb{S}_{\mathbb{C}^n} \to \mathbb{C}$ be a circled $L$-Lipschitz
function.
Consider the Haar probability measure on $\mathbb{U}(n)$.
Then for any
$x, y \in \mathbb{S}_{\mathbb{C}^n}$, $x \neq y$ and for
any $\lambda > 0$,
\[
\Pr_U[\lvert f(Ux) - f(Uy) \rvert > \lambda]
\leq 2 \exp(-\frac{\lambda^2 n}{8 L^2 \lVert x-y \rVert_2^2}).
\]
\end{fact}
The derandomisation in our paper is carried out by replacing the
Stinespring dilation unitary of a quantum channel,
which is chosen from the Haar measure in
\cite{aubrun_szarek_werner_2010_main},
with a unitary chosen uniformly at random
from a finite cardinality approximate unitary $t$-design for a
suitable value of $t$. The next few statements lead us to the definition
of an approximate unitary $t$-design.
\begin{definition}[\mbox{\cite[Definition~2.2]{low_2009}}]
A monomial in the entries of a matrix $U$ is of degree $(r,s)$ if
it contains $r$ conjugated elements and $s$ unconjugated elements. The
evaluation of monomial $M$ at the entries of a matrix $U$ is denoted
by $M(U)$. We call a monomial balanced if $r = s$,
and say that it has degree $t$ if it is of degree $(t,t)$.
A polynomial is said to be balanced of degree
$t$ if it is a sum of balanced monomials of degree at most $t$.
\end{definition}
\begin{definition}[\mbox{\cite[Definition~2.3]{low_2009}}]
A probability distribution $\nu$ supported on a finite set of $d \times d$
unitary matrices is said to be an exact unitary $t$-design if
for all balanced monomials $M$ of degree at most $t$,
$
\mathbb{E}_{U \sim \nu}[M(U)] = \mathbb{E}_{U \sim \mathrm{Haar}}[M(U)].
$
\end{definition}
\begin{definition}[\mbox{\cite[Definition~2.6]{low_2009}}]
A probability distribution $\nu$ supported on a finite set of $d \times d$
unitary matrices is said to be
an $\epsilon$-approximate unitary $t$-design if for all
balanced monomials $M$ of degree at most $t$
\[
\lvert
\mathbb{E}_{U \sim \nu}(M(U)) -
\mathbb{E}_{U \sim \mathrm{Haar}}(M(U))
\rvert \leq
\frac{\epsilon}{d^t}.
\]
\end{definition}
We will need the following fact.
\begin{fact}[\mbox{\cite[Lemma~3.4]{low_2009}}]
\label{fact:TvsHaar}
Let $Y: \mathbb{U}(n) \rightarrow \mathbb{C}$ be a balanced polynomial of
degree $a$ in the entries of the unitary matrix $U$ that is provided
as input. Let
$\alpha(Y)$ denote the sum of absolute values of the coefficients of $Y$.
Let $r$, $t$ be positive integers satisfying $2ar < t$.
Let $\nu$ be an $\epsilon$-approximate unitary $t$-design. Then
\[
\mathbb{E}_{U \sim \nu}[{\lvert Y_U\rvert}^{2r}] \leq
\mathbb{E}_{U \sim \mathrm{Haar}}[{\lvert Y_U \rvert}^{2r}]+
\frac{\epsilon \alpha(Y)^{2r}}{n^t}.
\]
\end{fact}
\section{Sharp Dvoretzky-like theorems via stratified analysis}
\label{sec:main}
In this section, we prove our main technical results viz. sharp
Dvoretzky-like theorems for Haar measure as well as approximate
$t$-designs using stratified analysis.
We start by proving the following two lemmas which are `baby stratified'
analogues of Fact~\ref{fact:LevyLipschitz} for Haar measure and
approximate unitary $t$-designs.
\begin{lemma}
\label{lem:expectationHaar}
Let $Y: \mathbb{S}_{\mathbb{C}^n} \rightarrow \mathbb{R}$ be a circled function with
global Lipschitz constant $L_1$. Suppose that
there exists a subset $\Omega \subseteq \mathbb{S}_{\mathbb{C}^n}$ such that
$Y$ restricted to $\Omega$
has a smaller Lipschitz constant $L_2$.
Let $x, y \in \mathbb{S}_{\mathbb{C}^n}$.
Let $Y_x := Y(Ux)$, $Y_y := Y(Uy)$ be two correlated random variables,
under the choice of a Haar random unitary $U$. Let $\lambda > 0$.
Then
\[
\Pr_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert > \lambda] \leq
2 \exp(-\frac{n \lambda^{2}}{8 L_2^2 \lVert x - y\rVert_2^2}) +
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c].
\]
\end{lemma}
\begin{proof}
By Fact~\ref{fact:extension}, there is a circled function $Y'$ that
agrees with $Y$ on $\Omega$ and is $L_2$-Lipschitz on all
of $\mathbb{S}_{\mathbb{C}^n}$.
Define correlated random variables $Y'_x$,
$Y'_y$ in the natural manner. Then
using Fact~\ref{fact:LevyLipschitz}, we get
\begin{eqnarray*}
\lefteqn{\Pr_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert > \lambda]} \\
& = &
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \in \Omega \times \Omega] \cdot
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert > \lambda|(Ux, Uy) \in \Omega \times \Omega
] \\
& &
{} +
\Pr_{U \sim \mathrm{Haar}}[
(Ux, Uy) \not \in \Omega \times \Omega
] \cdot
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert > \lambda|(Ux, Uy) \not \in \Omega \times \Omega
] \\
& = &
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \in \Omega \times \Omega] \cdot
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y'_x - Y'_{y}\rvert > \lambda|(Ux, Uy) \in \Omega \times \Omega
] \\
& &
{} +
\Pr_{U \sim \mathrm{Haar}}[
(Ux, Uy) \not \in \Omega \times \Omega
] \cdot
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert > \lambda|(Ux, Uy) \not \in \Omega \times \Omega
] \\
& \leq &
\Pr_{U \sim \mathrm{Haar}}[\lvert Y'_x - Y'_{y}\rvert > \lambda] +
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c] \\
& \leq &
2 \exp(-\frac{n \lambda^{2}}{8 L_2^2 \lVert x - y\rVert_2^2}) +
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c].
\end{eqnarray*}
This finishes the proof of the lemma.
\end{proof}
\begin{lemma}
\label{lem:expectationtdesign}
Let $Y: \mathbb{S}_{\mathbb{C}^n} \rightarrow \mathbb{R}$ be a
balanced polynomial of degree $a$ in entries of the vector $x \in \mathbb{C}^n$
that is
provided as input. Let
$\alpha(Y)$ denote the sum of absolute values of the coefficients of $Y$.
Suppose $Y$ has global Lipschitz constant $L_1$.
Suppose that
there exists a subset $\Omega \subseteq \mathbb{S}_{\mathbb{C}^n}$ such that
$Y$ restricted to $\Omega$
has a smaller Lipschitz constant $L_2$.
Let $x, y \in \mathbb{S}_{\mathbb{C}^n}$.
Let $Y_x := Y(Ux)$, $Y_y := Y(Uy)$ be two correlated random variables,
under the choice of a unitary $U$ chosen uniformly at random from an
$\epsilon$-approximate unitary $t$-design $\nu$.
Let $r$ be a positive integer satisfying $2ar \leq t$.
Let
$
0 < \epsilon <
\frac{
n^{t-r} (4 r L_2^2 \lVert x - y \rVert_2^2)^r
}{\alpha(Y)^{2r}}.
$
Then
\[
\mathbb{E}_{U \sim \nu}[\lvert Y_x - Y_{y}\rvert^{2r}] \leq
3 \left(\frac{4 r L_2^2 \lVert x - y\rVert_2^2}{n}\right)^{r} +
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c] \cdot
(L_1^2\lVert x- y \rVert_2^2)^{r}.
\]
\end{lemma}
\begin{proof}
Since
$Y_x -Y_y$ is a balanced polynomial in the entries of the unitary
matrix $U$, from Fact~\ref{fact:TvsHaar} we have
\[
\mathbb{E}_{U \sim \nu}[\lvert Y_x - Y_{y}\rvert^{2r}]
\overset{\mathrm{a}}{\leq}
\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert^{2r}] +
\frac{\epsilon \alpha(Y)^{2r}}{n^t}.
\]
By choosing
$\epsilon$ small enough to satisfy the constraint above, we get
$
\frac{\epsilon \alpha(Y)^{2r}}{n^t}
\overset{\mathrm{b}}{\leq}
\left(\frac{4 r L_2^2 \lVert x - y \rVert_2^2}{n}\right)^r.
$
Combining (a) and (b) gives
\[
\mathbb{E}_{U \sim \nu}[\lvert Y_x - Y_{y}\rvert^{2r}]
\overset{\mathrm{c}}{\leq}
\mathbb{E}_{U \sim \mathrm{Haar}}({\lvert Y_x - Y_{y}\rvert}^{2r}) +
\left(\frac{4 r L_2^2 \lVert x - y \rVert_2^2}{n}\right)^r.
\]
Now we find
$\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert^{2r}]$.
Since $Y$ is a balanced polynomial, it is circled.
By Fact~\ref{fact:extension}, there is a circled function $Y'$ such that
$Y'$ agrees with $Y$ on $\Omega$ and $Y'$ is $L_2$-Lipschitz on all
of $\mathbb{S}_{\mathbb{C}^n}$.
Define correlated random variables $Y'_x$,
$Y'_y$ in the natural manner. Then
\begin{eqnarray*}
\lefteqn{\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert^{2r}]} \\
& = &
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \in \Omega \times \Omega] \cdot
\mathbb{E}_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert^{2r}|(Ux, Uy) \in \Omega \times \Omega
] \\
& &
{} +
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \not \in \Omega \times \Omega] \cdot
\mathbb{E}_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert^{2r}|(Ux, Uy) \not \in \Omega \times \Omega]\\
& = &
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \in \Omega \times \Omega] \cdot
\mathbb{E}_{U \sim \mathrm{Haar}}[
\lvert Y'_x - Y'_{y}\rvert^{2r}|(Ux, Uy) \in \Omega \times \Omega
] \\
& &
{} +
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \not \in \Omega \times \Omega] \cdot
\mathbb{E}_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert^{2r}|(Ux, Uy) \not \in \Omega \times \Omega]\\
& \overset{\mathrm{d}}{\leq} &
\mathbb{E}_{U \sim Haar}[\lvert Y'_x - Y'_{y}\rvert^{2r}]+
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c] \cdot
(L_1^2\lVert x- y \rVert_2^2)^{r}.
\end{eqnarray*}
Now we find
$\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y'_x - Y'_{y}\rvert^{2r}]$
using Fact~\ref{fact:LevyLipschitz} and
Low's method \cite[Lemma~3.3]{low_2009}.
\begin{eqnarray*}
\lefteqn{\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y'_x - Y'_{y}\rvert^{2r}]} \\
& = &
\int_0^\infty
\Pr_{U \sim \mathrm{Haar}}[\lvert Y'_x - Y'_{y} \rvert^{2r}>\lambda] \,
d\lambda \
\;= \;
\int_0^\infty
\Pr_{U \sim \mathrm{Haar}}[\lvert Y'_x - Y'_{y} \rvert>\lambda^{1/(2r)}] \,
d\lambda \\
& \leq &
2 \int_0^\infty
\exp(-\frac{n \lambda^{1/r}}{8 L_2^2 \lVert x - y\rVert_2^2}) \,
d\lambda
\; \overset{\mathrm{e}}{\leq} \;
2 \left(\frac{4 r L_2^2 \lVert x - y\rVert_2^2}{n}\right)^{r}.
\end{eqnarray*}
Combining inequalities (d) and (e), we have
\[
\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert^{2r}] \leq
2 \left(\frac{4 r L_2^2 \lVert x - y\rVert_2^2}{n}\right)^{r} +
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c] \cdot
(L_1^2\lVert x- y \rVert_2^2)^{r}.
\]
Further combining with (c) gives us the desired conclusion of the lemma.
\end{proof}
We also need a so-called {\em chaining inequality} for probability
similar to Dudley's
inequality in geometric functional analysis
\cite{aubrun_szarek_werner_2010_main,pisier_1989}. The original
Dudley's inequality bounds the expectation of the supremum, over pairs
of correlated random variables, of the difference between them
in terms of an integral, over $\eta$, of
a certain function of the size of an $\eta$-net of $\mathbb{S}_{\mathbb{C}^n}$.
Our chaining lemma differs from it in two important respects.
First, instead of the expectation it bounds a tail probability of
the supremum, over pairs
of correlated random variables, of the difference between them.
Second, it replaces the integral by a finite summation
over $\eta$-nets of $\mathbb{S}_{\mathbb{C}^n}$ with geometrically decreasing
$\eta$. Despite the fancy name, our chaining lemma is a simple consequence
of the union bound of probabilities. Nevertheless, it is crucial to
proving our main result as it allows us to efficiently invoke
powerful measure concentration results in order to bound the variation of a
Lipschitz function on subspaces of $\mathbb{C}^n$.
\begin{lemma}[Chaining]
\label{lem:chaining}
Let $\{X_s\}_{s \in \mathcal{S}}$ be a family of correlated complex valued
random variables indexed
by elements of a compact metric space $\mathcal{S}$.
Let $\lambda, L_1 > 0$. The family is said to be $L_1$-Lipschitz if
for all $s, t \in \mathcal{S}$, $|X_s - X_t| \leq L_1 d(s,t)$ for
all points of the sample space.
Define $i_0$ to be the unique
integer such that the radius of $\mathcal{S}$ lies in the interval
$(2^{-i_0-1}, 2^{-i_0}]$. Define
$i_1 := \max\{i_0, \lceil \log \frac{2 L_1}{\lambda}\rceil\}$.
Let $p : \mathbb{Z} \rightarrow \mathbb{R}_+$ be a non-decreasing
function. Suppose the infinite series
$\sum_{i > i_0} \frac{\sqrt{|i| p(i)}}{2^i}$ is convergent with
value $C$.
Then,
\[
\Pr[\sup_{s,t \in \mathcal{S}} \lvert X_s-X_t\rvert > \lambda] \leq
\sum_{i = i_0+1}^{i_1+1}
\sum_{
(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i}:
d(u, u^\prime) < 2^{-i+2}
} \Pr [\lvert X_u - X_{u^\prime}\rvert >
\frac{\lambda \sqrt{|i| p(i)}}{4C \cdot 2^i}
],
\]
for a sequence of $2^{-i}$-nets $\mathcal{N}_i$, $i_0 \leq i \leq i_1$,
$|\mathcal{N}_{i_0}| = 1$, of $\mathcal{S}$.
\end{lemma}
\begin{proof}
For every $i \in \mathbb{Z}$, let $\mathcal{N}_i$ be a $2^{-i}$-net of $\mathcal{S}$.
Let $i_0$ be such that radius of $\mathcal{S}$
lies in $(2^{-(i_0+1)},2^{-i_0}]$. The net $\mathcal{N}_{i_0}$
consists of a single element, say $s_0$. For every $s \in \mathcal{S}$ and
$i \in \mathbb{Z}$, let $\pi_i(s)$ be an element of $\mathcal{N}_i$ satisfying
$d(s,\pi_i(s)) \leq 2^{-i}$. We have the following chaining equation
for every $s \in \mathcal{S}$:
\[
X_s =
X_{s_0} +
\left(\sum_{i = i_0}^{i_i} (X_{\pi_{i+1}(s)}-X_{\pi_{i}(s)})\right) +
(X_s - X_{\pi_{i_1 + 1}(s)}).
\]
Lipschitz property of the family implies that
\begin{eqnarray*}
\sup_{s,t \in \mathcal{S}} \lvert X_s-X_t\rvert
& \leq &
2 \sum_{i = i_0}^{i_1}
\sup_{s \in \mathcal{S}} \lvert X_{\pi_{i+1}(s)}-X_{\pi_{i}(s)} \rvert +
L_1 2^{-i_1} \\
& \leq &
2 \sum_{i = i_0}^{i_1}
\sup_{(u,u^\prime) \in \mathcal{N}_{i} \times \mathcal{N}_{i+1} : d(u,u')<2^{-i+1}}
\lvert X_u - X_{u'} \rvert +
L_1 2^{-i_1} \\
& \leq &
2 \sum_{i = i_0+1}^{i_1+1}
\sup_{(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i} : d(u,u')<2^{-i+2}}
\lvert X_u - X_{u'} \rvert +
\frac{\lambda}{2}.
\end{eqnarray*}
Now if
$\sup_{s,t \in \mathcal{S}} \lvert X_s-X_t\rvert > \lambda$, there must
exist an $i$, $i_0 + 1 \leq i \leq i_1 + 1$ such that
\[
\sup_{(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i} : d(u,u')<2^{-i+2}}
\lvert X_u - X_{u'} \rvert >
\frac{\lambda \sqrt{|i| p(i)}}{4C \cdot 2^i}.
\]
Applying the union bound on probability leads us to the conclusion
of the lemma.
\end{proof}
We now prove our sharp Dvoretzky-like theorem for subspaces chosen
from the Haar measure using stratified analysis.
\begin{theorem}
\label{thm:mainHaar}
Let $p : \mathbb{N} \rightarrow \mathbb{R}_+$ be a non-decreasing
function. Suppose the infinite series
$\sum_{i > 0} \frac{\sqrt{i p(i)}}{2^i}$ is convergent with
value $C$.
Let $f: \mathbb{S}_{\mathbb{C}^n} \rightarrow \mathbb{R}$ have
global Lipschitz constant $L_1$.
Let $L_2, c_1, c_2, c_3, \lambda > 0$.
Define $m := \lceil \frac{c_1 n \lambda^2}{L_2^2} \rceil$.
Suppose there is an increasing sequence of subsets
$\Omega_1 \subseteq \Omega_2 \subseteq \cdots$ of $\mathbb{S}_{\mathbb{C}^n}$
such that with probability at least $1 - c_2 e^{-c_3 m i}$,
a Haar random subspace of dimension
$m$ lies in $\Omega_i$ and $f$ restricted to $\Omega_i$ has Lipschitz
constant $L_2 \sqrt{p(i)}$.
Then there exists a constant $c$ depending on $c_3$, $C$,
$0 < c < 1$, such that for
$
m^\prime := c m
$
with probability at least $1- (c_2 + 1) 2^{-m'}$, a
subspace $W$ of dimension
$m^\prime$ chosen with respect to Haar measure
satisfies the property that
$\lvert f(w)- \mu \rvert < \lambda$
for all points $w \in W \cap \mathbb{S}_{\mathbb{C}^n}$.
\end{theorem}
\begin{proof}
In this proof $\mathbb{S}_{\mathbb{C}^{n}}$ denotes the unit $\ell_2$-length sphere in
$\mathbb{C}^n$ together with the origin point $0$. The radius of
$\mathbb{S}_{\mathbb{C}^{n}}$ is one which makes $i_0 = 0$ in
Lemma~\ref{lem:chaining}.
Consider a canonical embedding of $\mathbb{S}_{\mathbb{C}^{m'}}$ into
$\mathbb{S}_{\mathbb{C}^{m}}$ and further into $\mathbb{S}_{\mathbb{C}^n}$. Define
\[
B_i := \{U \in \mathbb{U}(n): \forall z \in \mathbb{S}_{\mathbb{C}^{m}}, Uz \in \Omega_i\}.
\]
For $s \in \mathbb{S}_{\mathbb{C}^{m'}}$, define the random variable
$Y_s := f(Us) - \mu$,
where the randomness arises solely from the choice of $U \in \mathbb{U}(n)$.
Then $\Pr_{U \sim \mathrm{Haar}}[B_i] \geq 1 - c_2 e^{-c_3 m i}$.
Let $i_1 := \lceil \log \frac{2 L_1}{\lambda} \rceil$.
Let $\mathcal{N}_i$, $i = 0, 1, \ldots, i_1$ be a sequence of
$2^{-i}$-nets in $\mathbb{S}_{\mathbb{C}^{m'}}$ of minimum cardinality,
where $\mathcal{N}_0 := \{0\}$
and $Y_0 := 0$. We can take $|\mathcal{N}_i| \overset{a}{\leq} 2^{2 (i+2) m'}$ by
Fact~\ref{fact:net}. By Lemma~\ref{lem:chaining}
\[
\Pr_{U \sim \mathrm{Haar}}[
\sup_{s,t \in \mathbb{S}_{\mathbb{C}^{m'}}}
\lvert Y_s -Y_t \rvert > \lambda
] \leq
2 \sum_{i=1}^{i_1+1}
\sum_{
(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i}:
\lVert u- u^\prime \rVert_2 < 2^{-i+2}
}
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y_u - Y_{u^\prime}\rvert >
\frac{\lambda \sqrt{i p(i)}}{4C \cdot 2^i}
].
\]
Applying Lemma~\ref{lem:expectationHaar} to the set $B_{i}$ gives, for
$u$, $u'$ satisfying $\lVert u- u^\prime \rVert_2 < 2^{-i+2}$,
\begin{eqnarray*}
\lefteqn{
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y_u - Y_{u^\prime}\rvert >
\frac{\lambda \sqrt{i p(i)}}{4C \cdot 2^i}
]
} \\
& \leq &
2 \exp\left(-\frac{n \lambda^2 i p(i)}{2^7 C^2 2^{2i} L_2^2 p(i)
\lVert u -u^\prime \rVert_2^2} \right) +
2 \Pr_{z \sim \mathrm{Haar}} [z \in \Omega_{i}^c] \\
& \leq &
2 \exp\left(-\frac{n i \lambda^2 }{2^9 C^2 L_2^2} \right) +
2 \Pr_{z \sim \mathrm{Haar}} [z \in \Omega_{i}^c] \\
\\
& \leq &
2 \exp\left(-\frac{im}{2^9 C^2} \right) +
2 c_2 \exp(-c_3 m i)
\; \leq \;
2 (c_2 + 1) \exp(-c_4 m i),
\end{eqnarray*}
for a constant $c_4$ depending only on $C$ and $c_3$.
This gives us
\begin{eqnarray*}
\lefteqn{
\Pr_{U \sim \mathrm{Haar}}[
\sup_{s,t \in \mathbb{S}_{\mathbb{C}^{m'}}}
\lvert Y_s -Y_t \rvert > \lambda
]
} \\
& \leq &
4 (c_2 + 1) \sum_{i=1}^{i_1+1}
\sum_{
(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i}:
\lVert u- u^\prime \rVert_2 < 2^{-i+2}
}
e^{-c_4 m i}
\;\leq\;
4 (c_2 + 1) \sum_{i=1}^{i_1+1}
|\mathcal{N}_{i-1}| \cdot |\mathcal{N}_{i}| \cdot e^{-c_4 m i} \\
& \leq &
4 (c_2 + 1) \sum_{i=1}^{i_1+1}
2^{4 m' (i+2)}
e^{-c_4 m i}
\;\leq\;
(c_2 + 1) 2^{-m'},
\end{eqnarray*}
where the third inequality follows from (a) and the fourth inequality
follows from the definition $m' := c m$ for an appropriate choice
of $c$ depending only on $c_4$. In other words, $c$ depends only on
$C$ and $c_3$.
Taking $t = 0$, we see that
with probability at least $1 - (c_2 + 1) 2^{-m'}$ over
the choice of a Haar random unitary, we have
that for all $s \in \mathbb{S}_{\mathbb{C}^{m'}}$, $|Y_s| \leq \lambda$.
This completes the proof of the theorem.
\end{proof}
\paragraph{Remark:}
The sets $\Omega_i$ and the Lipschitz constants $L_2 \sqrt{p(i)}$
for $1 \leq i \leq \lceil \log \frac{2 L_1}{\lambda} \rceil + 1$
formalise the idea of stratified analysis mentioned intuitively in the
introduction. As $i$ increases the relevant Lipschitz constant increases.
So we need a finer net i.e. a $2^{-i}$-net for the $i$th layer
$\Omega_i$ in order to control the variation of $f$ for subspaces
lying inside $\Omega_i$.
With exponentially high probability, we thus get a
Haar random subspace of dimension $m^\prime$, slightly smaller than $m$,
where $f$ is almost constant.
Note that the definition of $m$ involves only the smallest local
Lipschitz constant $L_2$. Thus the dimension of the space $m'$ that
we obtain is larger than what would be obtained by a naive
analysis which would be constrained by the
global Lipschitz constant $L_1$. Moreover, a naive analysis would not
give exponentially high probability, just an arbitrary constant close
one. These two properties underscore the power of our stratified
analysis. However, applying the stratified analysis to a concrete
function is not always straightforward. We need to define the
layers $\Omega_1, \Omega_2, \ldots, $ properly and show separately that
Haar random subspaces of dimension $m$ lie in $\Omega_i$ with
probability $1 - c_2 e^{-c_3 m i}$. But for several interesting functions
this can be done without much difficulty.
This will become clearer in
Section~\ref{sec:vonNeumannentropy} where we will show how to
recover Aubrun, Szarek and Werner's result for the Haar measure directly
from
Theorem~\ref{thm:mainHaar}, without having to apply a Dvoretzky-style
theorem twice in a messy fashion as in the original paper
\cite{aubrun_szarek_werner_2010_main}. Moreover, we get success probability
exponentially close to one unlike Aubrun, Szarek and Werner who could
get only a constant close to one. Furthermore, our
methods extend to approximate $t$-designs and allows us to prove
exponentially close to one probability even for that setting.
We now prove our sharp Dvoretzky-like theorem for subspaces chosen
from approximate $t$-designs using stratified analysis.
\begin{theorem}
\label{thm:maintdesign}
Let $p : \mathbb{N} \rightarrow \mathbb{R}_+$ be a non-decreasing
function. Suppose the infinite series
$\sum_{i > 0} \frac{\sqrt{i p(i)}}{2^i}$ is convergent with
value $C$.
Let $f: \mathbb{S}_{\mathbb{C}^n} \rightarrow \mathbb{R}$ be a balanced degree $`a'$
polynomial with global Lipschitz constant $L_1$.
Let $0 \leq L_2 \leq 1$, $c_1, c_2, c_3, \lambda > 0$.
Define $m := \lceil \frac{c_1 n \lambda^2}{L_2^2} \rceil$.
Suppose there is an increasing sequence of subsets
$\Omega_1 \subseteq \Omega_2 \subseteq \cdots$ of $\mathbb{S}_{\mathbb{C}^n}$
such that with probability at least $1 - c_2 e^{-c_3 m i}$,
a Haar random subspace of dimension
$m$ lies in $\Omega_i$ and $f$ restricted to $\Omega_i$ has Lipschitz
constant $L_2 \sqrt{p(i)}$.
Suppose
\[
0 < \epsilon <
\left(\frac{\lambda}{4 L_1}\right)^{2m} \cdot
\frac{n^{(2a-1)m} (L_2^2 p(1))^{m}}{\max\{\alpha(f)^{2m},1\}}.
\]
Then there exists a constant $c$ depending on $c_1$, $c_3$, $C$, $p(1)$,
$0 < c < 1$ such that for
\[
m^\prime :=
c m
\frac{\log \log \frac{C^2 L_1^2}{\lambda^2 p(1)}}
{\lceil \log \frac{C^2 L_1^2}{\lambda^2 p(1)} \rceil},
\]
with probability at least $1- (c_2+1) 2^{-m'}$, a
subspace $W$ of dimension
$m^\prime$ chosen under an $\epsilon$-approximate $(2am)$-design $\nu$
satisfies the property that
$\lvert f(w)- \mu \rvert < \lambda$
for all points $w \in W \cap \mathbb{S}_{\mathbb{C}^n}$.
\end{theorem}
\begin{proof}
In this proof $\mathbb{S}_{\mathbb{C}^{n}}$ denotes the unit $\ell_2$-length sphere in
$\mathbb{C}^n$ together with the origin point $0$. The radius of
$\mathbb{S}_{\mathbb{C}^{n}}$ is one which makes $i_0 = 0$ in
Lemma~\ref{lem:expectationtdesign}.
Consider a canonical embedding of $\mathbb{S}_{\mathbb{C}^{m'}}$ into
$\mathbb{S}_{\mathbb{C}^{m}}$ and further into $\mathbb{S}_{\mathbb{C}^n}$. Define
\[
B_i := \{U \in \mathbb{U}(n): \forall z \in \mathbb{S}_{\mathbb{C}^{m}}, Uz \in \Omega_i\}.
\]
For $s \in \mathbb{S}_{\mathbb{C}^{m'}}$, define the random variable
$Y_s := f(Us) - \mu$,
where the randomness arises solely from the choice of $U \in \mathbb{U}(n)$.
Then $\Pr_{U \sim \mathrm{Haar}}[B_i] \geq 1 - c_2 e^{-c_3 m i}$.
Let $i_1 := \lceil \log \frac{2 L_1}{\lambda} \rceil$.
Let $\mathcal{N}_i$, $i = 0, 1, \ldots, i_1$ be a sequence of
$2^{-i}$-nets in $\mathbb{S}_{\mathbb{C}^{m'}}$ of minimum cardinality,
where $\mathcal{N}_0 := \{0\}$
and $Y_0 := 0$. We can take $|\mathcal{N}_i| \overset{a}{\leq} 2^{2 (i+2) m'}$ by
Fact~\ref{fact:net}. By Lemma~\ref{lem:chaining}
\begin{equation}
\label{eq:chainingtdesign}
\Pr_{U \sim \nu}[
\sup_{s,t \in \mathbb{S}_{\mathbb{C}^{m'}}}
\lvert Y_s -Y_t \rvert > \lambda
] \leq
2 \sum_{i=1}^{i_1+1}
\sum_{
(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i}:
\lVert u- u^\prime \rVert_2 < 2^{-i+2}
}
\Pr_{U \sim \nu}[
\lvert Y_u - Y_{u^\prime}\rvert >
\frac{\lambda \sqrt{i p(i)}}{4C \cdot 2^i}
].
\end{equation}
Let $r$ be a positive integer such that $r (i_1+1) < m$.
Applying Lemma~\ref{lem:expectationtdesign} to the set $B_{i}$ gives, for
$u$, $u'$ satisfying $\lVert u- u^\prime \rVert_2 < 2^{-i+2}$,
\begin{eqnarray*}
\lefteqn{
\Pr_{U \sim \nu}[
\lvert Y_u - Y_{u^\prime}\rvert >
\frac{\lambda \sqrt{i p(i)}}{4C \cdot 2^i}
]
} \\
& = &
\Pr_{U \sim \nu}[
\lvert Y_u - Y_{u^\prime}\rvert^{2 r i} >
\left(\frac{\lambda^2 i p(i)}{2^4 C^2 2^{2i}}\right)^{r i}
]
\;\leq\;
\left(\frac{2^{2i+4} C^2}{\lambda^2 i p(i)} \right)^{r i}
\mathbb{E}_{U \sim \nu}[\lvert Y_u - Y_{u^\prime}\rvert^{2 r i}] \\
& \leq &
3 \left(\frac{2^{2i+4} C^2}{\lambda^2 i p(i)} \right)^{r i}
\left(
\left(
\frac{4 r i L_2^2 p(i) \lVert u - u^\prime \rVert_2^2}{n}
\right)^{r i}
+ c_2 e^{-c_3 m i} \cdot (L_1^2 \lVert u- u^\prime \rVert_2^2)^{r i}
\right) \\
& \leq &
3 \left(
\frac{2^{2i+6} C^2 r L_2^2 \lVert u- u^\prime \rVert_2^2}
{n \lambda^2}
\right)^{r i} +
3 c_2 e^{-c_3 m i}
\left(
\frac{2^{2i+4} C^2 L_1^2 \lVert u- u^\prime \rVert_2^2}{\lambda^2 i p(i)}
\right)^{r i} \\
& \leq &
\underbrace{
3 \left(
\frac{2^{10} C^2 r L_2^2}
{n \lambda^2}
\right)^{r i}
}_{=: \mathrm{I}} +
\underbrace{
3 c_2 e^{-c_3 m i}
\left(
\frac{2^{8} C^2 L_1^2}{\lambda^2 p(1)}
\right)^{r i}
}_{=: \mathrm{II}}.
\end{eqnarray*}
We now analyse the two terms in the above expression.
Take
\[
r :=
\frac{c_4 n \lambda^2}{2^{10} C^2 L_2^2} \cdot
\frac{1}{\lceil \log \frac{2^8 C^2 L_1^2}{\lambda^2 p(1)} \rceil}
\]
for a constant $c_4$, $0 < c_4 < 1$, $c_4$ depending only on
$C$, $c_1$, $c_3$, $p(1)$ chosen to be small enough so that
$r (i_1+1) < m$ and
$\frac{c_4 n \lambda^2}{2^{10} C^2 L_2^2} \leq \frac{c_3 m}{2}$.
Substitute $r$ back in I and II to get
\[
\mathrm{I}
\leq
3 \cdot 2^{-r i \log \log \frac{2^8 C^2 L_1^2}{\lambda^2 p(1)}},
~~~
\mathrm{II}
\leq
3 c_2 e^{-c_3 m i} 2^{\frac{c_3 m i}{2}}
<
3 c_2 e^{-c_3 m i / 2}.
\]
We choose
\[
m'' :=
r \log \log \frac{2^8 C^2 L_1^2}{\lambda^2 p(1)} <
\frac{c_3 m}{2}.
\]
This gives us
\[
\mathrm{I}
\leq
3 \cdot 2^{-m'' i},
~~~
\mathrm{II}
\leq
3 c_2 e^{-m'' i}.
\]
Thus, we have shown that
\[
\Pr_{U \sim \nu}[
\lvert Y_u - Y_{u^\prime}\rvert >
\frac{\lambda\sqrt{p(i)}}{4 C \cdot 2^i}
] \leq 3 (c_2 + 1) 2^{-m'' i}.
\]
Substituting above in Equation~\ref{eq:chainingtdesign}, we get
\begin{eqnarray*}
\lefteqn{
\Pr_{U \sim \nu}[
\sup_{s,t \in \mathbb{S}_{\mathbb{C}^{m^\prime}}}\lvert Y_s -Y_t \rvert > \lambda
]
} \\
& \leq &
2 \sum_{i=1}^{i_1+1}
\sum_{u,u^\prime \in \mathcal{N}_{i-1} \times \mathcal{N}_{i}:
\lVert u- u^\prime \rVert < 2^{-i+2}
} 3 (c_2 + 1) 2^{-m'' i} \\
& \leq &
6 (c_2+1) \sum_{i=1}^{i_1+1}
|\mathcal{N}_{i-1}| \cdot |\mathcal{N}_{i}| \cdot
2^{-m'' i}
\;\leq\;
6 (c_2+1) \sum_{i=1}^{i_1+1}
2^{4 m'(i+2)} 2^{-m'' i}
\;\leq\;
(c_2 + 1) 2^{-m'},
\end{eqnarray*}
if $m'$ is chosen as indicated above for a small enough constant
$c$, $0 < c < 1$, $c$ depending only on $c_4$, $c_1$, $C$ i.e.
$c$ depending only on $C$, $c_1$, $c_3$, $p(1)$.
Taking $t = 0$, we see that
with probability at least $1 - (c_2 + 1) 2^{-m^\prime}$ over
the choice of a uniformly random unitary from the approximate
$(2am)$-design, we have
that for all $s \in \mathbb{S}_{\mathbb{C}^{m'}}$, $|Y_s| \leq \lambda$.
This completes the proof of the theorem.
\end{proof}
\section{Strict subadditivity of minimum output von Neumann entropy for
approximate $t$-designs}
\label{sec:vonNeumannentropy}
We first apply Theorem~\ref{thm:mainHaar}
in order to directly recover Aubrun, Szarek and Werner's result
\cite{aubrun_szarek_werner_2010_main}
that channels with Haar random unitary Stinespring dilations
exhibit strict subadditivity of minimum output von Neumann entropy. In
fact, we
go beyond their result in the sense that we obtain exponentially
high probability close to one as opposed to constant
probability. After this warmup, we apply Theorem~\ref{thm:maintdesign}
in order to show that channels with approximate $n^{2/3}$-design
unitary Stinespring dilations exhibit strict
subadditivity of minimum output von Neumann entropy with exponentially high
probability close to one.
Let $k$ be a positive integer.
Consider the sphere $\mathbb{S}_{\mathbb{C}^{k^3}}$. Define the $k \times k^2$
matrix $M$ to be the rearrangment of a $k^3$-tuple from
$\mathbb{S}_{\mathbb{C}^{k^3}}$. Note that the $\ell_2$-norm on $\mathbb{C}^{k^3}$ is
the same as the Frobenius norm on $\mathbb{C}^{k \times k^2}$.
In Step~I, we define the
function $f: \mathbb{S}_{\mathbb{C}^{k^3}} \rightarrow \mathbb{R}$ as
$f(M) := \lVert M \rVert_\infty$. The function $f$ has global Lipschitz
constant $L_1 = 1$ since
\[
|f(M) - f(N)| \leq
\lVert M - N \rVert_\infty \leq
\lVert M - N \rVert_2.
\]
For large enough $k$ the mean $\mu$ of $f$, under the Haar measure, is less
than $2 k^{-1/2}$ \cite[Corollary~7]{aubrun_szarek_werner_2010_main}.
We use the notation of Theorem~\ref{thm:mainHaar}.
Define $L_2 := 1$, $p(i) := 1$ for all $i \in \mathbb{N}$. Then $C < 2$.
Define the layers $\Omega_1, \Omega_2, \ldots, $ to be all of
$\mathbb{S}_{\mathbb{C}^{k^3}}$. Let $j$, $4 \leq j \leq k$ be a positive integer.
Let $\lambda_j := \sqrt{\frac{j}{k}}$. Define $c_1 := 1$,
$m = k^2$, $c_2 := 0$, $c_3 := 1$. Trivially,
a Haar random subspace of dimension $m j$ lies in $\Omega_i$ with
probability at least $1 - c_2 e^{-c_3 m j i}$. Theorem~\ref{thm:mainHaar}
tells us that there is a universal constant $\hat{c}_1$ such that
for $m' := \hat{c}_1 k^2$, with probability at least
$1 - 2^{-m' j}$, a Haar random subspace $W$ of dimension $m' j$
satisfies
\[
\lVert M \rVert_\infty <
\frac{2}{\sqrt{k}} + \sqrt{\frac{j}{k}} <
2 \sqrt{\frac{j}{k}}
\]
for all $M \in W$.
In Step~II, we define the
function $f: \mathbb{S}_{\mathbb{C}^{k^3}} \rightarrow \mathbb{R}$ as
$f(M) := \lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2$.
The function $f$ has global Lipschitz
constant $L_1 = 2$ since
\begin{eqnarray*}
|f(M) - f(N)|
& \leq &
\lVert M M^\dagger - N N^\dagger \rVert_2
\; \leq \;
\lVert M M^\dagger - M N^\dagger \rVert_2 +
\lVert M N^\dagger - N N^\dagger \rVert_2 \\
& \leq &
\lVert M \rVert_\infty \lVert M^\dagger - N^\dagger \rVert_2 +
\lVert N^\dagger \rVert_\infty \lVert M - N \rVert_2 \\
& = &
(\lVert M \rVert_\infty + \lVert N \rVert_\infty)
\lVert M - N \rVert_2
\;\leq\;
2 \lVert M - N \rVert_2.
\end{eqnarray*}
The mean $\mu$ of $f$, under the Haar measure, is less
than $c_0 k^{-1}$ for a universal constant $c_0$
\cite[Corollary~7]{aubrun_szarek_werner_2010_main}.
We use the notation of Theorem~\ref{thm:mainHaar}.
Let $j$, $c_0 < j \leq k$ be a positive integer.
Define $L_2 := 4 \sqrt{\frac{j}{k}}$, $p(i) := i+3$ for all $i \in \mathbb{N}$.
Then $C \leq 4$.
Define the layers $\Omega_1, \Omega_2, \ldots, $ to be
the subsets
\[
\Omega_i :=
\left\{
M \in \mathbb{S}_{\mathbb{C}^{k^3}}:
\lVert M \rVert_\infty \leq 2 \sqrt{\frac{j(i+3)}{k}}
\right\}.
\]
It is easy to see that $f$ restricted to $\Omega_i$ has local
Lipschitz constant at most $L_2 \sqrt{p(i)}$.
Let $\lambda := \frac{j}{k}$. Define $c_1 := 16 \hat{c}_1$,
$m = \hat{c}_1 j k^2$, $c_2 := 1$, $c_3 := \ln 2$. By the previous
paragraph,
a Haar random subspace of dimension $m (i+3)$ lies in $\Omega_i$ with
probability at least
$1 - c_2 e^{-c_3 m (i+3)} \geq 1 - c_2 e^{-c_3 m i}$.
Theorem~\ref{thm:mainHaar}
tells us that there is a universal constant $\hat{c}_2$ such that
for $m' := \hat{c}_2 k^2$, with probability at least
$1 - 2^{-m' j}$, a Haar random subspace $W$ of dimension $m' j$
satisfies
\[
f(M) = \lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}\rVert_2 <
\frac{c_0}{k} + \frac{j}{k} <
\frac{2j}{k}
\]
for all $M \in W$. Setting $j = 1$ allows us to recover Aubrun, Szarek
and Werner's technical result \cite{aubrun_szarek_werner_2010_main} with
probability exponentially close to one viz.
with probability at least
$1 - 2^{-m'}$, a Haar random subspace $W$ of dimension $m'$
satisfies
$
\lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}\rVert_2 <
\frac{2}{k}
$
for all $M \in W$. We will now see how this implies the existence
of a channel with strictly subadditive minimum output von Neumann
entropy.
\begin{fact}
\label{fact:subadditivity}
Let $k$ be a positive integer.
Let $W$ be a Haar random subspace of dimension $m := \hat{c}_2 k^2$
chosen from the Hilbert space $\mathbb{C}^{k^3}$, where $\hat{c}_2$ is a
universal constant. Let $\Phi$ be the channel with output dimension
$k$ corresponding to the
subspace $W$. Then with probability at least $1 - 2^{-m}$ over the
choice of $W$,
\[
S_{\mathrm{min}}(\Phi)\geq
\log k - \frac{4}{k},
~~~
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi}) \leq
2 \log k - \frac{\hat{c}_2 \log k}{k} +
O \left(\frac{1}{k} \right).
\]
In other words,
$
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi}) <
S_{\mathrm{min}}(\Phi) + S_{\mathrm{min}}(\bar{\Phi})
$
for large enough $k$.
\end{fact}
\begin{proof}
The input dimension of the channel $\Phi$ is $\dim W = m$. The
Stinespring dilation
of the channel $\Phi$ is the $k^3 \times k^3$ unitary matrix
that defines the subspace $W$. The subspace $W$ is obtained by
taking the span of first $m$ columns of a Haar random unitary matrix.
Let $M$ be a unit $\ell_2$-norm vector in $\mathbb{C}^{k^3}$ rearranged as
a $k \times k^2$ matrix.
From Fact~\ref{fact:minoutputentropy}, we get
\[
S_{\mathrm{min}}(\Phi)\geq
\log k - k \max_{M \in W} \lVert M M^\dag - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2^2 \geq
\log k - \frac{4}{k}.
\]
And from Fact~\ref{fact:maxeigenvalue}, with $d = k^2$, we get
\begin{eqnarray*}
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi})
& \leq &
2 \log k -
\frac{m}{kd} \log k +
O \left( \frac{m}{kd} \log \frac{d}{m} + \frac{1}{k} \right) \\
& = &
2 \log k - \frac{\hat{c}_2 \log k}{k} +
O \left(\frac{1}{k} \right) \\
& < &
S_{\mathrm{min}}(\Phi) + S_{\mathrm{min}}(\bar{\Phi}),
\end{eqnarray*}
for large enough $k$.
\end{proof}
Thus we have shown
that for large enough $n$, Haar random $n \times n$ unitaries give rise
to channels
exhibiting strict subadditivity of minimum output von Neumann
entropy implying that classical Holevo capacity of quantum channels can
be superadditive.
In Step~III, we define the
function $f: \mathbb{S}_{\mathbb{C}^{k^3}} \rightarrow \mathbb{R}$ as
$f(M) := \lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2^2$ i.e.
this $f$ is the square of the $f$ defined in Step~II above.
Now, $f$ is a balanced polynomial of degree $a = 2$ and
$1 < \alpha(f) < k^{6}$ as can be seen by considering
$f(J)$ where $J$ is the $k \times k^2$ all ones matrix.
The function $f$ has global Lipschitz
constant $L_1 = 4$ since
\begin{eqnarray*}
|f(M) - f(N)|
& \leq &
|\lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2 -
\lVert N N^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2| \cdot
|\lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2 +
\lVert N N^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2| \\
& \leq &
(\lVert M \rVert_\infty + \lVert N \rVert_\infty)
(\lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2 +
\lVert N N^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2)
\lVert M - N \rVert_2 \\
& \leq &
4 \lVert M - N \rVert_2.
\end{eqnarray*}
The mean $\mu$ of $f$ under the Haar measure is less
than $c_0^2 k^{-2}$ for the same universal constant $c_0$
\cite[Corollary~7]{aubrun_szarek_werner_2010_main}.
We use the notation of Theorem~\ref{thm:maintdesign}.
Define $L_2 := 16 k^{-3/2} $, $p(i) := i^3$ for all $i \in \mathbb{N}$.
Then $C \leq 5$.
Define the layers $\Omega_1, \Omega_2, \ldots, $ to be
the subsets
\[
\Omega_i :=
\left\{
M \in \mathbb{S}_{\mathbb{C}^{k^3}}:
\lVert M \rVert_\infty \leq 2 \sqrt{\frac{i}{k}},
\lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}\rVert_2 < \frac{2i}{k}
\right\}.
\]
It is easy to see that $f$ restricted to $\Omega_i$ has local
Lipschitz constant at most $L_2 \sqrt{p(i)}$.
Let $\lambda := k^{-2}$. Define $c_1 := 2^8 \hat{c}_2$,
$m = \hat{c}_2 k^2 < \hat{c}_1 k^2$, $c_2 := 2$,
$c_3 := \ln 2$. By the previous two
paragraphs,
a Haar random subspace of dimension $m i$ lies in $\Omega_i$ with
probability at least
$1 - c_2 e^{-c_3 m i}$.
In particular,
a Haar random subspace of dimension $m$ lies in $\Omega_i$ with
probability at least
$1 - c_2 e^{-c_3 m i}$.
Let
\[
0 \leq \epsilon <
\left(\frac{1}{16 k^2}\right)^{2m} \frac{k^{9m} k^{-3m}}{k^{12m}} =
(4 k)^{-10 \hat{c}_2 k^2}.
\]
Theorem~\ref{thm:maintdesign}
tells us that there is a universal constant $\hat{c}_3$ such that
for
\[
m' := \hat{c}_3 k^2 \frac{\log \log k}{\log k},
\]
with probability at least
$1 - 3 \cdot 2^{-m'}$, a subspace $W$ of dimension $m'$ chosen from an
$\epsilon$-approximate $(4 \hat{c}_2 k^2)$-design $\nu$ satisfies
\[
f(M) = \lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}\rVert_2^2 <
\frac{c_0^2}{k^2} + \frac{1}{k^2} =
\frac{c_0^2 + 1}{k^2}
\]
for all $M \in W$.
We shall now see how this result gives us a channel with strict
subadditivity of minimum output von Neumann entropy.
\begin{theorem}
\label{thm:subadditivity}
Let $k$ be a positive integer.
Let $W$ be a subspace of dimension
$
m' := \hat{c}_3 k^2 \frac{\log \log k}{\log k}
$
chosen with uniform probability from a
$k^{-8 \hat{c}_2 k^2}$-approximate unitary $(4 \hat{c}_2 k^2)$-design
from the Hilbert space $\mathbb{C}^{k^3}$, where $\hat{c}_2$, $\hat{c}_3$ are
universal constants. Let $\Phi$ be the channel with output dimension
$k$ corresponding to the
subspace $W$. Then with probability at least $1 - 3 \cdot 2^{-m'}$ over the
choice of $W$,
\[
S_{\mathrm{min}}(\Phi)\geq
\log k - \frac{c_0}{k},
~~~
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi}) \leq
2 \log k - \frac{\hat{c}_3 \log \log k}{k} +
O \left(\frac{(\log \log k)^2}{k \log k} + \frac{1}{k} \right),
\]
for a universal constant $c_0$.
In other words,
$
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi}) <
S_{\mathrm{min}}(\Phi) + S_{\mathrm{min}}(\bar{\Phi})
$
for large enough $k$.
\end{theorem}
\begin{proof}
The input dimension of the channel $\Phi$ is $\dim W = m'$. The
Stinespring dilation
of the channel $\Phi$ is the $k^3 \times k^3$ unitary matrix
that defines the subspace $W$. The subspace $W$ is obtained by
taking the span of first $m'$ columns of the unitary matrix. This unitary
matrix is chosen uniformly at random from a
$k^{-8 \hat{c}_2 k^2}$-approximate
unitary $(4 \hat{c}_2 k^2)$-design.
Let $M$ be a unit $\ell_2$-norm vector in $\mathbb{C}^{k^3}$ rearranged as
a $k \times k^2$ matrix.
From Fact~\ref{fact:minoutputentropy}, we get
\[
S_{\mathrm{min}}(\Phi)\geq
\log k - k \max_{M \in W} \lVert M M^\dag - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2^2 \geq
\log k - \frac{c_0^2 + 1}{k}.
\]
And from Fact~\ref{fact:maxeigenvalue}, with $d = k^2$, we get
\begin{eqnarray*}
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi})
& \leq &
2 \log k -
\frac{m^\prime}{kd} \log k +
O \left( \frac{m'}{kd} \log \frac{d}{m'} + \frac{1}{k} \right) \\
& = &
2 \log k - \frac{\hat{c}_3 \log \log k}{k} +
O \left( \frac{(\log \log k)^2}{k \log k} + \frac{1}{k} \right) \\
& < &
S_{\mathrm{min}}(\Phi) + S_{\mathrm{min}}(\bar{\Phi}),
\end{eqnarray*}
for large enough $k$.
\end{proof}
Thus we have shown
that for large enough $n$, approximate unitary $n^{2/3}$-designs give rise
to channels
exhibiting strict subadditivity of minimum output von Neumann
entropy, implying that classical Holevo capacity of quantum channels can
be superadditive.
\paragraph{Remark:}
Observe that the counter example we get for additivity conjecture for
classical Holevo capacity of quantum channels, when the channel is chosen
from an approximate unitary $t$-design has weaker parameters
than a channel chosen from Haar random unitaries.
Nevertheless, as explained in the introduction our work
is the first partial derandomisation of a construction of quantum
channels violating additivity of classical Holevo capacity.
\section{Strict subadditivity of minimum output R\'{e}nyi $p$-entropy for
approximate $t$-designs}
\label{sec:Renyipentropy}
In this section, we apply Proposition~\ref{prop:poly} and
Theorem~\ref{thm:maintdesign}
in order to show that channels with approximate
$(n^{1.7} \log n)$-design
unitary Stinespring dilations exhibit strict
subadditivity of minimum output R\'{e}nyi $p$-entropy for
$p > 1$ with exponentially high
probability close to one.
Let $k$ be a positive integer.
Consider the sphere $\mathbb{S}_{\mathbb{C}^{k^3}}$. Define the $k \times k^{2}$
matrix $M$ to be the rearrangment of a $k^3$-tuple from
$\mathbb{S}_{\mathbb{C}^{k^3}}$. Note that the $\ell_2$-norm on $\mathbb{C}^{k^3}$ is
the same as the Frobenius norm on $\mathbb{C}^{k \times k^{2}}$.
Let $1 < p \leq 1.1$.
In Step~I, we define the function $f: \mathbb{S}_{\mathbb{C}^{k^3}} \rightarrow \mathbb{R}$
as $f(M) := \lVert M \rVert_{2p}$. The function $f$ has global Lipschitz
constant $L_1 = 1$ since
\[
|f(M) - f(N)| \leq
\lVert M - N \rVert_{2p} \leq
\lVert M - N \rVert_{2}.
\]
For large enough $k$ the mean $\mu$ of $f$, under the Haar measure,
is less than $2 k^{\frac{1}{2p} - \frac{1}{2}}$
\cite[Section~VIII]{aubrun_szarek_werner_2010},
\cite[Corollary~7]{aubrun_szarek_werner_2010_main}. We use the notation
of Theorem~\ref{thm:mainHaar}.
Define $L_2 := 1$, $p(i) := 1$ for all $i \in \mathbb{N}$. Then $C < 2$.
Define the layers $\Omega_1, \Omega_2, \ldots, $ to be all of
$\mathbb{S}_{\mathbb{C}^{k^3}}$. Let $j$, $4 \leq j \leq k$ be a positive integer.
Let $\lambda_j := j^{\frac{1}{2}} k^{\frac{1}{2p} - \frac{1}{2}}$.
Define $c_1 := 1$,
$m = k^{2 + \frac{1}{p}}$, $c_2 := 0$, $c_3 := 1$. Trivially,
a Haar random subspace of dimension $m j$ lies in $\Omega_i$ with
probability at least $1 - c_2 e^{-c_3 m j i}$. Theorem~\ref{thm:mainHaar}
tells us that there is a universal constant $\hat{c}_1$ such that
for $m' := \hat{c}_1 k^{2 + \frac{1}{p}}$, with probability at least
$1 - 2^{-m' j}$, a Haar random subspace $W$ of dimension $m' j$
satisfies
\[
\lVert M \rVert_{\infty} \leq
\lVert M \rVert_{2p} <
2 k^{\frac{1}{2p} - \frac{1}{2}} +
j^{\frac{1}{2}} k^{\frac{1}{2p} - \frac{1}{2}} <
2 j^{\frac{1}{2}} k^{\frac{1}{2p} - \frac{1}{2}}
\]
for all $M \in W$. In particular, with probability at least
$1 - 2^{-\hat{c}_1 j k^{\frac{4}{3p} + \frac{5}{3}} (\log k)^{-1}}$,
a Haar random
subspace $W$ of dimension
$\hat{c}_1 j k^{\frac{4}{3p} + \frac{5}{3}} (\log k)^{-1}$ satisfies
\[
\lVert M \rVert_{\infty} \leq
\lVert M \rVert_{2p} <
2 j^{\frac{1}{2}} k^{\frac{1}{2p} - \frac{1}{2}}
\]
for all $M \in W$.
Let $j$, $4 \leq j \leq k$ be a positive integer.
Define the function $f: [0, 1] \rightarrow [0, 1]$ as
$f(x) := x^p$.
Set $\epsilon := k^{-p}$ in Proposition~\ref{prop:poly}. Let
$n$ be the minimum positive odd integer satisfying
$
2 p k^p \sqrt{\ln k^{2p}} \leq
\frac{k^{-\frac{p}{n}} \sqrt{n}}{2};
$
$n < 2^7 p^3 k^{2p} \log k$.
Proposition~\ref{prop:poly} implies
that there is a polynomial $p(x)$ of degree at most
$2n + 1 < 2^{9} p^3 k^{2p} \log k$ such that
\begin{equation}
\label{eq:derivative}
\begin{array}{l l}
p(x) - 2 k^{-p} \leq x^p \leq p(x) + 3 k^{-p}, &
\forall x \in [0, 1], \\
|p'(x)| <
4 p (j+1)^{p-1} \sqrt{\ln k^{2p}}
k^{\frac{5}{3} - \frac{2}{3p} - p}, &
\forall x \in [0, j k^{\frac{2}{3p} - 1}], \\
|p'(x)| <
4 p (5j)^{p-1} \sqrt{\ln k^{2p}}
k^{2 - p - \frac{1}{p}}, &
\forall x \in (j k^{\frac{2}{3p} - 1}, 5 j k^{\frac{1}{p} - 1}], \\
|p'(x)| <
4 p \sqrt{\ln k^{2p}}, &
\forall x \in (5 j k^{\frac{1}{p} - 1}, 1].
\end{array}
\end{equation}
Also, Proposition~\ref{prop:poly} guarantees that
$\alpha(p(x)) < e^{2^7 p^3 k^{2p} \log k}$.
In Step~II, we define the function $f: \mathbb{S}_{\mathbb{C}^{k^3}} \rightarrow \mathbb{R}$
as $f(M) := \mathrm{Tr}\, [p(M M^\dag)]$, where $p$ is the polynomial defined in
Equation~\ref{eq:derivative}.
Now, $f$ is a balanced polynomial of degree
$a = 2n + 1 < 2^{9} p^3 k^{2p} \log k$ and
\[
\alpha(f) = \mathrm{Tr}\, [p(J J^\dag)] = k^3 \alpha(p(x)) <
e^{2^8 p^3 k^{2p} \log k},
\]
where $J$ is the $k \times k^2$ all ones matrix.
For a $k \times k$ matrix $X$, define
$\mathrm{Sing}(X)$ to be the $k \times k$ diagonal matrix consisting of the
singular values of $X$ arranged in decreasing order.
The function $f$ has global Lipschitz
constant $L_1 = 2^{4} p^{3/2} \sqrt{\log k}$ since
\begin{eqnarray*}
|f(M) - f(N)|
& = &
|\mathrm{Tr}\, [p(\mathrm{Sing}(M)^2)] - \mathrm{Tr}\, [p(\mathrm{Sing}(N)^2)]|
\; = \;
|\mathrm{Tr}\, [p(\mathrm{Sing}(M)^2)-p(\mathrm{Sing}(N)^2)]| \\
& \leq &
8 p^{3/2} \sqrt{\log k} \cdot
\lVert \mathrm{Sing}(M)^2 - \mathrm{Sing}(N)^2 \rVert_1 \\
& \leq &
8 p^{3/2} \sqrt{\log k} \cdot
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \cdot
\lVert \mathrm{Sing}(M) + \mathrm{Sing}(N) \rVert_2 \\`
& \leq &
2^{7/2} p^{3/2} \sqrt{\log k} \cdot
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \cdot
\sqrt{\lVert M \rVert_2^2 + \lVert N \rVert_2^2} \\
& \leq &
2^{4} p^{3/2} \sqrt{\log k} \cdot
\lVert M - N \rVert_2.
\end{eqnarray*}
Above, the first inequality follows from Equation~\ref{eq:derivative},
the second inequality is Cauchy-Schwarz and the last inequality
follows from \cite[Section~4]{Mirsky}.
By setting $j = 4$ in Step~I, we conclude that
the mean $\mu$ of $f$ under the Haar measure is less
than $2^{4p} k^{1-p}$.
We use the notation of Theorem~\ref{thm:maintdesign}.
Let $\lambda := k^{1 - p}$.
Define
\[
L_2 :=
2^{4p+3} p^{3/2}
\sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}},
\]
$p(i) := (i+4)^{2p-1}$ for all $i \in \mathbb{N}$.
Then $C \leq p^{2p}$.
Define the layers $\Omega_1, \Omega_2, \ldots, $ to be
the subsets
\[
\Omega_i :=
\left\{
M \in \mathbb{S}_{\mathbb{C}^{k^3}}:
\lVert M \rVert_{2p} \leq
2 (i+3)^{\frac{1}{2}} k^{\frac{1}{2p}-\frac{1}{2}}
\right\}.
\]
We will now show that $f$ restricted to $\Omega_i$ has local
Lipschitz constant at most $L_2 \sqrt{p(i)}$. Note that for any
$M \in \Omega_i$,
$
\lVert M \rVert_\infty \leq
2 (i+3)^{\frac{1}{2}} k^{\frac{1}{2p} - \frac{1}{2}}.
$
Let $B$ denote the number of singular values of $M$ larger than
$(i+3)^{\frac{1}{2}} k^{\frac{1}{3p} - \frac{1}{2}}$.
Let $b_1, \ldots b_k$ be the singular values of $M$ in descending order.
Then
\[
2^{2p} (i+3)^p k^{1-p} \geq
\lVert M \rVert_{2p}^{2p} \geq
\sum_{i=1}^B b_i^{2p} \geq
\left(\sum_{i=1}^B b_i^{2}\right)
(i+3)^{p-1} k^{\frac{5}{3} - \frac{2}{3p} - p},
\]
which gives
$
\sum_{i=1}^B b_i^{2} \leq
2^{2p} (i+3) k^{\frac{2}{3p} - \frac{2}{3}}.
$
Let $C$ denote the number of singular values of $N$ larger than
$(i+3)^{\frac{1}{2}} k^{\frac{1}{3p} - \frac{1}{2}}$.
Without loss of generality, $B \geq C$. Restricting $M$, $N$
to belong to $\Omega_i$, we get from Equation~\ref{eq:derivative} that
\begin{eqnarray*}
\lefteqn{
|f(M) - f(N)|
} \\
& = &
|
\mathrm{Tr}\, [
p(\mathrm{Sing}(M)^2) - p(\mathrm{Sing}(N)^2)
]
| \\
& \leq &
\sum_{i=1}^C |p(b_i^2) - p(c_i^2)| +
\sum_{i=C+1}^B |p(b_i^2) - p(c_i^2)| +
\sum_{i=B+1}^k |p(b_i^2) - p(c_i^2)| \\
& \leq &
8 p^{3/2} (5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sum_{i=1}^C |b_i^2 - c_i^2| \\
& &
{} +
8 p^{3/2} (5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sum_{i=C+1}^B |b_i^2 - c_i^2| \\
& &
{} +
8 p^{3/2} ((i+4))^{p-1} \sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}}
\sum_{i=B+1}^k |p(b_i^2) - p(c_i^2)| \\
& \leq &
8 p^{3/2} (5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sqrt{\sum_{i=1}^C (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=1}^C (b_i + c_i)^2} \\
& &
{} +
8 p^{3/2} (5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sqrt{\sum_{i=C+1}^B (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=C+1}^B (b_i + c_i)^2} \\
& &
{} +
8 p^{3/2} ((i+4))^{p-1} \sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}}
\sqrt{\sum_{i=B+1}^k (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=B+1}^k (b_i + c_i)^2} \\
& \leq &
2^{7/2} p^{3/2}
(5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sqrt{\sum_{i=1}^k (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=1}^C (b_i^2 + c_i^2)} \\
& &
{} +
2^{7/2} p^{3/2}
(5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sqrt{\sum_{i=1}^k (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=C+1}^B (b_i^2 + c_i^2)} \\
& &
{} +
2^{7/2} p^{3/2}
((i+4))^{p-1} \sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}}
\sqrt{\sum_{i=1}^k (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=1}^k (b_i^2 + c_i^2)} \\
& \leq &
2^{4} p^{3/2} 2^p
5^{p-1} (i+3)^{p - \frac{1}{2}}
\sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}} \cdot
k^{\frac{1}{3p} - \frac{1}{3}} \cdot
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \\
& &
{} +
2^{4} p^{3/2} 2^p
5^{p-1} (i+3)^{p - \frac{1}{2}}
\sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}} \cdot
k^{\frac{1}{3p} - \frac{1}{3}} \cdot
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \\
& &
{} +
2^{4} p^{3/2}
((i+4))^{p-1} \sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}}
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \\
& \leq &
2^{6} p^{3/2} 2^p
5^{p-1} (i+4)^{p - \frac{1}{2}}
\sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}} \cdot
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \\
& \leq &
2^{4p+3} p^{3/2}
(i+4)^{p - \frac{1}{2}}
\sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}} \cdot
\lVert M - N \rVert_2.
\end{eqnarray*}
This completes the proof of the claim above that $f$ restricted to
$\Omega_i$ has local Lipschitz constant at most $L_2 \sqrt{p(i)}$.
Define $c_1 := 2^{8p+6} p^3 \hat{c}_1$,
$m = \hat{c}_1 k^{\frac{4}{3p}+\frac{5}{3}} (\log k)^{-1}$, $c_2 := 1$,
$c_3 := \ln 2$. By Step~I,
a Haar random subspace of dimension $m i$ lies in $\Omega_i$ with
probability at least $1 - c_2 e^{-c_3 m i}$.
Let
\[
0 \leq \epsilon <
k^{\hat{c}_1 k^{2p + 3} (\log k)^{-1/2}} <
\left(\frac{k^{1 - p}}{4 L_1}\right)^{2m}
\frac{k^{3 (2a-1) m} 5^{(2p-1)m} L_2^{2m}}{\alpha(f)^{2m}}.
\]
Theorem~\ref{thm:maintdesign}
tells us that there is a universal constant $\hat{c}_3$ such that
for
\[
m' :=
\hat{c}_3 k^{\frac{4}{3p} + \frac{5}{3}} \frac{\log \log k}{(\log k)^2},
\]
with probability at least
$1 - 2 \cdot 2^{-m'}$, a subspace $W$ of dimension $m'$ chosen from an
$\epsilon$-approximate $(2am)$-design $\nu$
satisfies
\[
f(M) = \mathrm{Tr}\, [p(M M^\dag)] <
2^{4p} k^{1-p} + k^{1-p} <
2^{4p+1} k^{1-p}
\]
for all $M \in W$. By Equation~\ref{eq:derivative}, this implies that
\[
\mathrm{Tr}\, [(M M^\dag)^p] <
\mathrm{Tr}\, [p(M M^\dag)] + 3 k^{1-p} <
2^{4p+3} k^{1-p}
\]
for all $M \in W$. In other words,
$
\lVert M \rVert_{2p}^2 < 2^7 k^{\frac{1}{p} - 1}
$
for all $M \in W$.
We shall now see how this result gives us a channel with strict
supermultiplicativity of the $\lVert \cdot \rVert_{1 \rightarrow p}$-norm
or equivalently, strict
subadditivity of minimum output R\'{e}nyi $p$-entropy for any $p > 1$.
\begin{theorem}
\label{thm:supermultiplicativity}
Let $k$ be a positive integer. Let $1 < p \leq 1.1$.
Let $W$ be a subspace of dimension
$
m' :=
\hat{c}_3 k^{\frac{4}{3p} + \frac{5}{3}} \frac{\log \log k}{(\log k)^2}
$
chosen with uniform probability from a
$k^{\hat{c}_1 k^{5} (\log k)^{-1/2}}$-approximate unitary
$(2^{11} \hat{c}_1 k^{5.1})$-design
from the Hilbert space $\mathbb{C}^{k^3}$, where $\hat{c}_1$, $\hat{c}_3$ are
universal constants. Let $\Phi$ be the channel with output dimension
$k$ corresponding to the
subspace $W$. Then with probability at least $1 - 2 \cdot 2^{-m'}$ over the
choice of $W$,
\[
\lVert \Phi \rVert_{1 \rightarrow p} \leq
2^{7} k^{\frac{1}{p} - 1},
~~~
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow p} \geq
\hat{c}_3 k^{\frac{4}{3p} - \frac{4}{3}}.
\]
In other words,
$
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow p} >
\lVert \Phi \rVert_{1 \rightarrow p} \cdot
\lVert \bar{\Phi} \rVert_{1 \rightarrow p} \cdot
$
for large enough $k$. For $p > 1.1$, the channel $\Phi$ obtained
for $p = 1.1$ suffices to show supermultiplicativity.
\end{theorem}
\begin{proof}
The input dimension of the channel $\Phi$ is $\dim W = m'$. The
Stinespring dilation
of the channel $\Phi$ is the $k^3 \times k^3$ unitary matrix
that defines the subspace $W$. The subspace $W$ is obtained by
taking the span of first $m'$ columns of the unitary matrix. This unitary
matrix is chosen uniformly at random from a
$k^{\hat{c}_1 k^{5} (\log k)^{-1/2}}$-approximate unitary
$(2^{11} \hat{c}_1 k^{5.1})$-design. Note that
$
2 a m < 2^{11} \hat{c}_1 k^{5.1},
$
$
\epsilon > k^{\hat{c}_1 k^{5} (\log k)^{-1/2}},
$
where $a$, $m$ and $\epsilon$ are defined in Step~III above.
Let $M$ be a unit $\ell_2$-norm vector in $\mathbb{C}^{k^3}$ rearranged as
a $k \times k^2$ matrix.
From Equation~\ref{eq:redefpnorm}, we get
\[
\lVert \Phi \rVert_{1 \rightarrow p} =
\max_{M \in W: \lVert M \rVert_2 = 1}
\lVert M \rVert_{2p}^2 \leq
2^{7} k^{\frac{1}{p} - 1}.
\]
From Fact~\ref{fact:maxeigenvalue},
\[
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow p} \geq
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow \infty} \geq
\frac{m'}{k^3} =
\hat{c}_3 k^{\frac{4}{3p} - \frac{4}{3}}
\frac{\log \log k}{(\log k)^2} >
(\lVert \Phi \rVert_{1 \rightarrow p})^2
\]
for large enough $k$.
This shows the supermultiplicativity of the
$\lVert \cdot \rVert_{1 \rightarrow p}$-norm for $1 < p \leq 1.1$. For
$p > 1.1$, we use the fact that
$
\lVert \cdot \rVert_{1 \rightarrow \infty} \leq
\lVert \cdot \rVert_{1 \rightarrow p} \leq
\lVert \cdot \rVert_{1 \rightarrow 1.1}
$
to conclude the supermultiplicativity of
$\lVert \cdot \rVert_{1 \rightarrow p}$.
\end{proof}
Thus by setting $p = 1.1$, we see
that for large enough $n$ approximate unitary
$(n^{1.7} \log n)$-designs give
rise to channels
exhibiting strict subadditivity of minimum output R\'{e}nyi $p$-entropy
for any $p > 1$. Combined with the result of the previous section,
we can furthermore state
that for large enough $n$ approximate unitary
$(n^{1.7} \log n)$-designs give
rise to channels
exhibiting strict subadditivity of minimum output R\'{e}nyi $p$-entropy
for any $p \geq 1$.
\paragraph{Remarks:}
\noindent
1.\
In \cite{aubrun_szarek_werner_2010}, for channels obtained from
Haar random subspaces the lower bound on
$
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow p}
$
was of the order of
$
k^{\frac{1}{p} - 1},
$
whereas in our work it is
of the order of
$
k^{\frac{4}{3p} - \frac{4}{3}},
$
for channels obtained from approximate $t$-designs.
Hence the counter example we get for additivity of minimum output
R\'{e}nyi $p$-entropy of quantum channels, when the channel is
chosen
from an approximate unitary $t$-design has weaker parameters
than the Haar random channels of
\cite{aubrun_szarek_werner_2010}. Nevertheless, our work
is the first partial derandomisation of a construction of quantum
channels violating additivity of minimum output
R\'{e}nyi $p$-entropy, since it is possible to uniformly
sample a unitary from an exact
$(n^{1.7} \log n)$-design
using of the order of
$
n^{1.7} (\log n)^2
$
random bits versus $\Omega(n^2)$ random bits required to choose
a Haar random unitary to constant precision.
\smallskip
\noindent
2.\
It is possible to do the above counterexample on a sphere in $\mathbb{C}^{k^2}$.
However
in that case the number of random bits required to choose a
unitary from an exact design is larger than $k^4 \log k$, which is what a
Haar random unitary would require!
\section{Conclusion}
\label{sec:conclusion}
In this paper we have shown that a unitary chosen from an approximate
unitary $n^{2/3}$-design leads
to a quantum channel with superadditive classical Holevo capacity. In the
process of coming up
with such a channel we developed two new technical tools viz.
stratified analysis
of a sphere in $\mathbb{C}^n$ for Haar measure and unitary designs
(Theorems~\ref{thm:mainHaar}, \ref{thm:maintdesign}),
and approximation of any continuous monotonic function by
a polynomial of moderate degree (Proposition~\ref{prop:poly}).
The stratified analysis for the Haar measure was used to
recover in a simple fashion Aubrun, Szarek and Werner's counterexample
\cite{aubrun_szarek_werner_2010_main} for
additivity of minimum output Von Neumann entropy. The stratified
analysis for unitary designs was used to prove
counterexamples for additivity of minimum output von Neumann entropy and
R\'{e}nyi $p$-entropy for $p > 1$, when the unitary Stinespring dilation
of the channel is chosen from approximate unitary $t$-design for
suitable values of $t$. Choosing a unitary from these $t$-designs requires
less random bits than choosing from the Haar measure.
However the value of $t$ required is much larger than what is known to
be efficiently implementable by quantum circuits.
We believe our work results in a better understanding of the interplay
between geometric functional analysis and additivity questions in
quantum information theory, and our technical tools will find applications
to other problems in quantum information theory.
Our work represents a step in the quest for an efficient explicit
channel violating additivity of minimum output von Neumann entropy. This
is the major open problem in the area. Another problem left open is
whether there is a single channel that violates additivity of
minimum output R\'{e}nyi $p$-entropy for all $p \geq 1$.
|
{
"timestamp": "2019-04-18T02:18:48",
"yymm": "1902",
"arxiv_id": "1902.10808",
"language": "en",
"url": "https://arxiv.org/abs/1902.10808"
}
|
\section{Introduction}
\label{intro}
Warped disks are expected to occur in a large number of astrophysical
situations \citep[e.g.][]{Pringle1981,Pringle1999,Kingetal2013}.
Warping may occur due to external torques from various
sources. Binaries can provide such a torque on a circumstellar disk
from an external binary component
\citep[e.g.][]{PT1995,Larwoodetal1996,LO2000,Martinetal2009,Martinetal2011}
or a torque on a circumbinary disk form an internal binary
\citep[e.g.][]{Facchinietal2013,Lodato2013,Martin2017}. Around spinning black
holes, general relativistic Lense--Thirring precession may cause
warping \citep{BP1975} in X--ray binaries
\citep[e.g.][]{SF1996,WP1999, Ogilvie2001, Martinetal2007} and around
supermassive black holes \citep[e.g.][]{Herrnstein1996,Martin2008}.
Disks in AGN and in binary X-ray sources may be warped by the effects
of radiation pressure \citep{Pringle1996,OD2001}. Disks may also be
warped by a misaligned magnetic field \citep[e.g.,][]{Lai1999} or a planet \citep[e.g.][]{Nealon2018}.
The evolution of the warp in a disk depends upon how the \cite{SS1973}
viscosity $\alpha$ parameter compares with the disk aspect ratio
$H/R$. If $\alpha>H/R$, then the warp propagates diffusively through
the effects of viscosity. If $\alpha<H/R$, then pressure forces drive
the evolution and the warp propagates as a bending wave that travels
at half the sound speed, $c_{\rm s}/2$ \citep{PL1995,Pringle1999,LO2002}. In this
case, the viscosity is too small to damp the wave locally. For a
recent review of warped disks, see \cite{Nixon2016}.
One-dimensional (based on radius) models of disk tilt evolution offer
advantages over multi-dimensional models. They permit tracking the
evolution over long timescales with much less computational effort
than multi-dimensional models. They are also easier to interpret
physically. On the other hand, their applicability is limited by the
simplifications made to reduce the dimensionality. In any case, such
models can be compared with multi-dimensional models to obtain more
physical insight.
A one-dimensional model should ideally conserve angular momentum and
be valid for arbitrary tilts and warps (the derivative of the tilt
with respect to the logarithm of the radius). In the viscous regime,
\cite{Pringle1992} developed a set of intuitively-based
one-dimensional equations that satisfy these conditions.
\cite{Ogilvie1999} extended this analysis by directly working with the
fluid equations and obtained one-dimensional equations that apply for
even large warps. This analysis showed that the \cite{Pringle1992}
equations are valid for small warps and small $\alpha$, $H/r < \alpha
\ll 1$, but arbitrary tilts, with the extension that the effective
viscosity coefficients are constrained by the internal fluid
dynamics.
In application to disks around young stars, the wave--like regime is
of importance. One-dimensional linear disk evolution equations for
this regime typically assume that the warp is small and ignore disk
surface density evolution \citep[e.g.,][]{PL1995,
LO2000}. \cite{Ogilvie2006} analyzed the nonlinear dynamics of free
warps (imposed by initial conditions) in the absence of viscous
density evolution. However, as found in \cite{Bateetal2000},
significant density evolution can occur as the tilt evolves.
We are interested in the case that the disk is in good radial
communication so that the level of warping is small, as should apply
to protostellar disks. Therefore a linear analysis is often valid. We
are interested in the case that the disk tilt and surface density
change over the course of its evolution.
The goal of this work is to find a formulation
which describes the disk evolution correctly in both regimes, and manages to
connect the two. In Section~\ref{equations} we present two sets of warped disk equations, one valid in the viscous regime and one valid in the wave--like regime.
In Section~\ref{general}, we follow along the lines of \cite{Pringle1992} to extend the linear tilt evolution equations to apply to arbitrary tilts and account for viscous density evolution.
In Section~\ref{numerical} we
numerically solve the equations for an initially warped disk around a
single central object in the absence of any external torques. We draw
our conclusions in Section~\ref{conc}.
\section{Warped disk equations in the two regimes}
\label{equations}
Currently there are two sets of warped disk equations that describe
mutually exclusive regimes, the wave--like regime with $\alpha < H/R$
and the diffusive (or viscous) regime with $\alpha > H/R$. We provide
an overview of these two sets of equations in this Section. We
describe the disk as consisting of a set circular rings with spherical
radius $R$. We assume that the disk is in near Keplerian
rotation. \footnote{ Thus our results do not apply to strongly
non--Keplerian flows such as those that occur in accretion disks
close to the event horizon of a black hole.} The rings rotate with
Keplerian angular frequency $\Omega(R)=\sqrt{G M/R^3}$ about a central
object of mass $M$ and have a surface density $\Sigma(R)$. The disk
extends from inner radius $R_{\rm in}$ to outer radius $R_{\rm
out}$. The angular momentum per unit area of each ring is
\begin{equation}
\bm{L}=\Sigma R^2 \Omega \bm{l},
\label{eqL}
\end{equation}
where $\bm{l}$ is a unit vector. We consider a locally isothermal disk with aspect ratio $H/R$, where $H$ is the disk scale height.
Angular momentum transport in an accretion disk is driven by turbulent
eddies with a maximum size $H$, and maximum speed the sound speed,
$c_{\rm s}=H \Omega$. The azimuthal shear viscosity has the standard
form
\begin{equation}
\nu_1 = \alpha_1 \left(\frac{H}{R}\right)^2 R^2 \Omega
\end{equation}
\citep{SS1973}, where $\alpha_1 \simeq \alpha$, for dimensionless
parameter $\alpha<1$. The vertical shear viscosity is
\begin{equation}
\nu_2 = \alpha_2 \left(\frac{H}{R}\right)^2 R^2 \Omega,
\end{equation}
\citep{PP1983}, where $\alpha_2\simeq 1/(2\alpha)$ in the linear approximation.
\subsection{Wave--like limit equations}
In the wave--like limit, $\alpha <H/R$, we assume that the surface density does not evolve, $\partial \Sigma/ \partial t=0$. The evolution of a warped disk is described by two equations
\begin{align}
\frac{\partial \bm{G}}{\partial t}+ \omega \, \bm{l}\times \bm{G} + \alpha \Omega \bm{G}=\frac{\Sigma H^2 R^3\Omega^3}{4}\frac{\partial \bm{l}}{\partial R}
\label{lo1}
\end{align}
and
\begin{align}
\Sigma R^2 \Omega \frac{\partial \bm{l} }{\partial t} =
\frac{1}{R}\frac{\partial \bm{G}}{\partial R}+\bm{T},
\label{lo2}
\end{align}
where $\bm{G}$ is the internal disk torque and $\bm{T}$ is the external torque on the disk \citep[see equations~12 and~13 in][]{LO2000}.
The apsidal precession frequency in the plane of the disk is
\begin{equation}
\omega =\frac{\Omega^2-\kappa^2}{2 \Omega}
\end{equation}
with epicyclic frequency $\kappa$. We solve equations~(\ref{lo1}) and~(\ref{lo2}) in Section~\ref{wl} to compare to our solution to the generalised equations in the wave--like limit.
\subsection{Viscous limit equations}
In the viscous limit, $\alpha > H/R$, a warped disk is described by the evolution equation
\begin{align}
\frac{\partial \bm{L} }{\partial t} = &
\frac{3}{R}\frac{\partial}{\partial R} \left[\frac{R^{1/2}}{\Sigma} \frac{\partial}{\partial R} \left(\nu_1 \Sigma R^{1/2} \right) \bm{L} \right] \cr
& +\frac{1}{R}\frac{\partial}{\partial R} \left[
\left( \nu_2 R^2 \left|\frac{\partial \bm{l}}{\partial R} \right|^2-\frac{3}{2}\nu_1 \right) \bm{L} \right] \cr
& +\frac{1}{R}\frac{\partial}{\partial R} \left[
\frac{1}{2}\nu_2 R |\bm{L}|\frac{\partial \bm{l}}{\partial R} \right]
+\bm{T}
\label{viscous}
\end{align}
\citep{Pringle1992}. We solve equation~(\ref{viscous}) in Section~\ref{diff} to compare to our solution to the generalised equations in the viscous regime.
\section{Generalised warped disk equations}
\label{general}
We now show how it is possible to combine the above equations into
a single set that are valid in both the wave--like and the diffusive warp
propagation regimes. We follow the methods of \cite{PP1983} and \cite{Pringle1992}. We note that this is not a first principles derivation of the
evolution equations \citep[cf.,][]{Ogilvie1999, Ogilvie2006}. Conservation of mass is expressed as
\begin{equation}
\frac{\partial \Sigma}{\partial t}+\frac{1}{R}\frac{\partial}{\partial R}(R\Sigma v_R)=0,
\label{mass}
\end{equation}
where $v_R$ is the radial velocity. Conservation of angular momentum gives us
\begin{equation}
\frac{\partial \bm{L}}{\partial t }+\frac{1}{R}\frac{\partial }{\partial R}(\Sigma v_R R^3 \Omega \bm{l})
=\frac{1}{R}\frac{\partial \bm{G}}{\partial R} +\bm{T},
\label{angmom}
\end{equation}
where $\bm{G}$ is the internal disk torque and $\bm{T}$ is the
external torque on the disk. In equation (\ref{angmom}), we have
included the second term on the LHS compared to equation~(\ref{lo2}), in the wave--like equations, in order to enforce conservation of angular momentum as the disk density evolves.
\begin{figure*}
\centering
\includegraphics[width=7.5cm]{wave1cincsig.eps}
\includegraphics[width=7.5cm]{full0.eps}
\includegraphics[width=7.5cm]{full1.eps}
\includegraphics[width=7.5cm]{full10.eps}
\caption{Evolution of an initially warped disk around a single object
with no external torque with $\alpha=0.01$ and $H/R=0.1$ (in the
wave--like regime). The upper panels show the inclination and the
lower panels the surface density. Top left: The wave--like
equations~(\ref{lo1}) and~(\ref{lo2}). The other panels solve the full equations~(\ref{main}) and~(\ref{flux}) with $\beta=0$ (top right), $\beta=1$ (bottom left) and $\beta=10$ (bottom right). The times shown are every
$10\,P_{\rm in}$.}
\label{wavelike}
\end{figure*}
We take the dot product of equation~(\ref{angmom}) with $\bm{l}$ and subtract $R^2 \Omega$ times equation~(\ref{mass}) to obtain an equation for the radial velocity
\begin{equation}
v_R=\frac{ \partial \bm{G}/\partial R \cdot \bm{l}}{R\Sigma \, d (R^2 \Omega)/ d R}.
\label{vr}
\end{equation}
We substitute the radial velocity equation~(\ref{vr}) into the conservation of mass equation~(\ref{mass}) to obtain an equation for the surface density evolution
\begin{align}
\frac{\partial \Sigma }{\partial t} =
- \frac{1}{R} \frac{\partial }{\partial R}\left[ \frac{\partial \bm{G}/\partial R \cdot \bm{l} }{ d (R^2 \Omega)/ d R}\right].
\end{align}
Further, we substitute the radial velocity equation~(\ref{vr}) into the conservation of angular momentum equation~(\ref{angmom}) to obtain an equation for the evolution of the angular momentum in the disk
\begin{align}
\frac{\partial \bm{L}}{\partial t} =
& -\frac{1}{R} \frac{\partial }{\partial R}\left[ \frac{(\partial \bm{G}/\partial R \cdot \bm{l} )}{\Sigma \, d (R^2 \Omega)/d R}\bm{L}\right]
+\frac{1}{R}\frac{\partial \bm{G}}{\partial R} +\bm{T}.
\end{align}
Since the disk is in near--Keplerian rotation we take
$\Omega\propto R^{-3/2}$ and find
\begin{align}
\frac{\partial \Sigma}{\partial t}= -\frac{2}{R}\frac{\partial }{\partial R}\left[\frac{(\partial \bm{G}/\partial R \cdot \bm{l} )}{R\Omega}\right]
\label{densevol}
\end{align}
and
\begin{align}
\frac{\partial \bm{L} }{\partial t} =
-\frac{2}{R}\frac{\partial}{\partial R} \left[ \left(
\frac{( \partial \bm{G}/\partial R \cdot \bm{l})}{ \Sigma R\Omega} \right)\bm{L} \right] +
\frac{1}{R}\frac{\partial \bm{G}}{\partial R}+\bm{T}.
\label{main}
\end{align}
The key step to generalising the equations so that they are valid in
both diffusive and wave--like regimes is to now amend
equation~(\ref{lo1}) to read
\begin{align}
\frac{\partial \bm{G}}{\partial t}+ \omega \, \bm{l}\times \bm{G} &+
\alpha \Omega \bm{G}+\beta \Omega (\bm{G}\cdot \bm{l})\bm{l} = \cr
&\frac{\Sigma H^2
R^3\Omega^3}{4}\frac{\partial \bm{l}}{\partial R} - \frac{3}{2}
(\alpha+\beta) \nu_1 \Sigma R^2 \Omega^2 \bm{l}.
\label{flux}
\end{align}
To effect this generalisation we have found it necessary to introduce
two extra terms dependent on a new dimensionless parameter
$\beta$. The fourth term on the left hand side has the effect of
damping the component of disk torque $\bm{G}$ perpendicular to the
local disk plane. The addition of the final term on the right hand
side is to add an additional shear viscosity term. At this stage the
magnitude of $\beta$ is arbitrary, except that we shall require that
$\beta \gg \alpha$. We show the effects of different values for
$\beta$ in Section~\ref{numerical}.
\subsection{The new generalised equation in the two limits}
Equations~(\ref{main}), and~(\ref{flux}) provide a one-dimensional description of both the the disk surface density and the disk tilt. We now show that this generalised equation has the previous equations (Section~\ref{equations}) in both limits.
\begin{enumerate}
\item In the wave-like limit we have $\alpha < H/R \ll 1$. The
equations derived in this limit assumed that the surface density
did not change with time, because in this limit the wave-like warp
propagation happens on the shorter timescale than the viscous
evolution of the surface density. Thus, in this limit, the final
term on the right hand side of equation~(\ref{flux}) is
negligible. In addition the assumption that $\Sigma$ is independent
of time implies that $v_{\rm R} = 0$, unless there is an
external source of mass. Thus (equation~\ref{vr}) we may take
$\partial \bm{G}/\partial R \cdot \bm{l}=0$ and we may ignore the
fourth term on the left hand side. Given this,
equation~(\ref{flux}) now reduces to equation~(\ref{lo1}), as
required. We note, however, that the full solution to the new
equations allows for the evolution of the surface density also in
the wave-like regime. The degree to which the surface density
evolves depends on the magnitude of the new parameter $\beta$.
\item In the viscous limit $(\alpha > H/R)$, $\bm{G}$ evolves on a
viscous timescale and so $\partial \bm{G}/\partial t \ll \alpha
\Omega \bm{G}$ and we set $\partial \bm{G}/\partial t=0$.
Furthermore, we set $\omega=0$ provided that $\omega \ll \alpha
\Omega$ and we are left with
\begin{align}
\alpha \Omega \bm{G}+ &\beta \Omega (\bm{G}\cdot \bm{l})\bm{l} = \cr
&\frac{\Sigma H^2
R^3\Omega^3}{4}\frac{\partial \bm{l}}{\partial R} - \frac{3}{2}
(\alpha+\beta) \nu_1 \Sigma R^2 \Omega^2 \bm{l}.
\label{eq1}
\end{align}
We take the dot product of this with $\bm{l}$ to find an expression for
$\bm{G}\cdot \bm{l} $ and then substitute that into equation~(\ref{eq1}) to find
\begin{equation}
\bm{G}=\frac{1}{2}\nu_2 \Sigma R^3 \Omega \frac{\partial \bm{l}}{\partial R} - \frac{3}{2}\nu_1 \Sigma R^2 \Omega \bm{l}.
\label{viscouslimit}
\end{equation}
Substituting this equation for $\bm{G}$ into equation~(\ref{main}), we recover the viscous disk evolution equation~(\ref{viscous}), which is valid for $H/r < \alpha \ll 1$ \citep{Ogilvie1999}.
\end{enumerate}
\section{Numerical solutions}
\label{numerical}
We solve equations~(\ref{main}) and~(\ref{flux}) as an initial
value problem for $\bm{L}$, and $\bm{G}$ using finite differences.
The method is first--order explicit in time. We use Cartesian
coordinates and treat each component of the vectors separately. The
units in the code are defined with $G=M=1$, where $M$ is the mass of
the central object. The Keplerian orbital period at the inner disk
radius $R_{\rm in}=1$ is $P_{\rm in}=2\pi$. We take the boundary
conditions that $\bm{G}=\bm{0}$, $\Sigma=0$ and $\partial
\bm{l}/\partial R=0$ at $R=R_{\rm in}$ and $R=R_{\rm out}$. The
initial condition on $\bm{G}$ is always taken as
$\bm{G}(R,0)=\bm{0}$.
We consider the evolution of an initially warped disk around a single
central object. There is no external torque on the disk, so $\bm{T}=0$
and $\omega=0$. The disk extends from $R_{\rm in}=1$ up to $R_{\rm
out}=20$. We take the initial surface density of the disk to be
distributed as a simple power law with ends truncated at $R_{\rm in}$ and $R_{\rm out}$
\begin{equation}
\Sigma(R,0)=\Sigma_0 \left(\frac{R}{R_{\rm in}}\right)^{-1/2}\left[1-\left(\frac{R_{\rm in}}{R}\right)^{\frac{1}{2}}\right]\left[1-e^{R-R_{\rm out}}\right].
\end{equation}
The constant $\Sigma_0$ is arbitrary as the equations are linear in
$\Sigma$. Here we have scaled the total disk mass to be $0.001\,M$.
The first factor in brackets on the RHS is a power law that represents
a steady disk with $\nu_1 \Sigma \propto \,\rm const$ if mass is added
at the outer edge. The second and third factors enforce zero torque
($\Sigma=0$) inner and outer boundary conditions, respectively. Note
that the surface density is not in steady state since we do not add
material to the disk.
The initial tilt of the disk is described by
\begin{equation}
i(R,0)=10^\circ \,\left[\frac{1}{2}\tanh \left( \frac{R-R_{\rm warp}}{R_{\rm width}}\right)+\frac{1}{2}\right].
\end{equation}
Since the equations are linear in disk tilt, the normalisation of $i$ is arbitrary.
The disk has an inclination of zero at the inner disk edge, an
inclination of $10^\circ$ at the outer disk edge and a warp at radius $R_{\rm
warp}=10$ with a width of radius $R_{\rm width}=2$. There is no twist in
the disk. Since we do not have any torques to cause precession, the
disk remains untwisted throughout its evolution. Thus we consider only the inclination of
the disk and not the nodal precession angle.
\subsection{Wave--like propagation; $\alpha< H/R$}
\label{wl}
We consider the evolution of an initially warped disk with parameters
in the wave--like limit, $\alpha=0.01$ and $H/R=0.1$.
Figure~\ref{wavelike} shows the disk inclination and surface density
evolution for several cases. In the top left panel we solve the
wave--like warped disk equations~(\ref{lo1}) and~(\ref{lo2}) with a
fixed density distribution. The warp in the disk propagates both
inwards and outwards. The inwards propagating wave reflects off the
inner boundary and then begins to propagate outwards.
In the other panels of Fig.~\ref{wavelike} we solve the full disk
equations~(\ref{main}) and~(\ref{flux}) with different values for
$\beta$. In the top right panel we show the behaviour that occurs if
we do not introduce the parameter $\beta$. With $\beta=0$ the result
is that there appears to be unphysical evolution of the disk surface
density which occurs where the initial warp change was strongest, and
which continues long after the initial warp has propagated away.
The surface density anomaly shuld not keep growing at the position
of the initial tilt change, even when the tilt at that point has
evolved elsewhere. Furthermore, this behaviour is not seen in three
dimensional hydrodynamical simulations \citep[e.g.][]{Nealon2015}.
This unphysical behaviour was the reason for introducing the new
parameter $\beta$. The inclination evolution is very similar when we
solve the wave--like equations (top left panel) or the full equations
for $\beta \gtrsim 1$ (bottom panels). However, there is surface
density evolution when we solve the full equations and angular
momentum is conserved. The bottom two panels show that for $\beta
\gtrsim 1$, the surface density evolution is independent of the value
for $\beta$. There is slight difference between $\beta=1$ and
$\beta=10$ but we find no difference for even higher $\beta$ compared
to $\beta=10$.
\begin{figure}
\centering
\includegraphics[width=7.5cm]{diff10.eps}
\caption{Evolution of an initially warped disk around a single object
with no external torque with $\alpha=0.1$ and $H/R=0.01$ (in the
viscous regime). The upper panels show the inclination and the
lower panels the surface density.
The full equations are solved with $\beta=10$. The times shown are every
$500\,P_{\rm in}$ and as time advances the inclination at the inner
edge of the disk increases.}
\label{diffusive}
\end{figure}
\subsection{Diffusive warp propagation; $\alpha> H/R $}
\label{diff}
As a check, we consider the evolution of a disk with parameters in the
diffusive regime. We take $\alpha=0.1$ and
$H/R=0.01$. Fig.~\ref{diffusive} shows the disk inclination and
surface density evolution solving the full equations~(\ref{main})
and~(\ref{flux}) with $\beta=10$. We have also solved the diffusive equation~(\ref{viscous}) but find there is no difference between the two solutions and so we do not show this. In the diffusive regime there is no difference between solving the diffusive equations and the full equations that we have derived. The additional $\beta$ damping term has no effect in this limit.
\begin{figure}
\centering
\includegraphics[width=7.5cm]{int10.eps}
\caption{Evolution of an initially warped disk around a single object
with no external torque with $\alpha=0.1$ and $H/R=0.1$ (in the
intermediate regime). The full equations are solved with $\beta=10$. The upper panels show the inclination and the
lower panels the surface density. The times shown are every
$10\,P_{\rm in}$ and as time advances the inclination at the inner
edge of the disk increases.}
\label{intermediate}
\end{figure}
\subsection{Intermediate regime; $\alpha=H/R$}
Neither the wave-like equations nor the diffusive equations are able to model the evolution of a disk with $\alpha \approx H/R$. However, the full equations we have developed, equations~(\ref{main})
and~(\ref{flux}), can be used in this regime. Fig.~\ref{intermediate} shows the solution to the full equations with $\beta=10$ for a disk with $\alpha=0.1$ and $H/R=0.1$. The inner parts of the disk appear more diffusive in nature and the outer parts look more wave-like in the inclination evolution.
\section{Conclusions}
\label{conc}
We have introduced a new set of equations that describe the evolution of disk warp and of disk surface density in both low viscosity and high viscosity disks. We have shown that the two sets of equations agree with the equations for warp propagation previously derived in the two distinct regimes of low viscosity (wave-like warp propagation) and of high viscosity (diffusive warp propagation). In order to achieve this we have introduced a new dimensionless parameter $\beta$ which has the dominant effect of preventing unphysical evolution of surface density in the wave-like regime. We have not been able to determine the required magnitude of $\beta$ except to note that for $\beta\gg \alpha$ the unphysical evolution of surface density in the wave-like regime no longer occurs. In order to determine the value of $\beta$, and indeed to determine whether or not the new equations we present here provide an adequate description of warp evolution in general, it will be necessary to undertake a detailed analytic analysis \citep[c.f.][]{Ogilvie1999} and/or compare with detailed numerical simulations.
\section*{Acknowledgments}
RGM, SHL and AF acknowledge support from NASA through grant
NNX17AB96G. RN has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 681601). CJN is supported by the Science
and Technology Facilities Council (grant number ST/M005917/1).
\bibliographystyle{apj}
|
{
"timestamp": "2019-03-01T02:18:29",
"yymm": "1902",
"arxiv_id": "1902.11073",
"language": "en",
"url": "https://arxiv.org/abs/1902.11073"
}
|
\section{introduction}
Over the years there have been many attempts to solve some of the drawbacks of the Standard Model (SM) related to the presence of
a fundamental scalar boson (like the hierarchy problem, triviality, etc...). Some of the proposals along these lines are interesting
due to the fact that fundamental scalar bosons fit naturally into these models, as in supersymmetric models~\cite{su1,su2,su3}
and asymptotically safe SM extensions~\cite{sa00,sa11}. However, no signals of these theories have appeared so far. The Higgs particle found at the LHC~\cite{atlas,cms}
may be the first signal of a fundamental scalar boson, although the possibility that this boson is a composite one has not yet been discarded, and in this case some of the SM problems commented above may be alleviated.
Scalar bosons are essential to the mechanisms of chiral and gauge symmetry breaking in the SM, but it should be remembered that most
of what we have learned about the mechanisms of spontaneous symmetry breaking is based on the presence of composite or pair-correlated
scalar states, as happens in the Nambu-Jona-Lasinio model, QCD chiral symmetry breaking, and in the microscopic BCS theory of superconductivity.
For instance, chiral symmetry breaking is promoted in QCD by a nontrivial vacuum expectation value of a fermion
bilinear operator and the role of the Higgs boson is played by the composite $\sigma$ meson. These types of gauge theory models, dubbed technicolor (TC), were proposed
40 years ago~\cite{wei,sus} and reviewed in Refs.~\cite{far,mira}. The many variations of these models continue to be studied~\cite{an,sa,sa1,sa2,sa3,sa4,be},
but no phenomenologically viable model has been found so far.
It is clear that building SM extensions in order to solve unknown questions (like the origin of the fermionic mass spectra), is easier when we
deal with fundamental scalar bosons, than when the spontaneous symmetry breaking
is promoted by composite scalars, even if we are far from solving the problems related to the existence of fundamental scalar bosons. The difficulty in models with
composite scalar boson resides in knowing the dynamics of the non-Abelian gauge theory responsible for their formation.
We may say that the root of most TC problems lies in the way the ordinary fermions acquire their masses, which is
shown in Fig.1, where an ordinary fermion $f$ couples to a technifermion $F$ mediated by an extended technicolor (ETC) boson.
\begin{figure}[h]
\centering
\vspace*{-0.5cm}
\includegraphics[scale=0.45]{figETC.eps}
\vspace{-0.25cm}
\caption[dummy0]{Ordinary fermion mass $f$ in ETC models}
\label{fig1}
\end{figure}
\noindent
\par Assuming a standard non-Abelian TC self-energy ($\Sigma_{\T}$) given by~\cite{lane2}
\be
\Sigma_{\T} (p^2) \propto \frac{\mu_{\tt{TC}}^3}{p^2} \(\frac{p}{\mu_{\tt{TC}}}\)^{\gamma_m},
\label{eq1}
\ee
where $\mu_{\tt{TC}}$ is the characteristic TC dynamical mass at zero momentum (of order the Fermi mass) and $\gamma_m$ is the anomalous mass dimension (which depends on the TC coupling constant, and for an asymptotically free theory has a small value), the ordinary
fermion mass turns out to be
\be
m_f \propto \frac{\mu_{\tt{TC}}^3}{M_{\E}^2},
\label{eq2}
\ee
where $M_{\E}$ is the ETC gauge boson mass. In order to explain the top-quark mass we need a small $M_{\E}$ value, and since ETC is one interaction that changes flavor, the simplest model that we can imagine will inevitably lead to flavor changing neutral currents incompatible with the experimental data (among other problems).
Solutions to the above dilemma seem to require a large $\gamma_m$ value~\cite{holdom} leading to a TC self-energy with a harder momentum behavior, and many models along these lines can be found in the literature~\cite{lane0,appel,yamawaki,aoki,appelquist,shro,kura,yama1,yama2,mira2,yama3,mira3,yama4}. In particular, we may quote the work of Takeuchi~\cite{takeuchi} where the TC Schwinger-Dyson equation (SDE) was solved with the introduction of an four-fermion \textit{ad hoc} interaction, which can lead to the following expression for the TC self-energy:
\be
\Sigma_{\T}(p^2\rightarrow\infty)\propto \ln^{-\delta} (p^2/\mu_{\tt{TC}}^2),
\label{eq3}
\ee
where $\delta$ is a function of the many parameters of the model. The Takeuchi solution, when dominated by the four-fermion interaction,
is not different from the behavior of the self-energy when a bare mass is introduced into the theory, or from irregular SDE solution~\cite{lane2}.
\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{figETCfull.eps}
\vspace{-0.25cm}
\caption[dummy0]{The coupled system of SDEs for TC ($T\equiv$technifermion) and QCD ($Q\equiv$quark) including ETC and electroweak or other corrections. $G \,(g)$ indicates a technigluon (gluon).}
\label{fig2}
\end{figure}
Recently, we numerically solved the coupled TC [based on an $SU(2)$ group] and QCD gap equations~\cite{us1}, which are depicted in the Fig.2. It turned out that
both self-energies have the same asymptotic behavior as Eq.(\ref{eq3}). It is not difficult to understand the origin of such behavior. In Ref.~\cite{us2}
we analytically verified that the radiative corrections shown in Fig.2 act as an effective bare mass. In the case of ordinary quarks the second diagram ($b_2$)
on the right-hand of Fig.2 originates an effective mass due to TC condensation; on the other hand, the techniquarks obtain a tiny effective mass due to
QCD condensation [see diagram ($a_2$) in Fig.2], and an even larger mass due to the other diagrams [($a_3$) and ($a_4$)]. Therefore, the TC self-energy can be described by
\be
\Sigma_{\T}(p^2)\approx \mu_{\tt{TC}} \left[ 1+ \delta_1 \ln\left[(p^2+\mu^2_{\tt{TC}})/\mu^2_{\tt{TC}}\right] \right]^{-\delta_2} \,,
\label{eq4}
\ee
where $\delta_1$ and $\delta_2$ are parameters that will depend on the many possible SDE radiative corrections depicted in Fig.2; in particular, the dominant
correction to the technifermion masses will be generated by diagrams $(a_3)$ and $(a_4)$ of Fig.\ref{fig2}, and by diagram ($b_2$) in the case of ordinary fermion masses. We get a similar expression for ordinary quarks, and it should be noticed that the \textit{isolated} infrared TC and QCD self-energy
behavior is the traditional one [the one associated to the regular solution or Eq.(\ref{eq1})] with dynamical masses of order $\mu_{\tt{TC}} \approx O(1)$TeV and
$\mu_{\tt{QCD}}\approx 250$MeV, respectively, i.e., the coupled SDE system is a combination of the regular and irregular self-energy solutions~\cite{lane2}.
It is interesting to recall that such behavior is indeed that which minimizes the vacuum energy in gauge theories~\cite{us0}, and it is not different from Takeuchi's result
but rather originates from known interactions (QCD, for example).
The main consequence of the results of Refs. \cite{us1} and \cite{us2} [i.e., Eq.(\ref{eq4})] is that the dynamically generated masses will barely
depend on the ETC scale $M_\E$. In Ref.\cite{us1} we numerically verified that the ordinary quark masses behave as
\be
m_{\Q} \propto \lambda_E \mu_{\tt{TC}} [1+\kappa_1 \ln(M^2_{\E}/\mu_{\tt{TC}}^2)]^{-\kappa_2} \, ,
\label{eq5}
\ee
where $\lambda_E$ involves ETC couplings, a Casimir operator eigenvalue, and other constants, and $\kappa_i$ are related to the self-energies that enter in the calculation
of the generated masses, which is compatible with the quark mass computed with the help of Eq.(\ref{eq4}). Looking at Eq.(\ref{eq5}), it is clear that we can push the ETC scale up to the grand unification scale (or even the Planck scale) without large variations
of the $m_{\Q}$ values with $M_E$. It is also clear that the ordinary fermionic mass hierarchy will not arise from different $M_\E$ scales! The purpose of the present work it to
discuss how viable TC models can be built in this context, as well as to verify the phenomenological consequences of these models, and to show how that
they can be consistent with existing high-energy data.
It is important to note that the study of SDEs is very sophisticated, taking into account gluon-mass generation and possibly
confinement~\cite{g1,g2,g3,g4,g5} as well as complex vertex structures~\cite{g6,g7}. However, the solutions discussed in Refs.~\cite{us1,us2} and in this work are related to the asymptotic behavior produced by the effective mass of the coupled SDE, and are not affected by the infrared intricacies of the strongly interacting theories.
The paper is organized as follows. In Sec. II we present one specific TC model, which is just an example of the many models that can be built
along the lines described in that section. We discuss the fact that a horizontal symmetry is necessary in this scheme. In Sec. III we discuss how a composite scalar boson can be lighter than the typical composition scale
of the theory responsible for this particular state. In Sec. IV we determine the order of magnitude of pseudo-Goldstone masses. In Sec. V
we compare the value of the TC condensate in our model with the one expected in walking TC theories. Section VI contains a
brief discussion of possible experimental consequences of the models discussed in Sec. II, and in Sec. VII we discuss what can be expected
regarding the trilinear scalar coupling. Section VIII contains our conclusions.
\section{Building TC models}
In Ref.~\cite{us1} we briefly proposed one specific TC model, which will be detailed here. As will be discussed at the end of this section,
there is a large class of models that can be built along the
same lines as the model described here. The model discussed in Ref.~\cite{us1} is based on the following group structure
$$
SU(9)_U \otimes SU(3)_H \,\, ,
$$
where the $SU(9)_U$ group is a non-Abelian grand unified theory (GUT) containing the SM and a $SU(4)_{\tt{TC}}$ group. The $SU(3)_H$ group is a horizontal or family
symmetry that is important for generating the hierarchy of fermion masses.
There are several reasons for this particular choice.
First, the $SU(9)_U$ GUT will play the role of ETC, because the generated fermion masses will weakly depend on the GUT boson masses
(here acting as ``ETC"
boson masses) as
shown in Eq.(\ref{eq5}). This group also contains the standard $SU(5)_{\tiny \textsc{gg}}$ Georgi-Glashow GUT~\cite{gg}. Second, the $SU(4)_{\tt{TC}}$ group contained in the GUT will condense before QCD, generating an appropriate Fermi scale necessary
to break the electroweak group. Note that this choice is based on the most attractive channel (MAC) hypothesis~\cite{cor1,suss}, but it can be relaxed if the GUT breaking can
be promoted at very high energies, where even fundamental scalar bosons may be natural due to the presence of supersymmetry~\cite{su1,su2}. In this case we
could not neglect the possibility of a small TC group [perhaps $SU(2)$] that condenses at one mass scale larger than the QCD one. Third, the horizontal or family
symmetry is necessary to prevent the first- and second-generation ordinary fermions from coupling to TC. The third fermionic generation will obtain masses
due to diagrams like the one in Fig.1, and will be of order $\lambda_E \mu_{\tt{TC}}$, as described below.
The $SU(9)_U$ group has the following anomaly free fermionic representations~\cite{fra}:
\be
5\otimes[9,8]_i \oplus 1\otimes [9,2]_i \, ,
\ee
where $[\underline{8}]$ and $[\underline{2}]$ are antisymmetric under $SU(9)_U$, and $i=1,2,3$ is the horizontal index necessary for the replication
of the $SU(3)_H$ families. The decompositions of these representations under $SU(4)_{\tt{TC}}\otimes SU(5)_{\tiny \textsc{gg}}$ are
\br &&\hspace{-0.5cm}[\bf{9},\bf{2}]_i\nonumber\\
&&(1,10) = \left(\begin{array}{ccccc} 0 & \bar{u_{i}}_{B} & - \bar{u_{i}}_{Y} & -{u_{i}}_{R} & -{d_i}_{R} \\
-\bar{u_i}_{B} & 0 & \bar{u_i}_{R} & -{u_i}_{Y} & -{d_{i}}_{Y} \\ \bar{u_{i}}_{Y} & -\bar{u_{i}}_{R} & 0
& -{u_i}_{B} & -{d_{i}}_{B} \\ {u_i}_{R} & {u_i}_{Y} & {u_i}_{B} & 0 & \bar{e_i}\\
{d_i}_{R} & {d_i}_{Y} & {d_i}_{B} & -\bar{e_{i}} & 0\end{array}\right)\nonumber\\\nonumber \\ &&(4,5) =
\,\,\,\left(\begin{array}{c} {T_i}_{R} \\ {T_i}_{Y} \\ {T_i}_{B} \\ \bar{L_i}\\ \bar{N_i}
\end{array}\right)_{TC}\,\,\,,\,\,\,(\bar{6},1)= N_{i}\nonumber \\ \nonumber \\
&&\hspace{-0.5cm}[\bf{9},\bf{8}]_i\nonumber\\
&&(1,\bar{5}) =\,\,\, \left(\begin{array}{c} \bar{d_i}_{R} \\ \bar{d_i}_{Y} \\ \bar{d_i}_{B} \\
e_i \\ \nu_{e_i}
\end{array}\right)
\,\,\,\,\,\,(1,\bar{5}) = \left(\begin{array}{c} \bar{X}_{R_{k}} \\ \bar{X}_{Y_{k}} \\ \bar{X}_{B_{k}} \\
E_{k} \\ N_{E_{k}}
\end{array}\right)_i\nonumber\\ \nonumber \\\nonumber\\
&&(\bar{4},1)= \,\,\,\,\,\bar{T_i}_{\varepsilon}, L_i ,{N_i}_{L}. \nonumber \er
\noindent In the fermionic content of the above model, we identify the usual quarks as $Q = (u ,d)$, while $T$ corresponds to techniquarks and $(L , N)$ to technileptons, where $\varepsilon = R,Y,B$ is a color index, and $k=1...4$ indicates the generation number of exotic fermions that must be introduced in order to render the model anomaly free.
The $SU(3)_H$ quantum numbers must be assigned such that the quartet of technifermions that condenses in the MAC of the
product ${\bf \bar{4}\otimes 4}$ belongs to the ${\bf \overline{6}}$ representation of the horizontal group, whereas the QCD quark condensate (generated in the color product
${\bf \bar{3}\otimes 3}$), is formed in the triplet representation (${\bf 3}$) of $SU(3)_H$. This is nothing else than the horizontal symmetry scheme
with fundamental scalar bosons proposed in Refs.~\cite{h1,h2,h3,h4}, and it leads to a quark mass matrix in the horizontal group basis of the form
\br
m_q =\left(\begin{array}{ccc} 0 & m_1 & 0\\ m_1^* & 0 & 0 \\
0 & 0 & m_3
\end{array}\right),
\label{eq6} \er
\noindent where $m_1$ and $m_3$ indicate the first- and third-generation quark masses.
It is instructive to show the diagrams that lead to the different masses shown in Eq.(\ref{eq6}). For instance, let us assume that
$m_q$ is the mass matrix of charge $2/3$ quarks, where $m_3$ would be related to the top-quark mass. The diagrams responsible for
this mass are shown in Fig.3.
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{masstop2.eps}
\caption[dummy0]{Diagrams contributing to the top-quark mass.}
\label{fig3}
\end{figure}
\par In this figure the technifermions $T$ and $L$ [that condense in the ${\bf \overline{6}}$ of $SU(3)_H$] give masses to the $t$ quark whose interaction is mediated by one $SU(9)$ gauge boson. Apart from the logarithmic term appearing in Eq.(\ref{eq5}) this mass is
\be
m_3\approx 2 \lambda_9 \mu_{\tt{TC}} \, ,
\label{eq7}
\ee
where we can assume that $\lambda_9 \approx 0.1$, is the product of the $SU(9)$ coupling constant times some Casimir operator eigenvalue, the factor $2$ accounts both diagrams of Fig.3, and $\mu_{\tt{TC}}$ can be assumed to be of $O(1)$TeV. The $SU(9)$ interaction is playing
the role of the ETC interaction. These naive assumptions
will lead to a top-quark mass of approximately $200$GeV. The logarithmic term appearing in Eq.(\ref{eq5}) [and
neglected in Eq.(\ref{eq7})] slightly decreases the value of our rough estimate.
Note that the first and second charge $2/3$ quarks do not couple directly to the techniquarks
due to the different $SU(3)_H$ quantum numbers, and at this level they remain massless.
We can now see how the first-generation fermions obtain their masses. In Fig.4 we show the diagrams that are responsible for the $u$-quark mass.
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{massup2.eps}
\caption[dummy0]{Diagrams contributing to the light-quarks masses.}
\label{fig4}
\end{figure}
This quark does not couple to techniquarks at leading order, but does couple to other ordinary quark and itself due to the
bosons of the unified theory and the horizontal one. Its mass can be approximated from Eq.(\ref{eq5}) [as we did to obtain Eq.(\ref{eq7})] and
is given by
\be
m_1\approx \lambda_5 \mu_{\tt{QCD}} \, ,
\label{eq8}
\ee
where we can assume naively that the $SU(5)_{\tiny \textsc{gg}}$ factor $\lambda_5 \approx 0.1$ and $\mu_{\tt{QCD}}\approx 200$MeV, which gives
a mass of order $20$MeV. Here we do not introduce a factor of $2$ in Eq.(\ref{eq8}) due to the presence of the two diagrams in Fig.4, because the $c$-quark condensate (in the second diagram of Fig.4) may be smaller than the $u$ and $d$ condensates\footnote{Note that the self-energy and the condensate values are intimately connected, i.e., one is basically an integral of the other.
The $c$-quark self-energy appearing in Fig.4 will involve the same type of integral as the $c$-quark condensate.
It is known that the introduction of heavy quark masses act to diminish
the condensate value or the amount of chiral symmetry breaking ~\cite{x1}. For example, it has been determined for the $s$-quark that
$\<\bar{s}s\>/\<\bar{u}u\>=0.6\pm 0.1$ ~\cite{x2,x3}.
In Ref.~\cite{sa5} the same effect of a heavy fermion mass (e.g., fermion loops) was also observed as a factor that lowers the composite Higgs boson mass. Therefore, the second diagram of Fig.4 is
expected to have a smaller effect in the calculation of the first-generation quark masses.}.
In Eqs.(\ref{eq7}) and (\ref{eq8}) we probably overestimated the results when we neglected the logarithmic dependence on
the unified or ``ETC" boson masses. These are very simple calculations. To obtain better estimates we must solve the
coupled SDE and obtain good fits to the self-energies, which would give us reasonable values for the parameters $\delta_1$ and $\delta_2$ in the
approximate expression of Eq.(\ref{eq4}). It is clear that this is far beyond the scope of this work.
The mass of the second quark generation will necessarily involve the horizontal symmetry, where the coupling to techniquarks will appear
only at two-loop order. The $c$-quark mass will be generated by diagrams like the ones shown in Fig.5,
\begin{figure}[h]
\centering
\includegraphics[scale=0.55]{masscharm.eps}
\caption[dummy0]{Diagrams contributing to the $c$-quark mass.}
\label{fig5}
\end{figure}
and it is expected to be $1$ order of magnitude below the typical mass of the third quark generation, due to an extra factor $\lambda_{3H}\approx 0.1$ that contains the $SU(3)_H$ coupling constant. In this way, we verify that the horizontal or family symmetry is fundamental to generate a quark mass
matrix with the Fritzsch texture~\cite{f1,f2}
\br
m_q =\left(\begin{array}{ccc} 0 & m_1 & 0\\ m_1^* & 0 & m_2 \\
0 & m_2^* & m_3
\end{array}\right),
\label{eq9} \er
which has several good qualities of the experimentally known quark mass matrix.
Lepton masses will appear in the same way as quark masses. The $\tau$ lepton is the only one that will couple with techniquarks at leading order, due to the appropriate choice of quantum numbers of the horizontal symmetry. As a consequence, the mass matrix for the leptonic sector is similar to the one described above, although lepton masses should be naturally smaller
than quark masses, because quarks end up coupling to two different condensates and a larger number of diagrams contribute to their masses. It is not difficult to verify
the different number of SDEs between quarks and leptons that can be generated with the Feynman rules of the model described here.
We have not discussed the $SU(9)_U$ and horizontal symmetry breaking, which we just assume to happens at the unification scale $\Lambda_{{}_{SU(9)}}$,
which can possibly be
naturally promoted by fundamental scalar bosons. The breaking of the GUT symmetry can also be used to produce a larger splitting in the third fermionic generation.
For instance, if in the $SU(9)_U$ breaking (besides the Standard model interactions and the TC one) we
leave an extra $U(1)$ interaction, we could have quantum numbers such that only the top quark would be allowed to couple to the TC condensate at leading order.
In fact, the splitting ($S_{(t-b)}$) between the $t$ and $b$ quarks
\be
S_{(t-b)}= \frac{m_b}{m_t}\approx \frac{1}{40} \, ,
\ee
is quite large, and it is interesting that the $b$ quark and the $\tau$ lepton could couple at a larger order in the coupling constant [possibly $(\alpha_9^2)$], which could be accomplished by
this remaining $U(1)$ interaction that we referred to above. More sophisticated models in which large fermionic mass splittings and even neutrino masses can be generated
were presented in Refs.~\cite{as1,as2,as3,as4,as5}.
At this point, we hope that we have made clear the necessity of introducing a horizontal or family symmetry. It is necessary to prevent
the first and second generations of ordinary fermions from obtaining large masses that couple to TC at leading order. This symmetry can be a local one, but a
global symmetry is not necessarily discarded. If the family symmetry is local, its breaking can also happen at very high energies and (again) may even be
promoted by fundamental scalars at the GUT or Planck scale, producing feeble effects at lower energies.
When building a TC model the existence of
grand unification is also welcome. For example, in the model described here
a $SU(5)_{\tiny \textsc{gg}}$ gauge boson interaction is fundamental to give the electron a mass, which appears due to the electron coupling to the
first-generation quark, with exactly the same interaction that may mediate proton decay in the $SU(5)_{\tiny \textsc{gg}}$ theory. There are more diagrams
contributing to the first-generation quark masses than there are for the electron mass, which may explain why leptons are less massive than quarks.
Concerning the possible class of models presented here, it is also clear that a full and precise determination of the mass spectra is quite complex. Once a GUT involving the SM and TC is proposed, we also have to choose
the horizontal symmetry. The coupled SDE of such a model has to be solved by determining all self-energies with their specific infrared and ultraviolet expressions.
Of course, simple estimates can be made by approximating the calculation of each specific fermion mass diagram, by the product of the dynamical mass involved in
the diagram (TC or QCD) with the respective coupling constants and Casimir operator eigenvalues, as performed in Eqs.(\ref{eq7}) and (\ref{eq8}) where a logarithmic
term was neglected.
\section{Scalar mass}
The common lore about theories with a composite scalar boson is that its mass should be of the order of the dynamical mass scale that forms such particle.
This concept is related to the work of Nambu-Jona-Lasinio~\cite{nl} and was also discussed for the $\sigma$ meson in QCD~\cite{ds}, where the scalar composite mass appearing in one strongly interacting theory is given by
\be
m_\sigma = 2 \mu_{\tt{QCD}}\,\, .
\label{eq10}
\ee
Equation (\ref{eq10}) comes from the fact that at leading order the SDE for the quark propagator is similar to the homogeneous Bethe-Salpeter equation (BSE)
for a massless pseudoscalar bound state $\Phi_{BS}^P (p,q)|_{q \rightarrow 0}$ (the pion), and a scalar p-wave bound state $\Phi_{BS}^S (p,q)|_{q^2 = 4 \mu^2 }$
[the sigma meson or the $f_0(500)$~\cite{pdg}], i.e.,
\be
\Sigma (p^2) \approx \Phi_{BS}^P (p,q)|_{q \rightarrow 0} \approx \Phi_{BS}^S (p,q)|_{q^2 = 4 \mu_{\tt{QCD}}^2 }\,\,.
\label{eq11}
\ee
Equation (\ref{eq11}) tells us that in QCD the $\sigma$ meson must have a mass $2\mu_{\tt{QCD}}\approx 500$MeV. In TC we should expect a scalar boson with a mass of $2$TeV, which
is clearly not the case for the observed Higgs boson~\cite{atlas,cms}
There are two subtle points concerning the result of Eq.(\ref{eq10}) and the determination of the scalar composite mass. The first one is that Eq.(\ref{eq10})
was determined using the homogeneous BSE. There is nothing wrong with this. However this gives the right result if the fermionic self-energy that
enters into the BSE is a soft one. When the self-energy decreases slowly [as in Eq.(\ref{eq4})] the scalar mass is modified by the
normalization condition of the inhomogeneous BSE. This modification lowers the composite scalar mass as a consequence
of Eq.(\ref{eq4}).
The second point about Eq.(\ref{eq10}) that we would like to note is not exactly about the equation itself, but rather about the values of the
dynamical QCD and TC mass scales that arise at such a scale. The QCD dynamical mass scale is usually extracted from the hadronic spectra;
for instance, it is expected to be $1/3$ of the nucleon mass or $1/2$ of the sigma meson mass. However, it is not currently clear how much this spectra
is affected by gluons (or technigluons in the TC case) and mixing among different particles. These points will be discussed in the
following subsections.
\subsection{Normalization condition and the scalar mass}
The BSE normalization condition in the case of a non-Abelian gauge theory is given by \cite{lane2}
\br
2\imath q_{\mu}= \imath^2\!\!\int d^4\!p\, Tr\left\{{\cal P}(p,p+ q)\left[\frac{\partial}{\partial q^{\mu}}F(p,q)\right]{\cal P}(p, p+ q) \right\}\nonumber \\
-\imath^2\!\!\int d^4\!pd^4\!k \,Tr\left\{{\cal P}(k,k + q)\left[\frac{\partial}{\partial q^{\mu}}K'(p,k,q)\right]{\cal P}(p, p+ q)\right\} \nonumber
\label{eq11a}
\er
where
$$
K'(p,k,q) = \frac{1}{(2\pi)^4}K(p,k,q) \,\,\, ,
$$
$$
F(p,q) = \frac{1}{(2\pi)^4}S^{-1}(p+q) S^{-1}(p) \,\,\, ,
$$
where ${\cal P}(p, p + q)$ is a solution of the homogeneous BSE and $K(p,k,q)$ is the fermion-antifermion scattering kernel
in the ladder approximation. When the internal momentum $q_{\mu} \rightarrow 0$, the wave function ${\cal P}(p, p + q)$ can be determined only through
the knowledge of the fermionic propagator:
\be
{\cal P}(p) = S(p)\gamma_{5}\frac{\Sigma(p)}{F_{\Pi}}S(p) \,\, ,
\ee
\noindent where $\Sigma (p)$ will describe the technifermion self-energy and it should be noticed that $F_{\Pi}$ describes the technipion decay constant associated with $n_{d}$ technifermion doublets. If we identify $\Sigma(p^2) \equiv \mu_{\tt{TC}} f(p^2)$ we can write the normalization condition in the rainbow approximation as
\br
&&2i\left(\frac{F_{\pi}}{\mu_{\tt{TC}}}\right)^2 q_{\mu} = \frac{i^2}{(2\pi)^4}\times \nonumber \\
&& \left[\int d^4\!p\, Tr{\Big \{}S(p)f(p)\gamma_{5}S(p )\left[\frac{\partial}{\partial q^{\mu}}S^{-1}(p + q) S^{-1}(p)\right]\right. \nonumber \\
&& \left. S(p)f(p)\gamma_{5}S(p){\Big \}} + \frac{i^2}{(2\pi)^4}\int d^4\!pd^4\!k \,Tr{\Big \{} S(k)\right. \nonumber \\
&& \left.f(k)\gamma_{5}S(k)\left[\frac{\partial}{\partial q^{\mu}}K(p,k,q)\right]S(p)f(p)\gamma_{5}S(p){\Big \}}\right]. \nonumber \\
\label{eq11b}
\er
\par Equation (\ref{eq11b}) is quite complicated, but it can be separated into two parts:
\be
2i\left(\frac{F_{\Pi}}{\mu_{\tt{TC}}}\right)^2 q_{\mu} = I_\mu^{0} + I_\mu^{K} \,\,\, ,
\label{eq14a}
\ee
corresponding, respectively, to the two integrals on the right-hand side of Eq.(\ref{eq11b}). The fermion propagator given by
$S(p) = {1}/[{\not{\!\!p} - \Sigma(p)}]$ can be written as
\be
\frac{\partial}{\partial q^{\mu}}S^{-1}(p + q) = \gamma_{\mu} - \frac{\partial}{\partial q^{\mu}}\Sigma(p+q) \,\,\, ,
\ee
\noindent and the term $ \frac{\partial}{\partial q^{\mu}}\Sigma(p+q)$ in the above expression may be written as
\be
\frac{\partial \Sigma(p+q) }{\partial q^{\mu}} = (p + q)_{\mu} \frac{d\Sigma(Q^2)}{dQ^2}
\ee
\noindent where $Q^2 = (p + q)_{\mu}(p + q)^\mu$. Considering the angle approximation we transform the term $\frac{d\Sigma(Q^2)}{dQ^2}$ as
\be
\frac{d\Sigma(Q^2)}{dQ^2} = \frac{d\Sigma(p^2)}{dp^2}\Theta(p^2 - q^2) + \frac{d\Sigma(q^2)}{dq^2}\Theta(q^2 - p^2)
\ee
where $\Theta$ is the Heaviside step function. We can finally contract Eq.(\ref{eq14a}) with $q^\mu$ and compute it at $q^2=M_H^2$ in order to obtain
\br
M_{H}^2 = 4&&\mu_{\tt{TC}}^2{\Big\{}\frac{n_{f}N_{TC}}{8\pi^2}\int d^2\!p\frac{f^2(p)\Sigma(p)}{(p^2 + \Sigma^2(p))^2} \times \nonumber \\
&& \times \left(-p^2\frac{d\Sigma(p)}{dp^2} \right) \left(\frac{\mu_{\tt{TC}}}{F_{\Pi}}\right)^2 + \nonumber \\
&& + \,\,I^{K}(q^2 = M_H^2,f(p,k),g_{TC}^2(p,k)) {\Big \}},
\label{eq11c}
\er
where $n_f$ is the number of technifermions, $N_{TC}$ is the number of technicolors and $g_{TC}$ is the technicolor coupling constant.
An expression similar to Eq.(\ref{eq11c}) was already obtained by us in Ref.~\cite{usx}. In that work we just assumed (in a totally \textit{ad hoc} fashion)
a hard momentum behavior for the TC self-energy. The calculation here will differ not only in the origin of the self-energy but also in the approach
we follow to determine the value of $M_H$. Considering the work of Ref.~\cite{us2}
it becomes evident that the behavior of $M_{H}$ is a result that will fundamentally depend on the boundary conditions satisfied by the
coupled system described in Fig.2. In Eq.(\ref{eq11c}) the UV behavior of the term
\be
{\rm \bf (UV)}\,\,\,\,\,\,\,lim_{{}_{{}_{\hspace*{-0.5cm} p^2 \to \Lambda^2}}}\!\!\!-p^2\frac{d\Sigma(p)}{dp^2} ,
\label{newuv}
\ee
\noindent will be affected by the effective mass generated by the diagrams $(a_2)$, $(a_3)$, and $(a_4)$ in Fig.2. In Ref.~\cite{us2} we verified that
the UV behavior of the term in Eq.(\ref{newuv}) is modified as $\alpha_E$ is different or equal to zero, and we shall comment on this term later.
We compute $M_H$ by numerically solving the
differential coupled equations shown in Eqs.(11) and (12) of Ref.~\cite{us2} , fitting the resulting solutions (all fits with $R^2=0.98$), and inserting the fits into Eq.(\ref{eq11c}). We consider the TC gauge groups $SU(2)_{TC}$, $SU(3)_{TC}$ and $SU(4)_{TC}$, with $n_f=5$ fermions in the fundamental representation, $\mu_{\tt{TC}}=1$TeV, and use the MAC hypothesis to constrain the TC gauge coupling and Casimir eigenvalue. Hereafter, we follow
Refs.~\cite{us1,us2} and use a Casimir eigenvalue $C_E=1$ and gauge coupling constant $\alpha_E = 0.032$, which are quantities related to the ETC gauge theory.
Our results for $M_H$ are shown in Table I, where we can see that the normalization condition lowers the scalar mass by a factor of $O(1/10)$. The results are consistent with those of Ref.~\cite{usx} obtained with the naive assumption of an irregular solution for the TC self-energy. Therefore, the effect
of radiative corrections in coupled SDEs involving a TC theory act in order to produce a scalar composite boson with a mass compatible with that of
the observed Higgs boson.
\begin{table}[htbp]
\centering
\begin{tabular}{ccc}\hline \hline
SU(N) & $n_f$ & $M_H$(GeV) \\ \hline
2 & 5 & 105.3 \\
3 & 5 & 141.5 \\
4 & 5 & 148.8 \\
\hline \hline
\end{tabular}
\caption{The last column contains the composite scalar mass determined through Eq.(\ref{eq11c}), where we used the TC self-energy obtained by solving the coupled SDE system. The different factors and couplings of the gap equations are described in the text. }
\label{tbl:IneqFP}
\end{table}
\subsection{Dynamical mass scales and mixing}
The most precise quantity to constrain the dynamical mass scale in the QCD case is the
pion decay constant, which is a function of the quark self-energy. In the TC case the technipion decay constant is
related to the $W$ and $Z$ gauge boson masses. However, in both cases that quantity depends on the dynamical mass scale as well as the functional
expression for the self-energy. Therefore, we have some freedom in pinpointing the dynamical mass scale. Even the numerical determination of the self-energy
through SDE solutions includes the introduction of a cutoff and specific approximations. We conclude that the calculation of the scalar boson mass
depends on the functional form of the self-energy and on the dynamical mass scale. It is curious that in the past the scalar boson mass
was considered in order to constrain the dynamical mass scale, i.e., in QCD the scalar $\sigma$ meson mass has led to the usual value $\mu_{\tt{QCD}}\approx 250$MeV, which is also approximately
the value of the QCD mass scale ($\Lambda_{\tt{QCD}}$). The problem is that the result of Eq.(\ref{eq10}) is modified not only by the inhomogeneous
BSE condition, butalso by many other effects as we discuss in the following.
The dynamical QCD mass scale is also thought to be related to the nucleon mass, but even this is not certain since we do not know how much gluons
contribute to the nucleon mass~\cite{lorce}. It is also not yet clear how much of the sigma meson mass comes from mixing with heavier
quark-antiquark scalars and with glueballs~\cite{mi1,mi2,mi3,mi4,mi5,mi6}, and the same is true if we just exchange QCD with TC, which means that
the scales $\mu_{\tt{QCD}}$ and $\mu_{\tt{TC}}$ may be smaller than usually thought, leading to a smaller scalar
composite mass (i.e., the $\sigma$ and the ``Higgs" mass). The scalar mass can also be modified by the effect of radiative loop corrections due to
the presence of heavy fermions, as described in Ref.~\cite{sa5}.
These are not the only effects that modify the scalar mass and lead to a new relation between the scalar mass and the dynamical mass scale. There is still another effect that is intimately related to the type of dynamical
symmetry breaking model that we discussed in the previous section.
In Sec. II we discussed a model with two composite scalar states responsible for the chiral (and gauge) symmetry breaking: the scalars
belonging to the ${\bf \overline{6}}$ and ${\bf 3}$ representations of the horizontal group formed by technifermions and
quarks, respectively. The different scalars may mix among themselves due to electroweak or other interactions, as already pointed out in Ref.~\cite{us1}.
An order-of-magnitude estimate of these mixing diagrams is quite lengthy, but the most important
fact is that the scalar coupling to the electroweak bosons is going to be enhanced, when compared to this coupling
calculated when the TC self-energy is soft. Note that this effective coupling happens when scalars and $W$ bosons couple through a ordinary fermion or technifermion
loop. The $W$ coupling to fermions is the SM one, while the scalar composite coupling to ordinary fermions was shown by Carpenter \textit{et al}.~\cite{ca1,ca2} to be
proportional to $\frac{g_w}{2M_W}\Sigma$, where $\Sigma$ is the fermionic self-energy, which now is a slowly decreasing function of momentum and enhances
the effective coupling. If we denote a composite scalar by $\phi$, it is possible to show that the $\phi\phi WW$ effective coupling will
be proportional to~\cite{us3}
\be
\Gamma_{\phi\phi WW}\propto \frac{g_W^4\delta^{ab}}{M_W^2}\frac{g^{\mu\nu}}{32\pi^2}\int dq^2 \frac{\Sigma_\phi^2}{q^2},
\label{eq12}
\ee
where $\Sigma_\phi$ has to be substituted by the TC or QCD self-energy depending on which fermion is involved in the composite scalar. Of course, the
complete calculation of the mixing diagrams is quite model dependent, but, as commented in Ref.~\cite{us1}, the origin of this mixing is another
way to see how a full Fritzsch matrix pattern of fermion masses can be generated in the type of model that we are proposing here. It is due to
this type of coupling that the second-generation fermion masses are generated in models with fundamental scalar bosons~\cite{h1,h2,h3,h4}. Finally, in
the context where all SM symmetry breaking is promoted by composite scalars we cannot even say how much of the $\sigma$ [r $f_0 (500)$] meson mass is due to a possible mixing with a composite Higgs boson.
\section{Pseudo-Goldstone bosons}
In the condensation of the $SU(4)_{\tt{TC}}$ group a large number of Goldstone bosons are formed. Even if we consider other TC groups, only
three of the Goldstone bosons are absorbed in the SM gauge breaking, and regardless of the theory we may end up with several light composite
states resulting from the chiral symmetry breaking of the strong sector.
These pseudo-Goldstone bosons (or technipions) in the model
of Sec. II may have different quantum numbers.
They may be colored bosons $~\bar{Q}\gamma_5\lambda^a Q$, where $\lambda^a$ is a color group generator, charged bosons $~\bar{L}\gamma_5 Q$ and
neutral pseudo-Goldstone bosons $~\bar{N}\gamma_5 N$. These bosons receive masses through radiative corrections, and we will verify that,
as a consequence of the logarithmic TC self-energy, they will be heavier than usually thought, which is desired in view of the
stringent limits on light technipions~\cite{scs1}.
In Ref.~\cite{us1} we briefly commented that the technipion masses ($m_\Pi$) are enhanced in comparison with models where the TC self-energy does
not have the form of Eq.(\ref{eq4}). One of the arguments is quite simple: the technifermions obtain an effective mass ($m_F$) of several GeV
through diagrams ($a_3$) and ($a_4$) of Fig.2. Note that in our case the condensation effect is not soft, and the calculation of these diagrams will result in
a mass that is not different from those of the third ordinary fermionic family. In particular, in our model there will be several contributions
to these types of diagrams. Even the neutral technifermion $N$ will receive contributions from TC condensation mediated by the electroweak $Z$ boson,
and from QCD condensation due to $SU(9)$ GUT bosons. These masses, apart from small logarithmic terms, will be roughly of order
\be
m_F \approx \sum_{i}\lambda_i \mu_{\tt{TC}} \, ,
\label{eq13}
\ee
where $\lambda_i$ represents the product of some coupling constant times Casimir operator eigenvalue contained in any diagram of the type ($a_3$) or ($a_4$)
contributing to the technifermion mass. For the colored and charged technifermions we cannot even discard a mass as heavy or higher than the top-quark
mass. These masses will generate rather heavy technipions as can be verified using the Gell-Mann-Oakes-Renner relation
\be
m_\Pi^2 \approx m_F \frac{\<{ \bar{\psi}_T}\psi_T\>}{2F_{\Pi}^2} \, ,
\label{eq14}
\ee
where $\<{ \bar{\psi}_T}\psi_T\>$ is the TC condensate and $F_{\Pi}$ is the technipion decay constant. With $m_F$ of order of several GeV and standard values
for the condensate and technipion decay constant the technipion masses turn out to be of order of $100$ GeV or higher, as discussed in Ref.~\cite{us1}.
Another way to see that technipion masses are enhanced through the calculation of a diagram that was already shown in Ref.~\cite{us1}
(see Fig.4 of that reference). Any radiative boson exchange within a technipion modifying its mass will necessarily involve the technipion vertex
connecting it to technifermions ($\Gamma_{\Pi F}$). However this vertex is proportional to the technipion wave function $\Phi_{BS}^\Pi (p,q)$,
which at leading order is also related to the TC self-energy as
\be
\left.\Phi_{BS}^\Pi (p,q)\right|_{q\rightarrow 0} \approx \Sigma_T (p^2) \, ,
\label{eq15}
\ee
which is responsible for an enhancement of this radiative correction. An order-of-magnitude calculation of such a diagram was presented in Ref.~\cite{us1},
and we will comment later on the phenomenology of technipions with masses that are not very different from that of the Higgs boson.
\section{TC condensate}
In the previous section and throughout this work we have commented about the different condensates (TC and QCD), and it is interesting to make a connection between
the several studies about the TC condensate value based on walking TC~\cite{yama} and the one we are discussing here. The TC condensate at one high energy
scale $\Lambda$ is related to its value at another scale $\mu$ by
\be
\<{ \bar{\psi}_T}\psi_T\>_{\Lambda} = Z^{-1}_m\<{ \bar{\psi}_T}\psi_T\>_\mu \, ,
\ee
where $Z^{-1}_m$ is a renormalization constant which is given by
$$
Z^{-1}_m \sim \left(\frac{\Lambda}{\mu}\right)^{\gamma_m} \, ,
$$
where $\gamma_m$ is the condensate operator anomalous dimension.
It is possible to compare the condensate values for a theory where the anomalous dimension is perturbative and small at high energy,
i.e. $\gamma_m \to 0$ and the one with a nontrivial large anomalous dimension, for instance, in the extreme walking case where $\gamma_m \to 2$.
We can define the following ratio that measures the difference between condensates in the walking and nonwalking regimes:
\be
R_w = \frac{\<{ \bar{\psi}_T}\psi_T\>_{\Lambda}^{\gamma_m \to 2}}{\<{ \bar{\psi}_T}\psi_T\>_{\Lambda}^{\gamma_m \to 0}} \, ,
\label{rw}
\ee
Considering these extreme cases this ratio is proportional to
\be
\left. R_w\right|_{\gamma_m \to 2} \approx \left( \frac{\Lambda}{\mu}\right)^{2} \, ,
\label{rw2}
\ee
and this expression serves as an indicator of how much the theory is modified by the nontrivial anomalous dimension. This kind of relation can also be used to
verify how radiative corrections appearing in Fig.2 change the TC behavior.
The UV boundary conditions of the differential TC gap equations modified by the radiative corrections (as can be seen in Ref.~\cite{us2}) are given by
\be
\left. p^2\frac{d\Sigma(p)}{dp^2}\right|_{\Lambda \to \infty} = -a\int^{\Lambda^2}_0 dk^2\frac{\Sigma(k)}{k^2 + \Sigma^2(p)} \, ,
\ee
where $a$ is a factor involving the gauge coupling constant and Casimir operator eigenvalue related to the interaction that induces the radiative correction [e.g., constants related to one of the diagrams $(a_2)$, $(a_3)$ or
$(a_4)$ in Fig.2]. On the other hand, we recall that in an $SU(N)$ gauge theory the condensate can be represented by
\be
\<{ \bar{\psi}_T}\psi_T\>_{\Lambda} = -\frac{N}{4\pi^2}\int^{\Lambda^2}_0 dk^2\frac{\Sigma(k)}{k^2 + \Sigma^2(k)} \, .
\ee
These relations allow us to redefine the ratio shown in Eq.(\ref{rw}) where the condensate values are determined with and without radiative corrections, i.e., when they
are calculated with the coupled SDE system ($\alpha_E \neq 0$ ) and with the values of the isolated condensates ($\alpha_E = 0$),
\be
R_w^{rad.cor.} = \frac{\<{\bar{\psi}_T}\psi_T\>^{\alpha_E \neq 0}_{\Lambda}}{\<{\bar{\psi}_T}\psi_T\>^{\alpha_E =0}_{\Lambda }}
\approx
\frac{\left. p^2\frac{d\Sigma(p)}{dp^2}\right|^{\alpha_E\neq 0}_{\Lambda \rightarrow\infty} }{\left. p^2\frac{d\Sigma(p)}{dp^2}\right|^{\alpha_E=0}_{\Lambda \rightarrow \infty}} \, .
\label{rwcor}
\ee
We computed Eq.(\ref{rwcor}) by considering the solutions of the coupled and isolated SDE system in the case of the $SU(3)$ TC group, with
$\mu =1$ TeV, $\alpha_E =0.032$, $\alpha_{{_{TC}}} = 0.87$ and $C_{{}_{TC}} = 4/3 $. The self-energies were obtained in terms of
the variable $x = p^2/\mu^2$ for each ETC scale $M_E$, and the condensates were integrated from $x=10^2$ up to the UV cutoff $ x_\Lambda= \Lambda^2/\mu^2 \sim 10^7 $.
The ratio $R_w^{rad.cor.}$ was fitted with $R^2=0.999$ in the form $a_1[ln(M^2_E/\mu^2)]^{a_2}$ and the result is
\be
R_w^{rad.cor.} \propto 7.87 \times 10^6[ln(M^2_E/\mu^2)]^{-4.3} \, .
\label{eq14g}
\ee
If we consider the value of our cutoff ($\Lambda^2/\mu^2 = 10^7$), we can verify that the effect of the radiative correction is not exactly that of
the extreme walking case shown in Eq.(\ref{rw2}), but it is still quite large. We again see that the effect of radiative corrections is not that
different from the effect of the \textit{ad hoc} four-fermion interactions determined by Takeuchi~\cite{takeuchi}. Moreover, if we compute
the generated quark mass ($m_Q$) as a function of the TC condensate we obtain
\be
m_Q \approx \frac{\<{\bar{\psi}_T}\psi_T\>^{\alpha_E \neq 0}_{\Lambda}}{\Lambda^2} \approx C [ln(M^2_E/\mu^2)]^{-\kappa_2},
\label{eq14j}
\ee
where the constant $C \sim O(\mu)$. This behavior is consistent with that of Eq.(\ref{eq5}).
\section{Experimental constraints}
\subsection{$S$ parameter}
The $S$ parameter provides an important test for new physics beyond the Standard Model~\cite{pt}. This parameter can be described by the absorptive
part of the vector-vector minus axial-vector-axial-vector vacuum polarization in the following form in the case of a TC model with
new composite vector and axial-vector mesons with masses $M_V$ and $M_A$ and respective decay constants $F_V$ and $F_A$~\cite{pt}:
\be
S=4\int_0^\infty \frac{ds}{s} Im \overline{\Pi}(s)=4\pi \left[ \frac{F_V^2}{M_V^2}-\frac{F_A^2}{M_A^2} \right] \, .
\label{eqss}
\ee
An interesting analysis of the $S$ parameter in TC theories was performed in Ref.~\cite{asan} with the use of the Weinberg sum rules,
where the case of a conformal theory was considered. In our case, we have a TC model which is just a scaled QCD theory,
with effective masses due to the different SDE contributions shown in Fig.2, besides its dynamical mass of $O(1)$ TeV.
There is no reason to expect modifications of Eq.(\ref{eqss}) for this type of theory, as well as the simple extension to TC of the first and second Weinberg sum rules,
which are respectively
\be
F_V^2 - F_A^2 = F_\Pi^2 \, ,
\label{1sm}
\ee
and
\be
F_V^2 M_V^2 - F_A^2 M_A^2 =0 \, ,
\label{2sm}
\ee
which lead to
\be
S=4\pi F_\Pi^2 \left[ \frac{1}{M_V^2}+\frac{1}{M_A^2} \right] \, .
\label{sfin}
\ee
We can also apply the result of vector meson dominance to Eq.(\ref{sfin})~\cite{wei2}, implying that $M_A^2 = 2 M_V^2$. This relation
is not exact even in QCD, but by considering it we are at most overestimating the $S$ parameter, which is now be given by
\be
S\approx \frac{6\pi F_\Pi^2}{M_V^2} \, .
\label{s2}
\ee
The TC technipion decay constant is usually assumed to be $F_\Pi \approx 246$GeV.
To determine the value of $S$ shown in Eq.(\ref{s2}) we must have one
estimate of the vector-meson mass. It should be remembered that the vector-boson mass is quite large only due to the spin-spin part
of the hyperfine interactions. We can determine the vector-boson mass by using the hyperfine splitting calculation performed in the
heavy quarkonium context in Ref.~\cite{ei}
\be
M(^3S_1)-M(^1S_0)\approx \frac{8}{9} {\bar{g}}^2(0) \frac{|\psi (0)|^2}{\mu^2} \, ,
\label{eqvc}
\ee
where $M(^3S_1)$ and $M(^1S_0)$ describe the masses of vector and scalar lighter bosons, respectively. In Eq.(\ref{eqvc}), $|\psi (0)|^2$ is the meson wave
function at the origin, describing a vector boson formed by techniquarks with dynamical mass $\mu_{\tt{TC}}$. Equation (\ref{eqvc}) seems to be reasonable
even when the vector-boson constituents are light~\cite{sc}.
We make the following assumptions: 1) The TC theory has an infrared frozen coupling constant ${\bar{g}}^2(0)/4\pi \approx 0.5$, whose value can be similar to several determinations of this quantity in the QCD case (see, for instance, Ref.~\cite{usf}), 2) The lightest TC scalar boson has the same mass as the Higgs
boson found at the LHC, i.e., $M(^1S_0)=125$GeV, 3) The wave function is approximated by $|\psi (0)|^2 \approx \mu_{\tt{TC}}^3\approx 1$TeV$^3$, consistent
with the other BSE wave functions proportional to the dynamical fermion mass (see Eq.(\ref{eq11})). As a consequence, we obtain
a vector-boson mass $M_V\approx 5.71$TeV, leading to
\be
S \approx 0.035 \, ,
\label{sfu}
\ee
whose value has probably been overestimated but is still consistent with the experimental data ($S=0.02\pm 0.07$)~\cite{pdg}.
\subsection{Horizontal symmetry}
A necessary condition for the type of model that we are proposing here is the presence of the horizontal (or family) symmetry. This symmetry can be local, and
it is only necessary to enforce the connection between the TC sector and the third ordinary fermionic generation, i.e., the $t$ and $b$ quarks, the $\tau$, and
its neutrino. This symmetry in general leads to flavor violations at an undesirable level; however, in the scheme proposed here the masses of the horizontal
gauge bosons can be quite heavy, affecting only logarithmic corrections to the fermion masses, and not producing significant tree-level reactions
that may be severely constrained by the experimental data. On the other hand there are hints of $B$ decay anomalies~\cite{b1,b2,b3,b4,b5} which, if confirmed, could also set a mass scale for our horizontal symmetry.
One of the anomalies in $B$ decays appears in the measurement of the ratio between the branching fractions of the processes $B^0 \rightarrow K^{*0}\mu^+\mu^-$ and
$B^0 \rightarrow K^{*0}e^+e^-$, which in the small dilepton invariant mass region is given by
\be
R (K^*)=\frac{B^0 \rightarrow K^{*0}\mu^+\mu^-}{B^0 \rightarrow K^{*0}e^+e^-}= 0.66 {\substack{+0.11 \\ -0.07}}\pm 0.03 \, ,
\label{bdec}
\ee
which is around $2.2$ standard deviations away from the SM expectation.
If such deviation is confirmed in the future, it could be explained by a current-current interaction described by the following effective Lagrangian:
\be
L_h \propto \alpha_h \frac{\lambda_{bs}C^{\mu\mu}}{M_h^2} (\overline{s}\gamma_\nu P_L b)(\overline{\mu}\gamma^\nu \mu) \, ,
\label{lb}
\ee
where $\alpha_h$ is the horizontal gauge coupling, $\lambda_{bs}$ are mixing angles, $M_h$ is the horizontal gauge boson mass, and $C^{\mu\mu}$ is a Wilson
coefficient. If we naively assume the results of the $SU(3)_h$ horizontal model of Ref.~\cite{alo} for these several constants, we can roughly estimate that
$M_h$ should be greater than $10$TeV. However, this is only a guess because (as said repeatedly in the previous sections) the horizontal gauge boson
can be quite heavy, and this scale can be set to these masses only if the anomalies remain discrepant with the SM expectation. Otherwise, the
dependence on the factor $1/M_h^2$ in all observables of this kind will lessen experimental constraints originated from horizontal symmetries.
There are other possible flavor-changing neutral currents induced by the horizontal symmetry. For instance, the effective Lagrangian
\be
L_h \propto \alpha_h \frac{\lambda_{sd}}{M_h^2} (\overline{s}_L\gamma_\nu d_L)(\overline{s}_R\gamma^\nu d_R) \, ,
\label{sd}
\ee
is induced by one-gauge-boson exchange and contributes to the $K^0 - \bar{K}^0$ transition, which for $\lambda \approx 1/20$ requires $M_h \geq 200$TeV ~\cite{die}.
This contribution can be easily evaded in our type of model simply by increasing the horizontal gauge boson mass scale, which will not affect the mechanism
of ordinary fermion mass generation. Therefore, a careful scrutiny of the gauge symmetry
breaking of the horizontal group will only be necessary if the $B$ decay anomaly is confirmed.
\subsection{Technipion masses}
The LHC collaborations already have enough data to constrain the existence of light technipions~\cite{scs1}. Due to the fact that the technifermions acquire masses
of $O(100)$ GeV, the resulting pseudo-Goldstone bosons [i.e., those generated in the chiral breaking of the $SU(4)_{\tt{TC}}$ TC gauge group discussed in Sec. II]
may be heavier than the SM Higgs boson. Moreover, due to the choice of the horizontal symmetry quantum numbers the technipions will mainly couple to the third ordinary fermionic family,
i.e., $t$ and $b$ quarks and the $\tau$ lepton, in such a way that may easily evade the limits found in Ref.~\cite{scs1} obtained from data on the SM Higgs boson decaying
into $\gamma \gamma$ and $\tau^+ \tau^-$.
The colored and charged technipions will be quite heavy and are produced along with $t$ and $b$ quarks. In the case of the decay into $b$ quarks the branching ratio
may be reduced by a possible small coupling between this quark and the technipion, which will happen through the exchange of a rather heavy gauge boson, and
their signal could easily be buried in the background. This leaves us with the lightest technipions, which should be the
neutral ones ($\bar{N}\gamma_5 N$). In this case a neutral technipion may be produced through vector-boson fusion and decay through the weak $ZZ$ channel.
The discussion of the TC condensate in Sec. V can be used to estimate the neutral technipion mass ($m_\Pi$) in a different way than in Ref.~\cite{us1}.
As considered in Eq.(\ref{eq14j}) the neutral technifermion mass ($m_N$) in terms of the TC condensate generated by diagram $(a_4)$ of Fig.2 is given by
\be
m_N \sim \frac{\<{\bar{\psi}_T}\psi_T\>^{\alpha_E \neq 0}_{\Lambda}}{\Lambda^2} \, .
\ee
The above equation together with Eq.(\ref{eq14}) leads to the following estimate of the neutral technipion mass
\be
m_\Pi^2 \approx \frac{(\<{\bar{\psi}_T}\psi_T\>^{\alpha_E \neq 0}_{\Lambda})^2}{2F_{\Pi}^2\Lambda^2} \, .
\ee
Assuming $SU(3)_{{TC}}$ as the TC gauge group, $\<{\bar{\psi}_T}\psi_T\>^{\alpha_E =0} \sim \mu^3$ with $\mu = 1$ TeV,
$R_w^{rad.cor.} \approx 7.87 \times 10^6[ln(M^2_E/\mu^2)]^{-4.3}$ defined and appearing in Eqs.(\ref{rwcor}) and (\ref{eq14g}), we obtain
\be
m_\Pi \sim 160 \,\,\, GeV ,
\ee
which is a rough estimate for the smallest pseudo-Goldstone mass of our type of model, which has not yet been eliminated by the LHC data~\cite{scs1}.
The fact that in our type of model the technifermions couple preferentially to the third fermionic family and obtain a large effective mass due ETC interactions, and that their other couplings to ordinary fermions are always diminished by the exchange of a very heavy horizontal or GUT gauge boson makes the search for pseudo-Goldstone signals quite difficult. The main hope for detecting technipions may be the resonant production of the lightest neutral technipion and its decay into neutral weak bosons.
\section{Scalar boson trilinear coupling}
As already pointed out many years ago~\cite{ebo}, the measurement of the Higgs boson trilinear coupling is fundamental to determining the nature of
this particle. If the Higgs boson is a composite particle its trilinear coupling may deviate from the SM value of a fundamental scalar boson,
and its measurement can even provide a signal of the underlying theory forming the composite state~\cite{doff}.
In TC or any composite scalar model the scalar trilinear coupling is determined through its coupling to fermions.
Using Ward identities, we can show that the couplings of the scalar boson to fermions are \cite{ca2}
\be
G^{\sl a} (p+q,p) = -\imath \frac{g_{W}}{2M_{W}}
\left[\tau^{\sl a}\Sigma(p)P_R - \Sigma(p+q)\tau^{\sl a} P_L \right]
\label{fsc}
\ee
where $P_{R,L} = \frac{1}{2} (1 \pm \gamma_5 )$, $\tau^{\sl a}$ is a $SU(2)$
matrix, and $\Sigma$ is the fermionic self-energy in weak-isodoublet
space. As in Ref.\cite{ca2}, we assume that the scalar composite Higgs
boson coupling to the fermionic self-energy is saturated by the
top quark. We also do not differentiate between the two fermion momenta $p$ and $p+q$ since, in all situations of interest,
$\Sigma(p+q)\approx \Sigma(p)$. Therefore, the coupling between a composite Higgs boson and fermions at large momenta is given by
\be
\lambda_{{}_{Hff}}(p)\equiv G(p,p) \sim -\frac{g_{W}}{2M_{W}}\Sigma(p^2).
\label{fh}
\ee
The trilinear coupling of the composite scalar boson is determined by the diagram shown in Fig.6.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.5]{acop3H.eps}
\vspace{0.3cm}
\caption{The dominant contribution to the trilinear scalar coupling. The blobs in this figure represent the coupling of the composite scalar boson to fermions. The double lines represent
the composite scalar boson.}
\label{fig6}
\end{center}
\end{figure}
Assuming that the coupling of the scalar boson to the fermions is given by Eq.(\ref{fh}), we find that
\be
\lambda_{{}_{3H}} = \frac{3g^3_{W}}{64\pi^2}\left(\frac{3n_{F}}{M^3_{W}}\right)\int^{M^2_E}_{0}\frac{\Sigma^4(p^2)p^4dp^2}{(p^2 + \Sigma^2(p^2))^3}.
\label{tri}
\ee
\noindent where $n_{F}$ is the number of technifermions included in the model.
The SM trilinear scalar coupling value, according to the normalization of Ref.~\cite{malt}, is
\be
\lambda_{SM} = \frac{M^2_H}{2v^2}.
\label{norml}
\ee
Combined with the above normalization, the trilinear coupling of Eq.(\ref{tri}) leads to the following scalar trilinear coupling $\lambda$:
\be
\lambda = \frac{1}{6v}\lambda_{{}_{3H}} .
\label{nor2}
\ee
Considering Eqs.(\ref{tri}) and (\ref{nor2}), $v=F_{{}_{\Pi}}$, and the relation
$$
M^2_{W} = \frac{g^2_W F^2_{{}_{\Pi}}}{4}
$$
we obtain for the trilinear coupling
\be
\lambda = \frac{1}{16\pi^2}\left(\frac{3n_{F}}{F^4_{{}_{\Pi}}}\right)\int^{M^2_E}_{0}\frac{\Sigma^4(p^2)p^4dp^2}{(p^2 + \Sigma^2(p^2))^3},
\label{tri3}
\ee
which is the trilinear scalar composite coupling that can be compared to the SM coupling of Eq.(\ref{norml}).
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.45]{acopl3H2.eps}
\vspace{-2cm}
\caption{Experimental limits on the scalar boson trilinear coupling, and curves of the trilinear coupling value (\ref{tri3})) in the case of a composite scalar boson. }
\label{fig7}
\end{center}
\end{figure}
Using the results for the TC self-energy obtained in Ref.~\cite{us2} and Sec. III, which is dominated by diagrams ($a_1$) and ($a_4$) of Fig.2, we compute the trilinear coupling presented in Eq.(\ref{tri3}). A comparison of the trilinear composite coupling with the SM one is shown in Fig.7. The composite trilinear coupling does differ from the SM one,
but only a small amount. In Fig. 7 we also show the current LHC limits on this coupling obtained in Ref.~\cite{malt} from the $(b\bar{b}\gamma\gamma)$ signal, whose values are
$\lambda < -1.3\lambda_{SM} = -0.169$ (red region) and $\lambda > 8.7\lambda_{SM} = 1.13$ (green region). Figure 7 remind us that the actual result
for the scalar trilinear coupling does vary with $M_E$, and this variation should appear when the coupled gap equations are solved taking into account the
running of the ETC gauge coupling constant. Of course, this will introduce only a small variation in the curves of that figure.
The white region is not excluded yet, and this large region shows how difficult it is to differentiate one composite scalar boson from a fundamental one by just observing the specific coupling.
\section{Conclusions}
In Refs.~\cite{us1,us2} we called attention to the fact that the self-energies of strongly interacting theories are modified when we consider coupled SDEs including radiative
corrections. The effect of the radiative corrections is not very different from the \textit{ad hoc} introduction of effective four-fermion interactions, as verified many
years ago by Takeuchi~\cite{takeuchi}, and it leads to self-energies that decrease logarithmically with the momentum. This effect was reviewed in the Introduction of this work, where it was made clear that the usual TC model building has to be modified, where the ordinary fermion mass hierarchy is not related to different ETC gauge boson masses.
The presence of a horizontal symmetry is mandatory in the type of models envisaged in Sec. II. This symmetry is necessary to give masses to only the third generation of
ordinary fermions at leading order. The model discussed in Sec. II is based on the non-Abelian gauge group structure $SU(9)_U \otimes SU(3)_H$, where the $SU(9)_U$
group contains the SM, an $SU(5)_{\tiny \textsc{gg}}$ Georgy-Glashow GUT~\cite{gg}, and a $SU(4)_{\tt{TC}}$ group. The $SU(3)_H$ horizontal symmetry was introduced in
such a way that their fermionic quantum numbers allow only the third fermionic generation to be coupled to the technifermions. The other fermions remain massless at
leading order. However, the first-generation fermions obtain their masses due to the coupling with QCD, which also has a slowly decreasing self-energy. This is the most
interesting fact of our model: the different fermionic mass scales are dictated by the different strong interactions present in the model! We have shown some of the diagrams that
generate the different masses, and made rough estimates of their masses. We believe that a large number of theories can be built along the lines of the model of
Sec. II. Precise determinations of fermion masses in this type of model will demand a lengthy determination of SDE coupled equations, where different self-energies
can be fitted by equations like Eq.(\ref{eq5}).
The fact that the ETC interactions can be pushed to very high energies apparently seems to open
a path for a plethora of TC models capable of describing the ordinary fermionic mass spectra.
The determination of fermion masses will involve a delicate balance of different gauge group
theories for TC, ETC (or GUT), and horizontal symmetry. The ordinary fermion mass matrix calculation will involve the knowledge
of specific Casimir eigenvalues, which will depend on the different fermionic representations of the different gauge groups.
It will also involve the different coupling constant values of these theories at different scales, and the far more demanding
solutions of the coupled system of Schwinger-Dyson equations even with a minimum of approximations. Therefore, while
a new frontier arise, generic combination of gauge theories and
respective fermionic representations will not be able to explain the known fermionic spectra, meaning that an enormous engineering effort
will be necessary for a \textit{precise} calculation of ordinary fermion masses.
In Sec, III we discussed how the composite scalar boson may have a mass lighter than the characteristic mass scale of the theory that forms the composite particle.
This could explain how the observed Higgs boson mass, if composite, is smaller than the Fermi mass scale. Perhaps the most important factor regarding the mass value of the scalar
composite resides in the normalization
condition of the inhomogeneous BSE, which has to be taken into account when the self-energy is hard and not decaying as $1/p^2$. The normalization condition,
as shown by the results presented in Table I, is enough to lower the scalar mass by a factor of $1/10$. However, we have listed many other effects
that may also lower the scalar composite mass.
Section IV contains a brief discussion about pseudo-Goldstone boson masses. It is just a complementary discussion to the one already presented in Refs.~\cite{us1,us2},
indicating that their masses should be of the order of or higher than that of the observed Higgs boson. Moreover, the pseudo-Goldstone bosons couple at leading order
only to the third-generation fermions, which is another fact that will complicate their experimental observation.
In Sec. V we computed the TC condensate in the coupled SDE scenario. This calculation serves as a comparison with the enhancement that appears in the TC
condensate in walking TC theories. Although the mechanism is totally different, i.e., here the gauge theory is just a running theory, there is also one enhancement in the
condensates as a result of a logarithmically decreasing self-energy with the momentum. Again, it is possible to verify that the effect is not qualitatively different
from the \textit{ad hoc} inclusion of a four-fermion interaction, which is replaced by genuine radiative corrections of known interactions.
In Sec. VI we commented on possible experimental constraints on this type of model. The main point is that the ETC gauge boson masses may be pushed to very high
energies and unnatural flavor-changing events will be absent. The $S$ parameter will be of the expected order, and should not differ from the case of TC as a scaled QCD theory. Complementing the discussion of Sec. IV with what was presented in Sec. V, we estimated pseudo-Goldstone masses and verified that they cannot yet be seen
at the LHC according the analysis of Ref.~\cite{scs1}.
In Sec. VII we computed the trilinear scalar coupling and verified that a signal of compositeness is far from being observed with the present data~\cite{malt},
and this coupling does not differ by a large amount from the SM value in the case of a fundamental scalar boson. Finally, we may say that in the scenario
presented in this work there is a possibility that the SM gauge symmetry breaking promoted dynamically by composite scalar bosons is still alive.
\section*{Acknowledgments}
This research was partially supported by the Conselho Nacional de Desenvolvimento Cient\'{\i}fico e Tecnol\'ogico (CNPq)
under the grants 302663/2016-9 (A.D.) and 302884/2014 (A.A.N.).
|
{
"timestamp": "2019-03-18T01:18:32",
"yymm": "1902",
"arxiv_id": "1902.11072",
"language": "en",
"url": "https://arxiv.org/abs/1902.11072"
}
|
\section{Introduction}
Deep neural networks have been evolved to a general-purpose machine learning method with remarkable performance on practical applications~\citep{lecun2015deep}. Such models are usually over-parameterized, involving an enormous number (possibly millions) of parameters. This is much larger than the typical number of available training samples, making deep networks prone to overfitting~\citep{caruana2001overfitting}. Coupled with overfitting, the large number of unknown parameters makes deep learning models extremely hard and computationally expensive to train, requiring huge amount of memory and computation power. Such resources are often available only in massive computer clusters, preventing deep networks to be deployed in resource limited machines such as mobile and embedded devices.
To prevent deep neural networks from overfitting and improve their generalization ability, several
explicit and implicit regularization methods have been proposed. More specifically, explicit regularization strategies, such as weight decay involve $\ell_2$-norm regularization of the parameters \citep{nowlan1992simplifying, krogh1992simple}. Replacing the $\ell_2$ with $\ell_1$-norm has been also investigated \citep{scardapane2017group, zhang2016l1}. Besides the aforementioned general-purpose regularization functions, neural networks specific methods such as early stopping of back-propagation \citep{caruana2001overfitting}, batch normalization \citep{ioffe2015batch}, dropout \citep{srivastava2014dropout} and its variants --e.g., DropConnect \citep{wan2013regularization}-- are algorithmic approaches to reducing overfitting in over-parametrized networks and have been widely adopted in practice.
Reducing the storage and computational costs of deep networks has become critical for meeting the
requirements of environments with limited memory or computational resources.
To this end, a surge of network compression and approximation algorithms have recently been proposed in the context of deep learning. By leveraging the redundancy in network parameters, methods such as \citet{tai2015convolutional, cheng2015exploration, yu2017compressing, kossaifi2018tensor} employ low-rank approximations
of deep networks’ weight matrices (or tensors) for parameter reduction. Network compression methods in the frequency domain \citep{chen2016compressing} have also been investigated. An alternative approach for reducing the number of effective parameters in deep nets relies on sketching, whereby, given a matrix or tensor of input data or parameters, one first compresses it to a much smaller matrix (or tensor) by multiplying it by a (usually) random matrix with certain properties \citep{kasiviswanathan2017deep, daniely2016sketching}.
\pagebreak
A particularly appealing approach to network compression, especially for visual data\footnote{Most modern data is inherently multi-dimensional -color images are naturally represented by \(3\myrd\) order tensors, videos by \(4\myth\) order tensors, etc.)} (and other types of multidimensional and multi-aspect data) is tensor regression networks \citep{kossaifi2018tensor}.
Deep neural networks typically leverage the spatial structure of input data via series of convolutions, point-wise non-linearities, pooling, etc.
However, this structure is usually wasted by the addition, at the end of the networks' architectures, of a flattening layer followed by one or several fully-connected layers.
A recent line of study focuses on alleviating this using tensor methods. \citet{kossaifi2017tensor} proposed tensor contraction as a layer, to reduce the size of activation tensors, and demonstrated large space savings by replacing fully-connected layers with this layer. However, a flattening layer and fully-connected layers were still ultimately needed for producing the outputs. Recently, tensor regression networks \citep{kossaifi2018tensor} propose to replace flattening and fully-connected layers entirely with a tensor regression layer (TRL). This preserves the structure by expressing an output tensor as the result of a tensor contraction between the input tensor and some low-rank regression weight tensors. In addition, these allow for large space savings without sacrificing accuracy.
\citet{cao2017tensor} explore the same model with various low-rank structures on the regression weight tensor.
In this paper, we combine ideas from networks regularization, low-rank approximation of networks, and randomized sketching in a principled way and introduce a novel stochastic regularization term to the tensor regression networks. It consists of a novel randomized low-rank tensor regression, which leads to the stochastic reduction of the rank, either by a fixed percentage during training or according to a series of Bernoulli random variables.
This is akin to dropout, which, by randomly dropping units during training, prevents over-fitting. However, rather than dropping random elements from the \emph{activation} tensor, this is done on the regression weight tensor. We explore two schemes: (i) selecting random elements to keep, following a Bernoulli distribution and (ii) keeping a random subset of the \emph{fibers} of the tensor, with replacement.
We theoretically and empirically establish the link between CP TRL with the proposed regularizer and the dropout on the deterministic low-rank tensor regression.
To demonstrate the practical advantages of this method, we conducted experiments in image classification and phenotypic trait prediction from MRI. To this end, the CIFAR-100 and the UK Biobank brain MRI datasets were employed. Experimental results demonstrate that the proposed method i) improves performance in both classification and regression tasks, ii) decreases over-fitting, iii) leads to more stable training and iv) largely improves robustness to adversarial attacks and random noise.
One notable application of deep neural networks is in medical imaging, particularly magnetic resonance imaging (MRI). MRI analysis performed using deep learning includes age prediction for brain-age estimation \citep{cole2017predicting}. Brain-age has been associated with a range of diseases and mortality \citep{cole2017brain}, and could be an early predictor for Alzheimer's disease \cite{franke2012brain}. A more accurate and more robust brain age estimation can consequently lead to more accurate disease diagnoses.
We demonstrate a large performance improvement (more than 20\%) on this task using a 3D-ResNet with our proposed stochastically rank-regularized TRL, compared to a regular 3D-ResNet.
\section{Closely related work}
\textbf{Network regularization and dropout}. Several methods that improve generalization by mitigating overfitting have been developed in the context of deep learning. The interested reader is referred to the work of \citet{kukavcka2017regularization} and the references therein for a comprehensive survey of over 50 different regularization techniques for deep networks.
The most closely related regularization method to our approach is Dropout \citep{srivastava2014dropout}, which is probably the most widely adopted technique for training neural networks while preventing overfitting. Concretely, during dropout training each unit (i.e., neuron) is equipped with a binary Bernoulli random variable and only the network’s weights whose corresponding Bernoulli variables are sampled with value 1 are updated at each back-propagation step. At each iteration, those Bernoulli variables are re-sampled again and the weights are updated accordingly. The proposed regularization method can be interpreted as dropout on low-rank tensor regression, a fact which is proved in Section \ref{seq:cp-ssr}.
\textbf{Sketching and deep networks approximation}. \citet{daniely2016sketching} apply sketching to the input data in order to sparsify them and reduce their dimensionality. Subsequently they show any sparse polynomial function can be computed, on all sparse binary vectors, by a single layer neural network that takes a compact sketch of the vector as input. In contrast, \citet{kasiviswanathan2017deep}, approximate neural networks and apply a random sketching on weight matrices/tensors instead of input data and demonstrate that given a fixed layer input, the output of this layer using sketching matrices is an unbiased estimator of the original output of this layer and has bounded variance. As opposed to the aforementioned sketching methods for deep networks approximation, the proposed method applies sketching in the low-rank factorization of weights.
\textbf{Randomized tensor decompositions}.
Tensor decompositions exhibit high computational cost and low convergence rate when applied to massive multi-dimensional data. To accelerate computation, randomized tensor decompositions have been employed to scale tensor decompositions. A randomized least squares algorithm for CP decomposition is proposed by \citet{battaglino2018practical}, which is significantly faster than traditional CP decomposition. In \citep{erichson2017randomized}, CP is applied on a small tensor generated by tensor random projection of the high-dimensional tensor. The CP decomposition of the large-scale tensor is obtained by back projection of the CP decomposition of the small tensor. \citet{wang2015fast} introduce a fast yet provable randomized CP decomposition that performs randomized tensor contraction using FFT. Methods in \citep{sidiropoulos2014parallel, vervliet2014breaking} are highly computationally efficient algorithms for computing large-scale CP decompositions by applying randomization (random projections) into a set of small tensors, derived by subdividing a tensor into a set of blocks. Fast randomized algorithms that employ sketching for approximating Tucker decomposition have been also investigated \citep{tsourakakis2010mach, zhou2014decomposition}. More recently, a randomized tensor ring decomposition that employs tensor random projections has been developed in \citet{yuan2019randomized}. The most similar method to ours is that of \citet{battaglino2018practical}, where elements of the tensor are sampled randomly, and each factor of the decomposition updated in an iterative manner. By contrast, our method allows for end-to-end training, and applies randomization on the \emph{fibers} of the tensor, effectively randomizing the rank of the weight tensor.
\begin{figure*}
\begin{subfigure}[t]{0.4\textwidth}
\centering
\includegraphics[width=1\linewidth]{trl-diagram.pdf}
\caption{Tensor diagram of a TRL}
\label{fig:trl-diagram}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.50\textwidth}
\centering
\includegraphics[width=1\linewidth]{rnd-trl-diagram.pdf}
\caption{Tensor diagram of a SRR-TRL}
\label{fig:srr-trl-diagram}
\end{subfigure}
\caption{Tensor diagrams of the TRL (left) and our proposed SRR-TRL (right), with low-rank constraints imposed on the regression weights tensor using a Tucker decomposition. Note that the CP case is readily given by this formulation by additionally having the core tensor \mytensor{G} be super-diagonal, and setting \( \mymatrix{M} = \mymatrix{M}^{(0)} = \cdots = \mymatrix{M}^{(N)} = \mydiag(\myvector{\lambda}).\)}
\label{fig:trl-diagrams}
\end{figure*}
\section{Tensor Regression Networks}
In this section, we introduce the notations and notions necessary to introduce our stochastic rank regularization.
\paragraph{Notation:} We denote \(\myvector{v}\) vectors (1\myst order tensors) and \(\mymatrix{M}\) matrices (2\mynd order tensors). \(\myId\) is the identity matrix. We denote \(\mytensor{X}\) tensors of order \(N \geq 3\), and denote its element \((i, j, k)\) as \(\mytensor{X}_{i_0, i_1, \cdots, i_{N - 1}} \,\text{ or } \, \mytensor{X}(i_0, i_1, \cdots, i_{N - 1})\). A colon is used to denote all elements of a mode e.g. the mode-0 fibers of \(\mytensor{X}\) are denoted as \(\mytensor{X}_{\mycolon, i_1, \cdots, i_{N-1}}\). The transpose of \(\mymatrix{M}\) is denoted \(\mymatrix{M}\myT\).
Finally, for any \(i, j \in \myN, i < j,\, \myrange{i}{j}\) denotes the set of integers \(\{ i, i+1, \cdots , j-1, j\}\), and \(i \mydiv j\) the integer division of \(i\) by \(j\).
\paragraph{Tensor unfolding:}
Given a tensor,
\( \mytensor{X} \in \myR^{I_0 \times I_1 \times \cdots \times I_{N - 1}}\),
its mode-\(n\) unfolding is a matrix \(\mymatrix{X}_{[n]} \in \myR^{I_n, I_M}\),
with \(M = \prod_{\substack{k=0,\\k \neq n}}^{N - 1} I_k\)
and is defined by the mapping from element
\( (i_0, i_1, \cdots, i_N)\) to \((i_n, j)\), with
\[
j = \sum_{\substack{k=0,\\k \neq n}}^{N - 1} i_k \times \prod_{\substack{m=k+1,\\ m \neq n}}^{N - 1} I_m.
\]
\paragraph{Tensor vectorization:} Given a tensor,
\( \mytensor{X} \in \myR^{I_0 \times I_1 \times \cdots \times I_{N - 1}}\),
we can flatten it into a vector \(\text{vec}(\mytensor{X})\)
of size \(\left(I_0 \times \cdots \times I_{N - 1}\right)\)
defined by the mapping from element
\( (i_0, i_1, \cdots, i_{N - 1})\) of \(\mytensor{X}\) to element \(j\) of \(\text{vec}(\mytensor{X})\), with
\[ j = \sum_{k=0}^{N - 1} i_k \times \prod_{m=k+1}^{N - 1} I_m.\]
\paragraph{Mode-n product:}
For a tensor \(\mytensor{X} \in \myR^{I_0 \times I_1 \times \cdots \times I_{N - 1}}\) and a matrix \( \mymatrix{M} \in \myR^{J \times I_n} \), the n-mode product of a tensor
is a tensor of size
\(\left(I_0 \times \cdots \times I_{n-1} \times J \times I_{n+1} \times \cdot \times I_{N - 1}\right)\)
and can be expressed using unfolding of \(\mytensor{X}\)
and the classical dot product as:
\begin{equation}\nonumber
\left(\mytensor{X} \times_n \mymatrix{M}\right)_{[n]} = \mymatrix{M} \mytensor{X}_{[n]} \in \myR^{I_0 \times \cdots \times I_{n-1} \times R \times I_{n+1} \times \cdot \times I_{N - 1}}
\end{equation}
\paragraph{Generalized inner product:}
For two tensors \(\mytensor{X} \in \myR^{K_0 \times \cdots \times K_x \times I_0 \times \cdots \times I_{N - 1}}\) and \(\mytensor{Y} \in \myR^{I_0 \times \cdots \times I_{N - 1} \times L_0 \times \cdots \times L_y}\), we denote by
\(\myinner{\mytensor{X}}{\mytensor{Y}}_{N} \in \myR^{K_0 \times \cdots \times K_x\times I_{N - 1} \times L_0 \times \cdots \times L_y} \) the contraction of \(\mytensor{X} \) by \(\mytensor{W}\) along their \({N - 1}\) last (respectively first) modes.
\begin{equation}\nonumber
\myinner{\mytensor{X}}{\mytensor{Y}}_N =~\sum_{i_0=0}^{I_0}\sum_{i_1=0}^{I_1} \cdots \sum_{i_n=0}^{I_{N - 1}} \mytensor{X}_{\cdots, i_0, i_1, \cdots, i_n} \mytensor{Y}_{i_0, i_1, \cdots, i_n, \cdots}
\end{equation}
\paragraph{Kruskal tensor:}
Given a tensor \(\mytensor{X} \in \myR^{I_0 \times I_1 \times \cdots \times I_{N - 1}} \),
the Canonical-Polyadic decomposition (CP), also called PARAFAC, decomposes it into a sum of \(R\) rank-1 tensors. The number of terms in the sum, \(R\), is known as the rank of the decomposition. Formally, we find the vectors \( \mathbf{u}^{(0)}_{k} , \mathbf{u}^{(1)}_{k}, \cdots, \mathbf{u}^{(2)}_{k}\), for \(k = \myrange{0}{R-1}\) such that:
\begin{equation}
\mytensor{X}
= \sum_{k=0}^{R-1} \underbrace{ \myvector{u}^{(0)}_{k} \circ \myvector{u}^{(1)}_{k} \circ \cdots \circ \myvector{u}^{({N - 1})}_{k} }_{\text{rank-1 components}}
\end{equation}
These vectors can be collected in matrices, called factors or the decomposition. Specifically, we define, for each factor \( k \in \myrange{1}{{N - 1}}\), \(\mathbf{U}^{(k)} = \left[ \begin{matrix} \mathbf{u}^{(k)}_{0}, \mathbf{u}^{(k)}_{1}, \cdots, \mathbf{u}^{(k)}_{R-1} \end{matrix} \right].\) The magnitude of the factors can optionally be absorbed in a vector of weights \(\myvector{\lambda} \in \myR^R\), such that
\[
\mytensor{X}
= \sum_{k=0}^{R-1} \lambda_k \myvector{u}^{(0)}_{k} \circ \myvector{u}^{(1)}_{k} \circ \cdots \circ \myvector{u}^{({N - 1})}_{k}
\]
The decomposition can be denoted more compactly as \(
\mytensor{X} = \mykruskal{\mathbf{U}^{(0)}, \cdots, \mathbf{U}^{({N - 1})}}
\), or
\(
\mytensor{X} = \mykruskal{\myvector{\lambda};\, \mathbf{U}^{(0)}, \cdots, \mathbf{U}^{({N - 1})}}
\) if a weights vector is used.
\paragraph{Tucker tensor:}
Given a tensor \(\mytensor{X} \in \myR^{I_0 \times I_1 \times \cdots \times I_{N - 1}} \),
we can decompose it into a low rank core
\(\mytensor{G} \in \myR^{R_0 \times R_1 \times \cdots \times R_{N - 1}}\)
by projecting along each of its modes
with projection factors
\( \left( \mymatrix{U}^{(0)}, \cdots,\mymatrix{U}^{({N - 1})} \right) \), with \(\mymatrix{U}^{(k)} \in \myR^{R_k, I_k}, k \in (0, \cdots, {N - 1})\).
This allows us to write the tensor in a decomposed form as:
\begin{align}
\mytensor{X} &=
\mytensor{G} \times_0 \mymatrix{U}^{(0)}
\times_1 \mymatrix{U}^{(2)} \times
\cdots
\times_{N - 1} \mymatrix{U}^{({N - 1})} \nonumber \\
&= \mytucker{\mytensor{G}}{\mymatrix{U}^{(0)},
\cdots,
\mymatrix{U}^{({N - 1})}}
\end{align}
Note that the Kruskal form of a tensor can be seen as a Tucker tensor with a super-diagonal core.
\paragraph{Tensor diagrams:} In order to represent easily tensor operations, we adopt the tensor diagrams, where tensors are represented by vertices (circles) and edges represent their modes. The degree of a vertex then represents its order. Connecting two edges symbolizes a tensor contraction between the two represented modes. Figure~\ref{fig:trl-diagrams} presents a tensor diagram of the tensor regression layer and its stochastic rank-regularized counter-part.
\paragraph{Tensor regression layers (TRL):}
Let us denote by
\(\mytensor{X} \in \myR^{I_0 \times I_1 \times \cdots \times I_{N-1}} \)
the input activation tensor for a sample
and \( \myvector{y} \in \myR^{I_N}
\)
the label vector.
We are interested in the problem
of estimating the regression weight tensor
\( \mytensor{W} \in
\myR^{I_0 \times I_1 \times \cdots \times I_N}
\)
under some fixed low rank \( \left(R_0, \cdots, R_N\right)\):
\begin{align}\label{eq:trl}
\myvector{y} & = \myinner{\mytensor{X}}{\mytensor{W}}_N + \myvector{b} \nonumber \\
\text{with } &
\mytensor{W}
=
\mytensor{G}
\times_0 \mymatrix{U}^{(0)}
\times_1 \mymatrix{U}^{(1)}
\cdots
\times_N \mymatrix{U}^{(N)}
\end{align}
with \(\mytensor{G} \in \myR^{R_0 \times \cdots \times R_N} \),
\(\mymatrix{U}^{(k)} \in \myR^{I_k \times R_k}\) for each \(k\) in \(\myrange{0}{N}\)
and \(\mymatrix{U}^{(N)} \in \myR^{O \times R_{N}}\).
\section{Stochastic rank regularization}
In this section, we introduce the stochastic rank regularization (SRR). Specifically, we propose a new stochastic rank-regularization, applied to low-rank tensors in decomposed forms. This formulation is general and can be applied to any type of decomposition. We introduce it here, without loss of generality, to the case of Tucker and CP decompositions.
For any \(k \in \myrange{0}{N}\), let \(\mymatrix{M}^{(k)} \in \myR^{R_{0} \times R_{0}}\) be a sketch matrix (e.g. a random projection or column selection matrix) and, \(\mymatrix{\tilde U}^{(k)} = \mymatrix{U}^{(k)}(\mymatrix{M}^{(k)})\myT\) be a sketch of factor matrix \(\mymatrix{U}^{(k)}\), and \(\mytensor{\tilde G} = \mytensor{G} \times_0 \mymatrix{M}^{(0)} \times \cdots \times_{N} \mymatrix{M}^{(N)}\) a sketch of the core tensor \(\mytensor{G}\).
Given an activation tensor \(\mytensor{X} \in \myR^{I_0 \times \cdots \times I_{N - 1}}\) and a target label vector \(\myvector{y} \in \myR^{I_N}\), a stochastically rank regularized tensor regression layer is written from equation~\ref{eq:trl} as follows:
\begin{equation}\label{eq:rnd-trl}
\myvector{y} = \myinner{\mytensor{X}}{\mytensor{\tilde W}}_{N - 1}
\end{equation}
with \( \mytensor{\tilde W} \) being a stochastic approximation of Tucker decomposition, namely:
\begin{equation}\label{eq:rnd-weight}
\mytensor{\tilde W}
=
\mytensor{\tilde G}
\times_0 \mymatrix{\tilde U}^{(0)}
\times
\cdots
\times_{N} \mymatrix{\tilde U}^{(N)}
\end{equation}
Even though several sketching methods have been proposed, we focus here on SRR with two different types of binary sketching matrices, namely binary matrix sketching with replacement and binary diagonal matrix sketching with Bernoulli entries.
\subsection{SRR with replacement:}
In this setting, we introduce the SRR with binary sketching matrix (with replacement). We first choose \(\theta \in [0, 1]\).
Mathematically, we introduce the uniform sampling matrices
\(
\mymatrix{M}^{(0)} \in \myR^{R_{0} \times R_{0}},
\cdots,
\mymatrix{M}^{(N)} \in \myR^{R_{N} \times R_{N}}
\).
\(\mymatrix{M}_j\) is a uniform sampling matrix, selecting \(K_j\) elements, where \(K_j = R_j \mydiv \theta\).
In other words, for any \(i \in \myrange{0}{N}\), \(\mymatrix{M}^{(i)}\) verifies:
\begin{equation}\label{eq:sampling-matrix}
\mymatrix{M}^{(i)}(j, :) =
\begin{cases}
\myvector{0}
& \mbox{ if } j > K \\
\myId_m(r, :), m \in \myrange{0}{R_i}
& \mbox{ otherwise }
\end{cases}
\end{equation}
Note that in practice this product is never explicitly computed, we simply select the correct elements from \(\mytensor{G}\) and its corresponding factors.
\subsection{Tucker-SRR with Bernoulli entries}
In this setting, we introduce the SRR with diagonal binary sketching matrix with Bernoulli entries.
For any \(n \in \myrange{0}{N}\), let \(\myvector{\lambda}^{(n)} \in \myR^R_n\) be a random vector, the entries of which are i.i.d. Bernoulli(\(\theta\)), then a diagonal Bernoulli sketching matrix is defined as \( \mymatrix{M}^{(n)} = \text{diag}(\myvector{\lambda}^{(n)})\).
When the low-rank structure on the weight tensor \(\mytensor{\tilde W}\) of the TRL is imposed using a Tucker decomposition, the randomized Tucker approximation is expressed as:
\begin{equation}\label{eq:bernouilli-tucker-weight}
\begin{split}
\mytensor{\tilde W}
= & \,
\mytensor{G} \times_0 \mymatrix{M}^{(0)} \times \cdots \times_{N+1} \mymatrix{M}^{(N)}\\
&
\times_0 \left(\mymatrix{U}^{(0)}(\mymatrix{M}^{(0)})\myT \right)
\times
\cdots
\times_{N+1} \left(\mymatrix{U}^{(N)}(\mymatrix{M}^{(N)})\myT \right) \\
& = \mytucker{\mytensor{\tilde G}}{\mymatrix{\tilde U}^{(0)}, \cdots, \mymatrix{\tilde U}^{(N)}}
\end{split}
\end{equation}
The main advantage of considering the above-mentioned sampling matrices is that the products \(\mymatrix{\tilde U}^{(k)} = \mymatrix{U}^{(k)}(\mymatrix{M}^{(k)})\myT\) or \(\mytensor{\tilde G} = \mytensor{G} \times_0 \mymatrix{M}^{(0)} \times \cdots \times_{N} \mymatrix{M}^{(N)}\) are never explicitly computed, we simply select the elements from \( \mytensor{ G}\) and the corresponding factors.
Interestingly, in analogy to dropout, where each hidden unit is dropped independently with probability \(1 - \theta\), in the proposed randomized tensor decomposition, the columns of the factor matrices and the corresponding fibers of the core tensor are dropped independently and consequently the rank of the tensor decomposition is stochastically dropped. Hence the name \emph{stochastic rank-regularized} TRL of our method.
\subsection{CP-SRR with Bernoulli entries}\label{seq:cp-ssr}
An interesting special case of \ref{eq:rnd-weight} is when the weight tensor \(\mytensor{\tilde W}\) of the TRL is expressed using a CP decomposition.
In that case, we set \( \mymatrix{M} = \mymatrix{M}^{(0)} = \cdots = \mymatrix{M}^{(N)} = \mydiag(\myvector{\lambda}) \), with, for any \(k \in \myrange{0}{R}\), \(\lambda_k \sim \text{Bernoulli}(\theta).\)
Then a randomized CP approximation is expressed as:
\begin{equation}\label{eq:bernouilli-rnd-weight-long}
\begin{split}
\mytensor{\tilde W}
& =
\sum_{k=0}^{R-1} \mymatrix{\tilde U}^{(0)}_k \circ \cdots \circ \mymatrix{\tilde U}^{(N)}_k
\end{split}
\end{equation}
The above randomized CP decomposition on the weights is equivalent to the following formulation:
\begin{equation}\label{eq:bernouilli-rnd-weight}
\begin{split}
\mytensor{\tilde W}
= &
\sum_{k=0}^{R-1} \lambda_k \mymatrix{U}^{(0)}_k \circ \cdots \circ \mymatrix{U}^{(0)}_N \\
= &
\mykruskal{\myvector{\lambda};\, \mathbf{U}^{(0)}, \cdots, \mathbf{U}^{(N)}}
\end{split}
\end{equation}
This is easy to see by looking at the individual elements of the sketched factors. Let \(k \in \myrange{0}{N}\) and \(i_k \in \myrange{0}{I_k}, r \in \myrange{0}{R-1}\). Then
\(
\mymatrix{\tilde U}_{i_k, r}^{(k)} = \sum_{j=0}^{R-1} \mymatrix{U}_{i_k, j}^{(k)}\mymatrix{M}_{j, r}.
\)
Since \(\mymatrix{M} = \text{diag}(\myvector{\lambda})\), \myie \(\forall i, j \in \myrange{0}{R-1}, \mymatrix{M}_{ij} = 0\) if \(i \neq j\), and \( \lambda_i\) otherwise, we get
\(
\mymatrix{\tilde U}_{i_k, r}^{(k)} = \lambda_r \mymatrix{U}_{i_k, r}^{(k)}.
\)
It follows that \(\mytensor{\tilde W}_{i_0, i_1, \cdots, i_N} =
\sum_{r=0}^{R-1} \lambda_k \mymatrix{U}^{(0)}_k \circ \cdots \circ \lambda_k \mymatrix{U}^{(N)}_k.
\)
Since \(\lambda_r \in \{0, 1\},\) we have
\(
\mytensor{\tilde W}_{i_0, i_1, \cdots, i_N} =
\sum_{r=0}^{R-1} \lambda_k \left(\mymatrix{U}^{(0)}_k \circ \cdots \circ \mymatrix{U}^{(N)}_k\right).
\)
Based on the previous stochastic regularization, for an activation tensor \mytensor{X} and a corresponding label vector \(\myvector{y}\), the optimization problem for our tensor regression layer with stochastic regularization is given by:
\begin{equation}\label{eq:stochastic-problem}
\min_{\mymatrix{U}^{(0)}, \cdots, \mymatrix{U}^{(N)}}
\| \myvector{y} -
\frac{1}{\theta}
\myinner{
\mykruskal{\myvector{\lambda};\, \mathbf{U}^{(0)}, \cdots, \mathbf{U}^{(N)}}
}{\mytensor{X}}_{N - 1}
\|_F^2
\end{equation}
In addition, the above stochastic optimization problem can be rewritten as a deterministic regularized problem:
\begin{equation}\nonumber
\mathbb{E}_{ \, \lambda} \bm{ \big[ }\,
\min_{\mymatrix{U}^{(0)}, \cdots, \mymatrix{U}^{(N)}}
\| \myvector{y} -
\frac{1}{\theta}
\myinner{
\mykruskal{\myvector{\lambda};\, \mathbf{U}^{(0)}, \cdots, \mathbf{U}^{(N)}}
}{\mytensor{X}}_{N - 1}
\|_F^2
\,\bm{ \big] }
\end{equation}
\begin{equation}\nonumber
= \min_{\mymatrix{U}^{(0)}, \cdots, \mymatrix{U}^{(N)}} \;
\| \myvector{y} -
\myinner{
\mykruskal{\mathbf{U}^{(0)}, \cdots, \mathbf{U}^{(N)}}
}{\mytensor{X}}_{N - 1}
\|_F^2
\end{equation}
\begin{equation}\label{eq:deterministic-problem}
\hspace{5em} + \left(\frac{1 - \theta}{\theta}\right) \sum_{k=0}^{R-1} \left( \prod_{i=0}^N \| \mathbf{U}^{(i)}_{\mycolon, k} \right) \|_2^2
\end{equation}
This is easy to see by considering the equivalent rewriting of the above optimization problem, using the mode-\(N\) unfolding of the weight tensor. Equation~\ref{eq:stochastic-problem} then becomes:
\begin{equation}\nonumber
\min_{\mymatrix{U}^{(0)}, \cdots, \mymatrix{U}^{(N)}}
\| \myvector{y} -
\frac{1}{\theta}
\mymatrix{U}^{(N)} \text{diag}(\myvector{\lambda})
(\mymatrix{U}^{(-N)})\myT
\myvec(\mytensor{X})
\|_F^2
\end{equation}
with \(
\mymatrix{U}^{(-N)} = \left(\mathbf{U}^{(0)} \odot \cdots \odot \mathbf{U}^{(N)}\right).
\)
The result can then be obtained following \citet[Lemma~A.1]{mianjy2018implicit}. \section{Experimental evaluation}
In this section, we introduce the experimental setting, databases used, and implementation details. We experimented on several datasets, across various tasks, namely image classification and MRI-based regression.
All methods were implemented using PyTorch~\cite{paszke2017automatic} and TensorLy~\cite{kossaifi2016tensorly}.
\begin{figure*}[ht]
\centering
\subcaptionbox*{}{
\raisebox{50pt}{
\rotatebox[origin=t]{90}{Objective}
}
}
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{ICML_kruskal_25_25_25_15_200_0_001theta_1_0.pdf}
\caption{\textbf{\(\theta = 1\)}}
\label{fig:synthetic-3-2}
\end{subfigure}
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{ICML_kruskal_25_25_25_15_200_0_001theta_0_7.pdf}
\caption{\textbf{\(\theta = 0.7\)}}
\label{fig:synthetic-3-3}
\end{subfigure}
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{ICML_kruskal_25_25_25_15_200_0_001theta_0_4.pdf}
\caption{\textbf{\(\theta = 0.4\)}}
\label{fig:synthetic-3-6}
\end{subfigure}
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{ICML_kruskal_25_25_25_15_200_0_001theta_0_1.pdf}
\caption{\textbf{\(\theta = 0.1\)}}
\label{fig:synthetic-3-9}
\end{subfigure}
\caption{\textbf{Experiment on synthetic data:} loss of the TRL as a function of the number of epochs for the stochastic case (orange) and the deterministic version based on the regularized objective function (blue). As expected, both formulations are empirically the same.
}
\label{fig:synthetic}
\end{figure*}
\subsection{Numerical experiments}
In this section, we empirically demonstrate the equivalence between our stochastic rank regularization and the deterministic regularization based formulation of the dropout.
To do so, we first created a random regression weight tensor \( \mytensor{W} \) to be a third order tensor of size \((25 \times 25 \times 25)\), formed as a low-rank Kruskal tensor with \(15\) components, the factors of which were sampled from an i.i.d. Gaussian distribution.
We then generated a tensor of \(10000\) random samples, \mytensor{X} of size \((10000 \times 25 \times 25 \times 25)\), the elements of which were sampled from a Normal distribution. Finally, we constructed the corresponding response array \myvector{y} of size \(10000\) as: \( \forall i \in \myrange{1}{1500}, \myvector{y_i} = \myinner{\mytensor{X}_i}{\mytensor{W}}\).
Using the same regression weight tensor and same procedure, we also generated \(1000\) testing samples and labels.
We use this data to train a rank-\(15\) CP SRR-TRL, with both our Bernoulli stochastic formulation (equation ~\ref{eq:stochastic-problem}) and its deterministic counter-part (equation ~\ref{eq:deterministic-problem}). We train for \(500\) epochs, with a batch-size of \(200\), and an initial learning rate of \(10e-4\), which we decrease by a factor of \(10\) every \(200\) epochs. Figure~\ref{fig:synthetic} shows the loss function as a function of the epoch number. As expected, both formulations are identical.
\subsection{Image classification results on CIFAR-100}
In the image classification setting, we empirically compare our approach to both standard baseline and traditional tensor regression, and assess the robustness of each method in the face of adversarial noise.
\textbf{CIFAR-100} consists of 60,000 \(32 \times 32\) RGB images in 100 classes~\cite{krizhevsky2009learning}. We pre-processed the data by centering and scaling each image and then augmented the training images with random cropping and random horizontal flipping.
We compare the stochastic regularization tensor regression layer to full-rank tensor regression, average pooling and a fully-connected layer in an 18-layer residual network (ResNet) \cite{he2016deep}. For all networks, we used a batch size of \(128\) and trained for \(400\) epochs, and minimized the cross-entropy loss using stochastic gradient descent (SGD). The initial learning rate was set to \(0.01\) and lowered by a factor of \(10\) at epochs \(150\), \(250\) and \(350\). We used a weight decay (\(\mathrm{L}_{2}\) penalty) of \(10^{-4}\) and a momentum of \(0.9\).
\textbf{Results:}
Table~\ref{table:cifar100} presents results obtained on the CIFAR-100 dataset, on which our method matches or outperforms other methods, including the same architectures without SRR.
Our regularization method makes the network more robust by reducing over-fitting, thus allowing for superior performance on the testing set.
\begin{table}[ht]
\caption{Classification accuracy for CIFAR-100}
\label{table:cifar100}
\begin{center}
\begin{tabular}{ll}
\toprule
\multicolumn{1}{c}{\bf Architecture} & \multicolumn{1}{c}{\bf Accuracy}\\
\midrule
ResNet without pooling & 73.31 \%\\
ResNet &75.88 \%\\
ResNet with TRL &76.02 \%\\
ResNet with Tucker SRR &76.05 \%\\
ResNet with CP SRR & \textbf{76.19} \%\\
\bottomrule
\end{tabular}
\end{center}
\end{table}
A natural question is whether the model is sensitive to the choice of rank and $\theta$ (or drop rate when sampling with repetition). To assess this, we show the performance as a function of both rank and $\theta$ in figure~\ref{fig:surface}. As can be observed, there is a large surface for which performance remains the same while decreasing both parameters (note the logarithmic scale for the rank). This means that, in practice, choosing good values for these is not a problem.
\begin{figure*}[!htb]
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=1\linewidth]{surf_bernoulli_avgpool.png}
\caption{\textbf{Bernoulli SRR} }
\label{fig:surface-avgpool-drop-ranks}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=1\linewidth]{surf_repeat_avgpool.png}
\caption{\textbf{Repeat SRR}}
\label{fig:surface-repeat-drop-ranks}
\end{subfigure}
\caption{CIFAR-100 test accuracy as a function of the compression ratio (logarithmic scale) and the Bernoulli probability $\theta$ (left) or the drop rate (right). There is a large region for which dropping both the rank and $\theta$ does not hurt performance.}
\label{fig:surface}
\vspace{-0.3cm}
\end{figure*}
\textbf{Robustness to adversarial attacks:}
We test for robustness to adversarial examples produced using the Fast Gradient Sign Method \cite{kurakin2016adversarial} in Foolbox \cite{rauber2017foolbox}. In this method, the sign of the optimization gradient multiplied by the perturbation magnitude is added to the image in a single iteration. The perturbations we used are of magnitudes \(\lambda \times 10^{-3}, \lambda \in \{1, 2, 4, 8, 16, 32, 64, 128\} \).
In addition to improving performance by reducing over-fitting, our proposed stochastic regularization makes the model more robust to perturbations in the input, for both random noise and adversarial attacks.
We tested the robustness of our models to adversarial attacks, when trained in the same configuration.
In figure~\ref{fig:adversarial_attack-diagram}, we report the classification accuracy on the test set, as a function of the added adversarial noise. The models were trained \emph{without} any adversarial training, on the training set, and adversarial noise was added to the test samples using the Fast Gradient Sign method. Our model is much more robust to adversarial attacks.
Finally, we perform a thorough comparison of the various regularization strategies, the results of which can be seen in figure~\ref{fig:adversarial_attack}.
\begin{figure}[!htb]
\centering
\subcaptionbox*{}{
\raisebox{80pt}{
\rotatebox[origin=t]{90}{Classification accuracy (\%)}
}
}
\includegraphics[width=0.8\linewidth]{best_of_adversarial_safe.pdf}
\vspace{-0.2cm}
\caption{\textbf{Robustness to adversarial attacks} using Fast Gradient Sign attacks of various models, trained on CIFAR-100. Our stochastically rank-regularized architecture is much more robust to adversarial attacks, even though adversarial training was not used.}
\label{fig:adversarial_attack-diagram}
\end{figure}
\begin{figure*}[!htb]
\centering
\subcaptionbox*{}{
\raisebox{55pt}{
\rotatebox[origin=t]{90}{Classification accuracy (\%)}
}
}
\begin{subfigure}[t]{0.30\textwidth}
\centering
\includegraphics[width=1.08\linewidth]{dropoutTRL_foolbox.pdf}
\caption{FGS attack on Tucker TRL with different dropout rates on the tensor regression weights.}
\label{fig:adversarial_dropout_attack}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.30\textwidth}
\centering
\includegraphics[width=1.08\linewidth]{tucker_foolbox.pdf}
\caption{FGS attack on Bernoulli Tucker SRR-TRL with different drop rates.}
\label{fig:bernoulli-adversarial-tucker}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.30\textwidth}
\centering
\includegraphics[width=1.08\linewidth]{CP_82_foolbox.pdf}
\caption{FGS attack on CP SRR-TRL with different drop rates.}
\label{fig:bernoulli-adversarial-cp}
\end{subfigure}
\caption{\textbf{Robustness to adversarial attacks}, measured by adding adversarial noise to the test images, using the Fast Gradient Sign, on CIFAR-100 and Bernoulli drop. We compare a Tucker tensor regression layer with dropout applied to the regression weight tensor~\ref{fig:adversarial_dropout_attack} to our stochastic rank-regularized TRL, both in the Tucker (Subfig.~\ref{fig:bernoulli-adversarial-tucker}) and CP (Subfig.~\ref{fig:bernoulli-adversarial-cp}) case.}
\label{fig:adversarial_attack}
\end{figure*}
\subsection{Phenotypic trait prediction from MRI data}
In the regression setting, we investigate the performance of our SRR-TRL in a challenging, real-life application, on a very large-scale dataset. This case is particularly interesting since the MRI volumes are large 3D tensors, all modes of which carry important information. The spatial information is traditionally discarded during the flattening process, which we avoid by using a tensor regression layer.
\textbf{The UK Biobank brain MRI dataset} is the world's largest MRI imaging database of its kind \cite{sudlow2015uk}. The aim of the UK Biobank Imaging Study is to capture MRI scans of vital organs for \(100,000\) primarily healthy individuals by 2022. Associations between these images and lifestyle factors and health outcomes, both of which are already available in the UK Biobank, will enable researchers to improve diagnoses and treatments for numerous diseases. The data we use here consists of T1-weighted \(182\times218\times182\) MR images of the brain for \(7,500\) individuals captured on a 3 T Siemens Skyra system. \(5,700\) are used for training and rest are used to test and validate. The target label is the age for each individual at the time of MRI capture. We use skull-stripped images that have been aligned to the MNI152 template \cite{jenkinson2002improved} for head-size normalization. We then center and scale each image to zero mean and unit variance for intensity normalization.
\begin{table}[ht]
\caption{Classification accuracy for UK Biobank MRI. The ResNet with TRL and our stochastic rank-regularization performs better, while the baseline ResNet without average pooling did not train at all. The version \emph{with} average pooling did train but converged to a much worse performance.}
\label{table:mri}
\begin{center}
\begin{tabular}{ll}
\toprule
\multicolumn{1}{c}{\bf Architecture} & \multicolumn{1}{c}{\bf MAE}\\
\midrule
3D-ResNet without pooling & N/A \\
3D-ResNet & 3.23 years \\
3D-ResNet with TRL & 2.99 years \\
3D-ResNet with Tucker SRR & 2.96 years \\
3D-ResNet with CP SRR & \textbf{2.58 years} \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\textbf{Results:} For MRI-based experiments we implement an 18-layer ResNet with three-dimensional convolutions. We minimize the mean squared error using Adam \cite{kingma2014adam}, starting with an initial learning rate of \(10^{-4}\), reduced by a factor of 10 at epochs 25, 50, and 75. We train for 100 epochs with a mini-batch size of 8 and a weight decay (\(\mathrm{L}_{2}\) penalty) of \(5\times10^{-4}\).
As previously observed, our Stochastic Rank Regularized tensor regression network outperforms the ResNet baseline by a large margin, Table~\ref{table:mri}.
To put this into context, the current state-of-art for convolutional neural networks on age prediction from brain MRI on most datasets is an MAE of around 3.6 years \citep{cole2017predicting, herent2018brain}.
\textbf{Robustness to noise:}
We tested the robustness of our model to white Gaussian noise added to the MRI data. Noise in MRI data typically follows a Rician distribution but can be approximated by a Gaussian for signal-to-noise ratios (SNR) greater than \(2\) \cite{gudbjartsson1995rician}. As both the signal (MRI voxel intensities) and noise are zero-mean, we define \( \mathrm{SNR} = \frac{\sigma_{\mathrm{signal}}^2}{\sigma_{\mathrm{noise}}^2} \), where \( \sigma \) is the variance. We incrementally increase the added noise in the test set and compare the error rate of the models.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.9\linewidth]{mri_noise.pdf}
\vspace{-0.2cm}
\caption{Age prediction error on the MRI test set as a function of increased added noise.}
\label{fig:mri-noise}
\vspace{-0.3cm}
\end{figure}
The ResNet with SRR is significantly more robust to added white Gaussian noise compared to the same architectures without SRR (figure~\ref{fig:mri-noise}). At signal-to-noise ratios below 10, the accuracy of a standard ResNet with average pooling is worse than a model that predicts the mean of training set (MAE = 7.9 years).
Brain morphology is an important attribute that has been associated with various biological traits including cognitive function and overall health \citep{pfefferbaum1994quantitative,swan1998association}. By keeping the structure of the brain represented in MRI in every layer of the architecture, the model has more information to learn a more accurate representation of the entire input.
Additionally, the stochastic dropping of ranks forces the representation to be robust to confounds. This a particularly important property for MRI analysis since intensities and noise artifacts can vary significantly between MRI scanners \cite{wang1998correction}. SRR enables both more accurate and more robust trait predictions from MRI that can consequently lead to more accurate disease diagnoses.
\section{Conclusion}
We introduced the stochastic rank-regularized tensor regression networks. By adding rank-randomization during training, this renders the network more robust and lead to better performance. This also translates to more stable training, and networks less prone to over-fitting. The low-rank, robust representation also makes the network more resilient to noise, both adversarial and random. Our results demonstrate superior performance and convergence on a variety of challenging tasks, including MRI data and images.
\section*{Acknowledgements}
This research has been conducted using the UK Biobank Resource under Application Number 18545.
{\small
\bibliographystyle{icml2019}
|
{
"timestamp": "2019-03-01T02:01:57",
"yymm": "1902",
"arxiv_id": "1902.10758",
"language": "en",
"url": "https://arxiv.org/abs/1902.10758"
}
|
\section{Introduction}
Abdominal Aortic Aneurysm (AAA), an enlargement of the abdominal aorta with 50\% diameter over normal state, occurs increasingly often among old people \cite{sakalihasan2005abdominal}. The rupture of AAA brings in 85\%-90\% fatality rate \cite{kent2014abdominal}. Fenestrated Endovascular Aortic Repair (FEVAR) is a minimally invasive surgery for AAA, where a deployment catheter carrying a compressed stent graft is inserted via the femoral artery, advanced through the vasulature and deployed subsequently at the AAA position. Three typical stent grafts - iliac, fenestrated and thoracic stent graft are shown in Figure~\ref{fig:intro}(a), \ref{fig:intro}(b) and \ref{fig:intro}(c) respectively. In FEVAR, an accurate alignment of stent graft fenestrations or scallops (as shown in Figure~\ref{fig:intro}c) to aortic branches, i.e., renal arteries, is necessary for connecting branch stent grafts into aortic branches \cite{cross2012fenestrated}. Although several robot-assisted systems have been developed to facilitate the FEVAR procedure, i.e., the Magellan system (Hansen Medical, CA, USA), current navigation technique is still based on 2D fluoroscopic images which are insufficient for 3D-to-3D alignment. Either supplying 3D navigation for the AAA or fenestrated stent grafts would improve the navigation.
\begin{figure}[th]
\centering
\framebox{\parbox{3.3in}{\includegraphics[width = 3.3in]{./Introduction}}}
\caption{Illustration of iliac stent graft (a), thoracic stent graft (b), fenestrated stent graft (c), marker number and different stent segment status (d).}
\label{fig:intro}
\end{figure}
For 3D AAA navigation, a skeleton-based as-rigid-as-possible approach was proposed to adapt a 3D pre-operative AAA shape to intra-operative position of the deployment device from two fluoroscopic images for recovering the 3D AAA shape \cite{toth2015adaption}. A skeleton instantiation framework for AAA with a graph matching method and skeleton deformation was introduced to instantiate the 3D AAA skeleton from a single 2D fluoroscopic image \cite{zheng20183d}.
For offering 3D navigation for fenestrated stent grafts, many methods have been implemented. The 3D stent shape was recovered from a 2D X-ray image via registration and optimization in \cite{demirci20113d} but without estimation of the graft nor the angle or position of fenestrations or scallops. A 3D shape instantiation framework with stent graft modelling and Robust Perspective-n-Point (RPnP) method was proposed to instantiate the 3D shape of a fully-compressed stent graft \cite{zhoustent}. The work in \cite{zhoustent} was then used to recover the 3D shape of each stent segment (as shown in Figure~\ref{fig:intro}b), with customized markers, while Focal U-Net and graft gap (as shown in Figure~\ref{fig:intro}b) interpolation were proposed to semi-automatically segment customized markers and recover the whole 3D shape of fully-deployed stent grafts in \cite{zhou2018real_ral}. Equally-weighted Focal U-Net was also proposed for automatic marker segmentation in \cite{zhou2018towards_iros} to improve the automation of the 3D shape instantiation framework. However, the method by Zhou et al. could not instantiate the 3D shape of a partially-deployed stent segment, as the 3D marker references required by the RPnP method are unknown.
The method proposed in this paper aims to obtain the deformation pattern between partially-deployed and fully-deployed stent segment using deep learning based methods. General artificial neural networks can be applied to this task but with very large searching space of parameters. The relationship between each two markers is not uniform and the topological structure is non-Euclidean either. The classical convolutional kernel and thus the convolutional neural networks cannot be used for this problem. A novel convolution on an undirected simple graph called spectral graph convolution was described in \cite{shuman2012emerging}. A Graph Convolutional Network (GCN) with locally connected architecture was then proposed in \cite{bruna2013spectral} with $\mathcal{O}(n)$ parameter number for each layer based on the spectrum of graph Laplacian, which was validated on the MNIST dataset. Furthermore, a more efficient GCN with localized spectral convolution on a graph was proposed in \cite{defferrard2016convolutional}, reducing the parameter number to $\mathcal{O}(K)$ with improved performance on the MNIST dataset but computational complexity, where $K<n$ is the localized filter size. Another construction of GCN was also proposed in \cite{kipf2016semi} with first-order approximation of spectral graph convolutions for a large-scale architecture, but with less capacity for the same layer number compared to \cite{defferrard2016convolutional}.
An Adapted GCN based on the architecture in \cite{defferrard2016convolutional} is proposed for predicting 3D marker references of partially-deployed stent segment from 3D fully-deployed markers, which bridges the gap of utilizing the RPnP method for 3D shape instantiation of partially-deployed stent segment. The coarsening layers are removed and the softmax function at the network end is replaced with a linear mapping. The derived 3D marker references are integrated into a previously deployed 3D shape instantiation framework \cite{zhou2018real_ral}, with the customized marker placement, stent segment modelling and the RPnP method, to achieve 3D shape instantiation for partially-deployed stent segment, The pipeline is shown in Figure~\ref{fig:pipeline}. Three stent grafts with total 26 different stent segments were used for the validation. Details regarding the methodology and experimental setup are in Section~\ref{sec:method}. Results with an average angular error about $7^\circ$ and an average mesh distance error around 2mm are stated in Section~\ref{sec:result}. Discussion and conclusion are introduced in Section~\ref{sec:discussion} and Section~\ref{sec:conclusion} respectively.
\section{Methodology}
\label{sec:method}
In this section, we introduce the proposed Adapted GCN for predicting 3D marker references, while briefly introducing the stent segment modelling and 3D shape instantiation to facilitate the understanding of the whole framework. Experimental setup is also demonstrated.
\begin{figure}[th]
\centering
\framebox{\parbox{3.3in}{\includegraphics[width = 3.3in]{./pipeline}}}
\caption{Pipeline for shape instantiation of partially-deployed stent segment from a single fluoroscopic image and the 3D CT scan of fully-deployed stent graft}
\label{fig:pipeline}
\end{figure}
\subsection{Partially-deployed Stent Segment Modelling}
\label{sec:stent_model}
In practice, the parameters of stent segment, including the height and diameters at the fully-deployed and fully-compressed state, can be obtained via fenestrated stent graft and deployment catheter design. In this paper, as the stent grafts were experimented multiple times with compression and deployment, the practical parameters are different from the ideally designed ones and are measured manually.
In \cite{zhou2018real_ral}, a stent graft was modelled as a cylinder fitted by a series of concentric circles with a finite set of vertices $\mathcal{V}$ of coordinates $\textit{\textbf{V}}\in\mathbb{R}^{3\times(360h/0.1{\rm mm})}$. The coordinate of each circle vertex is defined as $(r {\rm cos}\theta~r {\rm sin}\theta~h)^\top$. In this paper, each partially-deployed stent segment is modelled as a cone with the diameters and the height of this segment.
Different from the fully-deployed stent segment in \cite{zhou2018real_ral}, the diameters of partially-deployed stent segments are not only decided by the designed deployed size but also the compression diameters $r_{\rm fc}\in\mathbb{R}_+$ and the gap width $w_{\rm g}\in\mathbb{R}_+$. In the experiments, one partially-deployed stent segment's diameter of its deployed side $r_{\rm pd}\in\mathbb{R}_+$ is set as the value designed for fully-deployed state $r_{\rm fd}\in\mathbb{R}_+$:
\begin{equation}
r_{\rm pd}:=r_{\rm fd}
\end{equation}
and the diameter of its compressed side $r_{\rm pc}\in\mathbb{R}_+$ is set as the minimal value between the deployed diameter, and the addition of compressed diameter and twice gap width:
\begin{equation}
r_{\rm pc}:={\rm min}\{r_{\rm fc}+2w_{\rm g}, r_{\rm fd}\}
\end{equation}
Using the diameters of the deployed side and the compressed side, a cone shape can be modelled for the partially-deployed stent segment
Following \cite{zhou2018real_ral}, these circle vertices are accumulated by connecting the neighbouring vertices regularly into triangular faces, resulting in a mathematically modelled stent segment mesh. Fenestrations or scallops are modelled by removing the corresponding vertices and triangular faces. The resolution of height $h$ was set as 0.1mm and that of rotation angle $\theta$ was set as $1^\circ$. A set of five customized markers are sewn on each stent segment. With known pre-operative 3D reference marker positions (3D marker references) and corresponding intra-operative 2D marker positions (2D marker references), the 3D intra-operative pose of marker set which is also the 3D intra-operative pose of the stent segment could be recovered by the RPnP method \cite{zhou2018real_ral}. Details regarding this part will be briefly introduced in Section~\ref{sec:instantiation}.
Unlike the work in \cite{zhoustent} and \cite{zhou2018real_ral} for fully-compressed and fully-deployed stent graft, where 3D marker references are known from computed tomography (CT) scan or stent graft design, 3D marker references for partially-deployed stent segment are unknown due to the unpredictability of the deployment process.
\subsection{Adapted GCN}
\label{sec:coordinate_prediction}
With known pre-operative 3D fully-deployed marker positions $\textit{\textbf{Y}}_{\rm f}^{\rm l}=(\textit{\textbf{y}}_{{\rm f}1}^{\rm l}~\cdots~\textit{\textbf{y}}_{{\rm f}5}^{\rm l})\in\mathbb{R}^{3\times5}$, an Adapted GCN for regressing pre-operative 3D marker references of partially-deployed stent segment $\textit{\textbf{Y}}_{\rm p}^{\rm l}\in\mathbb{R}^{3\times5}$ is proposed based on \cite{defferrard2016convolutional}. Original GCNs in \cite{defferrard2016convolutional} and \cite{kipf2016semi} were for classification tasks, while in this paper, the coarsening layers are removed and the softmax function at the network end is replaced by linear mapping.
\subsubsection{Data Pre-processing}
To focus the Adapted GCN training on learning the deformation between $\textit{\textbf{Y}}_{\rm f}^{\rm l}$ and $\textit{\textbf{Y}}_{\rm p}^{\rm l}$, in the training data, markers' coordinates for fully-deployed stent segment $\textit{\textbf{Y}}_{\rm f}^{\rm l}$ are standardized in local frame with the transformation:
\begin{equation}
\textit{\textbf{t}}_{\rm l}^{\rm g}:=\sum_{i=1}^5{({\textit{\textbf{y}}_{\rm f}^{\rm g}}_i)}
\end{equation}
\begin{equation}
\textit{\textbf{R}}_{\rm l}^{\rm g}:=
\begin{pmatrix}
\textbf{\textit{v}}_1/\|\textbf{\textit{v}}_1\|_2&\textbf{\textit{v}}_2/\|\textbf{\textit{v}}_2\|_2&\textbf{\textit{v}}_3/\|\textbf{\textit{v}}_3\|_2
\end{pmatrix}
\end{equation}
where $\textbf{\textit{v}}_1:={\textit{\textbf{y}}_{\rm f}^{\rm t}}_1$, $\textbf{\textit{v}}_2:=({\textit{\textbf{y}}_{\rm f}^{\rm t}}_1\times{\textit{\textbf{y}}_{\rm f}^{\rm t}}_2)$, $\textbf{\textit{v}}_3:= (\textbf{\textit{v}}_1 \times \textbf{\textit{v}}_2)$ and $\textit{\textbf{Y}}_{\rm f}^{\rm t}:=\textit{\textbf{R}}_{\rm l}^{\rm g} \textit{\textbf{Y}}_{\rm f}^{\rm l}$. $\times$ between two vectors represents the cross product. Then the transformation between global frame and local frame can be represented by:
\begin{equation}
\textit{\textbf{Y}}_{\rm f}^{\rm g}=\textit{\textbf{R}}_{\rm l}^{\rm g}\textit{\textbf{Y}}_{\rm f}^{\rm l}+\textit{\textbf{t}}_{\rm l}^{\rm g}\otimes(\textbf{1})_{1\times5}
\end{equation}
where, $\otimes$ is the kronecker product and $(\textbf{1})_{1\times5}$ is a $1\times5$ matrix consisting of $1$.
Before training the network, the ground truth of markers' coordinates for each partially-deployed stent segment in local frame $\textit{\textbf{Y}}_{\rm p}^{\rm l}$ is obtained by aligning the detected 3D markers' coordinates in global frame $\textit{\textbf{Y}}_{\rm p}^{\rm g}$ to the markers for corresponding fully-deployed stent segment in the local frame $\textit{\textbf{Y}}_{\rm f}^{\rm l}$ via singular value decomposition (SVD): $\textit{\textbf{U}}_{\rm svd}\Sigma{\textit{\textbf{V}}}_{\rm svd}=\textit{\textbf{Y}}_{\rm p}^{\rm g}{\textit{\textbf{Y}}_{\rm f}^{\rm l}}^{\top}$.
The aligned markers' coordinates for each partially-deployed stent segment is thus calculated with mapping $f:(\mathbb{R}^{3\times 5},\mathbb{R}^{3\times 5})\to\mathbb{R}^{3\times 5}$ defined as:
\begin{equation}
\textit{\textbf{Y}}_{\rm p}^{\rm l}=f(\textit{\textbf{Y}}_{\rm p}^{\rm g},\textit{\textbf{Y}}_{\rm f}^{\rm l}):=\textit{\textbf{R}}_{\rm p}^{\rm f}\textit{\textbf{Y}}_{\rm p}^{\rm g}+\textit{\textbf{t}}_{\rm p}^{\rm f}
\end{equation}
where
$
\textit{\textbf{R}}_{\rm p}^{\rm f}:=\textit{\textbf{V}}_{\rm svd}\textit{\textbf{U}}_{\rm svd}^\top
$ and
$
\textit{\textbf{t}}_{\rm p}^{\rm f}:=\sum_{i=1}^5{({\textit{\textbf{y}}_{\rm f}^{\rm l}}_i)}-\textit{\textbf{R}}_{\rm p}^{\rm f}\sum_{i=1}^5{({\textit{\textbf{y}}_{\rm p}^{\rm g}}_i)}
$
are the rotation matrix and translation vector of the transformation.
\subsubsection{Spectral Graph Convolution}
Different from conventional convolutional kernels used in Euclidean space, GCN employs spectral graph convolution on a graph \cite{shuman2012emerging}. The spectral graph Fourier transform and its inverse transform is defined as:
\begin{equation}
\tilde{\textit{\textbf{Y}}}=\mathcal{F}_\mathcal{G}(\textit{\textbf{Y}}):=\textit{\textbf{U}}^\top \textit{\textbf{Y}},\quad \textit{\textbf{Y}}=\mathcal{F}_\mathcal{G}^{-1}(\tilde{\textit{\textbf{Y}}})=\textit{\textbf{U}} \tilde{\textit{\textbf{Y}}}
\end{equation}
where $\mathcal{G}=(\mathcal{V},\mathcal{E},{\textit{\textbf{W}}})$ is an undirected simple graph with $n=5$ nodes, representing the coordinates of five customized markers, $\mathcal{V}$ is a finite set of $|\mathcal{V}|=n$ vertices, $\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$ is a set of edges, $\textit{\textbf{W}}\in\mathbb{R}^{n\times n}$ is the weighted adjancy matrix, $\textit{\textbf{Y}}$ is the coordinates' values defined on nodes, the fourier basis $\textit{\textbf{U}}$ is obtained by the eigenvector matrix of graph $\mathcal{G}$'s normalized Laplacian matrix $\textit{\textbf{L}}\in\mathbb{R}^{5\times5}$: $\textit{\textbf{L}}=\textit{\textbf{U}}\Lambda\textit{\textbf{U}}^{-1}$, where $\Lambda={\rm diag}(\lambda_0~\cdots~\lambda_{n-1})\in\mathbb{R}^{n}$ is the eigen values. The normalized Laplacian matrix is defined as:
\begin{equation}
\textit{\textbf{L}}:=\textit{\textbf{D}}^{-0.5}(\textit{\textbf{D}}-\textit{\textbf{W}})\textit{\textbf{D}}^{-0.5}
\end{equation}
where $\textit{\textbf{D}}\in\mathbb{R}^{n\times n}$ is the diagonal degree matrix. As normalized Laplacian matrix is semi-positive definite symmetric matrix, $\textit{\textbf{U}}^\top=\textit{\textbf{U}}^{-1}$. Spectral graph convolution on graph $\mathcal{G}$ could be defined as:
\begin{equation}
\label{eq:convolution}
(\textit{\textbf{g}}_\vartheta*\textit{\textbf{Y}})_\mathcal{G}:=\mathcal{F}^{-1}_\mathcal{G}(\mathcal{F}_\mathcal{G}\big(\textit{\textbf{g}}_\vartheta)\mathcal{F}_\mathcal{G}(Y)\big)=\textit{\textbf{U}}\tilde{\textit{\textbf{g}}}_\vartheta\textit{\textbf{U}}^\top\textit{\textbf{Y}}
\end{equation}
where $\tilde{\textit{\textbf{g}}}_\vartheta$ is defined as the convolutional kernel (also known as filter in \cite{defferrard2016convolutional}) and $\vartheta$ is the trainable parameters. A non-parametric kernel is defined as $\tilde{\textit{\textbf{g}}}_\vartheta(\Lambda)={\rm diag}(\vartheta)$ \cite{bruna2013spectral}, where $\vartheta\in\mathbb{R}^n$. There are also multiple approaches of parametrization for the localized filter, polynomial parametrization was introduced in \cite{defferrard2016convolutional}: $\tilde{\textit{\textbf{g}}}_\vartheta(\Lambda)=\sum_{k=0}^{K-1}{\vartheta_k (\Lambda)^k}$, where ${(\Lambda)^k}$ is the $k$ power of $\Lambda$. Because $\textit{\textbf{U}}^\top=\textit{\textbf{U}}^{-1}$ and this polynomial parametric kernel converts (\ref{eq:convolution}) into:
\begin{equation}
(\textit{\textbf{g}}_\vartheta*\textit{\textbf{Y}})_\mathcal{G}=\tilde{g}_\vartheta(\textit{\textbf{L}})\textit{\textbf{Y}}=\sum_{k=0}^{K-1}{\vartheta_k (\textbf{\textit{L}})^k}\textit{\textbf{Y}}
\end{equation}
where $K\in\mathbb{Z}_+$ represents the kernel size and $\vartheta\in\mathbb{R}^K$ implies the learning complexity to be reduced to $\mathcal{O}(K)$, compared with $\mathcal{O}(n)$ for non-parametric kernel.
Furthermore, recursive formulation for parametric kernel was introduced in \cite{defferrard2016convolutional} to reduce the computational time. The kernel is approximated by Chebyshev polynomials:
\begin{equation}
\tilde{\textit{\textbf{g}}}_\vartheta(\Lambda)=\sum_{k=0}^{K-1}{\vartheta_k T_k(\Lambda')}
\end{equation}
where $\Lambda'=2\Lambda/\lambda_{\rm max}-\textit{\textbf{I}}_n$, $\textit{\textbf{I}}_n$ is an unity matrix with size $n\times n$. $T_k(\Lambda')=2\Lambda'T_{k-1}(\Lambda')-T_{k-2}(\Lambda')$ is the recursive Chebyshev polynomials with $T_0(\Lambda')=1$ and $T_1(\Lambda')=\Lambda'$. This kernel is used in this paper and details can be found in \cite{hammond2011wavelets}.
\subsubsection{Network Architecture}
The number of five customized markers is shown in Figure~\ref{fig:intro}(d). An undirected simple graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\textit{\textbf{W}})$ with five nodes is constructed to represent the five markers' coordinates with weight adjacency matrix set referring to the distance scale:
\begin{equation}
\textit{\textbf{W}}=\begin{pmatrix}
0&e^{-(5/4)^2}&0&0&e^{-(5/8)^2}\\
e^{-(5/4)^2}&0&e^{-(5/4)^2}&0&0\\
0&e^{-(5/4)^2}&0&e^{-(5/4)^2}&0\\
0&0&e^{-(5/4)^2}&0&e^{-(5/4)^2}\\
e^{-(5/8)^2}&0&0&e^{-(5/4)^2}&0\\
\end{pmatrix}
\end{equation}
\begin{figure}[ht]
\centering
\framebox{\parbox{2.9in}{\includegraphics[width = 2.9in]{./network}}}
\caption{Network architecture of the proposed Adapted GCN.}
\label{fig:network}
\end{figure}
The network architecture is shown in Figure~\ref{fig:network}, where the input is ${\textit{\textbf{Y}}}_{\rm f}^{\rm l}+\epsilon$ and the output is $\hat{\textit{\textbf{Y}}}_{\rm p}^{\rm l}$, where $\epsilon\sim\mathcal{N}(0,0.1)$ is Gaussian noise. The mathematical expression for each two neighbouring layers can be written as:
\begin{equation}
\textit{\textbf{F}}^i=\sigma_i\big((\textit{\textbf{g}}_\vartheta*\textit{\textbf{F}}^{i-1})_\mathcal{G}\big)
\end{equation}
where $i\in[0,N+1]\cap\mathbb{Z}$, $\textit{\textbf{F}}^0$ is the input graph, $\textit{\textbf{F}}^{N+1}$ is the output graph, $\textit{\textbf{F}}^{1\sim N}$ are hidden layers, $N$ is the hidden layer number and $\sigma_i(\cdot)$ is the activation function for the $i^{\rm th}$ layer.
Eight hidden layers are used for the experiments, 32 channels are set in each hidden layer. Leaky ReLU is used as the activation function for non-linear mapping with 0.1 leaky rate for the input and the hidden layers. No non-linear activation function is used in the output layer. Chebyshev polynomial parametric kernel is used with an kernel size of 2 for each spectral convolutional layer.
\subsubsection{Loss Function and Optimization}
The root mean square error between the ground truth and the output coordinates is calculated as the loss function, with a regularization term of L2 norm of the weight matrix:
\begin{equation}
\mathcal{L}=\|\hat{\textit{\textbf{Y}}}_{\rm p}^{\rm l}-\textit{\textbf{Y}}_{\rm p}^{\rm l}\|_2+\alpha\|\vartheta\|_2
\end{equation}
Adam and Momentum Stochastic Gradient Descend (SGD) were compared for training the network. The optimization through Adam was hard to converge and hence Momentum SGD was used as the optimizer. The learning rate was set as 0.0001 and the learning momentum was set as 0.9. The L2 norm weight $\alpha$ was set as $5\times 10^{-4}$ and the batch size was set as 10.
As the RPnP method is only related to 3D reference marker shapes while is free to global 3D reference marker positions, the predicted 3D marker references $\hat{\textit{\textbf{Y}}}_{\rm p}^{\rm l}$ are also aligned to the local markers' coordinates of fully-deployed stent segment ${\textit{\textbf{Y}}}_{\rm f}^{\rm l}$ as $f(\hat{\textit{\textbf{Y}}}_{\rm p}^{\rm l},{\textit{\textbf{Y}}}_{\rm f}^{\rm l})$ for the transformation estimation of partially-deployed stent segments.
\subsection{3D Shape Instantiation}
\label{sec:instantiation}
With the predicted pre-operative 3D marker references from the Adapted GCN in Section~\ref{sec:coordinate_prediction} and manually labelled corresponding intra-operative 2D marker positions/references, following \cite{zhou2018real_ral}, the RPnP method \cite{li2012robust} is used to instantiate the 3D pose of intra-operative marker set including the rotation matrix $\hat{\textit{\textbf{R}}}_{\rm l}^{\rm g}\in\mathbb{R}^{3\times3}$ and translation vector $\hat{\textit{\textbf{t}}}_{\rm l}^{\rm g}\in\mathbb{R}^{3}$:
\begin{equation}
\hat{\textit{\textbf{Y}}}_{\rm p}^{\rm g}=\hat{\textit{\textbf{R}}}_{\rm l}^{\rm g}f(\hat{\textit{\textbf{Y}}}_{\rm p}^{\rm l},{\textit{\textbf{Y}}}_{\rm f}^{\rm l})+\hat{\textit{\textbf{t}}}_{\rm l}^{\rm g}\otimes(\textbf{1})_{1\times5}
\end{equation}
where $\hat{\textit{\textbf{Y}}}_{\rm p}^{\rm g}$ is the instantiated intra-operative 3D marker positions for partially-deployed stent segment. As markers are sewn on the stent segment, $\hat{\textit{\textbf{R}}}_{\rm l}^{\rm g}$ and $\hat{\textit{\textbf{t}}}_{\rm l}^{\rm g}$ are also the rotation matrix and translation vector for the partially-deployed stent segment. After moving the mathematically modelled stent segment mesh in Section~\ref{sec:stent_model} to the same local coordinate frame, $\hat{\textit{\textbf{R}}}_{\rm l}^{\rm g}$ and $\hat{\textit{\textbf{t}}}_{\rm l}^{\rm g}$ are applied for the stent segment transformation. After central point based correction, 3D shape instantiation of partially-deployed stent segment is achieved. More details could be found in \cite{zhou2018real_ral}.
\subsection{Experiment and Validation}
\label{sec:experiment}
\subsubsection{Marker Design}
Customized stent graft markers with five different shapes were designed based on commercially-used gold markers and were manufactured on a Mlab Cusing R machine (ConceptLaser, Lichtenfels, Germany) from SS316L stainless steel powder, as shown in Figure~\ref{fig:intro}(d) with their own numbers. The sizes are around 1$\sim$3 mm, similar to the commercial ones. Those five markers were sewn on each stent segment at five non-planar places.
\begin{figure}[ht]
\centering
\framebox{\parbox{3.3in}{\includegraphics[width = 3.3in]{./setup}}}
\caption{Illustration of the experimental setup with fixing an AAA phantom under the CT scan.}
\label{fig:setup}
\end{figure}
\subsubsection{Simulation of Surgery}
Three stent grafts were used in the experiments, including a iliac stent graft (Cook Medical, IN, USA) with five stent segments, 10$\sim$19mm diameters and total $90$mm height, a fenestrated stent graft (Cook Medical) with six stent segments, 22$\sim$30mm diameters and total 117mm height, and a thoracic stent graft (Medtronic, MN, USA) with 10 stent segments, 30mm diameter and total 179mm height. Five AAA phantoms were modelled from CT data scanned from patients and were printed on a Stratasys Object 3D printer (MN, USA) with VeroClear and TangoBlack colours. To simulate the practical situation in FEVAR where the fenestrated stent graft is customized to similar diameters to that of the AAAs, two suitable AAA target positions where their diameters are similar to that of the corresponding experiment stent graft were selected for each experiment stent graft, resulting in 6 experiments in total. The selected AAA phantom was fixed as shown in Figure~\ref{fig:setup}. In each experiment, a stent graft was compressed into a Captivia delivery catheter (Medtronic) with 8mm diameter, inserted into the selected phantom and deployed subsequently segment-by-segment from the proximal end to the distal end at the target AAA position.
\subsubsection{Data Collection}
A 3D CT scan and a 2D fluoroscopic image at the frontal plane were scanned for each partially-deployed stent graft using a GE Innova 4100 (GE Healthcare, Bucks, UK) system. The stent segments at the distal end and with odd indexes in the thoracic stent graft experiment were ignored to keep data balance. Thus, there are eight partially-deployed stent segments scanned by CT and flurorscopy in two different AAA phantoms for the iliac stent graft (segment number 1-4 and 5-8), 10 for the fenestrated stent graft (stent segment number 9-13 and 14-18), and eight for the thoracic stent graft (stent segment number 19-22 and 23-26). In addition, three CT scans were acquired for the three experiment stent grafts at fully-deployed state to supply 3D fully-deployed marker positions - $\textit{\textbf{Y}}_{\rm f}^{\rm l}$. In practical applications, this information can be obtained from stent graft designing.
\subsubsection{Marker Position Extraction}
\label{sec:marker_detection}
Although Equally-weighted Focal U-Net was proposed to potentially achieve automatic 2D marker segmentation and classification from intra-operative 2D fluoroscopic images. In this paper, the stent graft is in partially-deployed state which is different from the training data in \cite{zhou2018towards_iros} where the stent graft was in fully-deployed state. The segmentation and classification results of applying the trained model in \cite{zhou2018towards_iros} onto the fluoroscopic images in this paper is not accurate and unsatisfied. Hence the intra-operative 2D marker positions or references $\textit{\textbf{X}}^{\rm g}=(\textit{\textbf{x}}_1^{\rm g}~\cdots~\textit{\textbf{x}}_5^{\rm g})\in\mathbb{R}^{2\times5}$ were extracted manually via $Matlab^{\textregistered}$.
The shapes of 3D stents and 3D customized markers were segmented from CT scans via ITK-SNAP and the 3D central coordinates of customized markers $\textit{\textbf{Y}}^{\rm g}=(\textit{\textbf{y}}_1^{\rm g}~\cdots~\textit{\textbf{y}}_5^{\rm g})\in\mathbb{R}^{3\times5}$ were extracted using Meshlab.
\subsubsection{Data Augmentation}
Before training the Adapted GCN with the 3D marker positions of fully-deployed and partially-deployed stent segments, these coordinates were rotated and scaled to enlarge the training dataset. The rotations about three axises range from $-30^\circ$ to $30^\circ$ with the resolution of $3^\circ$. The scale ratios range from $0.2$ to $11.39$ with the geometric proportion of $1.5$.
\subsubsection{Criteria and Evaluation}
To evaluate the 3D marker references predicted by the proposed Adapted GCN, the aligned 3D marker reference prediction $\textit{\textbf{Y}}_{\rm p}^{\rm l}$ were compared to the ground truth of the aligned partially-deployed stent segment's marker positions $f(\hat{\textit{\textbf{Y}}}_{\rm p}^{\rm l},{\textit{\textbf{Y}}}_{\rm f}^{\rm l})$ via their mean distance error, ${\rm MDE}\big(\textit{\textbf{Y}}_{\rm p}^{\rm l},f(\hat{\textit{\textbf{Y}}}_{\rm p}^{\rm l},{\textit{\textbf{Y}}}_{\rm f}^{\rm l})\big)$, which is calculated as:
\begin{equation}
{\rm MDE}(\textit{\textbf{Y}}^1,\textit{\textbf{Y}}^2)=\frac{1}{n}\sum_{i=1}^{n}{\big\|\textit{\textbf{y}}_i^1-\textit{\textbf{y}}_i^2\big\|_2}
\end{equation}
where $\textit{\textbf{Y}}^{1}$ and $\textit{\textbf{Y}}^{2}$ can be two matrices of 3D or 2D marker coordinates with the same dimension number and the same point number.
To evaluate marker instantiation, the registered global markers' coordinates for each partially-deployed stent segment $\hat{{\textit{\textbf{Y}}}}_{\rm p}^{\rm g}$ are compared with the ground truth ${{\textit{\textbf{Y}}}}_{\rm p}^{\rm g}$ via ${\rm MDE}\big({{\textit{\textbf{Y}}}}_{\rm p}^{\rm g},\hat{{\textit{\textbf{Y}}}}_{\rm p}^{\rm g}\big)$ in 3D and the reprojected distance error ${\rm MDE}\big({{\textit{\textbf{X}}}}_{\rm p}^{\rm g},\hat{{\textit{\textbf{X}}}}_{\rm p}^{\rm g}\big)$ in 2D, where $\hat{{\textit{\textbf{X}}}}_{\rm p}^{\rm g}$ is the projected 2D coordinate from the estimated 3D global coordinate $\hat{{\textit{\textbf{Y}}}}_{\rm p}^{\rm g}$, calculated by $\hat{{\textit{\textbf{X}}}}_{\rm p}^{\rm g}=g(\hat{{\textit{\textbf{Y}}}}_{\rm p}^{\rm g})$ with mapping $g:\mathbb{R}^{3\times n}\to\mathbb{R}^{2\times n}$:
\begin{equation}
\label{eq:projection_points}
g(\textit{\textbf{Y}})=\begin{pmatrix}
\textit{\textbf{p}}_1^\top\textit{\textbf{Y}}^{\rm h}\oslash\textit{\textbf{p}}_3^\top\textit{\textbf{Y}}^{\rm h} \\ \textit{\textbf{p}}_2^\top\textit{\textbf{Y}}^{\rm h}\oslash\textit{\textbf{p}}_3^\top\textit{\textbf{Y}}^{\rm h}
\end{pmatrix}
\end{equation}
where $\textit{\textbf{P}}=\begin{pmatrix} \textit{\textbf{p}}_1 & \textit{\textbf{p}}_2 & \textit{\textbf{p}}_3 \end{pmatrix}^\top\in\mathbb{R}^{3\times4}$ is the projection matrix, $\oslash$ is Hadamard division, and $\textit{\textbf{Y}}^{\rm h}=(\textit{\textbf{y}}_1^{\rm h}~\cdots~\textit{\textbf{y}}_{\textit{n}^3}^{\rm h})=(\textit{\textbf{Y}}^\top~(\textbf{1})_{\textit{n}^3\times1})^\top\in\mathbb{R}^{{4}\times{\textit{n}^3}}$ is the homogeneous vector form of the 3D coordinates.
To evaluate 3D shape instantiation for each partially-deployed stent segment, the distance between the instantiated partially-deployed stent segment mesh and the corresponding ground truth was measured using Matlab function $point2trimesh$ \cite{point2trimesh}. Marker angle was estimated by the angle of the nearest vertex on the constructed stent segment. Mean absolute angle difference between the predicted markers and the ground truth was used to measure the angle error.
\subsubsection{Cross Validation}
Three-fold cross validations were performed along the division of stent graft. For example, for testing stent segments on iliac stent grafts, the data from the fenestrated and thoracic stent graft were used for the training.
\section{Results}
\label{sec:result}
In this section, the experimental results for the validation of the proposed method was illustrated including the 3D distance errors in the marker prediction, the 2D re-projected and 3D distance error in the marker instantiation, as well as the angular and the mesh error in the stent segment shape instantiation,
\subsection{Prediction of 3D Marker References}
\label{sec:result_GCN}
\begin{figure}[ht]
\centering
\framebox{\parbox{3.3in}{\includegraphics[width = 3.3in]{./deform_error}}}
\caption{Mean$\pm$std 3D distance of the initial variation and mean$\pm$std 3D distance error of 3D marker reference prediction with the proposed Adapted GCN.}
\label{fig:deform_error}
\end{figure}
The mean 3D distance between the prediction of 3D marker references and the ground truth, called Adapted GCN, and the initial mean 3D distance between the 3D fully-deployed markers and the ground truth, named initial variation, for the 26 partially-deployed stent segments are shown in Figure~\ref{fig:deform_error}. We can see that the mean 3D distance achieved by the Adapted GCN is significantly lower than the initial variation, especially for the fenestrated and thoracic stent graft (stent segment number 9$\sim$26), proving the efficiency of the proposed Adapted GCN on 3D marker reference prediction. The mean 3D distances achieved by the Adapted GCN for the iliac stent graft (stent segment number 1$\sim$8) are comparable to the initial variations. Because the diameter of the iliac stent graft is very close to that of the deployment catheter (due to limited experimental resources, we only got one available deployment catheter), and there is not much difference between the fully-deployed and partially-deployed state of the iliac stent graft.
\subsection{3D Marker Instantiation}
\begin{figure}[th]
\centering
\framebox{\parbox{3.3in}{\includegraphics[width = 3.3in]{./pose_example}}}
\caption{Comparison of instantiated intra-operative 3D marker positions and the 3D ground truth (a), and comparison of 2D projections of instantiated 3D markers and the 2D ground truth (b).}
\label{fig:pose_example}
\end{figure}
The predicted 3D marker references and the manually detected 2D marker references for partially-deployed stent segment are imported into the RPnP instantiation framework \cite{zhou2018real_ral} to recover the intra-operative 3D marker positions. The instantiated intra-operative 3D marker positions and their 2D projections are compared to the corresponding ground truth, with results shown in Figure~\ref{fig:pose_example}. We can see that the instantiated marker positions are very close to the ground truth in both 3D and 2D.
\begin{figure}[th]
\centering
\framebox{\parbox{3.3in}{\includegraphics[width = 3.3in]{./pose_error}}}
\caption{Mean$\pm$std 3D (a) and 2D projected (b) distance errors of the instantiated intra-operative marker positions with the ideal (red) and practical (blue) 2D marker references as the input 2D marker reference.}
\label{fig:pose_error}
\end{figure}
Due to the imaging error caused by the fluoroscopic system, 0.5$\sim$0.8mm deviation exists between the manually detected 2D marker references, named practical 2D marker references, and the projected 2D marker references from the ground truth 3D marker references, named ideal 2D marker references. Both of these two 2D marker references are used with the predicted 3D marker references to instantiate the intra-operative 3D marker positions. The 3D and 2D re-projected distance errors for the 26 partially-deployed stent segments are shown in Figure~\ref{fig:pose_error}. We can see that an average 2D distance error of 1.58mm and an average 3D distance error of 1.98mm are achieved respectively. The small accuracy gap in the Figure~\ref{fig:pose_error} between using practical and ideal 2D marker references indicates that the robustness of the instantiation framework to the imaging error introduced by the fluoroscopic system.
\begin{figure}[th]
\centering
\framebox{\parbox{3.3in}{\includegraphics[width = 3.3in]{./mesh_example}}}
\caption{Two comparison examples of instantiated meshes of partially-deployed stent segment and 3D makers from predicted 3D and practical 2D marker references, compared with the estimated stent segment ground truth and the 3D marker ground truth.}
\label{fig:mesh_example}
\end{figure}
\begin{figure}[th]
\centering
\framebox{\parbox{3.3in}{\includegraphics[width = 3.3in]{./stent_example}}}
\caption{Two comparison examples of instantiated meshes of partially-deployed stent segment from predicted 3D and practical 2D marker references, compared with the corresponding stent ground truth segmented from CT scan.}
\label{fig:stent_example}
\end{figure}
\subsection{3D Shape Instantiation of Partially-deployed Stent Segment}
As graft could not be imaged via CT, the ground truth of partially-deployed stent segment was estimated by registering the mathematical model in Section~\ref{sec:stent_model} onto the ground truth 3D marker references. Two comparison examples of the instantiated partially-deployed stent segment and the estimated ground truth are shown in Figure~\ref{fig:mesh_example}. Two comparison examples of the instantiated partially-deployed stent segment and the real ground truth represented by the CT stent scan are shown in Figure~\ref{fig:stent_example}. We can see that the reasonable 3D shape instantiation is achieved.
\begin{figure}[th!]
\centering
\framebox{\parbox{3.3in}{\includegraphics[width = 3.3in]{./mesh_error}}}
\caption{Mean$\pm$std angular and 3D mesh distance error of instantiated meshes of partially-deployed stent segment with ideal and practical 2D marker references as the input 2D marker reference.}
\label{fig:mesh_error}
\end{figure}
The mean angular error between the instantiated intra-operative 3D markers and the ground truth is shown in Figure~\ref{fig:mesh_error}(a). An average angular error of $7^\circ$ is achieved which is larger than the average angular error of $4^\circ$ in \cite{zhou2018real_ral}. This is reasonable, as 3D marker references in this paper are unknown and are predicted by training an Adapted GCN. The mean angular error for iliac stent graft (stent segment number 1$\sim$8) is larger than that for the fenestrated and thoracic stent graft (stent segment number 9$\sim$26) due to the same reason stated in Section~\ref{sec:result_GCN}. The mean distance error between the instantiated stent segment mesh and the ground truth is shown in Figure~\ref{fig:mesh_error}(b). An average distance error of 1$\sim$3mm is achieved which is comparable to the average distance error of 1$\sim$3mm in \cite{zhou2018real_ral}. The iliac stent graft (stent segment number 1$\sim$8) experiences lower mean distance error than the fenestrated and thoracic stent graft (stent segment number 9$\sim$26), as its size is smaller.
\begin{table*}[th]
\centering
\caption{The overall performance of marker reference prediction, marker instantiation and 3D shape instantiation on the six experiments, via mean 3D distance error (3D dist.), mean 2D projected distance error (2D dist.), angular error (Ang. error) and mesh distance error (Mesh dist.).}
\label{tab:results}
\begin{tabular}{ccccccccc}
\hline
\multicolumn{3}{c}{Stent graft} &iliac &iliac &fenestrated &fenestrated &thoracic &thoracic \\
\multicolumn{3}{c}{Stent segment number} &1-4 &5-8 &9-13 &14-18 &19-22 &23-26 \\
\hline
\multirow{2}{*}{Marker references} &\multirow{2}{*}{3D dist. (mm)}&Initial Variation &1.5152 &0.9772 &5.2585 &5.5062 &5.0397 &4.9839 \\ \cline{3-9}
& &Adapted GCN &1.2490 &1.2374 &1.6595 &1.8378 &1.5935 &1.4778 \\
\hline
\multirow{4}{*}{Marker instantiation} &\multirow{2}{*}{2D dist. (mm)} &Ideal 2D Marker Reference &1.3247 &1.8414 &1.3421 &2.1870 &1.3989 &1.2145 \\ \cline{3-9}
& &Practical 2D Marker Reference &1.3300 &1.8671 &1.3101 &2.2328 &1.2607 &1.4742 \\ \cline{2-9}
&\multirow{2}{*}{3D dist. (mm)} &Ideal 2D Marker Reference &1.8196 &1.8120 &2.0238 &2.1100 &2.2377&1.9398 \\ \cline{3-9}
& &Practical 2D Marker Reference &1.8505 &1.8085 &2.0629 &2.1285 &2.1495 &1.8948 \\
\hline
\multirow{4}{*}{Shape Instantiation}&\multirow{2}{*}{Ang. error ($^\circ$)} &Ideal 2D Marker Reference &10.9250 &7.1725 &5.2060 &7.8280 &6.3175&5.1775 \\ \cline{3-9}
& &Practical 2D Marker Reference &11.1625 &5.9375 &5.6240 &8.0560 &7.2200 &5.2250 \\ \cline{2-9}
&\multirow{2}{*}{Mesh dist. (mm)} &Ideal 2D Marker Reference &1.1530 &0.9841 &1.8910 &2.0721 &2.3562 &2.4084 \\ \cline{3-9}
& &Practical 2D Marker Reference &1.1688 &0.9992 &1.8803 &2.0800 &2.3579 &2.4122 \\
\hline
\end{tabular}
\end{table*}
Furthermore, the 3D distance error for 3D marker reference prediction, the 2D projected and 3D distance error for intra-operative 3D marker instantiation, the angular and distance error for 3D shape instantiation for partially-deployed stent segment for each experiment are shown with details in the Tab.~\ref{tab:results}.
For instantiating each stent segment on a computer with a CPU of $Intel^{\textregistered}$ Core(TM) i7-4790 @3.60GHz$\times$8, the computational time is around 7ms using Matlab. The 3D marker reference prediction in Tensorflow on a $Nvidia^{\textregistered}$ Titan Xp GPU costs around 0.8ms for each stent segment. The training of Adapted GCN takes approximately 5 hours. The implemented code was written based on the work of \cite{kipf2016semi}.
\section{Discussion}
\label{sec:discussion}
In this paper, a 3D shape instantiation approach based on a previously deployed framework \cite{zhou2018real_ral} is proposed for partially-deployed stent segment from a single intra-operative 2D fluoroscopic image. It is validated on three commonly used stent grafts with five different AAA phantoms. The mean distance errors of instantiated stent segments are around 1$\sim$3mm and the mean angular errors of instantiated markers are around $5^\circ \sim 11^\circ$.
Without knowing pre-operative 3D marker references, the Adapted GCN is introduced into the previous shape instantiation framework \cite{zhou2018real_ral} and achieves reasonable 3D marker reference prediction (an average 3D distance error of 1.5mm for the fenestrated and thoracic stent graft) from 3D fully-deployed markers. However, the 3D marker reference prediction for the iliac stent graft is insufficient. The diameter of deployment catheter used in the experiments is almost the same as that of the iliac stent graft, resulting in the partially-deployed 3D marker set shape is almost the same as the fully-deployed one. In the cross validation for the iliac stent graft, the Adapted GCN was trained on the fenestrated and thoracic stent graft data for learning partially-deployed deformation. The trained model would not be suitable for predicting 3D marker references for the iliac stent graft which did not experience obvious partially-deployed deformation.
In the training of the Adapted GCN, batch normalization and dropout were also explored, but these two methods decreased the accuracy. One potential reason for the batch normalization's performance is the network for regression tasks is sensitive to the scale of feature value and thus the usage of batch normalization in this task should be different. Future work is essential to confirm the feasebility of batch normalization and dropout in the proposed Adapted GCN.
The errors of 3D marker or shape instantiation with using ideal and practical 2D marker references are very similar in Figure~\ref{fig:pose_error} and Figure~\ref{fig:mesh_error}, implying that the proposed framework is insensitive to the imaging errors caused by the fluoroscopic system. Instantiating partially-deployed stent segment includes mainly three steps: marker segmentation which costs 0.1s on a Nvidia Titan Xp GPU \cite{zhou2018towards_iros}, 3D marker reference prediction which costs 0.8ms, and 3D shape instantiation which costs 7ms. The total computational time is less than 0.11s, which potentially could achieve real-time running as the typical frame rate for clinical usage is around 2 $\sim$ 5 frames per second.
In the future, this paper could be combined with the 3D shape instantiation for fully-deployed \cite{zhou2018real_ral} and fully-compressed \cite{zhou2018towards_iros} stent segment to build a system of real-time 3D shape instantiation for stent grafts at any states. The Equally-weighted Focal U-Net could be retrained and integrated into the instantiation framework for improving the automation.
\section{CONCLUSIONS}
\label{sec:conclusion}
A 3D shape instantiation framework for partially-deployed stent segment was proposed in this paper, including stent segment modelling, 3D marker reference prediction, 3D marker instantiation and 3D shape instantiation. Only a single fluoroscopic image with minimal radiation is required as the intra-operative input. The Adapted GCN is introduced to explore the variation pattern of 3D markers and to provide the 3D marker references for 3D marker instantiation. Compared with the previous relevant work, the proposed framework focuses on dealing with the difficulties of predicting the stent segment shape at the partially-deployed state and achieved a comparable accuracy.
\addtolength{\textheight}{-12cm}
\section*{Acknowledgement}
The authors would like to thank the support of NVIDIA Corporation for the donation of the Titan Xp GPU used for this research.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2019-03-01T02:19:10",
"yymm": "1902",
"arxiv_id": "1902.11089",
"language": "en",
"url": "https://arxiv.org/abs/1902.11089"
}
|
\section{Introduction}
Data commonly come in the form of ranking in preference survey such as voting and consumer surveys.
Asking people to rearrange items according to their preference,
we obtain the collection of rankings.
Several methods for ranked data have been proposed.
\cite{mallows1957non} proposed a parametric model, now called the Mallows model;
\cite{diaconis1989generalization}
developed a spectral analysis for ranked data;
Recently, the analysis of ranked data has gathered much attention in machine learning community (see \cite{liu2011learning,furnkranz2011preference}).
See Section \ref{section: literature}
for more details.
Partially ranked data is often observed
in real data analysis.
This is because one does not necessarily express his or her preference completely;
for example,
according to the election records of
the American Psychological Association collected in 1980,
one-third of ballots provided full preferences for five candidates,
and the rest provided only top-$t$ preferences with $t=1,2,3$
(see Section 2A in \cite{diaconis1989generalization});
Data are commonly of partially ranked in movie ratings
because respondents usually know only a few movie titles
among a vast number of movies.
Therefore,
analyzing partially ranked data efficiently
extends the range of application
of statistical methods for ranked data.
Partially ranked data is thought of as missing data.
We can naturally consider that there exists a latent complete ranking behind a partial ranking as discussed in \cite{lebanon2008non}.
The existing studies for partially ranked data make the Missing-At-Random (MAR) assumption,
that is,
an assumption that the missing mechanism generating partially ranked data is ignorable;
Under the MAR assumption,
\cite{busse2007cluster} and \cite{meilua2010dirichlet}
leverage an extended distance for partially ranked data;
\cite{lu2011learning} introduces a probability model for partially ranked data.
However, an improper application of the MAR assumption may lead to a relatively large estimation error as argued in the literature on missing data analysis (\cite{little2014statistical}).
In the statistical sense,
if the missing mechanism is non-ignorable,
using the MAR assumption
is equivalent to
using a misspecified likelihood function, which causes significantly biased parameter estimation and prediction.
In fact, \cite{marlin2009collaborative}
points out that there occurs a violation of the MAR assumption in music rankings.
This paper addresses learning the distribution of complete and partial rankings based on partially ranked data under a (possibly) non-ignorable missing mechanism.
Our approach includes
estimating a missing mechanism.
However, estimating a missing mechanism has an intrinsic difficulty.
Consider a top-$t$ ranking of $r$ items.
Length $t$ characterizes
the missing pattern generating a top-$t$ ranking from a complete ranking with $r$ items.
It requires $r!(r-2)$ parameters
to fully parameterize the missing mechanism
since $r!$ multinomial distributions with $r-1$ categories
models the missing mechanism.
Note that the number of complete rankings is $r!$.
A large number of parameters cause over-fitting
especially when the sample size is small.
To avoid over-fitting,
we introduce an estimation method
leveraging the recent graph regularization technique (\cite{hallac2015network}) together with the Expectation-Maximization (EM) algorithm.
The numerical experiments using simulation data as well as applications to real data
indicate that our proposed estimation method works well especially under non-ignorable missing mechanisms.
\subsection{Contribution}
In this paper, we propose estimators for the distribution of a latent complete ranking
and for a missing mechanism.
To this end,
we employ both a latent variable model and
a recently developed graph regularization.
Our proposal has two merits:
First, we allow a missing mechanism to be non-ignorable by fully parameterizing it.
Second, we reduce over-fitting due to the complexity of missing mechanisms by exploiting a graph regularization method.
\begin{figure}[h]
\centering
\includegraphics[width=7cm]{concept.pdf}
\caption{
A latent structure behind partially ranked data when the number of items is four:
A ranking is expressed as a list of ranked items.
The number located at the $i$-th position of a list represents the label of the $i$-th preference.
The top layer shows latent complete rankings with a graph structure.
A vertex in the top layer corresponds to a latent complete ranking.
A edge in the top layer is endowed by a distance between complete rankings.
The bottom three layers show partial rankings generated according to missing mechanisms.
An arrow from the top layer to the bottom three layers corresponds to a missing pattern.
A probability on arrows from a complete ranking to the resulting partial rankings corresponds to a missing mechanism.
}
\label{fig:concept}
\end{figure}
Our ideas for the construction of the estimators are two-fold:
First,
we work with a latent structure behind partially ranked data (see Figure \ref{fig:concept}).
This structure consists of the graph representing complete rankings (in the top layer)
and arrows representing missing patterns.
In this structure,
a vertex in the top layer represents a latent complete ranking;
An edge is endowed by a distance between complete rankings;
An arrow from the top layer to the bottom layers represents
a missing pattern;
A multinomial distribution on arrows from a complete ranking
corresponds to a missing mechanism.
Second,
we assume that two missing mechanisms become more similar as the associated complete rankings get closer to each other on the graph (in the top layer).
Together with both the restriction to the probability simplex and the EM algorithm,
these ideas are implemented by the graph regularization method (\cite{hallac2015network}) under the probability restriction.
In addition, we discuss the convergence properties of the proposed method.
The simulation studies as well as applications to real data
demonstrate that
the proposed method improves on the existing methods under non-ignorable missing mechanisms,
and
the performance of the proposed method is comparable to those of the existing methods
under the MAR assumption.
\subsection{Literature review}
\label{section: literature}
Relatively scarce is the literature on the inference for ranking-related data with (non-ignorable) missing data.
\cite{Marlin:2007:CFM:3020488.3020521} points out that the MAR assumption does not hold in the context of the collaborative filtering.
\cite{Marlin:2007:CFM:3020488.3020521} and \cite{marlin2009collaborative} propose two estimators based on missing mechanisms. These estimators show the higher performance both in prediction of rating and in the suggestion of top-$t$ ranked items to users than estimators ignoring a missing mechanism.
Using the Plackett-Luce and the Mallows models,
\cite{fahandar2017statistical} introduces a rank-dependent coarsening model for
pairwise ranking data.
This study is different from these studies in types of ranking-related data:
\cite{marlin2009collaborative} and \cite{Marlin:2007:CFM:3020488.3020521} discuss rating data; \cite{fahandar2017statistical} discusses pairwise ranking data; this study discusses partially ranked data.
Several methods have been proposed
for estimating distributions of partially ranked data
(\cite{beckett1993maximum,busse2007cluster,meilua2010dirichlet,lebanon2008non,jacques2014model,caron2014bayesian}).
These methods regard partially ranked data as missing data.
\cite{beckett1993maximum} discusses imputing items on missing positions of a partial ranking by employing the EM algorithm.
\cite{busse2007cluster} and \cite{meilua2010dirichlet} discuss
the clustering of top-$t$ rankings
by the existing ranking distances for top-$t$ rankings.
\cite{lebanon2008non} proposes a non-parametric model together with a computationally efficient estimation method for partially ranked data.
For the proposal, \cite{lebanon2008non} exploits the algebraic structure of partial rankings and utilizes the Mallows distribution as a smoothing kernel.
\cite{jacques2014model} proposes a clustering algorithm for multivariate partially ranked data.
\cite{caron2014bayesian} discusses Bayesian non-parametric inferences of top-$t$ rankings on the basis of the Plackett-Luce model.
\cite{caron2014bayesian} does not explicitly rely on the framework that regards partially ranked data as the result of missing data; However, the model discussed in \cite{caron2014bayesian} is equivalent to that under the MAR assumption.
Overall, all previous studies rely on the MAR assumption, whereas
our study is the first attempt to estimate the distribution of partially ranked data with a (possibly) non-ignorable missing mechanism.
We work with the graph regularization framework called Network Lasso (\cite{hallac2015network}).
Network Lasso employs the alternating direction method of multipliers (ADMM; see \cite{boyd2011distributed}) to solve a wide range of regularization-related optimization problems of a graph signal that cannot be solved efficiently by generic optimization solvers.
In addition, Network Lasso has a desirable convergence property,
cooperates with distributed processing systems,
and has been applied to various optimization problems on a graph.
We present an application of Network Lasso to missing data analysis.
In the application, we coordinate Network Lasso with the probability simplex constraint and the EM algorithm.
\subsection{Organization}
The rest of this paper is organized as follows.
Section 2 formulates a probabilistic model of partially ranked data based on a missing mechanism, and introduces a distance-based graph structure for a complete ranking.
Section 3 proposes the regularized estimator for both a latent complete ranking and a missing mechanism.
We also discuss the convergence properties of the proposed estimation procedure.
Section 4 demonstrates the result of simulation studies and real data analysis.
Section 5 concludes the paper.
The concrete algorithm of the proposed estimation procedure
and
The proof of the convergence property are
provided in Appendices A and B, respectively.
\section{Preliminaries}
\subsection{Notation}
\label{section: notation}
We begin with introducing notations for analyzing partially ranked data.
In this paper,
we identify a complete ranking of $r$ items $\{1,\ldots,r\}$
with a permutation that maps each item $i\in\{1,\ldots,r\}$ uniquely to a corresponding rank $\{1,\ldots,r\}$.
A top-$t$ ranking is a list of $t$ items out of $r$ items.
We identify a top-$t$ ranking with a permutation that maps an item in a subset of items uniquely to a corresponding rank in $\{1,\ldots,t\}$.
We denote by $S_{r}$ the collection of all complete rankings of $r$ items.
We denote by $\overline{S}_r$ the collection of all top-$t$ rankings with $t$ running through $t=1,\ldots,r-1$.
We denote by $t(\tau)$ the length of a partial ranking $\tau\in\overline{S}_{r}$.
A tuple of a complete ranking $\pi$ and length $t$ uniquely determines a top-$t$ ranking $\tau$:
$\pi^{-1}(i)=\tau^{-1}(i)$ for $i=1,\ldots,t$,
where $\pi^{-1}$ and $\tau^{-1}$ denote the inverses of $\pi$ and $\tau$, respectively.
We define the collection of complete rankings compatible with a given ranking $\tau\in \overline{S}_{r}$ as
\[
[\tau] = \{\pi\in S_r \mid \pi^{-1}(i) = \tau^{-1}(i)\ (i=1,\ldots,t(\tau))\}.
\]
\subsection{Partial rankings and a missing mechanism}
\label{section: missing}
Next we introduce
notations and terminologies for a probabilistic model with a missing mechanism.
A probabilistic model for top-$t$ ranked data with a general missing mechanism consists of
a probabilistic model of generating complete rankings
and that of a missing mechanism.
The joint probability of a complete ranking and a missing mechanism is decomposed as
\[
P(t,\pi)=P(t\mid \pi)P(\pi),
\]
where $P(\pi)$ determines how a complete ranking is filled,
and
$P(t \mid \pi)$ specifies a missing pattern conditioned on the latent complete ranking $\pi$.
Then, the probability of a top-$t$ ranking $\tau\in\overline{S}_{r}$ is obtained by marginalizing a latent complete ranking out:
\[
P(\tau)= \sum_{\pi\in[\tau]}P(t(\tau)\mid \pi)P(\pi).
\]
Now distributions $P(t\mid\pi)$ and $P(\pi)$ are parameterized by $\phi$ and $\theta$,
respectively;
hence $P(\tau;\theta,\phi)=\sum_{\pi\in[\tau]}P(t\mid\pi;\phi)P(\pi;\theta)$.
We call $\{P(\pi;\theta):\theta\in\Theta\}$ with a parameter space $\Theta$ a complete ranking model
and $\{P(t\mid\pi;\phi):\phi\in\Phi\}$
with a parameter space $\Phi$ a missing model.
We call $\theta$ a complete ranking parameter,
and call $\phi$ a missing parameter,
respectively.
Given the i.i.d.~observations $\tau_{(n)}=\{\tau_{1},\ldots,\tau_{n} \in \overline{S}_{r}\}$, we denote the negative log-likelihood function by
\begin{align}
L(\theta,\phi;\tau_{(n)}) := -\sum_{i=1}^n \log\left[\sum_{\pi\in[\tau_i]}P(t(\tau_i)\mid \pi;\phi)P(\pi;\theta)\right]. \label{likelihood}
\end{align}
\subsection{Distances, graph structures, and distributions on complete rankings}
\label{section: distance}
We end this section with introducing distances, graph structures, and distributions on complete rankings.
We endow the class $S_{r}$ of complete rankings with a distance structure as follows.
Since we identify the class $S_{r}$ as a the class of permutations,
we endow $S_{r}$ with the symmetric group structure and its group law $\circ$.
Using this identification, we leverage distances on symmetric groups for distances on $S_{r}$.
There exist a large number of distances on $S_{r}$ such as the Kendall distance,
the Spearman rank correlation metric, and the Hamming distance.
Among these, the Kendall distance (\cite{kendall1938new}) has been often used in statistics and machine learning.
The Kendall distance between two complete rankings $\pi_1$ and $\pi_2$, $d(\pi_{1},\pi_{2})$, is defined as
\[
d(\pi_{1},\pi_{2}):= \min_{\tilde{d}}\{\tilde{d}: \pi_{2}=a_{\tilde{d}}\circ\ldots\circ a_{1}\circ \pi_{1},\ a_{1},\ldots,a_{\tilde{d}}\in\mathcal{A}\},
\]
where $\mathcal{A}$ is the whole class of adjacent transpositions.
This distance is suitable for describing similarity between preferences because
the transform of a complete ranking by a single adjacent transposition is just the exchange of the $i$-th and $(i+1)$-th preferences.
In this paper, we focus on the Kendall distance $d$ as a distance on $S_{r}$.
There exists a one-to-one correspondence between the distance structure with the Kendall distance and a graph structure.
Set the vertex set $V=S_r$ and set the edge set $E=\{\{\pi,\pi'\}:\text{ there exists } b\in\mathcal{A} \text{ such that }\pi=b\circ \pi'\}$.
Then the distance structure $(S_{r},d)$ corresponds one-to-one to the graph $G=(V,E)$,
since the Kendall distance $d(\pi,\pi')$ is its minimum path length between the vertices corresponding to $\pi$,$\pi'$.
Remark that $E$ is rewritten as $E=\{\{\pi,\pi'\}: d(\pi,\pi')=1 \}$.
We introduce a well-known probabilistic model for complete rankings.
The Mallows model (\cite{mallows1957non}) is one of the most popular probabilistic models for complete rankings.
The Mallows model associated with the Kendall distance is defined as
\[
\left\{P(\pi;\sigma,c) = \frac{\exp\{-c d(\pi,\sigma)\}}{Z(c)}: c>0,\sigma\in S_{r}\right\},
\]
where $\sigma$ is a location parameter indicating a representative ranking, $c$ is a concentration parameter indicating a decay rate, and $Z(c)=\sum_{\pi\in S_r}\exp\{-c d(\pi,\sigma)\}$ is a normalizing constant that depends only on $c$.
The mixture model of $K\in\mathbb{N}$ Mallows distributions is defined as
\begin{align}
\left\{P(\pi;\bm{c},\bm{\sigma},\bm{w}) = \sum_{k=1}^K w_{k} \frac{\exp\{-c_k d(\pi,\sigma_k)\}}{Z(c_{k})}: c_{k}>0,\ \sigma_{k}\in S_{r},\ w_{k}> 0,\ \sum_{k=1}^{K}w_{k}=1\right\} \label{mallows_mixture}
\end{align}
where $\bm{c} = \{c_k\}_k,\bm{\sigma} = \{\sigma_k\}_k,\bm{w} = \{w_k\}_k$ represent the parameters of each mixture component.
The Mallows mixture model has been used for estimation and clustering analysis of ranked data (\cite{murphy2003mixtures,busse2007cluster}).
\section{Proposed Method}
\label{section: method}
In this section, we propose estimators for both complete ranking and missing models together with a simple estimation procedure.
Here we assume that the parameterization of the complete ranking and missing models is separable, that is, $\theta$ and $\phi$ are distinct.
We use the following missing model $\{P(t\mid \pi; \phi):\phi\in\Phi\}$ to allow a non-ignorable missing mechanism:
\begin{align*}
P(t\mid \pi; \phi) &= \phi_{\pi, t} \text{ for } t\in\{1,\ldots,r-1\} \text{ and } \pi\in S_{r},\\
\Phi&=\left\{\phi \in \mathbb{R}^{r!(r-1)}: \sum_{t=1}^{r-1}\phi_{\pi, t} = 1, \phi_{\pi,t}\geq 0, \pi \in S_{r}\right\}.
\end{align*}
We make no assumptions on a complete ranking model $\{P(\pi; \theta):\theta\in\Theta\}$.
\subsection{Estimators and estimation procedure}
\label{section: procedure}
We propose the following estimators for $\theta$ and $\phi$:
On the basis of i.i.d.~observations $\tau_{(n)}=\{\tau_{1},\ldots,\tau_{n} \in \overline{S}_{r}\}$,
\[
(\hat{\theta}(\tau_{(n)}),\hat{\phi}(\tau_{(n)}))=\mathop{\mathrm{argmin}}_{\theta\in\Theta, \phi\in\Phi} L_{\lambda}(\theta,\phi;\tau_{(n)}).
\]
Here $L_{\lambda}$ with a regularization parameter $\lambda>0$ is defined as
\[
L_{\lambda}(\theta,\phi;\tau_{(n)})= L(\theta,\phi;\tau_{(n)}) +\lambda\sum_{\{\pi,\pi'\}\in E}\|\phi_{\pi}-\phi_{\pi'}\|_2^2,
\]
where
$\phi_{\pi}$ with $\pi\in S_{r}$ denotes the vector $(\phi_{\pi, 0},\ldots,\phi_{\pi, (r-1)})$,
and recall that
$L(\theta,\phi;\tau_{(n)})$ is the negative log-likelihood function (\ref{likelihood})
and
$E$ is the edge set of the graph induced by the Kendall distance;
see Subsections \ref{section: missing}-\ref{section: distance}.
We conduct minimization in the definition of $\hat{\theta},\hat{\phi}$ using the following EM algorithm:
At the $(m+1)$-th step, set
\begin{align}
\theta^{m+1}&=\mathop{\mathrm{argmin}}_{\theta'\in\Theta} L(\theta';\tau_{(n)},q^{m+1}_{(n)}), \label{opttheta}\\
\phi^{m+1}&=\mathop{\mathrm{argmin}}_{\phi' \in\Phi} L_{\lambda}(\phi';\tau_{(n)},q_{(n)}^{m+1}), \label{optphi}
\end{align}
where for $i\in\{1,\ldots,n\}$,
\begin{align}
q_{i,\pi}^{m+1}&=
\begin{cases}
\phi^{m}_{\pi,t(\tau_{i})}P(\pi;\theta^{m})
\bigg{/}\sum_{\pi'\in[\tau_i]}\phi^{m}_{\pi',t(\tau_{i})}P(\pi';\theta^{m}) & (\pi\in[\tau_i]), \\
0 & (\pi\not\in[\tau_i]),
\end{cases}\label{optq}\\
q_{(n)}^{m+1}&:=\{q_{i,\pi}^{m+1}: i=1,\ldots,n , \pi\in S_{r} \},\\
L(\theta;\tau_{(n)},q^{m+1}_{(n)})&:=-\sum_{i=1}^n\sum_{\pi\in[\tau_i]}q^{m+1}_{i,\pi}\log P(\pi;\theta),\\
L_{\lambda}(\phi;\tau_{(n)},q^{m+1}_{(n)})&:=-\sum_{i=1}^n\sum_{\pi\in[\tau_i]}q^{m+1}_{i,\pi}\log\phi_{\pi', t(\tau_i)} +\lambda \sum_{\{\pi,\pi'\}\in E}\|\phi_{\pi}-\phi_{\pi'}\|_2^2. \label{likelihood_phi}
\end{align}
Consider minimizations (\ref{opttheta}) and (\ref{optphi}).
Minimization (\ref{opttheta}) depends on the form of a complete ranking model $P(\pi;\theta)$;
For example,
consider the Mallows model with $\theta=(\sigma,c)$ (see Section \ref{section: distance}).
In this case, we write down the minimization of $\theta$ at the $(m+1)$-th step as follows:
\begin{align*}
\sigma^{m+1}&=\mathop{\mathop{\mathrm{argmin}}}_{\tilde\sigma\in S_r}\sum_{i=1}^n\sum_{\pi\in S_r}q_{i,\pi}^{m+1} d(\pi,\tilde\sigma),\\
c^{m+1}&=\mathop{\mathrm{argmin}}_{\tilde{c}>0}\sum_{i=1}^n\sum_{\pi\in S_r}q_{i,\pi}^{m+1}\{\tilde{c}d(\pi,\sigma^{m+1}) + \log(Z(\tilde{c}))\}.
\end{align*}
See \cite{busse2007cluster} for more details.
Minimization (\ref{optphi}) in the $(m+1)$-th step is conducted using the following iteration:
At the $(l+1)$-th step, set
\begin{align}
\phi^{l+1}&=\mathop{\mathrm{argmin}}_{\tilde{\phi}\in\Phi }L_{\rho}(\tilde{\phi},\varphi^{l},u^{l}; q^{m+1}_{(n)}),\label{optvertex}\\
\varphi^{l+1}&=\mathop{\mathrm{argmin}}_{\tilde{\varphi}\in\mathbb{R}^{r!(r-1)^2}}L_{\rho}(\phi^{l+1},\tilde{\varphi},u^{l}; q^{m+1}_{(n)}),\label{optvarphi}\\
u^{l+1}&=u^{l}+(\phi^{l+1}-\varphi^{l+1}). \label{optu}
\end{align}
Here, $\varphi\in \mathbb{R}^{r!(r-1)^2}$ is the copy variable of $\phi$, $u\in \mathbb{R}^{r!(r-1)^2}$ is the dual variable, and $L_\rho$ is an augmented Lagrangian function with a penalty constant $\rho$ defined as
\begin{align}
L_{\rho}(\phi,\varphi,u;q^{m+1}_{(n)})&=-\sum_{\pi\in V}\sum_{t}[q^{m+1}_{\pi, t}\log \phi_{\pi, t}] \nonumber\\
&\quad+ \sum_{\{\pi,\pi'\}\in E}\left\{\lambda\|\varphi_{\pi, \pi'}-\varphi_{\pi',\pi}\|^2_2-\frac{\rho}{2}(\|u_{\pi, \pi'}\|_2^2+\|u_{\pi',\pi}\|_2^2)\right.\nonumber\\
&\quad\quad\quad\quad\quad\left.+\frac{\rho}{2}(\|\phi_{\pi}-\varphi_{\pi, \pi'}+u_{\pi, \pi'}\|_2^2+\|\phi_{\pi'}-\varphi_{\pi',\pi}+u_{\pi',\pi}\|_2^2)\right\},\label{lagrangian}
\end{align}
where
\begin{align}
q^{m+1}_{\pi, t}:=\sum_{i:t(\tau_i)=t}q_{i,\pi}^{m+1}, \label{qpit}
\end{align}
for all $\pi\in S_{r}$ and $t=1,\ldots,r-1$.
Note that $q^{m+1}_{\pi,t}=0$ when $\{i:t(\tau_i)=t\}$ is an empty set.
The detailed algorithm is provided in Appendix \ref{section: algorithm}.
\begin{remark}[Meaning of the penalty term]
We make the assumption that two complete rankings close to each other in the Kendall distance have smoothly related missing probabilities.
This assumption leads to adding a ridge penalty
\[
\sum_{\substack{\pi,\pi'\in S_r\\d(\pi,\pi')=1}}\|\phi_{\pi}-\phi_{\pi'}\|_2^2
=\sum_{\{\pi,\pi'\}\in E}\|\phi_{\pi}-\phi_{\pi'}\|_{2}^{2}
\]
to the negative log-likelihood function.
This assumption is reasonable because the Kendall distance measures the similarity of preferences expressed by two rankings.
\end{remark}
\begin{remark}[The proposed method under the MAR assumption]
For top-$t$ ranked data, the MAR assumption is expressed as
\[
P(t(\tau)\mid\pi;\phi) = P(t(\tau)\mid\pi';\phi),\ (\pi,\pi' \in [\tau]).
\]
Then under the MAR assumption, the negative log-likelihood function $L(\theta,\phi;\tau_{(n)})$ is decomposed as
\begin{align*}
L(\theta,\phi;\tau_{(n)})&=-\sum_{i=1}^n\log P(\tau_i\mid\pi\in[\tau_i];\phi)-\sum_{i=1}^n\log \left[\sum_{\pi\in[\tau_i]}P(\pi;\theta)\right]\nonumber\\
&=L(\phi;\tau_{(n)})+L(\theta;\tau_{(n)}),
\end{align*}
which indicates that the parameter estimation for $\phi$ is unnecessary for estimating $\theta$.
\end{remark}
\subsection{Convergence}
In this subsection,
we discuss theoretical guarantees for two procedures (\ref{opttheta})-(\ref{likelihood_phi}) and (\ref{optvertex})-(\ref{optu}).
It is guaranteed that
the sequence $\{L_{\lambda}(\theta^m,\phi^m;\tau_{(n)}):m=1,2,\ldots \}$ obtained using the procedure (\ref{opttheta})-(\ref{likelihood_phi}) monotonically decreases,
\[L_{\lambda}(\theta^{m},\phi^{m};\tau^{(n)}) \geq L_{\lambda}(\theta^{m+1},\phi^{m+1};\tau^{(n)}), \ m=1,2,\ldots,\]
because the procedure is just the EM algorithm.
Introduce a latent assignment variable $z_{(n)}=\{z_i\}_i$ ($z_i\in \mathbb{R}^{|S_r|}$ for every $i = 1,\ldots,n$. $z_{i\pi}=1$ if $\tau_i$ is the missing from $\pi$ and $z_{i\pi}=0$ otherwise).
Using $z_{(n)}$, we decompose the
likelihood function as follows:
\begin{align*}
&L_\lambda(\theta,\phi;\tau_{(n)},z_{(n)}) \\
&= -\sum_{i=1}^n\log\left[\prod_{\pi\in[\tau_i]}\{\phi_{\pi, t(\tau_i)}P(\pi;\theta)\}^{z_{i,\pi}}\right] +\lambda\sum_{\{\pi,\pi'\}\in E}\|\phi_{\pi}-\phi_{\pi'}\|_2^2.\\
&=\left\{-\sum_{i=1}^n\sum_{\pi\in[\tau_i]}z_{i,\pi}\log P(\pi;\theta)\right\}+\left\{-\sum_{i=1}^n\sum_{\pi\in[\tau_i]}z_{i,\pi}\log\phi_{\pi, t(\tau_i)} +\lambda\sum_{\{\pi,\pi'\}\in E}\|\phi_{\pi}-\phi_{\pi'}\|_2^2\right\}\\
&=L(\theta;\tau_{(n)},z_{(n)})+L_{\lambda}(\phi;\tau_{(n)},z_{(n)}).
\end{align*}
On the basis of the decomposition,
the standard procedure of the EM algorithm yields the iterative algorithm shown in (\ref{opttheta})-(\ref{likelihood_phi}).
Note that it depends on a complete ranking model $P(\pi;\theta)$
whether the convergent point of the sequence is a local maximum of $L(\theta,\phi;\tau_{(n)})$;
see Section 3 of \cite{mclachlan2007algorithm}.
Next, it is guaranteed that the sequence $\phi^{l+1}$ obtained using the procedure (\ref{optvertex})-(\ref{optu}) converges to the global minimum of $L_{\lambda} (\phi ; \tau_{(n)} , q_{(n)}^{m} )$ in the sense of $L_{\lambda}(\phi;\tau_{(n)},q^{m}_{(n)})$.
\begin{proposition}
\label{prop: convergence_admm}
The sequence $\{L_{\lambda}(\phi^l,\tau_{(n)},q^{m+1}_{(n)})\}_{l=1}^{\infty}$ converges to $\min_{\phi}L_{\lambda}(\phi,\tau_{(n)},q^{m+1}_{(n)})$.
\end{proposition}
The proof is provided in Appendix \ref{section: proof}.
The basis of the proof is reformulating the optimization problem (\ref{optphi}) as an instance of the alternating direction method of multipliers (ADMM; \cite{boyd2011distributed}:
We rewrite the problem (\ref{optphi}) as follows:
\begin{eqnarray}
\phi=&\mathop{\mathrm{argmin}}_{\phi'}&-\sum_{\pi\in V}\sum_{t}[q_{\pi, t}\log \phi'_{\pi, t}] +\lambda \sum_{\{\pi,\pi'\}\in E}\|\phi_{\pi}-\phi_{\pi'}\|_2^2\nonumber\\
&&\mathrm{s.t.}\ \sum_t \phi_{\pi, t} = 1 \ (\forall \pi\in V) \label{RFopt}
\end{eqnarray}
where $V$ is the vertex set of the graph defined in Section \ref{section: distance} and $q_{\pi, t}=\sum_{i:t(\tau_i)=t}\sum_{k=1}^Kq_{i,k,\pi}$.
Introducing a copy variable $\varphi$ on the edge set,
we recast the optimization problem (\ref{RFopt}) into an equivalent form:
\begin{eqnarray}
\phi,\varphi=&\mathop{\mathrm{argmin}}_{\phi',\varphi'}&-\sum_{\pi\in V}\sum_{t}[q_{\pi, t}\log \phi'_{\pi, t}] + \lambda\sum_{\{\pi,\pi'\}\in E}\|\varphi'_{\pi, \pi'}-\varphi'_{\pi',\pi}\|^2_2\label{optphidecomp}\\
&\mathrm{s.t.}&\phi'_{\pi}=\varphi'_{\pi, \pi'} \ (\forall \{\pi,\pi'\}\in E),\nonumber\\
&& \sum_t \phi'_{\pi, t} = 1 \ (\forall \pi\in V).\nonumber
\end{eqnarray}
Note that this reformulation follows the idea of \cite{hallac2015network}.
We employ ADMM to solve
the optimization of the sum of objective functions of splitted variables under linear constraints.
\section{Numerical experiments}
In this section,
we apply our methods to both simulation studies and real data analysis.
In simulation studies, we use the Mallows mixture models (\ref{mallows_mixture}) with two types of missing models.
In the real data analysis, we use the election records of the American Psychological Association collected in 1980.
\subsection{Performance measures}
We evaluate the performance of several estimators for $\theta$ and $\phi$ in estimating distributions of a latent complete ranking and of a partial ranking.
We measure the performance using the following total variation losses:
When the true values of a complete ranking and missing parameters are
$\theta$ and $\phi$, respectively,
the losses of estimators $\hat{\theta}$ and $\hat{\phi}$ are given as
\begin{align}
L_{\mathrm{par}}
&=L_{\mathrm{par}}(\theta,\phi;\hat{\theta},\hat{\phi})
=\sum_{\tau\in \overline{S}_{r}}
|P(\tau;\theta,\phi)-P(\tau;\hat{\theta},\hat{\phi})| \label{Lpar}\\
L_{\mathrm{comp}}
&=L_{\mathrm{comp}}(\theta,\hat{\theta})
=\sum_{\pi\in S_{r}}
|P(\pi;\theta)-P(\pi;\hat{\theta})|.\label{Lcomp}
\end{align}
Losses $L_{\mathrm{par}}$ and $L_{\mathrm{comp}}$ measure the estimation losses for partial and complete ranking distributions, respectively.
\subsection{Method comparison}
\label{section: comparison methods}
We compare our estimators with the estimator based on the maximum entropy approach proposed by \cite{busse2007cluster} and a non-regularized estimator, abbreviated by ME and NR, respectively.
In addition,
we use the proposed estimator with the regularization parameter selected using two-fold cross-validation based on $L_{\mathrm{par}}$.
We denote the proposed method introduced in section \ref{section: method} with the value of regularization parameter $\lambda$ as R$\lambda$ and that with the regularization parameter selected using cross-validation as RCV.
The maximum entropy approach (ME) uses an extended distance between top-$t$ rankings to introduce an exponential family distribution of a top-$t$ ranking.
From the viewpoint of missing data analysis, ME implicitly assumes the MAR assumption.
For this reason,
in the maximum entropy approach,
we estimate the missing model parameter $\phi$ by assuming homogeneous missing probabilities $P(t\mid \pi) = \phi_t\ (\forall \pi\in S_r)$ and using the maximum likelihood in the evaluation of the loss $L_{\mathrm{par}}(\theta,\phi)$.
The non-regularized estimator (NR) is the minimizer of the non-regularized likelihood function $L(\theta,\phi; \tau_{(n)})$.
The estimation based on the non-regularized likelihood function can be implemented straightforwardly.
\subsection{Stopping criteria}
In simulation studies,
we use the following stopping criteria and hyperparameters.
We terminate the iteration of the EM algorithm
when
the change of the likelihood of the observable distribution gets lower than $\epsilon = 1$.
We terminate the iteration of ADMM when both the primal and dual residuals got less than $\epsilon_p=\epsilon_d=1$
or when the number of iterations exceeded 100.
In addition,
to prevent being trapped in local minima due to the EM algorithm,
we use the following devices.
First, we use 10 different initial location parameters in the EM algorithm.
Second,
we make the value of the location parameter transit from the current to a different one in the first five iterations of the EM algorithm.
\subsection{Simulation studies}
\label{section: synthetic}
We conducted two simulation studies.
The data-generating models are as follows:
For complete ranking models,
we use the Mallows and Mallows mixture models.
For missing models,
we use a binary missing mechanism in which the possible missing patterns are only that no items are missing or that all but the first items are missing.
We parameterize missing models in such a way that there is a discrepancy between the distribution of a complete ranking generated by the latent Mallows model and the marginal distribution of a partial ranking restricted to $S^{(r-1)}_{r}:=\{\tau:t(\tau)=r-1,\tau\in \overline{S}_{r}\}$.
Note that $S^{(r-1)}_{r}$ is identical to $S_{r}$ as a set.
In each simulation,
we generate 100 datasets with sample size of $n=1000$.
We set the number of items to $r=5$.
\subsubsection{Tilting the concentration parameter}
\label{section: tilt concentration}
\begin{figure}[htbp]
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=45mm]{synG_10P8.pdf}
\end{center}
\subcaption{$c^{\ast}=0.8$}
\label{p8}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=45mm]{synG_10P10.pdf}
\end{center}
\subcaption{$c^{\ast}=1.0$}
\label{p10}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=45mm]{synG_10P12.pdf}
\end{center}
\subcaption{$c^{\ast}=1.2$}
\label{p12}
\end{minipage}
\caption{Boxplots of $L_{\mathrm{par}}$ in (\ref{Lpar}) for the dataset tilting the concentration parameter: The compared methods are the maximum entropy approach (ME), the non-regularized estimator (NR), the proposed methods (R1; R10; R100), and its version with cross-validation (RCV).
The results with different values of the concentration parameter $c^{\ast}$ are shown in (A)-(C).}
\label{synGP}
\end{figure}
\begin{figure}[htbp]
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synG_10C8.pdf}
\end{center}
\subcaption{$c^{\ast}=0.8$}
\label{fig:one}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synG_10C10.pdf}
\end{center}
\subcaption{$c^{\ast}=1.0$}
\label{fig:one}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synG_10C12.pdf}
\end{center}
\subcaption{$c^{\ast}=1.2$}
\label{fig:one}
\end{minipage}
\caption{Boxplots of $L_{\mathrm{comp}}$ in (\ref{Lcomp}) for the dataset tilting the concentration parameter: The compared methods are the maximum entropy approach (ME), the non-regularized estimator (NR), the proposed methods (R1; R10; R100), and its version with cross-validation (RCV).
The results with different values of the concentration parameter $c^{\ast}$ are shown in (A)-(C).}
\label{synGC}
\end{figure}
In the first simulation study,
we use the Mallows model and the missing model that tilts the concentration parameter $c$:
The missing model is parameterized by $c^{\ast}>0$ and $R\in [0,1]$ as
\[
\phi_{\pi} = ( \phi_{\pi,0},\ldots,\phi_{\pi,(r-1)})
= (1-C_{\pi}(c^{\ast},R),0,\ldots,0,C_{\pi}(c^{\ast},R)), \ \pi\in S_{r}
\]
where
\[
C_{\pi}(c^{\ast},R) =\min\left\{1,\frac{Z(c)}{Z(c^{\ast})}R\exp \{-(c^{\ast}-c) d(\pi,\sigma_0)\}\right\}.
\]
In this parameterization,
the parameter
$c^{\ast}$ specifies the degree of concentration of
the marginal distribution $P(\tau;\theta,\phi)$
of a partial ranking restricted to $S^{(r-1)}_{r}$:
If
$\{Z(c)/Z(c^{\ast})\}R\exp \{-(c^{\ast}-c) d(\pi,\sigma_0)\}\leq 1$,
$P(\tau;\theta,\phi)$ has the form of the Mallows distribution with the concentration parameter $c^{\ast}$:
\begin{align*}
P(\tau;\theta,\phi) &= \sum_{\pi'\in[\tau]}P(t=r\mid \pi';\phi)P(\pi';\theta)\\
&=C_{\pi}(c^{\ast},R)\frac{1}{Z(c)}\exp\{-c d(\pi,\sigma)\}\\
&= \frac{R}{Z(c^{\ast})}\exp\{-c^{\ast}d(\pi,\sigma_0)\},
\end{align*}
where $\pi(i)=\tau(i), i=1,\ldots,r-1$.
The parameter $0\leq R\leq 1$ specifies the proportion of partial rankings in $S^{(r-1)}_r$.
We set $c = 1$, $R=0.7$, and $c^{\ast} \in \{0.8,1,1.2\}$.
Figures \ref{synGP} and \ref{synGC} show the results.
When $c^{\ast}\neq 1$, the proposed methods outperform ME both in $L_{\mathrm{par}}$ and $L_{\mathrm{comp}}$.
When $c^{\ast}=1$, the proposed methods underperform compared to ME.
These results reflect that the setting with $c^{\ast}=1$ satisfies the MAR assumption, whereas the settings with $c^{\ast}\neq 1$ do not satisfy the MAR assumption.
For $L_{\mathrm{par}}$, the proposed methods outperform NR regardless of the values of $c^{\ast}$.
However, there are subtle distinctions in the values of $L_{\mathrm{comp}}$ of these methods.
The performance of the proposed method with the cross-validated regularization parameter (RCV) is comparable with that of the proposed method with the optimal regularization parameter both for $L_{\mathrm{par}}$ and $L_{\mathrm{comp}}$, indicating the utility of cross-validation.
\subsubsection{Tilting the mixture coefficient}
\label{section: tilt mixture}
In the second simulation study, we use the Mallows mixture model with two clusters and the missing model that tilts the mixture coefficient $w$.
We instantiate a missing model, in which missing probabilities depend on the cluster assignment $k\in\{1,\ldots,K\}$, such that $P(t\mid \pi,z_k=1)=P(t\mid z_k=1)=\phi_{k,t}$, where $z_k=1$ if and only if the assigned cluster is $k$ and $z_k=0$ otherwise.
Then,
the missing model is parameterized by $w^{\ast}\in [0,1]$ and $R\in [0,1]$ as
\[
\phi = \left(
\begin{array}{ccccc}
\phi_{1,1} & \ldots \phi_{1,(r-1)}\\
\phi_{2,1} & \ldots \phi_{2,(r-1)}
\end{array}
\right)=\left(
\begin{array}{ccccc}
1-C_{1}(w^{\ast},R) & 0 &\ldots & 0 & C_{1}(w^{\ast},R)\\
1-C_{2}(w^{\ast},R) & 0 &\ldots& 0 & C_{2}(w^{\ast},R)
\end{array}
\right),
\]
where
$C_{k}(w^{\ast},R) = (w^{\ast}_k / w_k)R.$
In this parameterization, the parameter $w^{\ast}$ determines the mixture coefficient of the marginal distribution $P(\tau;\theta,\phi)$ of a partial ranking restricted to $S^{(r-1)}_r$.
We set the parameter values as follows:
\begin{itemize}
\item $\bm\sigma=((1,2,3,4,5), (3,2,5,4,1))$, $\bm{c} = (1,1)$, and $w = (0.5,0.5)$;
\item $R=0.7$ and $w^{\ast}=\{(0.5,0.5),(0.6,0.4),(0.7,0.3)\}$.
\end{itemize}
In this simulation study,
we additionally use the classification error as the performance measure.
\begin{figure}[htbp]
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synC2_10P5.pdf}
\end{center}
\subcaption{$w^{\ast}_1=0.5$}
\label{fig:one}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synC2_10P6.pdf}
\end{center}
\subcaption{$w^{\ast}_1=0.6$}
\label{fig:one}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synC2_10P7.pdf}
\end{center}
\subcaption{$w^{\ast}_1=0.7$}
\label{fig:one}
\end{minipage}
\caption{Boxplots of $L_{\mathrm{par}}$ in (\ref{Lpar}) for the dataset tilting the mixture coefficient: The compared methods are the maximum entropy approach (ME), the non-regularized estimator (NR), the proposed methods (R1; R10; R100) and its version with cross-validation (RCV). The results with different values of the concentration parameter $w^{\ast}$ are shown in (A)-(C).}
\label{synCP}
\end{figure}
\begin{figure}[htbp]
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synC2_10C5.pdf}
\end{center}
\subcaption{$w^{\ast}_1=0.5$}
\label{fig:one}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synC2_10C6.pdf}
\end{center}
\subcaption{$w^{\ast}_1=0.6$}
\label{fig:one}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synC2_10C7.pdf}
\end{center}
\subcaption{$w^{\ast}_1=0.7$}
\label{fig:one}
\end{minipage}
\caption{Boxplots of $L_{\mathrm{comp}}$ in (\ref{Lcomp}) for the dataset tilting the mixture coefficient: The compared methods are the maximum entropy approach (ME), the non-regularized estimator (NR), the proposed methods (R1; R10; R100) and its version with the cross-validation (RCV). The results with different values of the concentration parameter $w^{\ast}$ are shown in (A)-(C).}
\label{synCC}
\end{figure}
\begin{figure}[H]
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synC2_10D5.pdf}
\end{center}
\subcaption{$w^{\ast}_1=0.5$}
\label{fig:one}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synC2_10D6.pdf}
\end{center}
\subcaption{$w^{\ast}_1=0.6$}
\label{fig:one}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{synC2_10D7.pdf}
\end{center}
\subcaption{$w^{\ast}_1=0.7$}
\label{fig:one}
\end{minipage}
\caption{Boxplots of the classification errors for the dataset tilting the mixture coefficient: The compared methods are the maximum entropy approach (ME), the non-regularized estimator (NR), the proposed methods (R1; R10; R100) and its version with the cross-validation (RCV). The results with different values of the concentration parameter $w^{\ast}$ are shown in (A)-(C).}
\label{synCD}
\end{figure}
Figures \ref{synCP}--\ref{synCD} show the results.
The proposed methods outperform ME when $w_1^{\ast}\neq 0.5$ in comparing $L_{\mathrm{par}}$;
when $w_1^{\ast}$ is $0.7$ in comparing $L_{\mathrm{comp}}$;
when $w_1^{\ast}\neq 0.5$ in comparing the classification error.
The proposed methods outperform NR both in comparing $L_{\mathrm{par}}$ and $L_{\mathrm{comp}}$ except when $w_1^{\ast}$ is $0.7$ for $L_{\mathrm{comp}}$.
As $w^{\ast}$ deviates from $0.5$, the classification error of ME increases.
On the other hand, the classification errors of the other methods decrease.
\subsection{Application to real data}
We apply the proposed method to real data.
We use the election records for five candidates collected by the American Psychological Association. Among the 15549 vote casts, only 5141 filled all candidates ($t=5,4$); 2108 filled $t=3$; 2462 filled $t=2$; and the rest filled only $t=1$.
\begin{figure}[H]
\centering
\includegraphics[width=50mm]{APA_CV_point.pdf}
\caption{Sample size dependency of the mean of $L_{\mathrm{par}}$ in (\ref{Lpar}) for the American Psychological Association dataset: The vertical axis represents the mean value of $L_{\mathrm{par}}$; and the horizontal axis represents the sample size $n$ on the log-scale. The results with $n=100, 500, 1000, 5000, 10000$ are shown. The compared methods are the maximum entropy approach (ME), the non-regularized estimator (NR), the proposed method with cross-validation (RCV).}
\label{apa2}
\end{figure}
\begin{figure}[H]
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{APA_CV_232.pdf}
\end{center}
\subcaption{$n=100$}
\label{fig:one}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{APA_CV_233.pdf}
\end{center}
\subcaption{$n=1000$}
\label{fig:one}
\end{minipage}
\begin{minipage}{0.32\hsize}
\begin{center}
\includegraphics[width=50mm]{APA_CV_234.pdf}
\end{center}
\subcaption{$n=10000$}
\label{fig:one}
\end{minipage}
\caption{Boxplots of $L_{\mathrm{par}}$ in (\ref{Lpar}) for the American Psychological Association dataset: The compared methods are the maximum entropy approach (ME), the non-regularized estimator (NR), the proposed method with cross-validation (RCV). The results with a different sample size $n$ of the train dataset are shown in (A)-(C).}
\label{apa}
\end{figure}
For comparison, we chose several pairs of train and test datasets randomly
to measure $L_{\mathrm{par}}$
since we do not have the true values of the model parameters nor the form of the model.
To see the dependence of the estimation performance on the sample size,
we used different sizes ($n=100, 500, 1000, 5000, 10000$) of the train datasets,
whereas we fixed the size of the test datasets to $n=3000$.
We sampled test datasets independently $30$ times
and sampled train datasets from the remaining data independently $30$ times
for each size.
In calculating $L_{\mathrm{par}}$,
we used the empirical distribution of the employed test dataset as the true distribution.
For a complete ranking model,
we made the use of the likelihood of the Mallows mixture model with the number of clusters set to 3 as in \cite{busse2007cluster}.
Since R1 performs poorly in terms of $L_{\mathrm{par}}$ according to the simulation study,
we eliminate R1 from the candidate of two-fold cross-validation.
Figures \ref{apa2} and \ref{apa}
show the result.
When the sample size is small ($n=100, 500$),
the proposed method is comparable to ME,
and
NR works poorly.
When the sample size is moderate ($n=1000$),
the proposed method outperforms
both ME and NR.
When the sample size is large ($n=5000, 10000$),
the proposed method outperforms ME,
and
it is comparable to NR.
These results indicate that
considering non-ignorable missing mechanisms
contributes to the improvement of the performance
when the sample size is sufficient,
while the graph regularization reduces over-fitting
when the sample size is insufficient.
\section{Conclusion}
We proposed a regularization method for partially ranked data to prevent modeling bias due to the MAR assumption and avoid over-fitting due to the complexity of missing models.
Our simulation experiments showed that the proposed method improves on the maximum entropy approach (\cite{busse2007cluster})
under non-ignorable missing mechanisms.
They also showed that the proposed method improves on the non-regularized estimator especially in estimating distribution of a partial ranking.
Our real data analysis suggested that
moderate or large sample sizes attribute the improvement by the proposed method
and
the proposed method is effective in reducing over-fitting.
We propose two main tasks for future work.
The first task is to improve the computational efficiency of our method since it was not a priority in this study.
Leveraging partial completion of items (instead of full completion)
might be effective for reducing the computational cost.
For this purpose, the distance of top-$t$ ranking described in \cite{busse2007cluster} might be beneficial for the construction of the graph.
The second task is to develop cross-validation or an information criterion
for inferring the distribution of a latent complete ranking.
In this study,
we employed cross-validation based on the distribution of a partial ranking.
When the distribution of a latent complete ranking is of interest, cross-validation based on the distribution of a latent complete ranking would be more suitable.
However, the construction of such cross-validation would be difficult because the empirical distribution of a latent complete ranking cannot be obtained directly,
which rises ubiquitously where one uses the EM algorithm for the estimation of latent variables.
There have been several derivations of information criteria comprising the distribution of latent variables (\cite{shimodaira1994new,cavanaugh1998akaike}).
We conjecture that these derivations would be useful for inferring partially ranked data.
|
{
"timestamp": "2019-03-01T02:12:49",
"yymm": "1902",
"arxiv_id": "1902.10963",
"language": "en",
"url": "https://arxiv.org/abs/1902.10963"
}
|
\section{Summary \label{S5}}
We present the determination of the
interstellar magnesium abundance as derived from the resonance
emission-line doublet Mg~{\sc ii} $\lambda$2797, $\lambda$2803 in
4189 SDSS spectra of low-metallicity emission-line star-forming
galaxies with redshifts $z$ $\sim$ 0.3 -- 1.05.
This
emission is detected in $\sim$35\% of the entire sample of low-metallicity
star-forming galaxies with redshifts $z$ $\ge$ 0.3 selected from SDSS DR14
\citep{Abolfathi2018}.
We study the dependence of the magnesium-to-oxygen and
magnesium-to-neon abundance ratios on metallicity.
We use the magnesium-to-neon ratio relative to the solar
value [Mg/Ne] instead of [Mg/O] in evaluation of magnesium depletion in the
interstellar medium because neon is a noble gas and does not
incorporate into dust. The dependence of [O/Ne] on metallicity is explained by
the coupling of a small amount of oxygen into dust grains. We derive
magnesium depletion of [Mg/Ne] $\simeq$ --0.4 at solar metallicity.
The global parameters of the magnesium sample such as the mass of the
stellar population,
star formation rate, and extinction coefficient C(H$\beta$) are
derived and compared with investigations of other authors.
More massive and more metal abundant
galaxies are found to have higher magnesium depletion.
Our data for interstellar magnesium-to-oxygen abundance ratios
relative to the solar value are in good agreement with
similar measurements made for Galactic
stars,
for giant stars in the Milky Way satellite dwarf galaxies, and with
low-metallicity DLAs.
\citet{Finley2017,Feltre2018} reported that the galaxies with Mg {\sc ii}
both in emission or in absorption are located along the star-forming galaxy main sequence
but their distribution shows a dichotomy with the dependence of SFR on
stellar mass of the galaxies.
We show that the Mg {\sc ii} emitting galaxies from
\citet{Finley2017} and \citet{Feltre2018}
in the SFR -- $M_\star$ relation are located in the region occupied by our
Mg {\sc ii} emitters with high EW(H$\beta$),
whereas Mg {\sc ii} absorbers of \citet{Feltre2018} and Fe {\sc ii}$^*$
emitters from \citet{Finley2017} are located in the region where our Mg emitters
with low EW(H$\beta$) are preferably located. This also confirms that
Mg {\sc ii}
emission has a nebular origin. In this case the presence of emission or
absorption is determined mainly by the mass of the old stellar population and
by the age of the present burst of star formation.
\acknowledgements
N.G.G. and Y.I.I. thank the Max-Planck
Institute for Radioastronomy, Bonn, Germany, for the hospitality.
They acknowledge support from the National Academy of Sciences of Ukraine
(Project No. 0116U003191).
This research has made use of the NASA/IPAC Extragalactic Database
(NED), which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics
and Space Administration.
Funding for the Sloan Digital Sky Survey (SDSS) has been
provided by the Alfred P. Sloan Foundation, the Participating Institutions,
the National Science Foundation, the U.S. Department of Energy, the National
Aeronautics and Space Administration, the Japanese Monbukagakusho, the
Max Planck Society, and the Higher Education Funding Council for England.
|
{
"timestamp": "2019-03-01T02:18:43",
"yymm": "1902",
"arxiv_id": "1902.11083",
"language": "en",
"url": "https://arxiv.org/abs/1902.11083"
}
|
\section{Introduction}
\label{sec:intro}
Accreting black holes are found to exist across a wide range of masses, from the stellar-mass remnants of stars (discoverable as the primaries of binary systems; black hole X-ray binaries, from here on BHBs), to their supermassive ($10^6$--$10^{10}~M_{\odot}$) analogues at the centres of galaxies (active galactic nuclei; AGN). Despite the disparity in mass, size-scale, and local environment, there is growing evidence that the physical nature of accretion flows around stellar-mass and supermassive black holes is mass-invariant, at least in the innermost regions (e.g., \citealt{Merloni2003,F04,kjf06,Plotkin2012}). This apparent scale-invariant property of accretion has led to the hypothesis that the diversity of AGN types arises due to the combination of observer viewing angle (e.g., \citealt{Urry1995}) and the evolving states of AGN (e.g., \citealt{Merloni2003,F04,kjf06}), akin to the changes we see occurring in BHBs (see, e.g., \citealt{Nowak1995,vanderKlis1995,RemMc2006,bel10}). However, tracking the long timescale evolution of individual AGN to observe such state changes is not possible due to the orders-of-magnitude difference in dynamical times compared to those of BHBs. We can instead further our understanding of the evolving properties of BHBs, determine the physical conditions under which state-changes occur, and then see if it can account for the phenomenology observed in different AGN types.
The spectral classification of BHB states can be loosely divided into two categories: soft and hard (see, e.g., \citealt{Nowak1995} for a review). In soft BHB states the X-ray spectrum is dominated by a soft (peak at $\sim1~\mathrm{keV}$) multi-temperature blackbody component, attributable to optically thick emission originating from a thin accretion disc \citep{SS1973}. In hard BHB states we instead see a spectrum dominated by hard power-law emission which has a more ambiguous origin. Some models adopt either static or inflow geometries, whilst others place the emission region within an outflow/jet. Inflow/static models include inverse Compton (IC) or synchrotron self-Compton (SSC) scattering within an optically thin `corona' or radiatively inefficient accretion flow (RIAF) in the inner regions of the accretion flow \citep{Lightman1974,Eardley1975,Shapiro1976,Haardt1993, ny94,Narayan1995,Esin1997}. Outflow models instead propose either SSC/IC or optically thin synchrotron from within a jet/outflow \citep{Markoff2001,mnw05, Yuan2005,Romero2008}. Understanding the interplay between these spectral components, determining which is the dominant mechanism at play, and explaining the connection between the accretion disc, corona and jet, is a key focus of recent targeted multiwavelength observing campaigns on BHBs (e.g. \citealt{Corbel2000,Corbel2003,Gandhi2008,Gandhi2010,Miller-Jones2012,Russell2013,Corbel2013,Russell2014}). X-ray variability studies show state classifications can also be made in the time domain, with low-rms/high-rms variability observed in soft/hard states respectively (see, e.g., \citealt{rm06} for a review). A full outlook on the structure and evolution of BHBs comes from studies in both the spectral the time-variability domains.
Targeted observing campaigns focused on BHB outbursts have led to an empirical correlation between their X-ray and radio fluxes (e.g. \citealt{Hannikainen1998,Corbel2000,Corbel2003,gfp03,Corbel2008,MillerJones2011,Corbel2013,Gallo2014}), and these correlations have been extended into the optical/NIR bands (e.g. \citealt{Russell2006}). The radio/X-ray correlation has been observed to cover several orders of magnitude in luminosity in the low hard states of some sources, such as GX~339$-$4 \citep{Corbel2000,Corbel2003,Corbel2013}, and V404~Cygni \citep{Corbel2008}, and the sources track the same correlation over different outbursts. So the radio/X-ray correlation is locked in as BHBs evolve through their hard states. These correlations indicate that the allocation of power between the physical components in BHBs, as a function of accretion rate, is an intrinsic property of hard state BHBs. In order to draw robust conclusions about the drivers of state changes and the nature of jet launching, we need to be able to reliably identify the source of the X-ray emission, and determine the exact nature of the connection between the X-ray-emitting regions and the jet radio core.
Further developing models of the accretion flow and how it interacts with the jet/outflow in BHBs requires a combination of broadband (radio-to-X-ray) timing and spectral information; these two pictures are seldom treated in unison however, unfortunately, despite the wealth of variability phenomena. However, \cite{Heil2015} developed a novel state classification method for BHBs which characterises the shape of the power spectrum of their X-ray light curves through the course of an outburst, analogous with the well-known hardness intensity diagram of BHB states (HID; \citealt{Homan2001,Homan2005,Belloni2004,Belloni2005}). A single variable, the power-spectral `hue', encodes the relation between two ratios of integrated power across individual frequency bands in Fourier space (see Figure~\ref{fig:heil}). One can use this information to track spectral properties alongside timing characteristics. Since the timing characteristics represent complementary changes in the system configuration over time, we would expect to see some consistency between the physical state of the inner accretion flow/jet and the hue.
In this paper we combine, for the first time, the X-ray variability classification scheme of \cite{Heil2015} with broadband spectral information to build a consistent picture of the evolution of the jet and inner accretion flow of GX~339$-$4. By probing the dominant spectral components and comparing model parameters with the evolution of its variability, we develop a somewhat quantitative description of changes to the accretion flow and jet during both the rise and decay of its outburst. We focus in particular on the relative dominance of the jet and corona in the X-ray band. In Section \ref{sec:gx339anddata} we present the radio-IR-optical-X-ray data compilation we use for model-fitting. In Section \ref{sec:model} we briefly discuss the outflow-dominated model used in our fits. In Section \ref{sec:method} we present our spectral-modelling method and the results of fits to X-ray and broadband (radio, IR-optical, X-ray) spectra, as well as the key parameter trends with variability properties of GX~339$-$4, and a brief consideration of high-energy pair processes in the jet. In Section \ref{sec:discussion} we discuss the significance of these parameter trends and comparisons with previous modelling of the broadband spectra GX~339$-$4. In Section \ref{sec:conclusions} we summarise our results and conclude.
\begin{figure*}
\vspace{-2.5cm}
\includegraphics[width=0.6\linewidth,angle=270]{powercolors_final.pdf}
\vspace{-2.5cm}
\caption{\textbf{Left}: A conceptual diagram representing a BHB power spectrum divided into frequency bins in log space. Two power-colour ratios are defined as PC1 = C/A, PC2 = B/D, where A, B, C, and D are the integral power across the defined frequency bands. \textbf{Right}: The power-colour hue diagram taken from Heil et al. (2015). The angular position in degrees (where $0^{\circ}$ corresponds to the semi-major axis at $45^{\circ}$ to the x-and-y axes) is defined as the hue, and the corresponding states are marked roughly; Soft, Hard, HIMS (hard intermediate state) and SIMS (soft intermediate state). Soft and hard states overlap in the top left of the diagram because their power-spectra have a similar shape, though the normalisations are different---hard states have stronger broad-band variability than soft states. A BHB will start from the top left of the diagram, follow a clockwise path during outburst back to its original position, and then move anti-clockwise through outburst decay back towards the hard state.}
\label{fig:heil}
\end{figure*}
\section{GX~339$-$4: physical characteristics and data selection}
\label{sec:gx339anddata}
GX~339$-$4 has been one of the most intensely studied BHBs since its discovery in 1973 \citep{Markert1973}, due primarily to its short X-ray duty cycle (going into outburst roughly every 2--3 years). As such we have extensive spectral and timing information of GX~339$-$4 covering multiple outbursts (7 with simultaneous radio/X-ray coverage; see \citealt{Corbel2013}), making it the ideal candidate for studies of how spectral properties (and the physical mechanisms behind them) track the time variability behaviour in BHBs. \\
\indent One caveat of conducting such studies on GX~339$-$4 is the lack of accuracy achieved in determining its physical properties. The most heavily cited and utilised mass function measurement is that obtained by \cite{Hynes2003} of $5.8\pm0.8~M_{\odot}$, and a later estimate included a lower limit of $7~M_{\odot}$ \citep{Munoz-Darias2008}. In contrast more recent near-infrared detections of absorption lines from the donor star of GX~339$-$4 indicate a mass function of $\sim1.91\pm0.08~M_{\odot}$ \citep{Heida2017}. Distance has also been difficult to determine\footnote{Though the recent Gaia survey \citep{Gaia} has already led to new distance estimates of many BHBs \citep{Gandhi2018}}, with early estimates finding a broad range from 6--15 kpc \citep{Hynes2004}, and best estimates giving $\sim8~\mathrm{kpc}$ \citep{Zdziarski2004}, based on a comparison of the redshifted spectral lines seen in GX~339$-$4 with those of stars in the Galactic bulge region at $D=8\pm2~\mathrm{kpc}$, and the high peculiar velocity of GX~339$-$4 ($v\sim 140~\mathrm{km~s^{-1}}$; \citealt{Hynes2004}). One of the most elusive physical properties of GX~339$-$4 has been the orbital inclination. With almost no model-independent consensus, we mostly rely on modelling of accretion disc reflection in the X-ray spectra to determine the inclination. Reflection modelling has derived inclination estimates over a large range: 15$^{\circ}$--50$^{\circ}$ \citep{Miller2006,Reis2008,Done2010,Plant2014,Plant2015,Garcia2015,Parker2016}. These values are not wholly reliable for two reasons: 1) these are model-dependent estimates that are degenerate with other key parameters of the reflection models, and 2) the disc inclination may not be equal to the orbital inclination of the binary (see, e.g., \citealt{Wijers1999,Maccarone2002,Begelman2006}). We nonetheless adopt the best estimates possible in order to model the data. We choose to fix the observational characteristics of GX~339$-$4 at distance $D=8~\mathrm{kpc}$ \citep{Zdziarski2004}, inclination $i=40^{\circ}$ (a rough average of the broad range of estimates), and mass $M_{\mathrm{BH}}=f(M)(1+q)^2/\sin^3i=9.8~M_{\odot}$, adopting the mass function of \cite{Heida2017} and assuming the mass of the donor star to black hole mass is $q=0.17$. \\
\indent GX~339$-$4 has a compact radio jet during the hard state \citep{Fender2001}, and the emission from this jet dominates up to IR \citep{Corbel2002} and possibly optical frequencies \citep{Gandhi2008,Gandhi2010,Gandhi2011,Casella2010}. Correlations between the optical/IR/X-ray light curves during various GX~339$-$4 outbursts indicate a physical connection between the regions near the black hole and the self-absorbed regions of jets at $\sim10^3$--$10^4~r_{\rm g}$, supported by recent optical and IR lags of $\sim100$~ms (with respect to X-ray) detected from the jet \citep{Gandhi2010,Kalamkar2016}; a roughly equivalent lag was detected in BHB V404~Cygni recently too \citep{Gandhi2017}. This lag between emission at high and low frequencies in BHBs is best interpreted as variations propagating through the jet. Such variations are thought to be associated with accretion rate fluctuations propagating through the disc \citep{Lyubarskii1997,Uttley2001}.
\subsection{Data}
\label{sec:data}
We compile data from 20 separate quasi-simultaneous (all observations within 24-hrs of one another), broadband observations of GX~339$-$4, covering the radio, near-IR/optical, and X-ray bands. Here we describe how the data were collected and reduced. In basic terms, the selection criteria are that there is quasi-simultaneous broadband coverage of GX~339$-$4 and that it is in its hard state, defined by its variability and spectral properties (hue and hardness ratio).
\subsubsection{X-ray data}
Data from the \textit{Rossi X-ray Timing Explorer (RXTE)} proportional counter array \citep[PCA]{Jahoda_2006a} and High-Energy Timing Experiment \citep[HEXTE]{Rothschild_1998a} were extracted using HEASOFT 6.16 following the standard procedure as described, e.g., in \cite{Grinberg_2013a}, in particular discarding data within 10 minutes of the South Atlantic Anomaly passages.\\
\indent For the PCA, we use data from the top xenon layer of proportional counter unit (PCU)~2 only since these data are best calibrated. We apply \texttt{PCACORR} calibration tool \citep{Garcia_2014a} to further improve the data quality. No HEXTE data are available for over half of our observations (Table~\ref{tab:data}) due to the failure of the rocking mechanisms of both HEXTE clusters late in the \textit{RXTE} mission lifetime. We extract cluster A and B data where available. We refrain from using the \texttt{HEXTECORR} calibration tool \citep{Garcia_2016a} on the HEXTE~B data as the improvement would only be marginal given the data quality.\\
\indent The PCA light curves are used to calculate the power-spectral hue of each observation (shown in Table~\ref{tab:data}), following the method of \cite{Heil2015}. Figure~\ref{fig:heil} shows how the PCU~2 light curves are used to calculate the power-spectral hue. A fourier transform is taken, and the resulting power spectrum is divided into four roughly even log-spaced frequency bands: A = 0.0039--0.031~Hz, B = 0.031--0.25~Hz, C = 0.25--2.0~Hz, D = 2--16~Hz. The ratios of integrated power between bands C/A and B/D are then taken, defining power-colour 1 (PC1) and power-colour 2 (PC2) respectively. Placed on a scatter plot of PC1 and PC2, the data follow an annulus. The clockwise angular position of each observation (with respect to a semi-major axis at 45$^{\circ}$ to the x-and-y axes) defines its power-colour hue. All our X-ray data have been pre-selected with $-20^{\circ}<$~hue~$<140^{\circ}$, which \cite{Heil2015} define as the hard state, and all have a hardness ratio $>0.75$. The corresponding hardness ratios for all the PCU~2 spectra are shown in Table~\ref{tab:data} and displayed in a hardness-intensity diagram in Figure~\ref{fig:HID} for clarity, although we note that those values have been calculated using counts collected by all 3 Xenon layers of the detector.
\begin{figure}
\includegraphics[width=\linewidth]{jan19_HID_by_year.pdf}\vspace{-0.5cm}
\caption{Hardness-Intensity diagram showing all the \textit{RXTE}-PCU 2 observations of GX~339$-$4 (gray points) and highlighting the 20 observations in our sample, divided by observation year, with the colours/symbols indicated in the key. Hardness ratios and intensities are calculated using all Xenon-layers of the PCU~2 detector (as opposed to utilizing only the top Xenon layer as in the sample of modelled data) in order to show absolute values in line with ratio measurements in the literature. Intensities are normalised by the peak average intensity of the source. Hard colour is defined as the ratio of source counts in the 8.6--18~keV to 5--8.6~keV energy bands respectively.}
\label{fig:HID}
\end{figure}
\begin{figure}
\vspace{-0.5cm}
\includegraphics[width=\linewidth]{jan19_radio_optical_v_xray.pdf}
\vspace{-0.5cm}
\caption{The radio (5.5~GHz and 8.8~GHz) and OIR (V, I, J and H bands) fluxes of all 20 quasi-simultaneous broadband observations of GX~339$-$4 in mJy alongside the X-ray PCA data fluxes [3--20~keV] in erg~cm$^{-2}$~s$^{-1}$. The fluxes are separated by observation years 2005, 2007, 2010 and 2011, covering three separate outbursts, with the key indicating the symbols and colours corresponding to each year.}
\label{fig:data_fluxes}
\end{figure}
\begin{figure}
\centering
\hspace{-0.5cm}
\includegraphics[width=1.05\linewidth]{jan19_BroadbandLightcurve.pdf}\vspace{-0.8cm}
\caption{The PCU~2 intensity (top), optical V band, and radio fluxes (bottom) as a function of MJD, divided by observation year, truncated in time to fit together on one plot. Each panel shows a 300 day snapshot of the PCU~2 light curve, with the following MJD ranges for each year: 53300--53600 (2005), 54100--54400 (2007), 55100--55400 (2010), 55400--55700 (2011).}
\label{fig:lcurves}
\end{figure}
\subsubsection{Radio/IR-Optical data}
\label{subsubsec:oir}
We select radio fluxes of GX~339$-$4 covering a 15-year period (1997--2012) resulting from observations made with the Australian Telescope Compact Array (ATCA) \citep{Corbel2013}, choosing only those observations falling within a 24-hr window of the corresponding X-ray observations. We then include optical and near-infrared fluxes resulting from observations of GX~339$-$4 made with the SMARTS 1.3 m telescope from 2002--2010, covering the \textit{V}, \textit{J}, \textit{I} and \textit{H} bands \citep{Buxton2012}. The magnitudes in all four bands are de-reddened assuming $n_\mathrm{H}=5\pm1\times10^{21}~\mathrm{cm}^{-2}$ \citep{Kong2002}, giving $E(B-V)=0.94\pm0.19$ \citep{PS1995}, such that $A_V=2.9\pm0.6$ \citep{Cardelli1989}. The flux density values quoted in Table~\ref{tab:data} are the de-reddened flux densities given by \cite{Buxton2012}. We reject SMARTS observations that fall outside the 24 hour window of the pre-selected quasi-simultaneous radio and X-ray observations. This selection criterion leaves us with 20 separate broadband quasi-simultaneous spectra of GX~339$-$4, covering the decay of its 2005 outburst, the peak and decay of its 2007 outburst, and the rise and decay of its 2010 outburst. A full description of the data is shown in Table~\ref{tab:data}, and see Figure~\ref{fig:data_fluxes} for a plot of the radio and IR/optical (OIR) fluxes against the X-ray fluxes of all 20 observations, and Figure~\ref{fig:lcurves} for X-ray lightcurves with optical and radio band fluxes showing the different outburst stages our datasets probe. One notices instantly that the first three observations during the 2011 outburst decay have notably lower OIR fluxes for their given X-ray fluxes than in all other observations, with a trend that deviates from an otherwise well-behaved correlation. \\
\indent Selecting quasi-simultaneous data with a 24-hr time-window coincidence across radio/OIR/X-ray bands in this way optimises the trade-off between the quantity of data we require for our modelling, and the information lost by neglecting source variability on short timescales. \cite{Gandhi2011} show that the mid-IR spectral slope is variable on timescales of $\sim20~\mathrm{minutes}$. We therefore highlight the uncertainties in the overall flux and spectral slope incurred by grouping data over the 24-hr time window, and simply note it as a caveat to our analysis.
\begin{table*}
\centering
\caption{The broadband quasi-simultaneous data of GX~339$-$4. Shown from left to right: (1) spectrum number, (2) MJD, (3) the 5.5~GHz and 8.8~GHz radio fluxes, (4) the OIR fluxes, bands $V$, $I$, $J$, and $H$, (5) the observational ID of the \textit{RXTE} observation, (6) the X-ray unfolded PCA data flux (model indepedent), 3--20~keV, (7) the hardness ratio, defined as the ratio of PCU~2 (all layers) source counts between the [8.6--18~keV]/[5--8.6~keV] bands, (8) the power-spectral hue, (9) HEXTE cluster A or B spectra included.}
\label{tab:data}
\begin{tabular}{@{}lcccccccc}
\hline
Spec. \# & MJD & $F_{\rm R}$ [mJy] & $F_{\rm IR/opt}$ [mJy] & ObsID & $F_{\rm X}$ & HR & Hue & HEXTE?\\
& (-245000) & 5.5 GHz & $V, I$ & & [$10^{-10}~\mathrm{erg}~\mathrm{s}^{-1}~\mathrm{cm}^{-2}$] & & [$^{\circ}$]\\
& & 8.8 GHz & $J, H$ & & [3--20 keV]\\
\hline
1 & 53485 & $4.39\pm0.06$ & $17\pm5$, $11\pm1$ & 90704-01-13-01 & $5.67\pm0.05$ & 0.88 & $42\pm13$ & A \& B\\
& & $4.23\pm0.08$ & $8.9\pm0.9$, $8.7\pm0.8$\\
2 & 53489 & $3.1\pm0.1$ & $17\pm6$, $12\pm1$ & 91095-08-06-00 & $3.79\pm0.02$ & 0.89 & $32\pm4$ & A \& B \\
& & $3.5\pm0.1$ & $11\pm1$, $11\pm1$\\
3 & 53490 & $2.88\pm0.08$ & $17\pm6$, $12\pm1$ & 91095-08-07-00 & $3.39\pm0.02$ & 0.87 & $22\pm3$ & A \& B\\
& & $3.3\pm0.1$ & $11\pm1$, $11\pm1$\\
4 & 53490 & $2.53\pm0.05$ & $17\pm5$, $11\pm1$ & 91105-04-17-00 & $3.45\pm0.07$ & 0.89 & $20\pm10$ & No\\
& & $2.94\pm0.07$ & $11\pm1$, $10\pm1$\\
5 & 53492 & $2.53\pm0.05$ & $19\pm6$, $13\pm1$ & 91095-08-09-00 & $2.95\pm0.03$ & 0.87 & $24\pm4$ & A \& B\\
& & $2.94\pm0.07$ & $12\pm1$, $11\pm1$\\
6 & 53496 & $1.42\pm0.09$ & $18\pm6$, $12\pm1$ & 90704-01-14-00 & $2.21\pm0.04$ & 0.89 & $9\pm6$ & No\\
& & $1.7\pm0.1$ & $11\pm1$, $10\pm1$\\
7 & 54135 & $19.5\pm0.3$ & $120\pm40$, $72\pm7$ & 92035-01-02-02 & $104.2\pm0.1$ & 0.79 & $87\pm5$ & B only\\
& & $17\pm1$ & $67\pm6$, $65\pm6$\\
8 & 54258 & $2.6\pm0.2$ & $16\pm5$, $12\pm1$ & 92704-03-26-00 & $3.02\pm0.05$ & 0.91 & $22\pm10$ & No\\
& & $2.6\pm0.2$ & $10\pm1$, $10\pm1$\\
9 & 54335 & $3.3\pm0.05$ & $26\pm8$, $20\pm2$ & 93409-01-05-03 & $7.33\pm0.05$ & 0.93 & $18\pm6$ & B only\\
& & $2.95\pm0.07$ & $17\pm2$, $20\pm2$\\
10 & 55240 & $6.17\pm0.06$ & $47\pm15$, $29\pm3$ & 95409-01-06-00 & $17.53\pm0.05$ & 0.95 & $347\pm3$ & No\\
& & $5.9\pm0.1$ & $31\pm3$, $31\pm3$\\
11 & 55260 & $7.2\pm0.1$ & $65\pm21$, $44\pm4$ & 95409-01-08-03 & $29.94\pm0.06$ & 0.92 & $359\pm2$ & No\\
& & $7.3\pm0.1$ & $37\pm4$, $39\pm4$\\
12 & 55263 & $8.24\pm0.05$ & $62\pm20$, $39\pm4$ & 95409-01-09-01 & $33.4\pm0.1$ & 0.91 & $358\pm3$ & No\\
& & $8.1\pm0.1$ & $40\pm4$, $39\pm4$ \\
13 & 55271 & $10.2\pm0.1$ & $92\pm29$, $54\pm5$ & 95409-01-10-03 & $41.53\pm0.07$ & 0.88 & $15\pm3$ & No\\
& & $11.3\pm0.1$ & $50\pm5$, $47\pm5$\\
14 & 55277 & $13.8\pm0.1$ & $99\pm32$, $58\pm6$ & 95409-01-11-02 & $57.0\pm0.2$ & 0.86 & $19\pm13$ & No\\
& & $15.45\pm0.06$ & $48\pm5$, $48\pm5$\\
15 & 55280 & $15.56\pm0.05$ & $88\pm28$, $56\pm5$ & 95409-01-11-03 & $63.7\pm0.2$ & 0.84 & $45\pm10$ & No\\
& & $18.59\pm0.05$ & $54\pm5$, $51\pm5$\\
16 & 55290 & $18.8\pm0.1$ & $N/A$, $54\pm5$ & 95409-01-13-00 & $83.2\pm0.1$ & 0.81 & $38\pm26$ & No\\
& & $21.1\pm0.2$ & $56\pm5$, $52\pm5$\\
17 & 55605 & $4.45\pm0.04$ & $7\pm2$, $4.2\pm0.4$ & 96409-01-07-03 & $6.32\pm0.05$ & 0.81 & $95\pm17$ & No\\
& & $4.17\pm0.05$ & $2.7\pm0.3$, $2.0\pm0.2$\\
18 & 55608 & $4.07\pm0.04$ & $9\pm3$, $5.3\pm0.5$ & 96409-01-07-02 & $5.02\pm0.05$ & 0.87 & $138\pm7$ & No\\
& & $3.87\pm0.05$ & $3.5\pm0.3$, $2.8\pm0.3$\\
19 & 55610 & $3.9\pm0.1$ & $10\pm3$, $6.1\pm0.6$ & 96409-01-07-04 & $4.05\pm0.05$ & 0.89 & $80\pm18$ & No\\
& & $4.0\pm0.1$ & $4.6\pm0.4$, $4.1\pm0.4$\\
20 & 55618 & $2.54\pm0.04$ & $16\pm5$, $11\pm1$ & 96409-01-09-00 & $1.84\pm0.03$ & 0.88 & $14\pm15$ & No\\
& & $2.95\pm0.05$ & $9.5\pm0.9$, $8.6\pm0.8$\\
\hline
\end{tabular}
\end{table*}
\section{The model}
\label{sec:model}
We use a semi-analytical, zonal jet model (see \citealt{mnw05,Maitra2009,Connors2017}). To calculate the dynamics, we assume the BHB launches a roughly isothermal jet that is accelerated to mildly relativistic velocities by internal pressure \citep{Crumley2017}. We refer the reader to \cite{Connors2017} for the most up-to-date details and changes within the model prior to the changes discussed below, and to Table~\ref{tab:params} for a description of the key physical parameters of the model. We make two key improvements upon previous implementations of the model. \\
\indent Firstly, the calculation of IC emission within the jet has been improved to include multiple scattering events rather than adopting a single-scattering treatment. This allows the model to treat cases in which the jet-base is initially optically thick ($\tau \gg 1$), such that the flux contribution from higher IC scattering orders may be significant. In BHBs we expect the IC-emitting regions to remain optically thin \citep{Haardt1993,Done2007}. However, even as the IC region approaches $\tau \sim 1$, the higher-energy emission may become relevant since the Compton-y parameter (the number of scatterings $\times$ the energy shift per scattering) for a single electron goes as $y_i=16\Theta_i^2 \mathrm{Max}(\tau_i^2,\tau_i)$, where $\Theta_i \equiv kT_i/m_ec^2$ is the dimensionless energy of the electron, with $i$ representing a single electron within the full population. In simple terms, a large $y_i$ results in efficient scattering (more than one scattering order) but can be achieved in the following ways and will result in different spectral shapes: a IC spectrum with high $\tau$ ($\gg 1$) and moderate $\Theta_i$ ($\le 1$) will be a smooth power law, whereas at moderate $\tau$ ($\sim 1$) and high $\Theta_i$ ($\ge 1$) the IC spectrum will appear bumpy due to the separation of scattering orders (see, e.g., \citealt{Ghisellini2013}). Details of the multiple IC calculation used in this work can be found in a forthcoming paper (Ceccobello et al., in preparation). \\
\indent Secondly, we have altered the jet height profile (z-profile) to improve the treatment of IC scattering in the first few zones of the jet. In all previous implementations of the model, a log scale is used between $z_{min}$ and $z_{max}$, where $z_{min} \sim 0.3~r_0$ and $z_{max}$ is a model parameter adjusted according to the source being modelled. Instead now we enforce $\Delta z = 2 r$ in all zones up to the cut-off of the Comptonising region (at $z_{cut}=100~r_0$), and space the remaining zones logarithmically up to $z_{max}$. In this way, we treat the input photon distribution for IC scattering as roughly isotropic without incurring any resolution-dependent errors, and without losing too much resolution in the effects of the jet profile at low heights. \\
\indent From here on we refer to this model as \texttt{agnjet}, thus maintaining consistency with its earlier applications to mildly-relativistic ($\gamma_{j}\sim~\mathrm{a~few}$) jets in Active Galactic Nuclei \citep{Markoff2015,Prieto2016, Connors2017,Crumley2017}.
\begin{table}
\centering
\caption{A list of the main input parameters of the \texttt{agnjet} model}
\label{tab:params}
\begin{tabular}{@{}cp{6cm}}
\hline
Parameter & Description \\
\hline
$N_{\rm j}$ ($L_{\rm Edd}$) & the normalised jet power.\\
$r_0$ and $h_0$ ($r_{\rm g}$) & the radius and height (length) of the jet nozzle. The height is fixed at $h_0=2r_0$, such that the nozzle is a cylinder. \\
$\Theta_{\rm e}$ ($kT_{\rm e}/mc^2$) & the electron temperature of the input distribution. \\
$\beta_{\rm e}$ & the ratio of electron to magnetic energy density, ${U_{\rm e}}/{U_{\rm B}}$. \\
$p$ & the power-law index of the accelerated electron distribution. \\
$z_{\mathrm{acc}}$ ($r_g$) & the distance from the black hole along the jet axis where particle acceleration into a power-law distribution first begins. \\
$n_{\mathrm{nth}}$ & the fraction of particles accelerated at a distance $z_{\rm acc}$ from the black hole along the axis of the jet. \\
$f_{\mathrm{sc}}$ & the scattering fraction, a measure of the efficiency with which electrons are accelerated at $z_{\mathrm{acc}}$, defined as ${\beta_{\mathrm{sh}}}^2/{\left(\lambda/R_{\mathrm{gyro}}\right)}$ where $\beta_{\mathrm{sh}}$ is the shock speed relative to the plasma, $\lambda$ is the scattering mean free path in the plasma at the shock region, and $R_{\mathrm{gyro}}$ is the gyroradius of the particles in the magnetic field. In reality we do not require a shock so this parameterisation can generally be seen as a measure of the acceleration efficiency, as it sets the maximum post-acceleration electron energy.\\
\hline
\end{tabular}
\end{table}
\section{Spectral fits}
\label{sec:method}
We perform all spectral fits in this work using the multiwavelength data analysis package \texttt{ISIS} \citep{Houck2000}, version 1.6.2-40. All models are forward-folded through the detector response matrices; when fitting to X-ray spectra this corresponds to the Proportional Counter Array (PCA) and High Energy X-ray Transmission Spectrometer (HEXTE) instrument responses, whereas data at all other wavelengths is assigned a "dummy" response equivalent to a detector of effective area = $1~\mathrm{m}^2$. Data at wavelengths outside the X-ray band are loaded into \texttt{ISIS} as flux measurements (shown in Table~\ref{tab:data}). We bin PCA spectra at a minimum signal-to-noise ratio of $S/N=4.5$, between energy limits of 3--45~keV or 3--20~keV depending on the availability of counts in the highest energy bins. A systematic error of 0.1\% is added to the PCA counts based on the improved calibration tool \texttt{PCACORR} \citep{Garcia_2014a}. We include HEXTE A/B spectra for the observations indicated in Table~\ref{tab:data}, and bin each at minimum signal-to-noise $S/N=4.5$ between energy limits 20--200~keV. At each stage of the fitting process we use the \texttt{ISIS} implementation of Markov Chain Monte Carlo (MCMC) parameter exploration routine \citep{Murphy2014}, based on the popular routine, \texttt{emcee}, developed by \cite{Foreman-Mackey2013}. In each case we initialize $50 \times n_{fp}$ walkers per free parameter, where $n_{fp}$ is the number of free parameters, and we run the MCMC chain until it has converged---we judge convergence as the point beyond which changes to the posterior probability distribution functions of the parameters are minimal, resulting in chains ranging in length between $10^3$--$10^4$ steps.
\subsection{X-ray spectral fits}
\label{subsec:initialxrayfits}
Before exploring broadband model fits to the quasi-simultaneous data of GX~339$-$4, we first fit phenomenological models to the available X-ray spectra in order to place prior constraints on nuisance parameters, allowing us to reduce the uncertainties in our broadband fits. These include the energy of the Gaussian iron emission line resulting from disc reflection, $E_{\mathrm{line}}$, and its corresponding line width, $\sigma_{line}$. We fix the interstellar Hydrogen column density to $n_\mathrm{H}=5 \times 10^{21}~\mathrm{cm^{-2}}$ based on previous X-ray spectral modelling of GX~339$-$4 \citep{Shidatsu2011,Garcia2015,Parker2016}, and on the cross-section adopted when correcting for extinction in the OIR \citep{Kong2002}. We consider 3 model classes, assigned according to the breadth of X-ray band coverage and the number of X-ray counts in the spectra:
\begin{itemize}
\item X1: \texttt{tbabs$\times$[powerlaw+gaussian]}
\item X2: \texttt{tbabs$\times$[reflect(powerlaw)+gaussian]}
\item X3: \texttt{tbabs$\times$[reflect(powerlaw$\times$highecut)+gaussian]}.
\end{itemize}
The reflection convolution model \texttt{reflect} is that of \cite{Magdziarz1995}, and we adopt this in preference to more recent reflection models (\texttt{RELXILL}; \citealt{Dauser2014,Garcia2014}, \texttt{REFLIONX}; \citealt{Ross1999,Ross2005}) since it convolves an arbitrary input spectrum, whereas the more recent models rely on robust model tables that are expensive to produce. The absorption model \texttt{tbabs} is described in \cite{Wilms2000}. We adopt the solar abundances of \cite{Wilms2000} and set the photo-ionisation cross-sections according to \cite{Verner1996}. Model~X1 is most likely to provide a sufficient fit to those X-ray spectra with low source counts, Model~X2 (a reflected power law) will apply when source counts are high enough to distinguish a break in the spectrum at $E \sim 10~\mathrm{keV}$, characteristic of a reflected X-ray spectrum, and Model~X3 applies to only one spectrum for which we see a clear visible cutoff in the spectrum. We perform Markov Chain Monte Carlo (MCMC) parameter exploration on each X-ray spectral fit in order to characterise the posterior probability distribution functions (PDF) of $E_{\mathrm{line}}$ and $\sigma_{line}$. We fix $E_{\mathrm{line}}$ and $\sigma_{line}$ based on these fits, and carry those values forward to our broadband spectral modelling described in Section \ref{subsec:bbmodelling}. \\
\indent Figure~\ref{fig:gamma_trend} shows the evolution of $\Gamma$, the power-law spectral index, against both the power-spectral hue and the unfolded data luminosity (assuming $D=8~\mathrm{kpc}$). There exists a clear dichotomy between the more luminous X-ray spectra of the 2010 outburst rise and 2007 single observation of its outburst, with the observations in the decay phases of 2005, 2007, and 2011. The spectrum appears to slightly soften with increasing luminosity/hue. The increase in power-spectral hue is coincident with increasing luminosity as the source evolves through its outburst, but with two distinct trends depending on whether the source is in the rise or decay of an outburst, making the plot appear like the mirror of the hardness-intensity diagram (see Figure~\ref{fig:hue_v_lx}). Many previous works find $\Gamma$ to be mostly constant during the rising hard state of GX~339$-$4 \citep{Wilms1999,Zdziarski2004,Plant2014,Garcia2015}, so the fact that we see a slight positive correlation may be related to the model treatment, in particular the reflection model used, as well as the treatment of the data. For example, \cite{Garcia2015} combine spectra across ranges of X-ray hardness, and use a different model for the X-ray reflection, which likely leads to contrasting photon indices. However, we note such a positive trend does agree with the broader trends seen in multiple BHBs (see e.g. \citealt{RemMc2006}), and is coincident with the narrowing and strengthening of broadband X-ray variability \citep{Heil2015}, and brighter radio jets \citep{Fender2006}.
\begin{figure}
\includegraphics[width=1.1\linewidth]{jan19_gamma_v_hue_and_lx_by_year.pdf}\vspace{-0.5cm}
\caption{The power-law spectral index ($\Gamma_{\mathrm{pl}}$, derived from initial spectral fits to all 20 X-ray spectra) against the unfolded data luminosity (left) between 3--20 keV and the power-spectral hue (right). The key shows how the data are divided by observation year.}
\label{fig:gamma_trend}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{jan19_hue_v_lx_by_year.pdf}\vspace{-0.5cm}
\caption{The Eddington-scaled X-ray luminosity of all X-ray spectra against the power-spectral hue derived from the light curves. The key shows how the data are divided by observation year. }
\label{fig:hue_v_lx}
\end{figure}
\subsection{Broadband spectral modelling}
\label{subsec:bbmodelling}
Next, we fit all 20 of the quasi-simultaneous broadband spectra energy distributions (SEDs) of GX~339$-$4 with two more physically motivated models, with the goal of tracking the trends in the physical parameters of the jet and corona or inner accretion flow. We find that the X-ray spectra are best fit by a coronal-like IC-scattering plasma, in which the scattering electrons are at temperatures of $kT_\mathrm{e}\sim$ hundreds of keV, and the plasma has optical depths in the range 0.1--1. We also find that the hotter jet electrons likely contribute a non-negligible flux in the X-ray band as a result of SSC.\\
\indent The two models we adopt are as follows:
\begin{itemize}
\item B1: an absorbed, reflected jet component + Gaussian line: \texttt{tbabs}$\times$[{\texttt{reflect}\texttt{(agnjet)+gaussian}}]\item B2: the sum of absorbed, reflected jet and coronal components + Gaussian line:\\ \texttt{tbabs}$\times$[\texttt{reflect}\texttt{(agnjet}\texttt{+nthcomp})\texttt{+gaussian}].
\end{itemize}
Here \texttt{agnjet} represents the jet and outer standard disc components, and \texttt{nthcomp} represents a spherical corona in the inner regions of the accretion flow \citep{Zdziarski1996,Zycki1999}, and thus B2 is only distinguishable from B1 through the additional coronal thermal Comtponisation component. Figure~\ref{fig:corona_jet} shows a diagram of the setup which represents spectral Model~B2. Whilst \texttt{agnjet} does in fact include a coronal-like jet base \citep{mnw05}, its treatment of SSC is purely relativistic, allowing only for photon-scattering electrons at $\Theta_{\rm e}\equiv{kT_e/m_ec^2}\ge1$ ($kT_e\ge511~\mathrm{keV}$)---this is due to the expectation that energy is dissipated to the electrons quite rapidly within the jet, giving rise to high synchrotron fluxes in the radio bands in regions further out along its axis; conservation arguments suggest similarly hot electrons ($\Theta_{\rm e}>1$) at the base of the jet (see e.g. \citealt{mnw05}). Popular models for the X-ray spectra observed in BHB hard states typically include \texttt{nthcomp} in which a thermal population of electrons at roughly $\Theta_{\rm e} \sim 0.02$--$0.2$ IC scatter the soft blackbody component of the accretion flow with seed photons temperatures in the range $kT_{\mathrm{BB}}\sim0.01$--$1~\mathrm{keV}$, set by the inner disc temperature (see e.g. \citealt{Haardt1993,Done2007} and references therein). Thus in our model-fitting treatment we choose to test a combination of both emission components in order to determine the relevant importance of each during an evolving outburst.
\begin{figure}
\includegraphics[width=0.4\textwidth,angle=270]{JetCorona.pdf}
\vspace{-1.2cm}
\caption{Diagram of a corona + jet model for a BHB, representing spectral Model~B2. The thin disc is truncated to radii on the order of 2--10~$r_{\rm g}$, and an optically thin compact corona exists within the inner accretion flow, with electron temperatures $kT_{e}\sim$ hundreds of keV. The jet electrons are relativistic, with $kT_e \ge 511~\mathrm{keV}$, and the plasma has bulk motion in the z-direction with $\gamma_j \sim$ on the order of 1 to a few. The observer sees emission from the jet in the form of synchrotron, SSC, and IC scattering of disc photons, as well as IC emission from the corona, and blackbody emission from the disc. The jet and coronal X-ray components irradiate the disc, resulting in a reflected X-ray spectrum.}
\label{fig:corona_jet}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{All_spectral_data_by_year.pdf}\vspace{-0.5cm}
\caption{All 20 broadband spectra split into panels based on observation year. The observed flux density is shown as a function of frequency, with the radio, OIR, and X-ray bands indicated. Unfolded fluxes are calculated independently of the spectral model.}
\label{fig:all_data}
\end{figure}
\begin{figure*}
\includegraphics[width=\textwidth]{All_spectral_fit_residuals_by_year_B1.pdf}
\caption{Standardised residuals ((data - model) / uncertainty) of fits to all 20 broadband spectra of GX~339$-$4 with Model~B1 (jet IC-dominated X-ray spectra): \texttt{tbabs $\times$ [reflect(agnjet)+gaussian]}, with typical $\chi^2_{R}\sim$ a few to 10s. Each panel shows fits to observations within 2005, 2007, 2010, and 2011 respectively. Radio, OIR, and X-ray data are labelled, and different symbols/colours indicate the broadband data-model residuals for each fit. The X-ray spectra are dominated by jet SSC and IC scattering of disc photons in the jet, as well as reflection off the disc.}
\label{fig:allfits_ssc}
\end{figure*}
\begin{figure*}
\includegraphics[width=\textwidth]{All_spectral_fit_residuals_by_year_B2.pdf}
\caption{Standardised residuals ((data - model) / uncertainty) of fits to all 20 broadband spectra of GX~339$-$4 with Model~B2 (coronal IC-dominated X-ray spectra): \texttt{tbabs $\times$ [reflect(agnjet+nthcomp)+gaussian]}, with typical $\chi^2_{R}\sim$1--2, with some outliers. Each panel shows fits to observations within 2005, 2007, 2010, and 2011 respectively. Radio, OIR, and X-ray data are labelled, and different symbols/colours indicate the broadband data-model residuals for each fit. The coronal emission (\texttt{nthcomp}) dominates the X-ray spectra, with SSC and IC scattering of disc photons in the jet contributing, and all emission reflecting off the disc. }
\label{fig:allfits_nthcomp}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.47\textwidth]{Spectral_fit_spec5.pdf}
\includegraphics[width=0.47\textwidth]{Spectral_fit_spec18.pdf}
\includegraphics[width=0.47\textwidth]{Spectral_fit_spec8.pdf}
\includegraphics[width=0.47\textwidth]{Spectral_fit_spec15.pdf}
\caption{An expanded view of the fit to broadband spectra 3 (top left), 18 (top right), 8 (bottom left) and 14 (bottom right) from GX~339$-$4 outburst decays in years 2005 and 2011, and outburst rises in years 2007 and 2010 respectively (see Table~\ref{tab:conf}). The top panels show the full broadband fit with Model~B1 (jet IC-dominated X-ray spectrum): \texttt{tbabs $\times$ [reflect(agnjet)+gaussian]}. The bottom panels show the same spectrum fit with Model~B2 (coronal IC-dominated X-ray spectrum): \texttt{tbabs $\times$ [reflect(agnjet+nthcomp)+gaussian]}. Radio data are marked with green squares, OIR data with orange triangles, and \textit{RXTE}-PCA, HXT A, and HXT B with blue, purple and dark blue circles respectively. The total broadband jet spectrum is shown with solid gray lines. The individual jet spectral components shown are SSC/IC (green dashed lines), pre-acceleration thermal synchrotron (blue dot-dot-dashed lines), post-acceleration synchrotron (red dot-dashed lines), and the accretion disc blackbody spectrum (black dotted line). The reflection component (Gaussian iron line included) is shown in orange, and the coronal component in the right-hand panels is shown as a brown triple-dot-dashed line. The solid black line shows the total absorbed model spectrum. The disc component of \texttt{agnjet} and coronal component of \texttt{nthcomp} normalisations are treated separately. In the panels beneath each fit we show the standardised $\chi$-residuals, (data-model)/uncertainty.}
\label{fig:specfits}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{jan19_Rf_v_hue_and_lx_by_year.pdf}
\caption{The maximum likelihood estimates of the reflection fraction, $R_{\mathrm{f}}$, as a function of Eddington-scaled X-ray data luminosity (left) and power-spectral hue (right). Hollow gray squares show the parameters derived from fits of Model~B1, and other symbols (indicated in the key) show fits of Model~B2.}
\label{fig:Rf}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.1\linewidth]{jan19_Lmodel_v_hue_and_lx_by_year.pdf}\vspace{-0.3cm}
\caption{The integrated radiative jet luminosity in the 3--100~keV X-ray band ($L_{\rm \gamma, jet}$) as a fraction of the total 3--100~keV model luminosity ($L_{\rm \gamma, total}$) (top), and the Bolometric coronal luminosity in ergs/s (bottom), shown as a function of Eddington-scaled X-ray data luminosity (left) and power-spectral hue (right). Luminosities are shown only for fits of Model~B2, divided by observation year. Uncertainties are shown only for $L_{\rm X}$ and hue. The jet radiative luminosity has not been corrected for Doppler beaming. }
\label{fig:Ljet}
\end{figure}
\indent We fix the location where particle acceleration starts to $\log_{10} [z_{\rm acc}] = 3.5$ since this is the approximate location at which a non-thermal population of electrons is generated in GX~339$-$4, according to the location of the variable self-absorption spectral break \citep{Markoff2003,Gandhi2011}. If we interpret a time lag of 100~ms \citep{Kalamkar2016} as being caused by the delay of plasma flow through the jet, this would imply a distance scale of $z \ge 0.1s \times \gamma_{\rm j}\beta_{\rm j} c \sim 10^3~r_{\rm g}$, where $\gamma_{\rm j}$ is the jet bulk Lorentz factor, and $\beta_{\rm j}$ is the jet bulk velocity. This distance is conservative given that the jet is assumed to travel at constant velocity---the jet could accelerate efficiently along its axis, as is the case with \texttt{agnjet}. It is preferable to keep $z_{\rm acc}$ fixed at this value since the data coverage provides limited constraints on its value, and the self-absorption break is variable on timescales shorter than 24 hours (e.g., \citealt{Gandhi2010}, and see Section~\ref{subsubsec:oir}). We fix the fraction of particles accelerated at $z_{\mathrm{acc}}$ to $n_{\rm nth}=0.1$, based on current studies of particle acceleration across mildly-relativistic shocks (e.g. \citealt{ss11}). We also set the power law index of the accelerated electrons to $p=2.2$, in accordance with typical values expected for the Fermi diffusive shock acceleration process (e.g., \citealt{Blandford1978,Drury1983}). Whilst it is desirable to allow $p$ to vary freely, since it influences other key model parameters (due to the energy supplied to the outer post-acceleration regions of the jet by the higher energy particles), it is not easily contrained due to the lack of data between the OIR and X-ray regions. We fix the scattering fraction (which determines the maximum energy to which particle are accelerated, thus setting an upper bound on the power law synchrotron cutoff) to $f_{\rm sc}=10^{-6}$ to ensure no significant direct contribution of optically thin synchrotron to the X-ray spectrum. This choice to suppress the X-ray synchrotron contribution is motivated by our objective to constrain the jet IC contribution to the X-ray spectrum of GX~339$-$4, and to limit degeneracies in tracking the jet properties in outburst. Also, a dominant jet synchrotron component in the X-rays likely predicts hard X-ray lags on timescales far shorter than those observed in GX~339$-$4 \citep{Nowak1999,Belloni2005,Altamirano2015}, based on short expected particle acceleration timescales within the jet \citep{Connors2017}. We note, on the other hand, that a non-negligible contribution in the X-ray band from synchrotron photons may be present without violating the observed lags.
The fundamental parameters of interest in \texttt{agnjet} are the normalised jet power, $N_{\rm j}$, the radius of the jet base, $r_0$, the dimensionless initial electron temperature, $\Theta_{\rm e}$, and the ratio of energy density between the electrons and magnetic field at the jet base, $\beta_{\rm e}$, all of which remain free parameters in the minimisation process.
We set the input photon distribution of \texttt{nthcomp} as a multi-temperature disc blackbody, and tie the disc temperature $T_{\mathrm{BB}}$ to the multi-temperature disc component within \texttt{agnjet}, $T_{\rm in}$, and allow the inner disc radius, $R_{\rm in}$, to vary between $\sim1.5$--10$r_{\rm g}$. Values higher than $10r_{\rm g}$ for the typical disc temperatures result in unrealistically high accretion rates, close to the Eddington rate for the given black hole mass, and lead to excessive soft X-ray fluxes which plainly disagree with the observations. The coronal electron temperature, $kT_{\mathrm{e,cor}}$, and spectral index, $\Gamma_{\mathrm{cor}}$ are free parameters of the model. The disc and coronal normalisations are treated separately, the disc being normalised by the black hole mass and distance parameters of \texttt{agnjet}, and coronal normalisation an independent counts normalisation constant inherent to the \texttt{nthcomp} model. As discussed in Section \ref{subsec:initialxrayfits}, we fix the centroid energy and width of the Gaussian iron line, $E_{\mathrm{line}}$ and $\sigma_\mathrm{line}$, according to the initial fits to each individual X-ray spectrum, after having fully explored the parameter distributions using MCMC parameter exploration.
After using minimization to converge as closely as possible on the global minimum of the fits to all 20 broadband spectra (characterized by the $\chi^2$ fit statistic), we initialise MCMC walkers around the maximum likelihood estimates of parameters in Model~B2, allowing the parameter search to explore the contributions of \texttt{agnjet} and \texttt{nthcomp}. Each MCMC chain is allowed to run for $10^3$--$10^4$ steps (on the basis of computational time constraints) such that the resultant posterior PDFs of the model parameters show coverage of the broad range intrinsic to models B1 and B2, and there is no longer significant evolution in those PDFs. We then discard the first 80\% of the MCMC chains, as this is well beyond the characteristic "burn-in" phase after which the chain is close to convergence, and the resultant walker distribution is populated enough to cover the parameter space.
Figure~\ref{fig:all_data} shows the full broadband spectrum from each observation, in unfolded flux space, colour-coded to show the evolution of each outburst in 4 separate panels, which we include to give clarity on the spectral evolution of GX~339$-$4 associated with all the datasets we model here.
Figures~\ref{fig:allfits_ssc} and \ref{fig:allfits_nthcomp} show the standardised data$-$model residuals from model fits to all 20 broadband spectra of GX~339$-$4. Specifically, Figure~\ref{fig:allfits_ssc} shows residuals in which IC emission from \texttt{agnjet} is the dominant X-ray spectral component, i.e., Model~B1. Figure~\ref{fig:allfits_nthcomp} shows residuals from fits to the same spectra in which the coronal IC emission of \texttt{nthcomp} dominates the X-ray spectra, i.e., Model~B2. The first thing we notice, is that Model~B2 (due to the additional presence of a corona, i.e., \texttt{nthcomp}) provides a much better fit to the X-ray spectra than Model~B1 in each case ($\chi^2_{R}\sim1$--2 for Model~B2 compared to $\chi^2_{R}\ge$ a few for Model~B1), due to the lower electron temperatures and higher optical depths inherent to the model: $kT_{\mathrm{e,cor}}/mc^2 \sim$ 0.02--0.4 (though we note that solutions permit coronal electron temperatures of up to $\sim1000$~keV, or $\Theta_{\rm e}\sim2$, see Table~\ref{tab:conf}). This is seen more clearly in Figure~\ref{fig:specfits} where fits to four spectra (one from each year of observation) are shown in more detail, emphasizing the model components. One can see that constraints provided by the radio and OIR data result in a non-negligible contribution from the jet in the X-ray band.
We find optical depths in the corona from all our B2 fits to be in the range $\tau \sim 0.1$--$1$, assuming the corona has a spherical geometry, and this is the key discriminator between the jet and coronal models we have considered. The electrons at the base of the jet in \texttt{agnjet} are strictly relativistic, and they remain quasi-isothermal throughout the jet. The optical depth in the jet base ranges between $\tau \sim10^{-4}$--$10^{-2}$. These conditions give rise to an emergent IC spectrum that is not a power law, but instead has significant curvature. This spectral curvature is a distinguishing feature of thermal Comptonisation (including SSC from a thermal particle distribution) with low optical depth ($\tau \ll 1$) and high electron temperature ($\Theta_{\rm e} \ge 1$). Due to the spectral curvature of the IC emission in \texttt{agnjet} in Model~B1, the reflection fraction ($R_{\mathrm{f}}$) systematically increases (see Figures~\ref{fig:specfits} and \ref{fig:Rf}). Thus fits with Model~B1, in which the jet IC emission dominates, require much stronger reflection than fits with Model~B2 in which the coronal IC emission dominates (see Figure~\ref{fig:Rf}). This increase in $R_{\mathrm{f}}$ is in disagreement with values derived from simpler X-ray spectral fits, and it is unlikely that a curved IC spectrum from the jet conspires with reflection to reproduce stable power law spectra over time. There is also a prominent apparent residual feature around 8--9~keV in most of the Model~B1 fits, which is clearly visible the residuals of all plots shown in Figure~\ref{fig:allfits_ssc}. This is a consequence of the curvature of the SSC component along with the dominance of the reflection component. In contrast, in fits of Model~B2, the power-law-like continuum of \texttt{nthcomp} fits well to the spectrum with the need for only a minimal reflection component and a Gaussian line to account for the line and added curvature.
We also notice in fits of Model~B2 (see Figure~\ref{fig:specfits}) that the presence of non-negligible IC emission from the jet (we find that the jet contributes a range of a few up to $\sim50$\% of the continuum flux in the 3--100~keV band) acts to skew the shape of the model coronal spectrum. The extent to which the jet contributes to the X-ray spectrum is illustrated quantitatively for all the fits in Figure~\ref{fig:Ljet}, which shows the 3--100~keV jet radiative luminosity $L_{\rm \gamma, jet}$ with respect to the total radiative luminosity of the model $L_{\rm \gamma, total}$, as a function of $L_{\rm X}$ and hue. The Bolometric coronal luminosity is also shown, which highlights the progressive brightening of the corona (ranigng from $<1\%$--$10\%$~$L_{\rm Edd}$) with $L_{\rm X}$ for clarity. We see no obvious trends in $L_{\rm \gamma, jet}$, but it is noticeable that the jet can have a significant contribution in the X-ray, and this is due to the jet dominating the OIR and radio fluxes. If the jet contributes to the X-ray spectrum, the corona may either have a softer or harder spectral shape than would be concluded if the jet were to be ignored. This possibility then opens up a myriad of interesting questions to explore regarding the contributions of each of these components to the X-ray variability of BHBs, in particular the hard X-ray lags (e.g., \citealt{Nowak1999,Belloni2005,Altamirano2015}).
The key results of our broadband model fits are that a coronal-like IC-scattering spectrum fits best to the data, whereby the electrons doing the scattering are at hundreds of keV, the plasma has optical depths in the range 0.1--1, and the photons being scattering originate in the disc with temperature around 0.1--1~keV. Such a scattering plasma produces the canonical power-law in the X-ray, with reflection reproducing the iron emission line and Compton reflection hump. However, SSC emission from hotter jet electrons within a plasma of optical depth in the range $\sim10^{-4}$--$10^{-2}$ likely has a non-negligible contribution in the X-ray, and this is constrained by the radio and OIR data, which can be well modelled by synchrotron from the hot jet electrons. Again we stress that any contribution of jet synchrotron emission to the X-ray bands has been suppressed, and a modelling treatment that includes that optically thin synchrotron component would add futher nuance to this conclusion.
\subsection{Jet parameter trends}
\label{subsec:global_trends}
We explore trends in the physical properties of the jet as a function of both the Eddington-scaled X-ray luminosity and variability properties (gauged by the power-spectral hue) of GX~339$-$4 during the different stages of its outburst rise and decay. Even though the coronal IC component dominates the X-ray spectra in Model~B2 (and Model~B2 provides superior fits to all our broadband datasets than Model~B1), the trends in key physical jet parameters (such as jet power) are similar to those found when fitting with Model~B1. This is because the radio and OIR data allow constraints on the jet physics. The main differences between the two models are firstly that the SSC emission from the jet is necessarily suppressed in Model~B2 to accommodate the dominant coronal component in the X-ray, and secondly that the reflection features are less prominent in Model~B2 (since the power-law like coronal spectrum accounts for most of the fit residuals in the X-ray).
Figure \ref{fig:jetpars} shows the maximum likelihood estimates (MLEs) of jet parameters $N_{\rm j}$, $r_0$, $\Theta_{\rm e}$ and $\beta_{\rm e}$ respectively as a function of $L_{\mathrm{X}}/L_{\mathrm{Edd}}$ and power-spectral hue. Table~\ref{tab:conf} shows the numerical values of the best-fit parameters of Model~B2, and their confidence limits (we show only the best-fit values of Model~B2 as it achieves better fits to all 20 broadband spectra).
The normalised jet power, $N_{\rm j}$, increases with increasing $L_{\mathrm{X}}$, and this is a clear trend despite the uncertainty on its value. We see that given the similar X-ray luminosities during the 2005 and 2011 outburst decays, $N_{\rm j}$ remains roughly constant as the hue decreases, until the source progresses further into the low hard state in the latter stages of outburst decay, at which point $N_{\rm j}$ decreases. This decrease is only seen in the 2005 decay, despite the similar X-ray luminosities between the 2005/2011 observations. This is likely due to the lower radio flux measured in the 2005 decay (see Figure~\ref{fig:data_fluxes}).
\begin{figure*}
\centering
\includegraphics[width=0.48\linewidth]{jan19_Nj_v_hue_and_lx_by_year.pdf}
\includegraphics[width=0.48\linewidth]{jan19_r0_v_hue_and_lx_by_year.pdf}
\includegraphics[width=0.48\linewidth]{jan19_Te_v_hue_and_lx_by_year.pdf}
\includegraphics[width=0.48\linewidth]{jan19_k_v_hue_and_lx_by_year.pdf}
\caption{The MLEs (and 90\% confidence limits) of the normalised jet power, $N_{\rm j}$ (top left), jet-base radius, $r_{\rm 0}$ (top right), electron temperature, $\Theta_{\rm e}$ (bottom left), and ratio of electron to magnetic energy density, $\beta_{\rm e}$ (bottom right), as a function of Eddington-scaled X-ray data luminosity (left) and power-spectral hue (right). Gray hollow squares show the parameters derived from fits of Model~B1, and other symbols (indicated in the key) show fits of Model~B2, divided according to observation year.}
\label{fig:jetpars}
\end{figure*}
The jet-base radius is poorly constrained across all fits, and has a broad range from 10s to 100s of $r_{\rm g}$. There is tentative evidence for lower values of $r_0$ during the 2011 outburst decay, likely due to the degeneracy inherent between $N_{\rm j}$ and $r_0$. A decrease in $N_{\rm j}$ is constrained by decreasing radio and OIR flux, and this independent constraint on $N_{\rm j}$ is accounted for by a decrease in $r_0$ in order to fit the X-ray spectrum.
Despite the systematically higher values of $\Theta_{\rm e}$ when the jet IC emission dominates the X-ray spectrum, in both cases the trends are similar: $\Theta_{\rm e}$ decreases slightly with increasing power-spectral hue. This is because $\Theta_{\rm e}$ is not solely constrained by the X-ray spectrum. The hardening of the optical spectra in all 20 of our datasets is modelled by thermal synchrotron emission from the optically thin regions of the jet. The optical hardening, alongside the contribution of synchrotron emission to the radio flux at larger distances in the jet, act to constrain $\Theta_{\rm e}$. In addition, $\Theta_{\rm e}$ appears lower during the early stages of the 2011 outburst decay than in all other fits. This constraint is determined by the lower OIR fluxes (relative to radio/X-ray fluxes) in the 2011 spectra, as shown in Figure~\ref{fig:data_fluxes}.
We see no clear global correlation between the plasma $\beta_{\rm e}$, and the power-spectral hue, or $L_{\rm X}$, except for an apparent increase at the highest hue values, i.e., as the broadband X-ray variability is becoming narrower. This trend appears to only exist in modelling of the 2011 outburst decay, and is likely a consequence of the lower OIR fluxes relative to the X-ray flux with respect to the other multiwavelength observations (see Figure~\ref{fig:data_fluxes})---the particle energy density increases with respect to its magnetic energy density, decreasing the relative synchrotron-to-SSC contribution to the broadband spectrum. The value of $\beta_{\rm e}$ ranges between $\sim$ 0.02--1 across all the fits, with most fits yielding $\beta_{\rm e} \sim 0.1$. The trend in the fitting process is for $\beta_{\rm e}$ to be pushed to values $<1$, i.e. a magnetically-dominated jet base, which is due to increases in $N_{\rm j}$, the jet power. $N_{\rm j}$ increases in accordance with the increase in radio flux irrespective of the jet's X-ray contribution, and $\beta_{\rm e}$ in theory decreases in order to reduce the electron density in the jet base (lower electron densities lead to a lower IC flux from the jet). $\beta_{\rm e}$ is also degenerate with $r_{\rm 0}$, such that a decrease in $r_{\rm 0}$ leads to higher electron energy densities, causing $\beta_{\rm e}$ to decrease in order to redistribute the available energy density to the magnetic field, re-normalising the IC contribution to the X-rays.
In summary, we see some evidence for parameter trends that provide a physical basis for the connection between the inner accretion flow (or corona) and the jet. In particular we see distinctions between outburst rise and decay, and these changes are well tracked by the broadband X-ray variability. The jet power, $N_{\rm j}$, increases with $L_{\rm X}$ as expected. The jet base radius, $r_{\rm 0}$, is poorly constrained, with some evidence for a drop at high values of the power-spectral hue. The jet-base electron temperature, $\Theta_{\rm e}$, decreases with power-spectral hue. The ratio of electron-to-magnetic energy density shows no broad correlation, but increases with power-spectral hue and $L_{\rm X}$ in the 2011 decay.
\subsection{Coronal parameter trends}
\label{subsubsec:corona_or_jet}
Any trends we may expect in the coronal properties are unsurprisingly dampened by the presence of non-negligible jet contributions to the X-ray spectrum. Nonetheless some patterns exist that are worth discussing briefly. \\
\indent Figure~\ref{fig:pars_v_hue_lx_nthcomp} shows the trends of the spectral index of the IC power-law in the corona, $\Gamma_{\mathrm{cor}}$, and the coronal electron temperature, $kT_{\mathrm{e,cor}}$, with $L_{\rm X}$ and hue. There is no observable trend between $kT_{\mathrm{e,cor}}$ and $L_{\rm X}$ or hue, any potential correlation is likely quenched by the fact that in most of the 20 GX~339$-$4 spectra the X-ray spectral coverage and photons counts are insufficient to constrain the cut-off energy, and the jet IC spectrum introduces significant scatter due to its high fractional contribution to the X-ray flux. There is a correlation between $\Gamma_{\mathrm{cor}}$ and hue and $L_{\rm X}$ during each outburst rise/decay (and striking monotonicity as a fucntion of hue, likely concurrent with X-ray hardness). Whilst a trend is expected based on our initial fits to the X-ray spectra (Section \ref{subsec:initialxrayfits}), there is added scatter in the slope again caused by the non-negligible contribution from IC emission in the jet base.
\begin{figure}
\centering
\includegraphics[width=0.53\textwidth]{jan19_pars_v_hue_and_lx_by_year.pdf}
\caption{The MLEs (and 90\% confidence limits) of the photon index ($\Gamma_{\mathrm{cor}}$; top) and electron temperature ($kT_{\mathrm{e,cor}}$; bottom) of \texttt{nthcomp} as a function of Eddington-scaled X-ray data luminosity (left) and power-spectral hue (right). The data are divided according to observation year and thus track separate outbursts.}
\label{fig:pars_v_hue_lx_nthcomp}
\end{figure}
\begin{table*}
\centering
\caption{The maximum likelihood estimates and 90\% confidence limits of fit-parameters of Model~B2 to all 20 broadband spectra of GX~339$-$4. Values are appropriately quoted to significance of the confidence limits, and thus may not match exactly the values show in Figures~\ref{fig:Rf}--\ref{fig:pars_v_hue_lx_nthcomp}. From left to right: (1) spectrum number, (2) $N_{\rm j}$, the normalised jet power, (3) $r_0$, the jet base radius, (4) $\Theta_{\rm e}$, the electron temperature in the base of the jet, (5) $\beta_{\rm e}$, the ratio of electron to magnetic energy density in the jet, (6) $T_{in}$, the inner disc temperature, (7) $R_{\rm in}$, the inner disc radius, (8) $\Gamma_{\mathrm{cor}}$, the photon index of the thermal Compton spectrum in the corona, (9) $kT_{\mathrm{e,cor}}$, the electron temperature in the corona, (10) $R_{\mathrm{f}}$, the reflection fraction, (11) chi-squared ($\chi^2$) over degrees of freedom (DoF). }
\begin{tabular}{@{}p{0.2cm}lllllllllr}
\hline
\# & $N_{\rm j}$ & $r_0$ & $\Theta_{\rm e}$ & $\beta_{\rm e}$ & $T_{\rm in}$ & $R_{\rm in}$ & $\Gamma_{\mathrm{cor}}$ & $kT_{\mathrm{e,cor}}$ & $R_{\mathrm{f}}$ & $\chi^2$/DoF\\
& [$10^{-3}$] & [$r_{\rm g}$] & & & [keV] & [$r_{\rm g}$] & & [keV] & & \\
\hline
\\
1 & $16^{+15}_{-1}$ & $230^{+10}_{-80}$ & $1.09^{+0.09}_{-0.08}$ & $0.11^{+0.01}_{-0.08}$ & $0.30^{+0.03}_{-0.18}$ & $2^{+6}_{-0}$ & $1.70^{+0.08}_{-0.04}$ & $160^{+780}_{-30}$ & $<0.3$ & 87/68 \\
2 & $31^{+15}_{-7}$ & $100^{+20}_{-8}$ & $1.76^{+0.03}_{-0.25}$ & $0.02^{+0.01}_{-0.01}$ & $0.19^{+0.04}_{-0.09}$ & $2.0^{+7.5}_{-0.3}$ & $1.64^{+0.04}_{-0.04}$ & $150^{+790}_{-20}$ & $<0.1$ & 106/101\\
3 & $39^{+5}_{-13}$ & $110^{+20}_{-10}$ & $1.6^{+0.1}_{-0.2}$ & $0.013^{+0.013}_{-0.002}$ & $0.15^{+0.08}_{-0.05}$ & $5^{+5}_{-3}$ & $1.62^{+0.04}_{-0.04}$ & $400^{+500}_{-300}$ & $<0.9$ & 101/91\\
4 & $11^{+9}_{-5}$ & $120^{+60}_{-30}$ & $1.8^{+0.4}_{-0.4}$ & $0.10^{+0.47}_{-0.07}$ & $0.18^{+0.10}_{-0.07}$ & $6^{+23}_{-5}$ & $1.7^{+0.3}_{-0.2}$ & $400^{+600}_{-300}$ & $<0.5$ & 54/30\\
5 & $5.7^{+1.1}_{-0.5}$ & $170^{+30}_{-30}$ & $1.9^{+0.3}_{-0.2}$ & $0.7^{+1.1}_{-0.3}$ & $0.15^{+0.06}_{-0.05}$ & $3^{+3}_{-1}$ & $1.78^{+0.22}_{-0.08}$ & $500^{+400}_{-400}$ & $0.10^{+0.17}_{-0.09}$ & 63/64\\
6 & $1.2^{+2.0}_{-0.3}$ & $80^{+60}_{-30}$ & $18^{+3}_{-6}$ & $0.13^{+0.19}_{-0.02}$ & $0.2^{+0.2}_{-0.1}$ & $10^{+0}_{-8}$ & $1.77^{+0.19}_{-0.09}$ & $300^{+700}_{-200}$ & $0.10^{+0.53}_{-0.08}$ & 16/29\\
7 & $50^{+12}_{-7}$ & $200^{+20}_{-70}$ & $1.9^{+0.1}_{-0.2}$ & $0.10^{+0.03}_{-0.03}$ & $0.13^{+0.04}_{-0.02}$ & $6^{+4}_{-4}$ & $1.82^{+0.02}_{-0.02}$ & $30^{+5}_{-4}$ & $0.24^{+0.05}_{-0.04}$ & 351/166\\
8 & $10^{+9}_{-5}$ & $92^{+83}_{-2}$ & $1.9^{+0.2}_{-0.5}$ & $0.12^{+0.56}_{-0.09}$ & $0.26^{+0.02}_{-0.15}$ & $3^{+7}_{-1}$ & $1.8^{+0.3}_{-0.2}$ & $24^{+440}_{-9}$ & $0.10^{+0.76}_{-0.07}$ & 38/32\\
9 & $7^{+8}_{-1}$ & $230^{+40}_{-40}$ & $2.1^{+0.1}_{-0.4}$ & $0.4^{+0.3}_{-0.3}$ & $0.14^{+0.13}_{-0.03}$ & $10^{+0}_{-8}$ & $1.65^{+0.06}_{-0.02}$ & $>100$ & $0.10^{+0.06}_{-0.09}$ & 116/71\\
10 & $20^{+6}_{-6}$ & $150^{+10}_{-10}$ & $2.3^{+0.2}_{-0.1}$ & $0.10^{+0.14}_{-0.04}$ & $0.23^{+0.08}_{-0.09}$ & $5^{+4}_{-3}$ & $1.52^{+0.08}_{-0.03}$ & $500^{+400}_{-400}$ & $0.05^{+0.08}_{-0.04}$ & 118/65\\
11 & $33^{+13}_{-8}$ & $120^{+20}_{-10}$ & $3.9^{+0.4}_{-0.7}$ & $0.018^{+0.009}_{-0.005}$ & $0.3^{+0.1}_{-0.1}$ & $5^{+5}_{-3}$ & $1.50^{+0.04}_{-0.02}$ & $<900$ & $0.4^{+0.2}_{-0.2}$ & 86/67\\
12 & $40^{+10}_{-20}$ & $150^{+20}_{-10}$ & $2.5^{+0.2}_{-0.2}$ & $0.03^{+0.10}_{-0.01}$ & $0.2^{+0.1}_{-0.1}$ & $5^{+4}_{-4}$ & $1.54^{+0.11}_{-0.03}$ & $500^{+500}_{-400}$ & $0.07^{+0.11}_{-0.06}$ & 163/65\\
13 & $22^{+4}_{-5}$ & $190^{+20}_{-10}$ & $2.6^{+0.1}_{-0.1}$ & $0.18^{+0.21}_{-0.06}$ & $0.21^{+0.09}_{-0.09}$ & $4^{+4}_{-2}$ & $1.59^{+0.09}_{-0.06}$ & $600^{+400}_{-400}$ & $0.08^{+0.08}_{-0.06}$ & 147/67\\
14 & $70^{+20}_{-20}$ & $200^{+30}_{-20}$ & $2.0^{+0.1}_{-0.2}$ & $0.04^{+0.04}_{-0.01}$ & $0.2^{+0.1}_{-0.1}$ & $4^{+5}_{-3}$ & $1.78^{+0.08}_{-0.07}$ & $500^{+500}_{-400}$ & $0.3^{+0.1}_{-0.1}$ & 150/65\\
15 & $71^{+9}_{-7}$ & $170^{+32}_{-3}$ & $2.05^{+0.08}_{-0.09}$ & $0.042^{+0.007}_{-0.008}$ & $0.35^{+0.02}_{-0.14}$ & $3^{+2}_{-1}$ & $1.75^{+0.08}_{-0.04}$ & $100^{+250}_{-60}$ & $0.1^{+0.2}_{-0.0}$ & 750/66\\
16 & $45^{+5}_{-6}$ & $108^{+28}_{-4}$ & $1.7^{+0.1}_{-0.1}$ & $0.14^{+0.05}_{-0.03}$ & $0.17^{+0.03}_{-0.06}$ & $3^{+6}_{-1}$ & $1.96^{+0.09}_{-0.1}$ & $300^{+700}_{-200}$ & $0.03^{+0.18}_{-0.00}$ & 200/66\\
17 & $8^{+3}_{-0}$ & $82^{+2}_{-26}$ & $1.00^{+0.02}_{-0.00}$ & $1.3^{+0.2}_{-1.0}$ & $0.18^{+0.05}_{-0.06}$ & $2.4^{+2.2}_{-0.8}$ & $1.97^{+0.13}_{-0.08}$ & $400^{+600}_{-300}$ & $0.3^{+0.3}_{-0.2}$ & 217/35\\
18 & $10^{+4}_{-1}$ & $56^{+5}_{-7}$ & $1.06^{+0.13}_{-0.05}$ & $0.4^{+0.2}_{-0.2}$ & $0.19^{+0.09}_{-0.08}$ & $10^{+0}_{-8}$ & $1.9^{+0.2}_{-0.2}$ & $90^{+840}_{-40}$ & $0.6^{+0.4}_{-0.4}$ & 87/35\\
19 & $13^{+7}_{-4}$ & $80^{+14}_{-18}$ & $1.2^{+0.1}_{-0.1}$ & $0.16^{+0.21}_{-0.10}$ & $0.23^{+0.06}_{-0.11}$ & $1.9^{+5.1}_{-0.2}$ & $1.75^{+0.24}_{-0.07}$ & $<200$ & $0.10^{+0.38}_{-0.09}$ & 15/34\\
20 & $25^{+15}_{-6}$ & $88^{+23}_{-5}$ & $1.87^{+0.07}_{-0.33}$ & $0.02^{+0.02}_{-0.01}$ & $0.22^{+0.07}_{-0.11}$ & $6^{+4}_{-4}$ & $1.7^{+0.3}_{-0.1}$ & $<100$ & $0.10^{+0.54}_{-0.08}$ & 40/30\\
\\
\hline
\end{tabular}
\label{tab:conf}
\end{table*}
\subsection{Pair processes?}
\label{subsec:pairs}
The importance of pair processes in jet models of bright hard state BHBs, GX~339$-$4 in particular, was explored by \cite{Maitra2009}, in which a previous version to the current \texttt{agnjet} model was fit to broadband spectra of GX~339$-$4. \cite{Maitra2009} made estimates of the pair production and annihilation rates and based on those rates, adjusted their modelling to an area of parameter space in which the influence of pairs on the particle distribution and resultant spectrum were negligible. Here we expand slightly on this approach by providing a more self-consistent estimate of the energy density of pairs by numerically calculating the resultant pair distribution due to the mutual interaction of each photon field in the jet (i.e. synchrotron, SSC as well as raw and IC-scattered disc photons).
We can calculate the particle distribution self-consistently with radiative and other cooling losses balanced with the source terms, which include pair injection. Pair injection and annihilation is calculated following the formalism of \cite{Mastichiadis1995} (their Equations 57 and 60), adopting the cross-sections and production rates provided by \cite{Coppi1990}. We calculate the energy density of pairs in the base of the jet self-consistently, with synchrotron losses included. The energy density of pairs depends strongly on the most energetic photons produced by SSC in the jet, as well as IC scattering of disc photons which is seen to produce high-energy X-rays (as shown in Figure~\ref{fig:specfits}). As such, for a given set of jet parameters, the relevance of pairs may depend on the disc parameters, $R_{\mathrm in}$ and $T_{\mathrm in}$. Figure~\ref{fig:pairs} shows the ratio of the energy density of pairs in the jet base to that of the input electron energy distribution, as a function of both $R_{\mathrm in}$ and $T_{\mathrm in}$, with all 20 fit solutions marked on the plot. The energy distribution of pairs and primary electrons is then shown for one particular fit solution. One can see that for the range of best fit values found, and for the full range of $R_{\mathrm in}$ and $T_{\mathrm in}$, the energy density of pairs is comparable to the input electron energy density. However, whilst the number density is on the order of the primary number density, the average energy of the secondaries is far lower ($\gamma_e\sim1$ compared with $\gamma_e\sim20$), and closer to the non-relativistic regime, and thus they will not contribute significantly to the observed emission. As discussed clearly in Section~\ref{sec:model}, \texttt{agnjet} is dynamically dominated by its initial rest mass energy density, and so the creation of pairs in the jet, though comparable in energy density to the primary electrons, will likely not alter the dynamics significantly enough to warrant a full calculation of its effects. Such a calculation is beyond the scope of this paper, and we leave the dynamical effects of pair production in the jet to future work.
\begin{figure*}
\includegraphics[width=0.49\linewidth]{upairs_rin_tin.pdf}
\includegraphics[width=0.49\linewidth]{pairs_dist.pdf}
\caption{{\bf Left:} Ratio of the energy density of electron-positron pairs to primary electrons in the base of the jet ($U_{\rm pairs}/U_{\rm prim}$), shown for a range of inner disc temperature ($T_{\rm in}$) and radius ($R_{\rm in}$). Red crosses show the values corresponding to the best fit to all 20 datasets, with a fit to a bright observation in 2010, MJD~55271 (spectrum~13), marked by the red triangle, the distribution for which is shown in the right hand figure. {\bf Right:} The energy distribution of primary electrons and secondary pairs in the base of the jet, showing the raw distribution and the absorbed one, generated using the same parameters indicated by the red triangle in the left hand figure.}
\label{fig:pairs}
\end{figure*}
\section{Discussion}
\label{sec:discussion}
\indent Previous modelling of GX~339$-$4 with older versions of \texttt{agnjet} proposed a significant contribution in the X-ray from optically thin non-thermal synchrotron emission, either dominating the full observable X-ray band, or solely the soft band ($<10$~keV), with jet IC dominating the harder emission \citep{Markoff2003,Maitra2009}. Here we have instead considered the case in which synchrotron emission is suppressed and the jet's X-ray contribution is almost entirely dominated by thermal SSC, with some contribution from IC-scattered disc photons. There can also be contributions in the X-ray from synchrotron-emitting non-thermal electrons in the jet base or inner accretion flow, given that both are collisionless, turbulent regions in which particle acceleration can occur. \cite{Connors2017} explore this scenario in modelling of Sgr~A*, the Galactic centre supermassive black hole, and A0620-00, a BHB in quiescence, and though they both have significantly lower X-ray luminosities than GX~339$-$4 ($L_{\rm X}/L_{\rm Edd} \sim 10^{-9}$), such a scenario cannot be ruled out in the case of GX~339$-$4. However, the millisecond-to-second timescale hard X-ray lags observed in the hard state of GX~339$-$4 (see, e.g., \citealt{Nowak1999,Belloni2005,Altamirano2015}) do not favour particle acceleration as being responsible for the delayed hard X-ray emission due to the rapid timescales predicted by various particle acceleration scenarios (see, e.g, \citealt{kc15,Connors2017}).
Thus IC emission is the most likely \textit{dominant} spectral component in the X-ray. BHB hard X-ray lags in the hard state \citep{Miyamoto1988,Kazanas1997,Nowak1999} are generally interpreted as a signature of the propagation of accretion rate fluctuations in the disc responding in the coronal hard emission through IC scattering of the disc photons (e.g., \citealt{Kotov2001}). It is quite apparent that jet SSC/IC scattering off hot electrons in a low-density plasma, such as the conditions presented in the model \texttt{agnjet}, is unlikely to reproduce such lags, due to the low number of IC scatterings, and the dominance of jet synchrotron photons as the input distribution for scattering. These arguments provide both a strong qualitative and quantitative argument for IC scattering in a corona of electron temperatures in the realm of hundreds of keV with optical depths on the order of 0.1--1 as the dominant X-ray emission component in GX~339$-$4. We highlight the allusions made by \cite{Nowak2005} and \cite{Wilms2006} to the presence of multiple hard X-ray components in the low/hard state, a scenario that has already been postulated/explored for GX~339$-$4 \citep{Fuerst2015}. Recent work on spectral-timing models of BHBs in the low/hard state also postulates that there are likely two Comptonisation regions in the accretion flow \citep{Mahmoud2018}, with the only distinction from our proposed geometry being that both components are part of the gas inflow.
In Section~\ref{subsec:global_trends} we outlined that the jet is always found to be magnetically-dominated, with $\beta_{\rm e}<1$ generally holding true in all our fits. We also note that the range of the jet-base magnetisation (defined as the ratio of magnetic enthalpy to rest mass, $\sigma=B^2/4 \pi n m_p c^2$, in a force-free magnetohydrodynamic plasma such that the gas pressure is neglected) derived from the best-fit parameters always lies in the range of $\sigma=1$--$2$ (so consistently of order unity such that the magnetic field is never sub-dominant). Whilst it is important to stress that the model \texttt{agnjet} does not allow dynamically important magnetic fields (i.e. high magnetisation), we can nonetheless conclude that our modelling is dynamically consistent given the final Lorentz factors are mildly relativistic. The magnetisation necessary for jet-launching based on recent simulations of black hole jets \citep{Tchekhovskoy2010,Tchekhovskoy2011,Sadowski2013} is typically higher (order 10). However, such simulations bias themselves toward Poynting-dominated jets due to difficulties with mass-loading into the jet. Thus given that the methods and regimes adopted by our modelling and jet-launching simulations are wholly different, it would be a misnomer to make a direct comparison.
Coronal models in the context of spectral softening in BHBs predict the inward progression of the optically thick accretion disc, leading to increased cooling in the corona, and thus a lower temperature and a softer spectrum \citep{Haardt1993,Ibragimov2005}. The question of during which part of the outburst the disc has extended down to the ISCO, remains a primary discussion point for BHBs in general, none more so than GX~339$-$4. Whilst most agree that in the brightest hard states the disc extends down to the ISCO (e.g. \citealt{Gierlinski2004,Penna2010}), \cite{Miller2006} claim the disc in GX~339$-$4 sits at the ISCO throughout the low/hard state. \cite{Done2007} strongly contest this and instead claim the disc is significantly truncated and gradually moves inwards during the rise of an outburst, with an ADAF at $r < R_{\rm in}$. \cite{Kara2019} recently showed, through reverberation mapping of the X-ray emitting regions of BHB MAXI~J1820$+$070, alongside spectral modelling of the iron K line, that the coronae of BHBs are likely contracting as the source brightens in the bright hard state, whilst the disc has already reached the ISCO. As discussed in Section~\ref{subsec:global_trends}, we see evidence for decreasing jet electron temperature during the evolution of an outburst (constrained by the full broadband spectrum), but no clear trend in the corona (which is mostly constrained by the X-ray spectrum). We also find, shown in Table~\ref{tab:conf}, that $R_{\rm in}$ is likely within 10~$r_{\rm g}$ during all observations, though we stress that our constraints are weak. However, we do not find evidence for the contraction of the jet base during outburst rise or decay. We thus propose that a complete understanding of the evolution of the X-ray emitting region does need to consider the jet-corona-disc system as a whole. Though a conclusion has yet to be reached on a ubiquitous answer to this debate, it certainly appears likely that the accretion discs of BHBs are not heavily truncated during the bright hard state.
Additionally, no interpretation has yet explained why transitions between the dominant optically thin inner flow and optically thick accretion disc occur over a broad range of X-ray luminosity in BHBs \citep[$L_X/L_{Edd} \sim 0.003$--0.2]{Done2003}. Observations indicate variations in the transition luminosity within the same source, and a tendency for sources to transition at higher luminosities in outburst rise than in decay \citep{Nowak1995,Maccarone2003,Done2007} (i.e., BHB hysteresis). We have been able to track changes to the plasma conditions in the jet base whilst postulating that a separate coronal component dominates the X-ray spectrum. There are indications of a distinction between the jet properties in outburst rise and decay (at the same X-ray hardness), and these changes appear to trace the shape of the broadband X-ray variability. Thus the broadband properties of the source can point to a way to understand the hysteresis of BHBs. Since the low OIR fluxes during the onset of outburst decay (see Figure~\ref{fig:data_fluxes}) likely indicate cooler jet electrons with respect to the outburst rise, this may be further evidence for distinct plasma conditions in the inner accretion flow between the two regimes.
Multiple broadband studies of GX~339$-$4 in the hard state and across both the hard-to-soft (outburst rise) and soft-to-hard (outburst decay) state transitions have concluded that emission from the jet dominates the spectrum in the radio-to-OIR bands \citep{Corbel2002,Homan2005,Russell2006,Gandhi2008,Coriat2009,Gandhi2011,Buxton2012}, and even perhaps into the UV bands \citep{Yan2012}. However, the nature of the emission is still uncertain. Whilst some claim the jet synchrotron break occurs in the mid-Infrared ($>10^{13}$~Hz; \citealt{Gandhi2011}), others conclude that the optically thick portion of the jet spectrum extends from the radio to beyond the V band ($>10^{14}$~Hz; \citealt{Coriat2009,Dincer2012,Buxton2012})---we note here that these conclusions are not all based on the same observations, and we may expect differences in the break location during different outbursts. A bias exists in our modelling, since we have fixed the location of particle acceleration in the jet, $z_{\rm acc}$. However, our results show that the flatter portion of the lower-frequency IR spectra and the bluer portion of the optical spectra can be modelled as a superposition of thermal and non-thermal jet synchrotron components, where the break frequency is always situated below the OIR bands. It should be noted that although several of our fits fail to capture the indices of the OIR spectra, in many cases the combination of thermal and non-thermal synchrotron emission can easily conspire to hide the jet break in the observed spectrum and successfully reproduce the optical up-turn.
We also find that during the decay of the 2011 outburst the jet break should be more pronounced due to the OIR dip relative to the radio flux, and the inversion of the OIR spectrum is well-modelled by thermal synchrotron. We cannot rule out the contribution from disc reprocessed emission during the soft-to-hard transition, but at these low X-ray luminosities ($\le 0.01~L_{\mathrm{Edd}}$) the jet spectrum is most likely to be dominating in the optical \citep{Gandhi2008}. Evidence for the optical emission of BHBs being dominated by synchrotron radiation in the jet within $\sim10^3~r_\mathrm{g}$ of the accreting black hole has now been seen in several BHBs, XTE~J1118$+$480 \citep{Kanbach2001}, GX~339$-$4 \citep{Gandhi2008,Gandhi2011}, the recently discovered transient MAXI~J1820$+$070 (see, e.g., \citealt{Townsend2018}) and V404~Cygni \citep{Gandhi2017}, with V404~Cygni showing confirmed activation of the self-absorbed radio jet alongside the onset of rapid optical variability. Our comprehensive modelling of GX~339$-$4 during both the rise and decay of multiple outbursts provides supporting evidence for a physical picture in which the jets of BHBs dominate the broadband spectrum at radio-to-OIR frequencies, and thus likely also contribute a non-negligible X-ray flux.
In a simplistic framework in which the corona is an outflowing, purely non-thermal plasma, to successfully explain the trend of increasing reflection fraction ($R_{\mathrm{f}}$) with X-ray power-law spectral slope ($\Gamma_{pl}$), we expect the bulk velocity of the corona ($\beta_j$) to decrease with increasing luminosity (see, e.g., \citealt{Beloborodov1999}). As noted by \cite{Done2007}, this disagrees with fundamental observations of BHB jet radio cores \citep{Fender2006}, where higher bulk velocities are observed at higher luminosities. However, a more complete outflow model with a physical connection between the bulk flow properties and dissipation of energy into the radiating electrons (beyond the physical jet model put forward in this work) points to other scenarios in which the correlation between $R_{\mathrm{f}}$ and $\Gamma_{pl}$ can be realised without violating requirements on the jet dynamics.
For example, in \texttt{agnjet} the electrons energies are in a Maxwell-J\"{u}ttner distribution with initial temperatures $\Theta_{\rm e} \ge 1$, and remain quasi-isothermal, cooling only in proportion to the jet acceleration in the z-direction ($T(z) = T_0[\gamma_j(z) \beta_j(z)]^{1-\Gamma}$, where $\Gamma = 4/3$ is the adiabatic index. The electrons in the outer regions of the jet must remain hot ($\Theta_{\rm e} \ge 1$) to reproduce the flat/inverted radio spectral index (and in our modelling particle acceleration occurs, so further energy has been dissipated into the electrons), but the electrons in the jet base may have low initial temperatures typical of coronae ($\Theta_{\rm e} \sim 0.2$), and heating can occur rapidly due to turbulence, shocks, thermal conduction or magnetic reconnection \citep{Quataert2000,Johnson2007,Sironi2015,Ressler2015,Rowan2017}. Our work here shows the importance of such a model. For example, the apparent decrease in jet-base electron temperature ($\Theta_{\rm e}$) with increasing power-spectral hue, i.e., as the source progresses through the hard state, agrees with the general consensus that as BHBs evolve through their outbursts the corona is cooling and becoming more compact \citep{Haardt1993,Ibragimov2005}. As discussed already, an unanswered question still exists as to the evolution of the coronal-disc setup, despite a recent breakthrough indicating that the disc remains close to the ISCO in the bright hard state \citep{Kara2019}. We argue that developing a clearer idea of how the corona and the jet interact may be a critical stepping stone in understanding the co-evolution of both with the accretion disc.
\section{Summary and conclusions}
\label{sec:conclusions}
We have combined a thermal IC-scattering corona (\texttt{nthcomp}: \citealt{Zdziarski1996,Zycki1999}) and a jet in broadband spectral modelling of GX~339$-$4, with two fundamental differences between the two IC scattering treatments: the input soft-photon distribution for the jet IC scattering (SSC + IC scattering of disc photons) in \texttt{agnjet} is dominated by thermal synchrotron photons, and the electrons are strictly relativistic ($\Theta_{\rm e} \ge 1$, $kT_e \ge 511~\mathrm{keV}$) within a plasma of low optical depth ($\tau\sim10^{-4}$--$10^{-2}$), whereas the input photons of the corona in \texttt{nthcomp} are disc blackbody photons at $T_{\mathrm{BB}} \sim$ 0.01--1 keV, and the electrons are typically on the order of $kT_e \sim10$s--$100$s of keV in a plasma of higher optical depth ($\tau\sim0.1$--$1$). Analogies to such a physical model can be found in many simulations of black hole accretion flows in which the inner flow is ADAF-like (geometrically-thick and optically thin) and the jet is a Poynting-dominated (we find jet-base magnetizations of order unity), low-density funnel launched via the Blandford-Znajek mechanism (see, e.g., \citealt{Mckinney2006,hk06,Tchekhovskoy2010,Tchekhovskoy2011,Sadowski2013}, and references therein). Only the cooler, higher optical depth coronal component of \texttt{nthcomp} can successfully reproduce the X-ray spectra of all 20 GX~339$-$4 datasets we modelled, and this is due to precisely the two identified model discriminants described. Given these conditions, the main results of our comprehensive modelling of GX~339$-$4 can be summarised in the following points:
\begin{itemize}
\item Even if IC scattering in the corona dominates the X-ray spectrum of GX~339$-$4 in the low/hard state, there will still likely be a non-negligible contribution from jet IC-scattered photons.
\item There are trends in the physical properties of the jet during both outburst rise and decay, even with the presence of a dominant coronal component, and these changes appear to show correlations with the shape of the broadband X-ray variability.
\end{itemize}
Addressing the former conclusion first, we find ratios of jet-to-corona continuum flux of a few to $\sim50\%$ in the 3--100~keV band across all fits. However we note that this conclusion is strongly model-dependent. The jet (\texttt{agnjet}) electrons are treated relativistically in a plasma at low optical depths ($\tau \le 0.01$). A treatment which includes cooler electrons in a region of higher optical depth, producing IC spectra with less curvature, would likely reduce the difference in spectral shape between the corona and jet base IC emission in our modelling (and in fact may return to a scenario where the `corona' is synonymous with the base of the jet). We cannot rule out a contribution to the X-ray from non-thermal optically thin jet synchrotron emission \citep{Markoff2003,Maitra2009}. We have artificially suppressed such a contribution in order to limit the degeneracies in our modelling, along with a strong argument for its non-dominance as a contribution in the X-ray emission of GX~339$-$4 (and other BHBs in the hard state)---a mix of synchrotron and IC jet emission, dominating the soft and hard X-rays respectively, struggles to explain the ubiquitous presence of hard X-ray lags (see, e.g., \citealt{Nowak1999,Belloni2005,Altamirano2015}). \\
\indent On the latter conclusion, by tracking the jet and coronal parameters as a function of both X-ray luminosity and the power-spectral hue (a simple characteriser of the shape of the broadband X-ray rms variability), we have shown some trends appear in the jet properties. As is expected, the jet power increases with X-ray luminosity, constrained primarily by the observed quasi-simultaneous radio flux. The jet-base electron temperature, $\Theta_{\rm e}$, can be seen to slightly decrease with increasing hue, thus coincident with the strengthening and narrowing of the broadband X-ray variability. The jet base is more compact with cooler electrons during the 2011 outburst decay of GX~339$-$4 as the shape of the X-ray variability strengthens and narrows. At lower values of the power-spectral hue, when the X-ray variability has a broader shape, we see no clear distinctions in the jet physics between outburst rise and decay. Our results point to a way of constraining the geometrical changes by linking the evolving X-ray variability in the inner regions to the plasma conditions further out in the jet.\\
\indent Determining the contribution of jet emission in the X-ray still remains a difficult task in the modelling of BHBs. The jet contribution must be quantified in order to better constrain the fraction of hard X-ray emission reflected off BHB accretion discs \citep{Ross1999,Ross2005,Dauser2010,Garcia2014,Garcia2015_2}, since if a significant fraction of the X-rays are beamed away from the disc, the emissivity profile along the disc is affected, and therefore the reflection fraction changes \citep{Dauser2013,WilkinsGallo2015} and the features relevant for determining the black hole spin and inner disc radius are altered. We will address the importance of the jet contribution to X-ray disc reflection in a forthcoming paper (Connors et al., in preparation). \\
\indent A significant caveat that all jet models so far suffer from is that the plasma conditions which determine the spectrum of the jet are disconnected from the jet dynamics. With \texttt{agnjet} for example, the velocity, particle density, and magnetic field profiles are pre-calculated dynamical quantities in the model, and the broadband spectrum follows from the radiative calculations, with cooling effects only incorporated into that resulting spectrum. An improved treatment would involve reducing the number of free parameters by physically linking the radiative calculations with the jet dynamics. Self-similar MHD solutions of a relativistic jet presented by \cite{Ceccobello2018} (building on work by \citealt{Polko2010,Polko2013,Polko2014}) provide the groundwork for such a treatment. By combining the plethora of dynamical jet solutions presented by \cite{Ceccobello2018} with radiative calculations such as those presented in this work, we shall in future be able to find more physically-realistic solutions for a given system (BHB or AGN) and perform model-fitting to retrieve more meaningful results with less degeneracies.
\section*{Acknowledgements}
We thank the referee for their useful comments which have gone a long way to improving this manuscript. DK thanks Maria Petropoulou for informative discussions regarding pair processes. RMTC also thanks Javier Garcia for useful discussions.
This research has made use of \texttt{ISIS} functions provided by ECAP/Remeis observatory and MIT (http://www.sternwarte.uni-erlangen.de/\texttt{ISIS}/).
RMTC is thankful for support from NOVA (Dutch Research School for Astronomy), and acknowledges funding from NASA grant No. No. 80NSSC177K0515. CC acknowledges support from the Netherlands Organisation for Scientific Research (NWO), grant Nr. 614.001.209. SM, DK, and ML acknowledge support from NWO VICI grant Nr. 639.043.513. VG is supported through the Margarete von Wrangell fellowship by the ESF and the Ministry of Science, Research and the Arts Baden-W\"urttemberg.
\bibliographystyle{mnras}
|
{
"timestamp": "2019-03-01T02:05:17",
"yymm": "1902",
"arxiv_id": "1902.10833",
"language": "en",
"url": "https://arxiv.org/abs/1902.10833"
}
|
\section{Introduction}
Constituent parsing is a core task in natural language processing (\textsc{nlp}), with a wide set of applications. Most competitive parsers are slow, however, to the extent that it is prohibitive of downstream applications in large-scale environments \cite{kummerfeldCorral}. Previous efforts to obtain speed-ups have focused on creating more efficient versions of traditional shift-reduce \cite{sagae2006best,zhang2009transition} or chart-based parsers \cite{collins1997three,charniak2000maximum}.
\newcite{zhu2013fast}, for example, presented a fast shift-reduce parser with transitions learned by a \textsc{svm} classifier. Similarly, \newcite{hall2014less} introduced a fast \textsc{gpu} implementation for \newcite{petrov2007improved}, and \newcite{ShenDistance2018} significantly improved the speed of the \newcite{stern2017minimal} greedy top-down algorithm, by learning to predict a list of syntactic distances that determine the order in which the sentence should be split.
In an alternative line of work, some authors have proposed new parsing paradigms that aim to both reduce the complexity of existing parsers and improve their speed. \newcite{vinyals2015grammar} proposed a machine translation-inspired sequence-to-sequence approach to constituent parsing, where the input is the raw sentence, and the `translation' is a parenthesized version of its tree. \newcite{GomVilEMNLP2018} reduced constituent parsing to sequence tagging, where only $n$ tagging actions need to be made, and obtained one of the fastest parsers to date. However, the performance is well below the state of the art \cite{DyerRecurrent2016,stern2017minimal,KitaevConstituencyACL2018}.
\paragraph{Contribution} We first explore different factors that prevent sequence tagging constituent parsers from obtaining better results. These include: high error rates when long constituents need to be closed, label sparsity, and error propagation arising from greedy inference. We then present the technical contributions of the work. To effectively close brackets of long constituents, we combine the relative-scale tagging scheme used by \newcite{GomVilEMNLP2018} with a secondary top-down absolute-scale scheme. This makes it possible to train a model that learns how to switch between two encodings, depending on which one is more suitable at each time step. To reduce label sparsity, we recast the constituent-parsing-as-sequence-tagging problem as multi-task learning ({\sc mtl}) \cite{caruana1997multitask}, to decompose a large label space and also obtain speed ups. Finally, we mitigate error propagation using two strategies that come at no cost to inference efficiency: auxiliary tasks and policy gradient fine-tuning.
\section{Preliminaries}
We briefly introduce preliminaries that we will build upon in the rest of this paper: encoding functions for constituent trees, sequence tagging, multi-task learning, and reinforcement learning.
\paragraph{Notation} We use $w$=$[w_0,w_1,...,w_n]$ to refer to a raw input sentence and bold style lower-cased and math style upper-cased characters to refer to vectors and matrices, respectively (e.g. $\vec{x}$ and $\vec{W}$).
\subsection{Constituent Parsing as Sequence Tagging}\label{section-linearization-of-trees}
\newcite{GomVilEMNLP2018} define a linearization function of the form $\Phi_{|w|}: T_{|w|} \rightarrow L^{(|w|-1)}$ to map a phrase structure tree with $|w|$ words to a sequence of labels of length $|w|-1$.\footnote{They (1) generate a dummy label for the last word and (2) pad sentences with a beginning- and end-of-sentence tokens.} For each word $w_t$, the function generates a label $l_t \in L$ of the form $l_t$=$(n_t,c_t,u_t)$, where:
\begin{itemize}
\item $n_t$ encodes the number of ancestors in common between between $w_t$ and $w_{t+1}$. To reduce the number of possible values, $n_t$ is encoded as the relative variation in the number of common ancestors with respect to $n_{t-1}$.
\item $c_t$ encodes the lowest common ancestor between $w_t$ and $w_{t+1}$.
\item $u_t$ contains the unary branch for $w_t$, if any.
\end{itemize}
Figure \ref{f-seq-lab-example} explains the encoding with an example.
\begin{figure}[hbtp]
\centering
\includegraphics[width=1\columnwidth]{images/running-example}
\caption{\label{f-seq-lab-example} A constituent tree linearized as by \newcite{GomVilEMNLP2018}.}
\end{figure}
\subsection{Sequence Tagging}\label{section-sequence-tagging}
Sequence tagging is a structured prediction task that generates an output label for every input token. Long short-term memory networks (\textsc{lstm}) \cite{hochreiter1997long} are a popular architecture for such tasks, often giving state-of-the-art performance \cite{Reimers:2017:EMNLP,yang2017ncrf}.
\paragraph{Tagging with \textsc{lstm}s} In \textsc{lstm}s, the prediction for the $i$th element is conditioned on the output of the previous steps. Let \textsc{lstm}$_\theta$($\vec{x}_{1:n}$) be a parametrized function of the network, where the input is a sequence of vectors $\vec{x}_{1:n}$, its output is a sequence of hidden vectors $\vec{h}_{1:n}$. To obtain better contextualized hidden vectors, it is possible to instead use bidirectional \textsc{lstms} \cite{schuster1997bidirectional}. First, a \textsc{lstm}$_\theta^l$ processes the tokens from left-to-right and then an independent \textsc{lstm}$_\theta^r$ processes them from right-to-left. The $i$th final hidden vector is represented as the concatenation of both outputs, i.e.
\textsc{bilstm}$_\theta(\vec{x},i)$ = $\textsc{lstm$_\theta^l$}(\vec{x}_{[1:i]}) \circ \textsc{lstm$_\theta^r$}(\vec{x}_{[|\vec{x}|:i]})$.
\textsc{bilstm}s can be stacked in order to obtain richer representations. To decode the final hidden vectors into discrete labels, a standard approach is to use a feed-forward network together with a softmax transformation, i.e. $P(y|\vec{h}_i)$ = $softmax(W \cdot \vec{h}_i + \vec{b})$.
We will use the \textsc{bilstm}-based model by \newcite{yang2017ncrf}, for direct comparison against \newcite{GomVilEMNLP2018}, who use the same model.
As input, we will use word embeddings, PoS-tag embeddings and a second word embedding learned by a character-based \textsc{lstm} layer. The model is optimized minimizing the categorical cross-entropy loss, i.e. $\mathcal{L}$ = $-\sum{log(P(y|\vec{h}_i))}$. The architecture is shown in Figure \ref{f-baseline-architecture}.
\begin{figure}[hbtp]
\centering
\includegraphics[width=1\columnwidth]{images/running-example-architecture.pdf}
\caption{\label{f-baseline-architecture} The baseline architecture used in this work.}
\end{figure}
\subsection{Multi-task Learning}\label{section-multitask}
Multi-task learning is used to solve multiple tasks
using a single model architecture, with task-specific classifier functions from the outer-most representations \cite{caruana1997multitask,collobert2008unified}. The benefits are intuitive: sharing a common representation for different tasks acts as a generalization mechanism and allows to address them in a parallel fashion. The \emph{hard-sharing} strategy is the most basic \textsc{mtl} architecture, where the internal representation is fully shared across all tasks. The approach has proven robust for a number of \textsc{nlp} tasks \cite{Bingel:ea:17} and comes with certain guarantees if a common, optimal representation exists \cite{Baxter:00}.
\newcite{dong2015multi} use it for their multilingual machine translation system, where the encoder is a shared gated recurrent neural network \cite{cho2014properties} and the decoder is language-specific. \newcite{plank2016multilingual} also use a hard-sharing setup to improve the performance of \textsc{bilstm}-based PoS taggers. To do so, they rely on \emph{auxiliary tasks}, i.e, tasks that are not of interest themselves, but that are co-learned in a \textsc{mtl} setup with the goal of improving the network's performance on the main task(s). We will introduce auxiliary tasks for sequence tagging constituent parsing later on in this work. A \textsc{mtl} architecture can also rely on \emph{partial sharing} when the different tasks do not fully share the internal representations \cite{duong2015low,ruder2017learning,rei2017semi} and recent work has also shown that hierarchical sharing (e.g. low-level task outputs used as input for higher-level ones) could be beneficial \cite{sogaard2016deep, sanh2018hierarchical}.
\subsection{Policy Gradient Fine-tuning}\label{section-policy-gradient}
Policy gradient (\textsc{pg}) methods are a class of reinforcement learning algorithms that directly learn a parametrized policy, by which an agent selects actions based on the gradient of a scalar performance measure with respect to the policy. Compared to other reinforcement learning methods, \textsc{pg} is well-suited to \textsc{nlp} problems due to its appealing convergence properties and effectiveness in high-dimensional spaces \cite{sutton2018reinforcement}.
Previous work on constituent parsing has employed \textsc{pg} methods to mitigate the effect of exposure bias, finding that they function as a model-agnostic substitute for dynamic oracles \citep{fried2018policy}. Similarly, \newcite{le2017tackling} apply \textsc{pg} methods to \newcite{chen2014fast}'s transition-based dependency parser to reduce error propagation. In this work, we also employ \textsc{pg} to fine-tune models trained using supervised learning. However, our setting (sequence tagging) has a considerably larger action space than a transition parser. To deal with that, we will adopt a number of variance reduction and regularization techniques to make reinforcement learning stable.
\section{Methods}\label{section-methods}
We describe the methods introduced in this work, motivated by current limitations of existing sequence tagging models, which are first reviewed. The source code will be released shortly
and a link will be provided with a forthcoming version of this manuscript.
\subsection{Motivation and Analysis}
For brevity, we limit this analysis to the English Penn Treebank (\textsc{ptb}) \cite{marcus1993building}. We reproduced the best setup by \newcite{GomVilEMNLP2018}\footnote{\url{https://github.com/aghie/tree2labels}},
which we are using as baseline, and run the model on the development set.
We below show insights for the elements of the output tuple $(n_t,c_t,u_t)$, where
$n_t$ is the number of levels in common between $w_t$ and $w_{t+1}$, $c_t$ is the non-terminal symbol shared at that level, and $u_t$ is a leaf unary chain located at $w_t$.
\paragraph{High error rate on closing brackets} We first focus on predicting relative tree levels ($n_t$). See Figure \ref{f-baseline-level-performance} for F-scores over $n_t$ labels. The sparsity on negative $n_t$s is larger than for the positive ones, and we see that consequently, the performance is also significantly worse for negative $n_t$ values, and performance worsens with higher negative values. This indicates that the current model cannot effectively identify the end of long constituents. This is a known source of error for shift-reduce or chart-based parsers, but in the case of sequence tagging parsers, the problem seems particularly serious.
\begin{figure}[hbtp]
\centering
\includegraphics[width=1\columnwidth]{images/base-level-performance.pdf}
\caption{\label{f-baseline-level-performance} F-score for $n_t$ labels on the \textsc{ptb} dev set using \newcite{GomVilEMNLP2018}.}
\end{figure}
\paragraph{Sparsity} The label space is large and sparse: the output labels are simply the possible values in the tuple $(n_t,c_t,u_t)$. An analysis over the \textsc{ptb} training set shows a total of 1423 labels, with 58\% of them occurring 5 or less times. These infrequent cases might be difficult to predict, even if some of the elements of the tuple are common.
\paragraph{Greedy decoding} Greedy decoding is prone to issues such as error propagation. This is a known source of error in transition-based dependency parsing \cite{qi2017arc}; in contrast with graph-based parsing, in which parsing is reduced to global optimization over edge-factored scores \cite{mcdonald2005non}.
In the case of \textsc{bilstm}-based sequence tagging parsers, for a given word $w_t$, the output label as encoded by \newcite{GomVilEMNLP2018} only reflects a relation between $w_t$ and $w_{t+1}$. We hypothesize that even if the hidden vector representations are globally contextualized over the whole sequence, the intrinsic locality of the output label also turns into error propagation and consequently causes a drop in the performance. These hypotheses will be tested in \S \ref{section-experiments}. In particular, we will evaluate the impact of the different methods intended to perform structured inference (\S \ref{section-local-predictions}).
\subsection{Dynamic Encodings}
\newcite{GomVilEMNLP2018} encode the number of common ancestors $n_t$, from the output tuple $(n_t,c_t,u_t)$, as the variation with respect to $n_{t-1}$. We propose instead to encode certain elements of a sentence using a secondary linearization function. The aim is to generate a model that can dynamically switch between different tagging schemes at each time step $t$ to select the one that represents the relation between $w_t$ and $w_{t+1}$ in the most effective way.
On the one hand, the relative-scale encoding is effective to predict the beginning and the end of short constituents, i.e. when a short constituent must be predicted ($|n_t| \leq 2$). On the other hand, with a relative encoding scheme, the F-score was low for words where the corresponding $n_t$ has a large negative value (as showed in Figure \ref{f-baseline-level-performance}). This matches a case where a long constituent must be closed: $w_t$ is located at a deep level in the tree and will only (probably) share a few ancestors with $w_{t+1}$. These configurations are encoded in a more sparse way by a relative scheme, as the $n_t$ value shows a large variability and it depends on the depth of the tree in the current time step. We can obtain a compressed representation of these cases by using a \emph{top-down absolute scale} instead, as any pair of words that share the same $m$ top levels will be equally encoded.
The absolute scale becomes however sparse when predicting deep levels.
Figure \ref{f-dynamic-example} illustrates the strengths and weaknesses of both encodings with an example, and how a dynamically encoded tree helps reduce variability on $n_t$ values.
In our particular implementation, we will be using the following setup:
\begin{itemize}
\item $\Phi_{|w|}: T_{|w|} \rightarrow L^{|w|-1}$, the relative-scale encoding function, is used by default.
\item $\Omega_{|w|}: T_{|w|} \rightarrow L'^{|w|-1}$ is the secondary linearization function that maps words to labels according to a top-down absolute scale. $\Omega$ is used iff: (1) $\Omega(w_{[t:t+1]})$ = $(n'_t,c'_t,u'_t)$ with $n'_t \leq 3$, i.e. $w_t$ and $w_{t+1}$ share at most the three top levels, and (2) $\Phi(w_{[t:t+1]})$ = $(n_t,c_t,u_t)$ with $n_t \leq -2 $, i.e. $w_{t}$ is at least located two levels deeper in the tree than $w_{t+1}$.\footnote{The values were selected based on the preliminary experiments of Figure \ref{f-baseline-level-performance}.}
\end{itemize}
\begin{figure}[hbtp]
\centering
\includegraphics[width=1\columnwidth]{images/dynamic_encodings.pdf}
\caption{\label{f-dynamic-example} A synthetic constituent tree where $n_t$ is encoded using a relative scheme, a top-down absolute scale, and an ideal dynamic combination. The relative scheme is appropriate to open and close short constituents, but becomes sparse when encoding the large ones, e.g. $n_t$ for the tokens `e', `h' and 'l'. The opposite problem is observed for the top-down absolute scheme (e.g. tokens from `a' to `d'). \textsc{d} combines the best of both encodings.}
\end{figure}
\subsection{Decomposition of the label space}
We showed that labels of the form $(n_t,c_t,u_t) \in L$ are sparse. An intuitive approach is to decompose the label space into three smaller sub-spaces, such that $n_i \in N$, $c_i \in C$ and $u_i \in U$. This reduces the output space from potentially $|N|\times|C|\times|U|$ labels to just $|N|+|C|+|U|$. We propose to learn this decomposed label space through a multi-task learning setup, where each of the subspaces is considered a different task, namely task$_N$, task$_C$ and task$_U$. The final loss is now computed as $\mathcal{L} = \mathcal{L}_n + \mathcal{L}_c + \mathcal{L}_u$.
We relied on a hard-sharing architecture, as it has been proved to reduce the risk of overfitting the shared parameters \cite{baxter1997bayesian}. A natural issue that arises is that the prediction of labels from different label sub-spaces could be interdependent to a certain extent, and therefore a hierarchical sharing architecture could also be appropriate. To test this, in preliminary experiments we considered variants of hierarchical sharing architectures. We fed the output of the task$_U$ as input to task$_N$ and/or task$_C$. Similarly, we tested whether it was beneficial to feed the output of task$_N$ into task$_C$, and viceversa. However, all these results did not improve those of the hard-sharing model. In this context, in addition to a generalization mechanism, the shared representation could be also acting as way to keep the model aware of the potential interdependencies that might exist between subtasks.
\subsection{Mitigating Effects of Greedy Decoding}\label{section-local-predictions}
We propose two ways to mitigate error propagation arising from greedy decoding in constituent parsing as sequence tagging: auxiliary tasks and policy gradient fine-tuning. Note that we want to optimize bracketing F-score {\em and} speed. For this reason we do {\em not}~explore approaches that come at a speed cost in testing time, such as beam-search or using conditional random fields \cite{Lafferty:ea:01} on top of our \textsc{lstm}.
\paragraph{Auxiliary tasks} Auxiliary tasks force the model to take into account patterns in the input space that can be useful to solve the main task(s), but that remain ignored due to a number of factors, such as the distribution of the output label space \cite{rei2017semi}. In a similar fashion, we use auxiliary tasks as a way to force the parser to pay attention to aspects beyond those needed for greedy decoding. We propose and evaluate two separate strategies:
\begin{enumerate}
\item Predict partial labels $n_{t+k}$
that are $k$ steps from the current time step $t$. This way we can jointly optimize at each time step a prediction for the pairs $(w_t, w_{t+1})$, \dots, $(w_{t+k}, w_{t+k+1})$. In particular, we will experiment both with previous and upcoming $n_k$'s, setting $|k|$=$1$.\label{enu-aux-task-1}
\item Predict the syntactic distances presented by \newcite{ShenDistance2018}, which reflect the order a sentence must be split to obtain its constituent tree using a top-down parsing algorithm \cite{stern2017minimal}. The algorithm was initially defined for binary trees, but its adaptation to \emph{n}-ary trees is immediate: leaf nodes have a split priority of zero and the ancestors' priority is computed as the maximum priority of their children plus one. In this work, we use this algorithm in a sequence tagging setup: the label assigned to each token corresponds to the syntactic distance of the lowest common ancestor with the next token. This is illustrated in Figure \ref{f-syntactic-distances}.
\begin{figure}[hbtp]
\centering
\includegraphics[width=1\columnwidth]{images/running-example-syntactic-distances.pdf}
\caption{\label{f-syntactic-distances} A constituent with syntactic distances attached to each non-terminal symbol, according to \newcite{ShenDistance2018}. Distances can be used for sequence tagging, providing additional information to our base encoding \cite{GomVilEMNLP2018}}
\end{figure}
\end{enumerate}
The proposed auxiliary tasks provide different types of contextual information. On the one hand, the encoding of the $n_t$s by \newcite{GomVilEMNLP2018} only needs to know about $w_t$ and $w_{t+1}$ paths to generate the label for the time step $t$. On the other hand, to compute the syntactic distance of a given non-terminal symbol, we need to compute the syntactic distances of its subtree, providing a more global, but also sparser context. For training, the loss coming from the auxiliary task(s) is weighted by $\beta$=0.1, i.e, the final loss is computed as $\mathcal{L} = \mathcal{L}_n + \mathcal{L}_c + \mathcal{L}_u + \beta\sum_{a}{\mathcal{L}_{a}} $.
\paragraph{Policy gradient fine-tuning}
Policy gradient training methods allow us to fine-tune our models with a tree-level objective, optimizing directly for bracketing F-score. We start off with a converged supervised model as our initial policy. The sequence labeling model can be seen as a functional approximation of the policy $\pi$ parametrized by $\theta$, which at timestep $t$ selects a label $l_t$=$(n_t,c_t,u_t)$\footnote{3 different labels in the \textsc{mtl} setting.} given the current state of the model's parameters, $s_t$.
The agent's reward, $R_{tree}$, is then derived from the bracketing F-score.
This can be seen as a variant of the \textsc{reinforce} algorithm \citep{williams1992simple} where the policy is updated by gradient ascent in the direction of:
\begin{equation}
\Delta_\theta log\pi(l_t|s_t; \theta)R_{tree}
\end{equation}
\paragraph{Baseline and Variance Reduction}
We use as baseline a copy of a pre-trained model where the parameters are frozen.
The reward used to scale the policy gradient can then be seen as an estimate of the advantage of an action $l_t$ in state $s_t$ over the baseline model. This is equivalent to $R_{tree} - B_{tree}$, where $R_{tree}$ is the bracketing F-score of a sequence sampled from the current policy and $B_{tree}$ is the the tree-level F-score of the sequence greedily predicted by the baseline. To further reduce the variance, we standardize the gradient estimate $\Delta_\theta$ using its running mean and standard deviation for all candidates seen in training so far. In initial experiments without these augmentations, we observed that fine-tuning with vanilla \textsc{pg}
often led to a deterioration in performance. To encourage exploration away from the converged supervised model's policy, we add the entropy of the policy to the objective function \citep{williams1991function}. Moreover, following \newcite{lillicrap2015continuous}, we optionally add noise sampled from a noise process $N$ to the policy. The gradient
of our full fine-tuning objective function takes the following form:
\begin{multline}
\Delta_\theta (log\pi(l_t|s_t; \theta) + N) (R_{tree} - B_{tree}) \\ + \beta \Delta_\theta H(\pi(s_t; \theta) + N)
\end{multline}
\noindent where $H$ is the entropy and $\beta$ controls the strength of the entropy regularization term.
\section{Experiments}\label{section-experiments}
We now review the impact of the proposed techniques on a wide variety of settings.
\paragraph{Datasets} We use the English Penn Treebank (\textsc{ptb}) \cite{marcus1993building} and the Chinese Penn Treebank (\textsc{ctb}) \cite{xue2005penn}. For these, we use the same predicted PoS tags as \newcite{DyerRecurrent2016}. We also provide detailed results on the \textsc{spmrl} treebanks \cite{seddah2014introducing},\footnote{Except for Arabic, for which we do not have the license.} a set of datasets for constituent parsing on morphologically rich languages. For these, we use the predicted PoS tags provided together with the corpora. To the best of our knowledge, we provide the first evaluation on the \textsc{spmrl} datasets for sequence tagging constituent parsers.
\paragraph{Metrics} We report bracketing F-scores, using the \textsc{evalb} and the \textsc{eval-spmrl} scripts. We measure the speed in terms of sentences per second.
\paragraph{Setup} We use \textsc{ncrf}pp \cite{yang2017ncrf}, for direct comparison against \citet{GomVilEMNLP2018}. We adopt bracketing F-score instead of label accuracy for model selection and report this performance as our second baseline. After 100 epochs, we select the model that faired best on the development set.
We use GloVe embeddings \cite{pennington2014glove} for our English models and \texttt{zzgiga} embeddings \cite{Liu2017InOrder} for the Chinese models, for a more homogeneous comparison against other parsers \cite{DyerRecurrent2016,Liu2017InOrder,Fer2018Faster}. ELMo \cite{peters2018deep} or BERT \cite{devlin2018bert} could be used to improve the precision, but in this paper we focus on keeping a good speed-accuracy tradeoff. For \textsc{spmrl}, no pretrained embeddings are used, following \newcite{KitaevConstituencyACL2018}. The models are run on a single CPU\footnote{Intel Core i7-7700 CPU 4.2 GHz} (and optionally on a consumer-grade GPU for further comparison) using a batch size of 128 for testing. Additional hyperparameters can be found in Appendix \ref{appendix-a}.
\subsection{Results}
Table \ref{table-ptb-dev-set} contrasts the performance of our models against the baseline on the \textsc{ptb} development set.
\begin{table}[htbp]
\tabcolsep=0.2cm
\begin{center}
\small
\begin{tabular}{lccc}
\hline
\bf Model & \bf F-score & \bf (+/-) & \bf Sents/s \\
\hline
\defcitealias{GomVilEMNLP2018}{G\'omez and Vilares (2018)}\citetalias{GomVilEMNLP2018} &89.70&-&109\\
Our baseline &89.77&(+0.07)&111\\
+ \textsc{de} & 90.22 &(+0.52)& 111 \\
+ \textsc{mtl} & 90.38 &(+0.68)& 130 \\
\hline
aux($n_{t+1}$)& 90.41 &(+0.71)&130\\
aux($n_{t-1}$) & 90.57&(+0.87)&130\\
aux(distances) & 90.55 &(+0.85)&130\\
\hline
+ \textsc{pg}&\bf{90.70}&(+1.00)&130\\
\hline
\end{tabular}
\end{center}
\caption{\label{table-ptb-dev-set} Results on the \textsc{ptb} dev set, compared against \newcite{GomVilEMNLP2018}. \textsc{de} refers to dynamic encoding and \textsc{mtl} to a model that additionally casts the problem as multi-task learning. Each auxiliary task is added separately to the baseline with \textsc{de} and \textsc{mtl}. Policy gradient fine-tunes the model that includes the best auxiliary task.}
\end{table}
\begin{table*}[!]
\begin{center}
\small
\begin{tabular}{l|c|cccccccc}
\hline
\bf Model & \bf \textsc{ctb} & \bf Basque & \bf French & \bf German & \bf Hebrew & \bf Hungarian & \bf Korean & \bf Polish & \bf Swedish \\
\hline
Our baseline& 88.57&87.93&81.09&87.83&89.27&88.85&83.51&92.60&80.11\\
\textsc{+de}&88.37&87.91&81.16&\bf 88.81&89.03&88.70&83.92&93.35&79.57\\
\textsc{+mtl}& 88.57&89.41&81.70&88.52&\bf 92.72& 89.73& 84.10& 93.81& 82.83\\
\hline
aux($n_{t+1}$)&88.73& 89.65&81.95&88.64&92.65&89.69& 84.09&93.86&82.82\\
aux($n_{t-1}$)&88.48& 89.47&81.77&88.58&92.53&89.71&84.13& 93.87&82.74\\
aux(distances)&88.51& 89.48& 82.02& 88.68& 92.66& 89.80& 84.20&93.83& 83.12\\
\hline
\textsc{+pg}&\bf 89.01&\bf 89.73&\bf 82.13&88.80&92.66&\bf89.86&\bf84.45&\bf93.93&\bf83.15\\
\hline
\end{tabular}
\end{center}
\caption{\label{table-other-dev-sets} Results on the \textsc{ctb} and \textsc{spmrl} dev sets}
\end{table*}
To show that the model which employs dynamic encoding is better (+0.52) than the baseline when it comes to closing brackets from long constituents, we compare their F-scores in Figure \ref{f-baseline-level-performance-2}.
When we recast the constituent-parsing-as-sequence-tagging problem as multi-task learning, we obtain both a higher bracketing F-score (+0.68) and speed (1.17x faster).
Fusing strategies to mitigate issues from greedy decoding also leads to better models (up to +0.87 when adding an auxiliary task\footnote{We observed that adding more than one auxiliary task did not translate into a clear improvement.} and up to +1.00 if we also fine-tune with \textsc{pg}). Note that including auxiliary tasks and \textsc{pg} come at a time cost in training, but not in testing, which makes them suitable for fast parsing.
\begin{figure}[hbtp]
\centering
\includegraphics[width=1\columnwidth]{images/base-level-performance-2.pdf}
\caption{\label{f-baseline-level-performance-2} F-score for $n_t$s on the \textsc{ptb} dev set, obtained by the \newcite{GomVilEMNLP2018} baseline (in blue, first bar for each $n_t$, already shown in Figure \ref{f-baseline-level-performance}) and our model with dynamically encoded trees (in orange, second bar).}
\end{figure}
Table \ref{table-other-dev-sets} replicates the experiments on the \textsc{ctb} and the \textsc{spmrl} dev sets. The dynamic encoding improves the performance of the baseline on large treebanks, e.g. German, French or Korean, but causes some drops in the smaller ones, e.g. Swedish or Hebrew. Overall, casting the problem as multitask learning and the strategies used to mitigate error propagation lead to improvements.
For the experiments on the test sets we select the models that summarize our contributions: the models with dynamic encoding and the multi-task setup, the models including the best auxiliary task, and the models fine-tuned with policy gradient.
\begin{table}[hbtp]
\tabcolsep=0.07cm
\begin{center}
\small
\begin{tabular}{lrrc}
\hline
\bf Model & \bf Sents/s&\bf Hardware & \bf F-score\\
\hline
\newcite{vinyals2015grammar}&120&Many CPU&{88.30}\\
\newcite{CoaCra2016}&168&1 CPU&88.60\\
\defcitealias{Fer2015Parsing}{Fern\'andez and Martins (2018)}\citetalias{Fer2015Parsing}&41&1 CPU&90.20\\
\newcite{zhu2013fast}&90&1 CPU&90.40\\
\newcite{DyerRecurrent2016}&17&1 CPU&91.20\\
\newcite{stern2017minimal}&76&16 CPU&91.77\\
\newcite{ShenDistance2018}&111&1 GPU&91.80\\
\newcite{KitaevConstituencyACL2018}&213&2 GPU&93.55\\
(single model)&&&\\
\newcite{KitaevConstituencyACL2018}&71&2 GPU&95.13\\
(with ELMo)&&&\\
\defcitealias{GomVilEMNLP2018}{G\'omez and Vilares (2018)}\citetalias{GomVilEMNLP2018}&115& 1 CPU&90.00\\
\hline
Our baseline &115&1 CPU& 90.06\\
\textsc{+de} &115&1 CPU& 90.19 \\
\textsc{+mtl} &132&1 CPU&90.36\\
+ best aux &132&1 CPU&90.59\\
\textsc{+pg}&132&1 CPU&90.60\\
\textsc{+pg}&942&1 GPU&90.60\\
\shadowed{\textsc{+pg} (no char emb)}&\shadowed{149}&\shadowed{1 CPU}&\shadowed{90.50}\\
\shadowed{\textsc{+pg} (no char emb)}&\shadowed{1267}&\shadowed{1 GPU}&\shadowed{90.50}\\
\hline
\end{tabular}
\end{center}
\caption{\label{table-sota-ptb} Comparison on the \textsc{ptb} test set}
\end{table}
\begin{table}[bpth]
\begin{center}
\small
\begin{tabular}{lc}
\hline
\bf Model & \bf F-score \\
\hline
\newcite{zhu2013fast}&83.2\\
\newcite{DyerRecurrent2016}&84.6\\
\newcite{Liu2017InOrder}&86.1\\
\newcite{ShenDistance2018}&86.5\\
\defcitealias{Fer2018Faster}{Fern\'andez and G\'omez-Rodr\'iguez (2018)}\citetalias{Fer2018Faster}&86.8\\
\defcitealias{GomVilEMNLP2018}{G\'omez and Vilares (2018)}\citetalias{GomVilEMNLP2018}&84.1\\
\hline
Our baseline&83.90\\
+\textsc{de}&83.98\\
+\textsc{mtl}&84.24\\
+best aux&85.01\\
\textsc{+pg}&85.61\\
\shadowed{\textsc{+pg} (no char emb)}&\shadowed{83.93}\\
\hline
\end{tabular}
\end{center}
\caption{\label{table-sota-chinese} Comparison on the \textsc{ctb} test set}
\end{table}
\begin{table*}[!]
\tabcolsep=0.07cm
\begin{center}
\small
\begin{tabular}{p{5.3cm}|ccccccccc}
\hline
\bf Model & \bf Basque & \bf French & \bf German & \bf Hebrew & \bf Hungarian & \bf Korean & \bf Polish & \bf Swedish&\bf Avg \\
\hline
\newcite{fernandez2015parsing}&85.90&78.75&78.66&88.97&88.16&79.28&91.20&82.80&84.21\\
\newcite{CoaCra2016}&86.24&79.91&80.15&88.69&90.51&85.10&92.96&81.74&85.67\\
\newcite{bjorkelund2014introducing} (ensemble)&88.24&82.53&81.66&89.80&91.72&83.81&90.50&85.50&86.72\\
\newcite{coavoux2017multilingual}&88.81&82.49&85.34&89.87&92.34&86.04&93.64&84.00&87.82 \\
\newcite{KitaevConstituencyACL2018}&89.71&\bf 84.06&\bf 87.69&90.35&\bf 92.69&\bf 86.59&93.69&84.35&\bf88.64 \\
\hline
Baseline&89.54&80.56&84.05&88.83&90.42&83.33&92.48&83.67&86.61\\
\textsc{+de}&89.56&80.69&84.64&88.80&90.02&83.67&93.20&83.40&86.75\\
\textsc{+mtl}&90.90&80.98&84.94&91.99&90.63&83.91&93.80&86.26&87.92\\
+best aux&\bf 91.23&81.27&84.95&\bf92.03&90.60&83.67&93.84&86.62&88.02\\
\textsc{+pg}&91.18&\bf81.37&84.88&\bf92.03&90.65&84.01&\bf93.93&\bf86.71&88.10\\
\shadowed{\textsc{+pg} (no char emb)}&\shadowed{90.14}&\shadowed{81.38}&\shadowed{85.09}&\shadowed{92.11}&\shadowed{90.34}&\shadowed{83.46}&\shadowed{93.87}&\shadowed{86.65}&\shadowed{87.88}\\
\hline
\end{tabular}
\end{center}
\caption{\label{table-sota-spmrl} Comparison on the test \textsc{spmrl} datasets (except Arabic)}
\end{table*}
Tables \ref{table-sota-ptb}, \ref{table-sota-chinese} and \ref{table-sota-spmrl} compare our parsers against the state of the art on the \textsc{ptb}, \textsc{ctb} and \textsc{spmrl} test sets. \newcite{GomVilEMNLP2018} also run experiments without character embeddings, to improve speed without suffering from a big drop in performance. For further comparison, we also include them as additional results (shadowed). In a related line, \newcite{smith2018investigation} show that for dependency parsing two out of three embeddings (word, postag and characters) can suffice.
\subsection{Discussion}
The results across the board show that the dynamic encoding has a positive effect on 6 out of 10 treebanks. Casting the constituent-parsing-as-sequence-labeling problem as \textsc{mtl} surpasses the baseline for all tested treebanks (and it leads to better parsing speeds too). Finally, by mitigating issues from greedy decoding we further improve the performance of all models that include dynamic encodings and multi-task learning.
On the \textsc{ptb}, our models are both faster and more accurate than existing sequence tagging or sequence-to-sequence models, which already were among the fastest parsers \cite{GomVilEMNLP2018,vinyals2015grammar}. We also outperform other approaches that were not surpassed by the original sequence tagging models in terms of F-score \cite{zhu2013fast,Fer2015Parsing}. On the \textsc{ctb} our techniques also have a positive effect. The baseline parses 70 sents/s on the \textsc{ctb}, while the full model processes up to 120. The speed up is expected to be larger than the one obtained for the \textsc{ptb} because the size of the label set for the baseline is bigger, and it is reduced in a greater proportion when the constituent-parsing-as-sequence-labeling problem is cast as \textsc{mtl}.
On the \textsc{spmrl} corpora, we provide the first evaluation of sequence labeling constituent parsers, to verify if these perform well on morphologically rich languages. We then evaluated whether the proposed techniques can generalize on heterogeneous settings.
The tendency observed for the original tagging models by \newcite{GomVilEMNLP2018} is similar to the one for the \textsc{ptb} and \textsc{ctb}: they improve other fast parsers, e.g. \newcite{CoaCra2016}, in 5 out of 8 treebanks and \newcite{Fer2015Parsing} in 7 out of 8, but their performance is below more powerful models. When incorporating the techniques presented in this work, we outperform the original sequence tagging models on all datasets. We outperform the current best model for Basque, Hebrew and Polish \cite{KitaevConstituencyACL2018} and for Swedish \cite{bjorkelund2014introducing}, which corresponds to the four smallest treebanks among the \textsc{spmrl} datasets. This indicates that even if sequence tagging models are conceptually simple and fast, they can be very suitable when little training data is available. This is also of special interest in terms of research for low-resource languages. Again, casting the problem as \textsc{mtl} reduces the parsing time for all tested treebanks, as reflected in Table \ref{table-spmrl-speeds}.
\begin{table}[bpth]
\tabcolsep=0.16cm
\begin{center}
\small
\begin{tabular}{l|ccc}
\hline
\multirow{2}{*}{\bf Dataset} &{\bf Baseline} &{\bf Full} &\bf \shadowed{\bf Full (no char)} \\
&{\bf speed} &\bf speed\textsubscript{(increase)}&\bf \shadowed{ speed\textsubscript{(increase)}}\\
\hline
Basque&179&223\textsubscript{ (1.25x)}&\shadowed{257}\textsubscript{ \shadowed{(1.44x)}}\\
French&76&91\textsubscript{ (1.20x)}&\shadowed{104}\textsubscript{ \shadowed{(1.37x)}}\\
German&70&100\textsubscript{ (1.43x)}&\shadowed{108}\textsubscript{ \shadowed{(1.54x)}}\\
Hebrew&44&102\textsubscript{ (2.32x)}&\shadowed{115}\textsubscript{ \shadowed{(2.61x)}}\\
Hungarian&93&134\textsubscript{ (1.44x)}&\shadowed{150}\textsubscript{ \shadowed{(1.61x)}}\\
Korean&197&213\textsubscript{ (1.08x)}&\shadowed{230}\textsubscript{ \shadowed{(1.17x)}}\\
Polish&187&253\textsubscript{ (1.35x)}&\shadowed{278}\textsubscript{ \shadowed{(1.49x)}}\\
Swedish&98&158\textsubscript{ (1.61x)}&\shadowed{187}\textsubscript{ \shadowed{(1.81x)}}\\
\hline
\end{tabular}
\end{center}
\caption{\label{table-spmrl-speeds} Comparison of speeds on the \textsc{spmrl} datasets}
\end{table}
\section{Conclusion}
We have explored faster and more precise sequence tagging models for constituent parsing. We proposed a multitask-learning architecture that employs dynamic encodings, auxiliary tasks, and policy gradient fine-tuning. We performed experiments on the English and Chinese Penn Treebanks, and also on the \textsc{spmrl} datasets. Our models improve current sequence tagging parsers on all treebanks, both in terms of performance and speed. We also report state-of-the-art results for the Basque, Hebrew, Polish, and Swedish datasets.
\section{Acknowlegments}
DV has received support from the European Research Council (ERC), under the European Union's Horizon 2020 research and innovation programme (FASTPARSE, grant agreement No 714150), from the TELEPARES-UDC project
(FFI2014-51978-C2-2-R) and the ANSWER-ASAP project (TIN2017-85160-C2-1-R) from MINECO, and from Xunta de Galicia (ED431B 2017/01). MA and AS are funded by a Google Focused Research Award.
\bibliographystyle{acl_natbib}
|
{
"timestamp": "2019-03-01T02:14:38",
"yymm": "1902",
"arxiv_id": "1902.10985",
"language": "en",
"url": "https://arxiv.org/abs/1902.10985"
}
|
\section{Introduction}
Building an AI capable of inferring the 3D structure and pose of an object from
a single image is a problem of immense importance. Training such a system using
supervised learning requires a large number of labeled images -- how to obtain
these labels is currently an open problem for the vision community.
Rendering~\cite{su2015render} is problematic as the synthetic images seldom
match the appearance and geometry of the objects we encounter in the
real-world. Hand annotation is preferable, but current strategies rely on
associating the natural images with an external 3D dataset (\eg
ShapeNet~\cite{DBLP:journals/corr/ChangFGHHLSSSSX15}, ModelNet~\cite{wu20153d}),
which we refer to as \emph{3D supervision}. If the 3D shape dataset does not
capture the variation we see in the imagery, then the problem is inherently
ill-posed.
\input{teaser.image.tex}
Non-Rigid Structure from Motion (NRS\emph{f}M\xspace) offers computer vision a way out of
this quandary -- by recovering the pose and 3D structure of an object category
\emph{solely} from hand annotated 2D landmarks with no need of 3D supervision.
Classically~\cite{bregler2000recovering}, the problem of NRS\emph{f}M\xspace has been applied
to objects that move non-rigidly over time such as the human body and face. But
NRS\emph{f}M\xspace is not restricted to non-rigid objects; it can equally be applied to
rigid objects whose object categories deform non-rigidly~\cite{kong2016sfc}.
Consider, for example, the four objects in Figure~\ref{fig:teaser}~(b), our
reconstructions from the visual object category ``chair''. Each object in
isolation represents a rigid chair, but the set of all 3D shapes describing
``chair'' is non-rigid. In other words, each object instance can be modeled as
a deformation from its category's general shape.
Current NRS\emph{f}M\xspace algorithms~\cite{kumar2018scalable, lee2016consensus,
chhatkuli2016inextensible} all suffer from the difficulty of processing
large-scale image sequences, limiting their ability to reliably model complex
shape variations. This additionally hinders their ability to generalize to
unseen images. Deep Neural Networks~(DNNs) are an obvious candidate to help
with such issue. However, the influence of DNNs has been most noticeable when
applied to raster representations~(\eg raw pixel
intensities~\cite{deng2009imagenet}). While DNNs have recently exhibited their
success to 3D point representations~(\eg point clouds)~\cite{qi2017pointnet,
huang2016point}, their use has not been explored in recovering poses and 3D
shapes from an ensemble of vector-based 2D landmarks.
\subsubsection{Contributions}
We propose a novel DNN to solve the problem of NRS\emph{f}M\xspace. Our employment of DNNs
moves from an opaque black-box to a transparent ``glass-box'' in terms of its
interpretability. The term ``black-box'' is often used as a critique of DNNs
with respect to the general lack of understanding surrounding the inner
workings. We demonstrate how the problem of NRS\emph{f}M\xspace can be cast as a
multi-layer block sparse dictionary learning problem. Through recent theoretical
innovations~\cite{papyan2017convolutional}, we then show how this problem can be
reinterpreted as a feed-forward DNN auto-encoder that can be efficiently solved
through modern deep learning environments.
Our deep NRS\emph{f}M\xspace is capable of handling hundreds of thousands of images and
learning large parameterizations to model non-rigidity. Our proposed approach is
completely unsupervised in a 3D sense, relying solely on the projected 2D
landmarks of the non-rigid object or object category to recover the pose and 3D
shape. Our approach dramatically outperforms state-of-the-art methods on a
number of benchmarks, and gets impressive qualitative reconstructions on the
problem of NRS\emph{f}M\xspace~--~examples of which are shown in Figure~\ref{fig:teaser}.
Moreover, the considerable capacity of modeling non-rigidity allows us to
efficiently apply it to unseen data. This facilitates an accurate 3D
reconstruction of objects from a single view with no aid of 3D ground-truth.
Finally, we propose a measure of model quality~(using coherence and trained
parameters), which improves the practical utility of our model in the real
world applications.
\section{Related Work}
\subsubsection{Non-rigid structure from motion}
NRS\emph{f}M\xspace is an inherently ill-posed problem since the 3D shapes can vary between
images, resulting in more variables than equations. To alleviate the
ill-posedness, various constraints are exploited including temporal
\footnote{Shapes deform continuously along the sequence of frames.}~\cite{
akhter2011trajectory, gotardo2011computing, kumar2016multi, kumar2018scalable},
and articulation\footnote{The distance of joints are somehow constant in
human skeleton.}~\cite{ramakrishna2012reconstructing} priors.
Dai~\etal~\cite{dai2014simple} pioneered the exploration of NRS\emph{f}M\xspace with minimum
assumptions. They proposed a low-rank model of non-rigidity and a factorization
algorithm recovering both cameras and 3D shapes with no need of additional priors.
The major drawback of this method is the low rank assumption, which highly
restricts the application to complex sequences. To solve this problem, Kong and
Lucey~\cite{kong2016prior} proposed to use an over-complete dictionary with
sparsity to model non-rigid objects and upgraded the factorization algorithm by
characterizing the uniqueness of dictionary learning. However, due to the
enormous parameter space, their method was sensitive to noise and thus had
limited utility in real world applications.
\subsubsection{Structure from category}
NRS\emph{f}M\xspace has often been criticized as solving a toy problem with few useful
applications beyond being a theoretical curiosity for computer vision. Recently,
Kong~\etal~\cite{kong2016sfc} proposed a novel concept---Structure from
Category~(S\emph{f}C\xspace)---directly connecting NRS\emph{f}M\xspace to inferring camera poses and 3D
structure within an ensemble of images stemming from the same object category.
The strength of this approach is the ability to solely use 2D landmarks without
3D supervision.
They provided a convex relaxation solution to this problem. However, the
proposed optimization algorithm could not be applied to large-scale images,
limiting its effectiveness for modeling complex shape variations.
\subsubsection{Single view human pose estimation}
Besides S\emph{f}C\xspace and NRS\emph{f}M\xspace, there is another task related to our work, that is
single view human pose estimation. A common solution is assuming that the human
body can be represented through a sparse dictionary.
Ramakrishna~\etal~\cite{ramakrishna2012reconstructing}
proposed to use a matching pursuit algorithm to estimate the sparse
representation. However, since the problem is not convex, their algorithm fails
when initialization is poor. Zhou~\etal~\cite{zhou20153d} proposed to utilize a
convex relaxation to alleviate sensitivities to initialization, but inevitably
introduce additional errors. Another drawback from Zhou~\etal~\cite{zhou20153d,
ramakrishna2012reconstructing} is its dependence on external 3D models
for estimating the model dictionary (\ie 3D supervision).
\section{Background}
\label{sec: scdnn}
Sparse dictionary learning can be considered as an unsupervised learning task
and divided into two sub-problems: (i) dictionary learning, and (ii)
sparse code recovery. Let us consider sparse code recovery problem, where we
estimate a sparse representation $\zv$ for a measurement vector $\xv$ given the dictionary~$\Wv$
\ie
\begin{equation}
\min_\zv \Vert \xv - \Wv\zv\Vert_2^2 \quad \st \Vert \zv \Vert_0 < \lambda,
\label{eq:sparse_coding}
\end{equation}
where $\lambda$ related to the trust region controls the sparsity of recovered
code. One classical algorithm to recover the sparse representation is
Iterative Shrinkage and Thresholding Algorithm
(ISTA)~\cite{daubechies2004iterative, rozell2008sparse,beck2009fast}. ISTA
iteratively executes the following two steps with $\zv^{[0]} = \zero$:
\begin{gather}
\vv = \zv^{[i]} - \alpha\Wv^T(\Wv\zv^{[i]} - \xv), \\
\zv^{[i+1]} = \argmin_{\uv} \frac{1}{2}\Vert \uv - \vv \Vert^2_2 + \tau\Vert\uv\Vert_1,
\end{gather}
which first uses the gradient of~$\Vert \xv - \Wv\zv\Vert_2^2$ to
update~$\zv^{[i]}$ in step size $\alpha$ and then finds the closest sparse
solution using an $\ell_1$ convex relaxation. It is well known in literature
that the second step has a closed-form solution using soft thresholding operator.
Therefore, ISTA can be summarized as the following recursive equation:
\begin{equation}
\zv^{[i+1]} = h_\tau\big(\zv^{[i]} - \alpha\Wv^T(\Wv\zv^{[i]} - \xv)\big),
\label{eq:ista}
\end{equation}
where $h_\tau$ is a soft thresholding operator and $\tau$ is related to
$\lambda$ for controlling sparsity.
Recently, Papyan~\cite{papyan2017convolutional} proposed to use ISTA and sparse
coding to reinterpret feed-forward neural networks. They argue that feed-forward
passing a single-layer neural network~$\zv = \relu(\Wv^T\xv - b)$ can be
considered as one iteration of ISTA when~$\zv~\ge~0, \alpha=1$ and~$\tau = b$.
Based on this insight, the authors extend this interpretation to feed-forward
neural network with~$n$ layers
\begin{equation}
\begin{aligned}
\zv_1 & = \relu(\Wv_1^T\xv - b_1) \\
\zv_2 & = \relu(\Wv_2^T\zv_1 - b_2) \\
& \quad \vdots \\
\zv_n & = \relu(\Wv_n^T\zv_{n-1} - b_n) \\
\end{aligned}
\end{equation}
as executing a sequence of single-iteration ISTA, serving as an approximate
solution to the multi-layer sparse coding problem: find~$\{\zv_i\}_{i=1}^n$,
such that
\begin{equation}
\begin{aligned}
\xv = \Wv_1\zv_1 & , \quad \Vert \zv_1 \Vert_0 < \lambda_1, \zv_1 \ge 0, \\
\zv_1 = \Wv_2\zv_2 & , \quad \Vert \zv_2 \Vert_0 < \lambda_2, \zv_2 \ge 0, \\
\vdots \quad \quad & , \quad \quad \vdots \\
\zv_{n-1} = \Wv_n\zv_n & , \quad \Vert \zv_n \Vert_0 < \lambda_n, \zv_n \ge 0, \\
\end{aligned}
\end{equation}
where the bias terms~$\{b_i\}_{i=1}^n$ (in a similar manner to $\tau$) are
related to~$\{\lambda_i\}_{i=1}^n$, adjusting the sparsity of recovered code.
Furthermore, they reinterpret back-propagating through the deep neural network
as learning the dictionaries~$\{\Wv_i\}_{i=1}^n$. This connection offers a novel
breakthrough for understanding DNNs. In this paper, we extend this to the block
sparse scenario and apply it to solving our NRS\emph{f}M\xspace problem.
\input{architecture.image.tex}
\section{Deep Non-Rigid Structure from Motion}
Under weak-perspective projection, NRS\emph{f}M\xspace deals with the problem of factorizing
a 2D projection matrix $\Wv\in\RR^{p\times 2}$ as the product of a 3D shape
matrix $\Sv\in\RR^{p\times 3}$ and camera matrix $\Mv\in\RR^{3\times 2}$.
Formally,
\begin{equation}
\Wv = \Sv\Mv,
\label{eq:proj}
\end{equation}
\begin{equation}
\Wv = \begin{bmatrix} u_1 & v_1 \\ u_2 & v_2 \\ \vdots & \vdots \\ u_p & v_p \end{bmatrix},~
\Sv = \begin{bmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ \vdots & \vdots & \vdots \\ x_p & y_p & z_p \end{bmatrix},~
\Mv^T\Mv = \Iv_2,
\end{equation}
where $(u_i, v_i), (x_i, y_i, z_i)$ are the image and world coordinates of the
$i$-th point. Due to the scale ambiguity between camera focal length and shape
size, we ignore camera scale. The goal of NRS\emph{f}M\xspace is to recover simultaneously
the shape~$\Sv$ and the camera $\Mv$ for each projection $\Wv$ in a given set
$\mathbb{W}$ of 2D landmarks. In a general NRS\emph{f}M\xspace including S\emph{f}C\xspace, this
set~$\mathbb{W}$ could contain deformations of a non-rigid object or various
instances from an object category.
\subsection{Modeling via multi-layer sparse coding}
\label{sec:mlscm}
To alleviate the ill-posedness of NRS\emph{f}M\xspace and also guarantee sufficient freedom
on shape variation, we propose a novel prior assumption on 3D shapes via
multi-layer sparse coding: The vectorization of $\Sv$ satisfies
\begin{equation}
\begin{aligned}
\sv = \Dv_1\psiv_1 & , \quad \Vert \psiv_1 \Vert_0 < \lambda_1, \psiv_1 \ge 0, \\
\psiv_1 = \Dv_2\psiv_2 & , \quad \Vert \psiv_2 \Vert_0 < \lambda_2, \psiv_2 \ge 0, \\
\vdots \quad \quad & , \quad \quad \vdots \\
\psiv_{n-1} = \Dv_n\psiv_n & , \quad \Vert \psiv_n \Vert_0 < \lambda_n, \psiv_n \ge 0, \\
\end{aligned}
\label{eq:mlsc}
\end{equation}
where $\Dv_1 \in \RR^{3p\times k_1}, \Dv_2 \in \RR^{k_1 \times k_2}, \dots,
\Dv_n \in \RR^{k_{n-1}\times k_n}$ are hierarchical dictionaries.
In this prior, each non-rigid shape is represented by a sequence of hierarchical
dictionaries and corresponding sparse codes. Each sparse code is determined by
its lower-level neighbor and affects the next-level. Clearly this hierarchy
adds more parameters, and thus more freedom into the system. We now show that
it paradoxically results in a more constrained global dictionary and sparse
code recovery.
\subsubsection{More constrained code recovery} In a classical single dictionary
system, the constraint on the representation is element-wise sparsity. Further,
the quality of its recovery entirely depends on the quality of the dictionary.
In our multi-layer sparse coding model, the optimal code not only
minimizes the difference between measurements~$\sv$ and~$\Dv_1\psiv_1$ along
with sparsity regularization~$\Vert \psiv_1 \Vert_0$, but also satisfies
constraints from its subsequent representations. This additional joint
inference imposes more constraints on code recovery, helps to control the
uniqueness and therefore alleviates its heavy dependency on the dictionary
quality.
\subsubsection{More constrained dictionary}
When all equality constraints are satisfied, the multi-layer sparse coding
model degenerates to a single dictionary system. From Equation~\ref{eq:mlsc},
by denoting~$\Dv^{(l)} = \prod_{i=1}^l \Dv_i$, it is implied that
$\sv = \Dv_1\Dv_2\dots\Dv_n\psiv_n = \Dv^{(n)}\psiv_n$.
However, this differs from other single dictionary models~\cite{zhu2014complex,
zhu2013convolutional, kong2016prior, kong2016sfc, zhou20153d} in terms that a
unique structure is imposed on~$\Dv^{(n)}$~\cite{sulam2017multi}. The
dictionary~$\Dv^{(n)}$ is composed by simpler atoms hierarchically. For
example, each column of $\Dv^{(2)} = \Dv_1\Dv_2$ is a linear combination of
atoms in $\Dv_1$, each column of $\Dv^{(3)} = \Dv^{(2)}\Dv_3$ is a linear
combination of atoms in $\Dv^{(2)}$ and so on. Such a structure results in a
more constrained global dictionary and potentially leads to higher quality with
lower mutual coherence~\cite{donoho2006stable}.
\subsection{Multi-layer block sparse coding}
Given the proposed multi-layer sparse coding model, we now build a conduit
from the proposed shape code~$\{\psiv_i\}_{i=1}^k$ to the 2D projected points.
From Equation~\ref{eq:mlsc}, we reshape vector~$\sv$ to a matrix~$\Sv\in\RR^{p\times3}$ such that $\Sv = \Dv^\sharp_1(\psiv_1 \otimes \Iv_3)$,
where $\otimes$ is Kronecker product and $\Dv_1^\sharp\in\RR^{p\times 3k_1}$ is
a reshape of $\Dv_1$~\cite{dai2014simple}.
From linear algebra, it is well known that~$\Av\Bv \otimes \Iv = (\Av\otimes \Iv)(\Bv\otimes\Iv)$
given three matrices~$\Av, \Bv$, and identity matrix $\Iv$. Based on this lemma, we can derive
that
\begin{equation}
\small
\begin{aligned}
\Sv = \Dv^\sharp_1(\psiv_1 \otimes \Iv_3) & , \quad \Vert \psiv_1 \Vert_0 < \lambda_1, \psiv_1 \ge 0, \\
\psiv_1\otimes \Iv_3 = (\Dv_2 \otimes \Iv_3)(\psiv_2\otimes \Iv_3) & , \quad \Vert \psiv_2 \Vert_0 < \lambda_2, \psiv_2 \ge 0, \\
\vdots \quad \quad & , \quad \quad \vdots \\
\psiv_{n-1} \otimes \Iv_3 = (\Dv_n \otimes \Iv_3)(\psiv_n\otimes \Iv_3) & , \quad \Vert \psiv_n \Vert_0 < \lambda_n, \psiv_n \ge 0. \\
\end{aligned}
\label{eq:mlbsc_s}
\end{equation}
Further, from Equation~\ref{eq:proj}, by right multiplying the camera
matrix~$\Mv\in\RR^{3\times2}$ to the both sides of Equation~\ref{eq:mlbsc_s}
and denote $\Psiv_i = \psiv_i \otimes \Mv$, we obtain that
\begin{equation}
\small
\begin{aligned}
\Wv = \Dv^\sharp_1 \Psiv_1 & , \quad \Vert \Psiv_1 \Vert_0^{(3\times 2)} < \lambda_1, \\
\Psiv_1 = (\Dv_2 \otimes \Iv_3)\Psiv_2 & , \quad \Vert \Psiv_2 \Vert_0^{(3\times 2)} < \lambda_2, \\
\vdots \quad \quad & , \quad \quad \vdots \\
\Psiv_{n-1} = (\Dv_n \otimes \Iv_3)\Psiv_n & , \quad \Vert \Psiv_n \Vert_0^{(3\times 2)} < \lambda_n, \\
\end{aligned}
\label{eq:mlbsc_w}
\end{equation}
where $\Vert \cdot \Vert_0^{(3\times 2)}$ divides the argument matrix into
blocks with size $3\times 2$ and counts the number of active blocks.
Since~$\psiv_i$ has active elements less than $\lambda_i$, $\Psiv_i$ has active
blocks less than $\lambda_i$, that is $\Psiv_i$ is block sparse.
This derivation demonstrates that if the shape vector $\sv$ satisfies the
multi-layer sparse coding prior described by Equation~\ref{eq:mlsc}, then its
2D projection~$\Wv$ must be in the format of multi-layer \emph{block} sparse
coding described by Equation~\ref{eq:mlbsc_w}. We hereby interpret NRS\emph{f}M\xspace
as a hierarchical \emph{block} sparse dictionary learning problem \ie
factorizing $\Wv$ as products of hierarchical dictionaries~$\{\Dv_i\}_{i=1}^n$
and block sparse coefficients~$\{\Psiv_i\}_{i=1}^n$.
\subsection{Block ISTA and DNNs solution}
\label{sec:architecture}
Before solving the multi-layer block sparse coding problem in Equation~\ref{eq:mlbsc_w},
we first consider the single-layer problem:
\begin{equation}
\min_{\Zv} \Vert \Xv - \Wv\Zv\Vert_F^2 \quad \st~\Vert \Zv \Vert_{0}^{(3\times2)} < \lambda.
\end{equation}
Inspired by ISTA, we propose to solve this problem by iteratively executing
the following two steps:
\begin{gather}
\Vv = \Zv^{[i]} - \alpha\Wv^T(\Wv\Zv^{[i]} - \Xv), \\
\Zv^{[i+1]} = \argmin_{\Uv} \frac{1}{2}\Vert \Uv - \Vv \Vert^2_F + \tau\Vert\Uv\Vert_{F1}^{(3\times2)},
\end{gather}
where $\Vert \cdot \Vert_{F1}^{(3\times2)}$ is defined as the summation of
Frobenius norm of each $3\times2$ block, serving as a convex relaxation of
block sparsity constraint. It is derived in~\cite{deng2013group} that the
second step has a closed-form solution computing each block separately by
$\small\Zv^{[i+1]}_j = (h_\tau(\Vert\Vv_j\Vert_F)/\Vert \Vv_j \Vert_F)\Vv_j$,
where the subscript $j$ represents the $j$-th block and $h_\tau$ is a soft
thresholding operator. However, soft thresholding the Frobenius norms for every
block brings unnecessary computational complexity. We show in
the supplementary material that an efficient relaxation is
$\Zv^{[i+1]}_j = h_{b_j}(\Vv_j)$, where $b_j$ is the threshold for the $j$-th
block, controlling its sparsity.
Based on this relaxation, a single-iteration block ISTA with step size
$\alpha=1$ can be represented by :
\begin{equation}
\Zv = h_{\bv} \big(\Wv^T\Xv\big) = \relu(\Wv^T\Xv - \bv\otimes\one_{3\times 2}),
\label{eq:singleBISTA}
\end{equation}
where $h_{\bv}$ is a soft thresholding operator using the $j$-th element $b_j$
as threshold of the $j$-th block and the second equality holds if $\Zv$ is
non-negative.
\subsubsection{Encoder}
Recall from Section~\ref{sec: scdnn} that the feed-forward pass through a deep
neural network can be considered as a sequence of single ISTA iterations and
thus provides an approximate recovery of multi-layer sparse codes. We follow
the same scheme: we first relax the multi-layer block sparse coding to be
non-negative and then sequentially use single-iteration block ISTA to solve it \ie
\begin{equation}
\begin{aligned}
\Psiv_1 & = \relu((\Dv^\sharp_1)^T\Wv - \bv_1\otimes\one_{3\times 2}), \\
\Psiv_2 & = \relu((\Dv_2 \otimes \Iv_3)^T\Psiv_1 - \bv_2\otimes\one_{3\times 2}), \\
& \quad \vdots \\
\Psiv_n & = \relu((\Dv_n \otimes \Iv_3)^T\Psiv_{n-1} - \bv_n\otimes\one_{3\times 2}), \\
\end{aligned}
\end{equation}
where thresholds $\bv_1, ..., \bv_n$ are learned, controlling the block sparsity.
This learning is crucial because in previous NRS\emph{f}M\xspace algorithms utilizing
low-rank~\cite{dai2014simple}, subspaces~\cite{zhu2014complex}
or compressible~\cite{kong2016prior} priors, the weight given to this prior
(\eg rank or sparsity) is hand-selected through a cumbersome cross validation
process. In our approach, this weighting is learned simultaneously with all
other parameters removing the need for any irksome cross validation process.
This formula composes the encoder of our proposed DNN.
\subsubsection{Decoder}
Let us for now assume that we can extract camera $\Mv$ and regular sparse
hidden code $\psiv_n$ from $\Psiv_n$ by some functions \ie $\Mv = \Fc(\Psiv_n)$
and $\psiv_n = \Gc(\Psiv_n)$, which will be discussed in the next section. Then
we can compute the 3D shape vector $\sv$ by:
\begin{equation}
\begin{aligned}
\psiv_{n-1} & = \relu(\Dv_n \psiv_n - \bv_n'), \\
& \quad \vdots \\
\psiv_1 & = \relu(\Dv_2 \psiv_2 - \bv_2'), \\
\sv & = \Dv^\sharp_1\psiv_1,
\end{aligned}
\end{equation}
Note we preserve the ReLU and bias term during decoding to further enforce
sparsity and improve robustness. These portion forms the decoder of our DNN.
\subsubsection{Variation of implementation}
The Kronecker product of identity matrix $\Iv_3$ dramatically increases the
time and space complexity of our approach. To eliminate it and make parameter
sharing easier in modern deep learning environments~(\eg TensorFlow, PyTorch),
we reshape the filters and
features and show that the matrix multiplication in each step of the encoder
and decoder can be equivalently computed via multi-channel $1\times1$
convolution~($*$) and transposed convolution~($*^T$) \ie
\begin{equation}
(\Dv_1^\sharp)^T\Wv = \dsf_1^\sharp *^T \wsf,
\end{equation}
where {\small $\dsf_1^\sharp \in \RR^{3\times1\times k_1 \times p},
\wsf\in\RR^{1\times2\times p}$}\footnote{The filter dimension is
height$\times$width$\times$\# of input channel$\times$\# of output channel.
The feature dimension is height$\times$width$\times$\# of channel.}.
\begin{equation}
(\Dv_{i+1}\otimes\Iv_3)^T\Psiv_i = \dsf_{i+1} *^T \Psi_{i},
\end{equation}
where {\small$\dsf_{i+1} \in \RR^{1\times1\times k_{i+1} \times k_i},
\Psi_i\in\RR^{3\times2\times k_i}.$ }
\begin{equation}
\Dv_i\psiv_i = \dsf_{i} * \psi_{i},
\end{equation}
where {\small$\dsf_{i} \in \RR^{1\times1\times k_{i} \times k_{i-1}},
\psi_i\in\RR^{1\times1\times k_i}.$}
\subsubsection{Code and camera recovery}
Estimating~$\psiv_n$ and~$\Mv$ from~$\Psiv_n$ is discussed
in~\cite{kong2016prior} and solved by a closed-form formula. Due to its
differentiability, we could insert the solution directly within our pipeline.
An alternative solution is using a relaxation \ie a fully connected layer
connecting $\Psiv_n$ and $\psiv_n$ and a linear combination among each blocks
of $\Psiv_n$ to estimate~$\Mv$, where the fully connected layer parameters and
combination coefficients are learned from data. In our experiments, we use the
relaxed solution and represent them via convolutions, as shown in
Figure~\ref{fig:architecture}, for conciseness and maintaining proper dimensions.
Since the relaxation has no way to force the orthonormal constraint on the
camera, we seek help from the loss function.
\input{nrsfm_mocap.table.tex}
\subsubsection{Loss function} The loss function must measure the
reprojection error between input 2D points $\Wv$ and reprojected 2D points
$\Sv\Mv$ while simultaneously encouraging orthonormality of the estimated
camera~$\Mv$. One solution is to use spectral norm regularization of~$\Mv$
because spectral norm minimization is the tightest convex relaxation of the
orthonormal constraint~\cite{zhou20153d}. An alternative solution is to hard
code the singular values of~$\Mv$ to be exact ones with the help of Singular
Value Decomposition~(SVD). Even though SVD is generally non-differentiable, the
numeric computation of SVD is differentiable and most deep learning packages
implement its gradients~(\eg PyTorch, TensorFlow). In our implementation and
experiments, we use SVD to ensure the success of the orthonormal constraint and
a simple Frobenius norm to measure reprojection error,
\begin{equation}
Loss = \Vert \Wv - \Sv\tilde{\Mv} \Vert_F, \quad \tilde{\Mv} = \Uv\Vv^T,
\end{equation}
where $\Uv\Sigmav\Vv^T = \Mv$ is the SVD of the camera matrix.
\section{Experiments}
We conduct extensive experiments to evaluate the performance of our deep
solution for solving NRS\emph{f}M\xspace and S\emph{f}C\xspace problems. Further, for evaluating
generalizability, we conduct an experiment applying the pre-trained DNN
to unseen data and reconstruct 3D human pose from a single view. Note that
in all experiments, our model has no access to 3D ground-truth except
qualitative and quantitative evaluations for comparison against the
state-of-art methods. A detailed description of our architectures is in the
supplementary material.
\subsection{NRS\textbf{\textit{f}}M on CMU Motion Capture}
We first apply our method to solving the problem of NRS\emph{f}M\xspace using the CMU motion
capture dataset\footnote{http://mocap.cs.cmu.edu/}. For evaluation on complex
sequences, we concatenate all motions of the same subject and select ten
subjects from CMU MoCap so that each subject contains tens of thousands of
frames. We randomly create orthonormal cameras for each frame to project the 3D
human joints onto images. We compare our method against state-of-the-art
NRS\emph{f}M\xspace works with code released online\footnote{
Paladini~\etal~\cite{paladini2009factorization} fails on all sequences and
therefore removed from the table. Works~\cite{taylor2010non, del2007non,
vicente2012soft, hamsici2012learning, chhatkuli2016inextensible,
gotardo2011kernel, lee2016consensus, gotardo2011non} did not release code.
Works~\cite{akhter2011trajectory, gotardo2011computing, kumar2018scalable,
kumar2016multi} use additional priors, say temporal continuity, and thus not
applicable.}~\cite{torresani2004learning, dai2014simple, kong2016prior}.
Since none of them are capable of scaling up to this number of frames, we
shuffle each sequence, divide them into mini batches
each containing 500 frames, feed each mini batch into baselines, and then
compute the mean error. Our model is trained on the entire sequence. For error
metrics, we use the shape error ratio defined as
$\small\frac{1}{\vert \Sc \vert}\sum_{\Sc} \frac{\Vert \Sv - \hat{\Sv} \Vert_F}{\Vert \hat{\Sv} \Vert_F},$
where $\hat{\Sv}$ is the 3D ground-truth and $\Sc$ is the set of all shapes;
as well as the mean point distance defined as
$\frac{1}{\vert \Sc \vert}\sum_{\Sc} \sum_i \frac{\Vert\Sv_i - \hat{\Sv}_i \Vert_2}{p},$
where $\Sv_i$ is 3D coordinates of $i$-th point on shape $\Sv$ and $p$ is the
number of points. Note that shapes are normalized to real-world sizes so that
each human skeleton is around 1.8 meters high, and the mean point distance is
computed in centimeters. The results are summarized in Table~\ref{tab:nrsfm}.
One can see that our method obtains impressive reconstruction performance and
outperforms others in every sequences. We randomly select a frame for each
subject and render the reconstructed human skeleton in
Figure~\ref{fig:nrsfm}~(a) to \ref{fig:nrsfm}~(j). To give a sense of the quality of
reconstructions when our method fails, we go through all ten subjects in a total
of 140,606 frames and select the frames with the largest errors as shown in
Figure~\ref{fig:nrsfm}(k) and \ref{fig:nrsfm}~(l). Even in the worst cases, our
method grasps a rough 3D geometry of human body instead of completely diverging.
\subsubsection{Noise performance} To analyze the robustness of our method, we
re-train the neural network for Subject 70 using projected points with
Gaussian noise perturbation. The results are summarized in Figure~\ref{fig:noise}.
The noise ratio is defined as $\Vert \text{noise} \Vert_F / \Vert \Wv \Vert_F$.
One can see that our method gets far more precise reconstructions even
when adding up to $20\%$ noise to our image coordinates compared to baselines
with no noise perturbation. This experiment clearly demonstrates the
robustness of our model and its high accuracy against state-of-the-art works.
\input{noise.plot.tex}
\subsubsection{Missing data}
Landmarks are not always visible from the camera owing to the occlusion by other
objects or itself. In the present paper, we focus on a complete measurement
situation not accounting for invisible landmarks. However, thanks to recent
progress in deep-learning-based depth map
reconstruction from sparse observations~\cite{chen2018estimating,
mal2018sparse, li2018depth, liao2017parse, cadena2016multi}, our central
pipeline of DNN can be easily adapted to handling missing data.
\subsection{S\textbf{\textit{f}}C on IKEA furnitures}
We now apply our method to the application of S\emph{f}C\xspace using IKEA dataset~\cite{lpt2013ikea,
wu2016single}. The IKEA dataset contains four object categories: bed, chair,
sofa, and table. For each object category, we employ all annotated 2D point
clouds and augment them with 2K ones projected from the 3D ground-truth
using randomly generated orthonormal cameras\footnote{Augmentation is utilized
due to limited valid frames, because the ground-truth cameras are
partially missing.}. We compare our method against the
baselines~\cite{dai2014simple, kong2016sfc} again using the shape error ratio
metric. The error evaluated on real images are reported and summarized into
Table~\ref{tab:sfc}. One can observe that our method outperforms baselines
with a large margin, clearly showing the superiority of our model. Table~\ref{tab:sfc}
from another perspective reveals the dilemma suffered by baselines of
restricting ill-possedness and modeling high variance of object category. For
qualitative evaluation, we randomly select frames from each object category
and show them in Figure~\ref{fig:sfc}. It shows that our model successfully
learns the intra-category shape variation and reconstructed landmarks
effectively depict the 3D geometry of objects.
\input{sfc.table.tex}
\subsection{Shape from single-view landmarks}
Even though almost all NRS\emph{f}M\xspace algorithms learn a shape dictionary from 2D
projections, none of them apply the learned dictionary to unseen data. This is
because all of them are facing the difficulty of handling large amount of
images and thus cannot generalize well. In this experiment, we show the
generalization of our learned dictionary by evaluating it using sequences invisible to training. Specifically,
we follow the same training and evaluation scheme in~\cite{zhou20153d},
training with Subject~86 in CMU MoCap and evaluating on Subject 13, 14 and 15.
We compare our model to methods for human pose
estimation~\cite{ramakrishna2012reconstructing, zhou20153d} following the same
error metrics in~\cite{zhou20153d}. It is worth mentioning that all baselines
learn shape dictionaries directly from 3D ground-truth, but our method learns
such dictionaries purely from 2D projections~(\ie no 3D supervision). Even in
such an unfair scenario, our method achieves competitive results as summarized
in Table~\ref{tab:recover}. This clearly demonstrates that our method
effectively learns the underlying geometry from pure 2D projections with no
need for 3D supervision, and the learned dictionaries generalize well to unseen
data.
\subsection{Coherence as guide}
As explained in Section~\ref{sec:mlscm}, every sparse code~$\psiv_i$ is
constrained by its subsequent representation and thus the quality of code
recovery depends less on the quality of the corresponding dictionary. However,
this is not applicable to the final code~$\psiv_n$, making it least constrained
with the most dependency on the final dictionary~$\Dv_n$. From this
perspective, the quality of the final dictionary measured by mutual
coherence~\cite{donoho2006stable} could serve as a lower bound of the entire
system. To verify this, we compute the error and coherence in a fixed interval
during training in NRS\emph{f}M\xspace experiments. We consistently observe strong
correlations between 3D reconstruction error and the mutual coherence of the
final dictionary. We plot this relationship in Figure~\ref{fig:coherence}. We
thus propose to use the coherence of the final dictionary as a measure of model
quality for guiding training to efficiently avoid over-fitting especially when
3D evaluation is not available. This improves the utility of our deep NRS\emph{f}M\xspace in
future applications without 3D ground-truth.
\input{recover_mocap.table.tex}
\input{coherence.plot.tex}
\section{Conclusion}
In this paper, we proposed multi-layer sparse coding as a novel prior
assumption for representing 3D non-rigid shapes and designed an innovative
encoder-decoder neural network to solve the problem of NRS\emph{f}M\xspace using no 3D
supervision. The proposed DNN was derived by generalizing the classical sparse
coding algorithm ISTA to a block sparse scenario. The proposed DNN architecture
is mathematically interpretable as a NRS\emph{f}M\xspace multi-layer sparse dictionary
learning problem. Extensive experiments demonstrated our superior performance
against the state-of-the-art methods and the impressive generalization to
unseen data. Finally, we propose to use the coherence of the final dictionary
as a generalization measure, offering a practical way to avoid over-fitting and
selecting the best model without 3D ground-truth.
\input{results.image.tex}
{\small
\bibliographystyle{ieee}
|
{
"timestamp": "2019-03-01T02:05:55",
"yymm": "1902",
"arxiv_id": "1902.10840",
"language": "en",
"url": "https://arxiv.org/abs/1902.10840"
}
|
\section{Introduction}
In atomic nuclei
there are numerous excited states that are originated from the
single-particle and collective motion of the constituent nucleons.
Thus it is useful to have a few representative quantities of the excited states.
The sum rule \cite{Bohigas1979267,Lipparini1989103} is a quantity which involves all the excited states, and contains important collective aspects of the properties of the excited states, such as the giant resonances \cite{Harakeh-Woude} and
the Nambu-Goldstone modes \cite{PhysRevC.92.034321,PhysRevC.97.034321}.
The energy-weighted sum rule is the most commonly used one among various energy moment of the sum rules.
Although it is a summation over all the excited states,
the Thouless theorem \cite{Thouless196178} allows us to evaluate the sum-rule value that is
the summation over all the excited states computed through the
random-phase approximation (RPA) using the expectation value of the double commutator of the Hamiltonian at the ground state computed within the self-consistent Hartree-Fock (HF) theory.
The theorem has been proven also for the Hartree-Fock-Bogoliubov (HFB) + quasiparticle RPA (QRPA)
\cite{PhysRevC.66.024309}, and the second RPA \cite{PhysRevC.90.024305,PhysRevC.35.1159}.
The double commutator of the Hamiltonian becomes simple for the isoscalar and isovector coordinate operators. In the zero-range Skyrme force, only the kinetic-energy term in the Hamiltonian contributes to the energy-weighed sum for an isoscalar coordinate operator, and the kinetic-energy term and momentum-dependent terms in the interaction contribute to the energy-weighted sum rule of an isovector coordinate operator.
Therefore, the Thouless theorem significantly reduces the computational costs of the energy-weighted sum rule, and is also useful for verifying the accuracy of the QRPA calculation.
Nuclear density functional theory (DFT) can be regarded as a starting point of the mean-field models \cite{RevModPhys.75.121,RevModPhys.88.045004}.
In nuclear DFT, the form of the energy density functional (EDF) is not given a priori.
Several EDFs based on the non-relativistic Skyrme and Gogny forces, and relativistic theory are widely used.
The EDF of these types can be derived from the corresponding effective interaction. In that case one can go back to the Hamiltonian (effective interaction) starting from the EDF.
However in general, there is no direct correspondence
to the effective interaction in the nuclear DFT,
if the EDF and its coupling constants are constructed directly
by reproducing a representative set of the experimental observables.
The existence of the Hamiltonian operator is not guaranteed.
Although the Thouless theorem has been applied widely within the framework of the nuclear DFT, to the best of our knowledge,
it has not been proven for the nuclear DFT where the EDF does not correspond to a Hamiltonian operator, and thus
the double-commutator expression cannot be justified.
This includes the case when
the EDF is constructed independent of the interactions (such as UNEDF functionals \cite{PhysRevC.82.024313,PhysRevC.85.024304,PhysRevC.89.054314,0954-3899-42-3-034024}).
Even the standard Skyrme HFB calculation is not carried out within the two-body and three-body Skyrme effective interaction.
Prescriptions used in the spin-orbit and tensor functional may break the correspondence with the Hamiltonian.
The standard Skyrme spin-orbit interaction has a single interaction strength $W_0$ and it determines the
isoscalar and isovector coupling constants of the spin-orbit functional.
In several Skyrme EDFs, additional parameter $b_4'$ is introduced to control the isovector property
of the spin-orbit functional \cite{Reinhard1995467}.
The tensor-density (spin-current density) terms appear from the momentum-dependent $t_1$ and $t_2$ terms of the Skyrme effective interaction even without including the tensor effective interactions ($t_e$ and $t_o$ terms).
However, because of the complicated treatment of the tensor-density terms in deformed nuclei, the contribution from this term is often neglected except for a few parameter sets such as SLy5 \cite{Chabanat1998231} and SkP \cite{Dobaczewski1984103}.
Moreover, the existence of the two-body density-dependent term will also lose the connection to the Hamiltonian operator (However the density-dependent force has been shown not to contribute on the Thouless theorem \cite{Bohigas1979267}).
Another issue is the treatment of the pairing interaction.
Except for the SkP interaction, the pairing interaction used in the standard Skyrme HFB calculation has a simple form and density dependence, and is independent with the particle-hole interaction, while in the mean-field approach starting from an effective interaction, the same interaction should provide the Hartree-Fock potential and pairing potential.
In our previous work \cite{PhysRevC.91.044323},
it has been numerically shown that in SLy4 EDF,
the inclusion of the time-odd current terms is necessary to recover the energy-weighted sum-rule values of the Thouless theorem,
and that other terms in the time-odd functional do not impact the values of the energy-weighted sum rule at all.
The time-odd current terms are necessary in order to satisfy the Galilean invariance of the EDF.
More generalized form of the EDF could be used in the future, and thus
it is desired to understand the applicability of the Thouless theorem in the nuclear DFT.
We note that Kerman-Onishi condition can be derived for nuclear EDF
from the transformation of the densities without assuming the Hamiltonian operator \cite{PhysRevC.88.034311,PhysRevC.84.064303},
and note that the lack of the relation with the Hamiltonian formalism can cause
problems when evaluating the energy of the quantum-number projected state within the nuclear DFT
\cite{PhysRevC.79.044318,PhysRevC.79.044318,PhysRevC.79.044319,PhysRevC.79.044320}.
The aim of this paper is to derive the expression for the energy-weighted sum rule
within the nuclear DFT without using the double commutator of the Hamiltonian.
We consider a fluctuation to the HFB state, and compare the fluctuation of the energy in two ways to derive the expression of the energy-weighted sum rule.
This derivation can be applied to the nuclear EDF which does not have a corresponding Hamiltonian operator.
This paper is organized as follows.
In Sec.~\ref{sec:EDF},
the nuclear EDF is introduced.
Section~\ref{sec:Thouless}
recapitulates the conventional derivation of the Thouless theorem based on the double commutator of the Hamiltonian, then presents the derivation for the nuclear EDF.
Section~\ref{sec:fam} summarizes the energy-weighted sum rule calculation based on complex-energy finite-amplitude method.
In Sec.~\ref{sec:sumrule}, energy-weighted sum-rule values of various multipole operators are numerically calculated using the complex-energy finite-amplitude method, and are compared with the values of the Thouless theorem derived for general nuclear EDFs.
Conclusions are given in Sec.~\ref{sec:conclusion}.
\section{Nuclear EDF \label{sec:EDF}}
We consider a general form of the nuclear EDF of the Skyrme type
that is quadratic in local densities (except for the density-dependent terms)
and can contain up to two spacial derivatives
but without neutron-proton mixing \cite{ActaPhysPolB27_45,PhysRevC.69.014316}.
The nuclear EDF has the following form
\begin{align}
E[\rho,\tilde{\rho}] &= \int d\bm{r} {\cal E}(\bm{r}),\\
{\cal E}(\bm{r}) &= \frac{\hbar^2}{2m}\tau_0(\bm{r}) + \sum_{k=0}^1 \chi_k(\bm{r}) + {\cal E}_{\rm Coul}(\bm{r}) +
\sum_{t=n,p} \tilde{\chi}_t (\bm{r}), \label{eq:EDF}
\end{align}
where the first term in Eq.~(\ref{eq:EDF}) is the isoscalar kinetic energy,
$\chi_k$ is the isoscalar ($k=0$) and isovector ($k=1$) particle-hole EDF,
${\cal E}_{\rm Coul}$ is the Coulomb energy functional, and
$\tilde{\chi}_t$ are the neutron ($t=n=1/2$) and proton ($t=p=-1/2$) pairing EDF.
Throughout this paper, we use the index $k$ to specify the isoscalar or isovector character, and the index $t$ for neutrons or protons.
The particle-hole EDF is given by its time-even and time-odd parts
\begin{align}
\chi_k(\bm{r}) &= \chi_k^{\rm even}(\bm{r}) + \chi_k^{\rm odd}(\bm{r}), \\
\chi_k^{\rm even}(\bm{r}) &=
C^\rho_k[\rho_0] \rho_k^2 + C^{\Delta\rho}_k \rho_k \Delta\rho_k
+ C^\tau_k \rho_k\tau_k + C_k^{J0} J_k^2\nonumber \label{eq:phEDF} \\
&\quad
+ C_k^{J1} \bm{J}_k^2
+ C_k^{J2}\underline{\sf J}_k^2
+ C^{\rho\nabla J}_k \rho_k\bm{\nabla}\cdot \bm{J}_k, \\
\chi_k^{\rm odd}(\bm{r}) &=
C_k^s[\rho_0] \bm{s}_k^2 + C_k^{\Delta s} \bm{s}_k \cdot \Delta \bm{s}_k
+ C_k^T \bm{s}_k\cdot\bm{T}_k
+ C_k^j \bm{j}_k^2 \nonumber \\
&\quad + C_k^{\nabla j} \bm{s}_k \cdot (\bm{\nabla}\times \bm{j}_k)
+ C_k^{\nabla s} (\bm{\nabla}\cdot \bm{s}_k)^2 \nonumber \\
&\quad + C_k^F \bm{s}_k \cdot \bm{F}_k. \label{eq:ppEDF}
\end{align}
The time-even part is composed with the particle-hole density $\rho_k$, kinetic density $\tau_k$, pseudoscalar, pseudovector, and pseudotensor densities $J_k$, $\bm{J}_k$, and $\underline{\sf J}_k$.
The time-odd parts are described with the spin density $\bm{s}_k$, spin-kinetic density $\bm{T}_k$, current density $\bm{j}_k$, and tensor-kinetic density $\bm{F}_k$. Definitions of these local densities are summarized in Appendix \ref{sec:densities}.
Some of the coupling constants $C^\rho_k$ and $C^s_k$ have isoscalar-density dependence.
In the Skyrme force, all the coupling constants are basically derived from the effective interactions, while in the UNEDF optimizations
\cite{PhysRevC.82.024313,PhysRevC.85.024304,PhysRevC.89.054314,0954-3899-42-3-034024},
only the time-even coupling constants are determined.
For the even-even systems with the time-reversal symmetry, the time-odd functionals turn on only in the linear response calculation.
The Coulomb functional is composed of the direct and exchange terms, and are functionals of the proton particle-hole density only [$\rho_p=(\rho_0 - \rho_1)/2$]
\begin{align}
{\cal E}_{\rm Coul}(\bm{r}) &= {\cal E}_{\rm dir}(\bm{r}) + {\cal E}_{\rm ex}(\bm{r}), \\
{\cal E}_{\rm dir}(\bm{r}) &= \frac{1}{2}e^2 \rho_p(\bm{r}) \int d\bm{r}' \frac{\rho_p(\bm{r}')}{|\bm{r}-\bm{r}'|}, \\
{\cal E}_{\rm ex}(\bm{r}) &= -e^2 \frac{3}{4}\left(\frac{3}{\pi}\right)^{\frac{1}{3}} \rho_p(\bm{r})^{\frac{4}{3}}.
\end{align}
The general form of the pairing EDF that is quadratic in local pair densities is given by
\begin{align}
\tilde{\chi}_t(\bm{r}) &= \tilde{C}^\rho_t[\rho_0] |\tilde{\rho}_t|^2
+ \tilde{C}^{\Delta\rho}_t {\rm Re}( \tilde{\rho}^\ast_t \Delta\tilde{\rho}_t)
+ \tilde{C}^\tau_t {\rm Re}(\tilde{\rho}^\ast_t \tilde{\tau}_t) \nonumber \\
&\quad + \tilde{C}^{J0}_t |\tilde{J}_t|^2
+ \tilde{C}^{J1}_t |\tilde{\bm{J}}_t|^2
+ \tilde{C}^{J2}_t |\underline{\tilde{\sf J}}_t|^2 \nonumber \\
&\quad + \tilde{C}^{\nabla J}_t {\rm Re} ( \tilde{\rho}^\ast_t \bm{\nabla}\cdot \tilde{\bm{J}}_t)
\end{align}
with the pair density $\tilde{\rho}_t$, kinetic pair density $\tilde{\tau}_t$, and tensor pair densities
$\tilde{J}_t$, $\tilde{\bm{J}}_t$, and $\underline{\tilde{\sf J}}_t$.
In most of the Skyrme EDFs, only the first term with an isoscalar-density dependence is used in the pairing EDF
\begin{align}
\tilde{C}^\rho_t[\rho_0] = \frac{V_t}{4}\left( 1 - \eta_t \frac{\rho_0(\bm{r})}{\rho_c}\right), \label{eq:pairforce}
\end{align}
where $V_t$ is the strength and $\eta_t$ controls the isoscalar-density dependence.
\section{Thouless theorem for energy-weighted sum rule \label{sec:Thouless}}
\subsection{Operator derivation}
First we recapitulate the conventional derivation of the Thouless theorem \cite{Thouless196178}
based on the discussion in Ref.~\cite{Bohigas1979267}.
We consider a system described by a Hamiltonian of the Skyrme interaction
\begin{align}
\hat{H} &= \hat{T}+\hat{V},\\
\hat{T} &= \frac{1}{2m} \sum_{i=1}^A \hat{\bm{p}}_i^2,
\end{align}
\begin{align}
\hat{V} &= \sum_{i<j}
t_0 (1 + x_0 \hat{P}^\sigma) \delta(\hat{\bm{r}}_{ij}) \nonumber\\
&\quad + \frac{t_1}{2}
( 1 + x_1\hat{P}^\sigma) [\hat{\bm{k}}^{\prime 2}\delta(\hat{\bm{r}}_{ij}) + \delta(\hat{\bm{r}}_{ij})\hat{\bm{k}}^2] \nonumber \\
&\quad + t_2 (1 + x_2 \hat{P}^\sigma )\hat{\bm{k}}'\cdot \delta(\hat{\bm{r}}_{ij})\hat{\bm{k}} \nonumber \\
&\quad + \frac{t_3}{6}( 1+x_3\hat{P}^\sigma) \rho^\gamma\left(\frac{\bm{r}_1+\bm{r}_2}{2}\right)\delta(\hat{\bm{r}}_{ij}) \nonumber \\
&\quad + \frac{t_e}{2}[ \hat{\bm{k}}^{\prime}\cdot \hat{\sf S}\cdot\hat{\bm{k}}^{\prime} \delta(\hat{\bm{r}}_{ij}) + \delta(\hat{\bm{r}}_{ij}) \hat{\bm{k}}\cdot\hat{\sf S}\cdot\hat{\bm{k}}] \nonumber \\
&\quad + t_o \hat{\bm{k}}^{\prime}\cdot \hat{\sf S} \delta(\hat{\bm{r}}_{ij})\cdot \hat{\bm{k}}\nonumber \\
&\quad + i W_0 (\hat{\bm{\sigma}}_i + \hat{\bm{\sigma}}_j)\cdot [ \hat{\bm{k}}^{\prime} \times \delta(\hat{\bm{r}}_{ij})\hat{\bm{k}}],
\end{align}
where $\hat{\bm{r}}_{ij}=\hat{\bm{r}}_i-\hat{\bm{r}}_j$,
$\hat{P}^\sigma=(1+\hat{\bm{\sigma}}_i\cdot\hat{\bm{\sigma}}_j)/2$ is the spin-exchange operator,
$\hat{\sf S}=3(\hat{\bm{\sigma}}_i\cdot\bm{e}_r)(\hat{\bm{\sigma}}_j\cdot\bm{e}_r) - \hat{\bm{\sigma}}_i\cdot\hat{\bm{\sigma}}_j$ is the tensor operator, and
\begin{align}
\hat{\bm{k}} &= \frac{1}{2i}( \bm{\nabla}_i - \bm{\nabla}_j), \\
\hat{\bm{k}}' &= -\frac{1}{2i}( \bm{\nabla}_i - \bm{\nabla}_j).
\end{align}
The energy-weighted sum rule of an operator $\hat{F}$ is expressed in terms of the double commutator of the Hamiltonian
\begin{align}
m_1(\hat{F}) &=
\sum_{\lambda, \Omega_\lambda>0} \Omega_\lambda | \langle \lambda|\hat{F}|0\rangle|^2 \nonumber \\
&= -\frac{1}{2}\langle\Psi_{\rm HFB}|
[[ \hat{H}, \hat{F}], \hat{F}]|\Psi_{\rm HFB}\rangle, \label{eq:m1}
\end{align}
where $|\Psi_{\rm HFB}\rangle$ is the HFB state, $|0\rangle$ is the QRPA correlated ground state, $|\lambda\rangle$ is the QRPA $\lambda$-th excited state with an excitation energy $\Omega_\lambda=E_\lambda - E_0$.
When the operator $\hat{F}$ is an isoscalar-coordinate type
\begin{align}
\hat{F}^{\rm IS}=\alpha \sum_{i=1}^A f(\hat{\bm{r}}_i),
\end{align}
it can be shown that the double commutator of the interaction term cancels,
and the contribution to the energy-weighted sum rule is from the momentum operator in the kinetic-energy term in the Skyrme interaction
\begin{align}
m_1(\hat{F}^{\rm IS})&=
-\frac{1}{2} \langle [[\hat{T}, \hat{F}^{\rm IS}], \hat{F}^{\rm IS}]\rangle
= \alpha^2 \frac{\hbar^2}{2m} \sum_{i=1}^A \langle [\bm{\nabla}f(\hat{\bm{r}}_i)]^2\rangle \nonumber \\
&= \alpha^2 \frac{\hbar^2}{2m} \int d\bm{r} [\bm{\nabla}f(\bm{r})]^2\rho_0(\bm{r}).
\label{eq:isewsr}
\end{align}
The momentum-independent terms with $t_0$ and $t_3$ are shown to commute with the coordinate operator.
The $t_1$ and $t_2$ terms can be written as
\begin{align}
\hat{V}_{t_1,t_2}&= \frac{1}{2} \sum_{i,j=1}^A \left(
\frac{t_1}{8\hbar^2} \{\hat{\bm{p}}_{ij}^2, \delta(\hat{\bm{r}}_{ij})\} + \frac{t_2}{4\hbar^2} \hat{\bm{p}}_{ij}\delta(\hat{\bm{r}}_{ij}) \hat{\bm{p}}_{ij}
\right) \nonumber \\
&= \frac{1}{8\hbar^2} \sum_{i,j=1}^A \biggl\{
\frac{t_1}{2}\left[ \hat{\bm{p}}_{ij}, [ \hat{\bm{p}}_{ij}, \delta(\hat{\bm{r}}_{ij})]\right] \nonumber \\
& \quad +
(t_1+t_2) \hat{\bm{p}}_{ij} \delta(\hat{\bm{r}}_{ij}) \hat{\bm{p}}_{ij} \biggr\}, \label{eq:Vpot}
\end{align}
where $\hat{\bm{p}}_{ij}=\hat{\bm{p}}_i - \hat{\bm{p}}_j$.
The first term is the second derivative of the $\delta$ function, and commutes with any coordinate operator.
The commutator with the second term is shown to be
\begin{align}
[\hat{V}_{t_!,t_2},\hat{F}^{\rm IS}] &=
\frac{1}{8\hbar^2} (t_1+t_2) \sum_{i,j,k=1}^A [ \hat{\bm{p}}_{ij}\delta(\hat{\bm{r}}_{ij})\hat{\bm{p}}_{ij}, f(\hat{\bm{r}}_k)] \nonumber \\
&= \frac{t_1+t_2}{8\hbar^2} \sum_{i,j,k=1}^A \{ \hat{\bm{p}}_{ij}, [\hat{\bm{p}}_{ij}, f(\hat{\bm{r}}_k)]\delta(\hat{\bm{r}}_{ij}) \} \nonumber \\
&= -i\frac{t_1+t_2}{4\hbar}\sum_{i,j=1}^A\left\{ \hat{\bm{p}}_{ij}, [\bm{\nabla}f(\hat{\bm{r}}_i)] \delta(\hat{\bm{r}}_{ij})\right\}\nonumber\\
&= 0,
\end{align}
as interchanging $i$ and $j$ changes the sign.
In the same way, one can derive that the commutators with the $t_e$, $t_o$, and $W_0$ terms become zero.
For the isovector operator
\begin{align}
\hat{F}^{\rm IV}=\sum_{i=1}^A \alpha_{t_i} f(\hat{\bm{r}}_i) \tau_3(t_i), \label{eq:iv}
\end{align}
where $\tau_3(t_i) = 2t_i$,
generally both the kinetic and interaction parts of the Hamiltonian
contribute to the energy-weighted sum rule \cite{Lipparini1989103}
\begin{align}
m_1(\hat{F}^{\rm IV}) &= -\frac{1}{2}\langle [[ \hat{T}+\hat{V}, \hat{F}^{\rm IV}], \hat{F}^{\rm IV}] \rangle \nonumber\\
&= m_1^{\rm kin}(\hat{F}^{\rm IV})\left[1 + \kappa(\hat{F}^{\rm IV})\right],
\end{align}
where $m_1^{\rm kin}(\hat{F}^{\rm IV})$ is the contribution from the kinetic energy
\begin{align}
m_1^{\rm kin}(\hat{F}^{\rm IV}) &= -\frac{1}{2}\langle[[ \hat{T},\hat{F}^{\rm IV}], \hat{F}^{\rm IV}]\rangle \nonumber \\
&= \frac{\hbar^2}{2m} \sum_{i=1}^A \alpha_{t_i}^2 \langle [\bm{\nabla} f(\hat{\bm{r}}_i)]^2\rangle \nonumber \\
&= \frac{\hbar^2}{2m} \int d\bm{r} [\bm{\nabla} f(\bm{r})]^2 [\alpha_n^2\rho_n(\bm{r}) + \alpha_p^2\rho_p(\bm{r})],
\label{eq:ivewsr}
\end{align}
and the enhancement factor $\kappa(\hat{F}^{\rm IV})$ shows the relative contribution of the interaction-energy term with respect to the kinetic part to the energy-weighted sum rule.
The potential contribution is from the second term in Eq.~(\ref{eq:Vpot}).
The spin-exchange parts with $x_1$ and $x_2$
also contribute with $\frac{1}{2}$ factor from $\hat{P}^\sigma$ operator, as the $\bm{\sigma}_i\cdot\bm{\sigma}_j$ part produces the spin density which is zero for even-even systems.
\begin{align}
m_1^{\rm kin} \kappa(\hat{F}^{\rm IV}) &= -\frac{1}{2}\langle[[\hat{V}, \hat{F}^{\rm IV}], \hat{F}^{\rm IV}] \rangle
\nonumber \\
&= \frac{t_1(2+x_1)+t_2(2+x_2)}{8} \times \nonumber \\
&\quad \sum_{i,j=1}^A\alpha_{t_i}\tau_3(t_i)\biggl\{ \alpha_{t_i}\tau_3(t_i)\langle [\bm{\nabla}f(\hat{\bm{r}}_i)]^2 \delta(\hat{\bm{r}}_{ij})\rangle \nonumber \\
&\quad - \alpha_{t_j} \tau_3(t_j) \langle \bm{\nabla}f(\hat{\bm{r}}_i)\cdot \bm{\nabla}f(\hat{\bm{r}}_j)\delta(\hat{\bm{r}}_{ij})\rangle \biggr\}\\
&=
\frac{t_1(2+x_1)+t_2(2+x_2)}{8}(\alpha_n+\alpha_p)^2 \nonumber \\
& \quad \times
\displaystyle \int d\bm{r} [\bm{\nabla}f(\bm{r})]^2 \rho_n(\bm{r})\rho_p(\bm{r}), \label{eq:ivkappa}
\end{align}
where we note that $\frac{1}{8}[t_1(2+x_1)+t_2(2+x_2)] = C^\tau_0 - C^\tau_1$.
These expressions for the energy-weighted sum rule are based on the operator expressions of the kinetic and interaction terms.
Strictly speaking in the nuclear EDF, because there is
no correspondence between the EDF and the Hamiltonian operator $\hat{H}$, we cannot derive Eqs.~(\ref{eq:isewsr}), (\ref{eq:ivewsr}), and (\ref{eq:ivkappa}) in the same manner.
In the next subsection the expressions for the energy-weighted sum rule are derived without
assuming the Hamiltonian operator.
\subsection{Derivation for nuclear EDF}
We consider a small fluctuation starting from an HFB state $|\Psi_{\rm HFB}\rangle$.
As the HFB state is a vacuum of quasiparticles, $\hat{a}_\mu|\Psi_{\rm HFB}\rangle=0$,
following the discussion in Sec.~10.2 of Ref.~\cite{Blaizot-Ripka},
such a fluctuation from the HFB state can be described by a quasiparticle-quasihole, quasiparticle-quasiparticle, and quasihole-quasihole densities.
The quasihole-quasihole and quasiparticle-quasihole densities are given by
\begin{align}
\overline{\kappa}_{\mu\nu} &= \langle \Phi'| \hat{a}_\nu \hat{a}_\mu|\Phi'\rangle, \label{eq:qhqhdensity} \\
\overline{\rho}_{\mu\nu} &= \langle \Phi'| \hat{a}_\nu^\dag \hat{a}_\mu|\Phi'\rangle,
\end{align}
where the state $|\Phi'\rangle$ includes a small fluctuation.
The coherent state representation of the state $|\Phi'\rangle$ gives that $\bar{\rho}$ is shown to be a higher order in $\bar{\kappa}$, $\bar{\rho}\sim (\bar{\kappa}\bar{\kappa}^\dag)$.
Therefore the small-amplitude expansion of the energy from the HFB state is given as an expansion with respect to $\bar{\kappa}$ and $\bar{\kappa}^\ast$
\begin{align}
E'[\bar{\kappa},\bar{\kappa}^\ast] =
E_0' + \frac{1}{2}
\begin{pmatrix} \bar{\kappa}^\ast & \bar{\kappa} \end{pmatrix}
\begin{pmatrix} A & B \\ B^\ast & A^\ast \end{pmatrix}
\begin{pmatrix} \bar{\kappa} \\ \bar{\kappa}^\ast \end{pmatrix}
+ O( |\bar{\kappa}|^3),
\end{align}
where $E_0'$ is the HFB value of the EDF (with particle-number constraint term),
$A$ and $B$ are the QRPA matrices given by
\begin{align}
A_{\rho\sigma,\mu\nu} &= \delta_{\rho\mu} \delta_{\sigma\nu} (E_\mu + E_\nu)
+ \frac{\partial^2 E'}{\partial \bar{\kappa}^\ast_{\rho\sigma}
\partial \bar{\kappa}_{\mu\nu}}, \\
B_{\rho\sigma,\mu\nu} &= \frac{\partial^2 E'}{\partial \bar{\kappa}^\ast_{\rho\sigma}
\partial \bar{\kappa}^\ast_{\mu\nu}},
\end{align}
with the quasiparticle energies $E$.
Let us suppose that this small fluctuation is given with a Hermitian operator $\hat{F}$
\begin{align}
|\Phi'\rangle = e^{i\eta\hat{F}}|\Psi_{\rm HFB}\rangle, \label{eq:perturbation}
\end{align}
where $\eta$ is a small real parameter.
The operator $\hat{F}$ is written in the quasiparticle representation as
\begin{align}
\hat{F} &= \langle\Psi_{\rm HFB}|\hat{F}|\Psi_{\rm HFB}\rangle \nonumber \\
&\quad + \sum_{\mu<\nu} \left\{ F^{20}_{\mu\nu} \hat{a}^\dag_\mu \hat{a}^\dag_\nu
+ F^{02}_{\mu\nu} \hat{a}_\nu \hat{a}_\mu \right\}
+ \sum_{\mu\nu} F^{11}_{\mu\nu} \hat{a}^\dag_\mu \hat{a}_\nu,
\label{eq:Fqp}
\end{align}
where $F^{02}=F^{20\ast}$. From Eqs.~(\ref{eq:perturbation}) and (\ref{eq:Fqp})
we can express the quasihole-quasihole densities $\bar{\kappa}$ in Eq. (\ref{eq:qhqhdensity}) in terms of the matrix element $F^{20}$ and $F^{02}$ as
\begin{align}
\overline{\kappa}_{\mu\nu} &=
\langle \Psi_{\rm HFB}|e^{-i\eta\hat{F}} \hat{a}_\nu \hat{a}_\mu e^{i\eta\hat{F}}|\Psi_{\rm HFB}\rangle = -i\eta F^{20}_{\mu\nu},\\
\overline{\kappa}^\ast_{\mu\nu} &=
\langle \Psi_{\rm HFB}|e^{-i\eta\hat{F}} \hat{a}^\dag_\mu \hat{a}^\dag_\nu e^{i\eta\hat{F}}|\Psi_{\rm HFB}\rangle = i\eta F^{02}_{\mu\nu}.
\end{align}
The energy of this state with the fluctuation $|\Phi'\rangle$ is given by
\begin{align}
E'[ -i\eta F^{20}, i\eta F^{02}]
= E_0' + \eta^2 m_1(\hat{F}) + O(\eta^3), \label{eq:qpexp}
\end{align}
where
\begin{align}
m_1(\hat{F}) = \frac{1}{2} \begin{pmatrix} F^{02} & -F^{20} \end{pmatrix}
\begin{pmatrix} A & B \\ B^\ast & A^\ast \end{pmatrix}
\begin{pmatrix} F^{20} \\ -F^{02} \end{pmatrix}. \label{eq:m1fromab}
\end{align}
Equation (\ref{eq:m1fromab}) is derived by applying the QRPA equations
\begin{align}
\begin{pmatrix} A & B \\ -B^\ast & -A^\ast \end{pmatrix}
\begin{pmatrix} X^\lambda \\ Y^\lambda \end{pmatrix}
= \Omega_\lambda
\begin{pmatrix} X^\lambda \\ Y^\lambda \end{pmatrix}
\end{align}
and the expression for the transition strength
\begin{align}
\langle \lambda | \hat{F} | 0\rangle = \sum_{\mu<\nu} \left( X^{\lambda\ast}_{\mu\nu} F^{20}_{\mu\nu}+ Y^{\lambda\ast}_{\mu\nu}
F^{02}_{\mu\nu} \right)
\end{align}
into Eq.~(\ref{eq:m1}) \cite{Ring-Schuck, Blaizot-Ripka, PhysRevC.79.054329}.
Equation (\ref{eq:qpexp}) shows that the energy-weighted sum rule $m_1(\hat{F})$ appears
as a second-order fluctuation of the total energy of the system where the fluctuation is produced by the operator $\hat{F}$
in the form of Eq. (\ref{eq:perturbation}).
\subsection{Local transformation of the densities}
When a Hamiltonian operator exists ($\hat{H}'=\hat{H}-\sum_{t=n,p} \lambda_t \hat{N}^t)$,
the energy of the perturbed state $|\Phi'\rangle$ is given by the expectation value of the Hamiltonian,
\begin{align}
\langle \Phi'| \hat{H}'| \Phi'\rangle
&= \langle \Psi_{\rm HFB}|e^{-i\eta \hat{F}} \hat{H}' e^{i\eta\hat{F}} |\Psi_{\rm HFB}\rangle\nonumber \\
&= \langle \Psi_{\rm HFB}| \hat{H}' + i\eta [\hat{H}',\hat{F}]
- \frac{\eta^2}{2}[[ \hat{H}', \hat{F}], \hat{F}] \nonumber \\
&\quad + O(\eta^3) |\Psi_{\rm HFB}\rangle.
\end{align}
By comparing the term proportional to $\eta^2$ with Eq.~(\ref{eq:qpexp}), we can derive the Thouless theorem
in the double commutator form
\begin{align}
m_1(\hat{F}) = - \frac{1}{2}\langle \Psi_{\rm HFB}| [[ \hat{H}', \hat{F}], \hat{F}]| \Psi_{\rm HFB}\rangle.
\end{align}
In the nuclear EDF, we regard Eq. (\ref{eq:perturbation}) as a transformation of the wave function.
For an isoscalar operator $\hat{F}^{\rm IS}$, this is nothing but a local
gauge transformation \cite{Blaizot-Ripka,PhysRevC.52.1827,PhysRevC.78.044326,PhysRevC.84.064303}.
The local gauge transformation changes the particle-hole and particle-particle density matrices as \cite{PhysRevC.69.014316}
\begin{align}
\hat{\rho}'(\bm{r} s,\bm{r}' s';t) &= e^{i\eta \alpha [f(\bm{r})- f(\bm{r}')]} \hat{\rho}(\bm{r} s,\bm{r}' s';t), \\
\hat{\tilde{\rho}}'(\bm{r} s,\bm{r}' s';t)&= e^{i\eta \alpha[f(\bm{r})+f(\bm{r}')]} \hat{\tilde{\rho}}(\bm{r} s,\bm{r}' s';t),
\end{align}
and the non-local densities transform as
\begin{align}
\rho'_t(\bm{r},\bm{r}') &= e^{i\eta\alpha[f(\bm{r}) - f(\bm{r}')]} \rho_t(\bm{r},\bm{r}'), \\
\bm{s}'_t(\bm{r},\bm{r}') &= e^{i\eta\alpha[f(\bm{r}) - f(\bm{r}')]} \bm{s}_t(\bm{r},\bm{r}'), \\
\tilde{\rho}'_t(\bm{r},\bm{r}') &= e^{i\eta\alpha[f(\bm{r}) + f(\bm{r}')]} \tilde{\rho}_t(\bm{r},\bm{r}'), \\
\tilde{\bm{s}}'_t(\bm{r},\bm{r}') &= e^{i\eta\alpha[f(\bm{r}) + f(\bm{r}')]} \tilde{\bm{s}}_t(\bm{r},\bm{r}').
\end{align}
In analogy with the Galilean transformation, a local momentum field can be defined as
\begin{align}
\bm{p}(\bm{r}) = \eta \alpha \bm{\nabla} f(\bm{r}).
\end{align}
The local densities in the EDF transform as
\begin{align}
\rho_k' &= \rho_k, \label{eq:rho} \\
\tau_k' &= \tau_k + 2 \bm{p}\cdot \bm{j}_k + \bm{p}^2 \rho_k, \label{eq:tau} \\
\bm{s}'_k &= \bm{s}_k, \\
\bm{T}'_k &= \bm{T}_k + 2 \bm{p}\cdot {\sf J}_k + \bm{p}^2 \bm{s}_k, \\
\bm{j}_k' &= \bm{j}_k + \bm{p}\rho_k, \\
\bm{F}'_k &= \bm{F}_k + \bm{p} J_k + {\sf J}_k \cdot \bm{p} + \bm{p}(\bm{p}\cdot \bm{s}_k), \\
{\sf J}_k' &= {\sf J}_k + \bm{p} \otimes \bm{s}_k, \\
\tilde{\rho}_t' &= e^{2i\eta\alpha f} \tilde{\rho}_t, \\
\tilde{\tau}_t' &= e^{2i\eta\alpha f} (\tilde{\tau}_t + i\bm{p}\cdot\bm{\nabla}\tilde{\rho}_t - \bm{p}^2 \tilde{\rho}_t), \\
\tilde{\sf J}_t' &= e^{2i\eta\alpha f}\tilde{\sf J}_t. \label{eq:Jpair}
\end{align}
For the local gauge invariant EDF, the transformation above does not change the EDF in Eq.~(\ref{eq:EDF}),
except for the kinetic-energy term. From Eq.~(\ref{eq:tau}), the kinetic-energy term transforms as
\begin{align}
E_{\rm kin}' &=
\frac{\hbar^2}{2m}\int d\bm{r} \tau_0' \nonumber \\
&=
\frac{\hbar^2}{2m}\int d\bm{r} \left[ \tau_0 + 2\eta\alpha(\bm{\nabla}f)\cdot\bm{j}_0 + \eta^2 \alpha^2(\bm{\nabla}f)^2\rho_0\right].
\end{align}
The term proportional to $\eta^2$ contributes to the energy-weighted sum rule of the isoscalar operator.
Then we have
\begin{align}
m_1(\hat{F}^{\rm IS}) = \frac{\hbar^2}{2m} \alpha^2 \int d\bm{r} [ \bm{\nabla} f(\bm{r})]^2 \rho_0(\bm{r})
\end{align}
for the local gauge invariant EDF.
This is the derivation of the Thouless theorem without using the Hamiltonian operator and double commutator.
Only the local gauge invariance property of the EDF is imposed in the derivation, and thus
the existence or absence of the spin, spin-orbit, and density-dependent terms both in the particle-hole and pairing channels does not contribute to the energy-weighted sum rule as long as the EDF is local gauge invariant.
As for the pairing channel, local gauge invariant pairing EDF does not contribute to the energy-weighted sum rule.
Such local gauge invariant EDFs are not limited to the ones with the isoscalar density dependence considered in Eq.~(\ref{eq:pairforce}), but including isovector density dependence \cite{PhysRevC.80.064301} and Fayans functional with particle-hole density-gradient dependence \cite{FAYANS199619}.
We can consider a general EDF that does not hold the local gauge invariance.
Without the local gauge invariance,
the transformation introduces additional terms, but when computing the energy of the transformed state,
the densities of an even-even nucleus are used.
Therefore any time-odd densities included in the transformed EDF vanish.
The contribution from the spin-orbit and tensor terms produce terms proportional to the spin density $\bm{s}$,
and thus they do not contribute to the energy-weighted sum as well.
The Coulomb functionals are written with the proton local densities only, and they are local gauge invariant.
Thus the possible contributions are from the $\rho_k\tau_k$ and $\bm{j}^2_k$ terms in the particle-hole EDF,
and ${\rm Re} \tilde{\rho}_t^\ast\tilde{\tau}_t$ and ${\rm Re} \tilde{\rho}_t^\ast\Delta\tilde{\rho}_t$ terms in the pairing EDF.
The particle-hole part and pairing part of the EDF transforms as
\begin{align}
\int d\bm{r}
\chi_k[\rho_k',\tau_k'\cdots] &=
\int d\bm{r} \biggl\{ \chi_k[\rho_k,\tau_k,\cdots] \nonumber \\
&\quad + (C^{\tau}_k + C^j_k) \bm{p}^2 \rho_k^2\biggr\} ,\\
\int d\bm{r} \tilde{\chi}_t[\tilde{\rho}'_t, \tilde{\rho}^{\prime\ast}_t, \tilde{\tau}'_t,\cdots, \rho_0']
&= \int d\bm{r} \biggl\{ \tilde{\chi}_t[\tilde{\rho}_t, \tilde{\rho}^{\ast}_t, \tilde{\tau}_t, \cdots, \rho_0] \nonumber \\
& \quad - (4 \tilde{C}_t^{\Delta\rho} + \tilde{C}_t^\tau )
\bm{p}^2 |\tilde{\rho}_t|^2\biggr\},
\end{align}
where we kept terms which are non-zero in time-reversal symmetric even-even systems.
The combinations of the coefficients $(C^{\tau}_k + C^j_k)$
and $(4\tilde{C}_t^{\Delta\rho} + \tilde{C}_t^\tau )$
show that these additional terms exist only when the local gauge symmetry of $\rho_k\tau_k-\bm{j}^2_k$ and/or
${\rm Re} (4\tilde{\rho}_t^\ast \Delta\tilde{\rho}_t-\tilde{\rho}_t^\ast\tilde{\tau}_t)$
is broken.
By taking the terms that are second order in $\eta$ and performing the integration, the Thouless theorem for the isoscalar operator in the nuclear EDF is derived
\begin{align}
m_1(\hat{F}^{\rm IS}) &= \alpha^2
\int d\bm{r} [ \bm{\nabla} f(\bm{r})]^2 \biggl\{ \frac{\hbar^2}{2m} \rho_0(\bm{r}) \nonumber \\
&\quad + \sum_{k=0}^1 (C^{\tau}_k + C^j_k) \rho_k(\bm{r})^2 \nonumber \\
&\quad - \sum_{t=n,p} (4 \tilde{C}_t^{\Delta\rho} + \tilde{C}_t^\tau) |\tilde{\rho}_t(\bm{r})|^2 \biggr\}.
\label{eq:EWSR-IS}
\end{align}
We note that in Ref.~\cite{Lipparini1989103} it is discussed that
the sum rule is obtained by the exact cancellation of the potential contribution to the effective mass ($\rho\tau$-term)
and the isoscalar current-current interaction in the RPA level
for the system with $N=Z$ and without spin-orbit interaction.
The present derivation based on the local gauge transformation gives a unified view,
including the contribution from the local gauge symmetry breaking of the isovector current terms
and pairing EDF, and that the local gauge symmetry breaking in the spin-orbit and tensor functionals are shown not to play any role for the energy-weighted sum rule of the isoscalar coordinate operators.
\subsection{Isovector operator}
The energy-weighted sum rule of the isovector operator
for nuclear EDF can be derived by generating the fluctuation using the
isovector operator in Eq.~(\ref{eq:iv}). Let us consider a corresponding transformation
with the isovector operator
\begin{align}
|\Psi'\rangle = \exp \left[
i \eta \sum_{i=1}^A \alpha_{t_i} f(\hat{\bm{r}}_i) \tau_3(t_i) \right]|\Psi_{\rm HFB}\rangle. \label{eq:ivtrans}
\end{align}
Because this transformation is not a local gauge transformation,
even the local gauge invariant EDF is not invariant under this transformation.
The density matrices transform with Eq.~(\ref{eq:ivtrans}) as
\begin{align}
\hat{\rho}'(\bm{r} s,\bm{r}' s';t) &= e^{i (2t)\eta\alpha_t[f(\bm{r})
-f(\bm{r}')]} \hat{\rho}(\bm{r} s,\bm{r}' s';t), \\
\hat{\tilde{\rho}}'(\bm{r} s,\bm{r}' s';t) &= e^{i(2t)\eta\alpha_t[f(\bm{r})
+f(\bm{r}')]} \hat{\tilde{\rho}}(\bm{r} s,\bm{r}' s';t).
\end{align}
Then non-local densities of neutrons and protons transform as
\begin{align}
\rho'_t(\bm{r},\bm{r}') &= e^{i(2t)\eta\alpha_t [f(\bm{r})
- f(\bm{r}')]} \rho_t(\bm{r},\bm{r}'), \label{eq:nlrhoiv1}\\
\bm{s}'_t(\bm{r},\bm{r}') &= e^{i(2t)\eta\alpha_t[f(\bm{r})
-f(\bm{r}')]} \bm{s}_t(\bm{r},\bm{r}'), \label{eq:nlrhoiv2}\\
\tilde{\rho}'_t(\bm{r},\bm{r}') &= e^{i(2t)\eta\alpha_t [f(\bm{r})
+ f(\bm{r}')]} \tilde{\rho}_t(\bm{r},\bm{r}'), \\
\tilde{\bm{s}}'_t(\bm{r},\bm{r}') &= e^{i(2t)\eta\alpha_t[f(\bm{r})
+f(\bm{r}')]} \tilde{\bm{s}}_t(\bm{r},\bm{r}').
\end{align}
Note that the indices in Eqs.~(\ref{eq:nlrhoiv1}) and (\ref{eq:nlrhoiv2}) are $t$.
We define local momentum fields of neutron and proton
\begin{align}
\bm{p}_t(\bm{r}) =(2t)\eta\alpha_t\bm{\nabla} f(\bm{r}).
\end{align}
The transformation in Eq.~(\ref{eq:ivtrans}) does not mix the neutron and proton phase. Therefore the isoscalar and isovector local densities transform as
\begin{align}
\rho_k' &= \rho_k, \\
\tau_0' &= \tau_0 + (\bm{p}_n + \bm{p}_p)\cdot \bm{j}_0 + \frac{1}{2}(\bm{p}_n^2 + \bm{p}_p^2) \rho_0 \nonumber \\
&\quad + (\bm{p}_n - \bm{p}_p)\cdot \bm{j}_1 + \frac{1}{2}(\bm{p}_n^2 - \bm{p}_p^2) \rho_1, \\
\tau_1' &= \tau_1 + (\bm{p}_n + \bm{p}_p)\cdot \bm{j}_1 + \frac{1}{2}(\bm{p}_n^2 + \bm{p}_p^2) \rho_1 \nonumber \\
&\quad + (\bm{p}_n - \bm{p}_p)\cdot \bm{j}_0 + \frac{1}{2}(\bm{p}_n^2 - \bm{p}_p^2) \rho_0, \\
\bm{s}_k' &= \bm{s}_k, \\
\bm{T}_0' &= \bm{T}_0 + (\bm{p}_n + \bm{p}_p)\cdot {\sf J}_0 + \frac{1}{2}(\bm{p}_n^2 + \bm{p}_p^2) \bm{s}_0 \nonumber \\
&\quad + (\bm{p}_n - \bm{p}_p)\cdot {\sf J}_1 + \frac{1}{2}(\bm{p}_n^2 - \bm{p}_p^2) \bm{s}_1, \\
\bm{T}_1' &= \bm{T}_1 + (\bm{p}_n + \bm{p}_p)\cdot {\sf J}_1 + \frac{1}{2}(\bm{p}_n^2 + \bm{p}_p^2) \bm{s}_1 \nonumber \\
&\quad + (\bm{p}_n - \bm{p}_p)\cdot {\sf J}_0 + \frac{1}{2}(\bm{p}_n^2 - \bm{p}_p^2) \bm{s}_0, \\
\bm{j}_0' &= \bm{j}_0 + \frac{1}{2} (\bm{p}_n + \bm{p}_p) \rho_0 + \frac{1}{2}(\bm{p}_n - \bm{p}_p)\rho_1, \\
\bm{j}_1' &= \bm{j}_1 + \frac{1}{2} (\bm{p}_n + \bm{p}_p) \rho_1 + \frac{1}{2}(\bm{p}_n - \bm{p}_p)\rho_0, \\
\bm{F}_0' &= \bm{F}_0 + \frac{1}{2}(\bm{p}_n + \bm{p}_p) J_0 + \frac{1}{2}(\bm{p}_n - \bm{p}_p) J_1 \nonumber \\
&\quad + \frac{1}{2} {\sf J}_0 \cdot (\bm{p}_n + \bm{p}_p) + \frac{1}{2} {\sf J}_1 \cdot (\bm{p}_n - \bm{p}_p) \nonumber \\
&\quad + \frac{1}{2} [ (\bm{p}_n\cdot \bm{s}_0) \bm{p}_n + (\bm{p}_p\cdot \bm{s}_0)\bm{p}_p \nonumber \\
& \quad + (\bm{p}_n\cdot \bm{s}_1) \bm{p}_n - (\bm{p}_p\cdot \bm{s}_1)\bm{p}_p],\\
\bm{F}_1' &= \bm{F}_1 + \frac{1}{2}(\bm{p}_n + \bm{p}_p) J_1 + \frac{1}{2}(\bm{p}_n - \bm{p}_p) J_0\nonumber \\
&\quad + \frac{1}{2} {\sf J}_1 \cdot (\bm{p}_n + \bm{p}_p) + \frac{1}{2} {\sf J}_0 \cdot (\bm{p}_n - \bm{p}_p) \nonumber \\
&\quad + \frac{1}{2} [ (\bm{p}_n\cdot \bm{s}_1) \bm{p}_n + (\bm{p}_p\cdot \bm{s}_1)\bm{p}_p \nonumber \\
&\quad + (\bm{p}_n\cdot \bm{s}_0) \bm{p}_n - (\bm{p}_p\cdot \bm{s}_0)\bm{p}_p],\\
{\sf J}_0' &= {\sf J}_0 + \frac{1}{2} (\bm{p}_n + \bm{p}_p)\otimes \bm{s}_0
+ \frac{1}{2}(\bm{p}_n - \bm{p}_p)\otimes \bm{s}_1, \\
{\sf J}_1' &= {\sf J}_1 + \frac{1}{2} (\bm{p}_n + \bm{p}_p)\otimes \bm{s}_1
+ \frac{1}{2}(\bm{p}_n - \bm{p}_p)\otimes \bm{s}_0, \\
\tilde{\rho}_t' &= e^{2i(2t)\eta\alpha_tf} \tilde{\rho}_t, \\
\tilde{\tau}_t' &= e^{2i(2t)\eta\alpha_tf}(\tilde{\tau}_t + i\bm{p}_t\cdot\bm{\nabla}\tilde{\rho}_t - \bm{p}_t^2 \tilde{\rho}_t), \\
\tilde{\sf J}_t' &= e^{2i(2t)\eta\alpha_t f}\tilde{\sf J}_t.
\end{align}
We then consider an EDF transformed with Eq.~(\ref{eq:ivtrans})
\begin{align}
E'[ -i\eta F^{20}, i\eta F^{02}]
&= \int d\bm{r} \biggl\{ \frac{\hbar^2}{2m} \tau_0'
+ \sum_{k=0}^1 \chi_k[ \rho_k', \tau_k', \cdots] \nonumber \\ &\quad +
\sum_{t=n,p} \tilde{\chi}_t[ \tilde{\rho}_t', \tilde{\rho}_t'^\ast, \tilde{\tau}_t', \cdots, \rho_0'] \biggr\} \nonumber\\
&\quad +O(\eta^3).
\end{align}
The kinetic-energy term transforms as
\begin{align}
\frac{\hbar^2}{2m} \tau_0' &=
\frac{\hbar^2}{2m}\left[ \tau_0 + (\bm{p}_n + \bm{p}_p)\cdot \bm{j}_0 + \frac{1}{2}(\bm{p}_n^2+ \bm{p}_p^2) \rho_0 \right. \nonumber \\
&\quad \left. + (\bm{p}_n - \bm{p}_p)\cdot \bm{j}_1 + \frac{1}{2}(\bm{p}_n^2 - \bm{p}_p^2) \rho_1 \right]. \label{eq:ivkin}
\end{align}
Again in the particle-hole part, as the time-reversal symmetry cancels most of the terms, only the terms from $\rho_k\tau_k$ and $\bm{j}_k^2$ generate time-even
contribution to the transformed EDF
\begin{align}
\int d\bm{r} \sum_{k=0}^1 \chi_k&[\rho_k',\tau_k',\cdots] \nonumber \\
=&
\int d\bm{r} \sum_{k=0}^1 \biggl\{ \chi_k[\rho_k,\tau_k, \cdots] \nonumber \\
&\quad + C_k^{\tau}\left[
\left( \rho'_k\tau'_k - \bm{j}_k^{\prime 2} \right) - \left( \rho_k\tau_k - \bm{j}_k^2 \right)\right]\biggr\},\nonumber\\
=& \int d\bm{r} \biggl\{ \sum_{k=0}^1 \chi_k[\rho_k,\tau_k, \cdots] \nonumber \\
&+\frac{1}{4}(C^{\tau}_0 - C^{\tau}_1)
(\bm{p}_n - \bm{p}_p)^2 (\rho_0^2 - \rho_1^2)\biggr\}. \label{eq:ivph}
\end{align}
The fluctuation of the pairing EDF does not contribute because the neutron and proton terms are independent in the pairing EDF.
By taking the terms proportional to $\eta^2$ from Eqs.~(\ref{eq:ivkin}) and (\ref{eq:ivph}), we have
\begin{align}
m_1(\hat{F}^{\rm IV}) &= \int d\bm{r} [\bm{\nabla}f(\bm{r})]^2 \biggl\{
\frac{\hbar^2}{2m}\left[\alpha_n^2 \rho_n(\bm{r}) + \alpha_p^2 \rho_p(\bm{r}) \right] \nonumber \\
&\quad +
(C_0^{\tau} - C_1^{\tau}) (\alpha_n + \alpha_p)^2 \rho_n(\bm{r})\rho_p(\bm{r})\biggr\} \nonumber \\
&= m_1^{\rm kin}(\hat{F}^{\rm IV})\left[ 1 + \kappa(\hat{F}^{\rm IV})\right].
\end{align}
The first term is the kinetic-energy contribution,
and the ratio to the second term defines the isovector enhancement factor $\kappa(\hat{F}^{\rm IV})$
\begin{align}
m_1^{\rm kin}(\hat{F}^{\rm IV}) &= \frac{\hbar^2}{2m} \int d\bm{r} [\bm{\nabla}f(\bm{r})]^2 \left[\alpha_n^2 \rho_n(\bm{r}) + \alpha_p^2 \rho_p(\bm{r}) \right],
\end{align}\begin{align}
\kappa(\hat{F}^{\rm IV}) &= \frac{2m}{\hbar^2} (C_0^{\tau} - C_1^{\tau})(\alpha_n+\alpha_p)^2 \nonumber \\
&\quad \times \frac{\displaystyle\int d\bm{r} [\bm{\nabla}f(\bm{r})]^2 \rho_n(\bm{r})\rho_p(\bm{r})}
{\displaystyle\int d\bm{r} [\bm{\nabla}f(\bm{r})]^2 \left[\alpha_n^2 \rho_n(\bm{r}) + \alpha_p^2 \rho_p(\bm{r})\right]}.
\end{align}
$\alpha_n=\alpha_p=1$ produces Eqs.~(6.32) and (6.38) in Ref.~\cite{Lipparini1989103}
\begin{align}
m_1^{\rm kin}(\hat{F}^{\rm IV}) &= \frac{\hbar^2}{2m} \int d\bm{r} [\bm{\nabla}f(\bm{r})]^2 \rho_0(\bm{r}),
\end{align}\begin{align}
\kappa &= \frac{8m}{\hbar^2} (C_0^{\tau} - C_1^{\tau})
\frac{\displaystyle\int d\bm{r} [\bm{\nabla}f(\bm{r})]^2 \rho_n(\bm{r})\rho_p(\bm{r})}
{\displaystyle\int d\bm{r} [\bm{\nabla}f(\bm{r})]^2 \rho_0(\bm{r})}.
\end{align}
We often use $\alpha_n=Z/A, \alpha_p=N/A$, especially for the dipole operator to remove the contribution of the center of mass motion.
In the case of the isovector dipole operators $f(\bm{r})=f^{\rm IV}_{1K}(\bm{r}) (K=0, 1)$,
we have a model-independent kinetic contribution (Thomas-Reiche-Kuhn sum rule \cite{Thomas1925,Ladenburg1923,Kuhn1925})
\begin{align}
m_1^{\rm kin}(\hat{F}^{\rm IV}_{1K}) &= \frac{\hbar^2}{2m} \frac{3}{4\pi} \frac{NZ}{A}, \\
\kappa^{\rm IV}_{1K} &= \frac{2m}{\hbar^2} \frac{A}{NZ} (C^{\tau}_0 - C^{\tau}_1)
\int d\bm{r} \rho_n(\bm{r}) \rho_p(\bm{r}).
\end{align}
If the EDF does not hold the local gauge invariance,
again all the additional terms to the energy-weighted sum rule in the particle-hole channel
come from $\rho_k\tau_k$ and $\bm{j}_k^2$ terms,
\begin{align}
\int d\bm{r} &\sum_{k=0}^1 \left( C^{\tau}_k \rho_k' \tau_k' + C^j_k \bm{j}_k^{\prime 2} \right) \nonumber \\
&= \int d\bm{r}\biggl\{ \sum_{k=0}^1 C^{\tau}_k \rho_k \tau_k
+ (C^{\tau}_0 - C^{\tau}_1) (\bm{p}_n - \bm{p}_p)^2 \rho_n \rho_p \nonumber \\
&\quad + (C^{\tau}_0 + C^j_0 + C^{\tau}_1 + C^j_1) (\bm{p}^2_n \rho_n^2 + \bm{p}^2_p \rho_p^2) \nonumber \\
&\quad + 2(C^{\tau}_0 + C^j_0 - C^{\tau}_1 - C^j_1) \bm{p}_n\cdot\bm{p}_p \rho_n \rho_p\biggr\}. \label{eq:ivrhotauminusj2}
\end{align}
The second term in the right hand side of Eq.~(\ref{eq:ivrhotauminusj2}) is the contribution to the enhancement factor. The third and fourth terms vanish when the EDF is local gauge invariant for $\rho_k\tau_k$ and $\bm{j}_k^2$ terms ($C_k^j= -C_k^\tau$).
The pairing EDF transforms as
\begin{align}
\tilde{\chi}_t[\tilde{\rho}_t',\tilde{\rho}_t^{\prime\ast}, \tau_t', \cdots, \rho_0']
&= \tilde{\chi}_t[\tilde{\rho}_t,\tilde{\rho}_t^\ast, \tau_t, \cdots, \rho_0] \nonumber \\
&\quad - (4 \tilde{C}_t^{\Delta\rho} + \tilde{C}_t^\tau) \bm{p}^2_t |\tilde{\rho}_t|^2,
\end{align}
and produces contributions from the local gauge symmetry breaking.
The energy-weighted sum rule of an isovector operator for the nuclear EDF is then given by
\begin{align}
m_1 & (\hat{F}^{\rm IV}) =
\int d\bm{r} [\bm{\nabla}f(\bm{r})]^2 \biggl\{
\frac{\hbar^2}{2m} \left[\alpha_n^2 \rho_n(\bm{r}) + \alpha_p^2 \rho_p(\bm{r}) \right] \nonumber \\
&\quad +
(C_0^{\tau} - C_1^{\tau}) (\alpha_n + \alpha_p)^2\rho_n(\bm{r})\rho_p(\bm{r}) \nonumber \\
&\quad +\sum_{k=0}^1(C^{\tau}_k + C^j_k) \left[\alpha_n \rho_n(\bm{r}) +(-1)^{k+1} \alpha_p \rho_p(\bm{r})\right]^2 \nonumber \\
&\quad - \sum_{t=n,p}(4 \tilde{C}_t^{\Delta\rho} + \tilde{C}_t^\tau) \alpha_t^2 |\tilde{\rho}_t(\bm{r})|^2\biggr\}.
\label{eq:EWSR-IV}
\end{align}
\section{Finite-amplitude method} \label{sec:fam}
To check the expressions for the energy-weighted sum rules for nuclear EDF derived in the
previous section, QRPA calculations based on the linear-response theory have been performed.
In this section the procedure to calculate the energy-weighted sum rule from the linear response theory is summarized.
The finite-amplitude method (FAM) for computing the linear response is performed \cite{nakatsukasa:024318,PhysRevC.84.014314}.
The FAM allows to perform a linear response within nuclear DFT
for a given external field $\hat{F}$ with a complex frequency $\omega$.
By solving the linearized time-dependent Hartree-Fock-Bogoliubov equations,
the strength function $S(\hat{F},\omega)$ can be numerically evaluated by an iterative method.
The strength function is written in terms of the QRPA energies and strengths as
\begin{align}
S(\hat{F},\omega) = - \sum_{\lambda (\Omega_\lambda>0)}
\left\{
\frac{ |\langle\lambda|\hat{F}|0\rangle|^2}{\Omega_\lambda - \omega}
+
\frac{ |\langle 0 |\hat{F}|\lambda\rangle|^2}{\Omega_\lambda + \omega}
\right\}.
\end{align}
We perform a contour integration in the complex-energy plane
to evaluate the energy-weighted sum rule numerically
\begin{align}
m_1(\hat{F}) = \frac{1}{2\pi i}\int_{A_1} \omega S(\hat{F},\omega) d\omega, \label{eq:A1}
\end{align}
where the integration path is taken to include all the positive-energy poles in the strength function.
We set the half counterclockwise arc $A_1$ from $\omega=-iR_{A_1}$ to $iR_{A_1}$ centered at the origin,
and a line on the imaginary axis from $\omega=iR_{A_1}$ to $\omega=-iR_{A_1}$ to encircle all the poles in the range of $0 < \Omega_\lambda < R_{A1}$. For a Hermitian operator $\hat{F}$ the integration along the imaginary axis vanishes, and Eq.~(\ref{eq:A1}) is derived.
We refer Ref.~\cite{PhysRevC.91.044323} for more detailed discussion on the complex-energy FAM for the sum rules.
\section{Comparison of sum-rule values} \label{sec:sumrule}
In the numerical comparison, we use the functionals based on the UNEDF1-HFB \cite{0954-3899-42-3-034024}
which contains only the time-even coupling constants in the particle-hole channel.
In the present calculation, we do not include the time-odd functionals except for the current terms $C^j_k\bm{j}_k^2$.
Therefore the functional does not correspond to a specific Hamiltonian operator, and breaks the local gauge invariance
in the spin-orbit terms.
For the comparison of the sum-rule values, we consider four EDFs with the isoscalar and isovector current terms.
\begin{enumerate}
\item isoscalar and isovector current terms in the local gauge invariant form ($C^j_0=-C^\tau_0$ and $C^j_1=-C^\tau_1$)
\item isovector current term only ($C^j_0=0, C^j_1=-C^\tau_1$)
\item isoscalar current term only ($C^j_0=-C^\tau_0, C^j_1=0$)
\item no current terms ($C^j_0=C^j_1=0$).
\end{enumerate}
We compute the energy-weighted sum rule of the monopole ($K=0$), dipole ($K=0$ and 1), quadrupole ($K=0$, 1, and 2), and
octupole ($K=0, 1, 2,$ and 3) operators of the isoscalar and isovector type.
Expressions for the energy-weighted sum rule of these operators in the cylindrical coordinate system are summarized in Appendix \ref{sec:multipole}.
We use $\alpha = Z/A$ for the isoscalar operators and $\alpha_n=Z/A$ and $\alpha_p=N/A$ for the isovector operators.
The calculations are performed with the HFBTHO code \cite{Stoitsov200543, Stoitsov20131592, PEREZ2017363} and its FAM extension
for the non-axial finite $K$ modes \cite{PhysRevC.92.051302}.
This version of the code uses linearized densities explicitly, and thus $\eta$ parameter in the FAM is not necessary in the numerical calculation.
$N_{\rm sh}=20$ harmonic-oscillator shells are used as the single-particle model space,
and $N_{\rm GH}=40$, $N_{\rm GL}=40$, and $N_{\rm leg}=80$ points are used for the Gauss quadratures.
60 MeV pairing window is employed.
In the FAM calculation the integration radius is set to $R_{A1}= 200$ MeV, and the half arc $A_1$ is discretized with $300$ points.
We compute $^{208}$Pb ground state as a representative case of the spherical state without pairing, and
$^{166}$Dy as a case with prolate deformation and pairing ($\beta=0.33$, $\Delta_n=0.64$ MeV, and $\Delta_p=0.58$ MeV).
Tables~\ref{table:208PbIS} and \ref{table:208PbIV} compare energy-weighted sum rule of $^{208}$Pb from the Thouless theorem [Eqs.~(\ref{eq:EWSR-IS}) and (\ref{eq:EWSR-IV})] with Eq.~(\ref{eq:A1}) of the complex-energy FAM,
while Tables~\ref{table:166DyIS} and \ref{table:166DyIV} are the same comparison but of $^{166}$Dy.
In $^{208}$Pb, the sum rule of different $K$ value gives the same value because of the spherical symmetry,
while in $^{166}$Dy, the sum-rule values depend on $K$ for the same multipole $L$ due to the ground-state deformation.
The agreement between the expressions from the Thouless theorem and the values from the complex-energy FAM is excellent.
From the ratio of the sum rules of the FAM to the value from the Thouless theorem,
the maximum discrepancy is about 0.7\% and 0.4\% for $^{208}$Pb of the isoscalar and isovector operators.
and 1.2\% and 0.5\% in $^{166}$Dy, respectively.
Because the current terms do not change the HFB state, the difference in the energy-weighted sum-rule value between the calculations with/without the current terms show the actual contribution of the local gauge symmetry breaking.
The effect of the isoscalar (isovector) current is much larger than the other in the sum rule of the isoscalar (isovector) multipole operator. This is because the contribution of the isoscalar (isovector) current term to the energy-weighted sum rule of the isoscalar operator is proportional to the isoscalar (isovector) density squared in Eq.~(\ref{eq:EWSR-IS}), and the isoscalar density is generally much larger than the isovector density. For the isovector multipole operator, as seen in Eq.~(\ref{eq:EWSR-IV}),
the contribution of the isovector current term is from an isoscalar-type density squared (in-phase with $\alpha_n$ and $\alpha_p$ weight factors), while that of the isoscalar current term is from an isovector-type (out of phase) density squared.
\begin{table*}[h]
\caption{
Energy-weighted sum rule of the isoscalar monopole (ISM), dipole (ISD), quadrupole (ISQ), and octupole (ISO) operators computed from Eq.~(\ref{eq:EWSR-IS}) in the HFB states and the complex-energy FAM for $^{208}$Pb. UNEDF1-HFB functional is employed.
Four choices for the isoscalar and isovector current coupling constants are listed.
The units are in MeV fm$^x$ where $x=4,2,4$, and $6$ for $L=0,1,2$, and $3$ modes, respectively.
\label{table:208PbIS}
}
\begin{ruledtabular}
\begin{tabular}{llcccccccccc}
&& \multicolumn{2}{c}{ISM($K=0$)} & \multicolumn{2}{c}{ISD($K=0$)} & \multicolumn{2}{c}{ISD($K=1$)} & \multicolumn{2}{c}{ISQ($K=0$)} & \multicolumn{2}{c}{ISQ($K=1$)} \\
&& HFB & FAM & HFB & FAM & HFB & FAM & HFB & FAM &HFB & FAM \\ \hline
$C_0^j=-C^\tau_0$ & $C_1^j=-C^\tau_1$ & 82947.5 & 83179.7 & 290910 & 293033 & 290910 & 293034 & 16501.9 & 16524.7 & 16501.9 & 16525.3 \\
$C_0^j=0$ & $C_1^j=-C^\tau_1$ & 86523.7 & 86762.4 & 301255 & 303436 & 301255 & 303436 & 17213.3 & 17233.7 & 17213.3 & 17233.2 \\
$C_0^j=-C^\tau_0$ & $C_1^j=0$ & 82578.8 & 82814.7 & 290909 & 291800 & 290909 & 291800 & 16428.5 & 16451.8 & 16428.5 & 16452.5 \\
$C_0^j=0$ & $C_1^j=0$ & 86155.1 & 86397.2 & 301254 & 302201 & 301254 & 302201 & 17140.0 & 17160.7 & 17140.0 & 17160.7 \\ \hline\hline
&& \multicolumn{2}{c}{ISQ($K=2$)} & \multicolumn{2}{c}{ISO($K=0$)} & \multicolumn{2}{c}{ISO($K=1$)} & \multicolumn{2}{c}{ISO($K=2$)} & \multicolumn{2}{c}{ISO($K=3$)} \\
&& HFB & FAM & HFB & FAM &HFB & FAM &HFB & FAM &HFB & FAM \\ \hline
$C_0^j=-C^\tau_0$&$C_1^j=-C^\tau_1$ & 16501.9 & 16524.8 & 1367530 & 1370530 & 1367530 & 1370530 & 1367530 & 1370530 & 1367530 & 1370530 \\
$C_0^j=0$&$C_1^j=-C^\tau_1$ & 17213.4 & 17233.6 & 1415080 & 1418200 & 1415080 & 1418200 & 1415080 & 1418200 & 1415080 & 1418200 \\
$C_0^j=-C^\tau_0$&$C_1^j=0$ & 16428.5 & 16452.1 & 1362040 & 1365110 & 1362040 & 1365120 & 1362040 & 1365110 & 1362040 & 1365110 \\
$C_0^j=0$&$C_1^j=0$ & 17140.0 & 17160.7 & 1409590 & 1412780 & 1409590 & 1412780 & 1409590 & 1412780 & 1409590 & 1412780
\end{tabular}
\end{ruledtabular}
\end{table*}
\begin{table*}[h]
\caption{Energy-weighted sum rule of the isovector multipole operators for $^{208}$Pb. The HFB value are evaluated using Eq.~(\ref{eq:EWSR-IV}).\label{table:208PbIV}}
\begin{ruledtabular}
\begin{tabular}{llcccccccccc}
&& \multicolumn{2}{c}{IVM($K=0$)} & \multicolumn{2}{c}{IVD($K=0$)} & \multicolumn{2}{c}{IVD($K=1$)} & \multicolumn{2}{c}{IVQ($K=0$)} & \multicolumn{2}{c}{IVQ($K=1$)} \\
&& HFB & FAM & HFB & FAM &HFB & FAM &HFB & FAM &HFB & FAM \\ \hline
$C_0^j=-C^\tau_0$&$C_1^j=-C^\tau_1$ & 146148 & 146576 & 293.309 & 292.541 & 293.309 & 292.530 & 29075.2 & 29112.3 & 29075.2 & 29111.4 \\
$C_0^j=0$&$C_1^j=-C^\tau_1$ & 146161 & 146592 & 293.328 & 292.560 & 293.328 & 292.547 & 29077.8 & 29115.1 & 29077.8 & 29114.1 \\
$C_0^j=-C^\tau_0$&$C_1^j=0$ & 131977 & 132456 & 260.286 & 260.392 & 260.286 & 260.394 & 26255.9 & 26296.8 & 26255.9 & 26296.8 \\
$C_0^j=0$&$C_1^j=0$ & 131989 & 132471 & 260.305 & 260.412 & 260.305 & 260.413 & 26258.4 & 26299.5 & 26258.4 & 26299.5\\ \hline\hline
&& \multicolumn{2}{c}{IVQ($K=2$)} & \multicolumn{2}{c}{IVO($K=0$)} & \multicolumn{2}{c}{IVO($K=1$)} & \multicolumn{2}{c}{IVO($K=2$)} & \multicolumn{2}{c}{IVO($K=3$)} \\
&& HFB & FAM & HFB & FAM &HFB & FAM &HFB & FAM &HFB & FAM \\ \hline
$C_0^j=-C^\tau_0$&$C_1^j=-C^\tau_1$ & 29075.2 & 29112.4 & 2304070 & 2310750 & 2304070 & 2310740 & 2304070 & 2310740 & 2304070 & 2310750 \\
$C_0^j=0$&$C_1^j=-C^\tau_1$ & 29077.8 & 29115.1 & 2304310 & 2311040 & 2304310 & 2311030 & 2304310 & 2311030 & 2304310 & 2311040 \\
$C_0^j=-C^\tau_0$&$C_1^j=0$ & 26255.9 & 26296.8 & 2116360 & 2122560 & 2116360 & 2122569 & 2116360 & 2122560 & 2116360 & 2122560 \\
$C_0^j=0$&$C_1^j=0$ & 26258.5 & 26299.5 & 2116600 & 2122850 & 2116600 & 2122850 & 2116610 & 2122850 & 2116600 & 2122850
\end{tabular}
\end{ruledtabular}
\end{table*}
\begin{table*}[h]
\caption{Energy-weighted sum rule of the isoscalar multipole operators for $^{166}$Dy.\label{table:166DyIS}}
\begin{ruledtabular}
\begin{tabular}{llcccccccccc}
&& \multicolumn{2}{c}{ISM($K=0$)} & \multicolumn{2}{c}{ISD($K=0$)} & \multicolumn{2}{c}{ISD($K=1$)} & \multicolumn{2}{c}{ISQ($K=0$)} & \multicolumn{2}{c}{ISQ($K=1$)} \\
&& HFB & FAM & HFB & FAM &HFB & FAM &HFB & FAM &HFB & FAM \\ \hline
$C_0^j=-C^\tau_0$&$C_1^j=-C^\tau_1$ & 60925.0 & 61189.1 & 307815 & 309873 & 152435 & 154273 & 14654.4 & 14683.9 & 13387.5 & 13413.9 \\
$C_0^j=0$&$C_1^j=-C^\tau_1$ & 63518.1 & 63785.8 & 319099 & 321158 & 157586 & 159336 & 15292.6 & 15322.3 & 13964.5 & 13991.0 \\
$C_0^j=-C^\tau_0$&$C_1^j=0$ & 60681.4 & 60944.4 & 307814 & 308693 & 152435 & 153668 & 14597.0 & 14626.5 & 13334.6 & 13361.0 \\
$C_0^j=0$&$C_1^j=0$ & 63274.5 & 63541.1 & 319098 & 319960 & 157565 & 158738 & 15235.1 & 15264.8 & 13911.6 & 13938.0 \\ \hline\hline
&& \multicolumn{2}{c}{ISQ($K=2$)} & \multicolumn{2}{c}{ISO($K=0$)} & \multicolumn{2}{c}{ISO($K=1$)} & \multicolumn{2}{c}{ISO($K=2$)} & \multicolumn{2}{c}{ISO($K=3$)} \\
&& HFB & FAM & HFB & FAM &HFB & FAM &HFB & FAM &HFB & FAM \\ \hline
$C_0^j=-C^\tau_0$&$C_1^j=-C^\tau_1$ & 9586.88 & 9609.07 & 1186720 & 1191410 & 1108940 & 1112760 & 890831 & 894645 & 578137 & 581155 \\
$C_0^j=0$&$C_1^j=-C^\tau_1$ & 9980.50 & 10002.8 & 1228800 & 1233500 & 1147780 & 1151630 & 920838 & 924757 & 596444 & 599449 \\
$C_0^j=-C^\tau_0$&$C_1^j=0$ & 9547.36 & 9569.49 & 1182180 & 1187100 & 1104770 & 1108830 & 887621 & 891577 & 575952 & 579003 \\
$C_0^j=0$&$C_1^j=0$ & 9940.97 & 9963.16 & 1224260 & 1229140 & 1143610 & 1147730 & 917628 & 921599 & 594258 & 597323
\end{tabular}
\end{ruledtabular}
\end{table*}
\begin{table*}[h]
\caption{Energy-weighted sum rule of the isovector monopole (IVM), dipole (IVD), quadrupole (IVQ), and octupole (IVO) operators for $^{166}$Dy. \label{table:166DyIV}}
\begin{ruledtabular}
\begin{tabular}{llcccccccccc}
&& \multicolumn{2}{c}{IVM($K=0$)} & \multicolumn{2}{c}{IVD($K=0$)} & \multicolumn{2}{c}{IVD($K=1$)} & \multicolumn{2}{c}{IVQ($K=0$)} & \multicolumn{2}{c}{IVQ($K=1$)} \\
&& HFB & FAM & HFB & FAM &HFB & FAM &HFB & FAM &HFB & FAM \\ \hline
$C_0^j=-C^\tau_0$&$C_1^j=-C^\tau_1$ & 105723 & 106141 & 234.288 & 234.797 & 234.288 & 234.599 & 25581.9 & 25628.4 & 23307.5 & 23349.6 \\
$C_0^j=0$&$C_1^j=-C^\tau_1$ & 105734 & 106154 & 234.306 & 234.815 & 234.306 & 234.618 & 25584.5 & 25631.2 & 23309.8 & 23352.0 \\
$C_0^j=-C^\tau_0$&$C_1^j=0$ & 95552.1 & 95979.6 & 208.108 & 208.912 & 208.108 & 208.758 & 23075.0 & 23125.3 & 21042.2 & 21085.9 \\
$C_0^j=0$&$C_1^j=0$ & 95562.4 & 95991.8 & 208.126 & 208.930 & 208.126 & 208.777 & 23077.5 & 23128.0 & 21044.5 & 21088.4 \\ \hline\hline
&& \multicolumn{2}{c}{IVQ($K=2$)} & \multicolumn{2}{c}{IVO($K=0$)} & \multicolumn{2}{c}{IVO($K=1$)} & \multicolumn{2}{c}{IVO($K=2$)} & \multicolumn{2}{c}{IVO($K=3$)} \\
&& HFB & FAM & HFB & FAM &HFB & FAM &HFB & FAM &HFB & FAM \\ \hline
$C_0^j=-C^\tau_0$&$C_1^j=-C^\tau_1$ & 16484.0 & 16518.8 & 2152290 & 2161030 & 1961990 & 1967530 & 1474910 & 1480650 & 942464 & 946689 \\
$C_0^j=0$&$C_1^j=-C^\tau_1$ & 16485.6 & 16520.5 & 2152540 & 2161290 & 1962220 & 1967780 & 1475070 & 1480390 & 942578 & 946813 \\
$C_0^j=-C^\tau_0$&$C_1^j=0$ & 14944.0 & 14980.2 & 1970800 & 1979920 & 1798850 & 1806540 & 1357670 & 1363970 & 871254 & 875830 \\
$C_0^j=0$&$C_1^j=0$ & 14945.5 & 14982.0 & 1971050 & 1980180 & 1799080 & 1805700 & 1357840 & 1364140 & 871368 & 875954
\end{tabular}
\end{ruledtabular}
\end{table*}
We then analyze the effect of the local gauge symmetry breaking of the pairing EDF on the energy-weighted sum rule
using the EDF which was discussed in Ref.~\cite{0954-3899-45-2-024004}.
The particle-hole part of the EDF is UNEDF1-HFB with the local gauge invariant current terms ($C^j_0=-C^\tau_0$ and $C^j_1=-C^\tau_1$)
and for the neutron pairing channel, in addition to a volume pairing term $\tilde{C}^\rho_n=V_n/4=-148.25$ MeV fm$^3$,
We include $\tilde{C}^{\Delta\rho}_n{\rm Re} (\tilde{\rho}_n^\ast \Delta\tilde{\rho}_n)$ and $\tilde{C}^\tau_n{\rm Re} (\tilde{\tau}^\ast_n \tilde{\rho}_n)$ terms.
Three cases are considered:
\begin{enumerate}
\item local gauge invariant pairing functional with $\tilde{C}^{\Delta\rho}_n=-20$ MeV fm$^5$ and $\tilde{C}^\tau_n=80$ MeV fm$^5$.
\item $\tilde{C}^{\Delta\rho}_n=0$ and $\tilde{C}^\tau_n=80$ MeV fm$^5$.
\item $\tilde{C}^{\Delta\rho}_n=-20$ MeV fm$^5$ and $\tilde{C}^\tau_n=60$ MeV fm$^5$.
\end{enumerate}
Because of the low accuracy of the FAM strength function at high excitation energy with large pairing strength,
we set a smaller integration radius $R_{A1}=100$ MeV in this calculation.
In Table \ref{table:pairing}, the energy-weighted sum rule for $^{120}$Sn for $K=0$ modes are compared.
The agreement is generally good.
The maximum discrepancy is about 2.4\% seen in the sum rule of the isovector monopole operator,
indicating the high-energy contribution to the energy-weighted sum rule above 100 MeV.
Comparing with the large effect of the local gauge symmetry breaking in the current terms,
the effect of the local gauge symmetry breaking of the pairing EDF is just about 1\% of the energy-weighted sum rule.
This is because the pairing contribution to the energy-weighted sum rule in Eqs.~(\ref{eq:EWSR-IS}) and (\ref{eq:EWSR-IV})
is proportional to the pair density squared, while the same contribution from the local gauge symmetry breaking in the particle-hole EDF is proportional to the density squared. The pairing density is about one order smaller than the particle-hole density, as we see that the contribution of the pairing energy to the total binding energy is much smaller than that of the particle-hole part.
Thus considering the accuracy of the calculation, it is not easy to detect the local gauge symmetry breaking of the pairing EDF from the isoscalar and isovector coordinate operators.
We note that the property of the HFB state (such as particle-hole and pair densities) changes with the pairing functionals, while they remain the same when we turn on and off the time-odd current coupling constants. The difference between, for example, the energy-weighted sum rule values computed with $\tilde{C}^{\Delta\rho}_n=-\tilde{C}^\tau_n/4$ and $\tilde{C}^{\Delta\rho}_n=0$ is not only from the direct contribution from the local gauge symmetry breaking of the pairing EDF.
\begin{table*}[h]
\caption{Energy-weighted sum rule of $K=0$ isoscalar and isovector operators computed for $^{120}$Sn. The HFB values are evaluated using Eqs.~(\ref{eq:EWSR-IS}) and (\ref{eq:EWSR-IV}). UNEDF1-HFB functional is used in the particle-hole channel, and three pairing functionals are employed.
} \label{table:pairing}
\begin{ruledtabular}
\begin{tabular}{lcccccccc}
& \multicolumn{2}{c}{ISM} & \multicolumn{2}{c}{ISD} & \multicolumn{2}{c}{ISQ} & \multicolumn{2}{c}{ISO} \\
& HFB & FAM & HFB & FAM & HFB & FAM & HFB & FAM \\ \hline
$\tilde{C}^{\tau}_n=-4\tilde{C}^{\Delta\rho}_n$ & 37612.3 & 37525.5 & 97823.2 & 98732.1 & 7482.72 & 7494.26 & 445873 & 447563 \\
$\tilde{C}^{\Delta\rho}_n=0$ & 37247.0 & 37085.9 & 97401.9 & 97633.7 & 7410.05 & 7415.64 & 440527 & 442097 \\
$\tilde{C}^{\tau}_n=-3\tilde{C}^{\Delta\rho}_n$ & 37807.7 & 37700.9 & 97261.3 & 98883.0 & 7521.59 & 7536.32 & 447205 & 449376 \\
\hline\hline
& \multicolumn{2}{c}{IVM} & \multicolumn{2}{c}{IVD} & \multicolumn{2}{c}{IVQ} & \multicolumn{2}{c}{IVO} \\
& HFB & FAM & HFB & FAM & HFB & FAM & HFB & FAM \\ \hline
$\tilde{C}^{\tau}_n=-4\tilde{C}^{\Delta\rho}_n$ & 59979.3 & 58634.3 & 171.491 & 171.583 & 11932.5 & 11859.2 & 679415 & 675736 \\
$\tilde{C}^{\Delta\rho}_n=0$ & 59627.8 & 58220.1 & 170.723 & 170.711 & 11862.6 & 11785.1 & 674162 & 670461 \\
$\tilde{C}^{\tau}_n=-3\tilde{C}^{\Delta\rho}_n$ & 60208.2 & 58798.6 & 171.957 & 172.069 & 11978.0 & 11907.4 & 681180 & 677783
\end{tabular}
\end{ruledtabular}
\end{table*}
\section{Conclusions \label{sec:conclusion}}
The expressions for the energy-weighted sum rule of the isoscalar and isovector coordinate operators
are derived for the case of the nuclear DFT where the EDF does not correspond to a Hamiltonian.
The importance of the local gauge invariance of the nuclear EDF for evaluating the energy-weighted sum rule of these operators is discussed.
For time-reversal symmetric even-even systems, the local gauge invariance of $\rho_k\tau_k - \bm{j}_k^2$ term
in the particle-hole channel and ${\rm Re}(4\tilde{\rho}^\ast_t\Delta\tilde{\rho}_t - \tilde{\rho}^\ast_t\tilde{\tau}_t)$ in the pairing channel
is responsible to obtain the energy-weighted sum-rule value of the conventional Thouless theorem, while the local gauge invariance of the other terms such as spin-orbit and tensor does not play any role for the energy-weighted sum rule of the multipole operators.
The expressions for the energy-weighted sum-rule values are compared with the QRPA calculations with the complex-energy FAM, and expressions we derived are both analytically and numerically justified.
The ratio of energy-weighted and inverse-energy-weighted sum rule is useful for estimating the giant resonance energy. The present derivation establishes the efficient evaluation of the sum-rule ratio for nuclear EDF that does not correspond to a Hamiltonian, as the dielectric theorem is available for the nuclear EDF to evaluate the inverse-energy-weighted sum rule \cite{PhysRevC.79.054329}.
The local gauge invariance of $\rho_k\tau_k-\bm{j}_k^2$ is related to the Galilean invariance, and thus almost all the practical nuclear EDFs should hold it. However, the present derivation of the Thouless theorem is also applicable to
other kinds of operators such as spin and isospin.
The energy-weighted sum rule of the spin operators is related to the spin-orbit and tensor energy terms
\cite{PhysRev.130.1525,ZAMICK198187}. It will be very useful to derive the expression for the energy-weighted sum rule of the spin and spin-multipole operators for better understanding of the spin-orbit and tensor terms in nuclear EDFs.
Extensions to non-Hermitian operators such as charge-exchange and pair transfer excitation, and the derivation of the cubic energy-weighted sum rule within the nuclear DFT are another challenging future subjects.
\section*{Acknowledgments}
Discussions with Markus Kortelainen and Witold Nazarewicz are acknowledged.
This work is supported by the JSPS KAKENHI Grant Number 16K17680 and 17H05194.
Numerical calculations were performed at the COMA (PACS-IX)
and Oakforest-PACS Systems through the Multidisciplinary Cooperative Research Program in Center for Computational Sciences, University of Tsukuba.
|
{
"timestamp": "2019-03-01T02:15:15",
"yymm": "1902",
"arxiv_id": "1902.11005",
"language": "en",
"url": "https://arxiv.org/abs/1902.11005"
}
|
\section{Introduction}
Many dynamical phenomena, including core collapse supernovae, the formation and subsequent cooling of proto-neutron stars, and both the electromagnetic and gravitational signals from neutron star mergers, depend sensitively on the neutron star equation of state (EOS) at densities where the EOS is not well understood. In addition, for these dynamical phenomena, there are two further complications. First, temperatures may range from below the Fermi temperature, for which ``cold" EOS suffice, to temperatures of up to 10-100~MeV in neutron star mergers (e.g., \citealt{Oechslin2007}). Second, the composition may range from nearly pure neutron matter to symmetric matter, with some dynamical timescales shorter than the timescale required to establish $\beta$-equilibrium. While astrophysical observations of stationary neutron stars probe the cold EOS in $\beta$-equilibrium and laboratory experiments constrain the hot EOS of symmetric matter, extrapolations between the two regimes remain difficult. (For a schematic representation of these various regimes, see Fig.~\ref{fig:phase}. For recent reviews, see e.g., \citealt{Lattimer2016, Ozel2016}.) Such extrapolations to arbitrary proton fraction and temperature add further uncertainty to the EOS and complicate numerical simulations of these phenomena.
In the zero-temperature limit, a large number of EOS have been calculated, ranging from purely nucleonic models (e.g., \citealt{Baym1971, Friedman1981, Akmal1998, Douchin2001}) to models incorporating quark degrees of freedom using state-of-the-art results from perturbative QCD \citep[e.g.,][]{Fraga2014}. Laboratory experiments and neutron-star observations do not yet have sufficient power to distinguish between these models. Furthermore, it is likely that these EOS do not span the full range of possible physics. This possibility has motivated the creation of a large number of parametric EOS, as were first introduced by \citet{Read2009} and \citet{Ozel2009}. These parametric models do not require a priori knowledge of the high-density nuclear physics governing the EOS and, hence, can be used to probe unknown physics from neutron star observations.
A much smaller number of EOS that self-consistently incorporate finite-temperature effects have been calculated to date. Among the most well-known of these are the LS model, which is based on finite-temperature compressible liquid drop theory with a Skyrme nuclear force \citep{Lattimer1991}; as well as the EOS of \citet{Shen1998a}, which was calculated using relativistic mean field (RMF) theory with a Thomas-Fermi approximation. More recently, the statistical model developed in \citet{Hempel2010} has been applied to an additional $\sim$10 combinations of RMF models and nuclear mass tables.
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\textwidth]{schematic_T_Yp.pdf}
\caption{\label{fig:phase} Cross-section of a phase diagram, containing temperature as a function of neutron excess, where neutron excess is defined as the difference between neutron and proton densities, $n_n$ and $n_p$, compared to the total baryon density. The approximate regimes probed by various terrestrial and astrophysical phenomena are indicated. The dense-matter EOS is primarily constrained by observations of neutron stars and by laboratory data from nuclei and nuclear experiments. Many dynamical phenomena, such as neutron star mergers, supernovae, and the cooling of proto-neutron stars, lie in the intermediate regions of parameter space where the temperature is non-zero and the matter can be at a variable proton fraction.}
\end{figure}
Just as parametrizations of the cold EOS have proven useful in representing a broader range of physics, so too would a parametric finite-temperature EOS be useful for incorporating EOS effects into supernova and merger calculations. To this end, many authors have employed so-called ``hybrid EOS," in which a thermal component for an ideal fluid is added to an arbitrary cold EOS to account for heating \citep{Janka1993}. The ideal-fluid thermal component is parametrized in terms of a simple adiabatic index as $P_{\rm th} = \epsilon_{\rm th} (\Gamma_{\rm th}-1)$, where $P_{\rm th}$ and $\epsilon_{\rm th}$ are the thermal pressure and energy density and $\Gamma_{\rm th}$ is the adiabatic index, which is assumed to be constant. Such an approach is computationally simple, but neglects the effect of degeneracy on the thermal pressure. At high densities and finite temperatures, part of the available energy acts to lift degeneracy, rather than contributing additional thermal support. This causes a net reduction in the thermal pressure at high densities, compared to the prediction for an ideal fluid.
The density-dependence of these thermal effects depends directly on the density-dependence of the nucleon effective mass, as has been shown for many EOS \citep{Constantinou2014,Constantinou2015}. \citet{Constantinou2015} performed a Sommerfeld expansion to approximate the thermal properties at next-to-leading order and showed that the expansion terms require both the effective mass and its derivatives. Given a complete expression for the density-dependence of the effective mass, they showed that this formalism can be used to accurately approximate the thermal properties of a wide variety of EOS. \citet{Constantinou2017a} later expanded this work and showed that the formalism can be used to recreate even models beyond mean field theory, such as the two-loop exchange model of \citet{Zhang2016}.
The strong dependence of thermal properties on the effective mass can also be seen in the behavior $\Gamma_{\rm th}$. For example, \citet{Constantinou2015} compared two EOS with similar zero-temperature properties but with different single-particle potentials, and hence different density-dependences in their nucleon effective masses. They found substantially different thermal properties for the two EOS and that a constant $\Gamma_{\rm th}$ model failed to describe either EOS. \citet{Zhang2016} also found a strong density-dependence in $\Gamma_{\rm th}$ for their two-loop exchange model. These results indicate that $\Gamma_{\rm th}$ has a significant density-dependence for a diverse range of analytic models, which is not captured in the constant $\Gamma_{\rm th}$ approximation of the hybrid EOS.
Neglecting the effect of degeneracy on the thermal pressure has important consequences for dynamical simulations as well. For example, \citet{Bauswein2010} compared the properties of a neutron star-neutron star merger that would be predicted by a hybrid EOS and by more realistic EOS. Specifically, they compared the \citet{Shen1998a} and \citet{Lattimer1991} EOS to hybrid EOS that were constructed from the zero-temperature versions of these same EOS with either $\Gamma_{\rm th}=1.5$ or 2. They found that using the hybrid EOS predicts post-merger frequencies from a hypermassive neutron star that are 50-250~Hz smaller than what is found with a realistic finite-temperature EOS. Moreover, the lifetime of the hypermassive remnant can deviate by a factor of two from the more realistic value and the post-collapse accretion disk mass around the resulting black hole can differ by up to 30\% when the simplified thermal effects are used \citep{Bauswein2010}. These results all suggest that it is indeed important to account for the effect of degeneracy on the thermal pressure when simulating neutron star mergers.
The Sommerfeld expansion results of \citet{Constantinou2015} can be used to explicitly correct a hybrid EOS to include degenerate effects, as long as the particle interactions and potentials of the cold EOS are known. However, requiring knowledge of the potentials of the cold EOS renders these corrections inapplicable to piecewise-polytropic EOS or other parametric forms of the EOS that are agnostic in their descriptions of the microphysics.
The goal of this paper is to develop a physically-motivated framework for incorporating the thermal pressure that maintains the wide applicability of the hybrid EOS approach. With such a model, it will be possible to robustly add thermal effects to any cold EOS in $\beta$-equilibrium, without having to make the simplifying assumptions of an ideal fluid at all densities. The framework we present in this paper is specific to neutron-proton-electron ($n$-$p$-$e$) matter, but could be generalized to include more exotic particles. We also include a symmetry-energy dependent correction that extrapolates the proton fraction away from $\beta$-equilibrium. The complete model thus allows us to build an EOS at finite-temperature and arbitrary proton fraction from any cold $n$-$p$-$e$~EOS in neutrinoless $\beta$-equilibrium, including piecewise-polytropic EOS. Moreover, the model is analytic and in closed-form and thus can be calculated efficiently in dynamical simulations.
We start in $\S$\ref{sec:overview} with a brief review of existing finite-temperature EOS and a discussion of the regimes in which thermal effects become important. In $\S$\ref{sec:outline}, we outline our model. We provide the symmetry-energy dependent extrapolation to arbitrary proton fraction in $\S$\ref{sec:Esym}. In $\S$\ref{sec:thermal}, we introduce our $M^*$-approximation of the thermal effects. We summarize the model in $\S$\ref{sec:boxes}, in which all of the relevant equations can be found in Boxes I and II. Finally, we quantify the performance of our model in $\S$\ref{sec:complete}. We find that with a relatively small set of parameters, our complete model is able to recreate existing finite-temperature EOS with introduced errors of $\lesssim$20\%, for densities above the nuclear saturation density.
\section{Overview of finite-temperature EOS}
\label{sec:overview}
Before introducing our new approximation for the pressure at arbitrary proton fraction and temperature, we will first briefly review the finite-temperature EOS that have been previously developed.
Two of the most widely-used finite-temperature EOS are the models of \citet[][hereafter LS]{Lattimer1991}, which is based on a finite-temperature liquid drop model with a Skyrme nuclear force, and \citet[][hereafter STOS]{Shen1998a} which is an RMF model that is extended with the Thomas-Fermi approximation. An additional eight EOS have been calculated with the framework of \citet[][hereafter, HS]{Hempel2010}, which is a statistical model that consists of an ensemble of nuclei and interacting nucleons in nuclear statistical equilibrium and, hence, goes beyond the single nucleus approximation that both LS and STOS assume. Each HS EOS represents the nucleons with an RMF model and additionally includes excluded volume effects. Of the RMF models that have been used with the HS method, six are nucleonic: TMA \citep{Toki1995}, TM1 \citep{Sugahara1994}, NL3 \citep{Lalazissis1997}, FSUGold \citep{Todd-Rutel2005}, IUFSU \citep{Fattoyev2010}, DD2 \citep{Typel2010}; while the models BHB$\Lambda \phi$ and BHB$\Lambda$ include hyperons with and without the repulsive hyperon-hyperon interaction mediated by the $\phi$ meson, respectively \citep{Banik2014}. Additionally, \citet{Steiner2013} created a set of two finite-temperature EOS, SFHo/x, that also used the statistical method of HS, but with new RMF parameterizations and constraints from neutron star observations. There are also the EOS of G. Shen, which are based on a virial expansion and nuclear statistical equilibrium calculations at low densities and RMF calculations at high densities, using the models FSUGold \citep{Shen2011} and NL3 \citep{Shen2011a}. Tables of these various EOS can be found on the website of M. Hempel,\footnote{https://astro.physik.unibas.ch/people/matthias-hempel/equations-of-state.html} \texttt{stellarcollapse.org}, and/or the \texttt{CompOSE} database.\footnote{https://compose.obspm.fr/home/}
More recently, several new finite-temperature EOS have been added to the \texttt{CompOSE} database. These include the SLY4-RG model, which is calculated in nuclear statistical equilibrium using a Skyrme energy functional \citep{Gulminelli2015,Raduta2018}, as well as chiral mean field theory models, which include hyperons as additional degrees of freedom \citep[e.g.,][]{Dexheimer2017}, generalized relativistic density functional models \citep[e.g.,][]{Typel2018}, and models calculated using a variational method applied to two- and three-body nuclear potentials \citep[e.g.,][]{Togashi2017}.
For the sake of simplicity in the following analysis, we will focus on a subset of these EOS and will include only models that are nucleonic. In particular, our sample will include STOS as well as the eight nucleonic EOS calculated with the HS method, to represent the models based on RMF theory. We will also include LS (with a compression modulus $K=220$~MeV) and SLY4-RG, to represent non-relativistic models with Skyrme nuclear forces.
In spite of the increasing number of finite-temperature EOS that have been calculated, they nevertheless span a relatively limited range of physics, especially when compared to the diversity of cold EOS models. In order to span a broader range of possible physics, many authors have used the so-called ``hybrid EOS," which assume that the thermal pressure is given simply by an ideal-fluid term that can be added to any cold EOS. The hybrid EOS were first introduced by \citet{Janka1993} and have been used in many subsequent works \citep[for recent reviews, see][]{Shibata2011, Faber2012,Baiotti2017, Paschalidis2017}. In these hybrid EOS, the thermal pressure is written as
\begin{equation}
\label{eq:Pthhyb}
P_{\rm th, hybrid}(n,T) = n E_{\rm th, hybrid}(n,T) ( \Gamma_{\rm th} - 1),
\end{equation}
where $E_{\rm th,hybrid}(n,T)$ is the thermal contribution to the energy per baryon, $n$ is the baryon number density, and $\Gamma_{\rm th}$ is the thermal adiabatic index and is constrained to be $1 \le \Gamma_{\rm th} \le 2$. In the hybrid approximation, $\Gamma_{\rm th}$ is assumed to be constant.
Following \citet{Etienne2008}, the hybrid temperature-dependence of $E_{\rm th,hybrid}$ is included as an ideal fluid plus a contribution from relativistic particles, i.e.,
\begin{equation}
\label{eq:Ethhyb}
E_{\rm th,hybrid}(n,T) = \frac{3}{2} k_B T + \frac{4 \sigma}{c}\frac{f_s}{ n} T^4,
\end{equation}
where $k_B$ is the Boltzmann constant, $T$ is the temperature, and $\sigma \equiv \pi^2 k_B^4 / [60 \hbar^3 c^2]$ is the Stefan-Boltzmann constant, with $\hbar$ the Planck constant and $c$ the speed of light. The parameter $f_S$ represents the number of ultra-relativistic species that contribute to the thermal pressure. For $k_BT\ll 2 m_e c^2$, where $m_e$ is the mass of an electron, photons will dominate and $f_S$=1. For $k_BT\gg 2 m_e c^2$, electrons and positrons become relativistic as well and yield $f_S = 1 + 2\times (7/8) = 11/4$. Finally, for $k_BT \gtrsim 10$~MeV, thermal neutrinos and anti-neutrinos appear, rendering $ f_s = 11/4 + 3 \times (7/8) = 43/8$. If right-handed neutrinos were to exist, this would become $f_s = 11/4+ 3\times2\times(7/8) = 8$.
We note that all 12 EOS discussed above neglect neutrinos in their calculations. The STOS EOS additionally neglects leptons and photons, which we add in wherever we use STOS in this paper. For the STOS thermal lepton and photon contribution, we use eq.~(\ref{eq:Ethhyb}) with the appropriate lepton density. For the cold lepton energy, we add the contribution for a degenerate gas of relativistic electrons. Because all the EOS neglect neutrinos, we will also neglect neutrinos in our comparisons and thus we will calculate $f_S$ only as
\begin{equation}
\label{eq:fs}
f_S =
\begin{cases}
1, & k_B T < 1~\text{MeV} , \\
11/4, & k_B T \ge~1 \text{MeV}.
\end{cases}
\end{equation}
We, therefore, account for the degrees of freedom introduced by the possible presence of ultra-relativistic positrons. However, throughout this paper, we will assume that the population of positrons is small and that their contribution to the pressure or energy at higher densities is negligible. If there were a scenario in which the population of positrons were significant compared to the electrons, one would have to explicitly account for the positrons in particle-counting as well as in imposing charge neutrality.
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\textwidth]{Tvsn_STOS_Yp01.pdf}
\caption{\label{fig:Tn} Phase diagram for regimes of interest in neutron star simulations. The blue shaded region represents the regime where the total pressure is dominated by the cold pressure, to within $1\%$, for the STOS EOS with proton fraction $Y_p=0.1$. The red shaded region represents the $T-n$ range where the thermal pressure is dominated by the ideal-fluid pressure ($P_{\rm th}=nk_B T$), to within $1\%$, for the same EOS and fixed $Y_p$. The white range in between these two extremes represents the phase space in which degenerate thermal effects are important. For comparison, the green line shows the profile of a hypermassive neutron star (HMNS) remnant 12.1~ms after a neutron star merger from the simulations of \citet{Sekiguchi2011} using the STOS EOS. The orange and purple lines show the profiles of a proto-neutron star (PNS) 200~ms after the bounce in a core-collapse supernova simulation and at the end of de-leptonization in the same simulation, both with a bulk version of the LS EOS \citep{Camelio2017}. }
\end{figure}
In order to highlight the regimes where a realistic finite-temperature EOS and the hybrid approximation differ, we show a phase diagram in Fig.~\ref{fig:Tn}. In this plot, we show various regions calculated for the EOS STOS, all at a fixed proton fraction of $Y_p=0.1$. The total pressure, $P_{\rm total}$, is thus calculated at $Y_p=0.1$ and a given temperature. The cold contribution, $P_{\rm cold}$, is calculated at the same $Y_p$ and at zero-temperature.\footnote{We note that, throughout this paper, we use the coldest HS calculation, performed at $k_BT=0.1$~MeV, as an approximation of the zero-temperature EOS. Even though the STOS EOS is calculated at $T=0$~MeV, we use the $k_BT=0.1$~MeV table as our cold component for this EOS as well, in order to maintain consistency with the HS set of EOS.} Finally, the thermal contribution, $P_{\rm th}$, is defined as $P_{\rm total} -P_{\rm cold}$ for the same proton fraction.
In this figure, the blue shaded region shows the regime where the total pressure is dominated by the cold pressure; there, the thermal pressure of STOS contributes $<1\%$ of the total pressure. The red shaded region represents the regime where the thermal pressure can be approximated by the ideal fluid pressure ($P_{\rm th, ideal}=nK_BT$), to within 1\%. The white region between these two extremes represents the range of parameter space in which the thermal pressure is important but the ideal-fluid approximation does not yet apply. In this white region, the effects of degeneracy on the thermal pressure cannot be neglected.
For comparison, we also show in Fig.~\ref{fig:Tn} the projected temperature-density profiles from three different simulations of relevant astrophysical phenomena. The green line shows the profile of a hypermassive neutron star remnant 12.1~ms after the merger of two 1.35~$M_{\odot}$ neutron stars, as simulated using the EOS STOS \citep{Sekiguchi2011}. The orange and purple lines both come from numerical simulations of the evolution of a proto-neutron star using a bulk-version of the LS EOS. The orange line gives the profile of the proto-neutron star at 200~ms after the core bounce, while the purple line shows the profile of the proto-neutron star at the end of the de-leptonization phase \citep{Camelio2017}. We note that these profiles are not necessarily calculated at $Y_p=0.1$, but we include them nevertheless to show the approximate relevant temperatures and densities for such phenomena.
In order to further explore the dependence on the proton fraction, we also calculated the regime where degeneracy dominates for increasing values of $Y_p$. We find that as the proton fraction increases towards $Y_p=0.5$, the white degeneracy region in Fig.~\ref{fig:Tn} shrinks, but still largely encompasses the shown profiles. We thus find that all of these simulations primarily probe the phase space where degenerate thermal effects are important. This suggests that using the hybrid approximation, instead of the full thermal pressure, may bias the outcomes from such simulations.
\section{Generic model of a finite temperature EOS}
\label{sec:outline}
In order to construct a finite-temperature EOS at arbitrary proton fraction, our model must be able to extrapolate from $\beta$-equilibrium to an arbitrary $Y_p$, as well as from cold matter to an arbitrary temperature. This will naturally introduce dials into our model that can be adjusted to represent a wide range of physics, based on the symmetry energy, its slope, and the strength of particle interactions we wish to include. Moreover, we will show that with a small set of parameters, the EOS that are currently in use in the literature can be replicated to high accuracy.
We start with our model in general terms, for which we will derive analytic expressions in the following sections. Our final model will be for the complete energy per baryon, $E(n,Y_p,T)$, separated into analytic, physically-motivated terms. A summary of the final equations can be found in Boxes I and II in $\S$\ref{sec:boxes}.
We can expand the energy per particle of nuclear matter, $E_{\rm nucl}$, about the neutron excess parameter, $(1-2 Y_p)$, to second order as
\begin{equation}
\label{eq:parabolic}
E_{\rm nucl}(n, Y_p, T) = E_{\rm nucl}(n, Y_p=\sfrac{1}{2}, T) + E_{\rm sym}(n, T)(1-2 Y_p)^2,
\end{equation}
where $E_{\rm nucl}(n, Y_p = \sfrac{1}{2}, T)$ represents the energy of symmetric nuclear matter and
\begin{equation}
E_{\rm sym}(n,T) \equiv \frac{1}{2} \frac{\partial^2 E_{\rm nucl}(n, Y_p, T)}{\partial (1-2 Y_p)^2} \biggr|_{Y_p=1/2}
\end{equation}
is the symmetry energy. The proton fraction is related to the overall baryon density, $n$, according to
\begin{equation}
\label{eq:Yp}
Y_p = \frac{n_p}{n} = \frac{N_p}{N_n + N_p},
\end{equation}
where $n_p$ is the proton density, $N_p$ is the total number of protons, and $N_n$ is the total number of neutrons. Throughout this paper, we enforce charge-neutrality, which requires that the proton and electron densities balance. Thus, the electron density, $n_e$, can be written as
\begin{equation}
\label{eq:ne}
n_e = Y_p n.
\end{equation}
Finally, by requiring that the baryonic components combine to give the total density $n$, we can write the neutron density as
\begin{equation}
n_n = (1-Y_p) n.
\end{equation}
We can further expand eq.~(\ref{eq:parabolic}) by separating the energy of cold, symmetric matter from its thermal contribution, i.e.,
\begin{align}
\begin{split}
\label{eq:intermidateE}
E_{\rm nucl}(n, Y_p, T) = & E_{\rm nucl}(n, Y_p=\sfrac{1}{2}, T=0) \\
+ & E_{\rm nucl,th}(n, Y_p=\sfrac{1}{2},T) \\
+ & E_{\rm sym}(n, T)(1-2 Y_p)^2.
\end{split}
\end{align}
Here and throughout the paper, we use the subscript ``\textit{th}" to indicate the thermal contribution to a variable, after the cold component has been subtracted.
In order to write the energy with respect to a cold EOS in $\beta$-equilibrium, as is often most relevant to start from in the study of neutron stars, we eliminate the cold, symmetric term in eq.~(\ref{eq:intermidateE}) to yield
\begin{align}
\begin{split}
E_{\rm nucl}(n, Y_p, T) = & E_{\rm nucl}(n, Y_{p,\beta}, T=0) \\
+ & E_{\rm nucl,th}(n, Y_p=\sfrac{1}{2},T) \\
+ & E_{\rm sym}(n, T)(1-2 Y_p)^2 \\
- & E_{\rm sym}(n, T=0)(1-2 Y_{p,\beta})^2,
\end{split}
\end{align}
where $Y_{p,\beta}$ represents the proton fraction of a zero-temperature system in $\beta$-equilibrium. We note that the proton fraction depends on the density, i.e., $Y_{p,\beta} =Y_{p,\beta}(n)$, but for simplicity we suppress this in our notation.
Finally, we must add the contribution of leptons and photons to this expression. The zero-temperature energy from relativistic degenerate electrons is given by
\begin{equation}
E_{\rm lepton}(n, Y_p, T=0) = 3 K Y_p (Y_p n)^{1/3},
\end{equation}
where the extra factor of $Y_p$ comes from our definition of $E$ as the energy per baryon, combined with eqs.~(\ref{eq:Yp}) and (\ref{eq:ne}). Here, $K\equiv (3 \pi^2)^{1/3} ( \hbar c/4)$. Additionally, there will also be a thermal contribution, $E_{\rm lepton,th}(n,Y_p,T)$, which we derive in $\S$\ref{sec:thermal}.
Thus, our skeletal model for the total energy is given by the following set of equations:
\begin{subequations}
\begin{align}
\begin{split}
\label{eq:fullE}
&E(n, Y_p, T) = E(n, Y_p, T=0) + E_{\rm th}(n, Y_p, T)
\end{split} \\
\begin{split}
\label{eq:Ecold}
\\& E(n, Y_p, T=0) = E(n, Y_{p,\beta}, T=0) \\
& \quad + E_{\rm sym}(n, T=0)\left[(1-2 Y_p)^2 - (1-2 Y_{p,\beta})^2\right] \\
& \quad + 3 K \left( Y_p^{4/3} - Y_{p,\beta}^{4/3} \right) n^{1/3}
\end{split} \\
\begin{split}
\label{eq:Eth}
& E_{\rm th}(n, Y_p, T) = E_{\rm nucl,th}(n, Y_p=\sfrac{1}{2},T) \\
& \qquad + E_{\rm lepton,~ th}(n, Y_p, T) \\
& \qquad + E_{\rm sym,th}(n, T)(1-2 Y_p)^2 .
\end{split}
\end{align}
\end{subequations}
From these relations, we can derive the pressure via the standard thermodynamic relation,
\begin{equation}
\label{eq:getP}
P \equiv -\frac{\partial U}{\partial V}\biggr \rvert_{N_q,S} = n^2 \left[ \frac{\partial E(n,T=0)}{\partial n}\right] \biggr \rvert_{Y_p, S}
\end{equation}
where $U$ is the total energy, $V$ is the volume, $N_q$ is the number of each species $q$, and $S$ is the total entropy. From eq.~(\ref{eq:Yp}), it is clear that evaluating these derivatives at constant $N_q$ is equivalent to evaluating them at constant $Y_p$. In this paper, we will mainly plot results in terms of pressure. We summarize the complete expressions for pressure in Box II of \S\ref{sec:boxes}.
While this set of expressions may seem to have a large number of terms, this separation allows these terms to be represented analytically. Moreover, as we will show, the parameters of each term are linked directly to physics on which there are experimental constraints and of which further constraints are the motivation of many observations of astrophysical neutron stars: namely, the value of the symmetry energy at the saturation density, the slope of the symmetry energy, and the strength of interactions between particles.
\section{Derivation of the cold symmetry energy in the Fermi Gas limit}
\label{sec:Esym}
We turn first to the symmetry energy correction term, $E_{\rm sym}(n, T)$ of eq.~(\ref{eq:parabolic}). The symmetry energy is defined as the per-nucleon difference in energy between symmetric matter and pure neutron matter. In other words, the symmetry energy represents the excess energy of matter with unequal numbers of protons and neutrons. In nuclear models, the symmetry energy is typically calculated as an expansion around the nuclear saturation density, for matter with $Y_p=1/2$. In eq.~(\ref{eq:parabolic}), we perform the expansion with respect to the proton fraction and, in the following section, will introduce a density-dependence to extrapolate beyond the saturation density, where the coefficients of our approximation are experimentally constrained. In this section, we will provide the approximation for $E_{\rm sym}(n,T)$ at zero-temperature. For the thermal contribution to the symmetry energy, which turns out to be negligible, see $\S$\ref{sec:thermal}.
It is particularly useful to parameterize the symmetry energy in terms of its separate kinetic and potential components at zero-temperature \citep[e.g.,][]{Tsang2009, Steiner2010}, modified by a parameter $\eta$ to account for short-range correlations due to the tensor force acting between a spin-triplet or isospin-singlet proton-neutron pair. These correlations can significantly reduce the kinetic symmetry energy to even a negative value at the saturation density, compared to the kinetic energy of an uncorrelated Fermi gas model \citep{Xu2011, Vidana2011, Lovato2011, Carbone2012, Rios2014,Hen2015}. In this framework, we parameterize the symmetry energy of eq.~(\ref{eq:Ecold}) as
\begin{equation}
\label{eq:Esym}
E_{\rm sym}(n, T=0)= \eta E_{\rm sym}^{\rm kin}(n)
+ \left[ S_0 - \eta E_{\rm sym}^{\rm kin}(n_{\rm sat}) \right] \left(\frac{n}{n_{\rm sat}}\right)^{\gamma},
\end{equation}
as in \citet{Li2015}. Here, $E_{\rm sym}^{\rm kin}(n)$ is the ``kinetic" symmetry energy, arising from the change in the Fermi energy of a gas at density $n$ as the relative proton/neutron fraction changes, $n_{\rm sat}=0.16$~fm$^{-3}$ is the nuclear saturation density,\footnote{We note that $n_{\rm sat}$ does vary slightly among the EOS in our sample, but we fix the value to $n_{\rm sat}=0.16$~fm$^{-3}$ in order to more easily compare the various EOS. We find that this does not significantly affect the results.} and the second term represents the ``potential" symmetry energy which accounts for the interactions between particles. Because the exact form of the potential symmetry energy is not well known, it is anchored at the saturation density by the magnitude of the overall symmetry energy, $S_0 \equiv E_{\rm sym}(n_{\rm sat})$, and is given an arbitrary density-dependence through the constant $\gamma$.
In contrast, the kinetic energy term can be calculated directly from the nuclear momentum distribution. The kinetic energy of a free Fermi gas is given simply by
\begin{equation}
\label{eq:fermiKE}
\frac{\varepsilon_{k,q}}{n} = \frac{3}{5} E_f(n_q)
\end{equation}
where $\varepsilon_{k,q}$ is the kinetic energy per particle, $q$ represents the particle (either a neutron or proton), and $E_f(n)$ is the Fermi energy,
\begin{equation}
E_f(n_q) = \frac{\hbar^2}{2 m} \left( 3\pi^2 n_q \right)^{2/3},
\end{equation}
in which $m$ is the mass of the relevant particle. For our approximation, we will neglect the small difference between the proton and neutron mass and simply take $m\approx m_n$, where $m_n$ is the neutron mass.
By taking the difference between symmetric matter and pure neutron matter, the kinetic symmetry energy as a function of the total density is then
\begin{align}
\begin{split}
\label{eq:Ekinsym}
E_{\rm sym}^{\rm kin}(n) &= \frac{3}{5} \left[ 2E_f\left(n_p = n_n =\frac{1}{2} n\right) - E_f(n_n = n) \right] \\
&= \frac{3}{5} \left(2^{1/3} -1 \right) E_f(n).
\end{split}
\end{align}
We can also eliminate the parameter $\eta$ in eq.~(\ref{eq:Esym}) by introducing the constant $L$, which is related to the overall slope at the saturation density via,
\begin{equation}
\label{eq:L}
L \equiv 3 n_{\rm sat} \left[ \frac{\partial E_{\rm sym}(n, T=0)}{\partial n} \right]\biggr\rvert_{n_{\rm sat}}.
\end{equation}
Combining eqs.~(\ref{eq:Esym}) and (\ref{eq:L}), we can solve for $\eta$ in terms of the quantities $S_0$ and $L$, which are constrained by nuclear physics experiments for matter near $Y_p=1/2$ \citep{Lattimer2013}. We find
\begin{equation}
\label{eq:eta}
\eta = \frac{5}{9} \left[ \frac{ L-3 S_0 \gamma}{\left(2^{1/3}-1\right)\left(2/3 - \gamma \right) E_f(n_{\rm sat})} \right] ,
\end{equation}
thereby leaving one free parameter, $\gamma$, which is constrained by nuclear experiments to lie in the range $\sim$~0.2 to 1.2 (see, e.g., Fig.\ 2 of \citealt{Li2015}; \citealt{Tsang2009}).
\begin{deluxetable}{llll}
\tabletypesize{\footnotesize}
\tablewidth{0.48\textwidth}
\tablecaption{\label{table:gamma} Symmetry energy parameters characterizing each EOS at $k_BT=0.1$~MeV. }
\tablehead{\\
\colhead{ EOS } &
\colhead{$S_0$ (MeV) } &
\colhead{$L$ (MeV) } &
\colhead{$\gamma$ }
}
\startdata
TM1 & 36.95 & 110.99 & 0.75 \\
TMA & 30.66 & 90.14 & 0.66 \\
NL3 & 37.39 & 118.49 & 0.62 \\
FSG & 32.56 & 60.43 & 1.11 \\
IUF & 31.29 & 47.20 & 0.52 \\
DD2 & 31.67 & 55.03 & 0.91 \\
STOS & 36.95 & 110.99 & 0.77 \\
SFHo & 31.57 & 47.10 & 0.41 \\
SFHx & 28.67 & 23.18 & -0.04\footnote{The inferred value for $\gamma$ for SFHx is highly sensitive to the density range that is included in the fit; see the discussion in the text for details.} \\
LS & 29.3 & 74.0 & 1.05 \\
SLY4-RG & 32.04 & 46.00 & 0.35
\enddata
\tablecomments{$S_0$ and $L$ are fixed to the values predicted for each EOS, while $\gamma$ is a fit parameter. All fits are performed for densities above $n\ge0.01$~fm$^{-3}$ and $n_{\rm sat}$=0.16~fm$^{-3}$}.
\end{deluxetable}
We thus have a complete expression for the symmetry energy that depends only on the three parameters $\gamma$, $S_0$, and $L$ which, in principle, can be constrained by nuclear experiments. We can now use this functional form to fit for $\gamma$, by combining it with the following relationship between the symmetry energy and $Y_{p,\beta}$ for charge-neutral $n$-$p$-$e$~matter in neutrinoless $\beta$-equilibrium,
\begin{equation}
\label{eq:YpBeta}
\frac{Y_{p,\beta}}{(1-2 Y_{p,\beta})^3} = \frac{64}{3 \pi^2 n} \left[ \frac{E_{\rm sym}(n, T=0)}{ \hbar c}\right]^3
\end{equation}
\citep[for a derivation of this relation, see, e.g.,][or Appendix A]{Blaschke2016}. When solved for $Y_{p,\beta}$, this becomes
\begin{equation}
\label{eq:YpBInv}
Y_{p,\beta} = \frac{1}{2} + \frac{(2 \pi^2)^{1/3}}{32} \frac{n}{\xi} \left\{ (2\pi^2)^{1/3} - \frac{\xi^2}{n} \left[\frac{\hbar c}{E_{\rm sym}(n,T=0)}\right]^3 \right\},
\end{equation}
where, for simplicity, we have introduced the auxilary quantity $\xi$, defined as
\begin{multline}
\label{eq:xi}
\xi \equiv \left[ \frac{E_{\rm sym}(n,T=0)}{\hbar c} \right]^2 \times \\
\left\{ 24 n \left[ 1+ \sqrt{ 1 + \frac{\pi^2 n}{288}\left(\frac{\hbar c}{E_{\rm sym}(n,T=0) }\right)^3}\right] \right\}^{1/3}.
\end{multline}
For each of the EOS in our sample, we stitch together a complete cold EOS at $\beta$-equilibrium from the publically-available tables at fixed $Y_p$, by requiring that $\mu_e + \mu_p - \mu_n = 0$, where $\mu_i$ is the chemical potential of each species. We then use the corresponding density-dependent proton fraction, $Y_{p,\beta}$, to fit for $\gamma$ using eqs.~(\ref{eq:Esym})-(\ref{eq:YpBeta}) and keeping $S_0$ and $L$ fixed for each EOS. We perform the fits using a standard least-squares method and limit the density range to $n \ge 10^{-2}$~fm$^{-3}$. In principle, eqs.~(\ref{eq:Esym})-(\ref{eq:YpBeta}) apply only to $n$-$p$-$e$~matter, which will be uniform only above $0.5 n_{\rm sat}$. However, in practice, we find a very small difference in the fits for $\gamma$ whether we include densities above $0.5n_{\rm sat}=0.08$~fm$^{-3}$ or whether we start the fits at a slightly lower but still astrophysically relevant cutoff of $n=10^{-2}$~fm$^{-3}$. We show the resulting fit values in Table~\ref{table:gamma}.
We note that the range of EOS provided in Table~\ref{table:gamma} is intentionally broad. While the symmetry energy parameters of some of these EOS disagree with the combined set of experimental constraints (see \citealt{Lattimer2013} for a recent review), or are in disagreement with certain theoretical considerations such as chiral effective field theory results for pure neutron matter \citep[see, e.g.,][]{Kruger2013}, they are all consistent with at least some experimental constraints on $S_0$ and $L$.
We find that $\gamma$ spans roughly the range of experimentally-allowed values, between 0.15 and 1.0, as expected, with the exception of SFHx. SFHx has an extremely low value of $L$, which makes the result of the fit highly sensitive to the density range that is included. For consistency, we still constrain the densities to $n \ge 10^{-2}$~fm$^{-3}$ for the fit to this EOS; however, the inferred value for $\gamma$ ranges from the reported value of $-0.04$ up to 0.18, depending on where the density cutoff is placed. Thus, the particular value for $\gamma$ for SFHx should be taken with some caution.
We have here used eq.~(\ref{eq:YpBeta}) to fit for $\gamma$ from the $\beta$-equilibrium proton fractions of realistic EOS. We wish to also emphasize that eq.~(\ref{eq:YpBeta}) can, of course, be used to calculate $Y_{p,\beta}$, given a choice of $S_0, L$, and $\gamma$. Once these three parameters are specified, eqs.~(\ref{eq:YpBInv})-(\ref{eq:xi}) can be used to calculate $Y_{p,\beta}$ for any EOS. As a result, all that is required of the cold EOS is knowledge of the run of pressure with density. This feature makes it possible to apply our model to piecewise polytropes or other families of parametric EOS that may not directly calculate $Y_{p,\beta}$.
\begin{figure}[ht]
\centering
\includegraphics[width=0.47\textwidth]{p_T0_Yp01_fromBeta_NL3_DD2.pdf}
\caption{\label{fig:coldYpcorr} Top: Pressure as a function of density for EOS NL3 and DD2, at $k_BT=0.1$~MeV and $Y_p=0.1$, as blue and orange diamonds, respectively. The solid lines show our model of the pressure, calculated using eqs.~(\ref{eq:Ecold}) and (\ref{eq:Esym}-\ref{eq:YpBeta}). Our model starts with the respective EOS in $\beta$-equilibrium and adds the appropriate symmetry energy and lepton corrections to extrapolate to $Y_p=0.1$. For $S_0, L$, and $\gamma$, we use the values listed in Table~\ref{table:gamma}. Bottom: Residuals between the true EOS at $Y_p=0.1$ and our model. We find that our model extrapolates from $\beta$-equilibrium to $Y_p=0.1$ reasonably well, especially at high densities where the model introduces an error of $\lesssim 1\%$ compared to using the full EOS.}
\end{figure}
We show an example of the performance of this model for $E_{\rm sym}(n, T=0)$ in Fig.~\ref{fig:coldYpcorr} for the EOS NL3 \citep{Lalazissis1997, Lalazissis1999} and DD2 \citep{Typel2010}. We show these two EOS as representative samples, with NL3 representing the family of EOS with larger $L$ values and DD2 representing the EOS with smaller symmetry energy slopes (see Table~\ref{table:gamma}). The top panel of Fig.~\ref{fig:coldYpcorr} shows the zero-temperature pressure predicted by NL3 and DD2 at $Y_p=0.1$ as blue and orange diamonds, respectively. The colored lines show our model: starting with the corresponding EOS in $\beta$-equilibrium, adding the symmetry energy correction of eqs.~(\ref{eq:Esym})-(\ref{eq:eta}), and correcting for the leptons, all according to eq.~(\ref{eq:Ecold}). For these models, we take the values of $S_0$, $L$, and $\gamma$ for each EOS from Table~\ref{table:gamma}. We note that we are plotting pressures, but could have similarly shown the energy. We use eq.~(\ref{eq:getP}) to convert the equations of this section to pressures; for the complete set of pressure expressions, see $\S$\ref{sec:boxes} and Box II.
The bottom panel of Fig.~\ref{fig:coldYpcorr} shows the residuals between our model and the pressure predicted by each EOS at $Y_p=0.1$. We find that our model performs very well at densities above $0.5~n_{\rm sat}$, with errors $\lesssim10\%$. At the highest densities, using our model compared to the full EOS introduces errors of only $\sim$1\%. The residuals for the other EOS in our sample are comparably small.
For $Y_p$=0.3, we find the residuals between our model and NL3 and DD2 are comparable to those shown in Fig.~\ref{fig:coldYpcorr}. We, therefore, conclude that this model reasonably captures the $Y_p$-dependence of the cold EOS, for a large range of $L$ values.
We thus have an expression for the symmetry energy at zero-temperature that depends only on $n, Y_p,$ $S_0$, $L$, and the narrowly-constrained parameter $\gamma$. There are two possible routes for creating a finite-temperature EOS with this framework. One possibility is to start from a cold, physically-motivated EOS, which will provide predicted values for $S_0$, $L$, and $Y_{p,\beta}$. In this case, eq.~(\ref{eq:YpBeta}) can be used to fit for $\gamma$. We have provided such fits for the EOS in our sample in Table~\ref{table:gamma}. Alternatively, a cold, parametric EOS can be chosen, for which the underlying physics are not specified. In this case, a user can freely specify $S_0$, $L$, and $\gamma$, which will uniquely specify $Y_{p,\beta}$. For the EOS in our sample, we find that this approach is able to accurately extrapolate from $\beta$-equilibrium to arbitrary proton fraction, introducing errors of $\lesssim10\%$ for densities of interest (above $0.5~n_{\rm sat}$), and errors of $\lesssim3\%$ at high densities.
\section{Thermal contribution to the energy}
\label{sec:thermal}
We now turn to the thermal energy, which was first defined in eq.~(\ref{eq:Eth}) as
\begin{align*}
\begin{split}
E_{\rm th}(n, Y_p, T) & = E_{\rm nucl, th}(n, Y_p=\sfrac{1}{2},T) \\
& + E_{\rm sym, th}(n,T) \times (1-2 Y_p)^2 \\
& + E_{\rm lepton, th}(n, Y_p, T) .
\end{split}
\end{align*}
It is useful to further divide the thermal energy into density regimes, over which the matter displays distinct behaviors. At the lowest densities, the contribution from relativistic leptons and photons dominates. At intermediate densities, an ideal-fluid description suffices. However, at high densities, matter can remain partially degenerate even at intermediate-to-high temperatures. In the high-density regime, some of the available energy goes into lifting the degeneracy of the particles rather than adding thermal support and, accordingly, the thermal pressure can dip well below the prediction for an ideal fluid. (See Fig.~\ref{fig:PthNL3} for the markedly different behaviors in thermal pressure across these three regimes.)
It is, therefore, convenient to write the thermal energy as
\begin{align}
\begin{split}
E_{\rm th}&(n, Y_p, T) = \\
& \begin{cases}
E_{\rm rel}(n,T), &\phantom{n_1<} n < n_1 \\
E_{\rm ideal}(T), &n_1 < n <n_2 \\
E_{\rm th, deg.}(n,Y_p=1/2,T) \\
\quad+ E_{\rm sym, th}(n, T) (1-2 Y_p)^2, &\phantom{n_1<} n>n_2
\end{cases}\\
\end{split}
\end{align}
where the relativistic component,
\begin{equation}
E_{\rm rel}(n,T) = \frac{4 \sigma}{c}\frac{f_s}{ n} T^4,
\end{equation}
and the ideal component,
\begin{equation}
E_{\rm ideal}(T) = \frac{3}{2} k_BT
\end{equation}
are given as in eqs.~(\ref{eq:Ethhyb}) and (\ref{eq:fs}). Here, $E_{\rm th, deg.}(n,Y_p=1/2,T)$ is the degenerate thermal energy of symmetric matter, which we introduce below. We note that, because the ideal-fluid and relativistic terms do not depend on the proton fraction, the symmetry-energy correction is only relevant in the degenerate regime. Finally, we define the first transition density, $n_1$, as the density at which the relativistic and ideal-fluid energies are equal. The second transition density, $n_2$, is the density at which the ideal-fluid energy is equal to the degenerate thermal energy, for a given temperature and proton fraction.
This piecewise expression of the thermal energy is convenient for later calculations of the thermal pressure and the sound speed. However, the discontinuities at the transition densities are artificial and will create problems in numerical simulations, potentially leading to undesired reflections of matter waves at density boundaries. Thus, whenever we actually implement the thermal energy or pressure, we use a smoothed version instead. This smoothed version is of the form
\begin{align}
\begin{split}
\label{eq:addinv}
E_{\rm th}&(n, Y_p, T) \approx E_{\rm rel}(n,T) \\
&+ \left[ E_{\rm ideal}(T)^{-1} + E_{\rm th, deg.}(n,Y_p,T)^{-1} \right]^{-1},
\end{split}
\end{align}
where we have added the latter two terms inversely to ensure that the ideal term dominates at intermediate densities and the degenerate term dominates at the highest densities. The smoothed approximation is also more computationally efficient than the piecewise version, as it does not require the calculation of transition densities, which will vary with the temperature and proton fraction.
In order to calculate the thermal energy in the degenerate regime, we consider the nucleons as a free Fermi gas. In that limit, the leading-order thermal energy of degenerate matter is given by
\begin{align}
\begin{split}
\label{eq:Ethq}
E_{\rm th,~q}^{\rm deg}(n, Y_q, T) &= a(Y_qn, M^*) \left(\frac{N_q}{N_p + N_n} \right)T^2,\\
& = a(Y_qn, M^*) Y_q T^2
\end{split}
\end{align}
for a single-species system of particle $q$.
For simplicity, we have introduced the level-density parameter $a$, which is defined as
\begin{equation}
\label{eq:a}
a(n_q, M^*) \equiv \frac{\pi^2 k_B^2}{2} \frac{ \sqrt{ \left( 3 \pi^2 n_q \right)^{2/3} (\hbar c)^2 + M^*(n_q)^2 } }{\left( 3 \pi^2 n_q \right)^{2/3} (\hbar c)^2 },
\end{equation}
where $M^*(n_q)$ is the Dirac effective mass of the relevant species at a specific density. \citep[For a complete derivation at next-to-leading order in temperature, see][]{Constantinou2015}.
As an example, the thermal nuclear energy for symmetric matter would be
\begin{align}
\begin{split}
E_{\rm th,~nucl}^{\rm deg}(n, T) &= \left[ \frac{a(n_p, M^*_{\rm p, SM})N_p + a( n_n, M^*_{\rm n, SM})N_n }{N_p + N_n}\right] T^2 \\
&= a(0.5 n, 0.5 M^*_{\rm SM}) T^2,
\end{split}
\end{align}
where the subscript SM stands for symmetric matter and, in the second line, we have used the fact that $n_n = n_p = 0.5n$ in symmetric matter. We have further made the approximation that the effective masses of neutrons and protons are comparable in symmetric matter and that the average of these two effective masses gives the overall effective mass of symmetric matter, i.e., $M^*_{\rm n, SM} \approx M^*_{\rm p, SM} \approx \sfrac{1}{2} M^*_{\rm SM}$.
By likewise defining the thermal energy per baryon for pure neutron matter (PNM), we can calculate the thermal contribution to the symmetry energy, as
\begin{align}
\label{eq:Esymth}
\begin{split}
&E_{\rm sym, th}(n, T) = \\
& \begin{cases}
0, & n < n_2 \\
\left[ a(n, M^*_{\rm PNM}) - a(0.5 n, 0.5 M^*_{\rm SM}) \right] T^2, & n > n_2,
\end{cases}
\end{split}
\end{align}
where the low-density limit of $E_{\rm sym, th}$ arises from the fact that both pure neutron matter and symmetric matter behave identically as ideal or relativistic fluids at $n < n_2$.
In principle, this symmetry energy term extrapolates the thermal energy of symmetric nuclear matter to arbitrary proton fraction. However, we find that including this term has a negligible effect on the results. In particular, making the approximation $E_{\rm th,~nucl}(n, Y_p, T)\approx E_{\rm th,~nucl}(n, Y_p=\sfrac{1}{2}, T)$ introduces an average error of $\lesssim 1\%$ in the total pressure across the density range of interest. We thus neglect the thermal correction to the symmetry energy for the remainder of the paper.
For leptons, the degenerate thermal pressure is even simpler. The effective mass of electrons is approximately constant, due to their small cross-sections of interaction. Hence, $M^*_e \approx m_e$. This allows us to write eq.~(\ref{eq:Ethq}) simply as
\begin{equation}
\label{eq:Ethelect}
E_{\mathrm{th,~}e^-}^{\rm deg}(n, Y_p, T) = a(Y_p n, m_e) Y_p T^2,
\end{equation}
where we have required that the electron fraction balance the proton fraction in order to satisfy the requirement of charge neutrality and we have used eq.~(\ref{eq:Yp}) to substitute $Y_p$. We note that in the presence of a significant population of positrons, the proton fraction in eq.~(\ref{eq:Ethelect}) should be replaced by the net lepton fraction.
With expressions for the degenerate and ideal fluid thermal terms in hand, we can now write a complete version of eq.~(\ref{eq:Eth}) for $E_{\rm th}$ as follows:
\begin{align}
\begin{split}
\label{eq:Ethfull}
& E_{\rm th}(n, Y_p, T) = \\
&
\begin{cases}
4 \sigma f_s T^4/ (c n), &\phantom{n_1<} n < n_1 \\
(3/2) k_B T, &n_1< n <n_2 \\
\left[ a(0.5 n, 0.5 M^*_{\rm SM}) + a(Y_p n, m_e) Y_p \right] T^2 , &\phantom{n_1<} n>n_2
\end{cases}
\end{split}
\end{align}
where we have neglected the thermal contribution to the symmetry energy, as discussed above.
We thus have a complete expression for the thermal energy of matter as a function only of the density, temperature, proton fraction, and the effective mass of the nucleons in symmetric matter.
\subsection{$M^*$-approximation}
\label{sec:Mstarapprox}
A full calculation of $E_{\rm th}$ using eq.~(\ref{eq:Ethfull}) requires knowledge of the Dirac effective masses in symmetric matter, and hence the scalar meson interactions and particle potentials of a particular EOS. We instead choose to express the Dirac effective mass with a physically-motivated yet computationally-simple approximation. At low densities, the effective mass must approach the dominant nucleon mass, while at higher densities, $M^*$ must decrease as particle interactions become important. We represent this behavior by introducing a power-law expression,
\begin{equation}
\label{eq:Meff}
M^*(n_q) = \left\{ (m c^2)^{-b} + \left[ mc^2 \left( \frac{ n_q }{n_0 }\right)^{-\alpha} \right]^{-b} \right\}^{-1/b},
\end{equation}
where $m$ is the nucleon mass (which we take to be the neutron mass, $mc^2=939.57$~MeV)\footnote{The EOS in our sample vary in their low-density limit of $M^*$ from 938$-$939.57~MeV. This parameter can easily be adjusted to any low-density value for $M^*$. For simplicity, however, we take it to simply be the neutron mass. We find that this simplification has a negligible effect on our results. } and $n_0$ is the transition density above which $M^*$ starts to decrease. The exponent $b$ determines the sharpness of the transition and $\alpha$ specifies the power-law slope at high densities.
We find that $b=2$ works well to represent the curvature connecting the low- and high-density regimes, and thus fix it to this value in the following analysis, leaving just two free parameters to describe the effective mass, $M^* = M^*(n_0,\alpha)$.
We fit the effective masses together at $k_BT=1, 10,$ and 47.9~MeV for nine of the EOS in our sample, using a standard least-squares method across the entire density range provided. We exclude the models LS and SLY4-RG here because the effective masses for these EOS are not currently published (but see $\S$\ref{sec:nonRMF} for a separate comparison with these models). The results of these fits are given in Table~\ref{table:Meff} for symmetric matter. For completeness, we also include in Table~\ref{table:Meff} the fits for pure neutron matter, which can be used to calculate $E_{\rm sym,th}(n,T)$ in eq.~(\ref{eq:Esymth}).
\begin{deluxetable}{lcccccc}
\tabletypesize{\footnotesize}
\tablewidth{0.48 \textwidth}
\tablecaption{\label{table:Meff} Parameters characterizing $M^{*}$, fit together at $k_BT=1, 10,$ and 47.9~Mev, for either pure neutron matter (PNM) or symmetric matter (SM). }
\tablehead{\\
\colhead{} &
\colhead{PNM ($Y_p$ = 0.01)} \hspace{-2cm} \vspace{-0.05cm} &
\colhead{} &
\colhead{SM ($Y_p $= 0.5)} \hspace{-2cm} \vspace{-0.05cm} &
\colhead{} \\\\
\colhead{ EOS } &
\colhead{$n_0$ (fm$^{-3}$) } &
\colhead{$\alpha$ } &
\colhead{$n_0$ (fm$^{-3}$) } &
\colhead{$\alpha$ }
}
\startdata
TM1 & 0.11 & 0.73 & 0.12 & 0.86 \\
TMA & 0.11 & 0.65 & 0.13 & 0.77 \\
NL3 & 0.10 & 0.90 & 0.11 & 1.08 \\
FSUGold & 0.10 & 0.61 & 0.11 & 0.72 \\
IUFSU & 0.11 & 0.72 & 0.12 & 0.85 \\
DD2 & 0.08 & 0.68 & 0.10 & 0.84 \\
STOS & 0.11 & 0.76 & 0.12 & 0.90 \\
SFHo & 0.21 & 0.82 & 0.22 & 0.89 \\
SFHx & 0.16 & 0.77 & 0.17 & 0.88 \\
\hline \\ Range & 0.08-0.21 & 0.61-0.90 & 0.10-0.22 & 0.72-1.08 \\ \hline \\
Mean & 0.12 & 0.74 & 0.13 & 0.87
\enddata
\tablecomments{We fix $b=2$ and $m=m_n$ in all fits.}
\end{deluxetable}
We show the performance of the fit for NL3 in Fig.~\ref{fig:Meff}. In this fit, we use the NL3 tables calculated at $k_BT=1, 10$, and 47.9~MeV (shown in purple, orange, and blue, respectively) with a proton fraction of $Y_p$=0.01, to emulate pure neutron matter (top panel), and $Y_p$=0.5, to represent symmetric matter (bottom panel). We show our approximation for $M^*$ as the black solid line. We find that the $M^*$-approximation accurately captures the behavior predicted by the full EOS, with fit parameters $n_0=0.10$~fm$^{-3}$ and $\alpha=0.90$ for $Y_p=0.01$ and fit parameters $n_0=0.11$~fm$^{-3}$ and $\alpha=1.08$ for $Y_p$=0.5.
\iffalse
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\textwidth]{nl3_Meff_allTs_Yp001.pdf}
\caption{\label{fig:Meff001} Dirac effective mass as a function of the number density, for NL3 at $Y_p$=0.01 (pure neutron matter) and $k_BT=1, 10$ and 47.9~MeV (in purple, orange, and blue, respectively). The symbols represent the effective mass predictions for the full version of NL3. The solid black line shows our approximation using eq.~(\ref{eq:Meff}). We find that, with fit parameters $n_0=0.10$~fm$^{-3}$ and $\alpha=0.90$, the $M^*$-approximation accurately reproduces the values predicted by the full EOS.}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.43\textwidth]{nl3_Meff_allTs_Yp05.pdf}
\caption{\label{fig:Meff05} Same as Fig.~\ref{fig:Meff001}, but for $Y_p=0.5$ (symmetric matter). We find that, with fit parameters $n_0=0.11$~fm$^{-3}$ and $\alpha=1.08$, the $M^*$-approximation again reproduces the values predicted by NL3 reasonably well, up to $\sim10~n_{\rm sat}$. At low temperatures, the discontinuity in the effective mass stems from the Maxwell construction used in the original EOS calculation to represent the phase transition to uniform nuclear matter. At high temperatures, this artifact disappears. }
\end{figure}
\fi
\begin{figure}[ht]
\centering
\includegraphics[width=0.485\textwidth]{nl3_Meff_allTs_Yp001_05.pdf}
\caption{\label{fig:Meff} Dirac effective mass as a function of the number density, for NL3 at $k_BT=1, 10$ and 47.9~MeV (in purple, orange, and blue, respectively) for $Y_p$=0.01 (pure neutron matter; top panel) and $Y_p$=0.5 (symmetric nuclear matter; bottom panel). The symbols represent the effective mass predictions for the full version of NL3. The solid black line shows our approximation using eq.~(\ref{eq:Meff}). We find that, with fit parameters $n_0=0.10$~fm$^{-3}$ and $\alpha=0.90$, the $M^*$-approximation accurately reproduces the values predicted by the full EOS for pure neutron matter. For symmetric nuclear matter with $n_0=0.11$~fm$^{-3}$ and $\alpha=1.08$, the $M^*$-approximation again reproduces the values predicted by NL3 reasonably well, up to $\sim10~n_{\rm sat}$. At low temperatures, the discontinuity in the effective mass stems from the Maxwell construction used in the original EOS calculation to represent the phase transition to uniform nuclear matter. At high temperatures, this artifact disappears.}
\end{figure}
As a brief aside, we note a discontinuity in the first derivative of $M^*$ at approximately half the nuclear saturation density for large $Y_p$ and low temperatures (seen most clearly in the purple stars in the bottom panel of Fig.~\ref{fig:Meff}, at $n_{\rm sat}/2 \approx 0.08~$fm$^{-3}$). This discontinuity is an artifact of the treatment of the first-order phase transition to uniform nuclear matter at these densities in the original EOS calculations.
There is an easily understood origin of this artifact. \citet{Lattimer1991}, \citet{Shen1998a}, and \citet{Hempel2010} all use a Maxwell construction to calculate the phase transition at approximately half the nuclear saturation density. At low proton fractions, where matter is approximately made up of a single species, the Maxwell construction works well to represent the phase transition. However, the Maxwell construction is invalid for multi-component species: When a system has more than one significant component, the Gibbs construction must instead be used \citep{Glendenning1992, Glendenning2000}. Because all EOS that are included in this section use the Maxwell construction, they all suffer from artifacts due to this choice at roughly half the saturation density, where the transition to uniform nuclear matter occurs.
Correcting these artifacts would require re-calculating all EOS with a different formalism and is beyond the scope of this paper. However, we note that at high temperatures ($k_BT\gtrsim$15~MeV), the non-uniform phase of matter disappears (see discussion around Fig.~5 in \citealt{Shen1998a}). Thus, we can avoid the issue altogether by performing our fit to $M^*$ at only the highest temperatures, when $Y_p$ is large. In practice, we find that whether we fit only the $k_BT=47.9$~MeV curve for $M^*$ or we fit the curves for all the temperatures together, the difference in the resulting parameters is small. We, therefore, choose to perform the fits to three temperatures ($k_BT=1, 10$ and 47.9~MeV) together and use the same method for both low and high proton fractions.
Returning to our discussion of the $M^*$ model, we note that the errors introduced by using our $M^*$-approximation are comparable to those shown in Fig.~\ref{fig:Meff} for the full set of nine EOS in this section. We thus conclude that our $M^*$-approximation reasonably captures the density-dependence of the Dirac effective mass, while greatly simplifying subsequent calculations.
Moreover, we find that the range of inferred fit parameters is relatively narrow. In particular, for a wide range of temperatures and EOS, we find that the transition density lies in the range $n_0 \in (0.08,0.22)$~fm$^{-3}$, with an average value of $\sim$0.13~fm$^{-3}$ for both pure neutron matter and symmetric matter. The power-law index characterizing the decay of $M^*$ is similarly well constrained, with $\alpha \in (0.61-0.90)$, with an average value of 0.74 for pure neutron matter; and $\alpha \in (0.72-1.08)$, with a slightly higher average value of 0.87 for symmetric matter. We find only a weak dependence of $n_0$ and $\alpha$ on the temperature, thus suggesting that these parameters could be treated as constants for use in numerical simulations.
\subsection{Performance of the $M^*$-approximation of thermal effects at fixed $Y_p$}
\label{sec:comparison}
We now turn to a comparison between the $M^*$-approximation of the thermal effects and the nine EOS listed in Table~\ref{table:Meff}. As in $\S$\ref{sec:Esym}, we make the comparison in terms of the pressure, rather than the energy, and use eq.~(\ref{eq:getP}) to convert between the two. The expressions for $P_{\rm th}(n, Y_p, T)$ are given in Box II in $\S$\ref{sec:boxes}. In particular, all results shown here use the smoothed approximation of the thermal pressure, as defined in eq.~(\ref{eq:addinvP}).
In order to focus specifically on the thermal pressure, we calculate the thermal contribution to the pressure from each realistic EOS in our sample by subtracting the cold component at the same $Y_p$.
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\textwidth]{Pth_lp_NL3_Yp01.pdf}
\caption{\label{fig:PthNL3} Smoothed thermal pressure as a function of density for the EOS NL3 with $Y_p=0.1$. The various colors are calculated at $k_B T$=1~MeV (purple), $k_B T$=10~MeV (orange), and $k_B T$=47.9 MeV (blue). The thermal pressure of the full EOS is shown as the symbols, while the solid lines represent the $M^*$-approximation of $P_{\rm th}$, using the fit parameters for NL3 from Table~\ref{table:Meff} ($n_0 =0.11$~fm$^{-3}, \alpha=1.08$). The dashed lines show the $\Gamma_{\rm th}=1.67$ hybrid approximation at each temperature. We find excellent agreement between the $M^*$-approximation and the full thermal pressure and find that the $M^*$-approximation offers a significant improvement over the hybrid EOS. }
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\textwidth]{n0_alpha_varied.pdf}
\caption{\label{fig:varyParams} The $M^*$-approximation of the thermal pressure at $k_BT=10$~MeV and $Y_p=0.5$, with intentionally extreme choices of the parameter values. The top panel shows the effect of varying $n_0$ for a fixed value of $\alpha=0.8$; the bottom panel shows the effect of varying $\alpha$ for fixed $n_0$=0.12~fm$^{-3}$. }
\end{figure}
\begin{figure*}[ht]
\centering
\includegraphics[width= \textwidth]{resids_Pth_Yp01_allEOS.pdf}
\caption{\label{fig:resids} Residuals between the smoothed $M^*$-approximation of the thermal pressure and the full results calculated for each EOS listed in Table~\ref{table:Meff}. From left to right, the panels are at $k_BT=1, 10$ and 47.9~MeV; all three panels are for $Y_p=0.1$. The various colors represent the different EOS. For comparison, we also include the residuals between the full EOS NL3 and the ideal-fluid approximation ($\Gamma_{\rm th}=1.67$) as the black dashed line. The vertical dotted line marks $n_{\rm sat}$. Our M$^*$-approximation of $P_{\rm th}$ produces residuals that are up to three orders of magnitude smaller than the ideal-fluid approximation. }
\end{figure*}
In general, we find excellent agreement between the $M^*$-approximation and the thermal pressures calculated from the full EOS. We show an example in Fig.~\ref{fig:PthNL3} for NL3. We find that our approximation of $P_{\rm th}$ closely recreates the full calculation for NL3 for nearly all densities and temperatures explored here. For comparison, we also include in Fig.~\ref{fig:PthNL3} the hybrid approximation with $\Gamma_{\rm th}=1.67$ as dashed lines.\footnote{We choose the relatively low value of $\Gamma_{\rm th}=1.67$ in order to minimize the residuals of the hybrid model. This value of $\Gamma_{\rm th}$ ensures the hybrid EOS matches an ideal fluid at intermediate densities. Larger values, as are more commonly used in numerical simulations, would cause the hybrid $P_{\rm th}$ to overestimate even the ideal regime.} The full thermal pressure agrees with the hybrid approximation only at intermediate densities. At the lowest densities, this value of $\Gamma_{\rm th}$ overestimates the contribution from relativistic species. At higher densities that are relevant for forming and merging neutron stars, particle interactions become important and the ideal-fluid approximation grossly overestimates the thermal pressure, remaining several orders of magnitude above the true thermal pressure.
In order to gain an intuitive understanding of the behavior of $P_{\rm th}$, we also explore an extreme range of the $M^*$ parameters. Specifically, in Fig.~\ref{fig:varyParams}, we zoom in on $P_{\rm th}$ at $k_BT=10$~MeV and $Y_p=0.5$ and show the effect of varying the parameters $n_0$ and $\alpha$ for symmetric matter. We intentionally take extreme values for the parameters, well beyond the ranges found in Table~\ref{table:Meff}, in order to emphasize that the variations between more realistic parameter choices will be small. Even for these unreasonable choices of values for $n_0$ and $\alpha$, we find that $P_{\rm th}$ approximates the full thermal pressure reasonably well and, in all cases, better than the ideal fluid approximation. Analyzing the specific dependences more closely, we see in Fig.~\ref{fig:varyParams} that the parameter $n_0$ controls the density at which the rise in the thermal pressure starts to slow. This corresponds to the density at which particle interactions become significant and degenerate thermal effects can no longer be ignored. The parameter $\alpha$, which controls the power-law slope of $M^*$, directly controls the height of the dip in $P_{\rm th}$. This makes intuitive sense: if particle interactions are stronger, $M^*$ decreases more rapidly, $\alpha$ will be larger, and the thermal pressure will deviate even more drastically from the ideal-fluid approximation as part of the free energy is taken up by those interactions.
Finally, we compare the $M^*$-approximation of the thermal pressure against the full sample of EOS listed in Table~\ref{table:Meff}. We show the corresponding residuals at three temperatures in Fig.~\ref{fig:resids} and find that the residuals are typically $\lesssim30\%$ at densities above $0.5~n_{\rm sat}$. For comparison, Fig.~\ref{fig:resids} also shows a sample set of residuals between the full thermal pressure from NL3 and the hybrid approximation ($\Gamma_{\rm th}=1.67$) as the black dashed line. We find that the $M^*$-approximation produces residuals that are up to three orders of magnitude smaller than the ideal-fluid approximation used in hybrid EOS, with only two additional parameters that are easy to specify.
\subsection{$M^*$-approximation for non-RMF models}
\label{sec:nonRMF}
We have so far only calculated the thermal pressures using the sub-sample of EOS for which there exist published tables of the effective masses. While this allowed us to directly test the performance of the $M^*$-approximation, this set of EOS happens to also be calculated exclusively with RMF models. In this section, we compare the $M^*$-approximation to the LS and SLY4-RG models, which are calculated using non-relativistic Skyrme energy functionals (see $\S$~\ref{sec:overview}). We also include here the two-loop exchange model of \cite{Zhang2016}, which is an extension of mean field theory. We note that the pressures of the \cite{Zhang2016} EOS are reported only at $Y_p=0$ and 0.5, which is why this EOS is not included in our full sample. As a result of these and other limitations in the publicly-available values for this EOS, all comparisons in this section are made at $Y_p=0.5$ and $T=20$~MeV. We also fix $n_0$ and $\alpha$ to the mean values for symmetric matter from Table~\ref{table:Meff} for all three EOS.
\begin{figure}[ht]
\centering
\includegraphics[width=0.45\textwidth]{resids_nonRMF_Yp05_T20.pdf}
\caption{\label{fig:nonRMF} Residuals between the smoothed $M^*$-approximation of the thermal pressure and the true EOS at $Y_p$=0.5 and $T=20$~MeV for three non-RMF models. For $n_0$ and $\alpha$, we use the mean fit values for symmetric matter from Table~\ref{table:Meff}. The dashed lines show the corresponding residuals between the true EOS and the hybrid approximation using $\Gamma_{\rm th}$=1.67, at the same proton fraction and temperature. The three EOS shown are LS (pink), SLY4-RG (green), and the two-loop model of \cite{Zhang2016} (``TL(sc)", blue). We find that, while the $M^*$-approximation produces slightly larger residuals for these EOS than for the RMF models, it nevertheless offers a significant improvement over the hybrid approximation at high densities. }
\end{figure}
Figure~\ref{fig:nonRMF} shows the residuals between the $M^*$-approximation of the thermal pressure and the true EOS for these three models. For comparison, this figure also shows the corresponding residuals between the hybrid approximation and the true EOS (dashed lines). In general, we find that the $M^*$-approximation of the thermal pressure results in larger residuals for these EOS compared to the RMF models, but that it still offers a significant improvement over the hybrid approximation at densities above $\sim n_{\rm sat}$.
We also compared the residuals at $T=50$~MeV and found that the $M^*$-approximation performed comparably to the hybrid approximation at this temperature. In fact, for densities between $n_{\rm sat}$ and $0.7$~fm$^{-3}$, the hybrid approximation produces slightly smaller residuals in the thermal pressure for these non-RMF models. In this regime, the hybrid approximation tends to over-estimate the thermal pressure for these models, while the $M^*$-approximation tends to under-estimate $P_{\rm th}$ by a similar degree. However, even in this case, the $M^*$-approximation still offers an appreciable improvement over the hybrid approximation at the highest densities, above $\sim0.7$~fm$^{-3}$.
\section{Putting it all together}
\label{sec:boxes}
We now summarize the equations and approximations that we have developed so far to represent the total energy per particle in Box I.
\begin{widetext}
\setlength{\fboxrule}{2pt}
\setlength{\fboxsep}{6pt}
\fbox{\parbox{ 0.9\textwidth}{
\textbf{ Box I: Total Energy Expressions for Finite-Temperature Dense Gas.} \\\\
The energy per particle of $n$-$p$-$e$~matter is given by
\begin{align*}
E(n, Y_p, T) & = \big( \text{Cold EOS in $\beta$-equilibrium}\big)
+ 3 K \left( Y_p^{4/3} - Y_{p,\beta}^{4/3} \right) n^{1/3} \\
& + E_{\rm sym}(n, T=0)\left[ (1-2 Y_p)^2 - (1-2 Y_{p,\beta})^2\right]
+
\begin{cases}
4 \sigma f_s T^4/ (c n), &\phantom{n_1<} n < n_1 \\
(3/2) k_B T, &n_1< n <n_2 \\
\left[ a(0.5 n, 0.5 M^*_{\rm SM}) + a(Y_p n, m_e) Y_p \right] T^2 , &\phantom{n_1<} n>n_2,
\end{cases}
\end{align*}
\\
where the symmetry energy is approximated as
\\
\begin{equation*}
E_{\rm sym}(n, T=0) = \eta E_{\rm sym}^{\rm kin}(n) + \left[ S_0 - \eta E_{\rm sym}^{\rm kin}(n_{\rm sat})\right]\left(\frac{n}{n_{\rm sat}}\right)^\gamma,
\end{equation*}
\begin{equation*}
E_{\rm sym}^{\rm kin}(n) = \frac{3 }{5} \left( 1 -2^{1/3} \right) E_f(n),
\end{equation*}
\begin{equation*}
\eta = \frac{5}{9} \left[ \frac{ L-3 S_0 \gamma}{\left(1-2^{1/3}\right)\left(2/3 - \gamma \right) E_f(n_{\rm sat})} \right],
\end{equation*}
\\
and the terms of the $M^*$-approximation are given by
\\
\begin{equation*}
a(n_q, M_q^*) \equiv \frac{\pi^2 k_B^2}{2} \frac{ \sqrt{ M_q^*(n_q)^2 + (3 \pi^2 n_q)^{2/3}(\hbar c)^2 } }{(3 \pi^2 n_q)^{2/3}(\hbar c)^2 }
\quad \text{and} \quad
M^*(n_q) = \left\{ (m c^2)^{-b} + \left[ mc^2 \left( \frac{ n_q }{n_0 }\right)^{-\alpha} \right]^{-b} \right\}^{-1/b}.
\end{equation*}
\begin{itemize}
\item The parameters $S_0$, $L$, and $\gamma \in (0.2-1.2)$ are freely specified; this will uniquely specify $Y_{p,\beta}$.
\item Alternatively, $S_0$, $L$, and $Y_{p,\beta}$ may be specified and the proton fraction may be fit for $\gamma$. We provide fits to $\gamma$ for eleven EOS in Table~\ref{table:gamma}.
\item We find that for $M^*_{\rm SM}$, $n_0 \sim 0.13$~fm$^{-3}$ and $\alpha \sim 0.9$ provide reasonable fits to most EOS.
\end{itemize}
}}
\end{widetext}
Using the expressions for the energy from Box I., we can derive the pressure via the standard thermodynamic relations of eq.~(\ref{eq:getP}), where the derivatives are evaluated at constant $Y_p$, $Y_{p,\beta}$, and $S$. The total entropy of the relativistic, ideal-fluid, and degenerate terms is given by
\begin{align}
\begin{split}
S(n,N_p, N_n,N_e, T) =
\begin{cases}
S_{\rm rel}, &\phantom{n_1<} n < n_1 \\
S_{\rm ideal} , &n_1< n <n_2 \\
S_{\rm deg} , &\phantom{n_1<} n>n_2,
\end{cases}
\end{split}
\end{align}
where $n_1$ and $n_2$ are the thermal energy transition densities, as defined in $\S$\ref{sec:thermal}.
The entropy of a gas of relativistic leptons and photons is given by
\begin{equation}
S_{\rm rel} = \frac{16 \sigma f_s}{3 c} \left( \frac{N_p + N_n}{n}\right) T^3.
\end{equation}
The entropy of a monatomic ideal fluid is given by the Sackur-Tetrode equation,
\begin{multline}
S_{\rm ideal} = \left( N_p + N_n + N_e \right) k_B \\
\times \left\{ \ln\left[ \left( \frac{N_p+N_n}{N_p + N_n + N_e} \right) n^{-1} \left(\frac{ m k_B T}{ 2\pi \hbar^2}\right)^{3/2} \right] + \frac{5}{2} \right\}.
\end{multline}
Finally, the entropy of a degenerate Fermi gas in our framework is given by
\begin{equation}
\label{eq:S}
S_q = 2 a_q N_q T
\end{equation}
for a particle $q$, so that the total entropy for the degenerate terms is
\begin{equation}
S_{\rm deg} =2 \left\{ a(0.5 n, 0.5 M^*_{\rm SM})[N_n + N_p] + a(Y_p n, m_e) N_e \right\} T.
\end{equation}
We summarize the resulting pressure equations in Box II.
\begin{widetext}
\setlength{\fboxrule}{2pt}
\setlength{\fboxsep}{6pt}
\fbox{\parbox{ 0.9\textwidth}{
\textbf{ Box II: Pressure Expressions for Finite-Temperature Dense Gas.} \\\\
The pressure of $n$-$p$-$e$~matter is given by
\begin{align*}
P(n, Y_p, T)& = \big( \text{Cold EOS in $\beta$-equilibrium}\big)
+ K \left( Y_p^{4/3} - Y_{p,\beta}^{4/3} \right) n^{4/3} \\
& + P_{\rm sym}(n, T=0)\left[ (1-2 Y_p)^2 - (1-2 Y_{p,\beta})^2 \right]
+
\begin{cases}
4 \sigma f_s T^4/ (3 c), &\phantom{n_1<} n < n_1 \\
n k_B T, &n_1< n <n_2 \\
-\left[ \frac{\partial a(0.5 n, 0.5 M^*_{\rm SM})}{\partial n} + \frac{\partial a(Y_p n, m_e)}{\partial n} Y_p \right] n^2 T^2 , &\phantom{n_1<} n>n_2,
\end{cases}
\end{align*}
\\
where $n_1$ and $n_2$ are the thermal energy transition densities for a particular temperature and proton fraction.
The symmetry pressure, corresponding to our model of the symmetry energy, is
\\
\begin{align*}
P_{\rm sym}(n, T=0) & = \frac{2\eta}{3} n E_{\rm sym}^{\rm kin}(n) +
\left[ S_0 - \eta E_{\rm sym}^{\rm kin}(n_{\rm sat})\right] \left( \frac{n}{n_{\rm sat}}\right)^{\gamma} \gamma n .
\end{align*}
\\
The full analytic expression for $Y_{p,\beta}$ is given in eq.~(\ref{eq:YpBeta}) and derived in Appendix A.
\\
The $M^*$-approximation derivatives are given by
\\
\begin{multline*}
\frac{\partial a(n_q, M^*)}{\partial n}\biggr\rvert_{Y_q} =
-\frac{2 a(n_q, M^*)}{3 n}
\left\{ 1 -\frac{1}{2}\left[ \frac{M^*(n_q)^2}{ M^*(n_q)^2+(3\pi^2 n_q)^{2/3}(\hbar c)^2} \right] \left( \frac{(3\pi^2 n_q)^{2/3}(\hbar c)^2}{M^*(n_q)^2}+ 3 \frac{\partial \ln [ M^*(n_q)]}{\partial \ln n}\biggr\rvert_{Y_q} \right) \right\},
\end{multline*}
\quad and
\begin{equation*}
\frac{\partial \ln[ M^*(n_q)]}{\partial \ln{n}} \biggr\rvert_{Y_q} = - \alpha \left[ 1- \left(\frac{M^*(n_q)}{Y_q mc^2}\right)^2 \right],
\end{equation*}
where, for symmetric matter, we replace $M^*({n_q})\rightarrow 0.5 M^*_{\rm SM}(0.5 n)$ and for the electrons, $M^*({n_q}) \rightarrow m_e$.
\begin{itemize}
\item As in Box I., there are five free parameters: $S_0, L, \gamma, n_0, \alpha$.
\item A user may freely specify $S_0$, $L$, and $Y_{p,\beta}$ and fit for $\gamma$. Alternatively, a user may specify $S_0$, $L$, and $\gamma$, which will uniquely specify $Y_{p,\beta}$.
\item We provide fits for $\gamma, n_0,$ and $\alpha$ for the EOS in our sample in Tables~\ref{table:gamma} and \ref{table:Meff}.
\end{itemize}
}}
\end{widetext}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\textwidth]{p_Ts_Yp01_fromBeta_noEsymTh_NL3_DD2.pdf}
\caption{\label{fig:pFull} Our approximation of $P$ and the EOS pressures predicted by NL3 and DD2 (in blue and orange, respectively). The EOS predictions are shown as the diamonds, while our model is shown as the solid lines. The three panels are at $Y_p=0.1$ and $k_BT=1, 10,$ or 47.9~MeV (from left to right). We find that our approximation is able to closely recreate the pressures predicted by NL3 and DD2 at densities above $n_{\rm sat}$ for all temperatures. }
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.98\textwidth]{resids_Ts_Yp01_fromBeta_noEsymTh_allEOS.pdf}
\caption{\label{fig:residsFull} Residuals between our approximation of $P$ and the EOS pressures predicted by the eleven EOS in our sample. The three panels are at $Y_p=0.1$ and $k_B T=1, 10,$ or 47.9~MeV (from left to right). The vertical dotted line marks $n_{\rm sat}$. }
\end{figure*}
The piecewise definitions of the thermal energy and pressure are mathematically convenient, but the sharp transitions are themselves unphysical, as discussed in $\S$\ref{sec:thermal}. We, therefore, instead implement the thermal pressure using a smoothed approximation of the form
\begin{equation}
\label{eq:addinvP}
P_{\rm th}(n, Y_p, T) \approx P_{\rm rel} + \left( P_{\rm ideal}^{-1} + P_{\rm deg}^{-1} \right)^{-1}.
\end{equation}
This smoothed approximation of the thermal pressure is used for the figures throughout this paper. We note that we use this separate smoothing for both the thermal pressure and the thermal energy (as in eq.~\ref{eq:addinv}) in order to keep the problem tractable. However, this is not mathematically exact since, formally, the energy is the proper thermodynamic function and the pressure should, ideally, be derived from the smoothed energy. Nevertheless, the errors introduced by the separate smoothing approximations will be limited to the regions close to the transition points. Physically, the mismatch between the approximate thermal energy and pressure will correspond to a small error in the sound speed in these regions, which we neglect for the present purposes.
Finally, we note that our model allows significant freedom in creating a new finite-temperature EOS. We have provided a set of parameters that correspond to physically-motivated EOS, but if one wishes to vary these parameters significantly, it will be useful to check that the resulting EOS is still physical. One requirement of a realistic EOS is that the sound speed remain sub-luminal at all densities and temperatures of interest. For this reason, we include in Appendix B a calculation of the sound speed for astrophysical merger scenarios.
\section{Complete model: Comparison of realistic EOS at arbitrary $Y_p$ and $T$}
\label{sec:complete}
In $\S$\ref{sec:Esym}, we found that our model is able to extrapolate from $\beta$-equilibrium to an arbitrary proton fraction with resulting errors of $\lesssim10\%$ at densities above 0.5~$n_{\rm sat}$. Similarly, in $\S$\ref{sec:comparison}, we showed that the $M^*$-approximation is able to reproduce the thermal pressure of realistic EOS, at fixed $Y_p$, to within $\sim30\%$ for a variety of EOS based on RMF theory. In this section, we quantify the performance of our complete model: starting with a cold EOS in $\beta$-equilibrium, and extrapolating to arbitrary temperature and proton fraction.
Figure~\ref{fig:pFull} shows an example of a complete model for NL3 and DD2 at three different temperatures. For our approximation, we start with the relevant cold EOS in $\beta$-equilibrium and add the corrections outlined in Box~II, to extrapolate the pressure to $Y_p=0.1$ and the three indicated temperatures. We take the values for $n_0, \alpha$, and $\gamma$ listed in Tables~\ref{table:gamma} and \ref{table:Meff} for each EOS. We show the results as the solid lines in Fig.~\ref{fig:pFull}, while the predictions of the full EOS are shown as the diamonds. We find close agreement between our approximation and the full pressures predicted by NL3 and DD2, especially at densities above $\sim0.5~n_{\rm sat}$.
Figure~\ref{fig:residsFull} shows the corresponding residuals between our approximation and the full EOS for NL3 and DD2, as well as the rest of our sample of EOS. For each EOS in this figure, we use the values for $n_0, \alpha$ and $\gamma$ listed in Tables~\ref{table:gamma} and \ref{table:Meff}, where possible. For LS and SLY4-RG, for which we do not have fit values for $n_0$ and $\alpha$, we use the average parameter values for symmetric matter in Table~\ref{table:Meff}. We find that our approximation works comparably well to recreate any of the EOS in our sample. Moreover, we find that for $n \gtrsim n_{\rm sat}$, the residuals are $\lesssim 20\%$ at all three temperatures.
For all the EOS in our sample, the error introduced by our model increases in the vicinity of $\sim 0.5 n_{\rm sat}$. This is a result of the break-down in the $E_{\rm sym}$ approximation at low densities. Our derivation of $E_{\rm sym}$ in $\S$\ref{sec:Esym} assumed uniform $n$-$p$-$e$~matter, but at densities below $\sim0.5~n_{\rm sat}$, the matter becomes inhomogeneous. Nevertheless, with the exception of LS, the errors at these densities are still typically $\lesssim50$\%.
We have thus verified that our model is able to recreate realistic EOS at relevant densities, with a simple set of parameters. The implications of this result are two-fold. First, this approximation can be used in lieu of more complicated calculations, to analytically represent the EOS that are commonly used in the literature with reasonable accuracy. Second, it implies that our approximation can be reliably used to create new finite-temperature EOS for $n$-$p$-$e$~matter that probe different physics through the choice of $n_0, \alpha, \gamma, S_0,$ and $L$. Our model allows further freedom in creating a new finite-temperature EOS through the choice of the cold, $\beta$-equilibrium EOS. We thus find that this model can span a broad range of possible physics, with parameters that are directly tied to the underlying physics and that can be integrated with minimal computational cost to a large array of numerical calculations.
\section{Conclusions}
In this paper, we have developed a general framework for calculating the pressure of neutron-star matter at arbitrary proton fraction and finite temperature. Our model is designed so that the corrections we have developed here can be added to any cold $n$-$p$-$e$~EOS in neutrinoless $\beta$-equilibrium. The model is based on a set of five physically-motivated parameters: $S_0, L, \gamma, n_0,$ and $\alpha$. The first three, $S_0$, $L$, and $\gamma$ characterize the symmetry energy and can be chosen to match a particular EOS or set of priors from laboratory experiments. The parameters $n_0$ and $\alpha$ are introduced through our $M^*$-approximation, where $n_0$ represents the density at which particle interactions become important and $\alpha$ characterizes the strength of those interactions. We find that the effective masses of nine realistic EOS can be well characterized by our $M^*$-approximation with a relatively narrow range of these parameters, with average values of $n_0\sim 0.13$~fm$^{-3}$ and $\alpha \sim 0.9$.
The complete model is able to extrapolate from cold matter in $\beta$-equilibrium to arbitrary proton fraction and temperature. We find that our model is able to recreate a sample of eleven realistic EOS with resulting errors of $\lesssim20\%$ at a variety of temperatures and proton fractions, above $n_{\rm sat}$. In particular, by including the effects of degenerate matter, our $M^*$-approximation reproduces the thermal pressure of realistic EOS with residuals that are several orders of magnitude smaller than the hybrid EOS that are commonly used in the literature.
In addition to providing a $1-3$ orders-of-magnitude improvement over the ideal-fluid approximation of the thermal pressure, this model also includes the effects of changing the proton fraction, which is particularly relevant in simulating the formation and cooling of proto-neutron stars.
The complete model can thus be used to accurately recreate the realistic EOS that are currently in use in the literature with a set of simple, analytic functions. Furthermore, the model can be used to calculate new finite-temperature EOS that span a wide range of underlying physics, following one of two possible paths. One possibility is to choose a physically-motivated cold EOS, which will provide predictions for the $\beta$-equilibrium proton fraction and symmetry energy parameters. These can then be used to fit for the free parameter $\gamma$, and then used to extrapolate to an arbitrary proton fraction. Alternatively, one can use a cold, parametric EOS that does not specify the microphysics. In this case, there is freedom to choose the symmetry energy parameters to probe entirely new physics. In either case, one can freely choose the interaction parameters to control the relative importance of thermal effects. All together, these possibilities will allow a new and wide range of physics to be robustly probed in studies of dynamical neutron star phenomena.
{\em{Acknowledgements.\/}} We thank Vasileios Paschalidis for useful discussions and comments on this work. CR is supported by the NSF Graduate Research Fellowship Program Grant DGE-1143953. FO and DP acknowledge support from NASA grant NNX16AC56G.
|
{
"timestamp": "2019-03-01T02:00:47",
"yymm": "1902",
"arxiv_id": "1902.10735",
"language": "en",
"url": "https://arxiv.org/abs/1902.10735"
}
|
\section{Preliminaries}\label{s:prelimin}
\subsection{Bounded cohomology}
Bounded cohomology was defined in a seminal paper of Gromov \cite{gromov82}.
Below we give basic definitions.
Let $G$ be a group.
A function $c \colon G^{n+1} \to \mathbb{R}$ is called homogeneous,
if for every $h \in G$ and every $g_0,\ldots,g_n\in G$ we have
$$c(g_0h,\ldots,g_nh) = c(g_0,\ldots,g_n).$$
Let us consider the space of bounded $n$-cochains:
$$\OP{C}^n_b(G) = \{c \colon G^{n+1} \to \mathbb{R}~|~c~\text{is homogeneous and bounded}\}.$$
The sequence $\{\OP{C}^n_b(G),d_n\}$ is a chain complex,
where $d$ is the ordinary coboundary operator $d_n \colon \OP{C}^n_b(G) \to \OP{C}^{n+1}_b(G)$.
The \textbf{bounded cohomology} of the group $G$ is the homology of this complex, i.e.,
$\OP{H}^n_b(G) = \OP{ker}(d_n)/\OP{im}(d_{n-1})$.
Note that $\OP{C}_b^n(G)$ is a subcomplex of the space of all homogeneous cochains,
thus we have a canonical map $\OP{H}_n^b(G) \to \OP{H}^n(G,\mathbb{R})$, called the \textbf{comparison map}.
The \textbf{exact bounded cohomology}, denoted $\OP{EH}_b^n(G)$, is defined to be the kernel of the comparison map.
A class belongs to $\OP{EH}_b^n(G)$ if it is a coboundary of a cochain,
but is never a coboundary of a bounded cochain.
On $\OP{C}^n_b(G)$ we have the supremum norm denoted by $\n\cdot\n_{sup}$.
This norm induces a semi-norm on $\OP{H}^n_b(G)$, i.e.,
if $c \in \OP{H}_b^n(G)$, then
$$\n c \n = \OP{min}\{\n a \n_{sup}~|~[a] = c\}.$$
Let $\OP{N}^n(G) \leq \OP{H}_b^n(G)$ be the linear subspace consisting of classes of zero norm.
The \textbf{reduced bounded cohomology} equals to the quotient
$$\rH^n_b(G) = \OP{H}_b^n(G)/\OP{N}^n(G).$$
Note that $\rH^n_b(G)$ with the induced norm, that we again denote by $\n\cdot\n$,
is a Banach space.
The \textbf{exact reduced bounded cohomology} is defined to be
$$\rEH_b^n(G)~=~\OP{EH}^n_b(G)/(\OP{EH}^n_b(G) \cap \OP{N}^n(G)).$$
Note that $\rEH_b^n(G) \leq \rH^n_b(G)$ and $\rEH_b^n(G)$ with the induced norm are Banach spaces.
\subsection{Measurable cocycles and induction on bounded cohomology}\label{s:induction}
Let $G$ be a topological group, $H$ a discrete group and $(X,\mu)$ a measurable space.
Suppose that $G$ acts on $X$ by measure preserving homeomorphisms.
A map $\gamma \colon G \times X \to H$ is called a \textbf{measurable cocycle}, if
for all $g \in G$ the map $\gamma(g,\cdot)$ is measurable and for all $g_1,g_2 \in G$ we have
$$\gamma(g_1g_2,x) = \gamma(g_1,g_2(x))\gamma(g_2,x)$$
for almost all $x \in X$. Now given a measurable cocycle $\gamma$, we produce a map
$$\OP{Ind}_b(\gamma) \colon \OP{H}_b^n(H) \to \OP{H}_b^n(G).$$
Let $c \in \OP{C}^n_b(H)$ and let $\gamma$ be a measurable cocycle.
First we define a map $\OP{Ind}'_b(\gamma)$ on the space of cocycles by the following formula:
$$
\OP{Ind}'_b(\gamma)(c)(g_0,g_1,\ldots,g_n) =
\int_{M} c(\gamma(g_0,x),\gamma(g_1,x),\ldots,\gamma(g_n,x)) d\mu(x).
$$
The next proposition shows that $\OP{Ind}'_b(\gamma)$ induces a map on the level of bounded cohomology
which we call $\OP{Ind}_b(\gamma)$.
\begin{proposition}\label{p:well.def}
Let $G$ be a group. Then the function
$$x \to c(\gamma(g_0,x),\gamma(g_1,x),\ldots,\gamma(g_n,x))$$
is a $\mu$-measurable function on $X$ for every $g_0,g_1,\ldots,g_n \in G$,
$\OP{Ind}'_b(\gamma)$ commutes with the coboundary $d$, and $\OP{Ind}'_b(\gamma)(c)$ is homogeneous.
\end{proposition}
\begin{proof}
The function $X \to H$ given by
$x \to (\gamma(g_0,x),\gamma(g_1,x),\ldots,\gamma(g_n,x))$
is $\mu$-measurable and the cochain $c$ is continuous ($H$ is discrete), thus their composition is $\mu$-measurable.
The fact that $\OP{Ind}'_b(\gamma)$ commutes with the coboundary $d$
follows directly from the definition of $\OP{Ind}'_b(\gamma)$.
For all $h ,g_0,\ldots \in G$, we have:
\begin{align*}
\OP{Ind}'_b(\gamma)(c)(g_0h,\ldots)& = \int_{X} c(\gamma(g_0h,x),\ldots) d\mu(x)\\
&= \int_{X} c(\gamma(g_0,h(x))\gamma(h,x),\ldots,) d\mu(x)\\
&= \int_{X} c(\gamma(g_0,h(x)),\ldots) d\mu(x)\\
&= \int_{X} c(\gamma(g_0,x),\ldots) dh^*\mu(x) = \OP{Ind}'_b(\gamma)(c)(g_0,\ldots).
\end{align*}
In the above we first used the cocycle condition, then the homogeneity of $c$
and that $\mu$ is $h$-invariant.
\end{proof}
We conclude this section by noting
that similar maps were used in the study of ordinary continuous cohomology of Lie groups \cite{guichardet} and
geometry of solvable and amenable groups \cite{sauer,shalom}.
\section{Definition of $\Gb$}
The goal of this section is to construct the homomorphism
$$\Gb\colon \OP{H}_b^\bullet(\pi_1(M)) \to \OP{H}_b^\bullet(\OP{Homeo}_0(M,\mu)).$$
In order to do that, we first define a certain measurable cocycle $\gamma$ and use it to define the above map.
\subsection{The cocycle}\label{s:cocycle}
Denote by $\C H_M=\OP{Homeo}_0(M)$ the connected component of the identity of the group of all compactly supported
homeomorphisms of $M$.
We fix a basepoint $z \in M$.
Let $\C H_{M,z}<\C H_M$ be the subgroup of all homeomorphisms in $\C H_M$ that fix $z$.
Consider the following fiber bundle
$$\C H_{M,z} \to \C H_M \xrightarrow{ev_z} M,$$
where $ev_z$ is the evaluation map at the basepoint $z$, i.e., if $g \in \C H_M$, then $ev_z(g) = g(z)$,
and $\C H_{M,z}$ is the fiber of $ev_z$. The group $\C H_{M,z}$ has an interesting quotient.
Namely, it is the group of connected components $\pi_0(\C H_{M,z})$ which
is closely related to $\pi_1(M)$.
Recall that $\pi_{M} = \pi_1(M)/Z(\pi_1(M))$,
where $Z(\pi_1(M))$ is the center of $\pi_1(M)$.
\begin{proposition}\label{p:delta}
There is an epimorphism $\delta \colon \C H_{M,z} \to \pi_{M}$.
\end{proposition}
\begin{proof}
Let us consider the long exact sequence of homotopy groups of the fibration
$$\C H_{M,z} \to \C H_M \xrightarrow{ev_z} M.$$
We have
$$\pi_1(\C H_M) \xrightarrow{ev_z^*} \pi_1(M,z) \to \pi_0(\C H_{M,z}) \to \pi_0(\C H_M) = 1.$$
We shall show that $\OP{im}(ev_z^*) \subset Z(\pi_1(M,z))$.
Indeed, let $g_t$, $t\in S^1$, be a loop in $\C H_M$ based at the identity and $[g_t] \in \pi_1(\C H_M)$.
Then $ev_z^*([g_t])$ is a loop based at $z$ represented by $g_t(z)$, $t\in S^1$.
Let $l_s$, $s\in S^1$, be an arbitrary loop in $M$ based at $z$.
We have that the image of the map $S^1 \times S^1 \to M$ given by $(t,s) \to g_t(l_s)$,
contains $g_t(z)$ and $l_s$, thus these loops commute in~$\pi_1(M,z)$.
We define $\delta$ to be the composition of the map
$$\C H_{M,z} \to \pi_0(\C H_{M,z}) \cong \pi_1(M,z)/\OP{im}(ev_z^*)$$ and
the quotient map $\pi_1(M,z)/\OP{im}(ev_z) \to \pi_{M}$.
\end{proof}
Let $\C H_{M,\mu} = \OP{Homeo}_0(M,\mu)$ and let $s \colon M \to \C H_M$ be a measurable section of $ev_z$,
i.e., $ev_z \circ s(x) = x$ for almost all $x \in M$.
We define a measurable cocycle:
$\gamma_s \colon \C H_{M,\mu} \times M \to \pi_M$
by the formula
$$\gamma_s(g,x) = \delta(s_{g(x)}^{-1} \circ g \circ s_x).$$
It follows from the formula, that $\gamma_s$ satisfies the cocycle condition.
\subsection{Example of a section $s$}\label{s:example}
Let us consider the following set:
$$ D = \OP{int}(\{~x \in M~|~\text{there exists a unique geodesic between}~z~\text{and}~x~\}).$$
The set $M\setminus D$ is called the cut locus of $M$. The Hausdorff dimension of $M\setminus D$ is at most
$\OP{dim}(M)-1$, see \cite{cut.locus}. Thus $\mu(M\setminus D) = 0$, and $\mu(D) = \mu(M)$.
We regard $s$ as a measurable map, hence it is enough to define $s$ on the full measure subset $D$.
Let $x \in D$. We define $s_x$ to be a point pushing map that transports $z$ to $x$ along the unique geodesic.
It can be done such that $s$ is continuous on $D$.
Let us now take a closer look at the cocycle $\gamma_s$ defined using a section $s$ as described in Subsection \ref{s:cocycle}.
Let $g \in \C H_M$ and $x \in D \cap g^{-1}(D) \subset M$.
The element $\gamma_s(g,x) \in \pi_{M}$ has a simple geometrical interpretation.
It can be constructed as follows: let $g_t$, $t \in [0,1]$, be any isotopy in $\C H_M$ connecting the identity with $g$.
Let $\alpha$ be the loop which is the concatenation of the unique geodesic from $z$ to $x$, the path $g_t(x)$, $t \in [0,1]$,
and the unique geodesic from $g(x)$ to $z$.
The loop $\alpha$ regarded as an element of $\pi_{M}$ is well defined and equals $\gamma_s(g,x)$.
\subsection{The definition of $\Gb$}
Let $\gamma_s \colon \C H_{M,\mu} \times M \to \pi_M$ be a measurable cocycle
given by a measurable section $s \colon M \to \C H_M$.
We define
$$\Gb = \OP{Ind}_b(\gamma_s) \colon \OP{H}_b^\bullet(\pi_M) \to \OP{H}_b^\bullet(\C H_{M,\mu}).$$
Note that the quotient map $\pi_1(M) \to \pi_{M}$ has an abelian kernel.
It follows from the mapping theorem \cite[Section 3.1]{gromov82} that a quotient map defines isometric isomorphisms
$\OP{H}_b^\bullet(\pi_{M}) \to \OP{H}_b^\bullet(\pi_1(M))$. Thus we can regard $\Gb$ as a following map:
$\Gb \colon \OP{H}_b^\bullet(\pi_1(M)) \to \OP{H}_{b}^\bullet(\C H_{M,\mu})$.
If $\C T_{M}<\C H_{M,\mu}$ is any subgroup, then we can compose the restriction map
$\OP{H}_b^\bullet(\C H_{M,\mu}) \to \OP{H}_b^\bullet(\C T_M)$ with
$\Gb$ and obtain the map
$$\Gb(\C T_{M}) \colon \OP{H}_b^\bullet(\pi_1(M)) \to \OP{H}_{b}^\bullet(\C T_{M}).$$
Usually we abuse the notation and write $\Gb$ instead of $\Gb(\C T_{M})$.
\subsection{Standard cohomology and exact bounded cohomology}\label{s:standard}
Let $s$ be a section defined in Subsection \ref{s:example} and $\gamma = \gamma_s$.
We show that for such $\gamma$, one can use the induction to define a map on the level of the ordinary cohomology.
Note that induction in the case of the ordinary cohomology is a slightly more delicate operation then for bounded cohomology.
Here the cocycles are not bounded and we need to show that the integral exists.
In order to show integrability we need to choose carefully the section $s$. Let $c \in \OP{C}^n(\pi_M)$. We define
$$\OP{Ind}'(\gamma)(c)(g_0,g_1,\ldots,g_n) =
\int_{M} c(\gamma(g_0,x),\gamma(g_1,x),\ldots,\gamma(g_n,x)) d\mu(x),$$
where $g_0,\cdots,g_n \in \C H_{M,\mu}$.
Let us show, that $\OP{Ind}'(\gamma)$ defines a map on the level of ordinary cohomology which we call $\OP{Ind}'(\gamma)$.
It follows from Proposition \ref{p:well.def} that the function $x \to c(\gamma(g_0,x),\ldots,\gamma(g_n,x))$ is measurable.
Integrability follows from Lemma \ref{l:fin.many.loops}, which shows that this function has essentially finite image.
Every measurable function with essentially finite image is integrable.
The fact that $d\OP{Ind}'(\gamma) = \OP{Ind}'(\gamma) d$ and that $\OP{Ind}'(\gamma)(c)$ is homogeneous
follows immediately from Proposition \ref{p:well.def}.
Thus for $\C T_M \leq \C H_{M,\mu}$ we have a well defined map
$$
\Gamma(\C T_M) \defeq i^* \circ \OP{Ind}'(\gamma) \colon \OP{H}^\bullet(\pi_M) \to \OP{H}^\bullet(\C T_M),
$$
where $i^* \colon \OP{H}^\bullet(\C H_{M,\mu}) \to \OP{H}^\bullet(\C T_M)$ is induced by the inclusion.
Any cocycle $\gamma$ can be extended to a cocycle
$$
\gamma' \colon \OP{Homeo}_0(M) \times M \to \pi_M,
$$
using the same formula as for $\gamma$, i.e., $\gamma'(g,x) = \delta(s_{g(x)}^{-1} \circ g \circ s_x)$.
Let $X$ be a measurable space and let $\gamma \colon X \to Y$ be a measurable function. We say that $\gamma$ has \textbf{essentially finite image},
if there exist a full measure set $Z \subset X$, such that $\gamma$ has a finite image on $Z$.
\begin{lemma}\label{l:fin.many.loops}
Let $f\in\OP{Homeo}_0(M)$. Then $\gamma'(f,\cdot\hspace{1px})\colon M\to\pi_M$ has essentially finite image.
\end{lemma}
\begin{proof}
Let $f\in\OP{Homeo}_0(M)$ and let Let $\{f_t\}$ be the isotopy between the identity and $f$.
The union of the supports $\bigcup_{t\in[0,1]}\OP{supp}(f_t)$ is a
compact subset of $M$. Recall that $M$ admits a complete Riemannian metric.
Hence there exists $r>0$ such that the geodesic ball $B_r(z)$ of
radius $r$ centered at $z$ contains $\bigcup_{t\in[0,1]}\OP{supp}(f_t)$.
Note that for each $x$ in the full measure subset of $M\setminus B_r(z)$ the element $\gamma'(f,x)$
is trivial in $\pi_M$. Hence it is enough to show that the
set $\{\gamma'(f,x)\}$ where $x$ belongs to the full measure
subset of $B_r(z)$ is finite in $\pi_1(B_r(z), z)$.
Here, since we have choosen the isotopy $\{f_t\}$, we can consider $\gamma'(f,x)$ as an element of $\pi_1(M,z)$.
The group $\OP{Homeo}_0(M)$ admits a fragmentation property with
respect to any open cover of $M$, see \cite[Corollary 1.3]{MR0283802}.
Hence the ball $B_r(z)$ can be covered with finite number of balls $B_i$ with the following property:
$f$ can be written as a product of homeomorphisms $h_i$
such that the support of $h_i$ lies in $B_i$. Since $M$ is a smooth manifold, for each $i$ there
exits a smooth ball $B'_i$, such that it is $\epsilon$-close to $B_i$ and such that
it is $\epsilon$-homotopic to $B_i$, see smooth approximation theorem \cite[Theorem 2.11.8]{MR1224675}.
Note that $\gamma'(f,x)$ satisfies a cocycle condition. It means that
$$\gamma'(f,x)=\gamma'(h_1\circ\ldots\circ h_n, x)=\gamma'(h_1,h_2\circ\ldots\circ h_n(x))\ldots\gamma'(h_n,x).$$
Hence it is enough to prove that the set $\{\gamma'(h_i,x)\}$ where $x$ belongs to the full
measure subset of $B_i$ is finite in $\pi_1(B_r(z), z)$.
The ball $B'_i$ is smooth, thus it has finite diameter $d_i$.
The group of homeomorphisms of a ball is connected.
Every path inside $B_i$ can be free $\epsilon$-homotoped
to a path in $B'_i$ and hence to a path whose Riemannian length is less than the diameter $d_i$.
Thus $\gamma'(h_i,x)$ can be represented by a path whose Riemannian length
is less than $d_i+2(r_i+\epsilon)$, where $r_i$ is a radius of a geodesic ball $B_{r_i}(z)$ which contains $B_i$.
By Milnor-Svarc lemma \cite{MR1744486} the word length of $\gamma'(h_i,x)$ is bounded in $\pi_1(B_r(z), z)$ and we are done.
\end{proof}
We have the following commutative diagram
$$
\begin{tikzcd}
\OP{H}^\bullet(\pi_M) \arrow[r, "\Gamma"] & \OP{H}^\bullet(\C T_{M})\\
\OP{H}_b^\bullet(\pi_M) \arrow[r, "\Gb"] \arrow[u] & \OP{H}_b^\bullet(\C T_M) \arrow[u]
\end{tikzcd}
$$
It follows that $\Gb$ restricts to the exact part of bounded cohomology.
$$
\EGb(\C T_M) \colon \OP{EH}_b^\bullet(\pi_M) \to \OP{EH}_b^\bullet(\C T_M).
$$
\begin{remark}\label{r:quasi}
$\OP{EH}_b^2(G)$ is the space of non-trivial homogeneous quasimorphisms on $G$ \cite[Chapter 2]{Calegari}, and
$\EGb(\C T_M) \colon \OP{EH}_b^2(\pi_M) \to \OP{EH}_b^2(\C T_M)$
is the map defined by Polterovich \cite{MR2276956}.
\end{remark}
\subsection{The reduced bounded cohomology}
It is straightforward to see, that $\Gb \colon \OP{H}_b^\bullet(\pi_1(M)) \to \OP{H}_b^\bullet(\C H_M)$ is a contraction map.
Hence it defines a well defined map on the reduced bounded and the reduced exact bounded cohomology.
$$\rGb(\C T_M) \colon \OP{\overbar{H}}_b^\bullet(\pi_1(M)) \to \OP{\overbar{H}}_b^\bullet(\C T_M),$$
$$\rEGb(\C T_M) \colon \OP{\overbar{EH}}_b^\bullet(\pi_M) \to \OP{\overbar{EH}}_b^\bullet(\C T_M).$$
\section{Proofs}\label{s:proofs}
Recall that $M$ is a complete Riemannian manifold of finite volume,
and $\C T_{M}$ is either $\OP{Homeo}_0(M,\mu)$ or $\OP{Diff}_0(M,\OP{vol})$.
If, in addition $M$ is a symplectic manifold, then $\C T_{M}$ could be also $\OP{Symp}_0(M,\omega)$.
Throughout this section we consider the map $\rEGb$ induced by $\gamma = \gamma_s$,
where $s$ is a section defined in Subsection~\ref{s:example}.
Before we proceed, let us give an outline of the proofs.
Assumptions of both Theorem \ref{t:surjects.on.F} and Theorem \ref{t:hyp.embed} imply
that there is an embedding $i \colon \OP{F}_2 \to \pi_M$ such that
in both cases $i^* \colon \rEH_b^\bullet(\pi_M) \to \rEH_b^\bullet(\OP{F}_2)$ is onto.
Indeed, in Theorem \ref{t:surjects.on.F} it is straightforward, and
in Theorem \ref{t:hyp.embed} we use the result in \cite{fps}, which in particular, implies that if $\OP{F}_2 \times K$
is hyperbolically embedded in $\pi_M$,
then one can extend a class in $\rEH_b^\bullet(\OP{F}_2 \times K)$ to $\rEH_b^\bullet(\pi_M)$.
That is why we require $\OP{F}_2 \times K$ to be hyperbolically embedded.
Given an element $c\in\rEH_b^\bullet(\pi_M)$, we look at the restriction $\rEGb(c)_{|\OP{F}_2}$ to a carefully embedded $\OP{F}_2\to\C T_M$.
The construction of the embedding is made such that there exists a non-zero real number $\Lambda$ so that $\Lambda i^*(c)$
and $\rEGb(c)_{|\OP{F}_2}$ are close in the norm. This
implies that $\rEGb(c)$ is non-trivial provided $i^*(c)$ is non-trivial.
Before showing the main technical lemma, we need to introduce some notations.
Let $i \colon~\OP{F}_2~\to~\pi_M$ be an embedding.
Denote by $a$ and $b$ the generators of $\OP{F}_2$.
A loop $\alpha$ in $M$ which is based at $z$, represents in a natural way an element in $\pi_M$.
If $\OP{dim}(M) = 2$ we assume that $i(a)$ and $i(b)$
can be represented by simple loops based at $z$.
In the next lemma we construct a family of maps $\rho_\epsilon \colon \OP{F}_2 \to \C T_M$ such that the diagram
$$
\begin{tikzcd}
\rEH^\bullet_b(\pi_M) \arrow[r, "\rEGb"] \arrow[d,"i^*"] &
\rEH^\bullet_{b}(\C T_{M}) \arrow[dl,"\rho_\epsilon^*"]\\
\rEH^\bullet_b(\OP{F}_2)
\end{tikzcd}
$$
is commutative up to scaling and a small error controlled by $\epsilon$.
\begin{lemma}\label{l:rep}
Assume that $M$, $\C T_{M}$ and $i \colon \OP{F}_2 \to \pi_M$ are as above.
Then there exists a family of representations $\rho_\epsilon \colon \OP{F}_2 \to \C T_{M}$, indexed by
$\epsilon \in (0,1)$, that satisfy the following property:
there exists a non-zero real number $\Lambda$, such that for every class $c \in \rEH_b^\bullet(\pi_M)$
$$\n\rho_\epsilon^*\rEGb (c)-\Lambda i^*(c)\n\xrightarrow{\epsilon \to 0} 0.$$
\end{lemma}
\begin{proof}
Let $m = \OP{dim}(M)$ and let $B^{m-1} \subset \mathbb{R}^{m-1}$ be the $m-1$ dimensional closed unit ball
and let $S^1 = \mathbb{R}/\mathbb{Z}$.
We fix $\epsilon \in (0,1)$ and define an isotopy $P^t_\epsilon \in \OP{Diff}(S^1 \times B^{m-1})$, $t \in [0,1]$
on $S^1\times B^{m-1}$ by the following formula
$$P^t_\epsilon(\psi,x)=(\psi+tf(\n x\n),x),$$
where $f$ is a smooth function such that
$f(t) = 1$ for $t \leq 1-\epsilon$ and $f(1) = 0$.
We call $P^t_\epsilon$ the finger-pushing isotopy and $P^1_\epsilon$ the finger-pushing map.
Note that $P^0_\epsilon = Id$ and that $P^1_\epsilon$ fixes point-wise the boundary of $S^1 \times B^{m-1}$
and fixes all points $(\psi,x)$ for which $\n x \n \leq 1-\epsilon$.
Let $g_0$ be the Riemannian metric which is the product of the standard euclidean Riemannian metrics on $B^{m-1}$ and $S^1$.
Then $g_0$ is a Riemannian metric on $S^1\times B^{m-1}$ that, by Fubini's theorem,
is preserved by the maps $P^t_\epsilon$ for every $t \in [0,1]$ and every $\epsilon \in (0,1)$.
In case $m=2k$, we construct in a similar way finger-pushing
isotopy $P^t_\epsilon \in \OP{Diff}(S^1\times B^1\times B^{2k-2})$, $t \in [0,1]$
which preserves the standard symplectic form
$dx\wedge dy+\sum_{i=1}^{k-1}dp_i\wedge dq_i$ on $S^1\times B^1\times B^{2k-2}$.
The precise construction is presented in \cite[proof of Theorem 1.3]{BK-frag}.
Recall that $a,b$ are generators of $\OP{F}_2$.
We represent $i(a)$ and $i(b)$ by loops $\alpha$ an $\beta$ in $M$ based at $z$
that are embedded and intersect only at $z$. Note that if $m=2$ we assumed that this is possible, and
if $m>2$ then any two elements of $\pi_M$ can be represented in this way.
Let $N(\alpha)$ be a closed tubular neighborhood of $\alpha$ and let $P^t_{\epsilon}(\alpha)$ be the isotopy
defined by pulling-back $P^t_{\epsilon}$ via
diffeomorphism $n_\alpha \colon N(\alpha) \to S^1 \times B^{m-1}$ and extending
by the identity outside $N(\alpha)$.
If $\C T_M = \OP{Homeo}_0(M,\mu)$ or $\OP{Diff}_0(M,\OP{vol})$, then by Moser trick $n_\alpha$ is chosen
such that $P^t_{\epsilon}(\alpha)$ preserves the $\OP{vol}$ form, and
if $\C T_M = \OP{Symp}_0(M,\omega)$, then by Moser trick $n_\alpha$ is chosen
such that $P^t_{\epsilon}(\alpha)$ preserves the symplectic form $\omega$.
Let
$$
A_{\epsilon}(\alpha) = n_\alpha^{-1}(\{(\psi,x)~|~\n x \n\leq 1-\epsilon\}),\qquad
B_{\epsilon}(\alpha) = N(\alpha)-A_{\epsilon}(\alpha).
$$
Note that $P^1_\epsilon(\alpha)$ fixes point-wise $A_{\epsilon}(\alpha)$.
Similarly we define $P^t_{\epsilon}(\beta)$, $A_{\epsilon}(\beta)$, $B_{\epsilon}(\beta)$.
The representation $\rho_{\epsilon} \colon \OP{F}_2 \to \C T_M$ is given by:
$$\rho_{\epsilon}(a) = P^1_{\epsilon}(\alpha),\qquad
\rho_{\epsilon}(b) = P^1_{\epsilon}(\beta).$$
Now we show that $\n\rho_\epsilon^*\rEGb ([c])-\Lambda i^*([c])\n\xrightarrow{\epsilon\to 0} 0$
for certain non-zero real number $\Lambda$ and every $[c]\in\rEH_b^\bullet(\pi_M)$.
To simplify the notation, we identify $\OP{F}_2$ with its image $i(\OP{F}_2)$.
First we investigate the values of $\gamma$ on elements of the form $\rho_{\epsilon}(w)$, where $w \in \OP{F}_2$.
Let $h_a \colon \OP{F}_2 \to \B \langle a \rangle$ be the
retraction onto the subgroup generated by $a$ that sends $b$ to the trivial element.
We define analogously $h_b \colon \OP{F}_2 \to \B \langle b \rangle$.
\begin{center}
\begin{tikzpicture}
\draw[pattern=north west lines, pattern color=gray!50!white, rounded corners, very thick, ] (-4,-2) rectangle (1,2);
\fill[white, rounded corners] (-3,-1) rectangle (0,1);
\draw[rounded corners, very thick] (-3,-1) rectangle (0,1);
\draw[pattern=north east lines, pattern color=gray!50!white, rounded corners, very thick] (0,-2) rectangle (5,2);
\fill[white, rounded corners] (1,-1) rectangle (4,1);
\draw[rounded corners, very thick] (1,-1) rectangle (4,1);
\draw (0.5,0) node {$\bullet$};
\draw (0.5,-0.2) node {$z$};
\draw (0.5,-1.5) node {$A_\epsilon$};
\draw (-3.4,-1.5) node {$A^a_\epsilon$};
\draw (4.4,-1.5) node {$A^b_\epsilon$};
\draw (-1.5,0) node {$\alpha$};
\draw (2.5,0) node {$\beta$};
\draw [->] (-1.10,-0.25) arc [radius=0.5, start angle=-30, end angle=240];
\draw [->] (2.90,-0.25) arc [radius=0.5, start angle=-30, end angle=240];
\draw (-4.5,1.8) node {$B_\epsilon$};
\draw (5.55,1.8) node {$B_\epsilon$};
\draw [->] (-4.5,2) arc [radius=0.3, start angle=150, end angle= 30];
\draw [<-] (5,2) arc [radius=0.3, start angle=150, end angle= 30];
\end{tikzpicture}
\end{center}
From the description of $\gamma$ in Subsection \ref{s:example},
we see that if $x$ belongs to the set $A_\epsilon\defeq A_{\epsilon}(\alpha)\cap A_{\epsilon}(\beta)$,
then $\gamma(\rho_{\epsilon}(w),x)$ is conjugated to $w$.
In the same way, if $x\in A^a_\epsilon \defeq A_{\epsilon}(\alpha) - N(\beta)$,
then $\gamma(\rho_{\epsilon}(w),x)$ is conjugated to $h_a(w)$ and
similarly, if $x\in A^b_\epsilon\defeq A_{\epsilon}(\beta) - N(\alpha)$,
then $\gamma(\rho_{\epsilon}(w),x)$ is conjugated to $h_b(w)$.
If $x\in B_\epsilon \defeq B_{\epsilon}(\alpha)\cup B_{\epsilon}(\beta)$,
then we do not have much control over the loops we get,
but this case is negligible if $\epsilon$ is small.
To sum up, we have:
$$
\gamma(\rho_{\epsilon}(w),x) =
\begin{cases}
e & x \in M-(N(\alpha) \cup N(\beta)),\\
u_xwu_x^{-1} & x \in A_{\epsilon} = A_{\epsilon}(\alpha) \cap A_{\epsilon}(\beta),\\
u_{a,x}h_a(w)u_{a,x}^{-1} & x \in A^a_\epsilon = A_{\epsilon}(\alpha) - N(\beta),\\
u_{b,x}h_b(w)u_{b,x}^{-1} & x \in A^b_\epsilon = A_{\epsilon}(\beta) - N(\alpha),\\
? & x \in B_\epsilon = B_{\epsilon}(\alpha) \cup B_{\epsilon}(\beta),
\end{cases}
$$
for some $u_x, u_{a,x}, u_{b,x} \in \pi_M$.
Let $n$ be a natural number and $[c] \in \rEH_b^n(\pi_M)$.
Without loss of generality, we can assume that $c(e,\ldots,e)~=~0$.
Denote by $\overbar{g}=(g_0,g_1,\ldots)\in\C T_{M}^n$.
We set $\gamma(\overbar{g},x)=(\gamma(g_0,x),\gamma(g_1,x),\ldots)$.
Let $\overbar{w} \in \OP{F}_2^n$. We have:
$$\rho_{\epsilon}^*\rEGb(c)(\overbar{w})=\rEGb(c)(\rho_\epsilon(\overbar{w}))=\int_{M} c(\gamma(\rho_{\epsilon}(\overbar{w}),x))d\mu(x).$$
Denote by $u.c(\overbar{w})=c(u\overbar{w}u^{-1})$, where $u\in\pi_M$.
We compute separately the integral on subsets
$A_{\epsilon}$, $A^a_{\epsilon}$, $A^b_{\epsilon}$ and $B_{\epsilon}$:
\begin{align*}\label{e:split}
\rho_{\epsilon}^* \rEGb (c)(\overbar{w}) =& \int_{A_\epsilon}u_x.c(\overbar{w})d\mu(x) +
\int_{A^a_\epsilon}u_{a,x}.c(h_a(\overbar{w}))d\mu(x)+\\
& \int_{A^b_\epsilon}u_{b,x}.c(h_b(\overbar{w}))d\mu(x)+
\int_{B_{\epsilon}}c(\gamma(\rho_{\epsilon}(\overbar{w}),x))d\mu(x).
\end{align*}
Recall that conjugation acts trivially on the cohomology. Thus we have $[u.c]=[c]$.
Both $\OP{Z}_b^n(G) = \OP{ker}(d_n)$
and $\rH_b^n(G)$ are Banach spaces and $[\cdot] \colon \OP{Z}_b^n(G) \to \rH_b^n(G)$ is a continuous linear map.
Thus we have:
\begin{align*}
[\int_{A_\epsilon}u_x.c(\overbar{w})d\mu(x)]&=[\sum_{u \in \pi_M}\mu(\{x \in A_\epsilon~|~u_x=u\})u.c(\overbar{w})]\\
&=\hspace{3px}\sum_{u \in \pi_M}\mu(\{x \in A_\epsilon~|~u_x=u\})i^*[u.c]=\mu(A_\epsilon)i^*([c]).
\end{align*}
Let $u.c_{|a}$ be the restriction of $u.c$ to the subgroup generated by the generator $a$.
The function $\overbar{w} \to c(uh_a(\overbar{w})u^{-1})$ is equal to the pull-back of the cocycle $u.c_{|a}$, namely:
$$c(uh_a(\overbar{w})u^{-1}) = h_a^*(u.c_{|a})(\overbar{w}).$$
Moreover, since $\rEH^n_b(\mathbb{Z})$ is trivial, then
$\overbar{w} \to c(uh_a(\overbar{w})u^{-1})$ defines the trivial class in $\rEH^n_b(\OP{F}_2)$.
It follows, that
$$
[\int_{A^a_\epsilon}u_{a,x}.c(h_a(\overbar{w}))d\mu(x)] =
\sum_{u \in \pi_M}\mu(\{x \in A_\epsilon~|~u_{a,x}=u\})h_a^*([u.c_{|a}]) = 0.
$$
The same holds for the integral over $A^b_\epsilon$. We denote
$$c^{\epsilon}_{res}(\overbar{w}) = \int_{B_{\epsilon}}c(\gamma(\rho_{\epsilon}(\overbar{w}),x))d\mu(x).$$
Note that $c^{\epsilon}_{res}$ is a cocycle on $\OP{F}_2$.
Now we can write:
$$
\rho^*_{\epsilon} \rEGb ([c]) = \mu(A_{\epsilon})i^*([c]) + [c^{\epsilon}_{res}].
$$
We have that
$$\n [c^{\epsilon}_{res}]\n \leq \mu(B_{\epsilon})\n c\n_{sup}.$$
Moreover, $\mu(A_{\epsilon}) \xrightarrow{\epsilon \to 0} \mu(N_a \cap N_b) \neq 0$ and
$\mu(B_{\epsilon})~\xrightarrow{\epsilon \to 0}~0$. It follows that:
$$
\n \rho^*_{\epsilon} \rEGb ([c])-\mu(N_a \cap N_b)i^*([c])\n \leq
[\mu(A_{\epsilon})-\mu(N_a \cap N_b)]\n i^*([c])\n +\mu(B_{\epsilon})\n c\n_{sup} .
$$
Hence $\n \rho^*_{\epsilon} \rEGb ([c])-\mu(N_a \cap N_b)i^*([c])\n \xrightarrow{\epsilon \to 0} 0.$
\end{proof}
\begin{remark}
In what follows we apply Lemma \ref{l:rep} for injective $i$.
However, Lemma \ref{l:rep} holds also for $i$ which is not an embedding.
Injectivity of $i$ was used only to simplify the notation,
when we identified $\OP{F}_2$ with its image $i(\OP{F}_2)$.
\end{remark}
Now we prove Theorem \ref{t:surjects.on.F} and Theorem \ref{t:hyp.embed}.
\begin{theorem}\label{t:surjects.on.F.strong}
Suppose that $p\colon\pi_M\to\OP{F}_2$ is a surjective homomorphism.
Then there is an injective homomorphism
$$\rEH_{b}^{\bullet}(\OP{F}_2) \hookrightarrow \rEH_{b}^{\bullet}(\C T_{M}).$$
\end{theorem}
\begin{proof}
Let $m = \OP{dim}(M)$.
If $m>3$, then we take $i\colon\OP{F}_2\to\pi_M$ to be any section of $p$.
If $m=2$, then it is easy to find two embedded loops based at $z$ and intersecting only at $z$, such that
they generate $\OP{F}_2$ and there is a retraction $\pi_1(M)\to\OP{F}_2$.
Instead of the original $p$, we use this retraction which from now on we call $p$.
Let $i$ be a section of this new $p$. We show that $\rEGb\circ p^*$ is an embedding.
The section $i$ satisfies the assumptions of Lemma \ref{l:rep}.
Let $\{\rho_\epsilon\}$ be the family of representations from Lemma \ref{l:rep}.
We have a diagram
$$
\begin{tikzcd}
\rEH^\bullet_b(\pi_M) \arrow[r, "\rEGb"] \arrow[d,shift left=.75ex, "i^*"]&
\rEH^\bullet_{b}(\C T_{M}) \arrow[ld,"\rho_\epsilon^*"]\\
\rEH^\bullet_b(\OP{F}_2) \arrow[u,shift left=.75ex,"p^*"]
\end{tikzcd}
$$
Note that $i^*\circ p^*=id$.
Suppose that $d~\in~\rEH^\bullet_b(\OP{F}_2)$ is a non-trivial class.
In the reduced cohomology it means that $\n d\n>~0$.
Let $c=p^*(d)$. Then we have that $\n i^*(c)\n=\n d\n>0$.
Since
$$\n\rho_\epsilon^*\rEGb(c)-\Lambda i^*(c)\n\xrightarrow{\epsilon\to 0} 0,$$
then for some small $\epsilon$ we have $\n\rho_\epsilon^*\rEGb(c)\n>0$.
It follows that
$$\rEGb (c)=\rEGb(p^*(d))\neq 0.$$
Thus $\rEGb\circ p^*$ is an embedding.
\end{proof}
\begin{remark}\label{r:int.volume}
A typical class in $\rEH_b^3(\OP{F}_2)$ is defined by choosing an isometric action $\rho$
of $\OP{F_2}$ on the $3$-dimensional hyperbolic space $\B H^3$
and defining a cocycle $\OP{vol}_{\rho}(a_1,\ldots,a_4)$ to be the
signed volume of the geodesic simplex $\Delta(\rho(a_1)x,\ldots,\rho(a_4)x)$,
where $x \in \B H^3$. For certain $\rho$, classes
defined by $\OP{vol}_{\rho}$ have a positive norm, see \cite{soma}.
The classes in $\rEH_b^3(\C T_M)$ which are constructed in
Theorem \ref{t:surjects.on.F.strong} have similar geometrical interpretation.
Namely, the value of $\rEGb(p^*(\OP{vol}_{\rho}))(f_1,\ldots,f_4)$ is the average value of
signed volumes of $\Delta(p\gamma(f_1,x),\ldots,p\gamma(f_4,x))$ over $M$.
Note that since every $\gamma(f_i,x)$ takes essentially
finitely many values, then this average is actually a finite sum
of weighted signed volumes of certain simplices in $\B H^3$.
Here the signed volume equals $\OP{vol}(\Delta)$ if $\Delta$
has the same orientation as $\B H^3$ and $-\OP{vol}(\Delta)$ otherwise.
\end{remark}
\begin{theorem}
Suppose $j \colon \OP{F}_2 \times K \to \pi_M$ is a hyperbolic embedding, where $K$ is finite.
Then
$$
\OP{dim} \rEH_{b}^\bullet(\C T_{M}) \geq \OP{dim} \rEH_{b}^\bullet(\OP{F}_2).
$$
\end{theorem}
\begin{proof}
It is easy to see, that if $\OP{F}_2$ embeds in $\pi_M$ when $\OP{dim}(M)=2$, then one can find a retraction
$\pi_M \to \OP{F}_2$. Thus if $\OP{dim}(M)=2$, the statement follows from Theorem \ref{t:surjects.on.F.strong}.
Let us assume, that $\OP{dim}(M)>2$.
Let $i\colon\OP{F_2}\to\pi_M$ be $j$ restricted to $\OP{F}_2\times\{e\}$.
Since $\OP{dim}(M)>2$, $i(a)$ and $i(b)$
can be represented by based loops that are disjoint everywhere except the base-point $z$.
Let $\{\rho_\epsilon\}_\epsilon$ be a family of maps
$\rho_\epsilon\colon\OP{F}_2\to\C T_{M}$ constructed in Lemma \ref{l:rep}.
We have
$$
\begin{tikzcd}
\rEH^\bullet_b(\pi_M) \arrow[r, "\rEGb"] \arrow[d,"i^*"] &
\rEH^\bullet_{b}(\C T_{M}) \arrow[ld,"\rho_\epsilon^*"]\\
\rEH^\bullet(\OP{F}_2)
\end{tikzcd}
$$
First we show that $\OP{ker}(\rEGb)\hspace{-2.5px}\subset\hspace{-1.5px}\OP{ker}(i^*)$.
There is a non-zero real number $\Lambda$ such that for every $c\in\rEH_b^\bullet(\pi_1(M))$
we have
$$\n \rho_\epsilon^*\rEGb (c)-\Lambda i^*(c)\n\xrightarrow{\epsilon \to 0} 0.$$
Let $c\in\rEH_b^\bullet(\pi_1(M))$ be such that $\rEGb(c)=0$. Then
$$\n\Lambda i^*(c)\n=\n \rho_\epsilon^* \rEGb (c)-\Lambda i^*(c)\n\xrightarrow{\epsilon \to 0} 0.$$
Hence $\n i^*(c)\n=0$ and $c\in\OP{ker}(i^*)$.
Thus $\OP{ker}(\rEGb)\hspace{-2.5px}\subset\hspace{-1.5px}\OP{ker}(i^*)$.
It follows that
$$\OP{dim} \rEH_{b}^\bullet(\C T_{M}) \geq \OP{dim}(\rEH_{b}^\bullet(\pi_1(M))/\OP{ker}(\rEGb))
\geq \OP{dim} (\rEH_{b}^\bullet(\pi_1(M))/\OP{ker}(i^*)).$$
In \cite{fps} it is shown, that if $j\colon\OP{F}_2\times K\to\pi_M$
is a hyperbolic embedding, then the map
$j^*\colon\rEH_b^\bullet(\pi_M)\to\rEH_b^\bullet(\OP{F}_2 \times K)$ is onto.
Using identification $\rEH_b^\bullet(\OP{F}_2 \times K)=\rEH_b^\bullet(\OP{F}_2)$,
we can write that $i^*=j^*$. Thus $i^*$ is onto and
$\rEH_{b}^\bullet(\pi_1(M))/\OP{ker}(i^*)=\rEH_{b}^\bullet(\OP{F}_2)$.
\end{proof}
\section {Questions and final remarks}
\begin{remark}\label{r:general}
The version of Theorem \ref{t:surjects.on.F} and Theorem \ref{t:hyp.embed} where $\rEH_b^\bullet$
is substituted with $\rH_b^\bullet$ holds in a more general setting.
Namely, it holds for a topological manifold $M$ equipped with
a regular finite Borel measure $\mu$ which is positive
on open sets and zero on nowhere dense sets,
and $\C T_M$ is the group of isotopic to the identity, not necessary compactly supported
measure-preserving homeomorphisms of $M$.
Similarly, $\C T_M$ can be the identity component of the group of volume-preserving diffeomorphisms
or symplectomorphisms.\end{remark}
\begin{remark}\label{r:extend}
Note that $\EGb \colon \OP{EH}_b^\bullet(\pi_1(M))\to\OP{EH}_b^\bullet(\OP{Diff}_0(M,\OP{vol}))$
does not factor through $\OP{Diff}_0(M)$
(similar argument works for homeomorphisms).
The reason is that if it is factored, then one can
construct non-trivial homogeneous quasimorphisms on $\OP{Diff}_0(M)$,
and it is known, that for many $M$, the group $\OP{Diff}_0(M)$ does not admit such quasimorphisms.
More precisely, let $M$ be such manifold,
for example a closed connected hyperbolic $3$-dimensional manifold.
Recall that $\OP{EH}_b^2(G)$ is the space of non-trivial homogeneous quasimorphisms on $G$.
It follows from \cite[Theorem 1.11]{MR2509711} that $\OP{EH}_b^2(\OP{Diff}_0(M)) = 0$.
Since $\OP{EH}_b^2(\pi_1(M))\neq 0$, it is easy to see, that
$\EGb \colon \OP{EH}_b^2(\pi_1(M))\to\OP{EH}_b^2(\OP{Diff}_0(M,\mu))$
is an embedding (see ,e.g., the proof for surfaces \cite[Theorem 2.5]{eq}),
and hence cannot factor through the trivial group.
\end{remark}
\begin{remark}
The assumption that homeomorphisms we deal with are isotopic to the identity can be dropped
for the price of substituting the group $\pi_1(M)$ with a mapping class group (such approach was used in \cite{eq} for surfaces).
Indeed, let $\delta^{ext} \colon \OP{Homeo}(M,z) \to \OP{MCG}(M,z)$ be the quotient map, where
$\OP{MCG}(M,z) = \pi_0(\OP{Homeo}(M,z))$ and $\OP{Homeo}(M,z)$
is the group of homeomorphisms of $M$ fixing $z$.
Consider a cocycle
$\gamma^{ext} \colon \OP{Homeo}(M,\mu) \times M \to \OP{MCG}(M,z)$,
given by the following formula $\gamma^{ext}(g,x) = \delta^{ext}(s_{g(x)}^{-1} \circ g \circ s_x)$.
This cocycle induces the following map
$\Gb^{ext} = \OP{Ind}_b(\gamma^{ext}) \colon \OP{H}_b^\bullet(\OP{MCG}(M,z)) \to \OP{H}_b^\bullet(\OP{Homeo}(M,\mu))$.
The problem with this approach is that almost nothing is known about
bounded cohomology of mapping class groups of manifolds of dimension greater than $2$.
\end{remark}
One source of non-trivial classes in $\OP{H}_b^n(\pi_1(M))$ is the following construction.
Let $M^n$~be a compact Riemannian manifold with negative sectional
curvature and let $\pi_1(M)$ act by deck-transformations on $\wt{M}$.
It is known, that there is a common bound for volumes of geodesic simplices in $\wt{M}$.
In the same spirit as in Remark \ref{r:int.volume}, one can define a non-trivial
class $[\OP{vol}_M]\in\OP{H}_b^n(\pi_1(M))$.
\begin{question}
Is the class $\Gb([\OP{vol}_M])\in \OP{H}_b^n(\OP{Homeo}_0(M,\mu))$ non-trivial?
\end{question}
A similiar, but in a sense more general question is for the ordinary cohomology.
Let $M^n$ be a closed Riemannian manifold such that $\pi_1(M)$ has a trivial center.
Then $\pi_1(M)=\pi_M$ and we have a map $\Gamma\colon\OP{H}^n(\pi_1(M))\to\OP{H}^n(\OP{Homeo}_0(M,\mu))$.
Let $f\colon M\to B\pi_1(M)$ be a map classifying the universal cover, i.e.,
a map that induces an isomorphism $\pi_1(f) \colon \pi_1(M)\to\pi_1(B\pi_1(M))$.
Let $[M]\in\OP{H}^n(M)$ be the fundamental class, then $f^*[M]\in\OP{H}^n(B\pi_1(M))=\OP{H}^n(\pi_1(M))$.
\begin{question}
Is the class $\Gamma(f^*[M])\in\OP{H}^n(\OP{Homeo}_0(M,\mu))$ non-trivial?
\end{question}
|
{
"timestamp": "2019-03-01T02:17:54",
"yymm": "1902",
"arxiv_id": "1902.11067",
"language": "en",
"url": "https://arxiv.org/abs/1902.11067"
}
|
\section{Introductory Facts}
Let us recall some notations and terminologies used in this paper. An ordered vector space $E$ is said to be {\em vector lattice} (or, {\em Riesz space}) if, for each pair of vectors $x,y\in E$, the supremum $x\vee y=\sup\{x,y\}$ and the infimum $x\wedge y=\inf\{x,y\}$ both exist in E. For $x\in E$, $x^+:=x\vee 0$, $x^-:=(-x)\vee0$, and $\lvert x\rvert:=x\vee(-x)$ are called the {\em positive} part, the {\em negative} part, and the {\em absolute value} of $x$, respectively. A vector lattice $E$ is called \textit{order complete} if $0\leq x_\alpha\uparrow\leq x$ implies the existence of $\sup{x_\alpha}$. A partially ordered set $A$ is called {\em directed} if, for each $a_1,a_2\in A$, there is another $a\in A$ such that $a\geq a_1$ and $a\geq a_2$ (or, $a\leq a_1$ and $a\leq a_2$). A function from a directed set $A$ into a set $E$ is called a {\em net} in $E$. A net $(x_\alpha)_{\alpha\in A}$ in a vector lattice $E$ is\textit{ order convergent} (or \textit{$o$-convergent}, for short) to $x\in E$, if there exists another net $(y_\beta)_{\beta\in B}$ satisfying $y_\beta \downarrow 0$, and for any $\beta\in B$ there exists $\alpha_\beta\in A$ such that $|x_\alpha-x|\leq y_\beta$ for all $\alpha\geq\alpha_\beta$. In this case, we write $x_\alpha\xrightarrow{o} x$. A vector $e\geq 0$ in a vector lattice $E$ is said to be a \textit{weak order unit} whenever the band generated by $e$ satisfies $B_e=E$, or equivalently, whenever for each $x\in E_+$ we have $x\wedge ne\uparrow x$; see more information for example (\cite{AB}, \cite{ABPO},\cite{AAyd},\cite{AGG}, \cite{Vul}, \cite{Za}).
A vector lattice $E$ under an associative multiplication is said to be a \textit{Riesz algebra} whenever the multiplication makes $E$ an algebra (with the usual properties), and in addition, it satisfies the following property: $xy\in E_+$ for every $x,y\in E_+$. A Riesz algebra $E$ is called \textit{commutative} if $xy=yx$ for all $x,y\in E$. A Riesz algebra $E$ is called \textit{$f$-algebra} if $E$ has additionally property that $x\wedge y=0$ implies $(xz)\wedge y=(zx)\wedge y=0$ for all $z\in E_+$. A vector lattice $E$ is called \textit{Archimedean} whenever $\frac{1}{n}x\downarrow 0$ holds in $E$ for each $x\in E_+$. Every Archimedean $f$-algebra is commutative; see Theorem 140.10 \cite{Za}. Assume $E$ is an Archimedean $f$-algebra with a multiplicative unit vector $e$. Then, by applying Theorem 142.1(v) \cite{Za}, in view of $e=ee=e^2\geq0$, it can be seen that $e$ is a positive vector. On the other hand, since $e\wedge x=0$ implies $x=x\wedge x= (xe)\wedge x= 0$, it follows that $e$ is a weak order unit. In this article, unless otherwise, all vector lattices are assumed to be real and Archimedean, and so $f$-algebras are commutative.
Recall that a net $(x_\alpha)$ in an $f$-algebra $E$ is called {\em multiplicative order convergent} (or shortly, {\em $mo$-convergent}) to $x\in E$ if $\lvert x_\alpha-x\rvert u\oc 0$ for all $u\in E_+$. Also, it is called {\em $mo$-Cauchy} if the net $(x_\alpha-x_{\alpha'})_{(\alpha,\alpha') \in A\times A}$ $mo$-converges to zero. $E$ is called {\em $mo$-complete} if every $mo$-Cauchy net in $E$ is $mo$-convergent, and is called {\em $mo$-continuous} if $x_\alpha\oc0$ implies $x_\alpha\fc 0$; see for detail information \cite{AAydn}. On the other hand, a net $(x_\alpha)$ in a Banach lattice $E$ is {\em unbounded norm convergent} (or {\em un-convergent}) to $x\in E$ if $\lvert \lvert x_\alpha-x\rvert \wedge u\rVert\to0$ for all $u\in E_+$; see \cite{Try}. We routinely use the following fact: $y\leq x$ implies $uy\leq ux$ for all positive elements $u$ in $f$-algebras. Moreover, an $f$-algebra $E$ which is at the same time a Banach lattice is called a {\em Banach lattice $f$-algebra} whenever $\lvert xy\rVert\leq \Vert x\rVert\Vert y\rVert$ holds for all $x,y\in E$. Motivated from above definitions, we give the following notion.
\begin{definition}
A net $(x_\alpha)$ in a Banach lattice $f$-algebra $E$ is said to be {\em multiplicative norm convergent} (or shortly, {\em $mn$-convergent}) to $x\in E$ if $\lvert \lvert x_\alpha-x\rvert u\rVert\to 0$ for all $u\in E_+$. Abbreviated as $x_\alpha\mc x$. If the condition holds only for sequences then it is called sequentially $mn$-convergence.
\end{definition}
\begin{remark}\ \label{rem 1 basic}
\begin{enumerate}
\item[(i)] For a net $(x_\alpha)$ in a Banach lattice $f$-algebra $E$, $x_\alpha\mc x$ implies $x_\alpha y\mc xy$ for all $y\in E$ because of $\lVert\lvert x_\alpha y-xy\rvert u\rVert=\lVert \lvert x_\alpha-x\rvert\lvert y\rvert u\rVert$ for all $u\in E_+$. The converse holds true in Banach lattice $f$-algebras with the multiplication unit. Indeed, assume $x_\alpha y\mc xy$ for each $y\in E$. Fix $u\in E_+$. So, $\lVert \lvert x_\alpha-x\rvert u\rVert=\lVert \lvert x_\alpha e-xe\rvert u\rVert\mc 0$.
\item[(ii)] In Banach lattice $f$-algebras, the norm convergence implies the $mn$-convergence. Indeed, by considering the inequality $\lVert \lvert x_\alpha-x\rvert u\rVert\leq \lVert x_\alpha-x\rVert\lVert u\rVert$ for any net $x_\alpha\mc x$, we can get the desired result.
\item[(iii)] If a net $(x_\alpha)$ is order Cauchy and $x_\alpha\mc x$ in a Banach lattice $f$-algebra then we have $x_\alpha\fc x$. Indeed, since order Cauchy norm convergent net is order convergent to its norm limit, we can get the desired result.
\item[(iv)] In order continuous Banach lattice $f$-algebras, both the (sequentially) order convergence and the (sequentially) $mo$-convergence imply the (sequentially) $mn$-convergence.
\item[(v)] In atomic and order continuous Banach lattice $f$-algebras, a sequence which is order bounded and $mn$-convergent to zero is sequentially $mo$-convergent to zero; see Lemma 5.1 \cite{DOT}.
\item[(vi)] For an $mn$-convergent to zero sequence $(x_n)$ in a Banach lattice $f$-algebra, there is a subsequence $(x_{n_k})$ which sequentially $mo$-converges to zero; see Lemma 3.11 \cite{GX}.
\end{enumerate}
\end{remark}
\begin{example}\label{int exam}
Let $E$ be a Banach lattice. Fix an element $x\in E$. Then the principal ideal $I_x=\{y\in E: \exists \lambda>0 \ \text{with} \ \lvert y\rvert\leq \lambda x\}$, generated by $x$ in $E$ under the norm $\lVert\cdot\rVert_\infty$ which is defined by $\lVert y\rVert_\infty=\inf\{\lambda>0:\lvert y\rvert\leq \lambda x\}$, is an $AM$-space; see Theorem 4.21 \cite{ABPO}.
Recall that a vector $e>0$ is called order unit whenever for each $x$ there exists some $\lambda>0$ with $\lvert x\rvert\leq \lambda e$. Thus, we have $(I_x,\lVert\cdot\rVert_\infty)$ is $AM$-space with the unit $\lvert x\rvert$. Since every $AM$-space with unit, besides being a Banach lattice, has also an $f$-algebra structure. So, we can say that $(I_x,\lVert\cdot\rVert_\infty)$ is a Banach lattice $f$-algebra. Therefore, for a net $(x_\alpha)$ in $I_x$ and $y\in I_x$, by applying Corollary 4.4 \cite{ABPO}, we get $x_\alpha \mc y$ in the original norm of $E$ on $I_x$ iff $x_\alpha\mc y$ in the norm $\lVert\cdot\rVert_\infty$. In particular, take $x$ as the unit element $e$ of $E$. Then we have $E_e=E$. Thus, for a net $(x_\alpha)$ in $E$, we have $x_\alpha \mc y$ in the $(E,\lVert\cdot\rVert_\infty)$ iff $x_\alpha \mc y$ in the $(E,\lVert\cdot\rVert)$.
\end{example}
\section{Main Results}
We begin the section with the next list of properties of $mn$-convergence which follows directly from the inequalities $\lvert x-y\rvert \leq \lvert x-x_\alpha\rvert +\lvert x_\alpha-y\rvert $ and $\lvert \lvert x_\alpha\rvert -\lvert x\rvert \rvert \leq\lvert x_\alpha-x\rvert $.
\begin{lemma}
Let $(x_\alpha)$ and $(y_\alpha)$ be two nets in a Banach lattice $f$-algebra $E$. Then the following holds true:
\begin{enumerate}
\item[(i)] $x_\alpha\mc x$ iff $(x_\alpha- x)\mc 0$ iff $\lvert x_\alpha-x\rvert \mc 0$;
\item[(ii)] if $x_\alpha\mc x$ then $y_\beta\mc x$ for each subnet $(y_\beta)$ of $(x_\alpha)$;
\item[(iii)] suppose $x_\alpha\mc x$ and $y_\beta\mc y$, then $ax_\alpha+by_\beta\mc ax+by$ for any $a,b\in \mathbb{R}$;
\item[(iv)] if $x_\alpha \mc x$ and $x_\alpha \mc y$ then $x=y$;
\item[(v)] if $x_\alpha \mc x$ then $\lvert x_\alpha\rvert \mc \lvert x \rvert$.
\end{enumerate}
\end{lemma}
The lattice operations in Banach lattice $f$-algebras are $mn$-continuous in the following sense.
\begin{proposition}\label{LO are $mn$-continuous}
Let $(x_\alpha)_{\alpha \in A}$ and $(y_\beta)_{\beta \in B}$ be two nets in a Banach lattice $f$-algebra $E$. If $x_\alpha\mc x$ and $y_\beta\mc y$ then $(x_\alpha\vee y_\beta)_{(\alpha,\beta)\in A\times B} \mc x\vee y$. In particular, $x_\alpha^+\mc x^+$.
\end{proposition}
\begin{proof}
Assume $x_\alpha\mc x$ and $y_\beta\mc y$. Then, for a given $\varepsilon>0$, there exist indexes $\alpha_0\in A$ and $\beta_0\in B$ such that $\lVert\lvert x_\alpha-x\rvert u\rVert\leq \frac{1}{2}\varepsilon$ and $\lVert\lvert y_\beta-y\rvert u\rVert\leq \frac{1}{2}\varepsilon$ for every $u\in E_+$ and for all $\alpha\geq\alpha_0$ and $\beta\geq\beta_0$. It follows from the inequality $\vert a\vee b-a\vee c\rvert\leq \lvert b-c\rvert$ in vector lattices that
\begin{equation*}
\begin{split}
\lVert\lvert x_\alpha \vee y_\beta - x\vee y\rvert u\rVert&\leq \lVert\lvert x_\alpha \vee y_\beta -x_\alpha \vee y\rvert u+\lvert x_\alpha \vee y- x\vee y\rvert u\rVert\\ &\leq \lVert\lvert y_\beta -y\rvert u\rVert+\lVert\lvert x_\alpha-x\rvert u\rVert\leq \frac{1}{2}\varepsilon+\frac{1}{2}\varepsilon=\varepsilon
\end{split}
\end{equation*}
for all $\alpha\geq\alpha_0$ and $\beta\geq\beta_0$ and for every $u\in E_+$. That is, $(x_\alpha\vee y_\beta)_{(\alpha,\beta)\in A\times B} \mc x\vee y$.
\end{proof}
The following proposition is similar to Proposition 2.7 \cite{AAydn}, and so we omit its proof.
\begin{proposition}
Let $E$ be a Banach lattice $f$-algebra, $B$ be a projection band of $E$ and $P_B$ be the corresponding band projection. Then $x_\alpha\mc x$ in $E$ implies $P_B(x_\alpha)\mc P_B(x)$ in both $E$ and $B$.
\end{proposition}
A positive vector $e$ in a normed vector lattice $E$ is called {\em quasi-interior point} iff $\lVert x-x\wedge ne\rVert\to 0$ for each $x\in E_+$. If $(x_\alpha)$ is a net in a vector lattice with a weak unit $e$ then $x_\alpha \uoc 0$ iff $\lvert x_\alpha\rvert\wedge e\oc 0$; see Lemma 3.5 \cite{GTX}, and for quasi-interior point case; see Lemma 2.11 \cite{DOT}. Analogously, we show the next result.
\begin{proposition}\label{quasi-interior point}
Let $0\leq(x_\alpha)_{\alpha\in A}\downarrow$ be a net in a Banach lattice $f$-algebra $E$ with a quasi-interior point $e$. Then $x_\alpha\mc 0$ iff $(x_\alpha e)$ norm converges to zero.
\end{proposition}
\begin{proof}
The forward implication is immediate because of $e\in E_+$. For the converse implication, fix a positive vector $u\in E_+$ and $\varepsilon>0$. Thus, for a fixed index $\alpha_1$, we have $x_\alpha\leq x_{\alpha_1}$ for all $\alpha\geq \alpha_0$. Then
$$
x_\alpha u\leq x_\alpha(u-u\wedge ne)+x_\alpha(u\wedge ne)\leq x_{\alpha_1}(u-u\wedge ne)+nx_\alpha e
$$
for all $\alpha\geq\alpha_1$ and each $n\in \mathbb{N}$. Hence, we get
$$
\lVert x_\alpha u\rVert\leq \lVert x_{\alpha_1}\rVert\lVert u-u\wedge ne\rVert+n\lVert x_\alpha e\rVert
$$
for every $\alpha\geq\alpha_1$ and each $n\in \mathbb{N}$. So, we can find $n$ such that $\lVert u-u\wedge ne\rVert<\frac{\varepsilon}{2\lVert x_{\alpha_1}\rVert}$ because $e$ is a quasi-interior point. On the other hand, it follows from $x_\alpha e\nc 0$ that there exists an index $\alpha_2$ such that $\lVert x_\alpha e\rVert<\frac{\varepsilon}{2n}$ whenever $\alpha\geq\alpha_2$. Since index set $A$ is directed set, there exists another index $\alpha_0\in A$ such that $\alpha_0\geq \alpha_1$ and $\alpha_0\geq \alpha_2$. Therefore, we get $\lVert x_\alpha u\rVert<\lVert x_{\alpha_0}\rVert\frac{\varepsilon}{2\lVert x_{\alpha_0}\rVert}+n\frac{\varepsilon}{2n}=\varepsilon$, and so $\lVert x_\alpha u\rVert\to 0$.
\end{proof}
\begin{remark}
A positive and decreasing net $(x_\alpha)$ in an order continuous Banach lattice $f$-algebra $E$ with weak unit $e$ is $mn$-convergent to zero iff $x_\alpha e\nc 0$. Indeed, it is known that $e$ is a weak unit iff $e$ is a quasi-interior point in an order continuous Banach lattice. Thus, by applying Proposition \ref{quasi-interior point}, we can get the desired result.
\end{remark}
\begin{proposition}\label{mn convergence positive}
Let $(x_\alpha)$ be a net in a Banach lattice $f$-algebra $E$. Then we have that
\begin{enumerate}
\item[(i)] $0\leq x_\alpha\mc x$ implies $x\in E_+$,
\item[(ii)] if $(x_\alpha)$ is monotone and $x_\alpha\mc x$ then $x_\alpha\oc x$.
\end{enumerate}
\end{proposition}
\begin{proof}
$(i)$ Assume $(x_\alpha)$ consists of non-zero elements and $mn$-converges to $x\in E$. Then, by Proposition \ref{LO are $mn$-continuous}, we have $x_\alpha=x_\alpha^+\mc x^+=0$. Therefore, we get $x\in E_+$.
$(ii)$ For order convergence of $(x_\alpha)$, it is enough to show that $x_\alpha\uparrow$ and $x_\alpha\mc x$ implies $x_\alpha\oc x$. For a fixed index $\alpha$, we have $x_\beta-x_\alpha\in X_+$ for $\beta\ge\alpha$. By applying $(i)$, we can see $x_\beta-x_\alpha\mc x-x_\alpha\in X_+$ as $\beta \to \infty$. Therefore, $x\geq x_\alpha$ for the index $\alpha$. Since $\alpha$ is arbitrary, $x$ is an upper bound of $(x_\alpha)$. Assume $y$ is another upper bound of $(x_\alpha)$, i.e., $y\geq x_\alpha$ for all $\alpha$. So, $y-x_\alpha\mc y-x\in X_+$, or $y\ge x$, and so $x_\alpha \uparrow x$.
\end{proof}
The $mn$-convergence passes obviously to any Banach lattice sub-$f$-algebra $Y$ of a Banach lattice $f$-algebra $E$, i.e., for any net $(y_\alpha)$ in $Y$, $y_\alpha\mc0$ in $E$ implies $y_\alpha\mc0$ in $Y$. For the converse, we give the following theorem which is similar to Theorem 2.10 \cite{AAydn}.
\begin{theorem}\label{$up$-regular}
Let $Y$ be a Banach lattice sub-$f$-algebra of an Banach lattice $f$-algebra $E$ and $(y_\alpha)$ be a net in $Y$. If $y_\alpha\mc 0$ in $Y$ then it $mn$-converges to zero in $E$ for both of the following cases hold;
\begin{enumerate}
\item[(i)] $Y$ is majorizing in $E$;
\item[(ii)] $Y$ is a projection band in $E$;
\end{enumerate}
\end{theorem}
It is known that every Archimedean vector lattice has a unique order completion; see Theorem 2.24 \cite{ABPO}.
\begin{theorem}
Let $E$ and $E^\delta$ be Banach lattice $f$-algebras with $E^\delta$ being order completion of $E$. Then, for a sequence $(x_n)$ in $E$, the followings hold true:
\begin{enumerate}
\item[(i)] If $x_n\mc 0$ in $E$ then there is a subsequence $(x_{n_k})$ of $(x_n)$ such that $x_{n_k}\fc 0$ in $E^\delta$;
\item[(ii)] If $x_n\mc 0$ in $E^\delta$ then there is a subsequence $(x_{n_k})$ of $(x_n)$ such that $x_{n_k}\fc 0$ in $E$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $x_n\mc 0$ in $E$, i.e., $\lvert x_n\rvert u\nc 0$ in $E$ for all $u\in E_+$. Now, let's fix $v\in E^\delta_+$. Then there exists $u_v\in E_+$ such that $v\leq u_v$ because $E$ majorizes $E^\delta$. Since $\lvert x_n\rvert u_v\nc 0$, by the standard fact in Exercise 13 \cite[p.25]{AB}, there exists a subsequence $(x_{n_{k}})$ of $(x_n)$ such that $(\lvert x_{n_{k}}\rvert u_v)$ order converges to zero in $E$. Thus, $\lvert x_{n_k}\rvert u_v\oc 0$ in $E^\delta$; see Corollary 2.9 \cite{GTX}. Then it follows from the inequality $\lvert x_{n_k}\rvert v\leq \lvert x_{n_k}\rvert u_v$ that we have $\lvert x_{n_k}\rvert v\oc 0$ in $E^\delta$. That is, $x_{n_k}\fc 0$ in the order completion $E^\delta$ because $v\in E^\delta_+$ is arbitrary.\\
For the converse, put $x_n\mc 0$ in $E^\delta$. Then, for all $u\in E^\delta_+$, we have $\lvert x_n\rvert u\nc 0$ in $E^\delta$. In particular, for all $w\in E_+$, $\lVert\lvert x_n\rvert w\rVert\to 0$ in $E^\delta$. Fix $w\in E_+$. Then, again by the standard fact in Exercise 13 \cite[p.25]{AB}, we have a subsequence $(x_{n_{k}})$ of $(x_n)$ such that $(x_{n_{k}})$ is order convergent to zero in $E^\delta$. Thus, by Corollary 2.9 \cite{GTX}, we see $\lvert x_{n_k}\rvert w\oc 0$ in $E$. As a result, since $w$ is arbitrary, $x_{n_k}\fc 0$ in $E$.
\end{proof}
Recall that a subset $A$ in a normed lattice $(E,\lvert \cdot\rVert)$ is said to be almost order bounded if, for any $\epsilon>0$, there is $u_\epsilon\in E_+$ such that $\lvert (|x|-u_\epsilon)^+\rVert=\lvert |x|-u_\epsilon\wedge|x|\rVert\leq\epsilon$ for any $x\in A$. For a given Banach lattice $f$-algebra $E$, one can give the following definition: a subset $A$ of $E$ is called a {\em $f$-almost order bounded} if, for any $\epsilon>0$, there is $u_\varepsilon\in E_+$ such that $\lVert|x|-u_\epsilon|x|\rVert\leq\epsilon$ for any $x\in A$. Similar to Proposition 3.7 \cite{GX}, we give the following work.
\begin{proposition}\label{$f$-almost}
Let $E$ be a Banach lattice $f$-algebra. If $(x_\alpha)$ is $f$-almost order bounded and $mn$-converges to $x$, then $(x_\alpha)$ converges to $x$ in norm.
\end{proposition}
\begin{proof}
Assume $(x_\alpha)$ is $f$-almost order bounded net. Then the net $(|x_\alpha-x|)$ is also $f$-almost order bounded. For any fixed $\varepsilon>0$, there exists $u_\varepsilon>0$ such that
$$
\lvert \lvert x_\alpha-x\rvert-u_\epsilon\lvert x_\alpha-x\rvert\rVert\leq\epsilon.
$$
Since $x_\alpha\mc x$, we have $\lvert \lvert x_\alpha-x\rvert u_\varepsilon\rVert\to 0$. Therefore, we get $\lvert x_\alpha-x\rVert\leq\varepsilon$, i.e., $x_\alpha\to x$ in norm.
\end{proof}
\begin{proposition}
In an order continuous Banach lattice $f$-algebra, every $f$-almost order bounded $mo$-Cauchy net converges $mn$- and in norm to the same limit.
\end{proposition}
\begin{proof}
Assume a net $(x_\alpha)$ is $f$-almost order bounded and $mo$-Cauchy. Then the net $(x_\alpha-x_{\alpha'})$ is $f$-almost order bounded and is $mo$-convergent to zero. Thus, it $mn$-converges to zero by order continuity. Hence, by applying Proposition \ref{$f$-almost}, we get the net $(x_\alpha-x_{\alpha'})$ converges to zero in the norm. It follows that the net $(x_\alpha)$ is norm Cauchy, and so it is norm convergent. As a result, we have that $(x_\alpha)$ $mn$-converges to its norm limit by Remark \ref{rem 1 basic}$(ii)$.
\end{proof}
The multiplication in $f$-algebra is $mo$-continuous in the following sense.
\begin{theorem}
Let $E$ be a Banach lattice $f$-algebra, and $(x_\alpha)_{\alpha \in A}$ and $(y_\beta)_{\beta \in B}$ be two nets in $E$. If $x_\alpha\mc x$ and $y_\beta\mc y$ for some $x,y\in E$ and each positive element of $E$ can be written as a multiplication of two positive elements then we have $x_\alpha y_\beta\mc xy$.
\end{theorem}
\begin{proof}
Assume $x_\alpha\mc x$ and $y_\beta\mc y$. Then $\lvert x_\alpha-x\rvert u\nc 0$ and $\lvert y_\beta-y\rvert u\nc 0$ for every $u\in E_+$. Let's fix $u\in E_+$ and $\varepsilon>0$. So, there exist indexes $\alpha_0$ and $\beta_0$ such that $\lVert\lvert x_\alpha-x\rvert u\rVert\leq \varepsilon$ and $\lVert\lvert y_\beta-y\rvert u\rVert\leq\varepsilon$ for all $\alpha\geq\alpha_0$ and $\beta\geq\beta_0$. Next, we show the $mn$-convergence of $(x_\alpha y_\beta)$ to $xy$. By considering the equality $\lvert xy\rvert =\lvert x\rvert \lvert y\rvert $, we have
\begin{eqnarray*}
\lVert\lvert x_\alpha y_\beta-xy\rvert u\rVert&=&\lVert\lvert x_\alpha y_\beta-x_\alpha y+x_\alpha y-xy\rvert u\rVert\\&\leq& \lVert \lvert x_\alpha-x+x\rvert\lvert y_\beta-y\rvert u\rVert+\lVert\lvert x_\alpha -x\rvert \ (\lvert y\rvert u)\rVert\\&\leq& \lVert\lvert x_\alpha-x\rvert \lvert y_\beta-y\rvert u\rVert+\lVert\lvert y_\beta-y\rvert (\lvert x\rvert u)\rVert+\lVert\lvert x_\alpha -x\rvert (\lvert y\rvert u)\rVert.
\end{eqnarray*}
The second and the third terms in the last inequality both order converge to zero as $\beta\to\infty$ and $\alpha\to \infty$ respectively because of $\lvert x\rvert u,\lvert y\rvert u\in E_+$, $x_\alpha\mc x$ and $y_\beta\mc y$. Now, let's show the $mn$-convergence of the first term of last inequality. There are two positive elements $u_1,u_2\in E_+$ such that $u=u_1u_2$ because the positive element of $E$ can be written as a multiplication of two positive elements. So, we get $\lVert\lvert x_\alpha-x\rvert \lvert y_\beta-y\rvert u\rVert=\lVert(\lvert x_\alpha-x\rvert u_1)(\lvert y_\beta-y\rvert u_2)\rVert\leq(\lVert\lvert x_\alpha-x\rvert u_1\rVert)(\lVert\lvert y_\beta-y\rvert u_2\rVert)$. Therefore, we see $\lvert x_\alpha-x\rvert \lvert y_\beta-y\rvert u\nc 0$. Hence, we get $x_\alpha y_\beta\mc xy$.
\end{proof}
We give some basic notions motivated by their analogies from vector lattice theory.
\begin{definition}\label{$mn$-notions}
Let $(x_\alpha)_{\alpha \in A}$ be a net in a Banach lattice $f$-algebra $E$. Then
\begin{enumerate}
\item[(1)] $(x_\alpha)$ is said to be {\em $mn$-Cauchy} if the net $(x_\alpha-x_{\alpha'})_{(\alpha,\alpha') \in A\times A}$ $mn$-converges to $0$,
\item[(2)] $E$ is called {\em $mn$-complete} if every $mn$-Cauchy net in $E$ is $mn$-convergent,
\item[(3)] $E$ is called {\em $mn$-continuous} if $x_\alpha\oc0$ implies that $x_\alpha\mc 0$,
\end{enumerate}
\end{definition}
\begin{lemma}\label{order and mn-convergence}
A Banach lattice $f$-algebra is $mn$-continuous iff $x_\alpha\downarrow 0$ implies $x_\alpha\mc 0$.
\end{lemma}
\begin{proof}
We show $mn$-continuity. Let $(x_\alpha)$ be an order convergent to zero net in a Banach lattice $f$-algebra $E$. Then there exists a net $z_\beta\downarrow 0$ in $E$ such that, for any $\beta$ there exists $\alpha_\beta$ so that $\lvert x_\alpha\rvert\leq z_\beta$, and so $\lVert x_\alpha\rVert\leq \lVert z_\beta\rVert$ for all $\alpha\geq\alpha_\beta$. Since $z_\beta\downarrow 0$, we have $z_\beta\mc 0$, i.e., for fixed $\varepsilon>0$, there is $\beta_0$ such that $\lVert z_\beta\rVert<\varepsilon$ for all $\beta\geq\beta_0$. Thus, there exists an index $\alpha_{\beta_0}$ so that $\lVert x_\alpha\rVert\leq \varepsilon$ for all $\alpha\geq\alpha_{\beta_0}$. Hence, $x_\alpha\mc 0$.
\end{proof}
\begin{theorem}\label{of-contchar}
Let $E$ be an $mn$-complete Banach lattice $f$-algebra. Then the following statements are equivalent:
\begin{enumerate}
\item[(i)] $E$ is $mn$-continuous;
\item[(ii)] if $0\leq x_\alpha\uparrow\leq x$ holds in $E$ then $(x_\alpha)$ is an $mn$-Cauchy net;
\item[(iii)] $x_\alpha\downarrow 0$ implies $x_\alpha\mc 0$ in $E$.
\end{enumerate}
\end{theorem}
\begin{proof}
$(i)\Rightarrow(ii)$ Take the net $0\leq x_\alpha\uparrow\leq x$ in $E$. Then there exists a net $(y_\beta)$ in $E$ such that $(y_\beta-x_\alpha)_{\alpha,\beta}\downarrow 0$; see Lemma 12.8 \cite{ABPO}. Thus, by applying Lemma \ref{order and mn-convergence}, we have $(y_\beta-x_\alpha)_{\alpha,\beta}\mc 0$. Therefore, the net $(x_\alpha)$ is $mn$-Cauchy because of $\lVert x_\alpha-x_{\alpha'} \rVert_{\alpha,\alpha'\in A}\leq\lVert x_\alpha-y_\beta\rVert+\lVert y_\beta-x_{\alpha'}\rVert$.
$(ii)\Rightarrow(iii)$ Put $x_\alpha\downarrow 0$ in $E$ and fix arbitrary $\alpha_0$. Thus, we have $x_\alpha\leq x_{\alpha_0}$ for all $\alpha\geq\alpha_0$, and so we can get $0\leq(x_{\alpha_0}-x_\alpha)_{\alpha\geq\alpha_0}\uparrow\leq x_{\alpha_0}$. Then it follows from $(ii)$ that the net $(x_{\alpha_0}-x_\alpha)_{\alpha\geq\alpha_0}$ is $mn$-Cauchy, i.e., $(x_{\alpha^{'}}-x_\alpha)\mc 0$ as $\alpha_0\le\alpha,\alpha^{'}\to\infty$. Since $E$ is $mn$-complete, there exists $x\in E$ satisfying $x_\alpha\fc x$ as $\alpha_0\le\alpha\to\infty$. It follows from Proposition \ref{mn convergence positive} that $x_\alpha\downarrow0$ because of $x_\alpha\downarrow$ and $x_\alpha\mc 0$, and so we have $x=0$. Therefore, we get $x_\alpha\mc 0$.
$(iii)\Rightarrow(i)$ It is just the implication of Lemma \ref{order and mn-convergence}.
\end{proof}
\begin{corollary}\label{of + f implies o}
Every $mn$-continuous and $mn$-complete Banach lattice $f$-algebra is order complete.
\end{corollary}
\begin{proof}
Suppose $E$ is $mn$-continuous and $mn$-complete a Banach lattice $f$-algebra. For $y\in E_+$, put a net $0\leq x_\alpha\uparrow\leq y$ in $E$. By applying Theorem \ref{of-contchar} $(ii)$, the net $(x_\alpha)$ is $mn$-Cauchy. Thus, there exists an element $x\in E$ such that $x_\alpha\mc x$ because of $mn$-completeness. Since $x_\alpha\uparrow$ and $x_\alpha\fc x$, it follows from Lemma \ref{mn convergence positive} that $x_\alpha\uparrow x$. Therefore, $E$ is order complete.
\end{proof}
The following proposition is an $mn$-version of \cite[Prop.4.2]{GX}.
\begin{proposition}
Let $E$ be an $mn$-continuous and $mn$-complete Banach lattice $f$-algebra. Then every $f$-almost order bounded and order Cauchy net is $mn$-convergent.
\end{proposition}
\begin{proof}
Let $(x_\alpha)$ be an $f$-almost order bounded $o$-Cauchy net. Then the net $(x_\alpha-x_{\alpha'})$ is $f$-almost order bounded and is order convergent to zero. Since $E$ is $mn$-continuous, $x_\alpha-x_{\alpha'}\mc 0$. By using Proposition \ref{$f$-almost}, we have $x_\alpha-x_{\alpha'}\nc 0$. Hence, we get that $(x_\alpha)$ is $mn$-Cauchy, and so it is $mn$-convergent because of $mn$-completeness.
\end{proof}
We now turn our attention to a topology on Banach lattice $f$-algebras. We show that $mn$-convergence in a Banach lattice $f$-algebra is topological. While $mo$- and $uo$-convergence need not be given by a topology, it was observed in \cite{DOT} that $un$-convergence is topological. Motivated from that definition, we give the following. Let $\varepsilon>0$ be given. For a non-zero positive vector $u\in E_+$, we put
$$
V_{u,\varepsilon}=\{x\in E:\lVert\lvert x\rvert u\lVert<\varepsilon\}.
$$
Let $\mathcal{N}$ be the collection of all the sets of this form. We claim that $\mathcal{N}$ is a base of neighborhoods of zero for some Hausdorff linear topology. It is obvious that $x_\alpha\mc0$ iff
every set of $\mathcal{N}$ contains a tail of this net, hence the $mn$-convergence is the convergence induced by the mentioned topology.
We have to show that $\mathcal{N}$ is a base of neighborhoods of zero. To show this we apply Theorem 3.1.10 \cite{R}. First, note that every element in $\mathcal{N}$ contains zero. Now, we show that for every two
elements of $\mathcal{N}$, their intersection is again in $\mathcal{N}$. Take any two set $V_{u_1,\varepsilon_1}$ and $V_{u_2,\varepsilon_2}$ in $\mathcal{N}$. Put $\varepsilon=\varepsilon_1\wedge\varepsilon_2$ and $u=u_1\vee u_2$. We show that $V_{u,\varepsilon}\subseteq V_{u_1,\varepsilon_1}\cap V_{u_2,\varepsilon_2}$. For any $x\in V_{u,\varepsilon}$, we have $\lVert \lvert x\rvert u\lVert<\varepsilon$. Thus, it follows from $ \lvert x\rvert u_1\leq \lvert x\rvert u$ that
$$
\lVert\lvert x\rvert u_1\lVert\leq\lVert\lvert x\rvert u\lVert<\varepsilon\leq\varepsilon_1.
$$
Thus, we get $x\in V_{u_1,\varepsilon_1}$. By a similar way, we also have $x\in V_{u_2,\varepsilon_2}$.
It is not a hard job to see that $V_{u,\varepsilon}+V_{u,\varepsilon}\subseteq V_{u,2\varepsilon}$, so that for each $U\in \mathcal{N}$, there is another $V\in \mathcal{N}$ such that $V+V\subseteq U$. In addition, one can easily verify that for every $U\in \mathcal{N}$ and every scalar $\lambda$ with $\lvert \lambda\rvert\leq 1$, we have $λU\subseteq U$.
Now, we show that for each $U\in \mathcal{N}$ and each $y\in U$, there exists $V\in \mathcal{N}$ with $y+V\subseteq U$. Suppose $y\in V_{u,\varepsilon}$. We should find $\delta>0$ and a non-zero $v\in E_+$ such that $y+V_{v,\delta}\subseteq V_{u,\varepsilon}$. Take $v:=u$. Hence, since $y\in V_{u,\varepsilon}$, we have $\lVert\lvert y\rvert u\rVert<\varepsilon$. Put $\delta:=\varepsilon-\lVert\lvert y\rvert u\rVert$.
We claim that $y+V_{v,\delta}\subseteq V_{u,\varepsilon}$. Let's take $x\in V_{v,\delta}$. We show that $y+x\in V_{u,\varepsilon}$. Consider the inuality $\lvert y+x\rvert u\leq\lvert y\rvert u+\lvert x\rvert u$. Then we have
$$
\lVert\lvert y+x\rvert u\rVert\leq \lVert\lvert y\rvert u\rVert+\lVert\lvert x\rvert u\rVert<\lVert\lvert y\rvert u\rVert+\delta=\varepsilon.
$$
Finally, we show that this topology is Hausdorff. It is enough to show that $\bigcap\mathcal{N}=\{0\}$. Suppose that it is not hold true, i.e., assume that $0\neq x\in V_{u,\varepsilon}$ for all non-zero $u\in E_+$ and for all $\varepsilon>0$. In particular, take $x\in V_{\lvert x\rvert,\varepsilon}$. Thus, we have $\lVert \lvert x\rvert^2\rVert<\varepsilon$. Since $\varepsilon$ is arbitrary, we get $\lvert x\rvert^2=0$, i.e., $x=0$ by using Theorem 142.3 \cite{Za}; a contradiction.
Similar to Lemma 2.1 and Lemma 2.2 \cite{KMT}, we have the following two lemmas. The proofs are analogous so that we omit them.
\begin{lemma}\label{mn-top 1}
$V_{u,\varepsilon}$ is either contained in $[-u,u]$ or contains a non-trivial ideal.
\end{lemma}
\begin{lemma}\label{mn-top 2}
If $V_{u,\varepsilon}$ is contained in $[-u,u]$, then $u$ is a strong unit.
\end{lemma}
\begin{proposition}
Let $E$ be a Banach lattice $f$-algebra. Then if a neighborhood of $mn$-topology is norm bounded, then $E$ has a strong unit.
\end{proposition}
\begin{proof}
Assume $V_{u,\varepsilon}$ is norm bounded for some $u\in E_+$ and $\varepsilon>0$. By applying Lemma \ref{mn-top 1}, we have $V_{u,\varepsilon}$ is contained in $[-u,u]$. So, by using Lemma \ref{mn-top 2}, $u$ is a strong unit.
\end{proof}
Consider Example \ref{int exam}. Then we know that $(I_x,\lVert\cdot\rVert_\infty)$ is a Banach lattice $f$-algebra. $I_x$ equipped with the norm $\lVert\cdot\rVert_\infty$ is lattice isometric to $C(K)$ for some compact Hausdorff space $K$, with $x$ corresponding to the constant one function $\mathsf{1}$; see for example Theorem 3.4 and Theorem 3.6. If $x$ is a strong unit in $E$ then $I_x=E$. It is easy to see that in this case $\lVert\cdot\rVert_\infty$ is equivalent to the original norm and $E$ is lattice and norm isomorphic to $C(K)$. It is easy to see that norm convergence implies $mn$-convergence, and so, in general, norm topology is stronger than $mn$-topology.
|
{
"timestamp": "2019-03-01T02:10:55",
"yymm": "1902",
"arxiv_id": "1902.10927",
"language": "en",
"url": "https://arxiv.org/abs/1902.10927"
}
|
\subsection{A: Model Hamiltonian}
In this section, we introduce the $f$-$f$ hopping integrals,
by which the $f_{l}$-orbital weight has momentum-dependence on the FS.
The obtained $f$-electron kinetic term is
\cite{S-Tazai-PRB2018}
\begin{eqnarray}
\hat{H}_{ff}=\sum_{\k l\sigma}E_{\k l}f^{\dagger}_{\k l\sigma}
f_{\k l\sigma}.
\end{eqnarray}
Here, we set $E_{\k1}\equiv E_{1}+\delta E_{\k}$ and $E_{\k2}\equiv E_{2}-\delta E_{\k}$.
To reproduce the $\k$-dependent $\delta E_{\k}$
shown in Fig.\ref{fig:appendix},
we introduce from the first to fifth neighbor hopping integrals
according to Refs.\cite{S-Yamakawa-FeSe,S-Tazai-PRB2018}.
The obtained momentum-dependence of $f_{l}$-orbital weight on the FS
is shown in Fig. 1 (b) in the main text.
As discussed in Ref. \cite{S-Tazai-PRB2018},
the RPA susceptibility is insensitive to $\delta E_{\k}$
since the $f_{l}$-orbital DOS, $D_l(\e)$,
is independent of $\delta E_{\k}$.
In the present study,
we verified that both $\chi$-VC and $U$-VC are also insensitive to
$\delta E_{\k}$.
\begin{figure}[htb]
\includegraphics[width=0.4\linewidth]{fig6.eps}
\caption{The FS with $f$-$f$ hopping.
Each number at $\k$ shows intra-orbital energy shift $\delta E_{\k}$.
}
\label{fig:appendix}
\end{figure}
In HF systems,
the quadrupole susceptibility remains small within the RPA.
To understand this result,
we examine the $(Q,Q')$ component of normalized Coulomb interaction:
\begin{eqnarray}
U^{Q,Q'}_0=(\vec{Q})^\dagger \hat{U}^0 \vec{Q}' .
\end{eqnarray}
TABLE \ref{tab:appendix} shows the diagonal component
$U^{Q}_0\equiv U^{Q,Q}_0$.
Since $U^{Q}_0$ for the EM channels
is much smaller than that for the MM channels,
the EM susceptibilities are small within the RPA.
Nonetheless of this fact,
EM susceptibilities strongly develop by considering the AL-VC,
since $X^{\rm AL}$ for the EM channel becomes large
when moderate MM fluctuations exist.
\begin{table}[htb]
\begin{tabular}{|c|c|c|c|c|c|} \hline
Q & C & $O_{20}$ & $H_{0}$ & $H_{z}$ & $O_{yz(zx)}$
\\ \hhline{|=|=|=|=|=|=|}
$U^{Q}_0$ & -1.3 & -0.18 & 0.17 & 0.34 & 0.27
\\ \hline
\end{tabular}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
Q & $J_{z}$ &$T_{z}$ &$D_{z(4)}$ & $J_{x(y)}$& $T_{x(y)}$ & $D_{x(y)}$
\\ \hhline{|=|=|=|=|=|=|=|}
$U^{Q}_0$ & 0.56 & 0.44 & 0.55 & 0.49 & 0.49 & 0.50 \\ \hline
\end{tabular}
\caption{Normalized Coulomb interaction for $Q$-multipole channel, $U^{Q}_0$.
}
\label{tab:appendix}
\end{table}
\subsection{B: Pseudospin representation of multipole operators}
Here, we list the multipole operators $\hat{Q}$
in the present CeCu$_2$Si$_2$ model,
which were already explained in Ref. \cite{S-Tazai-PRB2018}.
The EM (even-rank) operators
in the $4\times4$ matrix form are expressed as
\begin{eqnarray}A_{1}^{+}&&
\begin{cases}
\hat{C}&=\hat{\sigma}^{0}\hat{\tau}^{0} , \nonumber \\
\hat{O}_{20}&=\hat{\sigma}^{0} \left(2.00\hat{\tau}^{0}+3.00\hat{\tau}^{z} \right) , \nonumber \\
\hat{H}_{0}&=\hat{\sigma}^{0} \left(-5.73\hat{\tau}^{0}+11.5\hat{\tau}^{z}-12.8\hat{\tau}^{x} \right) ,
\end{cases} \nonumber \\ A_{2}^{+}&&
\begin{cases}
\hat{H}^{z}&=-19.8 \hat{\sigma}^{z} \hat{\tau}^{y} ,
\end{cases} \nonumber \\ E^{+}&&
\begin{cases}
\hat{O}_{yz}&=-3.87 \hat{\sigma}^{x} \hat{\tau}^{y} , \nonumber \\
\hat{O}_{zx}&=+3.87 \hat{\sigma}^{y} \hat{\tau}^{y} . \nonumber \\
\end{cases} \label{eqn:eleO}\\
\end{eqnarray}
The MM (odd-rank) operators are given by
\begin{eqnarray} A_{1}^{-}&&
\begin{cases}
\hat{D}_{4}&=+29.8i \hat{\sigma}^{0} \hat{\tau}^{y} , \nonumber \\
\end{cases} \nonumber \\ A_{2}^{-}&&
\begin{cases}
\hat{J}^{z}&= \hat{\sigma}^{z} \left(0.50\hat{\tau}^{0}+2.00\hat{\tau}^{z} \right) , \nonumber \\
\hat{T}^{z}&= \hat{\sigma}^{z} \left(9.00\hat{\tau}^{0}-1.50\hat{\tau}^{z} \right) , \nonumber \\
\hat{D}^{z}&= -29.8 \hat{\sigma}^{z} \hat{\tau}^{x} ,
\end{cases} \\
E^{-}&&
\begin{cases}
\hat{J}^{x}&= -1.12 \hat{\sigma}^{x} \hat{\tau}^{x} , \nonumber \\
\hat{J}^{y}&= -1.12 \hat{\sigma}^{y} \hat{\tau}^{x} ,\nonumber \\
\hat{T}^{x}&= \hat{\sigma}^{x} \left(3.75\hat{\tau}^{0}-3.75\hat{\tau}^{z} +5.03\hat{\tau}^{x}\right) , \nonumber \\
\hat{T}^{y}&= \hat{\sigma}^{y} \left(3.75\hat{\tau}^{0}-3.75\hat{\tau}^{z} +5.03\hat{\tau}^{x}\right) , \nonumber \\
\hat{D}^{x}&= \hat{\sigma}^{x} \left( 23.0\hat{\tau}^{0}-6.56\hat{\tau}^{z}-3.14\hat{\tau}^{x}\right) , \nonumber \\
\hat{D}^{y}&= \hat{\sigma}^{y} \left( 23.0\hat{\tau}^{0}-6.56\hat{\tau}^{z}-3.14\hat{\tau}^{x}\right) ,
\end{cases} \label{eqn:magneO} \\
\end{eqnarray}
where $\hat{\sigma}^{\mu}$ and $\hat{\tau}^{\mu}$($\mu=x,y,z$) are Pauli matrices for
the pseudo-spin and orbital basis, respectively.
$\hat{\sigma}^{0}$ and $\hat{\tau}^{0}$ are identity matrices.
The row and column of the Hermite matrix $\hat{Q}$ for each operator
is given as $L=(l,\s)$, where $l=1,2$ represents the $f$-orbital
and $\s=\uparrow,\downarrow$ represents the pseudo spin.
In the main text, we also introduce the vector representation
defined as $(\vec{Q})_\a= (\hat{Q})_{L,L'}$, where $\a=(L,L')$.
\subsection{C: Analytic expressions of vertex corrections}
From now on, we introduce the analytic expressions of
$\chi$-VC \cite{S-Tazai-CeB6} and $U$-VC \cite{S-Tazai-PRB2018}
due to AL diagrams.
First, we discuss the $\chi$-VCs,
whose diagrammatic expressions are shown in Fig. 2 (a) in the main text.
The expression for the AL1 term is given as
\begin{eqnarray}
X^{\rm{AL1}}_{\a \b}(q)=\frac{T}{2}\sum_{\a' \a'' \b' \b'' p}
C_{\a' \b''}^{\a} (q,p) V_{\a' \b'} (p-q) \nonumber \\
\times V_{\a'' \b''}(p) C_{\b' \a''}^{\b *} (\bar{q},\bar{p}),
\label{eqn:UALc}
\end{eqnarray}
where $p\equiv (\p, \omega_{j})$, $\bar{p}\equiv (\p,-\omega_{j})$,
and
$\hat{V}(q)\equiv u^{2}\hat{U}^0\hat{\chi}(q)\hat{U}^0+u\hat{U}^0$
is the dressed interaction given by the RPA.
The three-point vertex in Eq. (\ref{eqn:UALc}) is given as
\begin{eqnarray}
C^{EF}_{ABCD} (q,p) \equiv -T\sum_{k}G^{f}_{AF}(k-q)G^{f}_{EC}(k)
G^{f}_{DB}(k-p),
\end{eqnarray}
where $\hat{G}^{f}$ is the $f$-electron Green function.
Also, the expression for the AL2 term is given as
\begin{eqnarray}
X^{\rm{AL2}}_{\a \b}(q)&=&\frac{T}{2}\sum_{\a' \b' \a'' \b'' p}
C_{\a' \b''}^{'\a} (q,p) V_{\b'' \b'} (p-q) \nonumber \\
&\times& V_{\a'' \a'}(p) \tilde{C}_{\a'' \b'}^{'\b} (q,p),
\label{eqn:S-UALc}
\end{eqnarray}
where
\begin{eqnarray}
C^{'EF}_{ABCD} (q,p) \equiv -T\sum_{k}G^{f}_{BF}(k-q)G^{f}_{ED}(k)G^{f}_{CA}(k-q+p),
\nonumber \\
\tilde{C}^{'EF}_{ABCD} (q,p) \equiv -T\sum_{k}G^{f}_{AE}(k+q)G^{f}_{FC}(k)G^{f}_{DB}(k+q-p).
\nonumber
\end{eqnarray}
The total $\chi$-VC is given by
$\hat{X}^{\rm AL}=\hat{X}^{AL1}+\hat{X}^{AL2}$,
by subtracting the double counting
second order diagrams of order $u^2$.
Next, we explain the $U$-VC in the gap equation.
It is given as
\begin{eqnarray}
( {\hat \Lambda}_{kk'})_{LL'MM'} = \delta_{LM}\delta_{L'M'}+
({\hat L}_{kk'})_{LL'MM'}.
\label{eqn:defALc}
\end{eqnarray}
In the main text, we calculate the AL diagrams for ${\hat L}_{kk'}$.
It is expressed as
\begin{eqnarray}
\!\!(\hat{L}_{kk'})_{LL'MM'}\!\!\!\!\!\!&&=\!\frac{T}{2}\!\!\!\sum_{p,ABCDEF}
B_{ABCDEF}^{MM'} (k-k',p,k') \nonumber \\
&&\times V_{LACD} (k-k'+p) V_{BL'EF} (-p),
\label{eqn:UALc}
\end{eqnarray}
where
\begin{eqnarray}
&& \hspace{-30 pt}B_{ABCDEF}^{MM'} (q,p,k') =
G^{f}_{AB}(k'-p) \nonumber \\
&&\hspace{30 pt} \times
\left\{\! C^{''MM'}_{CDEF} (q,p)+
C^{''MM'}_{EFCD}(q,q+p) \right\}
\label{eqn:Bdef3}
\end{eqnarray}
and
\begin{eqnarray}
{C}^{''AB}_{CDEF} (q,p)\!\equiv
\!-T\sum_{k'}G^{f}_{CA}(k'+q)G^{f}_{BF}(k')G^{f}_{ED}(k'-p).
\nonumber \\
\end{eqnarray}
\subsection{D: Gap equation and retardation effect}
Here, we comment on the retardation effects.
In Fig.\ref{fig:chiorb}, we show the obtained paring interaction
on the FS defined as
$V^{\rm{sing}}_{\rm max} (\omega_{j})
\equiv \max_{\k,\k'} \{ V^{\rm{sing}}_{(\k,\pi T)(\k',\pi T+\omega_{j})}\}$.
The paring interaction is attractive (positive) at $\omega_{j}=0$,
whereas it becomes to repulsion for $\omega_{j}>0$.
For this reason, the gap function defined as
$\Delta(\epsilon_{n})\equiv \max_{\k} \{ \Delta (\k,\epsilon_{n})\}$
shows the sign-change as the function of $\e_{n}$,
as shown in the inset of Fig.\ref{fig:chiorb}.
This is a hallmark of the retardation effects
due to the strong $\omega_{j}$-dependence of the
EM (even-rank) fluctuation.
Since the depairing due to direct Coulomb interaction
is reduced by the retardation effect,
the fully-gapped $s$-wave superconductivity can be
stabilized in HF systems.
\begin{figure}[htb]
\includegraphics[width=.84\linewidth]{fig7.eps}
\caption{
Obtained paring interaction $V_{\rm max}^{\rm sing}(\omega_{j})$
and gap function $\Delta(\epsilon_{j})$ (inset)
as the function of Matsubara frequency.
Strong retardation effect is recognized.}
\label{fig:chiorb}
\end{figure}
|
{
"timestamp": "2019-03-01T02:13:07",
"yymm": "1902",
"arxiv_id": "1902.10968",
"language": "en",
"url": "https://arxiv.org/abs/1902.10968"
}
|
\section{Introduction}
\label{sec:introduction}
\IEEEPARstart{R}{e-identification} of animal individuals by unique natural markings in photo databases is an effective and non-invasive mark-recapture tool for monitoring populations \cite{into-photo-id-rays-sharks}.
Tracking population dynamics of animals such as manta rays is critical owing to their vulnerable conservation status, and economic importance in both ecotourism and fisheries \cite{review-manta-latest}.
These species cannot sustain heavy exploitation \cite{marine-croll}, and the trade of manta ray gill rakers is believed to be responsible for driving population declines upwards of 80\% in some locations \cite{marine-rohner}. Some species such as humpback whales are no longer threatened by commercial whaling. The conservation effort is now focused on the identification of individual humpback whales to better understand their use of breeding and feeding areas \cite{whales-conservation}.
Our research is focused on developing an automated system for visual re-identification of animals that bear unique natural
markings. We demonstrate the suitability of the proposed system on photo databases of manta ray belly patterns and humpback whale flukes. Manta rays have a unique spot pattern on their ventral surface that allows individuals to be distinguished from one another. The spot pattern is conserved throughout the animals life, much like a human fingerprint. Examples of spot patterns are shown in Fig.~\ref{fig:manta-intro} and Fig.~\ref{fig:manta-example}.
Humpback whales have patterns of black and white pigmentation and scars on the underside of their tails that are unique to each whale.
There are a number of factors that make animal re-identification
based on natural markings challenging.
Photo databases often rely on input from citizen scientists to fill in data gaps when researchers are not in the field.
This means image quality cannot be guaranteed as camera parameters, and the angle of image capture vary. Other factors include poor visibility (especially for underwater images), illumination, and small objects occluding the pattern on the animal.
\begin{figure}[!t]
\centering
\includegraphics[width=3.3in]{moskv1.pdf}
\caption{The proposed system learns embeddings for images from the database. The embeddings of the same individual are brought close together and embeddings of different manta rays are pushed further apart. A new query image is matched to the database by finding the closest points in the embedding space. The system learns embeddings that are invariant to viewing angle and illumination. Photo credit: Chris Garraway.}
\label{fig:manta-intro}
\end{figure}
The current state-of-the-art manta ray recognition system Manta Matcher \cite{manta_matcher} requires the user to manually align and normalize the 2D orientation of the manta
ray within the image, and select a rectangular region of interest containing the spot pattern.
The Manta Matcher works best with photos taken perpendicular to the manta's ventral pattern with no reflective particles in the water and in good lighting conditions.
In practice, these constraints limit the use of photos from citizen scientists and some marine biologists still do the identification manually using a handcrafted decision tree.
A common idea that has been applied to several species for recognizing individual animals is to search for an affine transformation matching patterns present in two distinct images
(lizards \cite{review-aphis}, arthropods \cite{review-aphis-antropodes}, sharks \cite{review-sharks}, turtles \cite{ review-turtles}).
However, this approach requires annotating body landmarks on each individual image in the same order.
This is not suitable for manta rays as we want to accept images of the animals in a wide range of poses
with no requirement that all body landmarks are clearly visible.
Convolutional neural networks (CNN) have been applied to the problem of animal identification as a classification problem \cite{review-pigs}, \cite{review-seals}, \cite{review-snolepard}, \cite{whale-kaggle}.
It means that the trained model is only able to identify the animals presented during training.
It is highly desirable to have a system that is not only capable of recognizing animals whose images have been used to train the neural network, but also capable
of recognizing animals whose images have been added to the database well after the network has been trained without requiring the re-training of the network on these new instances.
This paper focuses on this more challenging and less studied problem for animal re-identification
\begin{figure}[!t]
\centering
\includegraphics[width=3.3in]{moskv2.pdf}
\caption{The camera angle can vary dramatically. Here the same manta ray, named Eris, was photographed from two different viewpoints. A homography transformation is required to align the belly patterns of the two images. Photo credit: Chris Garraway.}
\label{fig:manta-example}
\end{figure}
In this work we focus on eliminating some constraints
of previous wildlife matching systems such as requirements for high image quality and a clear view of the animal markings in the image. We propose a solution inspired by advances in deep learning for face re-identification. Our approach uses a CNN to learn embeddings for images of animal markings in such a way that the distance between embeddings of the same individual is smaller than the distance between embeddings of this individual and other animals (see Fig.~\ref{fig:manta-intro}).
The main contribution of this work is a novel visual wildlife re-identification system with the following properties:
\begin{enumerate}
\item robustness to viewpoint changes, small occlusions and lighting conditions, and therefore ability to match images from citizen scientists;
\item re-identification of individuals never seen during training.
\end{enumerate}
The paper is organized as follows: in Section~\ref{sec:related-work} we discuss related work on re-identification. Our approach to learning embeddings is described in Section~\ref{sec:learning-embeddings}. The experimental setup and results are presented in Section~\ref{sec:experiments}.
\begin{figure*}[!t]
\centering
\includegraphics[width=0.95\textwidth]{moskv3.pdf}
\caption{System architecture. All images from the current database are passed through the trained CNN model to compute embeddings and fit a nearest neighbors classifier in an embedding space. At the prediction step, $k$ predictions are obtained by computing an embedding using trained CNN and finding the closest points from the database using the nearest neighbors classifier.}
\label{fig:train-predict}
\end{figure*}
\section{Related Work} \label{sec:related-work}
The techniques that have been proposed
for photo-identification of animal natural markings vary in the core methods used, amount of user involvement and ability
to be adapted to different species
We review solutions used in practice for different cases.
Matching
natural patterns has been approached by exhaustively generating two-dimensional affine transformations based on user provided key points and comparing each transformation of a candidate example with the examples stored in a repository \cite{review-aphis}, \cite{review-aphis-antropodes}, \cite{review-sharks}, \cite{review-turtles}. The algorithm was implemented in a solution called APHIS (Automated Photo-Identification Suite) and applied for re-identification of lizards \cite{review-aphis}, arthropods \cite{review-aphis-antropodes}, spotted raggedtooth sharks \cite{review-sharks} and turtle flippers \cite{review-turtles}. However, the method requires a user to select key points and identify the most distinctive spots for each image.
Some methods have been developed for specific species and, while performing well on these, are not easily transferable to other species. High-contrast colour patterns of humpback whale flukes \cite{review-whales} and dolphin dorsal fins \cite{review-dolphin} are matched by extracting hand-crafted features from corresponding segments obtained by overlaying a grid on a region of interest. This method is not robust to viewpoint changes.
Another approach identifies individual cetaceans from images showing the trailing edge of their fins by generating a representation of integral curvature of the nicks and notches along the trailing edge \cite{review-curvature}.
Current systems used in practice (Manta Matcher \cite{manta_matcher}, HotSpotter \cite{review-hotspotter}) are based on
automated extraction and matching of keypoint features using the Scale-Invariant Feature Transform (SIFT) algorithm \cite{review-sift} with different modifications and enhancements to work on specific cases. While the algorithm works well on images that clearly show the pattern of interest, it is not robust to large changes in
camera viewpoint, occlusions and variations in illumination.
The task of animal visual re-identification is related to the face recognition problem that has been extensively studied with deep learning in recent years \cite{parkhi2015deep, schroff2015facenet, sun2014deep}. The main idea is learning a function using a CNN that maps from a face image space to a space of a smaller dimension where the distance between the learned embedding vectors corresponds to a face similarity measure \cite{parkhi2015deep, chopra2005learning}. The network is trained on labelled image pairs or triplets to learn
a face similarity measure under which the
distance between the embeddings of faces from the same person is reduced as much as possible and that of the distance between embeddings of faces of different people is increased. The problem is then reduced to the nearest neighbours search problem in Euclidean space, which can be solved by efficient approximate nearest neighbours search algorithms \cite{wang2014learning}.
The difference between face verification and animal re-identification is that a face image is typically normalized to an upright position whereas a pattern on an animal body is not necessarily in a canonical position
and can appear at different angles.
See an example of the same manta ray
viewed from different vantage points in Fig.~\ref{fig:manta-example}. A robust identification system should be invariant to the pose of the object of interest and viewing angle. In our previous work \cite{dicta-paper}, we investigated
the difficulty of recognizing a set of artificially generated patterns subjected to various projective transformations
to simulate the variations in appearance of natural markings from different vantage points. This previous
study explored Siamese \cite{chopra2005learning} and Triplet \cite{wang2014learning} architectures with different loss functions for learning the homographic equivalence between patterns. It was concluded that these architectures with a relatively simple CNN in its core were suitable for pattern re-identification. The results were promising and we have now extended this approach to real images of animal markings in the wild.
\section{Learning embeddings} \label{sec:learning-embeddings}
Throughout the paper, we say that images from the same individual animal \textit{belong to the same class}.
Images of different individual animals are said to be from \textit{different classes}. The re-identification task can be formulated as a classification problem where the number of classes is in the order of thousands and not known in advance, and the number of examples for each class is small.
The following section gives an overview of the architecture of our re-identification system.
\subsection{System architecture}
The system, illustrated in Fig.~\ref{fig:train-predict}, consists of a CNN that produces embeddings for images and a k-nearest neighbors classifier in the embedding space.
During the learning phase, we train a CNN on a database of labelled images.
During the prediction phase, a new query image is fed to the network to produce an embedding.
The first $k$ animals in the embedding database that are closest to the embedding of the query image are returned.
Two outcomes are possible during the verification of the identity of the animals.
Either the marine biologist confirmes that the query image corresponds to one of the $k$ returned animals or the query image is considered to be from a never seen before animal. In the first case, the query image is added to the record of the recognized animal. In the second case, a new animal entry is created.
Over time, new images are added to the database but the CNN is not systematically retrained on the extended dataset.
The network is able to match against images that were in the database during training as well as against images added later.
\subsection{Model}
\begin{figure}[!t]
\centering
\includegraphics[width=3.3in]{moskv4.pdf}
\caption{Examples of a hard triplet, a semi-hard triplet and an easy triplet of embeddings. Anchor embedding $a$, its closest positive point $p$ (same class as $a$) and the red negative $n$ (different class from $a$) form a hard triplet as the negative is closer to the anchor than the positive example. The triplet $(a, p, n^{\prime})$ is a semi-hard triplet as the orange negative $n^{\prime}$ lies within the margin from the positive. The triplet contributes a positive value to the loss function. Whereas the triplet $(a, p, n^{\prime\prime})$ with the green negative $n^{\prime\prime}$ is an easy triplet because it contributes zero to the loss function.}
\label{fig:triplets-diagram}
\end{figure}
We have adapted a model proposed in FaceNet \cite{schroff2015facenet} that learns embeddings for faces by minimizing a triplet loss. Initially, it was claimed that representation learning with the triplet loss is inferior to a combination of classification and verification losses \cite{triplet-classif1, triplet-classif2}. However, modifications of the triplet loss (angular loss \cite{angular-loss}, magnet loss \cite{magnet-loss}) and smart triplet mining strategies (semi-hard \cite{schroff2015facenet}, batch-hard \cite{DBLP:in-defence-triplet, 100k-iden}) has proved that a model can successfully learn an end-to-end mapping between images and an embedding space.
The model consists of convolutional layers to extract features from an input image, a global pooling layer over feature maps and a fully connected layer to produce an embedding vector.
We compare different CNN architectures as a base network, see details in Section~\ref{sec:base-networks}.
The convolutional layers output a 3D array (e.g. $ 8 \times 8 \times 512$) that is then passed to a global pooling layer.
A global pooling layer takes the average of each feature map along the spatial axes
(e.g. a tensor $ 8 \times 8 \times 512$ is transformed into a tensor $ 1 \times 512$).
We follow FaceNet \cite{schroff2015facenet} and favor a global average pooling layer instead of a fully connected layer after convolutional layers. The global pooling layer makes the output of the network invariant to the size of the input images.
Moreover, as the layer has no parameters, overfitting is avoided \cite{network-in-network}.
The layer sums out the spatial information so it is more robust to spatial transformations of its input.
Pooled features maps are passed to a fully connected layer to produce an embedding vector.
\subsection{Loss function}
Our model is optimized using the triplet loss function \cite{wang2014learning} which accepts triplets of images. Let us define a triplet $ (I^a, I^{+}, I^{-}) $ where an image $I^a$ (\textit{anchor}) and an image $I^{+}$ (\textit{positive}) are from the same class and an image $I^{-}$ (\textit{negative}) is from a different class.
The function $ D $ between two input images $ I $ and $ J $ is defined as the Euclidean distance between their embeddings $f(I)$ and $f(J)$.
That is, $ D(I, J) \defeq \norm{f(I) - f(J) }$
The triplet loss function $\mathcal{L}$ encourages the squared distances between positive pairs of embeddings
to become smaller than the squared distances between negative pairs of embeddings by a given margin $ m $:
\begin{equation} \label{eq:triplet-loss}
\mathcal{L} \defeq \sum_{i=1}^{N} \max (0, m + D(I^a_i, I_i^{+})^2 - D(I_i^{+}, I_i^{-})^2)
\end{equation}
where \( N \) is the number of training triplets.
We also did experiments with the Siamese network architecture \cite{chopra2005learning} and a contrastive loss function \cite{hadsell2006dimensionality} over randomly generated pairs, however, the results were not as good as the results obtained with the triplet loss function.
\begin{figure*}[!t]
\centering
\subfloat[Images of acceptable quality]{\includegraphics[width=0.45\textwidth]{moskv5a.pdf}%
\label{fig:manta-accepted}}
\hfil
\subfloat[Images excluded from the dataset]{\includegraphics[width=0.45\textwidth]{moskv5b.pdf}%
\label{fig:manta-rejected}}
\caption{Not all images of manta rays are acceptable for training and testing the system.
An image is accepted if the belly pattern is visible (even at oblique angles and in muddy water), see examples in (a).
We accept more challenging images than other methods \cite{manta_matcher}. Examples of excluded images in (b): back of the manta, side view, poor underwater visibility. Photo credit: Lydie Couturier.}
\label{fig:manta-dataset}
\end{figure*}
\subsection{Example mining} \label{sec:methodology-sample-mining}
The strategy for selecting triplets for learning embeddings plays an equal or more important role than the loss \cite{sampling-matters}. Generating random triplets for training with the triplet loss would result in many triplets that are already in a correct position and contribute zero loss to (\ref{eq:triplet-loss}). Several strategies have been proposed to optimize training with the triplet loss function. Batch-hard triplet mining \cite{DBLP:in-defence-triplet} selects the hardest positive (the furthest example from the same class) and the hardest negative (the closest example from a different class) within a batch for each anchor image. Another technique, distance-weighted sampling \cite{sampling-matters}, selects a negative example with a probability function of the distance to the negative example.
We follow the semi-hard triplet mining strategy proposed by \cite{schroff2015facenet} as we found experimentally that this approach works better than batch-hard strategy for our application domain. The triplet loss is calculated over triplets that contribute positive value to the loss function.
In other words, these negative examples lie within a margin from the positive examples (see Fig.~\ref{fig:triplets-diagram}).
The selected triplets are not necessarily the hardest within a batch but they violate
the constraint $ D(I_i^{a}, I_i^{+})^2 + m < D(I_i^{a}, I_i^{-})^2 $.
The triplet mining strategies listed above require computing embeddings in order to select triplets.
This can be achieved by precomputing embeddings every $n$ steps using the most recent network checkpoint.
We adopt a more computationally efficient online mining strategy \cite{schroff2015facenet}
where triplets are generated on the fly after the embeddings have been computed
and before the evaluation of loss function and backpropagation phase.
\subsection{Evaluation methodology}
We evaluate the performance of the system by computing the following metrics:
\begin{itemize}
\item true positive rate on pairs from the test set;
\item top-$k$ accuracy on the test set ($k=1,5,10$).
\end{itemize}
\begin{figure}[!t]
\centering
\includegraphics[width=3.3in]{moskv6.pdf}
\caption{The user is required to draw a bounding box around the region containing the natural markings. Images are cropped to contain the pattern of interest only. This is the only input required from the user. Photo credit: Chris Garraway.}
\label{fig:manta-cropping}
\end{figure}
\subsubsection{Validation on pairs}
The network performance is evaluated on pairs generated from the test set using a method proposed in \cite{schroff2015facenet}. The set of pairs of images from a same class is denoted as $P_{+}$ and the set of all pairs from different classes is denoted as $P_{-}$.
Let us define the set of \textit{true accepts} $\text{TA}$ for a threshold $d$ as the set of correctly classified positive pairs with a threshold $d$:
\begin{equation*}
\text{TA}(d) \defeq \{ (i,j) \in P_{+}, \, \text{with} \, D(I_i, I_j) \leq d \}.
\end{equation*}
The set of \emph{false accepts} $\text{FA}$ is defined as the set of negative pairs that are incorrectly classified as positive with a threshold $d$:
\begin{equation*}
\text{FA}(d) \defeq \{ (i,j) \in P_{-}, \, \text{with} \, D(I_i, I_j) \leq d \}.
\end{equation*}
We calculate the \textit{true positive rate} $ \text{TPR} $ (or \textit{recall}) and the \textit{false acceptance rate} $\text{FAR}$ for a given threshold $d$ as:
\begin{equation*}
\text{TPR}(d) \defeq \frac{|\text{TA}(d)|}{|P_{+}|}, \quad \text{FAR}(d) \defeq \frac{|\text{FA}(d)|}{|P_{-}|}
\end{equation*}
Thanks to the relatively small size of the test datasets, all possible pairs are generated. The models are evaluated by plotting ROC curves and computing the area under the curve.
The models are compared with respect to the true positive rate $ \text{TPR} $ at the threshold $d$ when the false
acceptance rate $\text{FAR} = 0.01$.
\subsubsection{Accuracy evaluation on the test set} \label{sec:eval-real-life}
From a marine biologist's point of view, a reliable system should have at least 95\% top-10 accuracy.
The accuracy of re-identification depends on the number of matching images in the database for each query image.
We consider a realistic scenario where each query individual has two matching images in the database (our databases have at least 3 images for each individual). If there are more images per individual in the database, the task of re-identification becomes easier.
For training, the dataset is partitioned into a training set and a test set in such a way so each individual animal appears exclusively either in the training set or in the test set.
For testing, the database is made of the training set images plus $m=2$ random images for each test individual.
The rest of test images are used as query images.
The accuracy is averaged over all test individuals in multiple runs by moving different images from the test set to the database. We also analyze the effect of varying the number $m$ in Section \ref{sec:experiments-number-m}. A similar evaluation procedure has been performed in \cite{review-hotspotter, manta_matcher} on different datasets.
\section{Experiments}\label{sec:experiments}
\subsection{Datasets} \label{sec:experiments-dataset}
\subsubsection{Manta ray belly patterns}
The experiments have been conducted on a dataset of manta ray images from Project Manta (a multidisciplinary research program based at the University of Queensland, Brisbane, Australia). Images have been manually checked to select the ones that show a pattern on a belly with enough clarity to be recognized by a human. See some examples in Fig.~\ref{fig:manta-dataset} (left). The dataset is challenging as it contains photos of the patterns taken at oblique angles, in a muddy water or with some small occlusions (small fish, water bubbles).
Uninformative images
such as the view of the back of a manta, partial views or unclear patterns have been removed from the dataset.
See examples in Fig.~\ref{fig:manta-dataset} (right). Each image has been manually annotated with a bounding box around the pattern.
Then, each image has been cropped to the area inside the bounding box (Fig.~\ref{fig:manta-cropping}).
Manually highlighting the belly pattern region is the only input required from the user in our application.
The resulting dataset (see details in Table~\ref{tab:data-stats}) is partitioned into the training set (96 individuals) and the test set (24 individuals).
\begin{figure}[!t]
\centering
\includegraphics[width=3.3in]{moskv7.pdf}
\caption{Images from the whale dataset. Each row shows three images of the same fluke. List of photo credits is provided in acknowledgment.}
\label{fig:whale-dataset}
\end{figure}
\begin{table}[]
\renewcommand{\arraystretch}{1.5}
\caption{Statistics for the datasets}
\label{tab:data-stats}
\begin{tabular}{@{}lcccc@{}}
\hline
\multicolumn{1}{c}{} & \multicolumn{2}{c}{Manta rays} & \multicolumn{2}{c}{Whales} \\
\multicolumn{1}{c}{Stats} & Dataset & One fold & Dataset & One fold \\ \hline
Number of images & 1730 & $\sim$350 & 2908 & $\sim$550 \\ \hline
Number of individuals & 120 & 24 & 633 & 126 \\ \hline
Average \# images per ind. &\multicolumn{2}{c}{14} & \multicolumn{2}{c}{5} \\ \hline
Min \# images per ind. & \multicolumn{2}{c}{6} & \multicolumn{2}{c}{3} \\ \hline
\end{tabular}
\end{table}
\subsubsection{Humpback whale flukes}
The dataset of humpback whales flukes comes from the Happy Whale organization (happywhale.com) \cite{happywhale2017}, \cite{happywhale2018}.
The main challenge with this dataset is the small number of images per whale with two-thirds of the whales having one or two sightings.
For training and testing purposes we select individuals with a minimum of three images per whale resulting in a set of 2908 images for 633 unique whales, see Table~\ref{tab:data-stats}. Most of the images have already been cropped to include only the image of the fluke (see example images in Fig.~\ref{fig:whale-dataset}), however there are some noisy examples where the fluke is shown in the distance or text information
appears at the bottom. We did not do any cropping, although this may further improve the results.
The challenges encountered with the whale dataset are different from those of the manta ray dataset.
Although there is no large variation in pose or viewpoint,
there is a limited number of examples per individual, a combination of black-and-white and colour images, a variety of illumination conditions and some noisy images.
\subsection{Implementation details}
\subsubsection{Batch generation}
Training is performed on batches of $B \defeq P \times K$ images, where $P$ is a number of distinct classes in the batch and $K$ is a number of examples per class. During training, the whole batch is fed into the network and embeddings for the batch are computed. Embeddings are then combined into triplets based on pairwise distances according to the semi-hard triplet selection strategy discussed in Section~\ref{sec:methodology-sample-mining}.
We use batches of 15 classes with 5 images per class for manta rays and 3 images per class for whales as this is the maximum batch size that fits into the memory of the computer utilized in these experiments.
\subsubsection{Data augmentation}
Data augmentation is used extensively during training to increase the variety in the training set.
Transformations are applied on the fly so that at every epoch the network receives a new augmentation of the image.
For the manta ray dataset the following geometric transformations were used: rotation up to 90 degrees, horizontal and vertical flips,
small shifts up to 10 pixels and zooming in to 10 percent.
Most of the whale images have already a normalized view of the fluke upright. Therefore, only small rotation angles are used in data augmentation for whales.
\subsubsection{Base networks}
\label{sec:base-networks}
We compare convolutional layers of InceptionV3 \cite{inception-v2} and MobileNetV2 \cite{mobilenet-v2} as feature extractors to assess the influence of the CNN architecture on the performance of the system. One of the key differences between these two models is the number of parameters and operations. The smaller MobileNetV2 has 3.4 million parameters and 300 million multiply-adds operations \cite{inception-v2}. The bigger InceptionV3 has 23 million parameters and 5 billion multiply-adds per inference \cite{inception-v2}.
The convolutional layers of both networks have been initialized with weights pretrained on Imagenet \cite{imagenet-ILSVRC15}.
The input size of the network depends on the case study: the input images of whale flukes are resized to $224 \times 448$
because of the shape of the region of interest; the input images for manta ray pattern
are of shape $300 \times 300$ for InceptionV3 and $224 \times 224$ for MobileNetV2 (pretrained weights for MobileNetV2 are available only for some input sizes).
Images are preprocessed the same way as it was done for the model used for fine-tuning (pixel values are scaled from [0,255] to [-1,1]).
\subsubsection{Training} \label{sec:exp-train}
The Adam optimizer \cite{adam} is used for all experiments with a learning rate $10^{-5}$ and other hyperparameters with default values ($\beta_1 = 0.9$, $\beta_2 = 0.999$).
We used a learning rate $10^{-5}$ because higher values did not work well with the pretrained weights (the same has been observed in \cite{DBLP:in-defence-triplet} while training the pretrained network with the triplet loss).
In order to produce an accurate evaluation of the performance of the network, we perform k-fold cross-validation for the first experiment (Section~\ref{sec:ft-inception}).
All splits are done with respect to individuals so each individual appears only in training or test split.
Each dataset is split in five parts and five rounds of training are completed with four folds allocated for training and one fold for testing.
All experiments have been run on a cluster with two Tesla M40 24GB GPUs and 6 CPUs.
\subsection{Performance evaluation}
\subsubsection{Fine-tuning InceptionV3 based model} \label{sec:ft-inception}
\begin{figure}[!t]
\centering
\includegraphics[width=3.3in]{moskv8.pdf}
\caption{Top-k accuracy over 5 splits on manta ray patterns with \textit{Inception-Ft} configuration.}
\label{fig:bar-chart-acc}
\end{figure}
\begin{table}
\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{15pt}
\caption{Performance of \textit{Inception-Ft} model on humpback whales and manta rays datasets separately (metrics are averaged over 5 splits)}
\label{tab:exp-baseline}
\centering
\begin{tabular}{ccc}
\hline
\multicolumn{1}{l}{} & \multicolumn{2}{c}{Dataset} \\
Metrics & Humpback whales & Manta rays \\
\hline
Top-1 & 62.78\%$\pm$1.64 & 62.05\%$\pm$3.24 \\
\hline
Top-5 & 88.20\%$\pm$0.67 & 93.65\%$\pm$1.83 \\
\hline
Top-10 & 93.46\%$\pm$0.63 & 97.03\%$\pm$1.11\\
\hline
TPR & 73\% & 71\% \\
\hline
AUC & 0.980 & 0.966 \\
\hline
\end{tabular}
\end{table}
\begin{figure}[!t]
\centering
\includegraphics[width=3.3in]{moskv9.pdf}
\caption{Top-k accuracy significantly increases at the second prediction for all configurations.}
\label{fig:top-k-acc}
\end{figure}
We fine-tune models with InceptionV3 convolutional layers, a global pooling layer and a fully connected layer with 256 outputs on each dataset separately. We name this configuration \textit{Inception-Ft} (fine-tuned).
The metrics TPR and AUC are calculated over all possible pairs for the test fold when $\text{FAR} = 0.01$ (around 45,000 pairs for manta rays with approximately 2,000 positive pairs depending on the split; around 180,000 pairs with approximately 1,800 positive for the whales dataset). Top-k accuracy is computed for the query set where there are two matching images in the database for each query pattern. We also explore how the accuracy changes depending on the number of matches present in the database in Section~\ref{sec:experiments-number-m}.
The results of training over five splits on the manta ray dataset show that accuracy does not vary significantly over the splits, see Fig.~\ref{fig:bar-chart-acc}.
The top-1 accuracy is $62\%$ for both datasets and the top-10 accuracy is $93\%$ for humpback whales and $97\%$ for manta rays (Table~\ref{tab:exp-baseline}).
Moreover, the graph in Fig.~\ref{fig:top-k-acc} shows that the top-k accuracy increases sharply at the second prediction ($k=2$) and top-3 accuracy is over 90\% for \textit{Inception-Ft} configuration.
We cannot make a meaningful comparison with previous works as the results have been reported on different datasets and the source code is not publicly available.
Manta Matcher \cite{manta_matcher} demonstrates 50.97\% top-1 and 67.64\% top-10 accuracy on a dataset of 720 images of 265 different manta rays.
We think that our dataset is more challenging as it contains images taken from a wider variety of angles and illumination conditions (Fig.~\ref{fig:manta-accepted}).
The best results to our knowledge for re-identification of humpback whale flukes have been reported in \cite{review-curvature}.
The top-1 accuracy of 80\% has been achieved on a dataset of a similar size. However, the method is using integral curvature representation of the trailing edge of the flukes and is specifically designed for humpback whales. Our method is generic and not specialized for a particular species.
For the rest of the experiments we change one hyperparameter to evaluate its effect and all other parameters are kept unchanged;
the experiments are performed on one split of manta ray dataset.
\begin{table}
\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{15pt}
\caption{The larger model based on InceptionV3 demonstrates better performance than the smaller model based on MobileNetV2}
\label{tab:exp-models}
\centering
\begin{tabular}{ccc}
\hline
\multicolumn{1}{l}{} & \multicolumn{2}{c}{Base network} \\
Metrics & MobileNetV2 & InceptionV3 \\
\hline
Top-1 & 52.06\%$\pm$4.77 & 64.18\%$\pm$4.55 \\
\hline
Top-5 & 89.18\%$\pm$1.85 & 95.65\%$\pm$1.15 \\
\hline
Top-10 & 95.47\%$\pm$1.40 & 97.78\%$\pm$0.62 \\
\hline
TPR & 60\% & 73\% \\
\hline
AUC & 0.970 & 0.983 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{15pt}
\caption{Not normalized embeddings performs better than $l_2$-normalized}
\label{tab:exp-norm}
\centering
\begin{tabular}{ccc}
\hline
\multicolumn{1}{l}{} & \multicolumn{2}{c}{Embeddings} \\
Metrics & $l_2$-normalized & Not normalized \\
\hline
Top-1 & 48.72\%$\pm$4.06 & 64.18\%$\pm$4.55 \\
\hline
Top-5 & 88.50\%$\pm$1.62 & 95.65\%$\pm$1.15 \\
\hline
Top-10 & 91.57\%$\pm$1.75 & 97.78\%$\pm$0.62 \\
\hline
TPR & 61\% & 73\% \\
\hline
AUC & 0.959 & 0.983 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{6pt}
\caption{Accuracy is not sensitive to the dimension of the embedding space}
\label{tab:exp-embedding}
\centering
\begin{tabular}{cccc}
\hline
\multicolumn{1}{l}{} & \multicolumn{3}{c}{Embedding length} \\
Metrics & 128 & 256 & 512 \\
\hline
Top-1 & 64.46\%$\pm$3.40 & 64.18\%$\pm$4.55 & 65.75\%$\pm$4.80 \\
\hline
Top-5 & 95.33\%$\pm$1.08 & 95.65\%$\pm$1.15 & 94.67\%$\pm$1.61\\
\hline
Top-10 & 97.76\%$\pm$0.90 & 97.78\%$\pm$0.62 & 97.47\%$\pm$0.72\\
\hline
TPR & 72\% & 73\% & 70\% \\
\hline
AUC & 0.983 & 0.983 & 0.980 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{15pt}
\caption{Extensive augmentation (rotations up to $360^{\circ}$ and flips) of input images improve performance compared to only small amount of augmentation (rotations up to $10^{\circ}$)}
\label{tab:exp-aug}
\centering
\begin{tabular}{ccc}
\hline
\multicolumn{1}{l}{} & \multicolumn{2}{c}{Augmentation} \\
Metrics & Small augmentation & Extensive augmentation \\
\hline
Top-1 & 54.00\%$\pm$3.32 & 64.18\%$\pm$4.55 \\
\hline
Top-5 & 92.03\%$\pm$1.62 & 95.65\%$\pm$1.15 \\
\hline
Top-10 & 95.09\%$\pm$1.28 & 97.78\%$\pm$0.62 \\
\hline
TPR & 58\% & 73\% \\
\hline
AUC & 0.970 & 0.983 \\
\hline
\end{tabular}
\end{table}
\subsubsection{Influence of the base network}
We evaluate the effect of the model architecture by training two networks with convolutional layers from InceptionV3 and MobileNetV2. The larger model InceptionV3 demonstrates better performance in both validation on pairs ($\text{TPR}$ 73\% vs 60\%) and top-k accuracy (top-1 accuracy 64\% vs 52\%), see Table~\ref{tab:exp-models}. However, the difference in performance decreases for higher $k$ and top-10 accuracy is 97\% for Inception based and 95\% for MobileNet based networks.
The advantage of MobileNetV2 is a slightly faster execution but our system
does not have to work in real-time.
The rest of the experiments are continued with InceptionV3 convolutional layers.
\begin{figure}[!t]
\centering
\includegraphics[width=3.3in]{moskv10.pdf}
\caption{Accuracy of re-identification depending on the number of matching examples per query individual in the database.}
\label{fig:top-k-m-samples}
\end{figure}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.93\textwidth]{moskv11.pdf}
\caption{Three examples of correct predictions. All closest predictions share visual similarity to the query image. The pattern is correctly matched even for examples with a challenging viewpoint and illumination. Photo credits: Fabrice Jaine, Linda Earthwatch, John Lawson, Chris Garraway, Chris Garraway, Chris Kim, Maggie McNeil, Chris Garraway, Rebecca Fonskov, Kathy Townsend, Chris Dudgeon, Chris Garraway, Sarah Williamson, Ryan Jeffery, Amelia Armstrong, Ryan Jeffery, Josh Gransbury, Chris Garraway.}
\label{fig:preds-correct}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.93\textwidth]{moskv12.pdf}
\caption{Three examples of incorrect matching (no match within ten closest predictions). Two images on the right are the only matching examples for each query image in the database. The query images are difficult because of the oblique angle that limits the visibility of the whole pattern. Top-3 predictions share some visual similarity to the query image. Photo credits: John Gransbury, Ian Christie, Amelia Armstrong, Mark Gray, Kathy Townsend, Kathy Townsend, Graeme Haas, Gerard Smith, Amelia Armstrong, Fabrice Jaine, Michael Rowett, Mounties Earthwatch, Mark Atkinson, Lydie Couturier, Chris Garraway, Kathy Townsend, Chris Garraway, Deg Ed.}
\label{fig:preds-incorrect}
\end{figure*}
\subsubsection{Effect of embedding normalization} \label{sec:exp-norm}
FaceNet architecture \cite{schroff2015facenet} uses $l_2$-normalization whereas Hermans et al. \cite{DBLP:in-defence-triplet} argue that forcing the norm of the embedding to $1$ does not improve performance.
Our experiments demonstrate that restricting the embedding
space to a hypersphere decreases the accuracy and metrics for verification on pairs.
For example, top-1 accuracy drops from 64\% to 48\% when we apply $l_2$-normalization, TPR decreases from 73\% to 61\% see Table~\ref{tab:exp-norm}.
Therefore, the rest of the experiments was done without $l_2$-normalization.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.95\textwidth]{moskv13.pdf}
\caption{Visualization of embeddings computed for the manta ray test set (best viewed in colour) using t-SNE \cite{vandermaaten2008visualizing}. Embeddings for manta Telluno and manta Priapus form tight clusters and show that the learned representation is invariant to rotations, a viewing angle and small occlusions. Mixing between classes happens when the pattern has several sparse dots (manta Paw Paw and manta Nova; manta Kimba and manta Cousteau). Photo credits (in a clockwise order starting from manta Kimba: Steward Barry, Mark Gray, Fabrice Jaine, Nigel Marsh, Kathy Townsend, Lydie Couturier, Chris Garraway, Chris Garraway, Chris Garraway, Matt Prunty.}
\label{fig:tsne-manta}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.95\textwidth]{moskv14.pdf}
\caption{Visualization of embeddings computed for the test set for humpback whale flukes (best viewed in colour) using t-SNE \cite{vandermaaten2008visualizing}. Photo credit: Happywhale organization. List of photo credits is provided in acknowledgment.}
\label{fig:tsne-whales}
\end{figure*}
\subsubsection{Influence of embedding dimension}
We tested three values for the dimension of the embedding space, 128, 256 and 512. Averaged results are reported in Table~\ref{tab:exp-embedding}. The difference between achieved accuracy is statistically insignificant and we select dimension of 256 for all other experiments.
Experiments with smaller embedding spaces (dimensions 32 and 64) showed inferior performance compared to higher dimensional spaces.
\subsubsection{Effect of data augmentation}
We have investigated the effect of data augmentation on the manta ray dataset.
The pattern on a manta ray belly may appear at different angles so extensive data augmentation including full rotations and flips has been applied to \textit{Inception-Ft} model.
We train the same model with rotations to only 10 degrees and no flips to estimate the influence of data augmentation on the performance.
The experiment shows (Table~\ref{tab:exp-aug}) that the performance results of the tested model are lower when less augmentation is applied during training: top-1 accuracy drops significantly, 54\% vs 64\%, and $\text{TPR}$ has dropped to 58\% compared to 73\%.
This demonstrates that rotations and flips of training examples facilitate learning of pattern invariance to rotations.
However, the difference in top-5 and top-10 accuracy is less marked.
\subsubsection{Number of matching individuals in the database} \label{sec:experiments-number-m}
Previous experiments in this paper have been conducted under the condition that there are two matching images for each query individual in the database. This experiment compares accuracy for a different number of matching individuals (from one to five), see Fig.~\ref{fig:top-k-m-samples}. The fewer images in the database for each query individual, the more difficult it is for the network to find the right match.
The number of matching examples for an individual in the database is more important for top-1 than for top-5 or top-10 accuracy. Top-1 accuracy is around 45\% for only one matching image, it increases to 64\% for two matches and reaches 81\% when there are five images in the database for each individual. Top-10 accuracy reaches 98\% with at least three images per individual in the database which is beneficial for the practical application.
\subsubsection{Visualization of predictions}
Fig.~\ref{fig:preds-correct} shows three query images and top-5 predictions of the system.
All predictions share visual similarity with a query image.
Three examples of incorrect matches alongside with top-3 predictions and two matching examples from the database are shown in Fig.~\ref{fig:preds-incorrect}. These examples are challenging as the pattern is only partly visible because of the oblique angle.
We analyze the learned representation
with t-SNE \cite{vandermaaten2008visualizing}.
The t-SNE algorithm maps a high dimensional space into a two-dimensional while preserving the similarity between points. The t-SNE plot for the manta ray test set (see Fig.~\ref{fig:tsne-manta}) shows examples where embeddings for the same manta ray (manta Telluno, manta Priapus) are clustered together even when the viewpoint is different and small occlusions are present (water bubbles, small fish). Embeddings are less separated for the less distinguishable markings where a pattern consists of a small number of black marks placed sparsely (manta Paw Paw and manta Nova; manta Kimba and manta Cousteau). On the t-SNE plot for the humpback whales test set (see Fig.~\ref{fig:tsne-whales}) we observe that individuals are clustered together even when the fluke is visible from different distances (whale 120). The system is invariant to the pose of the fluke (whale 101, whale 21) and viewpoint position (whale 25). The mix between whales occurs for some totally black flukes (whales 136, 12, 72) or for the flukes with a similar colour pattern (whale 61 and whale 21).
\section{Conclusion}
We have presented a novel visual re-identification system for manta rays that is robust to viewpoint changes, variations in lighting and small occlusions. The results have been achieved by using a combination of InceptionV3 model, the semi-hard triplet mining strategy, the triplet loss function and an extensive geometric augmentation of the input images. The practical value of the system been demonstrated on a manta ray dataset and an humpback whale dataset.
The system requires the user to localize the region of interest by drawing a bounding box around it.
In the future, we plan to further improve the system by automating the localization of the patterns of interest.
One possible stategy is to train the network on auxiliary tasks like learning to predict the locations of specific body landmarks (tip of the wings and gills of manta rays, fluke tips and notch for whales). This would force the network to learn about the morphology of the animal.
This ability should help induce a better representation of the spatial position of the pattern with respect to the body.
\section*{Acknowledgment}
The authors would like to thank Project Manta \\ (https://sites.google.com/site/projectmantasite/home) and Happywhale organization (happywhale.com) for the datasets of images. Photo credits for published images of humpback whale flukes: Alethea Leddy, Barry Gutradt, Casey Clark, Channel Islands NMS Naturalist Corps, Colin Garland, Dale Frink, Fernando Arcas, JB, John Calambokidis, Kate Cummings, Kate Spencer, Mark Girardeau, Richard Jackson, Ryan Lawler, Traci Phillips.
Computational resources and services used in this work were provided by the HPC and Research Support Group, Queensland University of Technology, Brisbane, Australia.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2019-03-01T02:06:41",
"yymm": "1902",
"arxiv_id": "1902.10847",
"language": "en",
"url": "https://arxiv.org/abs/1902.10847"
}
|
\section{Introduction}
Let $A$ and $B$ be two finite subsets of the 2-dimensional sphere $S^2$ such that $|A|,|B| \ge 3$, and
let $f,\iota : (S^2,A) \rightarrow (S^2,B)$ be two maps of pairs:
$$
f : (S^2, A) \rightarrow (S^2,B)
$$
a branched covering with branch values contained in $B$, and
$$
\iota : (S^2,A) \hookrightarrow (S^2,B)
$$
a homeomorphism identifying domain and range, and $A$ with a subset of $B$.
The {\it Epstein deformation space} $\mathcal{D}_{f,\iota}$ is defined as the equalizer of the induced maps on Teichm\"uller spaces
$$
f^*,\iota^* : \sT_{(S^2,B)} \rightarrow \sT_{(S^2,A)}.
$$
In unpublished work from the 1990s, A. Epstein showed that if $f$ is not Latt\`es\footnote{See \cite{Milnor:Lattes} for definitions.}, then $\mathcal{D}_{f,\iota}$ is either empty or
a complex $|B| - |A|$ dimensional submanifold of $\sT_{(S^2,B)}$ \cite{Epstein}.
This generalizes a seminal result of W. Thurston who showed that in the {\it postcritically finite} case, when $ A = B$,
either $\mathcal{D}_{f,\iota}$ is empty, $f$ is a Latt\`es example, or $\mathcal{D}_{f,\iota}$ contains exactly one point \cite{DH93}.
In particular, $\mathcal{D}_{f,\iota}$ is always connected in the postcritically finite case.
When $f$ is not post-critically finite, then $\mathcal{D}_{f,\iota}$ need not be connected \cite{HK:rational}, but so far there is only
one known class of examples.
\begin{problem} What are necessary and sufficient conditions for $\mathcal{D}_{f,\iota}$ to be connected?
\end{problem}
The counter-example to connectedness in \cite{HK:rational} is part of a class of quadratic rational maps studied by
Milnor in \cite{Milnor:Per}. These are denoted $\mathrm{Per}_n(0)$ and
have one periodic $n$-cycle
containing a single critical point, and the behavior of the other critical value under iteration is unspecified. If in addition the extra critical
value does not lie in the first critical $n$-cycle, the rational map is said to be in $\mathrm{Per}_n(0)^*$.
To an element $F \in \mathrm{Per}_n(0)^*$ let $f,\iota : (S^2,A) \rightarrow (S^2,B)$ be
defined so that $f$ is the topological covering underlying $F$, $\iota$ is a identification of
domain and range of $f$ so that $A \subset B$, $A$ is the critical $n$-cycle, and $B$ is the union of $A$ and the extra critical point.
For $n=3$, $\sT_{(S^2,A)}$ is a singleton set, and hence
$\mathcal{D}_{f,\iota}$ is the entire space $\sT_{(S^2,B)}$.
For $n=4$, the work in \cite{HK:rational} shows that $\mathcal{D}_{f,\iota}$ is not connected, and in fact has infinitely many
connected components. This leads to the following question.
\begin{question} Does $\mathcal{D}_{f,\iota}$ have a natural connected closure?
\end{question}
The points in $\mathcal{D}_{f,\iota}$ are sometimes called the {\it dynamical points} in the space of complex structures
on $\sT_{(S^2,B)}$. The {\it augmented deformation space} $\mathcal{AD}_{f,\iota}$,
or {\it ideal dynamical points}, is the subset of the Bers
augmented deformation space $\mathcal{AT}_{(S^2,B)}$ (see \cite{Bers:ATS}) defined as the equalizer of extensions
$$
\widetilde f^*,\widetilde\iota^* : \mathcal{AT}_{(S^2,B)} \rightarrow \mathcal{AT}_{(S^2,A)}
$$
of $f^*$ and $\iota^*$.
At the time of this writing, it is not known whether $\mathcal{AD}_{f,\iota}$ is connected in the $\mathrm{Per}_4(0)^*$ case. Our goal in this paper is to
extend the techniques of \cite{HK:rational}, and apply them
to give a partial description of the structure of $\mathcal{AD}_{f,\iota}$ and the closure of $\mathcal{D}_{f,\iota}$
within it. We prove the following theorem.
\begin{theorem} \label{main-thm} For $(f,\iota)$ associated to an element of $\mathrm{Per}_4(0)^*$,
the closure of $\mathcal{D}_{f,\iota}$ in augmented deformation space $\mathcal{AD}_{f,\iota}$ is not connected.
\end{theorem}
\subsection{Some background and ideas behind the proofs}
The question of whether and when $\mathcal{D}_{f,\iota}$ is connected has roots in work of Thurston from the 1980s, in which he
showed that if
$$
F : \mathbb P^1 \rightarrow \mathbb P^1
$$
is a non-Latt\`es rational map from the complex projective line
to itself with a finite post-critical set
$$
\bigcup_{n=1}^\infty F^{(n)}(\mbox{Crit}_F),
$$
and $f : (S^2,P) \rightarrow (S^2,P)$ is the corresponding branched covering of pointed spheres with domain and range identified,
and postcritical set $P$,
then the lifting map on holomorphic markings defines a contracting map on Teichm\"uller space
$$
f^* : \sT_{(S^2,P)} \rightarrow \sT_{(S^2,P)}.
$$
Thus, $f^*$ has a unique fixed point, and hence $\mathcal{D}_{f,\iota}$ is a singleton set.
This classical result suggests that there could be a contracting flow in the general case
when the identification map $\iota: (S^2,A) \rightarrow (S^2,B)$ is a strict inclusion on $A$.
If such a flow exists, there are two possibilities: one is that $\mathcal{D}_{f,\iota}$ is connected, and the other is that some
points may be pushed out to the boundary of $\sT_{(S^2,B)}$. This suggests looking at dynamical elements
of the augmented Teichm\"uller space to find a natural connected completion of $\mathcal{D}_{f,\iota}$.
The proof in \cite{HK:rational}, that $\mathcal{D}_{f,\iota}$ can be disconnected translates the problem about flows on Teichm\"uller spaces
to one about the topology of algebraic varieties. A key ingredient is to define an intermediate covering $\sM_f$
of $\sT_{(S^2,B)} \rightarrow \sM_{(S^2,B)}$ that is a natural quotient of the space of marked rational maps
combinatorially equal to $f$. The projection $\sM_f \rightarrow \sM_{(S^2,B)}$ is a finite covering, and
hence has the structure of a quasi-projective variety.
In the case when $F \in \mathrm{Per}_4(0)^*$, the space $\sM_f$ is isomorphic to a Zariski
dense subset of $\mathbb P^1 \times \mathbb P^1$:
$$
\sM_f = \mathbb P^1 \times \mathbb P^1 \setminus \mathcal Z \cup \mathcal L
$$
where $\mathbb P^1 \times \mathbb P^1 \setminus \mathcal L = \sM_{(S^2,A)} \times \sM_{(S^2,A)}$, and $\mathcal Z$ indicates the locus
where the both critical points of $f$ lie in the same periodic cycle.
With respect to this parameterization, the image of $\mathcal{D}_{f,\iota}$ in $\sM_f$ is the diagonal
$$
\mathcal{V}_{f,\iota}= \{(x,x) \in \mathbb P^1 \times \mathbb P^1 \ | \ (x,x) \not\in \mathcal Z \cup \mathcal L\}.
$$
Let $L_f$ be the group of covering automorphisms of $\sT_{(S^2,B)}$ over $\sM_{f,\iota}$. It was shown in \cite{HK:rational} that
the projection of $\mathcal{D}_{f,\iota}$ on $\mathcal{V}_{f,\iota}$ is a regular covering with covering automorphism
group a proper subgroup $S_{f,\iota} \subset L_f$. In particular, $S_{f,\iota}$ acts transitively on fibers of $\mathcal{D}_{f,\iota} \rightarrow \mathcal{V}_{f,\iota}$.
Choose a basepoint $d_0 \in \mathcal{D}_{f,\iota}$ and let $v_0$ be its image in $\mathcal{V}_{f,\iota}$. Then
we can identify $L_f$ with $\pi_1(\sM_f,v_0)$ and $S_{f,\iota}$ with a subgroup of the fundamental group. Let $E_{f,\iota}$ be
the image of $\pi_1(\mathcal{V}_{f,\iota},v_0) \rightarrow \pi_1(\sM_f,v_0)$ induced by the inclusion map. Then
$E_{f,\iota}$ is the subgroup of $S_{f,\iota}$ that fixes the component of $\mathcal{D}_{f,\iota}$ containing $d_0$;
$E_{f,\iota}$ has infinite index in $S_{f,\iota}$; and hence $\mathcal{D}_{f,\iota}$ has infinitely many components.
In other words, the image of $\pi_1(\mathcal{V}_{f,\iota},v_0)$ is not sufficiently large in $\pi_1(\sM_f,v_0)$.
One way to try to rectify
this is to put both $\sM_f$ and $\mathcal{V}_{f,\iota}$ into a larger ambient space.
Let $\mathcal{AT}_{(S^2,B)}$ be the augmented Teichm\"uller space of $(S^2,B)$. Then $L_f$
extends to an action on $\mathcal{AT}_{(S^2,B)}$ giving a quotient $\mathcal{AM}_f$ and
a commutative diagram
$$
\xymatrix{
\sT_{(S^2,B)}\ar[d]_{/L_f} \ar@{^(->}[r] &\mathcal{AT}_{(S^2,B)}\ar[d]^{/L_f}\\
\sM_f \ar@{^(->}[r] &\mathcal{AM}_f.
}
$$
Similarly, $S_{f,\iota}$ acts on $\mathcal{AD}_{f,\iota}$ with quotient denoted $\mathcal{AV}_{f,\iota}$. The spaces
$\mathcal{AV}_{f,\iota}$ and $\mathcal{AM}_f$ have the advantage of being algebraic geometric sets, and can be studied
via singularity theory.
We define a connected pure 1-dimensional algebraic subset $X$ of a
blowup $\widetilde {\mathbb P^1 \times \mathbb P^1}$
of $\mathbb P^1 \times \mathbb P^1$ and an embedding $X \rightarrow \mathcal{AV}_{f,\iota}$ that is surjective except possibly
a finite set of points (Proposition~\ref{connectedness-prop}). By studying
properties of $X$ and its embedding in $\widetilde {\mathbb P^1 \times \mathbb P^1}$ we prove Theorem~\ref{main-thm}.
\subsection{Organization} In Section~\ref{teich-sec}, we give necessary definitions of Teichm\"uller and moduli spaces for
rational maps, and their augmented versions. In Section~\ref{topology-sec},
we prove some general properties of complements of plane algebraic curves
and their blowups.
In Section~\ref{proof-sec}, we apply these ideas to the $\mathrm{Per}_4(0)^*$ case, and prove Theorem~\ref{main-thm}.
\subsection{Further questions}
There are still many open questions along the lines of this investigation. So far, the $\mathrm{Per}_4(0)^*$ example is the only known case when $\mathcal{D}_{f,\iota}$ is disconnected. Are there others?
Is $\mathcal{AD}_{f,\iota}$ connected in the $\mathrm{Per}_4(0)^*$ case, and is $\mathcal{AD}_{f,\iota}$ connected in general?
The analysis in this paper suggest general approaches to these questions, which we leave for further investigation.
\subsection{Some comments on notation} This paper grew out of the ideas in \cite{HK:rational}, but some of the notation
has changed.
Most notably, we refer to the Teichm\"uller space $\mathcal T_f$ of
rational maps combinatorially equivalent to $f$, and its corresponding Moduli space $\mathcal M_f$.
Thus, Epstein's deformation space $\mathcal D_{f,\iota}$ will be considered as a subspace of $\mathcal T_f$ rather than of the isomorphic
space $\mathcal T_{(S^2,B)}$ (also known as the {\it parameter space}).
The space $\mathcal M_f$ may be more familiarly known as a connected component of a Hurwitz space of rational maps \cite{Ramadas:Hurwitz} and was
denoted by $\mathcal W_f$ in \cite{HK:rational}.
A smaller change is the use of $\mathcal D_{f,\iota}$ for the deformation space $\mathcal D_f$ to emphasize the dependence on both
the topological branched covering of pairs $f$ and the identification $\iota$. Similarly, we changed the notation of $S_f$
to $S_{f,\iota}$. These ease our transition from a discussion of deformation space to augmented deformation space.
\subsection{Acknowledgments} I would like to thank
X. Buff, S. Koch, C. McMullen, R. Ramadas, and L. D. Tr\'ang for helpful references and discussions, and
the anonymous referee for their careful reading and useful comments.
\section{Background definitions}
In this section, we recall definitions and properties of Teichm\"uller space, moduli space and deformation space
for rational maps and their augmented versions. For more details about the general theory see, for example, \cite{DMcompactification}, \cite{Bers:ATS},
\cite{HubKoch}, \cite{Ramadas:Hurwitz} and \cite{Selinger:Aug}. We also recall definitions of liftables $L_f$ and special liftables
$S_{f,\iota}$ from \cite{HK:rational}, and examine their extensions to the augmented spaces.
\subsection{Teichm\"uller spaces for rational maps}\label{teich-sec}
Let $A$ be a finite subset of 3 or more points of the topological 2-sphere $S^2$. The
{\it Teichm\"uller space}
$
\sT_{(S^2,A)}
$
of {\it holomorphic markings} on $(S^2,A)$ is the collection of orientation preserving homeomorphisms
$$
\phi : (S^2,A) \rightarrow (\mathbb P^1,\phi(A))
$$
defined up to post-composition by automorphisms of $\mathbb P^1$ (i.e., M\"obius transformations) and pre-composition by self-homeomorphisms of $S^2$ that are isotopic rel $A$ to the identity.
Similarly, given a finite branched covering of pairs $f : (S^2, A) \rightarrow (S^2, B)$, where $\infty > |A|, |B| \ge 3$ and
$B$ contains the branch values (or critical values) of $f$, we define the {\it Teichm\"uller space} $\sT_{f}$ of {\it holomorphic markings}
on $(f,A,B)$ as the set of commutative diagrams
$$
\xymatrix{
(S^2,A) \ar[r]^-{\psi}\ar[d]_f &(\mathbb P^1,\psi(A))\ar[d]^F\\
(S^2,B) \ar[r]^-\phi& (\mathbb P^1,\phi(B))
}
$$
also denoted by $(\phi,\psi,F)$,
where $F$ is a rational map, $\phi$ and $\psi$ are homeomorphisms. Two triples $(\phi,\psi,F)$ and $(\phi_1,\psi_1,F_1)$ in $\sT_f$ are
{\it equivalent} if there are homeomorphisms $\alpha: (S^2,A) \rightarrow (S^2,A)$ isotopic to the
identity map rel $A$ and $\beta: (S^2,B) \rightarrow (S^2,B)$ isotopic to the identity rel $B$, and biholomorphic maps $\mu,\nu : \mathbb P^1 \rightarrow \mathbb P^1$
so that the diagram
$$
\xymatrix{
(\mathbb P^1,\psi_1(A)) \ar[d]_{F_1}&(S^2,A) \ar[l]_-{\psi_1} \ar[d]_f\ar[r]^-{\alpha} &(S^2,A)\ar[d]_f\ar[r]^\psi &(\mathbb P^1,\psi(A))\ar[d]_F\\
(\mathbb P^1,\phi_1(B)&(S^2,B) \ar[l] _-{\phi_1} \ar[r]^-{\beta} &(S^2,B) \ar[r]^\phi &(\mathbb P^1,\phi(B))
}
$$
commutes, and $ \mu = \psi \circ \alpha \circ \psi_1^{-1}$ and $\nu = \phi \circ \beta \circ \phi_1^{-1}$.
By these definitions, $\sT_{f}$ comes with natural surjections
$$
\xymatrix{
&\sT_{f}\ar[dl]_q\ar[dr]^{p_U} \\
\sT_{(S^2,B)} &&\sT_{(S^2,A)}
}
$$
recording the holomorphic markings of the domain space
($p_U$) and the range space ($q$). Furthermore, $\sT_f$ can be thought of as the graph of a {\it lifting map}
$$
f^* : \sT_{(S^2,B)} \rightarrow \sT_{(S^2A)}
$$
defined by $f$. Thus, $q$ is an isomorphism of holomorphic spaces.
We create an iteration scheme from $f$ by partially identifying the domain $(S^2,A)$ and range $(S^2,B)$ of $f$. That is,
we fix a homeomorphism $\iota : S^2 \rightarrow S^2$ that restricts to an inclusion $\iota |_A : A \hookrightarrow B$.
Then we have a map of pairs
$$
\iota : (S^2,A) \rightarrow (S^2,B).
$$
Let $p_L = \iota^* \circ q$, where $\iota^* : \sT_{(S^2,B)} \rightarrow \sT_{(S^2,A)}$ is the map that
takes $\phi \in \sT_{(S^2,B)}$ to the class in $\sT_{(S^2,A)}$ defined by
$$
\phi \circ \iota : (S^2,A) \rightarrow (\mathbb P^1,(\phi \circ \iota)(A)) = (\mathbb P^1, \phi(A)).
$$
Then the elements of $\sT_f$ can be thought of as the holomorphic markings of the branched covering $f$, and $p_U$ and $p_L$ record
the induced marked holomorphic structures $(S^2,A)$ by the domain and range.
The
Epstein's {\it deformation space} $\mathcal{D}_{f,\iota}$ is
the subspace of $\sT_f$
consisting of holomorphic structures on the covering and base of $f$ that are equivalent relative to $A$. That
is, $\mathcal{D}_{f,\iota}$ consists of the triples $(\phi,\psi,F)$ so that
$\phi^{-1} \circ \psi$ is isotopic to the identity relative to $A$, or equivalently
$$
f^*\phi = \iota^*\phi.
$$
Another way to say this is that $\mathcal{D}_{f,\iota}$ is the equalizer in $\sT_f$ of the maps $p_U$ and $p_L$, that is
$$
\mathcal{D}_{f,\iota} = \{ \phi \in \sT_f \ | \ p_U(\phi, i) = p_L(\phi)\}.
$$
We think of $\mathcal{D}_{f,\iota}$ as the {\it dynamical Teichm\"uller space} for $(f,\iota)$.
\subsection{Moduli spaces}\label{moduli-sec}
Let $\sM_{(S^2,A)}$ be the space of
embeddings
$$
A \hookrightarrow \mathbb P^1
$$
up to post-composition by a M\"obius transformation. Then the restriction map $[\phi] \mapsto [\phi |_A]$
defines a regular covering map
$$
\sT_{(S^2,A)} \rightarrow \sM_{(S^2,A)}
$$
with covering automorphism group equal to the
{\it mapping class group} $\mathrm{Mod}(S^2,A)$
of orientation preserving homeomorphisms $h : S^2 \rightarrow S^2$ that
fix the points of $A$ up to isotopy rel $A$. (Unlike the case for Teichm\"uller spaces and moduli space of
general surfaces, $\mathrm{Mod}(S^2,A)$ on $\sT_{(S^2,A)}$ acts without fixed points.)
Similarly let $\sM_f$ be the space of commutative diagrams
$$
\xymatrix{
A\ar[d]_{f |_A}\ \ar@{^(->}^j [r] &\mathbb P^1\ar[d]^F\\
B\ \ar@{^(->}^i [r]&\mathbb P^1.
}
$$
where $(i,j,F)$ is defined up to modifications of the domain and range of $F$ by M\"obius transformations.
Then the map $(\phi,\psi, F) \mapsto (\phi |_B, \psi |_A, F)$ defines a covering map
$$
\sT_f \rightarrow \sM_f,
$$
with covering automorphism group $L_f \subset \mathrm{Mod}(S^2,B)$ called the subgroup of {\it liftables}
consisting of elements $h \in \mathrm{Mod}(S^2,B)$ such that for some $h' \in \mathrm{Mod}(S^2,A)$ the diagram
$$
\xymatrix{
(S^2,A)\ar[d]_-f \ar[r]^{h'} &(S^2,A)\ar[d]^-f \\
(S^2,B) \ar[r]^h &(S^2,B)
}
$$
commutes.
\begin{remark}
Since $A$ and $B$ are assumed to contain at least three points, $f: (S^2,A) \rightarrow (S^2,B)$ can have no non-trivial covering automorphisms.
In the degree 2 case, this follows from the fact that $f$ must have exactly two branch points and two branch values. Any other marked point would
lie in the unbranched part of the covering. Even in the case when the degree of $f$ is greater than 2, the fact that $f$ may be realized as a
rational map implies that all covering automorphisms must be conjugate to a M\"obius transformation. If there are at least three points in $A$,
then the M\"obius transformation must be the identity.
\end{remark}
By the definition of $L_f$ and since $f$ has no non-trivial covering automorphisms, $f$ defines a
unique lifting map
$$
f^\sharp: L_f \rightarrow \mathrm{Mod}(S^2,A)
$$
where $f^\sharp h = h'$.
The following Proposition was shown using slightly different language in \cite{K13} (cf. \cite{HK:rational} Proposition 2.2).
For the convenience of the reader, we include a proof below.
\begin{proposition} The map $\sT_{f} \rightarrow \sM_{f}$ is a regular covering with automorphism group $L_{f}$.
\end{proposition}
\begin{proof}
Let $h \in L_{f}$. Take any $(\phi,\psi,F) \in \sT_{f}$. Then we have
the commutative diagram
\begin{eqnarray}\label{modaction-eq}
\xymatrix{
(S^2,A)\ar[d]_f &(S^2,A) \ar[l]^{f^\sharp h} \ar[r]^{\psi}\ar[d]_f&(\mathbb P^1, \psi(A))\ar[d]^F\\
(S^2,B) & (S^2,B)\ar[l]^h \ar[r]^\phi &(\mathbb P^1,\mathcal B).
}
\end{eqnarray}
Since $h$ and $f^\sharp h$ fix $B$ and $A$, respectively,
$(\phi _B, \psi _A, F)$ and $(\phi\circ h, \psi\circ f^\sharp h, F)$ map to the same element in $\sM_{f}$.
Conversely, suppose $(\phi,\psi,F)$ and $(\phi_1,\psi_1, F_1)$ both map to equivalent elements in $\sM_{f}$.
Then by definition, there are embeddings $i : B \rightarrow \mathbb P^1$ and $j : A \rightarrow \mathbb P^1$ so
that $i = \phi |_B = \phi_1 |_B$, $j = \psi_1 |_A = \psi |_A$ and there are M\"obius transformations $\mu$ and $\nu$ such that
$F = \mu^{-1} \circ F_1 \circ \nu$. We can assume for simplicity that $F = F_1$ by replacing $(\phi_1,\psi_1,F_1)$ by
the equivalent element $(\mu^{-1} \circ\phi_1,\nu^{-1}\circ \psi_1 , F)$ in $\sT_{f}$.
Then the common image of $(\phi,\psi,F)$ and $(\phi_1,\psi_1,F)$ is some $(i,j,F)$ in $\sM_{f}$.
Thus the images of $\phi$ and $\phi_1$ are the same in $\sM_{(S^2,B)}$, the images of $\psi$ and $\psi_1$
are the same in $\sM_{(S^2,A)}$, and hence there are mapping classes $h \in \mathrm{Mod}(S^2,B)$ and $f^\sharp h \in \mathrm{Mod}(S^2,A)$
such that $\phi = h \circ \phi_1$ and $\psi = f^\sharp h \circ \psi_1$. The maps $h$ and $f^\sharp h$ complete a diagram of the form (\ref{modaction-eq}),
and thus $h \in L_{f}$.
\end{proof}
Let $\overline q : \sM_f \rightarrow \sM_{(S^2,B)}$ be the map sending $(i,j,F)$ to the equivalence class containing $i$ in $\sM_{(S^2,B)}$;
let $\overline p_U: \sM_f \rightarrow \sM_{(S^2,A)}$ be the map sending $(i,j,F)$ to the equivalence class containing $j$ in $\sM_{(S^2,A)}$;
let $\overline \iota^* : \sM_{(S^2,B)} \rightarrow \sM_{(S^2,A)}$ be the forgetful map sending an inclusion $i : B \rightarrow \mathbb P^1$
to $i \circ \iota |_A$; and
let $\overline p_L = \overline \iota^* \circ \overline q$.
Then we have the commutative diagram
$$
\xymatrix{
&\sT_{f} \ar[dl]_{\simeq}^q \ar[dr]^{p_U}\ar[dd]^{\rho}\\
\sT_{(S^2,B)}\ar[dd]_{/ \mathrm{Mod}(S^2,B)}\ar[dr]^{/L_f}&&\sT_{(S^2,A)}\ar[dd]^{/ \mathrm{Mod}(S^2,A)}\\
&\sM_{f} \ar[dl]_{\overline q}\ar[dr]^{\overline p_U}\\
\sM_{(S^2,B)} && \sM_{(S^2,A)}.
}
$$
All vertical arrows and the left three diagonal arrows are unbranched covering maps. The right diagonal arrows may not be surjective (see \cite{BEKP:Pullback}).
Let $\mathcal{V}_{f,\iota}$ be the image of the deformation space $\mathcal{D}_{f,\iota}$ in $\mathcal M_f$. Then we have
$$
\mathcal{V}_{f,\iota} = \mathrm{Eq}(\overline p_L, \overline p_U).
$$
\subsection{Stabilizer of deformation space}
Fix a basepoint $d_0 \in \mathcal{D}_{f,\iota}$, and let $m_0 \in \sM_{f,\iota}$ be the image of $d_0$ under the map
$\rho$. Then we have an identification
$$
\ell: \pi_1(\sM_{f,\iota},m_0) \rightarrow L_f,
$$
defined by the path-lifting theorem for coverings. That is, for each $\gamma \in \pi_1(\sM_{f,\iota},m_0)$, we lift $\gamma$
to a path $\gamma'$ on $\sT_{(S^2,B)}$ based at $d_0$, and let $\ell(\gamma)$ be the
(unique, since the covering is regular) mapping class that takes $d_0$ to the end point
of $\gamma'$.
\begin{proposition} The restriction of $\rho : \mathcal T_f \rightarrow \mathcal M_f$ to
$\mathcal{D}_{f,\iota}$ gives a covering map
$$
\rho_{\mathcal D} : \mathcal{D}_{f,\iota} \rightarrow \mathcal{V}_{f,\iota},
$$
and the image $E_{f,\iota,d_0}$ of
$$
\pi_1(\mathcal{V}_{f,\iota}, m_0) \rightarrow \pi_1(\sM_{f,\iota},m_0) \overset \ell\rightarrow L_f
$$
is the stabilizer of the connected component of $\mathcal{D}_{f,\iota}$ that contains $d_0$.
\end{proposition}
\begin{proof} We show that the projection $\rho_{\mathcal{D}_{f,\iota}}$ satisfies the path-lifting theorem.
Let $d_0 = (\phi_0,\psi_0,F_0)$, and let
$(i_t,j_t,F_t)$ be a path in $\mathcal{V}_{f,\iota}$ with $m_0 = (j_0,i_0,F_0)$. Let $\xi$ and $\eta$ be a representatives of the class
of $\phi_0$ and $\psi_0$ so that the diagram commutes
$$
\xymatrix{
S^2 \ar[d]_f \ar[r]^\eta &\mathbb P^1 \ar[d]^F\\
S^2 \ar[r]^\xi &\mathbb P^1,
}
$$
$\eta |_A = \psi_0 |_A$ and $\xi |_B = \psi |_B$.
Let $p_t : (S^2,B) \rightarrow (S^2,B_t)$ be any continuous family of homeomorphisms. Then
$$
(\xi \circ p_t |_{B_t}, \eta \circ p_t |_{A_t}, F_t) \sim (i_t, j_t, F_t)
$$
and $(\xi \circ p_t, \eta \circ p_t, F_t)$ is a lift of $(i_t,j_t,F_t)$ and lies in $\mathcal{D}_{f,\iota}$.
Thus, as a restriction of
an unbranched covering map, $\rho_{\mathcal D}$ is itself is a covering map.
The rest follows from basic covering space theory.
\end{proof}
The lifting map $f^\sharp : L_f \rightarrow \mathrm{Mod}(S^2,A)$ is uniquely defined
and satisfies the commutative diagram
$$
\xymatrix{
(S^2,A)\ar[d]_f \ar[r]^{f^{\sharp}h} & (S^2,A)\ar[d]^f\\
(S^2,B) \ar[r]^h &(S^2,B).
}
$$
For a homeomorphism $h : (S^2,B) \rightarrow (S^2,B)$, let $h_A$ be the element of $\mathrm{Mod}(S^2,A)$ defined by
ignoring the points of $B \setminus A$. Then $h_A$ is the isotopy class of $h$ defined up to homeomorphisms
isotopic to the identity rel $A$. Define
\begin{eqnarray*}
\iota^\sharp : \mathrm{Mod}(S^2,B) &\rightarrow& \mathrm{Mod}(S^2,A)\\
h&\mapsto& h_A.
\end{eqnarray*}
Let $S_{f,\iota} \subset L_f$ be the equalizer
$$
S_{f,\iota} = \mathrm{Eq}(f^\sharp,\iota^\sharp) \subset L_f.
$$
By the identification of $L_f$ with $\pi_1(\sM_{f,\iota},m_0)$, $S_{f,\iota}$ can equivalently be defined as the
equalizer of the homomorphisms
$$
(\overline p_L)_*, (\overline p_U)_* : \pi_1(\sM_{f,\iota},m_0) \rightarrow \pi_1(\sM_{(S^2,A)},a_0)
$$
where $a_0 = \overline p_L(m_0) = \overline p_U(m_0)$, where $m_0 \in \mathcal{V}_{f,\iota} = \mathrm{Eq} (\overline p_L,\overline p_U)$.
\begin{proposition} [\cite{HK:rational} Proposition 2.5] The stabilizer in $L_f$ of $\mathcal{D}_{f,\iota}$ equals $S_{f,\iota}$, and
$S_{f,\iota}$ acts transitively on the fibers of the covering
$$
\mathcal{D}_{f,\iota} \rightarrow \mathcal{V}_{f,\iota}.
$$
Thus the covering is regular and $S_{f,\iota}$ is the group of covering automorphisms.
\end{proposition}
\begin{corollary} \label{Defconsuff-cor} The deformation space $\mathcal{D}_{f,\iota}$ is connected if and only if $\mathcal{V}_{f,\iota}$ is connected and $S_{f,\iota} = E_{f,\iota,d_0}$.
\end{corollary}
In the case when $(f,\iota)$ is associated to an element of $\mathrm{Per}_4(0)^*$, $E_{f,\iota,d_0}$ has infinite index in $S_{f,\iota}$ (\cite{HK:rational}, Proposition 2.11).
\subsection{Augmented spaces}\label{augmented-sec}
By a {\it rational curve} $\mathcal C$ we mean a nodal curve with the following properties
\begin{enumerate}[\hspace{\parindent}(a)]
\item the irreducible components of $\mathcal C$ are isomorphic to $\mathbb P^1$, and
\item the fundamental group of $\mathcal C$ is trivial.
\end{enumerate}
A {\it pre-stable rational curve} $(\mathcal C,\mathcal A)$ is a rational curve $\mathcal C$ together with a finite set $\mathcal A$ contained in the complement of the nodes of $\mathcal C$.
The set of nodes of $\mathcal C$ union the points of $\mathcal A$ form
the {\it distinguished points} of $\mathcal C$.
A {\it stable rational curve} is a pre-stable rational curve with the following additional property:
\begin{enumerate}[\hspace{\parindent}(c)]
\item the number of distinguished points on each irreducible component of $\mathcal C$ is greater than or equal to $3$.
\end{enumerate}
For each component $C$ of a pre-stable rational curve $(\mathcal C,\mathcal A)$ there are three possibilities:
\begin{enumerate}
\item $C$ is stable, i.e., it contains at least 3 distinguished points;
\item $C$ is unstable, and contains two nodes; or
\item $C$ is unstable, and contains one node and zero or one point in $\mathcal A$.
\end{enumerate}
Let $\Sigma^{\mbox{pre}}_{(S^2,A)}$ be the set of pre-stable rational curves $(\mathcal C,\mathcal A)$ with a bijection
$A \rightarrow \mathcal A$,
and let $\Sigma_{(S^2,A)} \subset\Sigma^{\mbox{pre}}_{(S^2,A)}$ be the space of stable rational curves.
Define a map
$$
\mathfrak s : \Sigma^{\mbox{pre}}_{(S^2,A)} \rightarrow \Sigma_{(S^2,A)}
$$
sending $(\mathcal C,\mathcal A)$ to the result of contracting components of $\mathcal C$ using the following
rule:
in case (1) leave $C$ alone; and in case (2) and (3) contract $C$ to a point. If $C$ contains a point of $\mathcal A$, then mark the
image of the contraction by that point.
This map is well-defined since in case (2) there is at most one point of $\mathcal A$ in $C$. We call $\mathfrak s$ the {\it stabilization map}.
A {\it marking} of a pre-stable rational curve $(\mathcal C,\mathcal A) \in \Sigma^{\mbox{pre}}_{(S^2,A)}$ is a quotient map
$$
\phi : (S^2,A) \rightarrow (\mathcal C,\mathcal A),
$$
such that $\phi$ is a homeomorphism when restricted to $S^2 \setminus \gamma$ for some multi-curve $\gamma \subset S^2 \setminus A$,
$\phi$ restricts to a bijection $A \rightarrow \mathcal A$, and the components
of $\gamma$ are in one-to-one correspondence with the nodes of $\mathcal C$. The curve $\gamma$ is called the {\it contracting curve} for $\phi$.
We consider two markings {\it equivalent} if they are the same up to post-composition by automorphisms
of $\mathcal C$ and pre-composition by homeomorphisms $(S^2,A) \rightarrow (S^2,A)$ that are isotopic to the identity rel $A$.
The collection of markings of pre-stable and stable rational curves by $(S^2,A)$ is denoted by $\mathcal{AT}^{\mbox{pre}}_{(S^2,A)}$ and
$\mathcal{AT}_{(S^2,A)}$, respectively.
Post-composition by $\mathfrak s$ defines a surjection
$$
\mathcal{AT}^{\mbox{pre}}_{(S^2,A)} \rightarrow \mathcal{AT}_{(S^2,A)}.
$$
The space $\mathcal{AT}_{(S^2,A)}$ is called the {\it augmented Teichm\"uller space} of $(S^2,A)$.
There is a natural topology on $\mathcal{AT}_{(S^2,A)}$ such that points on
$\mathcal{AT}_{(S^2,A)} \setminus \sT_{(S^2,A)}$ are the limits of sequences points on $\sT_{(S^2,A)}$ for which the length of $\gamma$ tends to zero (see \cite{Bers:ATS} for more precise definitions).
\begin{remark}\label{AugTeichAuts-rem}
The mapping class group $\mathrm{Mod}(S^2,A)$ extends to actions on $\mathcal{AT}^{\mbox{pre}}_{(S^2,A)}$ and $\mathcal{AT}_{(S^2,A)}$. For points in $\mathcal{AT}^{\mbox{pre}}$ the stabilizer contains a copy of the group of
M\"obius transformations.
Given a point in $\mathcal{AT}_{(S^2,A)}$, the stabilitzer is the free abelian group of mapping classes generated by Dehn twists along
the components of the contracting curve.
\end{remark}
The action of $\mathrm{Mod}(S^2,A)$ on $\mathcal{AT}_{(S^2,A)}$ defines a branched covering
$$
\mathcal{AT}_{(S^2,A)} \rightarrow \mathcal{AM}_{(S^2,A)}
$$
where $\mathcal{AM}_{(S^2,A)}$ is the space of inclusions
$$
A \hookrightarrow \mathcal C
$$
of $A$ into a rational curve $\mathcal C$
up to holomorphic automorphism of $\mathcal C$ that do not permute components (cf. \cite{DMcompactification}).
We now define the augmented Teichm\"uller and moduli spaces of $f$.
A rational map $F : (\mathcal C_U,\mathcal A) \rightarrow (\mathcal C_L,\mathcal B)$ is {\it pre-admissible} if
\begin{enumerate}[\hspace{\parindent}(a)]
\item $F$ defines a surjective map from $\mathcal C_U$ to $\mathcal C_L$ of generically constant degree that maps nodes to nodes;
\item locally near each node of $\mathcal C_U$, $F$ has generically constant degree; and
\item $(\mathcal C_L,\mathcal B)$ is a stable rational curve and $(\mathcal C_U,\mathcal A)$ is pre-stable.
\end{enumerate}
We say that $F$ is {\it admissible} if in addition
\begin{enumerate}[\hspace{\parindent} (b)]
\item $(\mathcal C_U,\mathcal A)$ is stable.
\end{enumerate}
The {\it augmented Teichm\"uller space} $\mathcal{AT}_f$ for $f$ is the collection of holomorphic markings
\begin{eqnarray*}
\xymatrix {
(S^2,A) \ar[r]^\psi\ar[d]_f &(\mathcal C_U,\mathcal A)\ar[d]^F\\
(S^2,B) \ar[r]^\phi & (\mathcal C_L,\mathcal B)
}
\end{eqnarray*}
where the horizontal maps are markings in $\mathcal{AT}_{(S^2,A)}$ and $\mathcal{AT}_{(S^2,B)}$ respectively, and $F$ is a pre-admissible covering.
Here, as in the definition of $\sT_f$, we take $(\phi,\psi,F)$ up to the natural equivalences.
With this definition, the projection $\widetilde q : \mathcal{AT}_f \rightarrow \mathcal{AT}_{(S^2,B)}$ defines an isomorphism.
Let $\widetilde p_U, \widetilde p_L : \mathcal{AT}_f \rightarrow \mathcal{AT}_{(S^2,A)}$ be defined by
\begin{eqnarray*}
\widetilde p_L (\phi,\psi,F) &=& \mathfrak s(\phi)\\
\widetilde p_U(\phi,\psi,F) &=& \mathfrak s(\psi)
\end{eqnarray*}
\begin{proposition} The subgroup of liftables $L_f \subset \mathrm{Mod}(S^2,B)$ extends to an action on $\mathcal{AT}_{f}$.
\end{proposition}
\begin{proof} Let $\phi : (S^2,B) \rightarrow (\mathcal C_L,\mathcal B)$ be an element of $\mathcal{AT}_{(S^2,B)}$, and let $\gamma$ be the contracting multi-curve.
Since $f$ is unbranched outside of $B$, it follows that $f^{-1}(\gamma)$ is a multi-curve on $S^2 \setminus A$, and
since $h$ is liftable
$$
f^\sharp h(f^{-1}(\gamma)) = f^{-1}(h(\gamma)).
$$
Thus $h \in L_f$ takes $(\phi,\psi,F)$ to $(\phi \circ h, \psi \circ f^\sharp h, F)$ where $\phi \circ h$ and $ \psi \circ f^\sharp h$
contract the curves $h(\gamma)$ and $f^{-1}(h(\gamma))$ respectively.
\end{proof}
Let $\mathcal{AM}_f = \mathcal{AT}_f/L_f$. Then the points of $\mathcal{AM}_f$ are defined by diagrams
$$
\xymatrix{
A\ar[d]_{f|_A}\ar@{^(->}[r]^i&\mathcal C_U\ar[d]^F\\
B\ar@{^(->}[r]^j &\mathcal C_L
}
$$
where $(\mathcal C_L, \mathcal B)$ is stable, $(\mathcal C_U,\mathcal A)$ is pre-stable, and the inclusions $i,j$
are defined up to holomorphic automorphisms of $\mathcal C_U$ and $\mathcal C_L$.
We denote an element by $(i,j,F)$.
Let $\overline p_U, \overline p_L : \mathcal{AM}_f \rightarrow \mathcal{AM}_{(S^2,A)}$ be defined by
\begin{eqnarray*}
\overline p_L ([\phi,\psi,F]) &=& \mathfrak s\circ j\\
\overline p_U([\phi,\psi,F]) &=& \mathfrak s\circ i
\end{eqnarray*}
While the action of $L_f$ on $\sT_f$ has no fixed points and the quotient map $\sT_f \rightarrow \sM_f$ is a covering,
the quotient map $\mathcal{AT}_f \rightarrow \mathcal{AM}_f$ can have branch points.
Summarizing, we have a commutative diagram of augmented spaces:
$$
\xymatrix{
&\mathcal{AT}_{f} \ar[dl]_{\widetilde q}\ar[dr]^{\widetilde p_U}\ar[dd]^{\widetilde\rho}\\
\mathcal{AT}_{(S^2,B)}\ar[dd]_{/ \mathrm{Mod}(S^2,B)}\ar[dr]^{/L_f}&&\mathcal{AT}_{(S^2,A)}\ar[dd]^{/ \mathrm{Mod}(S^2,A)}\\
&\mathcal{AM}_{f} \ar[dl]_{\overline q}\ar[dr]^{\overline p_U}\\
\mathcal{AM}_{(S^2,B)} && \mathcal{AM}_{(S^2,A)},
}
$$
The augmented deformation space is defined to be the equalizer
$$
\mathcal{AD}_{f,\iota} = \mathrm{Eq}(\widetilde p_U,\widetilde p_L)
$$
and contains the deformation space $\mathcal{D}_{f,\iota}$. Let $\mathcal{AV}_{f,\iota} = \widetilde \rho(\mathcal{AD}_{f,\iota})$.
Then we have
$$
\mathcal{AV}_{f,\iota} = \mathrm{Eq}(\overline p_U,\overline p_L),
$$
\subsection{Stabilizer of augmented deformation space}\label{stab-sec}
In this section, we give a necessary and sufficient condition for $\mathcal{AD}_f$ to be connected. Unlike in the case
for Corollary~\ref{Defconsuff-cor}, $\mathcal{AD}_f \rightarrow \mathcal{V}_f$ is not an unbranched covering. To get around this
we use the notion of regular neighborhoods.
Recall that, for a simplicial subcomplex $V$ embedded in a manifold $X$,
a {\it regular neighborhood} $N(V)$ of $V$ is an open subset of $X$ containing $V$ that
has a deformation retract to $V$.
\begin{proposition} The stabilizer in $L_f$ of $\mathcal{AD}_{f,\iota}$ equals $S_{f,\iota}$, that is, if $g \in L_f$ is such that
$g(\alpha) = \alpha$ for all $\alpha \in \mathcal{AD}_{f,\iota}$, then $g \in S_{f,\iota}$.
\end{proposition}
\begin{proof} The stabilizer in $L_f$ of $\mathcal{AD}_{f,\iota}$ must be contained in $S_{f,\iota}$, since $\mathcal{D}_{f,\iota} \subset \mathcal{AD}_{f,\iota}$.
Let $h \in S_{f,\iota}$ and let $(\phi,\psi, F) \in \mathcal{AD}_{f,\iota}$. Then by definition $f^\sharp h = i^\sharp h$. Thus,
we have a commutative diagram
$$
\xymatrix{
(S^2,A) \ar[r]^{f^\sharp h}\ar[d]_f& (S^2,A) \ar[d]^f\ar[r]^\psi& (\mathcal C_U,\mathcal A)\ar[d]^F\\
(S^2,B) \ar[r]^h &(S^2,B) \ar[r]^\phi & (\mathcal C_L,\mathcal B)
}
$$
and we have
$$
\widetilde f^* (h([\phi])) = [\psi \circ f^\sharp h] = [\psi \circ\iota^\sharp h] =\widetilde \iota^*(h([\phi])).
$$
Thus $h$ stabilizes $\mathcal{AD}_{f,\iota}$.
\end{proof}
\begin{remark} Conversely, one can ask whether if $g \in L_f$ and $g(\alpha) = \alpha$ for some $\alpha \in \mathcal{AD}_{f,\iota}$, then does it follow that
$g \in S_{f,\iota}$? This is not true in general, since points in the boundary of $\mathcal{AD}_{f,\iota}$ have extra automorphisms that need not be
in $S_{f,\iota}$ (see Remark~\ref{AugTeichAuts-rem}).
\end{remark}
\begin{proposition} \label{suff-prop} Suppose there is a connected quasi-projective variety
$X$ with $\mathcal{V}_{f,\iota} \subset X \subset \mathcal{AV}_{f,\iota}$ such that
$X$ has a regular neighborhood $N(X) \subset \mathcal{AM}_f$ with the properties
\begin{enumerate}
\item $N(X) \cap \sM_{f}$ is connected,
and
\item the image
of the homomorphism induced by inclusion
$$
\pi_1(N(X) \cap \sM_f,m_0) \rightarrow \pi_1(\sM_f,m_0)
$$
contains $S_{f,\iota}$.
\end{enumerate}
Then $\mathcal{D}_{f,\iota}$ is contained in a connected component of $\mathcal{AD}_{f,\iota}$.
\end{proposition}
\begin{proof} Let $Y_0 \subset \mathcal{AT}_f$ be the connected component of the preimage of $X$ in $\mathcal{AT}_f$ containing $d_0$.
Let
$U$ be the connected component of the preimage of $N(X)$ in $\mathcal{AT}_f$ containing $d_0$. Then
since $N(X)$ has a retract to $X$, the set $U$ has a corresponding
retraction to a component of $\widetilde \rho^{-1}(X)$. This component is necessarily $Y_0$, since
$Y_0 \cap U \neq \emptyset$.
Since the image of $\pi_1(N(X) \cap \sM_f,m_0)$ in $L_f = \pi_1(\sM_f,m_0)$ contains $S_{f,\iota}$, it follows that the action of $S_{f,\iota}$ on $\mathcal{AT}_f$ preserves $U$ and hence $Y_0$.
Thus, we have
$$
\mathcal{D}_{f,\iota} \subset Y_0 \subset \mathcal{AD}_{f,\iota}.
$$
\end{proof}
\section{Blowups and topology of curve complements}\label{topology-sec}
In this section we study the topology of surface/curve pairs and the effect of blowups (see for example \cite{Fulton:AC} or \cite{Hartshorne}
for a review of elementary blowup theory for surfaces).
\subsection{Regular neighborhoods of algebraic curves on surfaces}\label{reg-sec}
Let $X$ be a smooth complex projective surface, and let $V \subset X$ be a pure codimensional one subvariety.
We first observe that we may assume that $V$ is a finite union of smooth curves with normal crossings.
Let $Q$ be the set of singular points on $V$. Then $V \setminus Q$ is a finite union of smoothly embedded
punctured Riemann surfaces in $X \setminus Q$.
By successively blowing up $X$ at the points of $Q$ (and at points of the preimages of $Q$),
it is possible to obtain a new projective surface $\widetilde X$ and a surjective morphism $\sigma: \widetilde X \rightarrow X$ such that
the preimage (or {\it total transform}) $\widetilde V = \sigma^{-1}(V)$ is a union of smoooth curves with normal crossings.
That is $(\widetilde X, \widetilde V)$
is locally isomorphic near a point of intersection on $\widetilde V$ to $(\mathbb C^2, \{x=0\} \cup \{y=0\})$.
Hereafter in this section we assume that $V$ has smooth components intersecting in normal crossing.
As before, let $Q$ be the set of intersections of $V$, and let $V_1,\dots, V_k$ be the irreducible components of $V$.
Since each $V_i$ is smooth there is an embedded tubular neighborhood $T(V_i) \subset X$ so that
for $i\neq j$, $T(V_i)$ only intersects $T(V_j)$ near points in $Q$ where $V_i$ and $V_j$ intersect,
and if $V_i$ and $V_j$ intersect at $q$, then
$$
N(q) = T(V_i) \cap T(V_j)
$$
is a neighborhood of $q$ so that $(N(q),V,q)$ is homeomorphic to
$$
(\{|x|<1\} \times \{|y|<1\}, \{x=0\} \cup \{y=0\}, (0,0)).
$$
For $i=1,\dots,k$, let
$$
V_i^c = V_i \setminus \bigcup_{q \in Q} N(q)
$$
and let $T(V_i^c)$ be the tubular neighborhood given by
$$
T(V_i^c) = T(V_i) \setminus \bigcup_{q \in Q} N(q).
$$
Let $T(V) = \bigcup_{i=1}^k T(V_i^c)$, called the {\it tubular neighborhood} of $V$.
Let $S(V_i^c)$ be the circle bundle over $V_i^c$ contained in the boundary of $T(V_i^c)$. Then $S(V_i^c)$ is an oriented 3-manifold
with torus boundary components corresponding to the intersections of $V_i$ with other components of $V$. In particular, if $V_i$ is
isomorphic to $\mathbb P^1$, then $S(V_i^c)$ is the complement of thickened Hopf links in the 3-sphere.
The {\it boundary manifold} of $V$ is given by
$$
S(V) = \bigcup_{i=1}^k S(V_i^c).
$$
This manifold and its embedding in $X \setminus V$ is uniquely determined up to homeomorphisms of $X$ that are isotopic to the identity rel $V$.
\begin{lemma} \label{contract-lem} The punctured tubular neighborhood $T(V) \setminus V$ has a deformation retraction to $S(V)$.
\end{lemma}
\begin{proof} For each $i$, $T(V_i^c) \setminus V_i^c$ has a deformation retract to $S(V_i^c)$ corresponding from the retraction of a punctured disk
to its boundary circle. Thus, we have only to consider what happens near the intersection points $q \in Q$. In $N(q)$ it is enough to show that
$$
\{ |x| < 1\} \times \{ |y| < 1\} \setminus \{x=0\} \cup \{y=0\}
$$
has a deformation retract to $\{|x|=1\} \times \{|y|=1\}$. Such a retraction is defined by the map
$$
((x,y),t) \mapsto \left (\frac{x}{1 + t(|x|-1)}, \frac{y}{1+t(|y|-1)}\right ).
$$
\end{proof}
\begin{remark}\label{graphmanifold-rem}
By its construction, $S(V)$ is naturally homeomorphic to a graph manifold (see, for example, \cite{Hemp:Res} for definitions) over the incidence graph
$\Gamma$ of the components of $V$;
this is the bipartite graph with vertices
$$
\{v_i \ | \ i=1,\dots k\}
$$
corresponding to the components of $V$, and edges between $v_i$ and $v_j$ for each $q \in Q$ such that $V_i$ and $V_j$ intersect
at $q$.
To each vertex $v_i$ associate
the manifold $S(V_i^c)$ and to each edge of $\Gamma$ between $v_i$ and $v_j$ associate the common torus boundary component of their associated vertex manifolds.
\end{remark}
\subsection{Regular neighborhoods and fundamental groups}\label{fund-sec}
In this section we prove an easy
variation of the Lefschetz hyperplane theorem \cite{AF:Lefschetz} and a useful corollary.
\begin{lemma}\label{surj-lem} Let $\mathfrak p : X \rightarrow \mathbb P^1$ be a smooth projective surface fibered over the complex projective line,
and let $V$ be a fiber.
Let $C \subset X$ be a pure codimension one algebraic subset none of whose components are fibers of $\mathfrak p$.
Then $V$ has a regular neighborhood $N(V)$ so that
$$
\pi_1(N(V) \setminus C) \rightarrow (X \setminus C)
$$
induced by inclusion is surjective.
\end{lemma}
\begin{proof} Let $P \subset \mathbb P^1$ be the set of points $p$ where $\mathfrak p$ restricted to $C$ drops in cardinality, i.e., at
least one intersection point of $C$ and the fiber above $p$ has higher multiplicity. Let $\mathfrak p^o$
be the restriction of $\mathfrak p$ to $X^o = \mathfrak p^{-1}(\mathbb P^1 \setminus P) \setminus C$. Then
$$
\mathfrak p^o: X^o \rightarrow \mathbb P^1 \setminus P
$$
is a fiber bundle, and the Zariski-Van Kampen theorem \cite{Kampen:Fund} \cite{Chen:Van} implies that for a general fiber $V'$ of $\mathfrak p^o$
$$
\pi_1(X \setminus C) \simeq \pi_1(V')/K
$$
where $K$ is the subgroup of $\pi_1(V')$ generated by the relations $\gamma^{-1} \beta (\gamma)$, where
$\beta$ ranges over automorphisms of $\pi_1(V')$ determined by the action of $\pi_1(\mathbb P^1 \setminus P)$ on
$V'$. In particular, the homomorphism
$$
\pi_1(V') \rightarrow \pi_1(X \setminus C)
$$
is surjective. Let
$N(V) = {(\mathfrak p^o)}^{-1}(U)$ where $U$ is a small neighborhood of $\mathfrak p (V)$.
Then $N(\widetilde V) \setminus C$ contains a general fiber of $\mathfrak p^o$, and
the claim follows.
\end{proof}
\begin{corollary}\label{connectedness-cor} Let $C \subset \mathbb C^2$ be an algebraic curve, and let $V \subset \mathbb C^2$ be a line not contained in $C$.
Then we can include $\mathbb C^2$ as a Zariski open subset of a smooth projective surface $X$, and find a sequence of blowups
$\sigma : \widetilde X \rightarrow X$
such that
\begin{enumerate}
\item $\sigma$ is an isomorphism over $\mathbb C^2 \setminus C$; and
\item the total transform $\widetilde V$ of the closure of $V$ in $X$
has a regular neighborhood $N(\widetilde V)$ in $\widetilde X$ such that
the map on fundamental groups induced by inclusion
$$
\pi_1(N(\widetilde V) \cap \mathbb C^2 \setminus C) \rightarrow \pi_1 (\mathbb C^2 \setminus C)
$$
is surjective.
\end{enumerate}
\end{corollary}
\begin{proof} Let $\mathfrak p : \mathbb C^2 \rightarrow \mathbb C$ be a linear projection so that $V$ is a fiber. Then
we can define completions $\mathbb C^2 \subset \mathbb P^1 \times \mathbb P^1$ and $\mathbb C \subset \mathbb P^1$, so that $\mathfrak p$
extends to a projection
$$
\overline {\mathfrak p}: \mathbb P^1 \times\mathbb P^1 \rightarrow \mathbb P^1
$$
so that the closure $\overline V$ of $V$ in $\mathbb P^1 \times \mathbb P^1$ is a fiber.
Let $\overline C$ be the union of the closure of $C$ in $\mathbb P^1 \times \mathbb P^1$, and the two lines in $\mathbb P^1 \times \mathbb P^1 \setminus \mathbb C^2$.
Let $\sigma : \widetilde {\mathbb P^1 \times \mathbb P^1} \rightarrow \mathbb P^1$ be sequence of blowups over points of $\overline C \cap \overline V$
such that the total transform $\widetilde V = \sigma^{-1}(\overline V)$ and the proper transform $\widehat C$ over $\overline C$
meet in normal crossings.
Let $N(\widetilde V)$ be a regular neighborhood of $\widetilde V$ as in Lemma~\ref{surj-lem}. Then
the homomorphism
$$
\pi_1(N(\widetilde V) \setminus \widehat C) \rightarrow \pi_1(\widetilde {\mathbb P^1 \times \mathbb P^1} \setminus \widehat C)
$$
induced by inclusion is surjective.
Since $\sigma$ is an isomorphism outside the preimage of $\overline C \cap \overline V$,
we have a commutative diagram
$$
\xymatrix{
\widetilde V\ar[d] \ar[r] &N(\widetilde V) \ar[d]\\
\overline V \ar[r] &N(\overline V)
}
$$
where the vertical arrows are quotient maps that contract the exceptional curves (or 1-dimensional fibers) over points on $\overline V \cap \overline C$.
It follows that the retraction of $N(\widetilde V) \setminus \widehat C$ to $\widetilde V \setminus \widehat C$ descends to a retraction
of $N(\overline V) \setminus C$ to $\overline V \setminus C$, and hence $N(\overline V) \setminus \overline C$ is a regular neighborhood of $\overline V \setminus \overline C$.
Finally, $P^1 \times \mathbb P^1 \setminus \overline C = \mathbb C^2 \setminus C$, so setting $N(V) = N(\overline V) \cap \mathbb C^2$, it follows
that $N(V)$ is a regular neighborhood of $V$ and
the homomorphism
$$
\pi_1(N(V) \setminus C) \rightarrow \pi_1(\mathbb C^2 \setminus C)
$$
defined by inclusion is surjective.
\end{proof}
Corollary~\ref{connectedness-cor} is used in our discussion of connectivity of augmented deformation space in Section~\ref{connectedness-sec}.
\begin{remark}\label{Hopf-rem}
The boundary manifold $S(\widetilde V)$ associated to $N(\widetilde V)$ in the previous proof
has the structure of a boundary manifold over the incidence
graph of the irreducible components of $\widetilde V$ (cf. Remark~\ref{graphmanifold-rem}). Furthermore,
each component of $\widetilde V$ is isomorphic to a line,
each vertex manifold is an $S^1$ fiber bundle over $S^2$ with a finite set of thickened fibers removed.
\end{remark}
\section{Application to the Main Example}\label{proof-sec}
Let $F \in \mathrm{Per}_4(0)^*$, and let
$f,\iota : (S^2,A) \rightarrow (S^2,B)$ be the underlying branched covering and identification
of domain and range so that
$A$ is the periodic 4 cycle, and $B = A \cup \{v\}$ where $v$ is the extra critical point.
We study the inclusion $\mathcal{D}_{f,\iota} \subset \mathcal{AD}_{f,\iota}$ by looking at their images $\mathcal{V}_{f,\iota}$ and $\mathcal{AV}_{f,\iota}$ in
$\sM_{f}$ and $\mathcal{AM}_{f}$.
\subsection{Parameterization of moduli space}\label{parameterization-sec}
We begin by embedding $\sM_f$ in $\mathbb P^1 \times \mathbb P^1$ as follows.
Consider an element of $\sM_f$ represented as a commutative diagram
$$
\xymatrix{
A\ \ar[d]_{f |_A} \ar@{^(->}[r]^i &\mathbb P^1\ar[d]^F\\
B\ \ar@{^(->}[r]^j &\mathbb P^1.
}
$$
By applying automorphisms of $\mathbb P^1$ on the right side, we can assume that
$$
i(B) = \{0,1,\infty, y,z\} \quad j(A) = \{0,1,\infty,x\},
$$
where
\begin{enumerate}[(i)]
\item $\infty$ and $z$ are the critical values of $f$;
\item $0$ is a critical point in $A$ with $f(0) = \infty$; and
\item $f(\infty) = 1$, $f(x) = 0$.
\end{enumerate}
The above data completely determines $F : \mathbb P^1 \times \mathbb P^1$ as a rational function in the variable $t$:
$$
F(t) = \frac{(t-x)(t-r)}{t^2} \qquad r = \frac{x + y - 1}{x -1}.
$$
It follows that in this example $z$ is determined by $x$ and $y$:
\begin{eqnarray}\label{z-eqn}
z = - \frac{(1 - 2x + x^2 - y)^2}{4 x (x-1)(x+y-1)}.
\end{eqnarray}
We have the following (see also, \cite{HK:rational}).
\begin{lemma} \label{par-lem} There is an identification
$$
\sM_f = \mathbb P^1 \times \mathbb P^1 \setminus \mathcal L \cup \mathcal Z,
$$
assigning $(i,j,F)$ to $(x,y)$,
where
$$
\mathcal L = \{x = 0\} \cup \{y = 0\} \cup \{x = 1\} \cup \{y = 1\} \cup \{x = \infty\} \cup \{y = \infty\}
$$
and $\mathcal Z$ is the closure in $\mathbb P^1 \times \mathbb P^1$ of the affine union of curves
$$
\{1-2x + x^2 - y =0 \}\cup \{x^2+y = 1\} \cup \{ x + y = 1\} \cup \{2xy+x^2-y-2x+1=0\}
$$
\end{lemma}
\begin{figure}[htbp]
\centering
\includegraphics[width=1.5in]{affine.pdf}
\caption{Picture of $\mathcal L \cup \mathcal Z$ in the affine plane. The lines in $\mathcal L$ are drawn in red.}
\label{affine-fig}
\end{figure}
By this parameterization, the image $\mathcal{V}$ of $\mathcal{D}_{f}$ in $\sM_{f}$ equals the diagonal
$$
\mathcal{V} = \{(x,y) \in \mathbb P^1 \times \mathbb P^1 \setminus \mathcal L \cup \mathcal Z \ | \ x = y\}.
$$
Figure~\ref{affine-fig} gives a picture\footnote{This figure was provided courtesy of Sarah Koch.} of the real part
of $\mathcal L \cup \mathcal Z$ in the affine open subset $\mathbb C^2 \subset \mathbb P^1 \times \mathbb P^1$, and Figure~\ref{ClosedW-fig} gives a picture near $(\infty,\infty)$.
\begin{figure}[htbp]
\centering
\includegraphics[width=1in]{nearinfinity.pdf}
\caption{Picture of $\mathcal L \cup \mathcal Z$ in $\mathbb P^1 \times \mathbb P^1$ near $(\infty,\infty)$. The line $x = y$ is drawn in black, and the lines $x = \infty$ and
$y = \infty$ are drawn in red.}
\label{ClosedW-fig}
\end{figure}
\subsection{The quotient of augmented deformation space}
Let $\mathcal{AM}_f$ be the quotient of $\mathcal{AT}_f$ by the action of $L_f$, and let $\mathcal{AV}_{f,\iota}$ be the image of $\mathcal{AD}_{f,\iota}$
in $\mathcal{AM}_f$. Our goal in this section is to concretely describe a subspace $X \subset \mathcal{AV}_{f,\iota}$ satisfying the properties in Proposition~\ref{suff-prop}.
First we recall that the elements of $\mathcal{AV}_{f,\iota}$ are the elements of $\mathcal{AM}_f$ that equalize the two maps
$$
p_L, p_U : \mathcal{AM}_f \rightarrow \mathcal{AM}_{(S^2,A)}.
$$
Each stratum of $\mathcal{AM}_f$ is described
by a partition of $\{0,1,\infty,y,z\}$ into two or three sets by an admissible multi-curve $\gamma$, as in Figure~\ref{partition-fig}.
Here, the empty multi-curve corresponds to the principal stratum $\sM_f \subset \mathcal{AM}_f$.
\begin{figure}[htbp]
\centering
\includegraphics[width=3.5in]{partition.pdf}
\caption{Possible partitions of five points by an admissible multi-curve.}
\label{partition-fig}
\end{figure}
The corresponding stable curves are shown in Figure \ref{stable5-fig}, with each $\mathbb P^1$, homeomorphic to $S^2$, is drawn as a line.
\begin{figure}[htbp]
\centering
\includegraphics[width=4in]{stable5.pdf}
\caption{The three types of stable curves for $(S^2,B)$ where $B$ has five elements.
The two left define one-dimensional strata, and the right defines a point stratum.}
\label{stable5-fig}
\end{figure}
For $\mathcal{AM}_{(S^2,A)}$ there are only two isomorphism types (shown in Figure~\ref{stable4-fig}). The left picture depicts points
belonging
to the main component
$\sM_{(S^2,A)}\subset\mathcal{AM}_{(S^2,A)}$, which is isomorphic to a thrice punctured sphere, while
the right picture depicts one of the three single point boundary points.
\begin{figure}[htbp]
\centering
\includegraphics[width=2.5in]{stable4.pdf}
\caption{The two topological homeomorphism types of stable curves for $(S^2,A)$ where $A$ has four elements.
}
\label{stable4-fig}
\end{figure}
The elements $\alpha \in \mathcal{AM}_f$ that lie in positive dimensional strata must be of the form (1) or (2)
in Figure~\ref{stable5-fig}. Those of type (1) lie in $\sM_f = \mathbb P^1 \times \mathbb P^1 \setminus \mathcal L \cup \mathcal Z$
and map under both $p_U$ and $p_L$ to elements of $\mathcal{AM}_{(S^2,A)}$ of type (I).
Those of type (2) divide into four subtypes: those that correspond to a partition of the form
\begin{enumerate}
\item [(2a)] $\{a,b,z\}\cup\{c,\infty\}$;
\item [(2b)] $\{a,\infty,z\}\cup\{b,c\}$;
\item [(2c)] $\{a,b,c\} \cup \{\infty,z\}$; or
\item [(2d)] $\{a,b,\infty\}\cup\{c,z\}$.
\end{enumerate}
For types (2a) and (2b), $p_L$ maps $\alpha$ to an element in $\mathcal{AM}_{(S^2,A)}$ of type (II),
while for types (2c) and (2d), $p_L$ maps $\alpha$ to one of type (I).
For types (2a) and (2d) $p_U$ maps $\alpha$ to an element of type (II), while for type
(2b) and (2c) $\alpha$ could apriori map to an element of type (I) or (II). This is because the critical
values $\infty$ and $z$ lie in the same component. Thus the rational map $F : \mathcal C_U \rightarrow \mathcal C_L$ must have two isomorphic
irreducible components in $\mathcal C_U$ lying over the unramified component in $\mathcal C_L$ upon which the distinguished points will be distributed.
From this we can reduce the types that can be in $\mathcal{AV}$ to (2a), (2d), as well as
(2b) and (2c) under the condition that
the preimage under $F$ of the distinguished points in the unramified component lie on the same component of $\mathcal C_U$.
In the allowable case of types (2a) and (2b), we see that for $p_L(\alpha)$ and $p_U(\alpha)$ to be equal the images of the two maps
in $\mathcal{AM}_{(S^2,A)}$ must give the partition
$$
\{0,1\} \cup \{x,\infty\}.
$$
Thus, the partition given in (2a) can only be
$$
\{0,1,z\} \cup \{y,\infty\}
$$
and for (2b) it can only be
$$
\{\infty, y, z\} \cup \{0,1\}.
$$
For type (2c), $p_L(\alpha)$ and $p_U(\alpha)$ must be equal, and $F$ defines the isomorphism on stable curves.
Taking into account the combinatorics of $f$, this implies equality of the cross ratios:
$(0,\infty;1,x)$ and $(\infty,1; x,0)$, which is false under the assumption that $x \notin \{0,1,\infty\}$.
Let $\mathcal A_1, \mathcal A_2 \subset \mathcal{AM}_f$ be the subsets corresponding to the partitions
$$
\mathcal A_1: \{\infty, y,z\} \cup \{0,1\},
$$
and
$$
\mathcal A_2: \{0,1,z\} \cup \{y,\infty\}.
$$
Then each of these is isomorphic to $\mathrm{Mod}_{0,4} \times \mathrm{Mod}_{(0,3)}$.
Furthermore, the closures of $\mathcal A_1$ and $\mathcal A_2$ in $\mathcal{AM}_f$ intersect at the point corresponding to the partition
$$
\{0,1\} \cup \{z\} \cup \{y,\infty\}.
$$
We have shown the following.
\begin{proposition} \label{connectedness-prop}
The pure 1-dimensional algebraic set $\mathcal{V} \cup \mathcal A_1 \cup \mathcal A_2 \subset \mathcal{AM}_f$ is contained in $\mathcal{AV}_{f,\iota}$, and its
complement is a finite set of points (possibly empty).
\end{proposition}
\begin{remark} We leave the question of whether $\mathcal{AV}_{f,\iota}$ is connected (in this case, and in general) to future study. \end{remark}
\subsection{Blowups}\label{blowup-sec}
In this section, we find a connected pure 1-dimensional algebraic subset $X \subset \mathcal{AV}$
that contains $\mathcal{V}$, and whose complement in $\mathcal{AV}$ is finite.
Let
$$
\sigma : \widetilde {\mathbb P^1 \times \mathbb P^1} \rightarrow \mathbb P^1 \times \mathbb P^1
$$
be sequence of blowups defined as follows.
First blowup the points $(0,0)$, $(1,1)$ and $(\infty,\infty)$ to get the exceptional curves $E_0,E_1,E_\infty$.
Next blowup the point of intersection $q \in E_\infty \cap \widehat L_y$, where
$\widehat L_y$ is the proper transform of $\{y=\infty\}$. Let $E_q$ be the
exceptional divisor. The union of curves is drawn in Figure~\ref{blowup2-fig} (compare Figure~\ref{affine-fig}).
\begin{figure}[htbp]
\centering
\includegraphics[width=2.5in]{blowup2.pdf}
\caption{The proper transform of $\mathcal V$ is drawn as a dotted line, and the exceptional curves $E_0,E_1,E_\infty$ and $E_q$
are drawn in red. Intersections with $\mathcal L$ are indicated as thickened black line, and intersections with $\mathcal Z$ are indicated with
a thin black line}
\label{blowup2-fig}
\end{figure}
\begin{lemma} The map $\sigma$ has the following properties:
\begin{enumerate}
\item $\sigma$ restricts to an isomorphism on $\widetilde {\mathbb P^1 \times \mathbb P^1} \setminus \sigma^{-1}(Q)$; and
\item the total transform $\widetilde \mathcal{V}$ and the proper transform $\widehat {\mathcal L \cup \mathcal Z}$ meet in normal crossing
singularities.
\item inclusion induces a surjection on fundamental groups
$$
\pi_1(\mathcal N(\widetilde \mathcal{V}) \cap \sM_f) \rightarrow \pi_1(\sM_f).
$$
\end{enumerate}
\end{lemma}
\begin{proof} Properties (1) and (2) follow from the definitions, and property (3) follows from Lemma~\ref{surj-lem}.
\end{proof}
Let
$$
X = (\widehat \mathcal{V} \cup E_\infty \cup E_q) \setminus \widehat {\mathcal L \cup \mathcal Z}.
$$
Then $X$ is connected since the punctures of $\widetilde \mathcal{V}$ at intersections with $\widehat {\mathcal L \cup \mathcal Z}$ occur
only in smooth points of $\widetilde \mathcal{V}$. Since $\sigma$ is an isomorphism over $\mathcal{V}$, there is a natural inclusion
$\nu: \mathcal{V} \hookrightarrow X$.
\begin{lemma} The inclusion $\mathcal{V} \hookrightarrow \mathcal{AV}$ induced by $\mathcal{D}_{f,\iota} \hookrightarrow \mathcal{AD}_{f,\iota}$ factors as
$\xi \circ \nu$ for some embedding
$$
\xi: X \hookrightarrow \mathcal{AV}.
$$
\end{lemma}
\begin{proof}
Recall from Proposition~\ref{connectedness-prop} the subset $\mathcal{V} \cup \mathcal A_1 \cup \mathcal A_2 \subset \mathcal{AV}$.
In what follows we define embeddings
\begin{eqnarray*}
\kappa_\infty: E_\infty\setminus \widehat {\mathcal L \cup \mathcal Z} &\rightarrow& \mathcal A_1\\
\kappa_q : E_q \setminus \widehat {\mathcal L \cup \mathcal Z} &\rightarrow&\mathcal A_2
\end{eqnarray*}
that together with the projection $\widehat V \rightarrow \mathcal{V}$
extend to define $\xi$.
When $\alpha \in \sM_f$ approaches a general point of $\mathcal L \cup \mathcal Z$, it corresponds to
two points in $\{0,1,\infty,y,z\}$ coming together. For the lines $(y=0$, $y=1$ and $y=\infty$, the pairs
$\{y,0\}$, $\{y,1\}$ and $\{y,\infty\}$ approach each other,
while simultaneously $\{x,1\}$, $\{1,\infty\}$ and $\{1,0\}$ approach each other.
Near the lines $x = 0$, $x=1$, $x=\infty$
the pairs $\{x,0\}$) (and $\{0,\infty\}$), $\{x,1\}$ (and $\{0,y\}$) and $\{x,\infty\}$ (and $\{0,1\}$) approach each other.
For $z$ approaching $0,1,\infty$ or $y$, we have
\begin{eqnarray*}
\{z,0\} &\mbox{near}& 1 - 2x + x^2 - y = 0; x \neq 0, 1; x + y \neq 1\\
\{z,1\} &\mbox{near}& x^2 + y = 1; x \neq 0; x+y \neq 1\\
\{z,\infty\} &\mbox{near}& x = 0, x=1, \mbox{or} \ x+y=1\\
\{z,y\} &\mbox{near}& 2xy + x^2 - y - 2x =0
\end{eqnarray*}
We choose local coordinates for a neighborhood of $E_\infty$ as follows. Let $\overline x = 1/x$ and $\overline y = 1/y$ be coordinates for an
open neighborhood of $p_\infty = (\infty,\infty)$ in $\mathbb P^1 \times \mathbb P^1$, so that in the coordinates $(\overline x,\overline y)$ we
have $p_\infty = (0,0)$. Let $u$ (and $\overline u = \frac{1}{u}$) be coordinates for $E_\infty$. Then a neighborhood of $E_\infty$
in $\widetilde {\mathbb P^1 \times \mathbb P^1}$ is isomorphic to the algebraic subset of $\mathbb C^2 \times \mathbb P^1$ defined by
$$
\{(\overline x,\overline y) \times u \in \mathbb C^2 \times \mathbb P^1\ | \overline x = u \overline y\},
$$
and
$$
E_\infty = \{(\overline x, \overline y) \times u \in \mathbb C^2 \times \mathbb P^1\ | \ \overline x = \overline y = 0\}.
$$
With respect to the $\overline x, \overline y$ coordinates, we have
\begin{eqnarray*}
z &=& - \frac{\overline x^4 \overline y^2 (1 - 2x + x^2 - y)^2}{4 \overline x^4\overline y^2x (x-1)(x+y-1)}\\
&=& - \frac{(\overline x^2\overline y - 2\overline x\overline y + \overline y - \overline x^2)^2}
{4 \overline x\overline y(1 - \overline x)(\overline y + \overline x -\overline x \overline y)}\\
\end{eqnarray*}
Using the identity $\overline x = u \overline y$, we have
\begin{eqnarray*}
z &=&- \frac{(u^2 \overline y^2 - 2 u\overline y + 1 - u^2 \overline y)^2}
{4 u \overline y(1-u\overline y)(1 + u (1 - \overline y))}.
\end{eqnarray*}
To each $(u, \overline y)$, associate the point in $\mathcal{AM}_{(S^2,B)}$
defined by
$$
([u,\infty ; y, z] , *)
$$
where the first component is the cross ratio of the points $u,\infty, y,z$ and the second
component is the unique triple of points in $\mathbb P^1$ up to automorphism. This defines a point in
$\mathcal{AM}_{(S^2,B)}$.
Since cross ratio is preserved under automorphisms of $\mathbb P^1$ we have
$$
[u,\infty: y, z] = [u\overline y,\infty, 1,z \overline y].
$$
As $\overline y$ approaches $0$, the cross ratio approaches
$$
[0,\infty; 1, -\frac{1}{4 u(1+u)}]
$$
and is degenerate only when $u = 0$ ($x = \infty$), $u = -1$ ($z = \infty$), $u = -\frac{1}{2}$ ($z = y$), or $u = \infty$ ($y = \infty$).
We thus have a well-defined embedding:
\begin{eqnarray*}
\kappa_\infty : E_\infty \setminus \widehat {\{\mathcal L \cup \mathcal Z\}} &\rightarrow& \mathcal A_1\\
u &\mapsto& ([u, \infty; y,z], *).
\end{eqnarray*}
Here $u$ corresponds to a line through $(\infty,\infty)$ in $\mathbb P^1 \times \mathbb P^1$, and we can think of the contracting curve $\gamma$
as a small loop on this line around $(\infty,\infty)$.
The intersection of $E_\infty$ with $E_q$ occurs at the point on $E_\infty$ corresponding to $(\overline y,\overline u) = (0,0)$,
and $E_q$ has
a neighborhood parameterized by $((\overline y, \overline u),v) \in \mathbb C^2 \times \mathbb P^1$ where $\overline y = v \overline u$.
Then
$\overline x = u \overline y = v$ and
\begin{eqnarray*}
z
&=& -\frac {(1 -2x + x^2 - y)^2}{4 x(x-1)(x+y-1)}\\
&=&
- \frac{(\overline u(1 -2\overline v + \overline v^2) - \overline v)^2}{4 \overline v (\overline v-1)(\overline v + \overline u \overline v - 1)}\\
\end{eqnarray*}
and as $\overline u$ goes to $0$,
$$
z = -\frac{\overline v}{4 (\overline v-1)^2} = -\frac{v}{4(1-v)^2}.
$$
Then we have
$$
[v,0;1,z] = [1,0; \overline v, \overline v z] = [1,0; \overline v, -\frac{1}{4(1-v)^2}].
$$
The cross ratio is degenerate when $v = \infty$ (where
$\widehat {\{y=\infty\}}$ meets $E_q$), $ v = 1$ (where $E_\infty$ and $E_q$ intersect),
and where $z = 0$ and $z = 1$ (corresponding to two distinct values of $v$ not equal to $1$ or $\infty$).
The points correspond to $v \in E_q \setminus \widehat {\mathcal L \cup \mathcal Z}$ except at the point
where $v = 1$.
For this point, consider the 2-component multicurve $\gamma$ on $S^2 \setminus B$
determined by two loops on the planes defined by
$E_\infty$ and $\widehat{\{y=\infty\}}$ around $(\overline y, \overline u) = (0,0)$.
The corresponding point $\alpha_0 \in \mathcal{AM}_f$ corresponds to the partition
$$
\{y,\infty\} \cup \{z\} \cup \{0,1\}.
$$
Define
\begin{eqnarray*}
\kappa_q : E_q \setminus \widehat {\{\mathcal L \cup \mathcal Z\}} &\rightarrow& \mathcal A_2\\
v &\mapsto& \left \{
\begin{array}{ll}
([0,1; \infty, \overline v], *) &\mbox{if $v \neq 1$}\\
\alpha_0 &\mbox{if $v = 1$}
\end{array}
\right .
\end{eqnarray*}
Then $\kappa_\infty$ and $\kappa_q$ extend to a morphism $\xi : X \rightarrow \mathcal{AV}$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{main-thm}] Fix $d_0 \in \mathcal{D}_{f,\iota}$, and let $\mathcal D_0$ be the connected component of
$\mathcal{D}_{f,\iota}$ that contains $d_0$. Let $\overline \mathcal D_0$ be the closure of $\mathcal D_0$ in $\mathcal{AD}_{f,\iota}$. Then
$\overline \mathcal D_0$ maps to $\widehat \mathcal{V}$ under the projection from $\mathcal{AT}_f \rightarrow \mathcal{AM}_f$.
Let $\widehat \mathcal D_0 \subset \mathcal{AT}_f$ be the connected component of the preimage of $\widehat \mathcal{V}$ that intersects $\mathcal D_0$.
Then, since $\widehat \mathcal{V}$ is closed in $X$ and
$X$ is connected, we can find a connected component $Y_0$ of the preimage of $X$ in $\mathcal{AT}_f$ that contains $\widehat D_0$,
and
$\widehat \mathcal D_0$ is the closure
in $Y_0$ of $\mathcal D_0$. That is, $\widehat \mathcal D_0 = \overline \mathcal D_0$.
We claim that there is an element
$$
g \in S_{f,\iota} \cap \mbox{Image}( \pi_1(N(X) \cap \sM_f) \rightarrow \pi_1(\sM_f))
\setminus \mbox{Image}(\pi_1(\widehat V) \rightarrow \pi_1(\sM_f)).
$$
Let $v_0$ be the image of $d_0$ in $\mathcal{V}$, and let $\gamma$ be a close path starting at $v_0$ and passing along $X$ to
a point near an intersection of $\widehat \mathcal Z$ with $E_\infty \cup E_q$, forming a small loop around that intersection point,
and returning along the original path back to $v_0$. Then $\gamma$ is not homotopic in $\mathcal{AM}_f$ to an closed
path on $\widehat V$ since $\widehat Z$ and $\widehat V$ do not intersect.
Let $g$ be the image of $\gamma$ in $\pi_1(\sM_f)$ after pushing off the boundary of $\mathcal{AM}_f$
into $\sM_f$.
Since $g \notin \mbox{Image}(\pi_1(\widehat V) \rightarrow \pi_1(\sM_f))$, $g$ does not preserve
$\widehat D_0$, and hence $g(\widehat \mathcal D_0)$ and $\widehat \mathcal D_0$ are disjoint. These are the closures of
$\mathcal D_0$ and $g(\mathcal D_0)$ in $\mathcal{AD}_{f,\iota}$ and the claim follows.
\end{proof}
\subsection{Connectivity of augmented deformation space}\label{connectedness-sec}
We finish this paper by giving a sufficient condition for $\mathcal{AD}_{f,iota}$ connected in the $\mathrm{Per}_4(0)^*$ case.
By Corollary~\ref{connectedness-cor}, we know that $\widetilde V$ has a regular neighborhood $N$ in $\widetilde {\mathbb P^1 \times \mathbb P^1}$ so
that
$$
\pi_1(N \cap \sM_f) \rightarrow \pi_1(\sM_f)
$$
is surjective.
As a closure of $\sM_f$, $\widetilde {\mathbb P^1 \times \mathbb P^1}$ is birationally equivalent to $\mathcal{AM}_f$ (but not isomorphic). By the
birational theory of
complex projective surfaces, there is a minimal smooth surface $Z$ with birational morphisms to $\mathcal{AM}_f$ and $\widetilde {\mathbb P^1 \times \mathbb P^1}$
$$
\xymatrix{
&Z\ar[dl]\ar[dr]\\
\mathcal{AM}_f &&\widetilde {\mathbb P^1 \times \mathbb P^1}.
}
$$
Lifting $U$ to $Z$ and projecting to $\mathcal{AM}_f$ gives a regular neighborhood $U'$ of $\overline \mathcal{V} \cup \mathcal K$, where $\mathcal K$ is a set
of boundary curves in $\mathcal{AM}_f$. Since $\sM_f$ is a smooth subset of both $\mathcal{AM}_f$ and $\widetilde {\mathbb P^1 \times \mathbb P^1}$ the
two projects of $Z$ are isomorphisms over $\sM_f$. Thus
$$
U' \cap \sM_f = U \cap \sM_f,
$$
and hence
$$
\pi_1(U' \cap \sM_f) \rightarrow \pi_1(\sM_f)
$$
is surjective. It follows that $U'$ has a connected preimage in $\mathcal{AT}_f$ an that the preimage of $\overline V \cup \mathcal K$ in $\mathcal{AT}_f$
is connected.
We have thus shown the following sufficient condition for $\mathcal{AD}_{f,\iota}$ to be connected.
\begin{proposition} If $\mathcal{AV}$ is connected and is Zariski dense in $\overline \mathcal{V} \cup \mathcal K$, then $\mathcal{AD}_{f,\iota}$ is connected.
\end{proposition}
\bibliographystyle{plain}
|
{
"timestamp": "2019-03-20T01:04:04",
"yymm": "1902",
"arxiv_id": "1902.10760",
"language": "en",
"url": "https://arxiv.org/abs/1902.10760"
}
|
\section{Conclusion and Future Work}
\label{sec:conclusion}
This paper presented a partitioning algorithm for semi-explicit and explicit MPC
that is applicable to multiparametric convex MINLPs. The algorithm is guaranteed
to converge given positivity of a novel cost function ``overlap'' metric. The
resulting MPC implementation is guaranteed real-time. Future work involves
proving stability of the resulting control law along the lines of
\cite{MunozDeLaPena2004,MunozDeLaPena2006}, and massively parallelizing the tree
exploration via the \texttt{map} parallel pattern \cite{McCool2012}.
\section{Illustrative Example}
\label{sec:example}
We apply Algorithm~\ref{alg:phase2} to minimum-fuel position control of a
spacecraft. A full description of the problem and of the MPC law is given in
\cite{Malyuta2019}. Here we provide a brief summary. Consider Clohessy-Wiltshire
out-of orbital plane dynamics:
\begin{equation}
\label{eq:cwh_ct}
\ddot z = -\omega_0^2z+u+w,
\end{equation}
where $\omega_o$~rad/s is the orbital rate. Let the state
$x=(z,\dot z)\in\mathbb R^2$, input $u\in\mathbb R$ and disturbance
$w\in\mathbb R$. \eqref{eq:cwh_ct} is discretized using a reaction control system
(RCS) input that can instantaneously induce a $\Delta v$ velocity increment
every $T_{\mathrm{s}}=100~\text{s}$,
i.e. $u(t)=\Delta v(kT_{\mathrm{s}})\delta_{\mathrm{D}}(t-kT_{\mathrm{s}})$
where $k\in\mathbb Z_+$ and $\delta_{\mathrm{D}}$ is the Dirac delta. The
disturbance models atmospheric drag, input-dependent input error and
state-dependent state estimate error. We use
$\Theta=[-10,10]~\text{cm}\times[-1,1]~\text{mm/s}$ which is robust controlled
invariant for \eqref{eq:cwh_ct}. Thruster impulse-bit imposes a non-convex lower
bound on the $\Delta v$ magnitude,
\begin{equation*}
\Delta v\in\{\Delta v\in\mathbb R : \underline{\Delta v}\le|\Delta v|\le\overline{\Delta v}\}\cup\{0\},
\end{equation*}
which is handled via mixed-integer programming by constraining the input to be
in one of three convex sets: $[-\overline{\Delta v},-\underline{\Delta v}]$,
$[\underline{\Delta v},\overline{\Delta v}]$ or $\{0\}$. Altogether,
\eqref{eq:minlp} is a mixed-integer second-order cone program (SOCP). Using a
3-step prediction horizon, the problem has
parameter dimension $p=2$ (the state) and commutation dimension $m=9$.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{progress}
\caption{Algorithm~\ref{alg:phase2} convergence plots. Cumulative closed
volume, cumulative closed leaf count and runtime are normalized by their
final values. The dotted reference line shows linear convergence.}
\label{fig:progress}
\end{figure}
\begin{table*}[t]
\centering
\begin{tabularx}{0.8\linewidth}{>{\hsize=1.7cm}CCC>{\hsize=0.4cm}CCCCCC}
Implementation & $\epsilon_{\mathrm{a}}$ & $\epsilon_{\mathrm{r}}$ & $\tau$ & $\lambda$ &
$T_{\mathrm{solve}}$
[min]
& $T_{\mathrm{query}}$ [$\mu$s] & $M$ [MB] \\ \hline \hline
Semi-explicit & $4.31\cdot 10^{-2}$ & $2.00$ & $12$ & $85$ & $1$ & $611$ & $0.06$ \\
Explicit & $4.31\cdot 10^{-2}$ & $2.00$ & $12$ & $85$ & $1$ & $106$ & $0.06$ \\
Semi-explicit & $1.05\cdot 10^{-2}$ & $1.00$ & $14$ & $532$ & $15$ & $630$ & $0.39$ \\
Explicit & $1.05\cdot 10^{-2}$ & $1.00$ & $14$ & $532$ & $14$ & $133$ & $0.39$ \\
Semi-explicit & $1.70\cdot 10^{-3}$ & $0.10$ & $17$ & $6844$ & $211$ & $695$ & $5.10$ \\
Explicit & $1.70\cdot 10^{-3}$ & $0.10$ & $17$ & $7726$ & $219$ & $188$ & $5.66$ \\
Semi-explicit & $1.76\cdot 10^{-4}$ & $0.05$ & $22$ & $56189$ & $1878$ & $491$ & $42.38$ \\
Explicit & $1.76\cdot 10^{-4}$ & $0.05$ & $22$ & $60593$ & $1852$ & $244$ & $45.26$ \\
\hline
\end{tabularx}
\caption{Numerical results for several $\epsilon$-suboptimality
settings. $\tau$ is the tree depth, $\lambda$ is the leaf count,
$T_{\mathrm{solve}}$ is the Algorithm~\ref{alg:phase2} runtime,
$T_{\mathrm{query}}$ is the median control input evaluation time and $M$ is
the storage memory requirement.}
\label{tab:results}
\end{table*}
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{.48\textwidth}
\centering
\includegraphics[width=1\linewidth]{evaltime}
\caption{MPC on-line evaluation time. Bars show the median while error bars
shown the minimum and maximum values.}
\label{fig:evaltime}
\end{subfigure}%
\hfill%
\begin{subfigure}[t]{.48\textwidth}
\centering
\includegraphics[width=1\linewidth]{fuel}
\caption{Relative over consumption of fuel with respect to implicit MPC due
to $\epsilon$-suboptimality. Implicit MPC uses $\approx 1.56$~mm of
``fuel''.}
\label{fig:fuel}
\end{subfigure}%
\caption{Comparison of the proposed semi-explicit and explicit implementations
to implicit MPC in terms of (\protect\subref{fig:evaltime}) on-line control
input computation time and (\protect\subref{fig:fuel}) total fuel
consumption over $3$ orbits.}
\end{figure*}
\longtxt{\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{response}
\caption{State and input responses for a coarse (blue) and a refined (red)
partition for semi-explicit and explicit implementations. Implicit MPC
response is in gray.}
\label{fig:response}
\end{figure}}
We use four increasingly coarse $\epsilon$-suboptimality settings. The relative
error is set to $\epsilon_{\mathrm{r}}=0.05,0.1,1,2$. For the absolute error, we
use the maximum optimal cost at the vertices of a scaled $\Theta$, i.e.
\begin{equation*}
\epsilon_{\mathrm{a}} = \max_{\theta\in\mathcal V(s\Theta)}V^*(\theta),
\end{equation*}
where $s=0.03,0.1,0.25,0.5$. We use Python 3.6.7 with CVXPY 1.0.14 \cite{cvxpy}
and Gurobi 8.1 \cite{gurobi} in Ubuntu 18.04.1 with a 3.60~GHz Intel i7-6850K
CPU and 64~GB of RAM.
Figure~\ref{fig:progress} shows convergence plots in terms of the closed leaf
count and volume fractions. A linear convergence is preferred since it is more
easily extrapolated to predict total runtime. The more refined partitions
favorably exhibit a more linear convergence. \longtxt{Since the curves in
Figure~\ref{fig:progress} stay mostly below the linear convergence reference,
the user can expect faster runtimes than predicted by a linear fit.}
Table~\ref{tab:results} summarizes the results. The data confirms that
$\tau=\mathcal O(\log(\psi^{-1}))$ using the proxy
$\psi\approx\epsilon_{\mathrm{a}}/\bar{\epsilon}_{\mathrm{a}}+\epsilon_{\mathrm{r}}/\bar{\epsilon}_{\mathrm{r}}$
where $\bar\epsilon_{\mathrm{a}}$ and $\bar\epsilon_{\mathrm{r}}$ are the
largest of the tested values. The leaf count is exponential in $\psi$, with the
semi-explicit partitions being generally coarser since
\eqref{eq:absolute_error_approx} and \eqref{eq:relative_error_approx} are easier
to satisfy than \eqref{eq:absolute_error_redefined} and
\eqref{eq:relative_error_redefined}. It follows that $T_{\mathrm{solve}}$ and
$M$ are also exponential in $\psi$.
Table~\ref{tab:results} and Figure~\ref{fig:evaltime} show statistics for the
on-line control input computation time $T_{\mathrm{query}}$. For implicit MPC,
$T_{\mathrm{query}}$ is the solution time of \eqref{eq:minlp}. For semi-explicit
MPC, it is the time to evaluate \eqref{eq:f_subopt_eval} and solve
\eqref{eq:nlp}. For explicit MPC, it is the evaluation time of
\eqref{eq:explicit_mpc_query}. As expected, explicit is the fastest and implicit
is the slowest. \longtxt{$T_{\mathrm{query}}$ is more variable for semi-explicit
than explicit MPC due to a non-trivial dependence on the number of iterations
required to solve the SOCP \eqref{eq:nlp}.} Even with the most refined
partition, the explicit implementation is up to two orders of magnitude faster
than implicit MPC. More importantly, real-time execution of both the explicit
and semi-explicit implementations is \textit{guaranteed}. Meanwhile, if we take
$T_{\mathrm{socp}}=1~\text{ms}$ as the approximate \eqref{eq:nlp} solution time,
then the worst-case runtime of implicit MPC is
$\approx 27 T_{\mathrm{socp}}=27~\text{ms}$ where $27=3^3$ is the number of
input subset choices over the 3-step prediction horizon.
Figure~\ref{fig:fuel} characterizes the extent of fuel consumption suboptimality
with respect to implicit MPC.
We see that the finer partitions induce a smaller fuel penalty, with down to
$<20\%$ fuel over-consumption. \longtxt{Figure~\ref{fig:response} shows the temporal
response for each implementation, comparing the coarsest to the most refined
partition. The input is actuated a lot more for the coarsest partition, hence
the higher fuel consumption.}
\longtxt{
\subsection{Improving the Storage Memory Requirement}
\label{subsec:improving_storage_memory_requirement}
Our binary tree implementation is not memory-optimized, so the $M$ values in
Table~\ref{tab:results} are far greater than necessary. Two more efficient
storage models are possible.
The first model stores the unique simplex vertices and each ``left'' child
node references its $p+1$ vertices via integer indices. For explicit MPC,
``right'' child leaf nodes also reference their vertices. Lastly, all leaves
store their $\epsilon$-suboptimal commutation in the semi-explicit case, or
the $\epsilon$-suboptimal decision vectors at each vertex in the explicit
case. Assuming a perfect binary tree and that $\Theta$ is a simplex, the
semi-explicit and explicit storage memory requirements are, respectively:
\begin{align*}
M_1^{\mathrm{se}}
&\approx
\lambda(\mu_{\mathrm{f}}p+\mu_{\mathrm{i}}(p+1)+\mu_{\mathrm{b}}m)+
\mu_{\mathrm{f}}p^2, \\
M_1^{\mathrm{e}}
&\approx
\lambda(\mu_{\mathrm{f}}p+(p+1)(\frac{3}{2}\mu_{\mathrm{i}}+\mu_{\mathrm{f}}\hat n))+
\mu_{\mathrm{f}}p^2,
\end{align*}
where $\mu_{\mathrm{f}}$, $\mu_{\mathrm{i}}$ and $\mu_{\mathrm{b}}$ are the
floating point, integer and Boolean sizes, and $\hat n\le n$ is the decision
vector part that is necessary for control input computation (e.g. the first
control input for MPC). In the case of the last two rows of
Table~\ref{tab:results}, we have
$M_1^{\mathrm{se}}\approx 1.98~\text{MB}\ll M$ and
$M_1^{\mathrm{e}}\approx 3.35~\text{MB}\ll M$ using \texttt{float64},
\texttt{uint32} and \texttt{char} types. The memory requirement for the actual
binary trees differs by $<0.01~\text{MB}$ from these theoretical values.
This storage model, however, has the downside of requiring to invert a
$\mathbb R^{p\times p}$ matrix on-line in order to evaluate in
$\theta\in\mathcal R$ in \eqref{eq:f_subopt_eval} or
\eqref{eq:explicit_mpc_query}. The matrix inverse, however, can be computed
off-line and stored instead of the cell vertices. Again assuming a perfect
binary tree and that $\Theta$ is a simplex, the storage memory requirements
for this second model are:
\begin{align*}
M_2^{\mathrm{se}}
&\approx
\lambda\mu_{\mathrm{f}}p(p+1)+\lambda m\mu_{\mathrm{b}}, \\
M_2^{\mathrm{e}}
&\approx
\frac{3}{2}\lambda\mu_{\mathrm{f}}p(p+1)+\lambda(p+1)\hat n\mu_{\mathrm{f}}.
\end{align*}
Again for the last two rows of Table~\ref{tab:results}, we have
$M_1^{\mathrm{se}}<M_2^{\mathrm{se}}\approx 3.05~\text{MB}\ll M$ and
$M_1^{\mathrm{e}}<M_2^{\mathrm{e}}\approx 5.55~\text{MB}\ll M$. While
greater economy is possible by eliminating further redundancy in the stored
vertices and optimal decision vectors, these two storage models show that the
memory required by the resulting partition need not be prohibitive. For
example, the partition can be stored on a NOR external memory device
\cite{Micron2011} and interfaced to a microprocessor. }
\section{Extension to Explicit MPC}
\label{sec:extensions}
This section modifies Algorithm~\ref{alg:phase2} for an explicit implementation
where no optimization happens on-line. First, we recognize that a convex
combination of vertex optimal decision vectors lower bounds $\bar V_\delta$.
\begin{theorem}
\label{theorem:cvx_comb_decision_vector}
Let $(\mathcal R,\delta)$ be a cell of the $f_\delta^\epsilon$ partition. Let
$v_i\in\mathcal V(\mathcal R)$ be the $i$-th vertex and $x_i\in\mathbb R^n$ the
associated optimal decision vector for \eqref{eq:nlp} where
$\theta=v_i$. Consider $\theta\in\mathcal R$ such that
$\theta=\sum_{i=1}^{|\mathcal V(\mathcal R)|}\alpha_iv_i$ with
$\alpha_i\in\mathbb R$.
Let
\begin{equation}
\label{eq:cvx_comb_x}
x^*\triangleq\sum_{i=1}^{|\mathcal V(\mathcal R)|}\alpha_ix_i
\end{equation}
and $\hat V_\delta(\theta)\triangleq f(\theta,x^*,\delta)$. Then $x$ is
feasible for \eqref{eq:nlp} and
\begin{equation}
\label{eq:f_le_V}
\hat V_\delta(\theta) \le \bar V_\delta(\theta).
\end{equation}
\shorttxt{
\begin{proof}
For feasibility, exploit that $g(\cdot,\cdot,\delta)$ and
$h(\cdot,\cdot,\delta)$ in \eqref{eq:nlp} are affine. For
\eqref{eq:f_le_V}, exploit that $f(\cdot,\cdot,\delta)$ in \eqref{eq:nlp}
is convex. See detailed proof in \cite{Malyuta2019c}.
\end{proof}
}
\longtxt{ \begin{proof}
Since $g(\cdot,\cdot,\delta)$ and $h(\cdot,\cdot,\delta)$ are affine,
\begin{align*}
g(\theta,x^*,\delta) =
\sum_{i=1}^{|\mathcal V(\mathcal R)|}\alpha_ig(v_i,x_i,\delta) = 0, \\
h(\theta,x^*,\delta) =
\sum_{i=1}^{|\mathcal V(\mathcal R)|}\alpha_ih(v_i,x_i,\delta) \in\mathcal K,
\end{align*}
so $x$ is feasible for \eqref{eq:nlp}. Next, since $f(\cdot,\cdot,\delta)$
is convex,
\begin{equation*}
\hat V_\delta(\theta) \le
\sum_{i=1}^{|\mathcal V(\mathcal R)|}\alpha_if(v_i,x_i,\delta)=
\sum_{i=1}^{|\mathcal V(\mathcal R)|}\alpha_iV_\delta^*(v_i) = \bar V_\delta(\theta).\qedhere
\end{equation*}
\end{proof}}
\end{theorem}
Theorem~\ref{theorem:cvx_comb_decision_vector} gives a direct expression
\eqref{eq:cvx_comb_x} for the $\epsilon$-suboptimal decision vector, which can
be evaluated as long as the $x_i$'s are known. To this end, let
Algorithm~\ref{alg:phase2} store the closed leaves with this information,
i.e. $(\mathcal R,\delta,\{x_i\}_{i=1}^{|\mathcal V(\mathcal R)|})$.
It remains to ensure that $x^*$ is $\epsilon$-suboptimal. By definition, this
requires
$\hat V_\delta(\theta)-V^*(\theta)\le\max\{\epsilon_{\mathrm{a}},\epsilon_{\mathrm{r}}V^*(\theta)\}$
where $V^*$ considers \textit{all} $\delta\in\mathbb I^m$. To this end, the
tractable absolute and relative error over-approximators are updated to also
check for $\epsilon$-suboptimality with respect to $\delta$ itself:
\begin{align}
\label{eq:absolute_error_redefined}
\hat e_{\mathrm{a}}(\mathcal R)
&= \max_{\theta\in\mathcal R,\delta'\in\mathbb I^m}
\bar V_\delta(\theta)-V_{\delta'}^*(\theta),
\tag{$\hat{\text{E}}^{\mathcal R}_{\mathrm{a}}$} \\
\label{eq:relative_error_redefined}
\hat e_{\mathrm{r}}(\mathcal R)
&= \frac{\hat e_{\mathrm{a}}(\mathcal R)}{
\min_{\theta\in\mathcal R}V^*(\theta)}.
\tag{$\hat{\text{E}}^{\mathcal R}_{\mathrm{r}}$}
\end{align}
We then use \eqref{eq:absolute_error_redefined} and
\eqref{eq:relative_error_redefined} instead of \eqref{eq:absolute_error_approx}
and \eqref{eq:relative_error_approx} in Algorithm~\ref{alg:phase2}. The
following theorem guarantees that $\hat V_\delta(\theta)$, and hence $x^*$ in
\eqref{eq:cvx_comb_x}, is $\epsilon$-suboptimal.
\begin{theorem}
$\hat V_\delta(\theta)$ obtained using \eqref{eq:absolute_error_redefined} and
\eqref{eq:relative_error_redefined} in Algorithm~\ref{alg:phase2} is
$\epsilon$-suboptimal.
\shorttxt{
\begin{proof}
Exploit \eqref{eq:f_le_V} and that
$\hat e_{\mathrm{a}}(\mathcal R)\le\epsilon_{\mathrm{a}}$ or
$\hat e_{\mathrm{r}}(\mathcal R)\le\epsilon_{\mathrm{r}}$. See detailed proof in \cite{Malyuta2019c}.
\end{proof}
}
\longtxt{ \begin{proof}
Using \eqref{eq:f_le_V}, the absolute error component of
\eqref{eq:epsilon_suboptimality} holds:
\begin{equation*}
\hat V_\delta(\theta)-V^*(\theta)\le
\bar V_\delta(\theta)-V^*(\theta)\le \hat e_{\mathrm{a}}(\mathcal R)
\le\epsilon_{\mathrm{a}}.
\end{equation*}
The relative error component of \eqref{eq:epsilon_suboptimality} follows
from:
\begin{equation*}
\hat e_{\mathrm{a}}(\mathcal R)=
\hat e_{\mathrm{r}}(\mathcal R)\min_{\theta\in\mathcal R}
V^*(\theta)\le\epsilon_{\mathrm{r}}V^*(\theta).\qedhere
\end{equation*}
\end{proof}}
\end{theorem}
Given a parameter $\theta$, the partition can be used to obtain the
$\epsilon$-suboptimal decision vector directly via a tree query:
\begin{equation}
\label{eq:explicit_mpc_query}
x^* = \sum_{i=1}^{|\mathcal V(\mathcal R)|}\alpha_ix_i\text{ such that }
\theta = \sum_{i=1}^{|\mathcal V(\mathcal R)|}\alpha_iv_i\in\mathcal R,~v_i\in\mathcal V(\mathcal R).
\end{equation}
\section{Future Work}
\label{sec:future}
Two possible directions of future research exist. First, the authors of
\cite{MunozDeLaPena2004,MunozDeLaPena2006} prove that stability of implicit
robust MPC is maintained by their explicit suboptimal control law, albeit
without a mixed-integer component. Because our definition of
$\epsilon$-suboptimality is a hybrid of their absolute and relative error
bounds, it shall be interesting to prove a similar result. Second,
lines~\ref{alg:phase2:line:depth_first}-\ref{alg:phase2:line:end} are can run in
parallel for different open leaves. As a result, Algorithm~\ref{alg:phase2} is
amenable to the highly parallelizable \texttt{map} pattern \cite{McCool2012}. We
aim to write a massively parallel implementation of Algorithm~\ref{alg:phase2}
in order to handle systems with higher dimensional parameter vectors.
\section{Introduction}
\label{introduction}
This paper develops a provably convergent algorithm for generating semi-explicit
and explicit real-time implementations of multiparametric convex mixed-integer
nonlinear programs (MINLPs) to within an arbitrarily small suboptimality
tolerance.
The algorithm is particularly well suited for model predictive control
(MPC). Two important types of MPC are hybrid and robust MPC
\cite{Mayne2014}. The former handles systems with discrete switches or piecewise
affinely-approximated nonlinearities like chemical powerplants, pipelines and
aerospace vehicles \cite{Bemporad1999a,Blackmore2012}. The latter handles
uncertain systems \cite{Bemporad2007,Malyuta2019}. When combined, the two
formulations require the solution of a MINLP. If the system dynamics are fast,
real-time implementation is hampered by the worst-case exponential runtime
complexity of mixed-integer solvers \cite{Bemporad1999a}.
Several approaches have been proposed to nevertheless attain real-time
performance. By leveraging the polynomial runtime complexity of convex solvers,
the technique of successive convexification is able to solve nonlinear programs
in real-time \cite{Mao2017,Bonalli2019}. Recently, the method was extended to
handle binary decision making via state-triggered constraints
\cite{Szmuk2019b}. Hence, at least some MINLPs may be solved in real-time via
successive convexification. However, this is a local method which may not always
converge to a feasible solution.
A more traditional method of ensuring real-time MPC performance while
guaranteeing convergence and global optimality has been to make the optimal
solution ``explicit'' by moving the optimization
off-line.
Various explicit MPC methodologies have been proposed
\cite{Alessio2009,Bemporad2006,Pistikopoulos2012}. For MPC laws more complicated
than linear or quadratic programs, exact explicit solutions are generally not
possible due to non-convexity of common active constraint sets
\cite{Bemporad2006b}. Instead, approximate solutions have been proposed via
local linearization \cite{Pistikopoulos2007a,Oberdieck2017} or via optimal cost
bounding by affine functions over simplices
\cite{Bemporad2006b,MunozDeLaPena2004,MunozDeLaPena2006} or hyperrectangles
\cite{Johansen2004}. An approximate explicit solution to mixed-integer quadratic
programs has been proposed based on difference-of-convex programming
\cite{Alessio2006} and for MINLPs based on local linearization and primal/master
subproblems \cite{Dua1998,Rowe2003}.
Our primary contribution is a novel algorithm for approximate semi-explicit and
explicit implementation of multiparametric convex MINLPs.
The former queries a binary tree for an integer solution and solves a single
convex program on-line. The latter queries a binary tree for a solution directly,
without on-line optimization.
To the best of our knowledge, we are also
the first to prove convergence of the partitioning scheme by using a novel
definition of an optimal cost ``overlap'' which is a fundamental determinant of
convergence speed and partition complexity.
The paper is organized as follows. Section~\ref{sec:problem_formulation} defines
the class of MPC laws that our algorithm can handle. Section~\ref{sec:phase2}
then presents the algorithm, followed by its convergence and complexity
properties in Section~\ref{sec:properties}. The algorithm is extended to an
explicit implementation in Section~\ref{sec:extensions}. The validity of the
approach is demonstrated through a spacecraft robust position control example in
Section~\ref{sec:example}. Section~\ref{sec:conclusion} concludes with future
research directions.
\textit{Notation}: $\mathbb I\triangleq\{0,1\}$ is the binary set and
$\mathbb B\triangleq\{x:\|x\|_2\le 1\}$ is the unit ball. Matrices are uppercase
(e.g. $A$), scalars, vectors and functions are lowercase (e.g. $x$), and sets
are calligraphic uppercase (e.g. $\mathcal S$). The scalar $\ell$ is a
placeholder. ${\mathop \mathrm{co}}\mathcal S$, $\mathcal S^{\mathrm{c}}$, $\partial\mathcal S$,
$\mathrm{vol}(\mathcal S)$ and $\mathcal V(\mathcal R)$ denote respectively the
convex hull, complement, boundary, volume and vertices of $\mathcal S$. The
cardinality of a countable set $\mathcal S$ is $|\mathcal S|$. Given
$\mathcal A\subseteq\mathbb R^n$, $s\in\mathbb R$ and $b\in\mathbb R^n$,
$\mathcal A+b\triangleq\{a+b\in\mathbb R^n:a\in\mathcal A\}$ and
$s\mathcal A\triangleq\{sa: a\in\mathcal A\}$.
\section{Phase I: Computing $f_\delta$}
\label{sec:phase1}
This section presents a standalone algorithm for computing $f_\delta$. The main
idea is to generate a coarse polytopic partition
$\mathcal R=\{(\mathcal R_i,\delta_i)\}_{i=1}^{|\mathcal R|}$ such that
$\Theta=\bigcup_{i=1}^{|\mathcal R|}\mathcal R_i$ and where each cell
$\mathcal R_i$ is associated with a fixed commutation $\delta_i$. We then have:
\begin{equation}
\label{eq:feasible_commutation_function}
f_\delta(\theta)=\delta_i\text{ such that }\theta\in\mathcal R_i.
\end{equation}
\begin{lemma}
\label{lemma:convexity}
For any fixed $\delta\in\mathbb I^m$, $\Theta^*_\delta$ is a convex set and
$V^*_\delta$ is a convex function.
\begin{proof}
Suppose $\theta',\theta''\in\Theta^*_\delta$. Let
$\alpha',\alpha''\in [0,1]$, $\alpha'+\alpha''=1$ and
$\theta=\alpha'\theta'+\alpha''\theta''$. Since $g,h$ are affine in their
first argument and $\mathcal K$ is a convex cone:
\begin{equation*}
\begin{split}
g(\theta,x,\delta)=
\alpha'g(\theta',x,\delta)+\alpha''g(\theta'',x,\delta)=0, \\
h(\theta,x,\delta)=
\alpha'h(\theta',x,\delta)+\alpha''h(\theta'',x,\delta)\in\mathcal K,
\end{split}
\end{equation*}
so $\theta\in\Theta^*_\delta$ which is thus a convex set. Next,
(\ref{eq:nlp}) is a minimization of $f$ over a convex set in $x$ which
preserves convexity in $\theta$ by the joint convexity property of
$f$. Thus, $V^*_\delta$ is a convex function.
\end{proof}
\end{lemma}
\begin{algorithm}
\centering
\begin{algorithmic}[1]
\State $\mathcal R\gets\emptyset$, $\bar\Theta\gets \Theta$
\For{$\delta\in 2^{\mathbb I^m}$}
\State $\mathcal R\gets\{(\mathcal R',\delta):
\mathcal R'\in\bar\Theta\cap\Theta^*_\delta\}\cup\mathcal R$
\State $\bar\Theta\gets\bar\Theta\setminus\Theta^*_\delta$
\If{$\bar\Theta=\emptyset$}
\State STOP
\EndIf
\EndFor
\caption{Brute force $f_\delta$ computation.}
\label{alg:phase1bf}
\end{algorithmic}
\end{algorithm}
Since $\Theta^*_\delta$ is convex, an arbitrarily precise inner approximation of
it can be found \cite{Dueri2016}. A conceptually trivial method for generating
$\mathcal R$ is given by Algorithm~\ref{alg:phase1bf}. The set difference and
intersection operations in Algorithm~\ref{alg:phase1bf} are element-wise
\cite{Baotic2009}. The idea is to exploit the ability to inner approximate
$\Theta^*_\delta$ to procedurally ``cover'' $\Theta$.
The filling problem is combinatorial, however, such that in the worst case all
$2^m$ possible values $\delta$ are needed to fill $\Theta$. Furthermore,
accurate polytopic inner approximation of $\Theta^*_\delta$ in
higher-dimensional spaces than about $\mathbb R^4$ suffers from excessive vertex
count \cite{Dueri2016}. Algorithm~\ref{alg:phase1bf} is therefore undesirable as
it makes no attempt to avoid either the combinatorial or the verticial
complexities. Instead, we propose Algorithm~\ref{alg:phase1} in which the first
problem is tackled by a volume-maximizing heuristic and the second problem by
inner-approximating $\Theta^*_\delta$ with a regular simplex.
\begin{algorithm}
\centering
\begin{algorithmic}[1]
\State Create empty tree with open leaf $\Theta$ as root
\State $\mathcal S\gets\mathtt{delaunay}(\mathcal V(\Theta))$
\State Add child open leaves $\mathcal S_i$
$\forall\mathcal S_i\in\mathcal S$
\label{alg:phase1:line:init}
\While{any open leaf exists}
\label{alg:phase1:line:whileloop}
\State $\mathcal R\gets\text{the first open leaf}$
\label{alg:phase1:line:getfirstopenleaf}
\State $\hat\delta\gets\text{solve (\ref{eq:maxvol}) for
$\mathcal D=\mathcal R$}$
\label{alg:phase1:line:findmaxvolcommutation}
\If{(\ref{eq:maxvol}) for $\mathcal D=\mathcal R$ infeasible}
\State STOP, $\Theta^*\setminus\Theta\ne\emptyset$
\label{alg:phase1:line:stop}
\ElsIf{(\ref{eq:nlp}) feasible for $\delta=\hat\delta$,
$\forall\theta\in\mathcal V(\mathcal R)$}
\label{alg:phase1:line:vertexfeasibilitytest}
\State Replace leaf with closed leaf $(\mathcal R,\hat\delta)$
\label{alg:phase1:line:closeleaf}
\Else
\State
$\bar v,\bar v'\gets\arg\max_{v,v'\in\mathcal V(\mathcal R)}\|v-v'\|_2$
\label{alg:phase1:line:longestedge}
\State $\mathcal S\gets\mathtt{triangulate}(\mathcal R,(\bar v+\bar v')/2)$
\label{alg:phase1:line:splitlongestedge}
\State Add child open leaves $\mathcal S_i$ $\forall\mathcal S_i\in\mathcal S$
\label{alg:phase1:line:addnewregion}
\EndIf
\EndWhile
\caption{Phase I computation of $f_\delta$.}
\label{alg:phase1}
\end{algorithmic}
\end{algorithm}
Algorithm~\ref{alg:phase1} relies on a reformulation of (\ref{eq:nlp}) as the
following NLP shooting problem:
\begin{equation}
\label{eq:shooting}
\@ifnextchar[{\@with}{\@without}[
g(\theta,x,\delta)=0 \\
&&& h(\theta,x,\delta)\in\mathcal K \\
&&& \theta=c^{\mathcal D}+\alpha d\in\mathcal D,\text{ }\alpha\in\mathbb R_+,
]{\alpha^*_\delta(d,\mathcal D)}{\max}{x,\theta,\alpha}{\alpha}
\tag{S$_{\delta,d}^{\mathcal D}$}
\end{equation}
where $\mathcal D\subseteq\Theta^*_\delta$ is the shooting domain,
$c^{\mathcal D}$ is the domain's centroid and $d$ is the shooting
direction. Since (\ref{eq:shooting}) is a convex program, it can be solved
efficiently \cite{Dueri2017}. The algorithm also requires the following bilevel
MINLP:
\begin{equation}
\label{eq:maxvol}
\@ifnextchar[{\@with}{\@without}[
\delta\in\mathbb I^m,
]{\hat \delta(\mathcal D)}{\arg\max}{\delta}{
\textstyle{\sum}_{i=1}^{|\mathcal V(\mathcal D)|}\alpha^*_\delta(d_i^{\mathcal D},\mathcal D)
}
\tag{V$^{\mathcal D}$}
\end{equation}
where $\mathcal V(\mathcal D)$ is the set of vertices of domain $\mathcal D$ and
$d^{\mathcal D}_i\triangleq v_i-c^{\mathcal D}$ is the direction from the centroid to the
$i$-th vertex. Problem (\ref{eq:maxvol}) finds a commutation $\hat\delta$ which
maximizes the average ``travel'' along each centroid-vertex ray and is a
heuristic for finding the volume-maximizing commutation for set $\mathcal C$ in
Algorithm~\ref{alg:phase1}. Although bilevel programs are generally difficult to
solve \cite{Lofberg2012}, (\ref{eq:maxvol}) is a MINLP that is solved
efficiently via branch and bound \cite{Fletcher1998}.
Algorithm~\ref{alg:phase1} creates a simplicial partition of $\Theta$ as
follows. The partition is stored as a tree whose leaves are cells
$(\mathcal S,\delta)$ storing the set $\mathcal S$ and associated commutation
$\delta$. Non-leaf nodes in the tree store just the sets and make evaluating
(\ref{eq:feasible_commutation_function}) more efficient (see
Section~\ref{subsec:complexity}). Open leaves become non-leaf nodes at the next
while-loop iteration while closed leaves are no longer considered in the
computation. We refer to nodes directly by their contents, i.e. $\mathcal S$ for
open and $(\mathcal S,\delta)$ for closed leaves. On
line~\ref{alg:phase1:line:init} the tree root is initialized to $\Theta$ and,
since generally $\Theta$ is not a simplex, Delaunay triangulation is first
applied. The tree is then iterated on line~\ref{alg:phase1:line:whileloop} in a
depth-first manner until no open leaves are left. By doing a depth-first search,
Assumption~\ref{assumption:positiveoverlap} in
Section~\ref{subsec:phase1convergence} is disproved more quickly in case that it
does not hold, such that the algorithm fails with less wasted time. Thus, the
first open leaf $\mathcal R$ in the tree data structure is selected on
line~\ref{alg:phase1:line:getfirstopenleaf}.
Lines~\ref{alg:phase1:line:findmaxvolcommutation}-\ref{alg:phase1:line:addnewregion}
partition $\mathcal R$ into at most $p+2$ cells. First, a volume-maximizing
$\hat\delta$ is found. If (\ref{eq:maxvol}) is infeasible, however, it means
$\theta=c^{\mathcal R}$ is infeasible for (\ref{eq:minlp}) which implies that
$\Theta$ contains regions outside $\Theta^*$. Section~\ref{subsec:Theta}
discussed possible actions in this case. On the other hand, if (\ref{eq:maxvol})
is feasible then we want to check if $\mathcal R\subseteq\Theta^*_{\hat\delta}$,
in which case the whole region can be added as a closed leaf
$(\mathcal R,\hat\delta)$. This check can be efficiently performed via
Lemma~\ref{lemma:containment}.
\begin{lemma}
\label{lemma:containment}
$\mathcal R\subseteq\Theta^*_{\hat\delta}$ $\Leftrightarrow$ (\ref{eq:nlp}) is
feasible $\forall\theta\in\mathcal V(\mathcal R)$ and $\delta=\hat\delta$.
\begin{proof}
$(\Rightarrow)$ Since $\mathcal R\subseteq\Theta^*_{\hat\delta}$,
$\theta\in\mathcal V(\mathcal R)\Rightarrow\theta\in\Theta_{\hat\delta}^*$. Since
$\Theta_{\hat\delta}^*$ is the fixed-commutation feasible parameter set,
(\ref{eq:nlp}) is by definition feasible. $(\Leftarrow)$ Any $\theta$ such
that (\ref{eq:nlp}) is feasible satisfies, by definition,
$\theta\in\Theta^*_{\hat\delta}$. By Lemma~\ref{lemma:convexity}, since
$\Theta^*_{\hat\delta}$ is convex,
${\mathop \mathrm{co}}\{\theta\in\mathcal V(\mathcal R)\}\equiv\mathcal R\subseteq\Theta^*_{\hat\delta}$.
\end{proof}
\end{lemma}
If Lemma~\ref{lemma:containment} fails, $\mathcal R$ is split into two smaller
simplices along its longest edge on
line~\ref{alg:phase1:line:splitlongestedge}. As explained in
Section~\ref{subsec:phase1convergence}, this yields a volume reduction that
necessarily leads to convergence if Assumption~\ref{assumption:positiveoverlap}
holds.
\section{Suboptimal Map Computation}
\label{sec:phase2}
This section presents an algorithm for computing $f_\delta^\epsilon$. We target
a semi-explicit MPC implementation where $f_\delta^\epsilon$ yields a $\delta$
that is used to solve the convex problem \eqref{eq:nlp}
on-line.
Recent research proposed an algorithm \cite[Algorithm~2]{Malyuta2019a}\xspace for computing $f_\delta$ as a
coarse simplicial partition
$\mathcal R=\{(\mathcal R_i,\delta_i)\}_{i=1}^{|\mathcal R|}$ that is stored as
a binary tree where $(\mathcal R_i,\delta_i)$ are the leaves. We have
$\mathcal R_i\subseteq\Theta_{\delta_i}^*$ and
$\Theta=\bigcup_{i=1}^{|\mathcal R|}\mathcal R_i$. To compute
$f_\delta^\epsilon$, we shall refine this partition until $\delta_i$ becomes
$\epsilon$-suboptimal within $\mathcal R_i$. $f_\delta^\epsilon$ can then be
recovered using:
\begin{equation}
\label{eq:f_subopt_eval}
f_\delta^\epsilon(\theta) = \delta_i\text{ such that }\theta\in\mathcal R_i.
\end{equation}
Let $(\mathcal R,\delta)$ be some partition cell. The algorithm must determine
whether $\delta$ is $\epsilon$-suboptimal in $\mathcal R$. This is encoded in
the following absolute and relative error bilevel MINLPs:
\begin{align}
\label{eq:absolute_error}
e_{\mathrm{a}}^*(\mathcal R)
&= \max_{\theta\in\mathcal R,\delta'\in\mathbb I^m\setminus\{\delta\}}
V_\delta^*(\theta)-V_{\delta'}^*(\theta),
\tag{E$^{\mathcal R}_{\mathrm{a}}$} \\
\label{eq:relative_error}
e_{\mathrm{r}}^*(\mathcal R)
&= \max_{\theta\in\mathcal R,\delta'\in\mathbb I^m\setminus\{\delta\}}
(V_\delta^*(\theta)-V_{\delta'}^*(\theta))/V_{\delta'}^*(\theta).
\tag{E$^{\mathcal R}_{\mathrm{r}}$}
\end{align}
If $e_{\mathrm{a}}^*(\mathcal R)\le\epsilon_{\mathrm{a}}$ or
$e_{\mathrm{r}}^*(\mathcal R)\le\epsilon_{\mathrm{r}}$ then $\delta$ is
$\epsilon$-suboptimal in $\mathcal R$. Both problems above, however, are
non-convex so their optimal solutions are not readily computable. As a remedy,
we formulate tractable upper bounds that are sufficient to guarantee
$\epsilon$-suboptimality.
We focus first on $e_{\mathrm{a}}^*(\mathcal R)$.
\begin{definition}
\label{definition:cost_affine_over_approximator}
Let $v_i\in\mathcal V(\mathcal R)$ be the $i$-th vertex of $\mathcal R$ and
let $\theta=\sum_{i=1}^{|\mathcal V(\mathcal R)|}\alpha_iv_i$ with
$\alpha_i\in\mathbb R$.
The affine over-approximator of $V^*_\delta(\theta)$ over $\mathcal R$ is:
\begin{equation}
\label{eq:over_approximator}
\bar V_\delta(\theta)\triangleq\sum_{i=1}^{|\mathcal V(\mathcal R)|}\alpha_iV^*_\delta(v_i).
\end{equation}
\end{definition}
\begin{proposition}
\label{proposition:over_approximator}
The affine over-approximator (\ref{eq:over_approximator}) satisfies
$V^*_{\delta}(\theta)\le\bar V_{\delta}(\theta)$ $\forall\theta\in\mathcal R$
and with equality at the vertices.
\begin{proof}
Since $V^*_{\delta}$ is convex \cite[Lemma~1]{Malyuta2019a}, Jensen's
inequality implies $V^*_\delta(\theta)\le\bar V_\delta(\theta)$.
\end{proof}
\end{proposition}
Consider now the following convex MINLP:
\begin{equation}
\label{eq:absolute_error_approx}
\bar e_{\mathrm{a}}(\mathcal R)
=
\max_{\theta\in\mathcal R,\delta'\in\mathbb I^m\setminus\{\delta\}}
\bar V_\delta(\theta)-V^*_{\delta'}(\theta)
\tag{$\bar{\text{E}}^{\mathcal R}_{\mathrm{a}}$}.
\end{equation}
\longtxt{
\begin{lemma}
\label{lemma:absolute_error}
The absolute error satisfies
$e^*_{\mathrm{a}}(\mathcal R)\le \bar e_{\mathrm{a}}(\mathcal R)$.
\begin{proof}
Let $\theta\in\mathcal R$ and $\delta'\in\mathbb I^m\setminus\{\delta\}$. By
Proposition~\ref{proposition:over_approximator},
$V^*_\delta(\theta)\le\bar V_\delta(\theta)\Rightarrow V^*_\delta(\theta)-V^*_{\delta'}(\theta)\le\bar V_\delta(\theta)-V^*_{\delta'}(\theta)$. The
result follows directly.
\end{proof}
\end{lemma}
}
\begin{theorem}
\label{theorem:absolute_error_check_sufficiency}
$\bar e_{\mathrm{a}}(\mathcal R)\le\epsilon_{\mathrm{a}}$ implies
$e_{\mathrm{a}}^*(\mathcal R)\le\epsilon_{\mathrm{a}}$.
\longtxt{
\begin{proof}
By Lemma~\ref{lemma:absolute_error},
$e_{\mathrm{a}}^*(\mathcal R)\le\bar e_{\mathrm{a}}(\mathcal R)\le\epsilon_{\text{a}}$.
\end{proof}}
\shorttxt{
\begin{proof}
Exploit Proposition~\ref{proposition:over_approximator}. See detailed proof in \cite{Malyuta2019c}.
\end{proof}
}
\end{theorem}
A tractable alternative to
(\ref{eq:relative_error}) was given in \cite{MunozDeLaPena2004}:
\begin{equation*}
\label{eq:relative_error_approx}
\bar e_{\mathrm{r}}(\mathcal R) = \frac{
\bar e_{\mathrm{a}}(\mathcal R)
}{
\displaystyle
\min_{\theta\in\mathcal R,\delta'\in\mathbb I^m\setminus\{\delta\}}V^*_{\delta'}(\theta)
}.
\tag{$\bar{\text{E}}^{\mathcal R}_{\mathrm{r}}$}
\end{equation*}
\longtxt{
\begin{lemma}
\label{lemma:relative_error}
The relative error satisfies
$e_{\mathrm{r}}^*(\mathcal R)\le\bar e_{\mathrm{r}}(\mathrm{R})$.
\begin{proof}
The result follows from Lemma~\ref{lemma:absolute_error} and the following
sequence of (conservative) over-approximations:
\begin{align*}
e_{\mathrm{r}}^*(\mathcal R)
&= \max_{\theta\in\mathcal R,\delta'\in\mathbb I^m\setminus\{\delta\}}
\frac{V_\delta^*(\theta)-V_{\delta'}^*(\theta)}{V_{\delta'}^*(\theta)} \\
&\le \max_{\theta\in\mathcal R,\delta'\in\mathbb I^m\setminus\{\delta\}}
\frac{V_\delta^*(\theta)-V_{\delta'}^*(\theta)}{
\displaystyle
\min_{\theta'\in\mathcal R,\delta''\in\mathbb I^m\setminus\{\delta\}}
V_{\delta''}^*(\theta')} \\
&= \frac{e_{\mathrm{a}}^*(\mathcal R)}{
\displaystyle
\min_{\theta\in\mathcal R,\delta'\in\mathbb I^m\setminus\{\delta\}}
V_{\delta'}^*(\theta)} \\
&\le \frac{\bar e_{\mathrm{a}}(\mathcal R)}{
\displaystyle
\min_{\theta\in\mathcal R,\delta'\in\mathbb I^m\setminus\{\delta\}}
V_{\delta'}^*(\theta)} = \bar e_{\mathrm{r}}(\mathcal R).
\end{align*}
\end{proof}
\end{lemma}
}
\begin{theorem}
\label{theorem:relative_error_check_sufficiency}
$\bar e_{\mathrm{r}}(\mathcal R)\le\epsilon_{\mathrm{r}}$ implies
$e_{\mathrm{r}}^*(\mathcal R)\le\epsilon_{\mathrm{r}}$.
\longtxt{\begin{proof}
By Lemma~\ref{lemma:relative_error},
$e_{\mathrm{r}}^*(\mathcal R)\le\bar e_{\mathrm{r}}(\mathcal R)\le\epsilon_{\text{r}}$.
\end{proof}}
\shorttxt{
\begin{proof}
Minimize the denominator in \eqref{eq:relative_error} independently and
exploit Theorem~\ref{theorem:absolute_error_check_sufficiency}. See detailed proof in \cite{Malyuta2019c}.
\end{proof}
}
\end{theorem}
Hence the computationally tractable (\ref{eq:absolute_error_approx}) and
(\ref{eq:relative_error_approx}) can be used to guarantee
$\epsilon$-suboptimality.
When both $\bar e_{\mathrm{a}}(\mathcal R)>\epsilon_{\mathrm{a}}$ and
$\bar e_{\mathrm{r}}(\mathcal R)>\epsilon_{\mathrm{r}}$, however,
(\ref{eq:absolute_error_approx}) and (\ref{eq:relative_error_approx}) are
indecisive due to Theorems~\ref{theorem:absolute_error_check_sufficiency} and
\ref{theorem:relative_error_check_sufficiency} being merely sufficient.
Section~\ref{sec:properties} shows that a sound remedy is to further subdivide
$\mathcal R$.
\begin{algorithm}[t]
\centering
\begin{algorithmic}[1]
\State Open all leaves of the tree output by \cite[Algorithm~2]{Malyuta2019a}\xspace
\While{any open leaf exists}
\label{alg:phase2:line:while}
\State $(\mathcal R,\delta)\gets\text{the most recent open leaf}$
\label{alg:phase2:line:depth_first}
\State $\bar e_{\mathrm{a}}(\mathcal R)\gets\text{solve
(\ref{eq:absolute_error_approx})}$
\label{alg:phase2:line:suboptimalitycheckstart}
\State $\bar e_{\mathrm{r}}(\mathcal R)\gets\text{solve
(\ref{eq:relative_error_approx})}$
\State $\text{infeasible}\gets\text{(\ref{eq:absolute_error_approx}) is
infeasible}$
\State $\text{$\epsilon$-suboptimal}\gets\text{$\bar e_{\mathrm{a}}(\mathcal R)\le\epsilon_{\mathrm{a}}$ or
$\bar e_{\mathrm{r}}(\mathcal R)\le\epsilon_{\mathrm{r}}$}$
\label{alg:phase2:line:suboptimalitycheck}
\If{infeasible or $\epsilon$-suboptimal}
\State Close leaf
\label{alg:phase2:line:suboptimalitycheckend}
\Else
\State $\delta^*\gets\text{solve \eqref{eq:better_delta_selection}}$
\label{alg:phase2:line:find_better_commutation}
\If{\eqref{eq:better_delta_selection} feasible}
\If{\eqref{eq:curvature_constraint} holds for $\Theta_\sigma=\mathcal R$}
\label{alg:phase2:line:check_curvature_constraint}
\State Change open leaf to $(\mathcal R,\mathcal \delta^*)$
\label{alg:phase2:line:add_R_without_split}
\Else
\State $\mathcal S_1,\mathcal S_2\gets\Call{split}{\mathcal R}$
\label{alg:phase2:line:split_R_opt1}
\State Add child open leaves $(\mathcal S_1,\delta^*)$ and $(\mathcal S_2,\delta^*)$
\label{alg:phase2:line:split_R_opt2}
\EndIf
\Else
\State $\mathcal S_1,\mathcal S_2\gets\Call{split}{\mathcal R}$
\label{alg:phase2:line:split_R_subopt}
\State Add child open leaves $(\mathcal S_1,\delta)$ and $(\mathcal S_2,\delta)$
\label{alg:phase2:line:end}
\EndIf
\EndIf
\EndWhile
\Function{split}{$\mathcal R$}
\State
$\bar v,\bar v'\gets\arg\max_{v,v'\in\mathcal V(\mathcal R)}\|v-v'\|_2$
\label{alg:phase2:line:edgesplitstart}
\State $v_{\mathrm{mid}}\gets (\bar v+\bar v')/2$
\State $\mathcal S_1\gets{\mathop \mathrm{co}}\{(\mathcal V(\mathcal R)\setminus\{\bar v\})
\cup\{v_{\mathrm{mid}}\}\}$
\label{alg:phase2:line:S1}
\State $\mathcal S_2\gets{\mathop \mathrm{co}}\{(\mathcal V(\mathcal R)\setminus\{\bar v'\})
\cup\{v_{\mathrm{mid}}\}\}$
\label{alg:phase2:line:S2}
\State \textbf{return} $\mathcal S_1$, $\mathcal S_2$
\label{alg:phase2:line:edgesplitend}
\EndFunction
\caption{Computation of $f_\delta^{\epsilon}$.}
\label{alg:phase2}
\end{algorithmic}
\end{algorithm}
Algorithm~\ref{alg:phase2} computes $f_\delta^\epsilon$ as follows.
First, all leaves of the tree output by \cite[Algorithm~2]{Malyuta2019a}\xspace are re-opened for further
partitioning. A ``closed'' leaf refers to a cell that will be a leaf in the
final tree while an ``open'' leaf will be further partitioned at the next
iteration.
We iterate through the tree depth-first on lines~\ref{alg:phase2:line:while} and
\ref{alg:phase2:line:depth_first}.
Lines~\ref{alg:phase2:line:suboptimalitycheckstart}-\ref{alg:phase2:line:end}
carry out the main work of subdividing the cell $(\mathcal R,\delta)$. First,
Theorems~\ref{theorem:absolute_error_check_sufficiency} and
\ref{theorem:relative_error_check_sufficiency} are used to check
$\epsilon$-suboptimality on
lines~\ref{alg:phase2:line:suboptimalitycheckstart}-\ref{alg:phase2:line:suboptimalitycheckend}. If
$\epsilon$-suboptimality is verified, the leaf is closed on
line~\ref{alg:phase2:line:suboptimalitycheckend}. Note that if
\eqref{eq:absolute_error_approx} fails then $\delta$ is the
only feasible commutation in $\mathcal R$ and is certainly optimal.
If, however, $\epsilon$-suboptimality cannot by guaranteed then we update
$(\mathcal R,\delta)$ on
lines~\ref{alg:phase2:line:find_better_commutation}-\ref{alg:phase2:line:end}. This
either splits $\mathcal R$ and/or improves the choice of $\delta$. The following
convex MINLP attempts to find a more optimal commutation:
\begin{equation}
\label{eq:better_delta_selection}
\@ifnextchar[{\@with}{\@without}[
\bar V_{\delta}(\theta)-\epsilon\ge V_{\delta'}^*(\theta), \\
&&& \epsilon \ge \max\{\epsilon_{\mathrm{a}},
\epsilon_{\mathrm{r}}\min_{\mathclap{\substack{\bar\theta\in\mathcal R,\\
\delta''\in\mathbb I^m\setminus\{\delta\}}}}V_{\delta''}^*(\bar\theta)\}, \\
&&& \theta\in\mathcal R, \\
&&& \delta'\in\{\delta''\in\mathbb I^m\setminus\{\delta\}:
\mathcal R\subseteq\Theta_{\delta''}^*\}.
]{\delta^*}{\arg\max}{\theta,\delta',\epsilon}{\epsilon}
\tag{D$_{\delta}^{\mathcal R}$}
\end{equation}
The first two constraints in \eqref{eq:better_delta_selection} search for a
$\delta^*$ with respect to which $\delta$ is ``the most''
suboptimal. Notwithstanding the use of $\bar V_\delta$ to retain convexity, we
shall see in Lemma~\ref{lemma:delta_rejection_forever} that once $\mathcal R$ is
small enough, $\delta^*$ is guaranteed to be better than $\delta$. The third
constraint of \eqref{eq:better_delta_selection} prevents a ``greedy'' choice of
$\delta^*$ by requiring it to be feasible everywhere in the simplex. Thanks to
\cite[Lemma~2]{Malyuta2019a}, this constraint is simple to implement by
requiring \eqref{eq:nlp} to be feasible
$\forall\theta\in\mathcal V(\mathcal R)$.
To avoid any unnecessary partitioning, we check on
line~\ref{alg:phase2:line:check_curvature_constraint} if $V_\delta^*$ varies
over $\mathcal R$ by less than the $\epsilon$-suboptimality threshold. If so,
the proof of Theorem~\ref{theorem:convergence} assures that $\mathcal R$ need
not be subdivided further as far as $\delta$ is concerned and we can simply
change $\delta$ to $\delta^*$.
If \eqref{eq:curvature_constraint} does not hold, however, then to guarantee
convergence $\mathcal R$ must be subdivided. On lines
\ref{alg:phase2:line:edgesplitstart}-\ref{alg:phase2:line:edgesplitend} this is
done by splitting $\mathcal R$ into two simplices at the midpoint of its longest
edge.
If \eqref{eq:better_delta_selection} is infeasible then $\mathcal R$ must be too
big for there to exist a single $\epsilon$-suboptimal $\delta$ within it. In the
formal terms of Section~\ref{subsec:convergence}, $\mathcal R$ is larger than
the optimal cost ``overlap''. Again, the remedy is to split $\mathcal R$ in half
on lines~\ref{alg:phase2:line:split_R_subopt} and \ref{alg:phase2:line:end}.
\section{Problem Formulation}
\label{sec:problem_formulation}
Our algorithm can handle the following multiparametric conic MINLPs, previously
defined in \cite{Malyuta2019a}:
\begin{equation}
\label{eq:minlp}
\@ifnextchar[{\@with}{\@without}[
g(\theta,x,\delta)=0, \\
&&& h(\theta,x,\delta)\in\mathcal K, \\
&&& \delta\in\mathbb I^m,
]{V^*(\theta)}{\min}{x,\delta}{f(\theta,x,\delta)}
\tag{P$_\theta$}
\end{equation}
where $\theta\in\mathbb R^p$ is a parameter, $x\in\mathbb R^n$ is a decision vector
and $\delta\in\mathbb I^m$ is a binary vector called the \textit{commutation}. The
cost function $f:\mathbb R^p\times\mathbb R^n\times\mathbb R^m\to\mathbb R$ is jointly
convex and the constraint functions
$g:\mathbb R^p\times\mathbb R^n\times\mathbb R^m\to\mathbb R^{l}$ and
$h:\mathbb R^p\times\mathbb R^n\times\mathbb R^m\to\mathbb R^d$ are affine in their first
two arguments. The functions can be nonlinear in the last argument. The convex
cone $\mathcal K=\mathcal C_1\times\cdots\times\mathcal C_{q}\subset\mathbb R^d$ is
a Cartesian product of $q$ convex cones. Examples include the positive orthant,
the second-order cone and the positive semidefinite cone. Let us also define a
fixed-commutation multiparametric conic NLP:
\begin{equation}
\label{eq:nlp}
\@ifnextchar[{\@with}{\@without}[
g(\theta,x,\delta)=0, \\
&&& h(\theta,x,\delta)\in\mathcal K,
]{V^*_\delta(\theta)}{\min}{x,\delta}{f(\theta,x,\delta)}
\tag{P$_{\theta}^\delta$}
\end{equation}
which corresponds to (\ref{eq:minlp}) where $\delta$ has been assigned a
specific value. Let $\Theta^*\subseteq\mathbb R^p$ and
$\Theta_\delta^*\subseteq\Theta^*$ denote respectively the parameter sets for
which \eqref{eq:minlp} and \eqref{eq:nlp} are feasible. Define the following
three maps similarly to \cite{Bemporad2006b,Malyuta2019a}.
\begin{definition}
The optimal map $f_\delta^*:\Theta^*\to\mathbb I^m$ associates
$\theta\in\Theta^*$ to any optimal commutation of (\ref{eq:minlp}), that is
any $\delta\in\{\delta\in\mathbb I^m:V^*(\theta)=V^*_\delta(\theta)\}$.
\end{definition}
\begin{definition}
The feasible map $f_\delta:\Theta^*\to\mathbb I^m$ associates
$\theta\in\Theta^*$ to a commutation such that (\ref{eq:nlp}) is feasible.
\end{definition}
\begin{definition}
\label{definition:suboptimal_commutation_function}
The suboptimal map $f_\delta^{\epsilon}:\Theta^*\to\mathbb I^m$ associates
$\theta\in\Theta^*$ to an $\epsilon$-suboptimal commutation $\delta$ such that
\begin{equation}
\label{eq:epsilon_suboptimality}
V^*_\delta(\theta)-V^*(\theta)\le\max\{\epsilon_{\mathrm{a}},
\epsilon_{\mathrm{r}}V^*(\theta)\},
\end{equation}
where $\epsilon_{\mathrm{a}}$ and $\epsilon_{\mathrm{r}}$ are the absolute and
relative errors.
\end{definition}
This paper presents an algorithm for computing $f_{\delta}^\epsilon$ over a
subset $\Theta\subseteq\Theta^*$. It is assumed that $\Theta$ is a convex and
full-dimensional polytope in vertex representation. We refer to
\cite[Section~IV-C]{Malyuta2019a} for how one might choose $\Theta$.
\section{Properties}
\label{sec:properties}
\subsection{Convergence}
\label{subsec:convergence}
This section proves in Theorem~\ref{theorem:convergence} that
Algorithm~\ref{alg:phase2} converges if
Assumption~\ref{assumption:positive_overlap} holds.
Without loss of generality, let us restrict the discussion to feasible
commutations in $\Theta$:
\begin{equation}
\Delta \triangleq \{\delta\in\mathbb I^m : \Theta_\delta^*\cap\Theta\ne\emptyset\}.
\end{equation}
\begin{definition}
\label{definition:overlap}
The \textnormal{overlap} is the largest $\gamma\in\mathbb R_+$ such that
$\forall\theta\in\Theta$, $\exists\delta\in\Delta$ which is
$\epsilon$-suboptimal in
$(\gamma\mathbb B+\theta)\setminus\Theta^{\mathrm{c}}$.
\end{definition}
\begin{assumption}
\label{assumption:positive_overlap}
The overlap is positive, i.e. $\gamma>0$.
\end{assumption}
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{overlap}
\caption{Illustration of ``overlap'' in
Definition~\ref{definition:overlap}. In (a) the overlap is positive thanks
to local continuity and in (b) it is positive because the downward jump from
$V_\delta^*$ to $V_{\delta'}^*$ is not too high. In (c) the jump is too
high, causing zero overlap.}
\label{fig:overlap}
\end{figure}
Previous research \cite[Definition~4]{Malyuta2019a} considered an overlap
$\kappa\in\mathbb R_+$ between
$\Theta_\delta^*$ sets. Because we require \cite[Algorithm~2]{Malyuta2019a}\xspace to converge, we need
$\kappa>0$ \cite[Theorem~1]{Malyuta2019a}. This implies that
$\forall\theta\in\Theta$ $\exists\delta$ feasible, so we know
$\gamma\in\mathbb R_+$ exists.
The overlap considered in this work is between sets where a certain commutation
is $\epsilon$-suboptimal. This is a non-trivial intrinsic property of
\eqref{eq:minlp}.
Assumption~\ref{assumption:positive_overlap} is not readily verifiable, but the
convergence of Algorithm~\ref{alg:phase2} implies that $\gamma>0$ thanks to
Theorem~\ref{theorem:convergence} below.
Figure~\ref{fig:overlap} illustrates just three overlap
possibilities. \shorttxt{We now define a ``variability'' metric for
$V_\delta^*$, which influences convergence speed.} \longtxt{In the simplest
case, local continuity gives positive local overlap as shown in
Figure~\hyperref[fig:overlap]{\getrefnumber{fig:overlap}a} and formalized in
the following proposition.
\begin{proposition}
If $V^*$ is continuous at $\theta\in\Theta\setminus\partial\Theta$ and
$\exists\delta\in\Delta$ optimal at $\theta$ such that $V_\delta^*$ is
continuous at $\theta$, then the overlap at $\theta$ is positive.
\begin{proof}
Consider the following function based on \eqref{eq:epsilon_suboptimality}:
\begin{align*}
f :~ &\Theta'_r\triangleq\{\theta'\in\mathbb R^p:\|\theta'-\theta\|_2<r\} \to \mathbb R, \\
&\theta' \mapsto
V_\delta^*(\theta')-V^*(\theta')-\max\{\epsilon_{\mathrm{a}},
\epsilon_{\mathrm{r}}V^*(\theta')\}.
\end{align*}
Because of local continuity of $V^*$ and $V_\delta^*$, $\exists r>0$ such
that $f$ is continuous. We also have
$f(\theta)\in\mathcal U\triangleq\{z\in\mathbb R : z<0\}$. Since $\mathcal U$ is an open
set, $f^{-1}(\mathcal U)\subseteq\Theta'_r$ is an open set
\cite{Royden1988}. Hence $\exists \gamma>0$ such that
$f(\theta')\subset\mathcal U$ for all
$\theta'\in \gamma\mathbb B+\theta$. Therefore $\delta$ is
$\epsilon$-suboptimal in $\gamma\mathbb B+\theta$.
\end{proof}
\end{proposition}
More complicated cases of positive overlap exist. For example,
Figures~\hyperref[fig:overlap]{\getrefnumber{fig:overlap}b} and
\hyperref[fig:overlap]{\getrefnumber{fig:overlap}c} illustrate that if the
discontinuity in $V^*$ is small enough, a positive overlap may still exist. We
now define a ``variability'' metric for $V_\delta^*$, which influences
convergence speed.}
\begin{definition}
\label{definition:precision_ball}
The \textnormal{variability} for a given $\delta\in\Delta$ is the largest
constant $\sigma_\delta\in\mathbb R_{++}$ such that
\begin{equation}
\label{eq:curvature_constraint}
\max_{\theta\in\Theta_{\sigma}}V_\delta^*(\theta)-
\min_{\theta\in\Theta_{\sigma}}V_\delta^*(\theta)<\max\{\epsilon_{\mathrm{a}},
\epsilon_{\mathrm{r}}\min_{\theta\in\Theta_{\sigma}}V^*(\theta)\},
\end{equation}
where
$\Theta_{\sigma}\triangleq(\sigma_\delta\mathbb B+\theta_o)
\setminus(\Theta_\delta^*)^{\mathrm{c}}$
for any $\theta_o\in\Theta_\delta^*$.
\end{definition}
If $V_\delta^*$ changes little, $\sigma_\delta$ will be relatively large. On the
other hand, a $V_\delta^*$ with high gradients will yield a small
$\sigma_\delta$. It shall be useful to define a \eqref{eq:minlp} ``condition
number'':
\begin{equation}
\label{eq:overall_condition_number}
\psi \triangleq \min\{\gamma,\min_{\delta\in\Delta}(\sigma_\delta)\}.
\end{equation}
In proving convergence, we want to be sure that Algorithm~\ref{alg:phase2} does
not ``oscillate'' forever between $\delta$ choices. This is guaranteed by the
following lemma.
\begin{lemma}
\label{lemma:delta_rejection_forever}
Consider the cell $(\mathcal R,\delta)$. Suppose that
$\mathcal R\subseteq(\psi\mathbb B+\theta_o)$ for some $\theta_o\in\Theta$. If a
more optimal $\delta^*$ is found on
line~\ref{alg:phase2:line:find_better_commutation} of
Algorithm~\ref{alg:phase2} then $\mathcal R$ will never again be associated
with $\delta$, i.e. the cell $(\mathcal R,\delta)$ will not re-appear.
\shorttxt{
\begin{proof}
Show that
$\min_{\theta\in\mathcal R}V_{\delta^*}^*(\theta)<\min_{\theta\in\mathcal R}V_{\delta}^*(\theta)$
and that this is contradicted if $\delta$ is re-associated with
$\mathcal R$. See detailed proof in \cite{Malyuta2019c}.
\end{proof}
}
\longtxt{
\begin{proof}
We begin by showing that
\begin{equation}
\label{eq:lemma4_0}
\min_{\theta\in\mathcal R}V_{\delta^*}^*(\theta)<
\min_{\theta\in\mathcal R}V_{\delta}^*(\theta).
\end{equation}
Due to the first two constraints of \eqref{eq:better_delta_selection},
$\exists\tilde\theta\in\mathcal R$ such that
\begin{equation}
\label{eq:lemma4_1}
\bar V_\delta(\tilde\theta)-
\max\{\epsilon_{\mathrm{a}},\epsilon_{\mathrm{r}}
\min_{\mathclap{\substack{\bar\theta\in\mathcal R,\\
\delta''\in\mathbb I^m\setminus\{\delta\}}}}V_{\delta''}^*(\bar\theta)\}
\ge V_{\delta^*}^*(\tilde\theta).
\end{equation}
Since
$\mathcal R\subset(\sigma_\delta\mathbb B+\theta_o)\setminus(\Theta_\delta^*)^{\mathrm{c}}=\Theta_\sigma$,
we have the following set of inequalities thanks to
\eqref{eq:curvature_constraint}:
\begin{align}
\bar V_\delta(\tilde\theta)
&\le \max_{\theta\in\mathcal R}V_\delta^*(\theta)
\le \max_{\theta\in\Theta_\sigma}V_{\delta}^*(\theta) \nonumber \\
&< \min_{\theta\in\Theta_\sigma}V_\delta^*(\theta)+\max\{\epsilon_{\mathrm{a}},
\epsilon_{\mathrm{r}}\min_{\theta\in\Theta_\sigma}V^*(\theta)\} \nonumber \\
&\le \min_{\theta\in\mathcal R}V_\delta^*(\theta)+\max\{\epsilon_{\mathrm{a}},
\epsilon_{\mathrm{r}}
\min_{\mathclap{\substack{\bar\theta\in\mathcal R,\\
\delta''\in\mathbb I^m\setminus\{\delta\}}}}
V_{\delta''}^*(\theta)\}. \label{eq:lemma4_2}
\end{align}
Using \eqref{eq:lemma4_2} in \eqref{eq:lemma4_1}, we have
\begin{equation*}
\min_{\theta\in\mathcal R}V_\delta^*(\theta)> V_{\delta^*}^*(\tilde\theta),
\end{equation*}
hence \eqref{eq:lemma4_0} must hold. Since
$\mathcal R\in(\psi\mathbb B+\theta_o)$, Algorithm~\ref{alg:phase2} can only
ever finish the iteration on
line~\ref{alg:phase2:line:suboptimalitycheckend} or
\ref{alg:phase2:line:add_R_without_split}, i.e. $\mathcal R$ is not
subdivided. Instead, future iterations may associate it with more optimal
commutations. Consider now some future iteration where the cell is
$(\mathcal R,\delta')$. By recursively applying \eqref{eq:lemma4_0}, it is
clear that:
\begin{equation}
\label{eq:lemma4_5}
\min_{\theta\in\mathcal R} V_{\delta'}^*(\theta)<\cdots
<\min_{\theta\in\mathcal R} V_{\delta^*}^*(\theta)<
\min_{\theta\in\mathcal R}V_\delta^*(\theta).
\end{equation}
By contradiction,
suppose that $\delta$ is chosen on
line~\ref{alg:phase2:line:find_better_commutation}, which means that
$\exists\tilde\theta\in\mathcal R$ such that
\begin{equation}
\label{eq:lemma4_3}
\bar V_{\delta'}(\tilde\theta)-
\max\{\epsilon_{\mathrm{a}},\epsilon_{\mathrm{r}}
\min_{\mathclap{\substack{\bar\theta\in\mathcal R,\\
\delta''\in\mathbb I^m\setminus\{\delta'\}}}}V_{\delta''}^*(\bar\theta)\}
\ge V_{\delta}^*(\tilde\theta).
\end{equation}
In the same way that we obtained \eqref{eq:lemma4_2}, we have:
\begin{equation}
\label{eq:lemma4_4}
\bar V_{\delta'}(\tilde\theta)
< \min_{\theta\in\mathcal R}V_{\delta'}^*(\theta)+\max\{\epsilon_{\mathrm{a}},
\epsilon_{\mathrm{r}}
\min_{\mathclap{\substack{\bar\theta\in\mathcal R,\\
\delta''\in\mathbb I^m\setminus\{\delta'\}}}}
V_{\delta''}^*(\theta)\}.
\end{equation}
Using \eqref{eq:lemma4_4} in \eqref{eq:lemma4_3}, we have
\begin{equation*}
\min_{\theta\in\mathcal R}V_{\delta'}^*(\theta)>
V_{\delta}^*(\tilde\theta)~\Rightarrow~
\min_{\theta\in\mathcal R}V_{\delta'}^*(\theta)>
\min_{\theta\in\mathcal R}V_{\delta}^*(\theta),
\end{equation*}
which contradicts \eqref{eq:lemma4_5}, hence $\delta$ cannot be more optimal
than $\delta'$ and thus cannot be re-associated with $\mathcal R$.
\end{proof}}
\end{lemma}
The next lemma assures that Algorithm~\ref{alg:phase2} decreases the partition
cell size until Lemma~\ref{lemma:delta_rejection_forever} applies. This shall be
instrumental to the convergence proof in Theorem~\ref{theorem:convergence}.
\begin{lemma}
\label{lemma:simplex_volume_decreases}
Let $(\mathcal R_k,\delta_k)$ be the leaf chosen at the $k$-th call of
line~\ref{alg:phase2:line:depth_first} of Algorithm~\ref{alg:phase2}. If
Assumption~\ref{assumption:positive_overlap} holds and the algorithm does not
terminate sooner, there will be a $k$ large enough such that
$\mathcal R_k\subset(\psi\mathbb B+\theta_o)$ for some $\theta_o\in\Theta$.
\shorttxt{
\begin{proof}
If $\mathcal R_k$ is not made smaller, iterations must always finish on
line~\ref{alg:phase2:line:add_R_without_split}. This implies after
$|\Delta|$ iterations that
$\mathcal R_k\subset(\psi\mathbb B+\theta_o)$. See detailed proof in \cite{Malyuta2019c}.
\end{proof}
}
\longtxt{ \begin{proof} An iteration can finish in one of four ways: on
line~\ref{alg:phase2:line:suboptimalitycheckend},
\ref{alg:phase2:line:add_R_without_split},
\ref{alg:phase2:line:split_R_opt2} or \ref{alg:phase2:line:end}. The first
way removes a leaf from consideration, so need not be considered. In the
latter two cases, $\mathcal R_k$ is split along its longest edge. It can
be shown that this halves its volume such that, as long as some iterations
finish on lines~\ref{alg:phase2:line:split_R_opt2} or
\ref{alg:phase2:line:end}, $\exists k$ large enough such that
$\mathcal R_k\subset(\psi\mathbb B+\theta_o)$. Hence, if a large enough $k$
does \textit{not} exist, it must be because iterations persistently finish
on line~\ref{alg:phase2:line:add_R_without_split}. Without loss of
generality, suppose that $\mathcal R_k$ is the last open simplex left. The
sequence of open leaves seen by the algorithm is then
$(\mathcal R_k,\delta_k)$, $(\mathcal R_{k+1},\delta_{k+1})$,
$(\mathcal R_{k+2},\delta_{k+2})$, etc. where
$\mathcal R_j\subset(\sigma_{\delta_j}\mathbb B+\theta_o)$ $\forall j\ge k$
and $\mathcal R_k=\mathcal R_j$ $\forall j\ge k$. However, since
$|\Delta|$ is finite, after a finite number of iterations this implies
that
$\mathcal R_k\subset(\min_{\delta\in\Delta}(\sigma_\delta)\mathbb B+\theta_o)$. Furthermore,
generally $\mathcal R_k\subset(\gamma\mathbb B+\theta_o)$ since otherwise
\eqref{eq:better_delta_selection} could fail and an iteration would finish
on line~\ref{alg:phase2:line:end}. Hence
$\mathcal R_k\subset(\psi\mathbb B+\theta_o)$.
%
%
%
%
%
%
%
\end{proof}}
\end{lemma}
\begin{theorem}
\label{theorem:convergence}
Algorithm~\ref{alg:phase2} converges if and only if
Assumption~\ref{assumption:positive_overlap} holds.
\begin{proof}
Let $(\mathcal R_k,\delta_k)$ be the leaf chosen at the $k$-th call of
line~\ref{alg:phase2:line:depth_first} and suppose that
Assumption~\ref{assumption:positive_overlap} holds. By
Lemma~\ref{lemma:simplex_volume_decreases}, if the algorithm did not already
terminate then there will be a $k$ large enough such that
$\mathcal R_k\subset(\psi\mathbb B+\theta_o)$ for some $\theta_o\in\Theta$. By
Definition~\ref{definition:overlap}, $\exists\delta\in\Delta$ that is
$\epsilon$-suboptimal in $\mathcal R_k$. By
Lemma~\ref{lemma:delta_rejection_forever}, if $\delta_k$ is not
$\epsilon$-suboptimal then it will be replaced with a more optimal
commutation and never again be re-associated with
$\mathcal R_k$. Furthermore, we will have that
\eqref{eq:better_delta_selection} is feasible and that
\eqref{eq:curvature_constraint} holds. Hence, every iteration will exit
through line~\ref{alg:phase2:line:suboptimalitycheckend} or
\ref{alg:phase2:line:add_R_without_split}. Since $|\Delta|$ is finite,
eventually only a single commutation will be left to consider, which must
then be the $\epsilon$-suboptimal one. We refer to this as ``monotonic''
convergence to the $\epsilon$-suboptimal commutation.
If Assumption~\ref{assumption:positive_overlap} does not hold,
$\exists\theta\in\Theta$ such that if $\theta\in\mathcal R_k$ then
$\exists\delta$ $\epsilon$-suboptimal in $\mathcal R_k$ if and only if
$\mathcal R_k=\{\theta\}$. Since this occurs for $k=\infty$, the algorithm
does not converge.
\end{proof}
\end{theorem}
Theorem~\ref{theorem:convergence} suggests that if $\psi$ is large, simplices
need to be partitioned relatively little until their small size assures
monotonic convergence to an $\epsilon$-suboptimal $\delta$. We note that high
cost function gradients decrease $\psi$ while coarser tolerances (i.e. higher
$\epsilon_{\mathrm{a}}$ and $\epsilon_{\mathrm{r}}$) increase $\psi$. While the
former are a property of \eqref{eq:minlp}, the latter are adjustable by the
user.
\subsection{Complexity}
\label{subsec:complexity}
The main result of this section is
Corollary~\ref{corollary:onlinefullcomplexity} which states that the on-line MPC
implementation has polynomial runtime complexity. We assume that
Assumption~\ref{assumption:positive_overlap} holds
\begin{theorem}
\label{theorem:tree_depth_complexity}
The depth $\tau$ of the tree output by Algorithm~\ref{alg:phase2} is
$\mathcal O(p^2\log(\psi^{-1}))$.
\begin{proof}
Consider an initial cell $(\mathcal R,\delta)$ output by \cite[Algorithm~2]{Malyuta2019a}\xspace and let
$\mathcal R_k\subset\mathcal R$ be a child simplex of $\mathcal R$ after $k$
subdivisions. In the worst case, the iteration history of
Algorithm~\ref{alg:phase2} can be viewed in two stages. By
Lemma~\ref{lemma:simplex_volume_decreases}, the first stage reduces the size
of $\mathcal R_k$ until $\mathcal R_k\subset(\psi\mathbb B+\theta_o)$ for some
$\theta_o\in\Theta$. By Lemma~\ref{lemma:delta_rejection_forever}, the
second stage iterates through all possible $\delta\in\Delta$ until the last
remaining commutation value, which must be $\epsilon$-suboptimal. Only the
first stage increases the tree depth because once
$\mathcal R_k\subset(\psi\mathbb B+\theta_o)$, \eqref{eq:better_delta_selection}
is feasible and \eqref{eq:curvature_constraint} holds, hence all subsequent
iterations exit through lines~\ref{alg:phase2:line:suboptimalitycheckend} or
\ref{alg:phase2:line:add_R_without_split}. The tree depth is hence governed
by the condition number $\psi$. It was shown in
\cite[Theorem~2]{Malyuta2019a} that, assuming $l_0$ is the longest edge
length of $\mathcal R$, the approximate required number of subdivisions is
\begin{equation}
\label{eq:tree_depth}
\tau = \left\lceil\frac{p(p+1)\log_2(l_0/\psi)}{2}\right\rceil,
\end{equation}
which yields
$\tau=\mathcal O(p^2\log(\psi^{-1}))$.
\end{proof}
\end{theorem}
\begin{theorem}
\label{theorem:onlineevalcomplexity}
The on-line evaluation complexity of $f_\delta^\epsilon$ via
\eqref{eq:f_subopt_eval} is $\mathcal O(p^3)$.
\begin{proof}
The result was proved in \cite[Theorem~3]{Malyuta2019a}.
\end{proof}
\end{theorem}
\begin{corollary}
\label{corollary:onlinefullcomplexity}
The semi-explicit implementation of \eqref{eq:minlp} has runtime complexity
$\mathcal O(p^3+n^\ell)$.
\begin{proof}
The semi-explicit implementation evaluates $f_\delta^\epsilon$ via
\eqref{eq:f_subopt_eval} and solves one convex program (\ref{eq:nlp}). The
latter operation is polynomial time $\mathcal O(n^\ell)$ \cite{Dueri2017}
which, together with Theorem~\ref{theorem:onlineevalcomplexity}, yields an
overall complexity $\mathcal O(p^3+n^\ell)$.
\end{proof}
\end{corollary}
Corollary~\ref{corollary:onlinefullcomplexity} stands in contrast to
implementing \eqref{eq:minlp} directly with a mixed-integer solver, which has an
exponential runtime $\mathcal O(n^\ell 2^m)$. Note that typically $p\ll m$, so
$p^3\ll 2^m$.
\section{Acknowledgment}
The authors would like to thank Martin Cacan, David S. Bayard, Daniel P. Scharf,
Jack Aldrich and Carl Seubert of the NASA Jet Propulsion Laboratory, California
Institute of Technology, for their helpful insights and discussions.
\bibliographystyle{ieeetr}
|
{
"timestamp": "2019-03-05T02:05:40",
"yymm": "1902",
"arxiv_id": "1902.10994",
"language": "en",
"url": "https://arxiv.org/abs/1902.10994"
}
|
\subsection{Fully Convolutional Block}
LSTM-FCN and ALSTM-FCN comprise of a fully convolutional block and an LSTM/Attention LSTM block. The FCN block has three stacked temporal convolutional blocks with the number of filters defined as 128, 256, and 128. Fig \ref{fig:cnn_ablation} depicts a visual representation of a single sample from the \textit{UMD} dataset after transformation via a random filter selected from each of the convolutional blocks.
As can be noticed, a randomly selected filter from the first CNN block is applying a form of noise reduction that is learned via gradient descent, whereas two subsequent randomly selected filters from the later layers are transforming the data to be far more inconsistent. Based on our analysis of a few filters on various datasets, we conclude that the CNN filters in all layers act as feature extractors and transform the data into separable classes. The model learns the parameters of these transformations on its own via stochastic gradient descent. If a dataset sample requires the removal of noise, it is learned by a few filters of the first CNN layer. It is challenging to postulate what type of transformation is occurring in each filter, as the model transforms the data differently for each of the datasets, on the basis of random initialization of the convolution kernels and order of stochastic gradient descent updates. However, the filter parameters are learned such that their objective is to transform the data into separable classes.
In order to empirically demonstrate that the LSTM-FCN and ALSTM-FCN models are learning to separate the classes better, we examine the features from the FCN block by applying them to a tuned linear SVM classifier. The results are summarized in Table \ref{tab:svm_table}. The linear SVM classifier that is applied on the features extracted from the FCN block is better in 41 datasets (LSTM-FCN model) and 45 datasets (ALSTM-FCN model) as compared to when the tuned linear SVM classifier is applied on to the raw signal. Based on this knowledge, we conclude that the FCN block is transforming the data into separable classes.
\subsection{LSTM/ALSTM Recurrent Block}
Due to the dimensional shuffle that is applied before the LSTM block, the features extracted by LSTM block by itself do not contribute significantly to the overall performance. When these features are applied onto a tuned linear SVM classifier, the classifier is better in only 19 datasets (for the LSTM block) and 4 datasets (for the ALSTM block) as compared to when the tuned linear SVM classifier is applied to the raw input dataset. The above indicates that the LSTM, by itself, is not separating the data into linear separable classes.
\input{svm_table.tex}
\subsection{LSTM/ALSTM Concatenated With FCN Block} \label{concatlayers}
Nevertheless, when the features of the LSTM block/ALSTM block are concatenated with the CNN features, we obtain a more robust set of features that can better separate the classes of the dataset. The above insight is statistically validated by applying the concatenated features to a single layer perceptron classifier which accepts the extracted features as input (due to the fact that the data is transformed into separable classes). The training scheme of all perceptron models is kept consistent with how we train all LSTM-FCN and ALSTM-FCN models, as detailed in Section \ref{Experiments}. Results, shown in Table \ref{tab:mlp_table}, show that the features from of the LSTM/ALSTM block coupled with the features from the FCN block improve the model performance.
\input{mlp_table.tex}
For the ALSTM-FCN model, the ALSTM features joined with the FCN features outperform the features from the ALSTM block or the FCN block on 49 datasets, yielding to a p-value of 1.34\textit{e-08} when a Wilcoxon Signed-rank test \cite{wilcoxon1964some} is applied. Similarly, the LSTM features joined with the FCN features in the model LSTM-FCN outperform the features from the LSTM block or the FCN block on 54 datasets, yielding to a p-value of 1.22\textit{e-08}. The \textit{Dunn-Sidak} \cite{abdi2007bonferroni} corrected significant alpha value is 0.02.
It is evident that when applying the LSTM block (with dimension shuffle) and the FCN block parallelly, the blocks augment each other, and force each other to detect a set of features which when combined, yield an overall better performing model. In other words, the LSTM block attached with the FCN block statistically helps improve the overall performance of the model providing informative features that in conjunction with the FCN features, are useful in separating the classes further.
\subsection{Dimension Shuffle vs No Dimension Shuffle}
Another ablation test performed is to check the impact dimension shuffle has on the overall behavior of the model. The dimension shuffle transposes the input univariate time series of $N$ time steps and $1$ variable into a multivariate time series of $N$ variables and $1$ time step. In other words, when dimension shuffle is applied to the input before the LSTM block, the LSTM block will process only $1$ time step with $N$ variables.
In this ablation test, LSTM-FCN with dimension shuffle is compared to LSTM-FCN without dimension shuffle on all 128 UCR datasets using a cell size of 8, 64, 128 (yielding to a total of $128 \times 3 = 384$ experiments). LSTM-FCN with dimension shuffle outperforms LSTM-FCN without dimension shuffle on 258 experiments, ties in 27 experiments, and performs worse in 99 experiments. For the experiments when LSTM-FCN with dimension shuffle outperforms LSTM-FCN without dimension shuffle, the accuracy improved on average by $6.00 \%$. Conversely, for the experiments when LSTM-FCN with dimension shuffle performs worse than LSTM-FCN without dimension shuffle, the accurcy is worse by an average of $5.26 \%$. A Wilcoxson signed-rank test results in a p-value of $3.69E-17$, indicating a statistical difference in performance where LSTM-FCN with dimension shuffle performs better. This result is contrary to what most people would hypothesize. LSTM-FCN without dimension shuffle overfits the UCR datasets in more instances than LSTM-FCN with dimension shuffle. This is because the LSTM block without dimension shuffle by itself performs extremely well. The FCN block and LSTM block without the dimension shuffle does not benefit each other.
Another critical fact to note is that the LSTM-FCN with dimension shuffle processes the univariate time series in one time step. The gating mechanisms of the LSTM-FCN is only being applied on a single time step. This attributes to why LSTM with dimension shuffle by itself performs poorly. However, as noticed in Section \ref{concatlayers}, when applying the LSTM block with dimension shuffle and the FCN block parallelly, the blocks augment each other, while improving its overall performance. To the best of our knowledge, we believe the LSTM block with a dimension shuffle acts as a regularizer to the FCN block, forcing the FCN block to improve its performance.
\subsection{Replacing LSTM with GRU, RNN, and a Dense Layer}
Since the usage of the LSTM block when applying dimension shuffle to the input is atypical, we replace the LSTM block with a GRU block (8, 64, 128 cells), basic RNN block (8, 64, 128 cells), and a Dense block with a sigmoid activation function (8, 64, 128 units) on all 128 datasets (total of 384 experiments on each model). We chose the sigmoid activation function for the Dense block, instead of the standard Rectifying Linear Unit (ReLU) activation, as we wish to compare the effectiveness of the gating effect exhibited by the 3 gates of the LSTM. The majority of the gates of the LSTM use the sigmoid activation function. Therefore, we construct the Dense block to also use the same. The input to the GRU block, RNN block, and Dense block had a dimension shuffle applied onto it. Replacing the LSTM block of LSTM-FCN with a GRU block was first proposed by \textit{Elsayed et. al} \cite{elsayed2018deep}. Table \ref{tab:substitute} summarizes a Wilcoxson signed-rank test when LSTM-FCN with dimension shuffle is compared to GRU-FCN, RNN-FCN, and Dense-FCN.
\input{model_replace.tex}
The Wilcoxson signed-rank test depicts LSTM-FCN with dimension shuffle to statistically outperform GRU-FCN, RNN-FCN, Dense-FCN. Surprisingly, the model to perform most similar to LSTM-FCN with dimension shuffle is Dense-FCN. LSTM-FCN outperforms Dense-FCN in 231 experiments, ties in 35 experiments and performs worse in 118 experiments.
An interesting observation is that GRU-FCN does not statistically outperform Dense-FCN. Based on our 384 experiments, GRU-FCN outpeforms Dense-FCN in 160 experiments, ties in 49 experiments, while performing worse in 175 experiments. As a disclaimer, we performed each of these experiments only once, therefore there may be some deviation when run multiple times due to the inherent variance of training using random initialization. However, due to the sample size of 384, we believe the variance will not be significant to result in a different conclusion.
\subsection{Temporal Convolutions}
The input to a Temporal Convolutional Network is generally a time series signal. As stated in \textit{Lea et al.}\cite{Lea_2016}, let $X_{t} \in \mathbb{R}^{F_0}$ be the input feature vector of length $F_0$ for time step $t$ for $0 < t \leq T$. Note that the time T may vary for each sequence, and we denote the number of time steps in each layer as $T_l$. The true action label for each frame is given by $y_t \in \{1, . . . , C\}$, where C is the number of classes.
Consider $L$ convolutional layers. We apply a set of 1D filters on each of these layers that capture how the input signals evolve over the course of an action. According to \textit{Lea et al.} \cite{Lea_2016}, the filters for each layer are parameterized by tensor $W^{(l)} \in \mathbb{R}^{F_l \times d \times F_{l-1}} $ and biases $b^{(l)} \in \mathbb{R}^{F_l}$, where $l \in \{1, . . . , L\}$ is the layer index and $d$ is the filter duration. For the $l$-th layer, the $i$-th component of the (unnormalized) activation ${\mathbf {\hat{E}}}^{(l)}_{t} \in \mathbb{R}^{F_{l}}$ is a function of the incoming (normalized) activation matrix $E^{(l-1)} \in \mathbb{R}^{F_{l-1} \times T_{l-1}}$ from the previous layer
\begin{equation}
{\mathbf {\hat{E}}}_{i, t}^{(l)} = f\left(b_{i}^{(l)} + \sum_{t'=1}^{d} \left< W_{i, t', .}^{(l)}, E_{., t+d-t'}^{(l-1)} \right> \right)
\end{equation}
for each time $t$ where $f(\cdot)$ is a Rectified Linear Unit.
We use Temporal Convolutional Networks as a feature extraction module in a Fully Convolutional Network (FCN) branch. A basic convolution block consists of a convolution layer, followed by batch normalization \cite{ioffe2015batch}, followed by an activation function, which can be either a Rectified Linear Unit or a Parametric Rectified Linear Unit \cite{Trottier2016}.
\subsection{Recurrent Neural Networks}
\def{\mathbf x}{{\mathbf x}}
\def{\cal L}{{\cal L}}
Recurrent Neural Networks, often shortened to RNNs, are a class of neural networks which exhibit temporal behaviour due to directed connections between units of an individual layer. As reported by \textit{Pascanu et al.} \cite{pascanu2013construct}, recurrent neural networks maintain a hidden vector $\mathbf h$, which is updated at time step $t$ as follows:
\begin{equation}
\mathbf h_t = \tanh(\mathbf W\mathbf h_{t-1} + \mathbf I\mathbf {\mathbf x}_t),
\end{equation}
tanh is the hyperbolic tangent function, $\mathbf W$ is the recurrent weight matrix and $\mathbf I$ is a projection matrix. The hidden state $\mathbf h$ is used to make a prediction
\begin{equation}
\mathbf y_t = \text{softmax}(\mathbf W\mathbf h_{t-1}),
\end{equation}
softmax provides a normalized probability distribution over the possible classes, $\sigma$ is the logistic sigmoid function and $\mathbf W$ is a weight matrix. By using $\mathbf h$ as the input to another RNN, we can stack RNNs, creating deeper architectures
\begin{equation}
\mathbf h_t^{l} = \sigma(\mathbf W\mathbf h_{t-1}^{l} + \mathbf I\mathbf h_t^{l-1}).
\end{equation}
\subsection{Long Short-Term Memory RNNs}
\def{\mathbf x}{{\mathbf x}}
Long short-term memory recurrent neural networks are an improvement over the general recurrent neural networks, which possess a vanishing gradient problem. As stated in \textit{Hochreiter et al.}\cite{hochreiter1997long}, LSTM RNNs address the vanishing gradient problem commonly found in ordinary recurrent neural networks by incorporating gating functions into their state dynamics. At each time step, an LSTM maintains a hidden vector $\mathbf h$ and a memory vector $\mathbf m$ responsible for controlling state updates and outputs. More concretely, \textit{Graves et al.} \cite{graves2012supervised} define the computation at time step $t$ as follows :
\begin{equation}
\begin{split}
& \mathbf g^u = \sigma(\mathbf W^u\mathbf h_{t-1} + \mathbf I^u{\mathbf x}_t ) \\
& \mathbf g^f = \sigma(\mathbf W^f\mathbf h_{t-1} + \mathbf I^f{\mathbf x}_t) \\
& \mathbf g^o = \sigma(\mathbf W^o\mathbf h_{t-1} + \mathbf I^o{\mathbf x}_t) \\
& \mathbf g^c = \tanh(\mathbf W^c\mathbf h_{t-1} + \mathbf I^c{\mathbf x}_t) \\
& \mathbf m_t = \mathbf g^f \odot \mathbf m_{t-1}\mathbf \ + \ \mathbf g^u \odot
\mathbf g^c \\
& \mathbf h_t = \tanh(\mathbf g^o \odot \mathbf m_t)
\end{split}
\end{equation}
where $\sigma$ is the logistic sigmoid function, $\odot$ represents elementwise multiplication, $\mathbf W^u, \mathbf W^f, \mathbf W^o, \mathbf W^c$ are recurrent weight matrices and $\mathbf I^u, \mathbf I^f, \mathbf I^o, \mathbf I^c$ are projection matrices.
While LSTMs possess the ability to learn temporal dependencies in sequences, they have difficulty with long term dependencies in long sequences. The attention mechanism proposed by \textit{Bahdanau et al.} \cite{bahdanau2014neural} can help the LSTM RNN learn these dependencies.
\subsection{Attention Mechanism}
The attention mechanism is a technique often used in neural translation of text, where a context vector $C$ is conditioned on the target sequence $y$. As discussed in \textit{Bahdanau et al.}\cite{bahdanau2014neural}, the context vector $c_i$ depends on a sequence of \textit{annotations} $(h_1, ..., h_{T_{x}})$ to which an encoder maps the input sequence. Each annotation $h_i$ contains information about the whole input sequence with a strong focus on the parts surrounding the $i$-th word of the input sequence.
The context vector $c_i$ is then computed as a weighted sum of these annotations $h_i$:
\begin{equation}
c_i = \sum_{j=1}^{T_x} \alpha_{ij}h_j.
\end{equation}
The weight $\alpha_{ij}$ of each annotation $h_j$ is computed by :
\begin{equation}
\alpha_{ij} = \frac{exp(e_{ij})}{\sum_{k=1}^{T_x} exp(e_{ik})}
\end{equation}
where $e_{ij} = a(s_{i-1}, h_j)$ is an \textit{alignment model}, which scores how well the input around position $j$ and the output at position $i$ match. The score is based on the RNN hidden state $s_{i−1}$ and the $j$-th annotation $h_j$ of the input sentence.
\begin{figure*}
\center
\fbox{
\includegraphics[width=0.75\linewidth]{"with_attention".pdf}
}
\center
\caption{The LSTM-FCN architecture. LSTM cells can be replaced by Attention LSTM cells to construct the ALSTM-FCN architecture.}
\label{fig:arch}
\end{figure*}
\textit{Bahdanau et al.}\cite{bahdanau2014neural} parametrize the alignment model $a$ as a feedforward neural network which is jointly trained with all the other components of the model. The alignment model directly computes a soft alignment, which allows the gradient of the cost function to be backpropagated.
\section{Introduction}
\input{introduction.tex}
\section{Experiments}
\label{Experiments}
\input{experiments_intro.tex}
\section{Dataset Ablation Test}
\input{results.tex}
\section{Model Ablation Tests}
\input{ablation.tex}
\section{Conclusion \& Future Work}
\label{conclusion}
\input{conclusion.tex}
\section*{Acknowledgment}
The authors would like to thank all the researchers that helped create and clean the data available in the updated UCR Time Series Classification Archive. Sustained research in this domain would be much more challenging without their efforts.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2019-03-05T02:04:29",
"yymm": "1902",
"arxiv_id": "1902.10756",
"language": "en",
"url": "https://arxiv.org/abs/1902.10756"
}
|
\section{introduction}
The nature of AGNs is still an open question in astrophysics.
AGNs contain a broad wavelength band emission,
from radio to very high energy (VHE) band.
Blazars, as a very extreme subclass of AGNs,
show rapid and high variability,
high and variable polarization,
variable and strong $\gamma$-ray emission and even superluminal motion
(\citealt{Fan2013a};
\citealt{Fan2013b}).
These extreme observational properties of blazars are due to the fact that
they host a relativistic jet pointing to the observer
(\citealt{Blandford1979}).
Blazars
have two subclasses,
namely BL Lacertae objects (BL Lacs) and
flat spectrum radio quasars (FSRQs).
BL Lacs show weak or no emission lines while
FSRQs show strong emission line features.
The classifications of blazars based on the spectral energy distributions (SEDs) can be found in
\citet{Padovani1995},
\citet{Nieppola2006},
\citet{Abdo2010a},
\citet{Fan2016},
\citet{Lin2018} and
\citet{ZF2019}.
The $\gamma$-ray emissions of blazars have caught astronomers' attention to
investigate the mechanism of the high energetic $\gamma$-ray emissions.
There have been two generations of $\gamma$-ray experiment,
EGRET (the Energetic Gamma-Ray Experiment Telescope, on-board the $Compton\ Gamma-Ray\ Observatory$) and
{\textit{Fermi-LAT}} ({\textit{Fermi}} Large Area $Gamma-Ray\ Space\ Telescope$),
which provide us good opportunities to detect strong $\gamma$-ray sources.
Based on the observations of EGRET, correlation analyses between the $\gamma$-ray emissions
and those at lower energy bands have been performed to study the beaming effect (
\citealt{Dondi1995};
\citealt{Xie1997};
\citealt{Fan1999};
\citealt{Cheng2000}, and reference therein).
{\textit{Fermi-LAT}}, a successor to ERGET,
detected more than 1000 blazars (see
\citealt{Abdo2010b};
\citealt{Nolan2012};
\citealt{Acero2015};
\citealt{Ackermann2015}).
The strong $\gamma$-ray emissions in blazars suggest the existence of
a relativistic beaming effect,
which is discussed in many papers (see \citealt{Arshakian2010};
\citealt{Fan2013a};
\citealt{Fan2013b};
\citealt{Fan2014};
\citealt{Fan2017};
\citealt{FanJi2014};
\citealt{Giovannini2014};
\citealt{Giroletti2012};
\citealt{Kovalev2009};
\citealt{Massaro2013a};
\citealt{Massaro2013b};
\citealt{Pushkarev2010};
\citealt{Savolainen2010};
\citealt{Xiao2015};
\citealt{Pei2016};
\citealt{Yang2017};
\citealt{Yang2018a};
\citealt{ZF2018}).
The $\gamma$-ray emissions are also used to estimate the beaming boosting factors (Doppler factors) for some $\gamma$-ray loud sources
(\citealt{Mattox1993};
\citealt{Dondi1995};
\citealt{vonMontigny1995};
\citealt{Cheng1999};
\citealt{Fan1999};
\citealt{Fan2013a};
\citealt{Fan2013b};
\citealt{Fan2014};
\citealt{Fan2005}).
\citet{Yang2018b} studied the effective spectral index properties, then suggested that synchrotron self-Compton (SSC) model could explain the main process for highly energetic $\gamma$ rays in BL Lacs.
Moreover, the DArk Matter Particle Explorer (DAMPE),
was successfully launched into a sun-synchronous orbit at the altitude of 500 km
on 2015 December 17th from the Jiuquan launch base.
DAMPE offers a new opportunity for advancing our knowledge of cosmic rays,
dark matter,
and gamma-ray astronomy as well (\citealt{Chang2017}).
This marks a new generation of astrophysics,
which has bound particles physics and astronomy together tightly.
For blazars,
the Doppler factor ($\delta = [\Gamma (1- \beta \rm{cos} \phi)]^{-1}$) is an important parameter,
where $\Gamma = (1 - \beta ^{2})^{1/2}$ is a bulk Lorentz factor,
$\beta$ is the jet speed in units of the speed of light,
and $\phi$ is a viewing angle between the jet and the line-of-sight.
The Doppler factor is a key quantity in jets since it determines how
much flux densities are boosted and timescales compressed in the observer frame.
The Doppler factor,
although a crucial parameter in the blazar paradigm dictating all of the observed properties of blazars,
is very difficult to estimate since there is no direct determining method for either $\beta$ or $\phi$.
For this reason,
many indirect methods have been proposed in order to estimate $\delta$,
which usually involves different energetic
(e.g.,
\citealt{Ghisellini1993};
\citealt{Mattox1993};
\citealt{Fan2013a};
\citealt{Fan2014})
and/or causality arguments
(\citealt{Lahteenimaki1999};
\citealt{Hovatta2009};
\citealt{Jorstad2005};
\citealt{Jorstad2017};
\citealt{Liodakis2018}) or
fitting the spectral energy distribution (SED,
\citealt{Ghisellini2014};
\citealt{Zhang2012};
\citealt{Chen2018}) of $\gamma$-ray emitting blazars.
\citet{Lahteenimaki1999} proposed to estimate the Doppler factor
($\delta_{\rm var}$) using radio flux density variations.
They obtained the timescales for radio emissions,
assumed the timescales to represent the emission size,
and got a brightness temperature ($T_{\rm B}^{\rm ob}$).
If the intrinsic brightness temperature is assumed to be
$T_{\rm B}^{\rm in}=5\times 10^{10} \rm{K}$
and the difference between the two brightness temperatures is from the beaming effect,
then a variation Doppler factor,
$\delta_{\rm var} = (T_{\rm B}^{\rm ob}/T_{\rm B}^{\rm in})^{1/3}$ can be estimated.
This method was used to estimate a large sample with longer coverage of radio observations
(see \citealt{Hovatta2009}).
\citet{Fan2009} and
\citet{Savolainen2010} also adopted that method to estimate the Doppler factor in the radio band.
Furthermore, the Doppler factor was also estimated for the $\gamma$-ray loud blazars,
\citet{Fan2013a} and \citet{Fan2014} suggested a Doppler factor can be expressed as
$$\delta \geq[1.54 \times 10^{-3} (1+z)^{4+2\alpha} (\frac{d_{\rm L}}{ \rm{Mpc}})^{2} (\frac{\Delta T}{\rm{hr}})^{-1} (\frac{F_{\rm{1 KeV}}}{\rm{\mu Jy}}) (\frac{E_{\rm{\gamma}}}{\rm{GeV}})^{\alpha}]^{\frac{1}{4+2\alpha}}$$
here $\Delta T$ is the time scale in units of hour,
$\alpha$ is the X-ray spectral index,
$F_{\rm{1 KeV}}$ is the flux density at 1 KeV in units of $\rm{\mu Jy}$,
$E_{\rm{\gamma}}$ is the energy in units of GeV,
at which the $\gamma$-rays are detected,
and $d_{\rm L}$ is luminosity distance in units of Mpc.
Superluminal motion is also an interesting observational property of blazars.
Thanks to the very large baseline interferometry (VLBI) established by Europe, Canada, United States, Russia and so on,
with high angular resolution at milliarcsecond,
many AGNs show interesting observational results,
some compact radio sources consist of more than one component,
and some of these components seem to be separating at apparent velocities being greater than the speed of light.
A parameter, $\beta_{\rm app}(= v / c)$,
is introduced to value the apparent velocity.
If $\beta_{\rm app} > 1$,
then it is called to be superluminal,
and this kind of sources are called superluminal sources.
The first apparent superluminal motion was observed from 3C 279,
a component moving away from the quasar core at nearly ten times the speed of light was detected.
\citet{VC1994} listed 66 extragalactic sources,
from which they found multi-epoch VLBI internal proper motions,
then investigated several modifications to a simple relativistic beam concept
and its statistical effects on apparent velocity.
They also checked the distribution of $\beta_{\rm app}$ for lobe-selected and
core-selected quasars respectively and obtained the $\beta_{\rm app}$ for different
object categories to be in general agreement with an AGN unification model.
In 1996, \citeauthor{Fan1996} compiled a sample of 48 superluminal motion sources to investigate the beaming effect,
found that the core dominance is an indicator of the orientation of the emission,
and proposed that the superluminal motion and beaming effect are probably the same things.
\citet{Kellermann2003} presented a sample of 96 superluminal sources to
study the nature of the relativistic beaming effect in blazars and
their surrounding environment of the massive black holes.
They found that most of the blazars show an outward flow away from the centre core
while a few sources show the opposite direction of features,
and there is no simple correlation between timescale of flux changes and apparent velocities.
In 2008, \citeauthor{ZF2008} collected an up-to-date sample of 123 superluminal sources
including 84 quasars,
27 BL Lac objects and 12 galaxies,
calculated the apparent velocities for each source,
and found that the radio emissions are strongly boosted by the beaming effect and
the superluminal motion is the same thing as the beaming effect in AGNs.
For the details of kinematics in superluminal motion,
\citet{Britzen2008} presented a detailed kinematic analysis of
the complete flux-density limited Caltech-Jodrell Bank Flat-spectrum
(named CJF \footnote{http://www.mpifr-bonn.mpg.de/staff/sbritzen/cjf.html}) sources.
CJF survey computed 2D kinematic models based on the optimal model-fitting parameters
of multi-epoch VLBA observations,
then investigated possible correlations between the apparent proper motions
and some other parameters in AGN jets.
They found a strong correlation between the 5 GHz luminosity and apparent velocity.
Based on the data of MOJAVE \footnote{http://www.physics.purdue.edu/MOJAVE/}
(Monitoring of Jets in Active galactic nuclei with VLBA Experiments) sample,
\citet{Lister2009} discussed the jet kinematics of a complete flux-density-limited sample of
135 radio-loud AGNs resulting from a 13-years program,
investigated the structure and evolution of parsec-scale jet phenomena,
and found there is an overwhelming tendency to display outward motions,
only eight inward moving components.
\citet{Lister2013} studied 200 AGNs parsec-scale jet orientation variations and superluminal motion,
found a general trend of increasing apparent speed with distance down the jet
for both radio galaxies and BL Lac objects.
Since {\textit{Fermi-LAT}} was launched in 2008, the 4-year catalogue includes 1591 AGNs.
Although AGNs are the main detection result by {\textit{Fermi-LAT}}, while there are many AGNs are not detected
by Fermi. Why are some AGNs detected by {\textit{Fermi-LAT}} and others are not? To answer this question,
we compile a large superluminal sample (189 FDSs and 102 non-FDSs) and
use them to make a comparison between the FDS and the non-FDS sources and do some statistical analyses.
This work is arranged as follows:
In section 1, we introduce our superluminal sample,
in section 2, we will give results,
and discussions and conclusions are presented in sections 3 and 4.
Through this paper,
a $\Lambda$-CDM model with $\Omega_{\rm \Lambda} \simeq 0.7$,
$\Omega_{\rm M} \simeq 0.3$ and $\Omega_{\rm K} \simeq 0.0$,
and $H_{\rm 0} = 73 ~{\rm{km \, s^{-1}}} {\rm{Mpc^{-1}}}$ is adopted.
\section{Samples and Results}
\subsection{Samples}
From the available literature,
we compile 291 sources with superluminal motions,
including 189 ( 142 FSRQs, 39 BL Lacs, 5 galaxies and 2 uncertain type blazar candidates (BCU) and 1 unknown type of AGN without a known redshift )
{\textit{Fermi}} detected superluminal sources (FDS) and
102 ( 98 FSRQs, 1 BL Lac and 12 galaxies and 1 unknown type of AGN without a known redshift )
{\textit{non-Fermi}} detected superluminal (non-FDS) sources,
where {\textit{Fermi}} detected sources mean these sources are detected by {\textit{Fermi-LAT}} telescope and listed in the {\textit{Fermi}} AGN catalogues.
There are 816 components for the 189 FDS sources in total,
30 of them have just one component.
In the present sample,
we also include the $\gamma$-ray emission source,
0007+106 (III ZW 2),
which was classified as an $\gamma$-ray source by \citet{Liao2016}.
All the FDS sources are listed in Table \ref{Tab-FDS-simp}.
\begin{deluxetable}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}
\tabletypesize{\scriptsize}
\rotate
\tablecaption{Superluminal Sources detected by {\textit{Fermi-LAT}}}
\tablewidth{0pt}
\tablehead{
\colhead{ FGL name }&
\colhead{ Class } &
\colhead{ redshift } &
\colhead{ $\delta_R$ } &
\colhead{ Ref } &
\colhead{ $m_o$ } &
\colhead{ Ext }&
\colhead{ S$_R$ }&
\colhead{ S$_X$ }&
\colhead{ $\Gamma_{\gamma}$ }&
\colhead{ F$\gamma$ }&
\colhead{ $\mu$ }&
\colhead{ comp }&
\colhead{ Ref } &
\colhead{ $\beta$ }\\
\colhead{ Other name }&
\colhead{ }&
\colhead{ }&
\colhead{ }&
\colhead{ }&
\colhead{ $magnitude $ }&
\colhead{ }&
\colhead{ $mJy$ }&
\colhead{ $1 \times 10^{-12} ~cgs$ }&
\colhead{ }&
\colhead{ $ph/cm^{2}/s$ }&
\colhead{ $\mu as/yr$ }&
\colhead{ }&
\colhead{ }&
\colhead{ }\\
\colhead{(1) }&
\colhead{(2) }&
\colhead{(3) }&
\colhead{(4) }&
\colhead{(5) }&
\colhead{(6) }&
\colhead{(7) }&
\colhead{(8) }&
\colhead{(9) }&
\colhead{(10) }&
\colhead{(11) }&
\colhead{(12) }&
\colhead{(13) }&
\colhead{(14) }&
\colhead{(15) }
}
\startdata
0007+106 & G & 0.089 & 2.51 & L18 & 15.8 & 0.227 & 98 & 6.14 & & & 204 $\pm$ 12 & 1 & MOJAVE & 1.197$\pm$0.069 \\
III ZW 2 & & & & & & & & & & & 269 $\pm$ 50 & 4 & MOJAVE & 1.58$\pm$0.29 \\ \hline
1FGL J1159.4-2149 & F & 0.927 & & & 17.8 & 0.104 & 386 & 0 & & & 9.3 $\pm$ 3.8 & 1 & MOJAVE & 0.46$\pm$0.19 \\
1157-215 & & & & & & & & & & & 83 $\pm$ 22 & 2 & MOJAVE & 4.1$\pm$1.1 \\
& & & & & & & & & & & 26.2 $\pm$ 8.0 & 3 & MOJAVE & 1.30$\pm$0.40 \\ \hline
1FGL J1245.8-0632 & F & 1.286 & & & 19.6 & 0.068 & 551 & 0 & & & 358 $\pm$ 53 & 1 & MOJAVE & 22.5$\pm$3.4 \\
1243-072 & & & & & & & & & & & 17.6 $\pm$ 8.9e & 3 & MOJAVE & 1.11$\pm$0.56 \\
& & & & & & & & & & & 40 $\pm$ 16 & 4 & MOJAVE & 2.51$\pm$1.00 \\ \hline
2FGLJ2148.2+0659 & F & 0.999 & 15.6 & H09 & 15.9 & 0.186 & 2590 & 1.46 & 2.77 & 3.90E-10 & 59.0 $\pm$ 1.8 & 2a & MOJAVE & 3.092$\pm$0.096 \\
2145+067 & & & & & & & & & & & 50.1 $\pm$ 9.1 & 3a & MOJAVE & 2.63$\pm$0.48 \\
& & & & & & & & & & & 49.2 $\pm$ 3.3 & 5 & MOJAVE & 2.58$\pm$0.17 \\
& & & & & & & & & & & 59.3 $\pm$ 3.8 & 7 & MOJAVE & 3.11$\pm$0.20 \\
& & & & & & & & & & & 27.6 $\pm$ 3.8 & 8 & MOJAVE & 1.45$\pm$0.20 \\ \hline
3FGL J0006.4+3825 & F & 0.229 & & & 17.6 & 0.205 & 572 & 0.75 & 2.617 & 6.06E-10 & 9$\pm$38 & C1 & CJF & 0.12$\pm$0.52 \\
0003+380 & & & & & & & & & & & 135$\pm$37 & C2 & CJF & 1.85$\pm$0.51 \\
& & & & & & & & & & & 145 & C3 & CJF & 1.99$\pm$0 \\
& & & & & & & & & & & 336 & C4 & CJF & 4.6$\pm$0 \\ \hline
\enddata
\label{Tab-FDS-simp}
\tablecomments{Only five objects were shown here and the whole Table will be given in the electronic version.
column (1) gives the {\textit{Fermi}} name (other name),
column (2) classification, F stands for FSRQs, B for BL Lacs, Sy for Seyfert galaxies, Un for unknown type AGNs,
column (3) redshift,
column (4) Doppler factor,
column (5) reference for Doppler factor,
column (6) apparent magnitude ($R_{\rm{mag}}$) from BZCAT \footnote{http://www.asdc.asi.it/bzcat/} (Massaro et al. 2009),
column (7) Galactic extinction ($A_{\rm R}$) from NED,
column (8) flux density at 1.4 GHz from BZCAT,
column (9) X-ray flux in the 0.1-2.4 KeV band from BZCAT,
column (10) $\gamma$-ray photon spectral index,
column (11) $\gamma$-ray photon flux arrange 1-100 GeV (Acero et al. 2015),
column (12) proper motion $\mu$ in microarcsecond per year,
column (13) components for proper motion,
column (14) reference for proper motion,
column (15) apparent velocity, $\beta_{\rm app}$.
}
\end{deluxetable}
For the 102 non-FDS sources ( 88 FSRQs, 1 BL Lac, 12 galaxies and 1 unknown type of AGN) with 400 components totally, 17 of them have just one component, they are in Table \ref{Tab-non-FDS-simp},
\begin{deluxetable}{c|c|c|c|c|c|c|c|c|c|c|c|c}
\tabletypesize{\scriptsize}
\rotate
\tablecaption{Superluminal Sources not detected by {\textit{Fermi-LAT}}}
\tablewidth{0pt}
\tablehead{
\colhead{ FGL name }&
\colhead{ Class } &
\colhead{ redshift } &
\colhead{ $\delta_R$ } &
\colhead{ Ref } &
\colhead{ $m_o$ } &
\colhead{ Ext }&
\colhead{ S$_R$ }&
\colhead{ S$_X$ }&
\colhead{ $\mu$ }&
\colhead{ comp }&
\colhead{ Ref } &
\colhead{ $\beta$ }\\
\colhead{ Other name }&
\colhead{ }&
\colhead{ }&
\colhead{ }&
\colhead{ }&
\colhead{ $magnitude$ }&
\colhead{ }&
\colhead{ $mJy$ }&
\colhead{ $1 \times 10^{-12} ~cgs$ }&
\colhead{ $\mu as/yr$ }&
\colhead{ }&
\colhead{ }&
\colhead{ }\\
\colhead{(1) }&
\colhead{(2) }&
\colhead{(3) }&
\colhead{(4) }&
\colhead{(5) }&
\colhead{(6) }&
\colhead{(7) }&
\colhead{(8) }&
\colhead{(9) }&
\colhead{(10) }&
\colhead{(11) }&
\colhead{(12) }&
\colhead{(13) }
}
\startdata
0003-066 & B & 0.347 & 5.1 & H09 & 17.9 & 0.087 & 2051 & 0.82 & 191 $\pm$ 15 & 2 & MOJAVE & 4.09$\pm$0.33 \\
& & & & & & & & & 250 $\pm$ 39 & 3 & MOJAVE & 5.36$\pm$0.83 \\
& & & & & & & & & 50.4 $\pm$ 5.3 & 4a & MOJAVE & 1.08$\pm$0.11 \\
& & & & & & & & & 100 $\pm$ 16 & 5 & MOJAVE & 2.15$\pm$0.35 \\
& & & & & & & & & 54 $\pm$ 11 & 6a & MOJAVE & 1.16$\pm$0.24 \\
& & & & & & & & & 330.4 $\pm$ 9.8 & 8a & MOJAVE & 7.08$\pm$0.21 \\
& & & & & & & & & 287 $\pm$ 25 & 9 & MOJAVE & 6.14$\pm$0.53 \\
& & & & & & & & & 116 $\pm$ 23 & 14 & MOJAVE & 2.48$\pm$0.50 \\ \hline
0010+405 & F & 0.255 & & & & & & & 428 $\pm$ 40 & 1 & MOJAVE & 6.92$\pm$0.64 \\
& & & & & & & & & 2 $\pm$ 16e & 2 & MOJAVE & 0.04$\pm$0.26 \\
& & & & & & & & & 2.9 $\pm$ 4.3e & 3 & MOJAVE & 0.047$\pm$0.070 \\
& & & & & & & & & 1.1 $\pm$ 3.1e & 4 & MOJAVE & 0.018$\pm$0.050 \\ \hline
0014+813 & F & 3.366 & & & 15.9 & 0.425 & 693 & 0.77 & 4$\pm$12 & C1 & B08 & 0.39$\pm$1.18 \\
& & & & & & & & & 86$\pm$15 & C2 & B08 & 8.49$\pm$1.48 \\
& & & & & & & & & 111$\pm$18 & C3 & B08 & 10.95$\pm$1.78 \\ \hline
0016+731 & F & 1.781 & 7.9 & H09 & 18.2 & 0.735 & 1136 & 0.11 & 106.2 $\pm$ 4.4 & 1a & MOJAVE & 8.23$\pm$0.34 \\ \hline
0022+390 & F & 1.946 & & & & & & & 113$\pm$43 & C1 & B08 & 8.51$\pm$3.24 \\
& & & & & & & & & 71$\pm$57 & C2 & B08 & 5.35$\pm$4.29 \\ \hline
\enddata
\label{Tab-non-FDS-simp}
\tablecomments{Only five objects were shown here and the whole Table will be given in the electronic version. column (1) name,
column (2) classification, F stand for FSRQs, B for BL Lacs, Sy for Seyfert galaxies, G for galaxies,
column (3) redshift,
column (4) Doppler factor,
column (5) reference for Doppler factor,
column (6) apparent magnitude ($R_{\rm mag}$) from BZCAT,
column (7) Galactic extinction ($A_{\rm R}$) from NED,
column (8) flux at 1.4 GHz from BZCAT,
column (9) X-ray flux in the 0.1-2.4 KeV band from BZCAT,
column (10) proper motion $\mu$ microarcsecond per year,
column (11) components for proper motion,
column (12) reference for proper motion,
column (13) apparent velocity, $\beta_{\rm app}$.
}
\end{deluxetable}
\subsection{Results}
\subsubsection{Proper Motion and Apparent Velocity Distribution}
For a proper motion ($\mu$), an apparent velocity ($\beta_{\rm app}$) can be computed by,
\begin{center}
\begin{equation}
\beta_{\rm app} = {\frac {\mu} {H_{\rm 0}} {\int_1^{1+z} {\frac{1} {\sqrt{ {\Omega_{\rm M}} x^3+1-{\Omega_{\rm M}} } } }dx}}.
\end{equation}
\end{center}
Hence, we calculate apparent velocities from its given proper motions if its apparent velocities are not given by MOJAVE
(\citealt{Lister2013}; \citealt{Lister2016})
or other reference literature for the sources in Tables \ref{Tab-FDS-simp} and \ref{Tab-non-FDS-simp}. Then we compare their maximum proper motion ($\mu^{\rm max}$) and maximum apparent velocity ($\beta^{\rm max}_{\rm app}$), averaged proper motion ($\mu^{\rm mean}$) and averaged apparent velocity ($\beta^{\rm mean}_{\rm app}$) between FDS and non-FDS sources. The corresponding uncertainty of averaged value is expressed by an error-transmission format:
$$\sigma = \sqrt{ \sum_{i=1}^{n} (\frac{\partial f}{\partial x_{i} } )^{2} \sigma_{x_{i}}^{2}}$$
The distributions of the maximum and averaged proper motion and the apparent velocity for FDS and non-FDS sources are shown in Figures \ref{mu-dis}-\ref{beta-dis}, where the red stepped line is for FDS, and the blue stepped line for non-FDS sources.
\begin{figure}
\centering
\includegraphics[width=6in]{mu-dis.eps}
\caption{The distribution of maximum (left panel) / mean (right panel) proper motion ($\mu^{\rm max}$ / $\mu^{\rm mean}$ (mas/yr)). The upper panel figures are their histograms of FDS and non-FDS sources, the lower panel figures are their cumulative probability distributions (CPD).}
\label{mu-dis}
\end{figure}
It is found that the maximum proper motion is distributed from 0.018 to 2.510 with a mean value of $\langle \mu^{\rm max}_{\rm FDS} \rangle = 0.361 \pm 0.037 \ {\rm mas} \cdot {\rm yr^{-1}}$ for the FDS sources, and from 0.021 to 2.941 with $\langle \mu^{\rm max}_{\rm non-FDS} \rangle = 0.224 \pm 0.027 \ {\rm mas} \cdot {\rm yr^{-1}}$ for the non-FDS sources. When a Kolmogorov-Smirnov (K-S) test is adopted to the two distributions, the probability for the two distributions to come from the same distribution is $p=6.2 \times 10^{-5}$, see Figure \ref{mu-dis}.
The mean proper motion is distributed from 0.011 to 1.853 with $\langle \mu^{\rm mean}_{\rm FDS} \rangle = 0.233 \pm 0.021 \ {\rm mas} \cdot {\rm yr^{-1}}$ for the FDS sources and from 0.015 to 2.193 with $\langle \mu^{\rm mean}_{\rm non-FDS} \rangle = 0.158 \pm 0.015 \ {\rm mas} \cdot {\rm yr^{-1}}$ for the non-FDS sources, and $p=1.6 \times 10^{-4}$, see Figure \ref{mu-dis}.
\begin{figure}
\centering
\includegraphics[width=6in]{beta-dis.eps}
\caption{The distribution of maximum (left panel) / mean (right panel) apparent velocity ($\beta_{\rm app}^{\rm max}$ / ($\beta_{\rm app}^{\rm mean}$). The upper panel figures are their histograms of FDS and non-FDS sources, the lower panel figures are their cumulative probability distributions (CPD).}
\label{beta-dis}
\end{figure}
From the calculations, it is found that the maximum apparent velocity $\beta^{\rm max}_{\rm app}$ is distributed from 1.04 to 39.70 with $\langle \beta_{\rm FDS}^{\rm max} \rangle = 12.36 \pm 1.64$ for the FDS sources and from 1.08 to 58.04 with $\langle \beta_{\rm non-FDS}^{\rm max} \rangle = 8.75 \pm 1.32$ for non-FDS sources, and $p=3.3 \times10^{-7}$, see Figure \ref{beta-dis}.
For the mean apparent velocity, $\beta^{\rm mean}_{\rm app}$ is distributed from 0.53 to 34.80 with $\langle \beta_{\rm FDS}^{\rm mean} \rangle = 8.17 \pm 0.94$ for the FDS sources and from 0.61 to 29.33 with $\langle \beta_{\rm non-FDS}^{\rm mean} \rangle = 5.99 \pm 0.78$ for non-FDSs,
and $p = 1.4 \times 10^{-4}$, see Figure \ref{beta-dis}.
\subsubsection{Correlations between Proper Motion and Redshift}
From the data listed in Tables \ref{Tab-FDS-simp} and \ref{Tab-non-FDS-simp}, when a linear regression fitting is adopted to the proper motion and redshift, we have
$${\rm log} \mu^{\rm max}_{\rm FDS} = -(0.35 \pm 0.10) {\rm log} z - (0.95 \pm 0.03)$$
with a correlation coefficient $r = -0.26$ and a chance probability of $p = 4.0 \times 10^{-4}$ for the FDSs, and
$${\rm log} \mu^{\rm max}_{\rm non-FDS} = -(0.14 \pm 0.11) {\rm log} z - (0.96 \pm 0.04)$$
with $r = -0.13$ and $p = 20 \%$ for the non-FDSs, see Figure \ref{mu-z}.
\begin{figure}
\centering
\includegraphics[width=5in]{mu-z.eps}
\caption{The plot of the maximum proper motion (log$\mu^{\rm max}$) against redshift (log$z$). The black plus stands for FDS sources and blue circle stands for non-FDS sources, black solid line and blue solid line stand for the best fitting results for FDS sources and non-FDS.}
\label{mu-z}
\end{figure}
\subsubsection{Correlations between Apparent Velocity and Redshift}
For the apparent velocity ($\beta_{\rm app}$), we have following results for the maximum and average values of apparent velocity ($\beta_{\rm app}$),
$${\rm log} \beta^{\rm max}_{\rm FDS} = (0.36 \pm 0.09) {\rm log} z +(0.76 \pm 0.03), ~r= 0.28~{\rm and}~p= 1.4 \times 10^{-4} , ~{\rm and}$$
$${\rm log} \beta^{\rm mean}_{\rm FDS} = (0.30 \pm 0.09) {\rm log} z +(0.55 \pm 0.03), ~r= 0.24~{\rm and}~p= 1.0 \times 10^{-3}, $$
for the 186 FDSs (3 excluded sources without redshift from NED ), and \\
$${\rm log} \beta^{\rm max}_{\rm non-FDS} = (0.59 \pm 0.10) {\rm log} z +(0.73 \pm 0.03), ~r= 0.50~{\rm and}~p= 1.6 \times 10^{-7}, ~{\rm and}$$
$${\rm log} \beta^{\rm mean}_{\rm non-FDS} = (0.54 \pm 0.11) {\rm log} z +(0.55 \pm 0.03), ~r= 0.46~{\rm and}~p= 2.8 \times 10^{-6}, $$
for the 101 non-FDS as shown in Figure \ref{beta-z}.
\begin{figure}
\centering
\includegraphics[width=6in]{beta-z.eps}
\caption{The plot of apparent velocity (log$\beta_{\rm app}$) against redshift (log$z$) for FDS and non-FDS sources. The dot stands for FSRQs, triangle for BL Lacs and square for Galaxies including Seyfert galaxies and normal galaxies, (a): maximum apparent velocity against redshift of FDS, (b): mean apparent velocity against redshift of FDS, and (c): maximum apparent velocity against redshift of non-FDS, (d): mean apparent velocity against redshift of non-FDS. Black solid lines in this figure represent its corresponding best linear fitting results.}
\label{beta-z}
\end{figure}
\subsubsection{Correlation between Apparent Velocity and $\gamma$-Ray Luminosity}
For the $\gamma$-ray sources,
the integral flux ($f$) in units of $\rm GeV \cdot cm^{-2}\cdot s^{-1}$,
can be expressed in the form (\citealt{Fan2013b})
$$f={N_{(E_{\rm L}\sim E_{\rm U})}({\frac{1}{E_{\rm L}}-\frac{1}{E_{\rm U}}}){\rm ln}{\frac{E_{\rm U}}{E_{\rm L}}}, ~{\rm if}~ \alpha_{\rm ph} = 2,~{\rm otherwise}}$$
\begin{equation}
{f={N_{(E_{\rm L}\sim E_{\rm U})}\frac{1-\alpha_{\rm ph}}{2-\alpha_{\rm ph}}\frac{(E_{{\rm U}}^{2-\alpha_{\rm ph}}-E_{{\rm L}}^{2-\alpha_{\rm ph}})}{(E_{\rm U}^{1-\alpha_{\rm ph}}-E_{\rm L}^{1-\alpha_{\rm ph}})}}}
\end{equation}
here ${N_{(E_{\rm L}\sim E_{\rm U})}}$ is the integral photons in the energy range of $E_{\rm L}$ and $E_{\rm U}$.
In this work, $E_{\rm L}$ and $E_{\rm U}$ are corresponding to 1 GeV and 100 GeV respectively.
Then, we calculate the $\gamma$-ray luminosity ($L_{\gamma}$) in units of $\rm erg\cdot s^{-1}$ by
\begin{equation}
{L_{\rm \gamma}=4\pi d^{2}_{\rm L}(1+z)^{\alpha_{\rm ph}-2}f}
\end{equation}
here, $L_{\rm \gamma}$ is the $\gamma$-ray luminosity, $d_{\rm L} = \frac{c(1+z)}{H_{\rm 0}}\int^{1+z}_{1}\frac{1}{\sqrt{\Omega_{\rm M}x^{3}+1-\Omega_{\rm M}}} dx$ is a luminosity distance, $(1+z)^{(\alpha_{\rm{ph}}-2)}$ stands for a K-correction, $\alpha_{\rm{ph}}$ for $\gamma$-ray photon spectral index.
Figure \ref{beta-L} shows the maximum and average apparent velocity against the $\gamma$-ray luminosity. The dash curved upper envelope is described by \citet{Cohen2007}, the upper envelope of this distribution traces out a single source of a given bulk Lorentz factor and intrinsic luminosity in the ($L$, $\beta_{\rm app}$) plane as the viewing angle $\phi$ changes. Such an aspect curve is plotted in Figure \ref{beta-L} for a jet with a bulk Lorentz factor of 42 and an intrinsic luminosity of log$L_{\rm in}$= 42, assuming Doppler boosting by a factor of $\delta^{3}$, by these formulas:
$\delta = \frac{1}{\Gamma(1-\beta {\rm cos} \phi)},
\beta_{\rm app} = \frac{\beta {\rm sin} \phi}{1-\beta {\rm cos} \phi}, L=L_{\rm in}\delta^{3}.$
\begin{figure}
\centering
\includegraphics[width=5in]{beta-L.eps}
\caption{The plot of apparent velocity ($\beta_{\rm app}$) against the logarithmic $\gamma$-ray luminosity (log$L_{\rm \gamma}$) for FDS sources. The dot stands for FSRQs, triangle for BL Lacs and square for galaxies. The dashed curve represents an envelope with fixed $\Gamma=42$ and ${\rm log} L_{\rm in}=42$.}
\label{beta-L}
\end{figure}
\subsubsection{Flux Density-Flux Density Correlations}
The multi-wavelength (radio, optical and X-ray) data are from the BZCAT,
the optical magnitude is made galactic extinction correction and then transferred into optical flux density.
The optical,
radio and X-ray flux densities are also K-corrected by $(1+z)^{\alpha-1}$,
where $\alpha$ ($F_{\nu} \propto \nu^{-\alpha}$) is the spectral index in the given band.
For the spectral indexes,
we adopt $\alpha_{\rm r}=0$ for radio band (\citealt{Donato2001}, \citealt{Abdo2010a}),
while for optical band,
$\alpha_{\rm o}=0.5$ for BLs and $\alpha_{\rm o}=1$ for the rest of the sources as did by
\citet{Donato2001},
$\alpha_{\rm X}=0.78$ for FSRQs,
$\alpha_{\rm X}=1.30$ for BLs and
$\alpha_{\rm X}=1.05$ for BCUs from \citet{Fan2016}.
For any two bands, we have \\
$${\rm log}S_{\rm r} = (0.28 \pm 0.04) {\rm log}F_{\rm o}+ (6.15 \pm 0.49), ~r=0.51~{\rm and}~p=5.9 \times 10^{-14},$$
$${\rm log}F_{\rm x} = (0.46 \pm 0.04) {\rm log}F_{\rm o}+ (5.46 \pm 0.51), ~r=0.64~{\rm and}~p=6.8 \times 10^{-22}, ~{\rm and}~$$
$${\rm log}F_{\rm x} = (0.53 \pm 0.08) {\rm log}S_{\rm r}- (1.65 \pm 0.22), ~r=0.46~{\rm and}~p=1.0 \times 10^{-10}$$
for FDSs, and \\
$${\rm log}S_{\rm r} = (0.28 \pm 0.07) {\rm log}F_{\rm o}+ (6.21 \pm 0.82), ~r=0.38~{\rm and}~p=4.0 \times 10^{-4},$$
$${\rm log}F_{\rm x} = (0.40 \pm 0.06) {\rm log}F_{\rm o}+ (4.63 \pm 0.78), ~r=0.47~{\rm and}~p=1.8 \times 10^{-5}, ~{\rm and}~$$
$${\rm log}F_{\rm x} = (0.51 \pm 0.11) {\rm log}S_{\rm r}- (1.70 \pm 0.30), ~r=0.50~{\rm and}~p=3.7 \times 10^{-6}$$
for non-FDSs. These corresponding results are shown in Figure \ref{F-F}.
For the subclasses of BL Lacs and FSRQs,
their correlations are shown in Table \ref{Tab-F-F} for FDSs and non-FDSs.
\begin{table*}
\centering
\caption{Flux density-Flux density correlation analysis results}
\label{Tab-F-F}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
~Type~& ~band~ & ~ Sample ~ & $~ a+ \Delta a ~$ & $~ b+\Delta b ~$ & $~ N ~$ & $~ r ~$ & $~ p ~$ \\ \hline
& & whole & $0.28 \pm 0.04$ & $6.15 \pm 0.49$ & 186 & 0.51 & $5.9 \times 10^{-14}$ \\ \cline{3-8}
& ${\rm log}S_{\rm r} ~vs ~ {\rm log}F_{\rm o}$ & BL Lac & $0.19 \pm 0.06$ & $5.08 \pm 0.70$ & 36 & 0.49 & $2.5 \times 10^{-3}$ \\ \cline{3-8}
& & FSRQ & $0.34 \pm 0.05$ & $7.00 \pm 0.64$ & 144 & 0.49 & $3.3 \times 10^{-10}$ \\ \cline{2-8}
& & whole & $0.46 \pm 0.04$ & $5.46 \pm 0.51$ & 180 & 0.64 & $6.8 \times 10^{-22}$ \\ \cline{3-8}
FDS & ${\rm log}F_{\rm X} ~vs~ {\rm log}F_{\rm o}$ & BL Lac & $0.45 \pm 0.10$ & $5.51 \pm 1.19$ & 36 & 0.61 & $6.7 \times 10^{-5}$ \\ \cline{3-8}
& & FSRQ & $0.41 \pm 0.05$ & $4.87 \pm 0.60$ & 139 & 0.59 & $1.4 \times 10^{-14}$ \\ \cline{2-8}
& & whole & $0.53 \pm 0.08$ & $-1.65 \pm 0.22$ & 179 & 0.46 & $1.0 \times 10^{-10}$ \\ \cline{3-8}
& ${\rm log}F_{\rm X} ~vs~ {\rm log}S_{\rm r}$ & BL Lac & $0.58 \pm 0.34$ & $-1.51 \pm 0.95$ & 35 & 0.29 & $9.4 \%$ \\ \cline{3-8}
& & FSRQ & $0.52 \pm 0.07$ & $-1.72 \pm 0.20$ & 138 & 0.52 & $3.9 \times 10^{-11}$ \\ \hline
& ${\rm log}S_{\rm r} ~vs~ {\rm log}F_{\rm o}$ & whole & $0.28 \pm 0.07$ & $6.21 \pm 0.82$ & 82 & 0.38 & $4.0 \times 10^{-4}$ \\ \cline{3-8}
& & FSRQ & $0.28 \pm 0.07$ & $6.22 \pm 0.86$ & 77 & 0.43 & $1.0 \times 10^{-4}$ \\ \cline{2-8}
non-FDS & ${\rm log}F_{\rm X} ~vs~ {\rm log}F_{\rm o}$ & whole & $0.40 \pm 0.06$ & $4.63 \pm 0.78$ & 76 & 0.47 & $1.8 \times 10^{-5}$ \\ \cline{3-8}
& & FSRQ & $0.36 \pm 0.07$ & $4.22 \pm 0.85$ & 71 & 0.54 & $1.2 \times 10^{-6}$ \\ \cline{2-8}
& ${\rm log}F_{\rm X} ~vs~ {\rm log}S_{\rm r}$ & whole & $0.51 \pm 0.11$ & $-1.70 \pm 0.30$ & 76 & 0.50 & $3.7 \times 10^{-6}$ \\ \cline{3-8}
& & FSRQ & $0.51 \pm 0.11$ & $-1.68 \pm 0.30$ & 71 & 0.49 & $1.5 \times 10^{-5}$ \\ \hline
\end{tabular}
\end{table*}
\begin{figure*}
\centering
\includegraphics[width=6in]{F-F.eps}
\caption{Plots for optical magnitude (log$F_{\rm o}$), radio flux density (log$S_{\rm r}$), X-ray flux (log$F_{\rm x}$), Left-hand panel for FDS sources, right-hand panel for non-FDS sources. Upper panel for radio against optical, middle panel for X-ray against optical and lower panel for X-ray against radio. Dot stands for FSRQs, triangle for BL Lacs and square for galaxies. The solid lines stand for best linear fitting results.}
\label{F-F}
\end{figure*}
\section{Discussions}
As a subclass of AGNs,
blazar has many extreme observation properties,
which may be attributed to a relativistic beaming effect.
We compile proper motions, $\mu$ for 1216 components for 291 sources,
including 189 FDS and 102 non-FDS sources.
Then we collect and calculate the corresponding apparent velocity ($\beta_{\rm app}$),
get their multi-wavelength data,
and Doppler factors from available references (BZACT, MOJAVA,
{\textit{Fermi}}-3rd catalogue, and so on),
make comparisons between FDSs and non-FDSs,
and then investigate some statistical correlations.
\subsection{Averaged Values for Superluminal Motions}
The averaged maximum values of proper motion for FDSs and non-FDSs are
$\langle \mu^{\rm max}_{\rm FDS} \rangle=0.361 \pm 0.037$ and $\langle \mu^{\rm max}_{\rm non-FDS} \rangle=0.224 \pm 0.027$;
while the averaged mean values of proper motion are
$\langle \mu^{\rm mean}_{\rm FDS} \rangle=0.233 \pm 0.021$ and
$\langle \mu^{\rm mean}_{\rm non-FDS} \rangle=0.158 \pm 0.015$ respectively.
The averaged maximum values of apparent velocity for FDS and non-FDS sources are
$\langle \beta^{\rm max}_{\rm FDS}\rangle=12.36 \pm 1.64$ and
$\langle \beta^{\rm max}_{\rm non-FDS} \rangle=8.75 \pm 1.32$;
the averaged mean values are
$\langle \beta^{\rm mean}_{\rm FDS} \rangle=8.17 \pm 0.94$ and
$\langle \beta^{\rm mean}_{\rm non-FDS} \rangle=5.99 \pm 0.78$.
Based on MOJAVE 1.5 Jy flux density-limited samples,
\citet{Lister2016} get a less than 0.02 \% probability that the LAT and non-LAT sub-samples
come from the same parent distribution according to K-S tests.
When a K-S test is adapted to these distributions of FDSs and non-FDSs,
we find that probability for the distributions of FDSs and non-FDSs
to be from the same distribution is less than $1.6 \times 10^{-4}$ as shown in
Figure \ref{mu-dis}-\ref{beta-dis},
suggesting FDSs and non-FDSs should be from two different distributions.
Our result confirms the result by
\citet{Lister2016}.
We can say that FDSs have a larger proper motion and apparent velocity than do non-FDSs.
These results are consistent with other results.
\citet{Jorstad2001} indicated that the sources with $\gamma$-ray emission
show greater apparent velocities ($\beta_{app}$) than these sources without $\gamma$-ray emissions.
\citet{Lister2009}, \citet{Lister2016} and
\citet{Piner2012} also confirmed this result,
which means that the $\gamma$-ray sources are highly beamed.
Our result based on the largest superluminal sample also confirms their results.
\subsection{Correlations}
\citet{VC1994} collected 66 sources with proper motions,
they showed that the proper motion decreases with increasing redshift (see also,
\citealt{Cohen2005}).
In our previous work
(\citealt{ZF2008}),
we collected a sample of 123 superluminal sources,
investigated the correlation between proper motion and redshift,
and got ${\rm log} \mu \sim {-0.28} {\rm log} z$. In this paper,
we obtained ${\rm log} \mu^{\rm max} =-( 0.28 \pm 0.07 ) {\rm log} z -( 0.95 \pm 0.02 )$
with $r = 0.22$ and a chance probability of $p = 1.2 \times 10^{-4}$ for the whole sample.
The result indicates clearly that the proper motion decreases with the increasing redshift,
which is consistent with the results from others'
(\citealt{VC1994},
\citealt{Cohen2005} and
\citealt{ZF2008}).
When we consider FDS and non-FDS separately,
there is an anti-correlation with a slope of $-0.35$ for the FDSs,
and an anti-correlation tendency with a slope of $-0.14$ for the non-FDSs.
When we investigate linear correlations between apparent velocities (log$\beta_{\rm app}$) and redshift (log$z$),
slopes are 0.36 and 0.59 are obtained from the log$\beta_{\rm app}$-log$z$ fitting for FDS and non-FDS
sources respectively,
which means the apparent velocity increases with redshift.
Obviously, for non-FDS,
the positive correlation is much better which with a chance probability is $p = 1.6 \times 10^{-7}$.
\citet{Lister2009} plotted maximum apparent velocity against redshift for their sample and
showed that the maximum superluminal velocity increases with redshift.
In the MOJAVE survey,
the minimum detectable luminosity rises sharply with redshift,
creating a classical Malmquist bias and the high redshift sources have higher apparent luminosities,
which they achieved primarily via Doppler boosting
(\citealt{Lister2009}).
Our result is consistent with theirs.
\citet{Kellermann2007} quoted their result of apparent velocity and luminosity and
indicated that there are no low luminosity sources with fast motions.
The high luminosity sources show a wider range of apparent velocity.
In Figure \ref{beta-L}, we also find such a tendency that the brighter $\gamma$-ray sources
show a higher apparent velocity.
It is due to a beaming effect because a source with a higher velocity suggests
a corresponding higher Doppler factor,
and a higher Doppler boosting results in a higher luminosity.
From Fig. \ref{beta-L},
we can see a tendency for higher apparent velocity source to have higher $\gamma$-ray luminosity.
However, we can also see that some luminous $\gamma$-ray source have also low apparent
velocity. Actually, there is an envelope between apparent velocity and $\gamma$-ray luminosity in Figure \ref{beta-L},
which is similar to that seen in the Caltech-Jodrell Bank Flat Spectrum (CJF) survey
(\citealt{Vermeulen1995}),
the 2 cm Survey
(\citealt{Kellermann2004}),
and the MOJAVE survey
(\citealt{Cohen2007};
\citealt{Lister2009};
\citealt{Piner2012}).
This upper envelope is not due to selection effects, although its precise physical origin is unclear. \citet{Lister2009} speculated that such an envelope may arise because of an intrinsic relation between jet speed and luminosity in the parent population.
\subsection{Flux-Flux Correlations}
The mutual correlations between fluxes were investigated in literature.
\citet{Fan1994} compiled 52 X-ray selected BL Lacs to study the mutual correlations among radio, X-ray and optical data and found closely mutual correlations.
\citet{Dondi1995} studied correlations between $\bar{L}_{\rm \gamma}$ versus $\bar{L}_{\rm r}$, $\bar{L}_{\rm o}$ and $\bar{L}_{\rm X}$, for a sample of quasars detected by EGRET, obtained positive correlations, and found that $\bar{L}_{\rm \gamma}$ correlates closer with $\bar{L}_{\rm r}$ than with $\bar{L}_{\rm o}$ or $\bar{L}_{\rm X}$.
In the present work, we investigate mutual correlations for radio, optical and X-ray fluxes for both FDS and non-FDS sources. For our flux-flux correlation analysis, we can see positive correlations and that there is no significant difference in slopes and intercepts between FDSs and non-FDSs.
In 2010, \citeauthor{Abdo2010a} used quasi-simultaneous data to calculate the
SEDs for a sample of 48 LAT Bright AGN sample sources (LBAS).
From their multwavelength data, we get following corresponding results,
$${\rm log}S_{\rm r}=(0.13 \pm 0.06) {\rm log}F_{\rm o}+(4.37 \pm 0.71), ~r=0.37~{\rm and}~p=1.2 \%, $$
$${\rm log}F_{\rm X}=(0.58 \pm 0.09) {\rm log}F_{\rm o}+(7.10 \pm 1.06), ~r=0.80~{\rm and}~p=3.1 \times 10^{-7}, and $$
$${\rm log}F_{\rm X}=(0.68 \pm 0.23) {\rm log}S_{\rm r}-(1.87 \pm 0.69), ~r=0.36~{\rm and}~p=1.7 \%. $$
We can see that the results based on the quasi-simultaneous data are similar to our results.
The emissions from AGNs are mainly from the jet,
however there are contaminations from the host galaxy (especially for BL Lac objects)
as well as the big blue bump (especially for FSRQs) in the optical band,
together with the contribution from the accretion system in the X-ray domain.
Since the sample considered here is all superluminal, their jet emission should
be strongly boosted, which make the contaminations in the optical and X-ray bands
be relatively small.
\citet{Ghisellini1989} proposed that the bulk velocity of the plasma increases
with distance from the core and synchrotron X-rays are weakly beamed,
while optical and radio emissions are more strongly beamed.
\citet{Fan1993} proposed an empirical frequency dependent Doppler factor:
$\delta_{\rm \nu}=\delta_{\rm o}^{1+1/8~ {\rm log} ({\nu}_{\rm o}/{\nu})}$,
where $\delta_{\rm o}$ is the optical Doppler factor,
then $\delta_{\rm X} \sim \delta_{\rm o}^{0.5}$,
$\delta_{\rm r} \sim \delta_{\rm o}^{1.5}$,
$\delta_{\rm X}$ and $\delta_{\rm r}$ are the X-ray and radio Doppler factors
(\citealt{Fan1993}).
In a beaming model,
the observed emission,
$f^{\rm ob}$,
is strongly boosted,
namely, $f^{\rm ob} = \delta^{p} f^{\rm in}$,
here $f^{\rm in}$ is the intrinsic emission in the source frame,
$\delta$ is the Doppler factor,
$p = 3 + \alpha$ is for a moving compact source or
$p = 2 + \alpha$ for a continuous jet
(\citealt{Lind1985}).
Here $ p = 2 + \alpha$ is used as we did before
(\citealt{Xiao2015}).
In the work, Doppler factors are available for 151 sources,
which makes it possible for us to investigate the mutual correlations
for the intrinsic (de-beamed) radio, optical and X-ray flux emissions.
For the 151 sources, we
have \\
$${\rm log}S_{\rm r}=(0.16 \pm 0.04) {\rm log}F_{\rm o}+(4.86 \pm 0.47), ~r=0.34~{\rm and}~p=1.5 \times 10^{-7}, $$
$${\rm log}F_{\rm X}=(0.47 \pm 0.04) {\rm log}F_{\rm o}+(5.61 \pm 0.47), ~r=0.60~{\rm and}~p=9.6 \times 10^{-23}, $$
$${\rm log}F_{\rm X}=(0.42 \pm 0.07) {\rm log}S_{\rm r}-(1.41 \pm 0.23), ~r=0.36~{\rm and}~p=5.4 \times 10^{-8}, $$
for the observed data; and \\
$${\rm log}S_{\rm r}^{\rm de-beamed}=(0.79 \pm 0.03) {\rm log}F_{\rm o}^{\rm de-beamed}+(12.02 \pm 0.42), ~r=0.88~{\rm and}~p= 2.2 \times 10^{-74}, $$
$${\rm log}F_{\rm X}^{\rm de-beamed}=(0.48 \pm 0.02) {\rm log}F_{\rm o}^{\rm de-beamed}+(5.80 \pm 0.28), ~r=0.86~{\rm and}~p= 7.4 \times 10^{-63}, $$
$${\rm log}F_{\rm X}^{\rm de-beamed}=(0.53 \pm 0.02) {\rm log}S_{\rm r}^{\rm de-beamed}-(1.56 \pm 0.03), ~r=0.85~{\rm and}~p= 4.8 \times 10^{-62}, $$
for the intrinsic data.
It is clear that the correlations between any two bands become much closer when the beaming effect is removed.
The result indicates that the beaming effect affects the observed broad band correlations.
The comparison results are shown in Figure \ref{de-F-F}.
\begin{figure}
\centering
\includegraphics[width=6in]{de-F-F.eps}
\caption{Mutual correlations between two bands flux densities. Left-hand panel for the observed data, right-hand panel for the removed beaming effect data. The upper panel, (a): radio flux density against optical flux density, (b): de-beamed radio flux density against de-beamed optical flux density; the middle panel, (c): X-ray flux density against optical flux density, (d): de-beamed X-ray flux density against de-beamed optical flux density; and the lower panel, (e): X-ray flux density against radio flux density, (f): de-beamed X-ray flux density against de-beamed radio flux density. The black plus for FDS sources and the blue circle for non-FDS sources and solid lines stand for best linear fitting results.}
\label{de-F-F}
\end{figure}
\subsection{Basic parameters for Jets}
\subsubsection{Doppler Factor, Lorentz Factor and Viewing Angle}
Doppler factors are important but not easy to estimate, although several methods were proposed. In the present work, we collect Doppler factor for 229 sources (151 FDS and 78 non-FDS) and list them in Column 4 in Tables \ref{Tab-FDS-simp} and \ref{Tab-non-FDS-simp}.
In a beaming model, the Lorentz factor ($\Gamma$) and viewing angle ($\phi$) can be obtained from $\delta$ and $\beta_{\rm app}$:
$$\Gamma =\frac{\beta_{\rm app}^{2}+\delta^{2}+1}{2\delta},
{\rm tan}\phi =\frac{2\beta_{\rm app}}{\beta_{\rm app}^{2}+\delta^{2}-1}.$$
From the Doppler factors in Tables \ref{Tab-FDS-simp} and \ref{Tab-non-FDS-simp}, we have,
$$\langle \delta^{\rm FDS} \rangle= 17.23 \pm 12.54 ~{\rm and}~ \langle \delta^{\rm non-FDS} \rangle= 9.48 \pm 8.86, $$
a K-S test result shows that
the probability for the two distributions to be from the same one
is $7.5 \times 10^{-7}$.
For the sources with available Doppler factors, we can calculate $\Gamma$ and $\phi$, and get their
mean values:
$$\langle \Gamma^{\rm FDS} \rangle= 21.41 \pm 21.59 ~{\rm and}~ \langle \Gamma^{\rm non-FDS} \rangle= 13.57 \pm 12.75, $$
and
$$\langle \phi^{\rm FDS} \rangle= 5.64^{\circ} \pm 9.69^{\circ} ~{\rm and}~ \langle \phi^{\rm non-FDS} \rangle= 8.94^{\circ} \pm 7.77^{\circ},$$
with chance probabilities being $p = 5.2 \times 10^{-5}$ and $p = 1.5 \times 10^{-7}$ respectively,
which show significant difference in Lorentz factor and viewing angle between FDS and non-FDS sources.
\citet{Savolainen2010} obtained that for the photons arriving to us at an angle $\phi$, the jet flow is at an angle $\phi_{co}$ in the co-moving frame:
\begin{equation}
\phi_{\rm co} = {\rm arccos}(\frac{{\rm cos} \phi-\beta}{1-\beta {\rm cos} \phi})
\end{equation}
then, from the obtained $\Gamma$ and $\phi$, we can get $\phi_{co}$, and their mean values are
$$\langle \phi_{\rm co}^{\rm FDS} \rangle= 82.76^{\circ} \pm 46.83^{\circ} ~{\rm and}~ \langle \phi_{\rm co}^{\rm non-FDS} \rangle= 93.68^{\circ} \pm 43.21^{\circ}$$
with a K-S test result of $p = 15.8 \%$ suggesting no clear difference in co-moving viewing angle between FDSs and non-FDSs.
From K-S test results, we can see that there are significant differences in Doppler factor,
viewing angle and Lorentz factor between FDSs and non-FDSs.
FDSs show higher Doppler factors, higher Lorentz factor, and smaller viewing angle than do non-FDSs. The fact that FDS sources have larger Doppler factor also confirmed by \citet{Lister2015} who used the MOJAVE sample as well. However, the co-moving viewing angles show no clear difference between FDSs and non-FDSs. It means that FDS and non-FDS jets have similar cone in the comoving frame.
Above analysis results suggest that the difference between {\textit{Fermi}} detected superluminal sources
and {\textit{non-Fermi}} detected superluminal sources comes from their difference beaming effect
with FDSs being strongly beamed than non-FDSs. The superluminal source 0007+106 (III ZW2)
was not listed in the 3FGL, but it was classified as a $\gamma$-ray source by \citet{Liao2016}.
So, we propose that the superluminal source is a $\gamma$-ray candidate and
the $\gamma$-ray source should be a superluminal source.
Our sample gives that $\langle \beta^{\rm max}_{\rm non-FDS} \rangle=8.75 \pm 1.32$ for non-FDS sources, and $\langle \beta^{\rm max}_{\rm FDS}\rangle=12.36 \pm 1.64$ for the FDS sources. If we take non-Fermi sources with $\beta^{\rm max}_{\rm non-FDS} > \langle \beta^{\rm max}_{\rm FDS}\rangle + 5\sigma$ as the candidate of $\gamma$-ray emitter, then there are 6 $\gamma$-ray emitting candidates, and they are 0153+744, 0208-512, 0536+145, 0552+398, 2223+210, 2351+456.
\begin{comment}
\subsubsection{Doppler Factor and Apparent Velocity}
For a beaming model, we have
\begin{equation}
\beta_{\rm app}=(2\delta \Gamma-\delta^{2}-1)^{1/2}
\end{equation}
and, let $\Gamma = \delta$, then $\beta_{\rm app}^{\rm max}=(\delta^{2}-1)^{1/2}$ (\citealt{Ghisellini1993}). So the limitation of Doppler factor can be computed by an apparent velocity as
\begin{equation}
\delta=((\beta_{\rm app}^{\rm max})^{2}+1)^{1/2}.
\end{equation}
Through this paper the maximum value of apparent velocities provided by a non-FDS source: 0153+744 ($\mu=0.698 \pm 0.189 ~ mas \cdot yr^{-1}$), $\beta_{\rm app}^{\rm max}=58.04 \pm 15.71$, suggesting a Doppler factor $\delta=58.05 \pm 1.02$.
When we plot $\delta$ verus $\beta_{\rm app}$ in Figure \ref{beta-delta}, we use it to calculate an Lorentz factor $\Gamma=58.05$ with the formula $\Gamma = \delta = (\beta_{\rm app}^{2}+1)^{1/2}$. In Figure \ref{beta-delta}, the dotted line corresponding to $\Gamma=58.05$, the sources lie on the line should have $\Gamma =\delta =58.05$. The two dashed curves with $\Gamma =1.5$ and $100$ and the two dash-dotted curves with $\phi = 1^{\circ}$ and $40^{\circ}$ respectively. These two pairs of curves compass almost every sources in our sample that means the most of our superluminal blazars have a viewing angle ($\phi$) less than $40^{\circ}$ and an Lorentz factor ($\Gamma$) less than 58.05.
\begin{figure}
\centering
\includegraphics[width=5in]{beta-delta.eps}
\caption{Plot of apparent velocity (log$\beta_{\rm app}^{\rm max}$) against Doppler factor (log$\delta$). Solid line corresponds to viewing angle ${\rm sin} \phi = 1/ \Gamma$ (Ghisellini et al. 1993), the dotted line stand for the maximum $\Gamma$ from this paper, two dashed curves stand for the correlation between $\beta_{\rm app}$ and $\delta$ when fixed by $\Gamma =1.5$ and $100$ and the two dash-dot curves stand for the correlation when the viewing angle are fixed by $\phi = 1^{\circ}$ and $40^{\circ}$. These two pairs of curves have embraced the majority of points. The blank symbol stands for FDS sources, and full symbol stands for non-FDS sources.}
\label{beta-delta}
\end{figure}
The correlation between co-moving viewing angle and Doppler factor is anti-correlated for both FDS and non-FDS sources as shown in the lower-left panel of Figure \ref{angle-L&D}, this is a self-consistent for the beaming boost with a smaller viewing angle corresponding to a higher Doppler factor.
The correlation between the Lorentz factor and viewing angle is shown in the Figure \ref{angle-L&D} too. Let $\delta=\Gamma$, above equation gives $\Gamma=1/{\rm sin}\phi$, which is shown in the upper-right panel of the figure with a solid line. The sources with $\delta < \Gamma$ locate in the upper area upon the solid line, while sources with $\delta > \Gamma$ locate in left-hand below area. The correlation between the Lorentz factor and co-moving viewing angle is shown in the lower-right panel of Figure \ref{angle-L&D}. It shows that when a co-moving viewing angle is in the range of $\phi_{\rm co}=[0^{\circ}, 120^{\circ}]$, the Lorentz factor is less than 30, beyond that, Lorentz factor increase quickly with co-moving viewing angle.
\begin{figure}
\centering
\includegraphics[width=6in]{angle-L&D.eps}
\caption{Upper left panel: Viewing angle ($\phi$) versus Doppler factor ($\delta$); Lower left panel: Co-moving viewing angle ($\phi_{\rm co}$) versus Doppler factor ($\delta$). Upper right panel: Viewing angle ($\phi$) versus Lorentz factor ($\Gamma$); Lower right panel: Co-moving viewing angle ($\phi_{\rm co}$) versus Lorentz factor ($\Gamma$). The blank symbols are FDS sources and full symbols for non-FDS sources. These solid curves in both upper panels with the same condition of $\Gamma=1/{\rm sin}\phi$.}
\label{angle-L&D}
\end{figure}
\end{comment}
\subsubsection{$\gamma$-ray Luminosity and Viewing Angle}
For $\gamma$-ray luminosity ($L_{\rm \gamma}$) and viewing angle ($\phi$), there is a significant anti-correlation,
$${\rm log}\phi=-(0.23 \pm 0.04) {\rm log}L_{\rm \gamma}+(11.14 \pm 1.93),$$
with $r=-0.38~{\rm and}~p= 2.2 \times 10^{-6}$. The best fitting result is shown in Figure \ref{phi-L} with a solid line. The results imply that the more luminous $\gamma$-ray sources have smaller viewing angles.
\begin{figure}
\centering
\includegraphics[width=5in]{phi-L.eps}
\caption{Correlation between observed $\gamma$-ray luminosity and viewing angle of FDS sources. The solid line shows the best linear fitting result of all the sources.}
\label{phi-L}
\end{figure}
However, in a beaming model, the observed flux density, $f^{\rm ob}$ is correlated with the intrinsic flux density, $f^{\rm in}$, by $f^{\rm ob} = \delta^{p} f^{\rm in}$, where $p=3+\alpha$ for a discrete case and $p=2+\alpha$ for a continuous case. So, for the luminosity, we have $L^{\rm ob} =\delta^{4+\alpha} L^{\rm in}$ for a discrete case, or $L^{\rm ob} =\delta^{3+\alpha} L^{\rm in}$ for the continuous case. For sources with available $\delta$, we can get their intrinsic luminosity and co-moving viewing angle, which show \\
$${\rm log}\phi_{\rm co}=(0.09 \pm 0.01) {\rm log}L_{\rm \gamma}^{\rm in}-(1.73 \pm 0.48),$$
with $r=0.69~{\rm and}~p= 7.3 \times 10^{-22}$, the result is shown in Figure \ref{phico-de-L} with a solid line.
\begin{figure}
\centering
\includegraphics[width=5in]{phico-de-L.eps}
\caption{Correlation between intrinsic $\gamma$-ray luminosity and co-moving viewing angle of FDS sources. The solid line shows the best linear fitting result.}
\label{phico-de-L}
\end{figure}
Obviously, there is a positive correlation between co-moving viewing angle and intrinsic luminosity,
which suggests that the luminous intrinsic $\gamma$-ray luminosity corresponds to a wider co-moving viewing angle.
If the emission per solid angle in the comoving is similar for {\textit{Fermi-LAT}} sources,
then the larger co-moving angle corresponds to larger solid angle frame,
then the emission is stronger.
From Figures \ref{beta-L}-\ref{phico-de-L}, we can see that FDS and non-FDS sources are different. FDSs show higher proper motion ($\mu$), higher apparent velocity ($\beta_{\rm app}$), higher Doppler factor ($\delta$), higher Lorentz factor ($\Gamma$) and smaller viewing angle ($\phi$) than non-FDSs.
We can say that FDSs have stronger beaming effect than do non-FDSs.
\section{Conclusions}
The 291 superluminal sources had 1216 components.
We collected their multi-wavelength data and Doppler factors and calculated the
apparent velocity for each component,
the Lorentz factor and viewing angle (co-moving angle) for those with an available Doppler factor,
calculate the $\gamma$-ray luminosity for the 3GFL sources.
Subsequently, we investigated the relationships between FDS and non-FDS sources.
Our main conclusions are summarized as follows:
1. FDS sources show higher proper motion, apparent velocity, Doppler factor, Lorentz factor and smaller viewing angles than non-FDS sources. For superluminal sources, FDSs are more beamed.
2. The intrinsic (de-beamed) fluxes show much closer mutual correlations among radio, optical and X-ray bands than the observed data.
3. A higher apparent velocity source has the tendency to exhibit higher $\gamma$-ray luminosity,
and the superluminal sources may be the $\gamma$-ray emission candidates;
moreover $\gamma$-ray sources with no known superluminal velocity can be superluminal candidates.
4. The $\gamma$-ray brighter source shows a smaller viewing angle,
which suggest a strong Doppler effect.
High energetic $\gamma$-ray emissions per solid angle are probably similar in the co-moving frame;
thus the de-beamed $\gamma$-ray luminosity is positively correlated with the co-moving viewing angle.
\begin{acknowledgements}
This work is partially supported by the National Natural Science
Foundation of China (NSFC 11733001, NSFC U1531245, NSFC 10633010, NSFC 11173009,NSFC 11403006), Natural Science Foundation of Guangdong Province
(2017A030313011), supports for Astrophysics Key Subjects of Guangdong Province and Guangzhou City, and Science and Technology Program of Guangzhou (201707010401). We also thank the MOJAVE team and Purdue University for use the precious data of kinematic details. We thank the referees for the useful comments and constructive suggestions.
\end{acknowledgements}
|
{
"timestamp": "2019-03-04T02:12:33",
"yymm": "1902",
"arxiv_id": "1902.10764",
"language": "en",
"url": "https://arxiv.org/abs/1902.10764"
}
|
\section{INTRODUCTION}\label{INTRODUCTION}
\sloppy
Although through-water depth determination from aerial imagery is a much more time consuming and costly process, it is still a more efficient operation than ship-borne sounding methods and underwater photogrammetric methods \cite{Agrafiotis2018} in the shallower (less than 10 m depth) clear water areas. Additionally, a permanent record is obtained of other features in the coastal region such as tidal levels, coastal dunes, rock platforms, beach erosion, and vegetation. This is true, even though many alternatives for bathymetry \cite{Menna2018} have arose since. This is especially the case for the coastal zone of up to 10m depth, which concentrates most of the financial activities, is prone to accretion or erosion, and is ground for development, where there is no affordable and universal solution for seamless underwater and overwater mapping. Image-based techniques fail due to wave breaking effects and water refraction, and echo sounding fails due to short distances.
At the same time bathymetric LiDAR with simultaneous image acquisition is a valid, albeit expensive alternative, especially for small scale surveys. In addition, despite the fact that the image acquisition for orthophotomosaic generation in land is a solid solution, the same cannot be said for the shallow water seabed. Despite the accurate and precise depth map provided by LiDAR, the sea bed orthoimage generation is prohibited due to the refraction effect, leading to another missed opportunity to benefit from a unified seamless mapping process.
\newpage
\subsection{Description of the problem}\label{sec:Description of the problem}
Even though UAVs are well established in monitoring and 3D recording of dry landscapes and urban areas, when it comes to bathymetric applications, errors are introduced due to the water refraction. Unlike in-water photogrammetric procedures where, according to the literature \cite{Lavest2000}, thorough calibration is sufficient to correct the effects of refraction, in through-water (two-media) cases, the sea surface undulations due to waves \cite{Fryer1985,Okamoto1982} and the magnitude of refraction that differ at each point of every image, lead to unstable solutions \cite{Agrafiotis2015,Georgopoulos2012}. More specifically, according to Snell’s law, the effect of refraction of a light beam to water depth is affected by water depth and angle of incidence of the beam in the air/water interface. The problem becomes even more complex when multi view geometry is applied.
\begin{figure}[ht!]
\begin{center} \includegraphics[width=0.95\columnwidth]{figures/1.jpg}
\caption{The geometry of two-media photogrammetry for the multiple view case}
\label{fig:figure1}
\end{center}
\end{figure}
In Figure \ref{fig:figure1} the multiple view geometry which applies to the UAV imagery is demonstrated: there, the apparent depth \textit{C} is calculated by the collinearity equation. Starting from the apparent (erroneous) depth of a point \textit{A}, its image-coordinates $a_{1}$, $a_{2}$, $a_{3}$…, $a_{n}$, can be backtracked in images $O_{1}$, $O_{2}$ , $O_{3}$, …, $O_{n}$ using the standard collinearity equation. If a point has been matched successfully in the photos $O_{1}$, $O_{2}$ , $O_{3}$, …, $O_{n}$, then the standard collinearity intersection would have returned the point \textit{C}, which is the apparent and shallower position of point A and in the multiple view case is the adjusted position of all the possible red dots in Figure \ref{fig:figure1}, which are the intersections for each stereopair.
Thus, without some form of correction, refraction produce an image and consequently a point cloud of the submerged surface which appears to lie at a shallower depth than the real surface. In literature, two main approaches to correct refraction in through-water photogrammetry can be found; analytical or image based.
In this work, a new approach to address the systematic refraction errors of point clouds derived from SfM-MVS procedures is introduced. The developed technique is based on machine learning tools which are able to accurately recover shallow bathymetric information from UAV-based imaging datasets, leveraging several coastal engineering applications. In particular, the goal was to deliver image-based point clouds with accurate depth information by learning to estimate the correct depth from the systematic differences between image-based products and (the current gold-standard for shallow waters) LiDAR point clouds. To this end, a Linear Support Vector Regression model was employed and trained to predict the actual depth \textit{Z} from the apparent depth of a point, $Z_{o}$ from the image-based point cloud.
The rest of the paper is organized as follows: Subsection \ref{sec:Related work} presents the related work regarding refraction correction and the use of SVMs in bathymetry determination. In Section \ref{sec:DATASETS}, datasets used are described while in Section \ref{sec:PROPOSED METHODOLOGY} the proposed methodology is described and justified. In Section \ref{sec:TESTS AND EVALUATION} the tests performed and the evaluations carried out are described. Section \ref{sec:CONCLUSIONS} concludes the paper.
\subsection{Related work}\label{sec:Related work}
Refraction effect has driven scholars to suggest several models for two-media photogrammetry, most of which are dedicated to specific applications. Two-media photogrammetry is divided into through-water and in-water photogrammetry. The through-water term is used when the camera is above the water surface and the object is underwater, hence part of the ray is traveling through air and part of it through water. It is most commonly used in aerial photogrammetry \cite{Skarlatos2018,Dietrich2017} or in close range applications \cite{Georgopoulos2012,Butler2002}. It is argued that if the water depth to flight height ratio is considerably low, then water refraction is unnecessary. However, as shown in the literature \cite{Skarlatos2018}, the water depth to flying height ratio is irrelevant, in cases ranging from drone and unmanned aerial vehicle (UAV) mapping to full-scale manned aerial mapping. In these cases water refraction correction is necessary.
\subsection{Bathymetry Determination using Machine Learning}\label{sec:Bathymetry Determination using Machine Learning}
Even though the presented approach here is the only one dealing with UAV imagery and dense point clouds resulting from the SfM-MVS processing, there is a small number of single image approaches for bathymetry retrieval using satellite imagery. Most of these methods are based on the relation between the reflectance and the depth. These approaches exploit a support vector machine (SVM) system to predict the correct depth \cite{Wang2018,Misra2018}. Experiments there showed that the localized model reduced the bathymetry estimation error by 60\% from an RMSE of 1.23m to 0.48m. In \cite{Mohamed2016} a methodology is introduced using an Ensemble Learning (EL) fitting algorithm of Least Squares Boosting (LSB) for bathymetric maps calculation in shallow lakes from high resolution satellite images and water depth measurement samples using Echo-sounder. The retrieved bathymetric information from the three methods was evaluated using Echo Sounder data. The LSB fitting ensemble resulted in an RMSE of 0.15m where the PCA and GLM yielded RMSE’s of 0.19m and 0.18m respectively over shallow water depths less than 2m. Except from the primary data used, the main difference between the work presented here and the work presented in these articles, is that they test and evaluate their proposed algorithms on percentages of the same test site and at very shallow depths while here two different test sites are used.
\section{DATASETS}\label{sec:DATASETS}
The proposed methodology has been applied in real-world applications in two different test sites for verification and comparison against bathymetric LiDAR data. In the following paragraphs, the results of the proposed methodology are investigated and evaluated. The initial point cloud used here can be created by any commercial photogrammetric software (such as Agisoft’s Photoscan©, used in this study) following standard process, without water refraction compensation. However, wind affects the sea surface with wrinkles and waves. Taking this into account, the water surface needs to be as flat as possible, so that to have best sea bottom visibility and follow the assumption of flat-water surface. In case of a wavy sea surface, errors would be introduced \cite{Okamoto1982,Agrafiotis2015} without any form of correction \cite{Chirayath2016} applied and the relation of the real and the apparent depths will be more scattered, affecting to some extent the training and the fitting of the model. Furthermore, water should not be turbid enough to have a clear bottom view. Obviously, water turbidity and water visibility are additional restraining factors. Just like in any photogrammetric project, sea bottom must present pattern, meaning that photogrammetric bathymetry might fail in sandy or seagrass sea bed. However, since normally, a sandy bottom does not present any abrupt height differences and detailed forms, and provided measures to eliminate the noise of the point cloud in these areas are taken, results would be acceptable, even in a less dense point cloud, due to matching difficulties.
\subsection{Test sites and available data}\label{sec:Test sites and available data}
In order to facilitate the training and the testing of the proposed approach, ground truth data of the seabed depth were required, together with the image-based point clouds. To facilitate this, ground control points (GCPs) were measured in land and used to georeference the photogrammetric data with the LiDAR data. The common system used is the Cyprus Geodetic Reference System (CGRS) 1993, to which the LiDAR data were already georeferenced.
\subsubsection{Amathouda Test Site}\label{sec:Amathouda Test Site}
The first site used is Amathouda (Figure \ref{fig:figure2} upper image), where the seabed reaches a maximum depth of 5.57 m. The flight was executed with a Swinglet CAM fixed-wing UAV with an Canon IXUS 220HS camera having 4.3mm focal length, 1.55$\mu$m pixel size and 4000$\times$3000 pixels format. A total of 182 photos were acquired, from an average flight height of 103 m, resulting in 3.3 cm average GSD.
\subsubsection{Agia Napa Test Site}\label{sec:Agia Napa Test Site}
The second test site is in Agia Napa (Figure \ref{fig:figure2} lower image), where the seabed reaches the depth of 14.8m. The flight here executed with the same UAV. In total 383 images were acquired, from an average flight height of 209m, resulting in 6.3cm average ground pixel size.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.95\columnwidth]{figures/21.jpg}
\caption{The two test sites. Amathouda (top) and Ag. Napa (bottom). Yellow triangles represent the GCPs positions.}
\label{fig:figure2}
\end{center}
\end{figure}
Table \ref{fig:t1}(presents the flight and image-based processing details of the two different test sites. There, it can be noticed that the two sites have a different average flight height, indicating that the suggested solution is not limited to specific flight heights. That means that a trained model on an area may be applied on another area, having the flight and image-based processing characteristics of the datasets used.
\begin{table}[ht!]
\begin{center} \includegraphics[width=1.0\columnwidth]{figures/t1.jpg}
\caption{Flight and image-based processing details regarding the two different test sites}
\label{fig:t1}
\end{center}
\end{table}
\subsubsection{Data pre-processing}\label{sec:Data pre-processing}
To facilitate the training of the proposed bathymetry correction model, data were pre-processed. Since the image-based point cloud was denser, than the LiDAR point cloud, it was decided to reduce the number of the points of the first one. To that direction the number of the image-based point clouds were reduced to the number of the LiDAR point clouds, for the two test sites. This way, for each position \textit{X}, \textit{Y} of the seabed two depths are corresponding: the apparent depth $Z_{o}$ and the LiDAR depth \textit{Z}. Consequently, outlier data were removed from the dataset. At this stage of the pre-processing, outliers were considered points having $Z_{o}$ $\geq$ \textit{Z} since this is not valid when the refraction phenomenon is present. Moreover, points having $Z_{o}$ $\geq$ 0m were also removed since they might cause errors in the training process. After being pre-processed, the datasets were used as follows: due to availability of a lot of reference data in Agia Napa test site, the site was split in two parts having different characteristics: Part I having 627.522 points (Figure \ref{fig:figure3}(left) in the red rectangle on the left, Figure \ref{fig:figure5}(top left)) and Part II having 661.208 points (Figure \ref{fig:figure3}(left) in the red rectangle on the right, Figure \ref{fig:figure5}(top right)). Amathouda dataset (Figure \ref{fig:figure3}(middle) and Figure \ref{fig:figure5}(bottom left)) was not split since the available points were much less and quite scattered (Figure \ref{fig:figure3}(right)). The distribution of the \textit{Z} and $Z_{o}$ of the points is presented in Figure \ref{fig:figure3}(right) the Agia Napa dataset is presented with blue colour, while the Amathouda dataset is presented with orange colour.
\subsubsection{LiDAR Reference data}\label{sec:LiDAR Reference data}
LiDAR point clouds of the submerged areas were used as reference data for training and evaluation of the developed methodology. These point clouds were generated with the RIEGL LMS Q680i (RIEGL Laser Measurement Systems GmbH, 3580 Horn, Austria), an airborne LiDAR system. This instrument uses the time-of-flight distance measurement principle of infrared nanosecond pulses for topographic applications and of green (532nm) nanosecond pulses for bathymetric applications. Table \ref{fig:t3} presents the details of the LiDAR data used.
\begin{table}[ht!]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figures/t2.jpg}
\caption{LiDAR data specifications}
\label{fig:t2}
\end{center}
\end{table}
\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=1.85\columnwidth]{figures/3.jpg}
\caption{The two test areas from the Agia Napa test site are presented (left) with blue colour: Part I on the left and Part II on the right. The Amathouda test site is presented in the middle with orange colour. The distribution of the \textit{Z} and $Z_{o}$ values for each dataset is presented (right) as well.}
\label{fig:figure3}
\end{center}
\end{figure*}
Even though the specific LiDAR system can offer point clouds with accuracy of 20mm in topographic applications according to the manufacturers, when it comes to bathymetric applications the system’s range error range is in the order of +/-50-100mm for depths up to 4m, similar to other conventional topographic airborne scanners \cite{Steinbacher2012}. According to the literature LiDAR bathymetry data can be affected by significant systematic errors that lead to much greater errors. In \cite{Skinner2011} the average error in elevations for the wetted river channel surface area was -0.5\% and ranged from -12\% to 13\%. In \cite{Bailly2010} authors detected a random error of 0.19m-0.32m for the riverbed elevation from the Hawkeye II sensor. In \cite{Fernandez2014} the standard deviation of the bathymetry elevation differences calculated reaches 0.79m, with 50\% of the differences falling between 0.33m to 0.56m. However, according to the authors it appears that most of these differences are due to sediment transport between observation epochs. In \cite{Westfeld2017} authors report that the RMSE of the lateral coordinate displacement is 2.5\% of the water depth for the smooth, rippled sea swell. Assuming a mean water depth of 5m leads to a RMSE of 12cm. If a light sea state with small wavelets assumed, results with an RMSE of 3.8\% which corresponds to 19cm in 5m water are expected. It becomes obvious that wave patterns can cause significant systematic effects in bottom coordinate locations. Even for very calm sea states, the lateral displacement can be up to 30cm at 5m water depth \cite{Westfeld2017}.
Considering the above, authors would like to highlight here that in the proposed approach, LiDAR point clouds are used for training the suggested model, since this is the State-of-the-Art method used for shallow water bathymetry of large areas \cite{Menna2018}, even though in some cases the absolute accuracy of the resulting point clouds is deteriorated. These issues do not affect the principle of the main goal of the presented approach which is to systematically solve the depth underestimation problem, by predicting the correct depth, as proved in the next sections.
\section{PROPOSED METHODOLOGY}\label{sec:PROPOSED METHODOLOGY}
A Support Vector Regression (SVR) method is adopted in order to address the described problem. To that direction, data available from two different test sites, characterized by different type of seabed and depths are used to train, validate and test the proposed approach. The Linear SVR model was selected after studying the relation of the real (\textit{Z}) and the apparent ($Z_{o}$) depths of the available points (Figure \ref{fig:figure3}(right)). Based on the above, the SVR model fits according to the given training data: the LiDAR (\textit{Z}) and the apparent depths ($Z_{o}$) of many 3D points. After fitting, the real depth can be predicted in the cases where only the apparent depth is available. In the performed study the relationship of the LiDAR (\textit{Z}) and the apparent depths ($Z_{o}$) of the available points rather follows a linear model and as such, a deeper learning architecture was not considered necessary.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figures/4.jpg}
\caption{The established correlations based on a simple Linear Regression and SVM Linear Regression models, trained on Amathouda and Agia Napa datasets.}
\label{fig:figure4}
\end{center}
\end{figure}
The use of a simple Linear Regression model was also examined, fitting tests were performed in the two test sites and predicted values were compared to the LiDAR data. However, this approach was rejected since the predicted models were producing larger errors than the ones produced by the SVM Linear Regression and they were highly dependent on the training dataset and its density, being very sensitive to the noise of the point cloud. This is explained by the fact that the two regression methods differ only in the loss function where SVM minimizes hinge loss while logistic regression minimizes logistic loss and logistic loss diverges faster than hinge loss being more sensitive to outliers. This is apparent also in Figure \ref{fig:figure4}, where the predicted models using a simple Linear Regression and an SVM Linear Regression trained on Amathouda and Agia Napa [I] datasets are plotted. In the case of training on the Amathouda dataset, it is obvious that the two predicted models (lines in red and cyan colour) differ considerably as the depth increases, leading to different depth predictions. However, in the case of the models trained in Agia Napa [I] dataset, the two predicted models (lines in magenta and yellow colour) are overlapping, also with the predicted model of the SVM Linear Regression, trained on Amathouda. These results suggest that the SVM Linear Regression is less dependent on the density and the noise of the data and ultimately the more robust method, predicting systematically reliable models, outperforming simple Linear Regression.
\subsection{Linear SVR}\label{sec:Linear SVR}
SVMs can also be applied to regression problems by the introduction of an alternative loss function \cite{Smola1996}. The loss function must be modified to include a distance measure. In this paper, a Linear Support Vector Regression model is used exploiting the implementation of \cite{Pedregosa2011}. The problem is formulated as follows: consider the problem of approximating the set of depths:
\begin{equation}
D = \{(Z_{0}^1, Z^1), ..., (Z_{0}^l, Z^l)\}, \hspace{5mm} Z_{0} \in R^n, \hspace{5mm} Z \in R
\label{equ:1}
\end{equation}
with a linear function
\begin{equation}
f(Z_{0}) = \langle w,Z_{0} \rangle + b
\label{equ:2}
\end{equation}
The optimal regression function is given by the minimum of the functional,
\begin{equation}
\phi(w,Z_{0}) = \frac{1}{2}\|w\|^2+c\sum_{i}(\xi_{i}^- + \xi_{i}^+)
\label{equ:3}
\end{equation}
Where \textit{c} is a pre-specified positive numeric value that controls the penalty imposed on observations that lie outside the epsilon margin ($\varepsilon$) and helps to prevent overfitting (regularization). This value determines the trade-off between the flatness of \textit{f}($Z_{o}$) and the amount up to which deviations larger than $\varepsilon$ are tolerated, and $\xi_i-$, $\xi_i+$ are slack variables representing upper and lower constraints of the outputs of the system, \textit{Z} is the real depth of a point \textit{X}, \textit{Y} and $Z_{o}$ is the apparent depth of the same point \textit{X}, \textit{Y}. Based on the above, the proposed framework is trained using the real (\textit{Z}) and the apparent ($Z_{o}$) depths of a number of points in order to predict the real depth in the cases where only the apparent depth is available.
\section{TESTS AND EVALUATION}\label{sec:TESTS AND EVALUATION}
\subsection{Training, Validation and Testing}\label{sec:Training, Validation and Testing}
In order to evaluate the performance of the developed model in terms of robustness and effectiveness, six different training sets were formed from two test sites of different seabed characteristics and then validated against 13 different testing sets.
\subsubsection{Agia Napa and Amathouda datasets}\label{sec:Agia Napa and Amathouda datasets}
The first and the second training approaches are using 5\% and 30\% of the Agia Napa Part II dataset respectively in order to fit the Linear SVR model and predict the correct depth over the Agia Napa Part I and Amathouda test sites. The third and the fourth training approaches are using 5\% and 30\% of the Agia Napa Part I dataset respectively in order to fit the Linear SVR model and predict the correct depth over the Agia Napa Part II and Amathouda test sites. The fifth training approach is using 100\% of the Amathouda dataset in order to fit the Linear SVR model and predict the correct depth over the Agia Napa Part I, the Agia Napa Part II and their combination. The \textit{Z}-$Z_{o}$ distribution of the points used for this training can be seen in Figure \ref{fig:figure5}(bottom left). It is important to notice here that the maximum depth of the training dataset is 5.57m while the maximum depth of the testing datasets is 14.8m and 14.7m respectively.
\begin{figure}
\begin{center}
\includegraphics[width=.23\textwidth]{figures/51.png}
\includegraphics[width=.23\textwidth]{figures/52.png}
\includegraphics[width=.23\textwidth]{figures/53.png}
\includegraphics[width=.23\textwidth]{figures/54.png}
\caption{The \textit{Z}-$Z_{o}$ distribution of the used datasets: the Agia Napa Part I dataset over the full Agia Napa dataset (top left), The Agia Napa Part II dataset over the full Agia Napa dataset (top right), Amathouda dataset (bottom left), The merged dataset over the Agia Napa and Amathouda datasets (bottom right).}
\label{fig:figure5}
\end{center}
\end{figure}
\subsubsection{Merged dataset}\label{Merged dataset}
Finally, a sixth training approach is performed by creating a virtual dataset containing almost the same number of points from each of these two datasets. The \textit{Z}-$Z_{o}$ distribution of this “merged dataset” is presented in Figure \ref{fig:figure5}(bottom right). In the same figure the \textit{Z}-$Z_{o}$ distribution of the Agia Napa dataset and Amathouda dataset are presented in blue and yellow colour respectively. This dataset was generated using the total of the Amathouda dataset points and 1\% of the Agia Napa Part II dataset.
\subsection{Evaluation of the results}\label{sec:Evaluation of the results}
Figure \ref{fig:figure6} demonstrates four of the predicted models: the black coloured line represents the predicted model trained on the Merged Dataset, the cyan coloured line represents the predicted model trained on the Amathouda Dataset, the red coloured line represents the predicted model trained on the Agia Napa Part I [30\%] Dataset, and the green coloured line represents the predicted model trained on the Agia Napa Part II [30\%] Dataset.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figures/6.jpg}
\caption{The \textit{Z}-$Z_{o}$ distribution of the employed datasets and the respective predicted linear models}
\label{fig:figure6}
\end{center}
\end{figure}
It is obvious that despite the scattered points which lie away from these lines, the models achieve to follow the \textit{Z}-$Z_{o}$ distribution of most of the points. It is important to highlight here that the differences between the predicted model trained on the Amathouda dataset (cyan line) and the predicted models trained on Agia Napa datasets are not remarkable, even though the maximum depth of Amathouda dataset is 5.57m and the maximum depth of Agia Napa datasets is 14.8m and 14.7m respectively. The biggest difference observed between the predicted models is between the predicted model trained on Agia Napa [II] dataset (green line) and the predicted model trained on the Merge dataset (black line): 0.45m at 16.8m depth, or a 2.7\% of the real depth. In the next paragraphs the results of the proposed method are evaluated in terms of cloud to cloud distances. Additionally, cross sections of the seabed are presented to highlight the high performance of the proposed methodology and the issues and differences observed between the tested and ground truth point clouds.
\subsubsection{Multiscale Model to Model Cloud Comparison}\label{sec:Multiscale Model to Model Cloud Comparison}
To evaluate the results of the proposed methodology, the initial point clouds of the SfM-MVS procedure and the point clouds resulted from the proposed methodology were compared with the LiDAR point cloud using the Multiscale Model to Model Cloud
Comparison (M3C2) \cite{Lague2013} in Cloud Compare freeware (Cloud Compare, 2019) to demonstrate the changes and the differences that are applied by the presented depth correction approach. The M3C2 algorithm offers accurate surface change measurement that is independent of point density \cite{Lague2013}. In Figure \ref{fig:figure7}(top) and Figure \ref{fig:figure7}(bottom), the distances between the reference data and the original image-based point clouds are increasing as the depth increases. These comparisons make clear that the refraction effect cannot be ignored in such applications. In both cases demonstrated in Figure \ref{fig:figure7}(top) and Figure \ref{fig:figure7}(bottom), the Gaussian mean of the differences is significant reaching 0.44 m (RMSE 0.51m) in the Amathouda test site and 2.23m (RMSE 2.64m) in the Agia Napa test site. Since these values might be considered ‘negligible’ in some applications, it is important to stress that in the Amathouda test site more than 30\% of the compared image-based points present a difference of 0.60-1.00m from the LiDAR points, while in Agia Napa, the same percentage presents differences of 3.00-6.07m, i.e. 20\% - 41.1\% percent of the real depth.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.95\columnwidth]{figures/7.jpg}
\caption{The initial M3C2 distances between the (reference) LiDAR point cloud and the image-based point clouds derived from the SfM-MVS. Figure \ref{fig:figure7}(top) presents the M3C2 distances of Agia Napa and Figure \ref{fig:figure7}(bottom) the initial distances for Amathouda test site.}
\label{fig:figure7}
\end{center}
\end{figure}
Figure \ref{fig:figure8} presents the cloud to cloud distances (M3C2) between the LiDAR point cloud and the point clouds resulted from the predicted model trained on each dataset. Table \ref{fig:t3} presents the results of each one of the 13 tests performed with every detail. There, a great improvement is observed. More specifically, in Agia Napa [I] test site, the initial 2.23m mean distance is reduced to -0.10m while in Amathouda, the initial mean distance of 0.44m is reduced to -0.03m, including outlier points such as seagrass that are not captured in the LiDAR point clouds for both cases or are caused due to point cloud noise again in areas with seagrass or poor texture. It is important also to note that the large distances between the clouds observed in Figure \ref{fig:figure7} disappear. This improvement is observed in every test performed proving that the proposed methodology based on Machine Learning achieves great reduction of the errors caused by the refraction in the seabed point clouds. In Figure \ref{fig:figure8}, it is obvious that the larger differences between the predicted and the LiDAR depths are observed in some specific areas, or areas with same characteristics. In more detail, the lower-left area of Agia Napa Part I test site and the lower-right area of Agia Napa Part II test site, have constantly larger error than other areas of the same depth. This can be explained by their position in the photogrammetric block, since these are areas far for from the control points, situated in the shore and they are in the outer area of the block. However, it is noticeable that these two areas, present smaller deviation from the LiDAR point cloud, when the model is trained in Amathouda test site, a totally different and shallower test site. Additionally, areas with small rock formations are also presenting large differences. This is attributed to the different level of detail in these areas between the LiDAR point cloud and the image-based one, since LiDAR average point spacing is about 1.1m. These small rock formations in many cases lead M3C2 to detect larger distances in these parts of the site and are responsible for the increased Standard Deviation of the M3C2 distances (Table \ref{fig:t3}).
\begin{table*}[ht!]
\begin{center}
\includegraphics[width=1.9\columnwidth]{figures/t3.jpg}
\caption{The results of the comparisons between the predicted models for all the tests performed. }
\label{fig:t3}
\end{center}
\end{table*}
\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=1.9\columnwidth]{figures/8.jpg}
\caption{The cloud to cloud (M3C2) distances between the LiDAR point cloud and the recovered point clouds after the application of the proposed approach. The first, the second and the third row of the figure demonstrate the calculated distance maps and their colour scales for the Agia Napa (Part I and Part II) and Amathouda test sites respectively}
\label{fig:figure8}
\end{center}
\end{figure*}
\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=1.9\columnwidth]{figures/9.jpg}
\caption{Indicative cross-sections (X and Y axis having the same scale) from the Agia Napa (Part I) region after the application of the proposed approach when trained with 30\% from the Part II region. The blue line corresponds to water surface while the green one corresponds to LiDAR data. The cyan line is the recovered depth after the application of the proposed approach, while the red line corresponds to the depths derived from the initial uncorrected image-based point cloud.}
\label{fig:figure9}
\end{center}
\end{figure*}
\subsubsection{Seabed cross sections}\label{sec:Seabed cross sections}
Several differences observed between the image-based point clouds and the LiDAR data that are not due to the proposed depth correction approach. Cross sections of the seabed were generated with main aim to prove the performance of the proposed method, excluding differences between the compared point clouds. In Figure \ref{fig:figure9} the footprint of a representative cross section is demonstrated together with three parts of the section. These parts highlight the high performance of the algorithm and the differences between the point clouds, reported above. In more detail, in the first and the second part of the section presented, it can be noticed that even if the corrected image-based point cloud is almost matching the LiDAR one on the left and the right side of the sections, in the middle parts, errors are introduced. These are mainly caused by coarse errors which though are not related to the depth correction approach. However, in the third part of the section, it is obvious that even when the depth reaches 14m, the corrected image-based point cloud matches the LiDAR one, indicating a very high performance of the proposed approach. Excluding these differences, the corrected image-based point cloud presents deviations of less than 0.05m (0.36\% remaining error at 14m depth) from the LiDAR point cloud.
\subsubsection{Fitting Score}\label{sec:Seabed cross sections}
Another measure to evaluate the predicted model in cases where a percentage of the dataset has been used for training and the rest percentage has been used for testing is by computing the coefficient $R^{2}$ which is the fitting score and is defined as
\begin{equation}
R^2 = 1- \frac{\sum(Z_{true}-Z_{predicted})^2}{\sum(Z_{true}-Z_{true.mean})^2}
\label{equ:4}
\end{equation}
The best possible score is 1.0 and it can also be negative (Pedregosa et al., 2011). $Z_{true}$ is the real value of the depth of the points not used for training while the $Z_{predicted}$ is the predicted depth for these points, using the model trained on the rest of the points. The fitting score is calculated only in cases where a percentage of the dataset is used for training. Results in Table \ref{fig:t3} highlight the robustness of the proposed depth correction framework.
\section{CONCLUSIONS}\label{sec:CONCLUSIONS}
In the proposed approach, based on known depth observations from bathymetric LiDAR surveys, an SVR model was developed able to estimate with high accuracy the real depths of point clouds derived from conventional SfM-MVS procedures. Experimental results over two test sites along with the performed quantitative validation indicated the high potential of the developed approach and the wide field for machine and deep learning architectures in bathymetric applications. It is proved that the model can be trained on one area and used on another one, or indeed on many other, having different characteristics and achieving results of very high accuracy. The proposed approach can be used also in areas were LiDAR data of low density are available, in order to create a denser seabed representation. The methodology is independent from the UAV system used, also the camera and the flight height and there is no need for additional data i.e. camera orientations, camera intrinsic etc. for predicting the correct depth of a point cloud. This is a very important asset of the proposed method in relation to the other state of the art methods used for overcoming refraction errors in seabed mapping. The limitations of this method are mainly imposed by the SfM-MVS errors in areas having texture of low quality (e.g. sand and seagrass areas). Limitations are also imposed due to incompatibilities between the LiDAR point cloud and the image-based one. Among ohers, the different level of detail imposed additional errors in the point cloud comparison and compromise the absolute accuracy of the method. However, twelve out of thirteen different tests (Table \ref{fig:t3}) proved that the proposed method meets and exceeds the accuracy standards generally accepted for hydrography established by the International Hydrographic Organization (IHO), where in its simplest form, the vertical accuracy requirement for shallow water hydrography can be set as a total of $\pm$25cm (one sigma) from all sources, including tides \cite{Guenther2000}.
\section*{ACKNOWLEDGEMENTS}\label{ACKNOWLEDGEMENTS}
Authors would like to acknowledge the Dep. of Land and Surveys of Cyprus for providing the LiDAR reference data, and the Cyprus Dep. of Antiquities for permitting the flight over the Amathouda site and commissioning the flight over Ag. Napa. Also, authors would like to thank Dr. Ioannis Papadakis for the discussions on the physics of the refraction effect.
{
\begin{spacing}{0.99}
\normalsize
|
{
"timestamp": "2019-04-19T02:15:15",
"yymm": "1902",
"arxiv_id": "1902.10733",
"language": "en",
"url": "https://arxiv.org/abs/1902.10733"
}
|
\section{Conclusion and Future Work}
In this study, we propose an architecture, IaKNN, to forecast the motion of surrounding dynamic obstacles,
in which we make the first attempt to generate a tractable quantity from complex traffic scene yielding a new interaction-aware motion model.
Extensive experiments show that IaKNN outperforms the baseline models in terms of effectiveness on the NGSIM dataset.
Further work will be carried out to extend IaKNN to a probabilistic formulation and
combine IaKNN with a maneuver-based model in which road topology and more of the traffic information are taken into account a priori.
\section{Experiments}
\subsection{Dataset} \label{dataset}
We evaluate our approach IaKNN on two public datasets,
namely US Highway 101 (US-101) and Interstate 80 (I-80) of the NGSIM program.
Each dataset contains $(x, y)$-coordinates of vehicle trajectories in a real highway traffic with 10Hz sampling frequency over a 45-min time span.
The 45-min dataset consists of three 15-min segments of mild, moderate and congested traffic conditions.
We follow the experimental settings that were proposed by existing studies \cite{deo2018multi,deo2018convolutional} and
combine US-101 and I-80 into one dataset.
As a result, the dataset involves 100,000 frames of raw data.
We construct the multi-agent training traffic scene in the following construction procedure.
Firstly, we align the raw data by its timestamps.
Secondly, we form a multi-agent traffic scene by picking one host vehicle and including five closest vehicles on its traffic lane or two adjacent traffic lanes.
Finally, we set the window size for extraction as 7 seconds to generate the training scenes.
\subsection{Evaluation Metrics}
Two metrics, namely the \textit{root-mean-square error} (RMSE) and \textit{negative log-likelihood} (NLL), are used to measure the effectiveness of IaKNN.
In particular, the first 2-seconds trajectories and the rest 5-seconds trajectories are used as past trajectories and the ground truth in a 7-seconds multi-agent training traffic scene, respectively.
\begin{itemize}[leftmargin=10pt]
\item RMSE: the root mean squared sum accumulated by the displacement errors over the predicted positions and real positions during the prediction time horizon.
\item NLL: the sum of the negative log probabilities of the predicated positions against the ground-truth positions during the prediction time horizon (we consider a predicted position to be correct if its distance from the ground-truth one is bounded by a small threshold and wrong otherwise).
\end{itemize}
\subsection{Baselines}
The following baseline models will be compared with our model IaKNN.
\begin{itemize}[leftmargin=10pt]
\item \emph{Constant Velocity} (\textit{CV}): Model of the primary kinematic equation with constant velocity.
\item \emph{Vanilla-LSTM} (\textit{V-LSTM}): Model of Seq2Seq. It is from a sequence of past trajectories to a sequence of future trajectories \cite{park2018sequence}.
\item \emph{Social LSTM} (\textit{S-LSTM}): Model of LSTM-based neural network with a \textit{social pooling} for pedestrian trajectory prediction. As demonstrated in \cite{alahi2016social}, the model performs consistently better than traditional models such as the linear model, the collision avoidance model and the social force model. Therefore, we do not compare these traditional methods in our experiments.
\item \emph{Convolutional Social Pooling-LSTM} (\textit{CS-LSTM}): Maneuver based motion model which will generate a multi-modal predictive distribution \cite{deo2018convolutional}.
\item \emph{IaKNN-NoFL}: The proposed method IaKNN without the filter layer.
\end{itemize}
\subsection{Implementation Details}
The default length of the past trajectories is two seconds and
the time horizon of the predicted trajectories is one to five seconds.
The default number of hidden units in LSTMs in the interaction layer and filter layer is set to 32 and all LSTM weight matrices are initialized using a uniform distribution over $[-0.001, 0.001]$.
The weight matrices for other layers are set with the Xavier initialization.
The biases are initialized to zeros.
Additionally, in the training process, we adopt the Adam stochastic gradient descent with hyper-parameters $\beta_1 = 0.9$, $\beta_2 = 0.99$ and set the initial learning rate to be 0.001.
In order to avoid the gradient vanishing, a maximum gradient norm constraint is set to 5.
For the parameters of baselines, we follow the original settings in the open source code.
The experiments are conducted on a machine with Intel(R) Xeon(R) CPU E5-1620 and one Nvidia GeForce GTX 1070 GPU.
\begin{figure}[htpb]
\begin{tabular}{c c}
\begin{minipage}{4cm}
\centering
\includegraphics[width=4cm]{rmse}
\end{minipage}
&
\begin{minipage}{4cm}
\centering
\includegraphics[width=4cm]{nll}
\end{minipage}
\end{tabular}
\caption{The RMSE and NLL of \emph{CV, V-LSTM, S-LSTM, CS-LSTM, IaKNN-NoFL, and IaKNN}. For each evaluation metric, we plot its average for the prediction time horizon from 1s to 5s.}
\vspace{-4mm}
\label{effect}
\end{figure}
\subsection{Result Analysis}
The performance results of baseline methods and our method on the traffic scene are shown in Figure \ref{effect}.
We compute the RMSE and NLL for all traffic scenes and plot the average for the prediction time horizon from 1s to 5s.
Clearly, the naive \textit{CV} produces the highest prediction errors.
\textit{V-LSTM, S-LSTM} and \textit{CS-LSTM} perform similarly in terms of both metrics
which is mainly because they are all LSTM-based neural networks.
Additionally, \textit{S-LSTM, CS-LSTM}, and \textit{IaKNN-NoFL} perform better than \textit{V-LSTM}, especially in RMSE,
and this is mainly because they take into account the interactive effects for modeling.
IaKNN outperforms all other baseline methods in terms of both metrics. Specifically, we observe that it outperforms the best baseline \textit{CS-LSTM} with about 20\% improvement.
This may be explained by the fact that the filter layer in our IaKNN model estimates the underlying trajectories from both the interaction-aware trajectories and the dynamic trajectories in a traffic scene and
the interaction layer has done a good job in capturing the interactive effects among the surrounding vehicles.
The combination of the deep neural network and probabilistic filter makes our model more applicable for the real-time trajectory prediction in the traffic scene.
\subsection{Case Studies}
We illustrate the results by showing the predicted trajectories generated by IaKNN and the real ones in the two lane-change traffic scenarios in Figure \ref{case}.
The results demonstrate the effectiveness of IaKNN.
Clearly, we observe that in general the predicted trajectories are very close to the real ones in the figure.
Moreover, we notice that due to the interactive effects between vehicles in the scenario,
some vehicles have a strong intention to increase their safe distances.
The predicted trajectories are more prone to confirm their intentions.
\begin{figure}[htpb]
\begin{tabular}{c}
\begin{minipage}{8cm}
\centering
\includegraphics[width=8cm]{73}
\end{minipage}
\\
\\
\begin{minipage}{8cm}
\centering
\includegraphics[width=8cm]{91}
\end{minipage}
\end{tabular}
\caption{Case studies of the prediction result by IaKNN: The predicted trajectories and the real ones are drawn in blue and green color, respectively. For each vehicle, we plot its future 2s trajectory.}
\vspace{-4mm}
\label{case}
\end{figure}
\section{Acknowledgements}
{\footnotesize
\bibliographystyle{named}
\section{Introduction}
One hardcore technique for autonomous vehicles is that of forecasting the motion of surrounding dynamic obstacles effectively
since it benefits the on-road motion planning which is a core component in the control system.
In fact, on the motion planning layer of the Apollo open platform~\cite{fan2018baidu},
on-road dynamic obstacles would become technically static
when their motion has been predicted and
the planning with static obstacles has been adequately solved~\cite{gu2013focused,mcnaughton2011parallel}.
The problem of forecasting the motion of surrounding dynamic obstacles for autonomous driving has many real challenges,
e.g., heavy noise in sensor data, complex traffic scenes and intractable interactive effects among the dynamic obstacles.
Existing methods could be categorized into three classes,
namely the physics-based motion model~\cite{liu2005learning}, the maneuver-based motion model~\cite{frazzoli2005maneuver}, and the interaction-aware motion model \cite{lefevre2014survey}.
The physics-based motion model is based on the basic kinematic and dynamic models from physics (e.g., Newton's Laws of Motion).
The maneuver-based motion model is designed for a particular maneuver,
where the future trajectory of a vehicle is predicted
by searching the trajectories which have been clustered a priori.
The interaction-aware motion model is one that captures the interactive effects among vehicle drivers in a traffic scene
by predicting the trajectories of multiple vehicles that are close to one another collectively.
Among those interaction-aware models,
many adopt deep models \cite{alahi2016social,gupta2018social,deo2018convolutional,kuefler2017imitating,bhattacharyya2018multi,ma2018trafficpredict}.
\if 0
physics interaction;
- physics; interaction;
-
"interaction physics
physics"
- kalman filter + neural network;
- independent;
"social force";
\fi
In this paper,
we propose a model called \emph{Interaction-aware Kalman Neural Networks} (IaKNN) for foresting the motion of surrounding dynamic obstacles effectively.
Specifically, IaKNN is a multi-layer architecture consisting of three layers, namely an interaction layer, a motion layer and a filter layer.
The interaction layer is a deep neural network with multiple convolution layers standing before the LSTM encoder-decoder architecture.
Fed with the past trajectories of vehicles that are close to one another,
this layer extracts the ``accelerations'' that capture not only those raw acceleration readings but also the interactive effects among vehicles in the form of social force (which is a measure of internal motivation of an individual in a social activity in sociology and has been used for studying the motion trajectories of pedestrians \cite{helbing1995social}).
The extracted accelerations are called \emph{interaction-aware accelerations}.
The motion layer is similar to the existing physics-based motion model
which transforms accelerations to trajectories by using kinematics models.
Here, instead of feeding the motion layer with the accelerations read from sensors directly, we feed with those interaction-aware accelerations that are outputted by the interaction layer and call the transformed trajectories \emph{interaction-aware trajectories}.
The filter layer consists of mainly a Kalman filter for optimally estimating the future trajectories based on the interaction-aware trajectories outputted by the motion layer.
The novelty in this layer is that
we incorporate two LSTM neural networks~\cite{hochreiter1997long} for learning the time-varying process and measurement noises
that would be used in the update step of the Kalman filter, and
this is the first of its kind for trajectory prediction.
\if 0
In the filter layer, we establish a \emph{time-varying multi-agent Kalman neural networks} to estimate the underlying trajectories.
In particular, two LSTM neural networks \cite{hochreiter1997long} are implemented to learn the time-varying process and measurement noise, inspired by the work of \cite{coskun2017long}.
\fi
In summary, our IaKNN model enjoys the merits of both the physics-based model (the motion layer) and the interaction-based model (the interaction layer) and employs neural-network-based probabilistic filtering for accurate estimation (the filter layer).
In experiments, we evaluate IaKNN on the \emph{Next Generation Simulation} (NGSIM) dataset \cite{colyar2007us} and the empirical results demonstrate the effectiveness of IaKNN.
In summary, the major contributions of this paper are listed as follows:
\begin{itemize}[leftmargin=10pt]
\item We propose to capture the interactive effects among vehicles with interaction-aware accelerations and then use them in kinematics models for trajectory prediction.
\item We propose to learn the time-varying process and measurement noises with LSTM neural networks in a Kalman filter, which, to the best of our knowledge, is the first neural network-based probabilistic filtering algorithm for real-time trajectory prediction.
\item We perform extensive experiments on the NGSIM dataset, which show that IaKNN consistently outperforms the state-of-the-art methods in terms of effectiveness.
\end{itemize}
\section{Methodology}\label{Method}
In this section, we present our architecture \emph{interaction-aware Kalman neural networks} (IaKNN).
Figure~\ref{figure1} gives an overview of the architecture,
where the notations are explained as follows.
$\mathcal{A}^{\mathcal{S}}$ is the portfolio of interaction-aware accelerations outputted by the interaction layer.
$\mathcal{T}$ is the portfolio of interaction-aware trajectories computed by the motion layer,
and
$\mathcal{S}$ and $\mathcal{V}$ are the state and the control of the Kalman filter in the filter layer, respectively,
where $\mathcal{R}$ and $\mathcal{Q}$ are the noise covariance matrices,
both learned by LSTM neural networks.
Besides, in this paper,
$t_0$, $t$, $L'$ and $L$ represent the starting time, current time, observation time horizon and prediction time horizon, respectively, where $t_0 \leq t \leq t_0+L'$.
In the following, we present three layers of IaKNN,
namely the interaction layer, the motion layer and the filter layer,
in Section~\ref{Interaction}, Section~\ref{Motion}, and Section~\ref{Filter}, respectively.
\subsection{Interaction Layer}\label{Interaction}
In the interaction layer, we aim to extract the \emph{interaction-aware accelerations} $\mathcal{A}^{\mathcal{S}}$ from the past traffic environment observations $\mathcal{O}_t$.
\subsubsection{Interaction-aware Acceleration}
Normally, the motion of a vehicle would be determined by its own accelerations.
Nevertheless, in a multi-agent system which we target in this paper,
the situation is much more complex
since drivers of vehicles would be affected by those of other vehicles that are nearby (or they would interact with one another).
For example, a vehicle would be forced to slow down if another vehicle nearby tries to cut the lane in the front.
In fact, the motion of vehicles is determined by not only their physical accelerations but also the interactive effects among vehicles.
Inspired by the classical social force model \cite{helbing2000simulating},
which models the intention of a driver to avoid colliding with dynamic or static obstacles,
we propose to extract those accelerations
such they capture both the raw accelerations recorded and the interactions among vehicles nearby.
We call them the \emph{interaction-aware accelerations} and denote them by $\mathcal{A}^{\mathcal{S}}$.
Specifically, at timestamp $t$, traffic environment observations $\mathcal{O}_{t}$ includes a sequence of recorded accelerations $a_{t_0:t}$, widths $w_{t_0:t}$, lengths $l_{t_0:t}$, and relative distances $d_{t_0:t}$ of agents in the system.
By following \cite{helbing1995social},
we compute the so-called \emph{repulsive interaction forces} $e_{t_0:t} := \text{exp}\big((v^i +v^j)\cdot \Delta t - d^{ij}\big)_{t_0:t}$, where superscripts $i$ and $j$ represent two vehicles that are close to each other and
include them in $\mathcal{O}_{t}$.
Thus, the interaction operator formula at timestamp $t$ is written in details as,
\begin{align*}
\mathcal{A}_{t}^{\mathcal{S}} = \textbf{Interaction}_{\{\mathcal{W}, b\}}\big(a_{t_0:t}, w_{t_0:t}, l_{t_0:t}, d_{t_0:t}, e_{t_0:t}\big).
\end{align*}
The interaction layer is implemented as a neural network
as presented in Figure \ref{figure2}.
\subsubsection{Operator Representation}
At timestamp $t$, the interaction layer in an operator formula is written as,
\begin{align*}
\textbf{Interaction}_{\{\mathcal{W}, b\}}: \mathcal{O}_{t} \longmapsto \mathcal{A}_{t}^{\mathcal{S}},
\end{align*}
where $\mathcal{O}_{t}$ is a portfolio of past environmental observations from $t_0$ to $t$ and $\mathcal{A}_{t}^{\mathcal{S}}$ is the portfolio of the interation-aware accelerations from $t+1$ to $t+L$.
\begin{figure}[tp]
\centering
\includegraphics[width=0.5\textwidth]{interaction.pdf}
\caption{Illustration of \textbf{Interaction Layer}: The architecture of interaction layer is an LSTM encoder-decoder. In the encoder, we build convolutional layers (CNN) regarded as a social tensor extractor, fully-connected layers (FCN) regarded as a mixer of the social features, and merge the deep features into the encoder LSTM. In the decoder, the decoder LSTM outputs the predicted accelerations.}
\vspace{-2mm}
\label{figure2}
\end{figure}
\subsection{Motion Layer}\label{Motion}
In the motion layer, we aim to calculate the \emph{interaction-aware trajectories} $\mathcal{T}$ based on the \emph{interaction-aware accelerations} $\mathcal{A}^\mathcal{S}$ from the interaction layer.
The main intuition of the motion layer comes from the primary kinematic equation which establishes a relationship among \textit{position, time and velocity} by Newtonian physics.
Our strategy is to use higher-order derivatives of a position for better forecasting. Specifically, let $p_t$ be the position of a dynamic obstacle at timestamp $t$. We write $p_t$ with the Taylor expansion as follows.
\begin{align}
p_t = p_{t-1} + v_{t-1}\cdot \Delta t + \frac{1}{2} a_{t-1}\cdot \Delta t^2 + O\Big(\Delta t^3\Big) \label{taylor}
\end{align}
where $v_{t-1}$ represents the velocity at timestamp $t-1$,
$a_{t-1}$ represents the acceleration at timestamp $t-1$, and
the Big-O term captures all remaining terms which would be ignored.
Moreover, we replace the acceleration term $a_t$ with an {\it interaction-aware acceleration} $\mathcal {A^S}$
which is derived from the environment observations.
\if 0
and
the velocity term with an {\it interaction-aware velocity} $v_t^{\mathcal{S}}$ which is computed based on $\mathcal {A^S}$ as follows.
\[
{\mathcal V}^{\mathcal{S}}:= \int{\mathcal{A}^{\mathcal{S}}}dt,
\]
The trajectories that consist of the positions $p_t$,
denoted by $\mathcal{T}$, are called the \emph{interaction-aware trajectories}
since they are based on the interaction-aware accelerations.
Equivalently, we could write $\mathcal{T}$ as follows.
\[
{\mathcal T}:= \int{{\mathcal V}^{\mathcal{S}}}dt = \int{\int{\mathcal{A}^{\mathcal{S}}}dt}dt,
\]
\fi
We specify the velocity term $v$ in Equation~\ref{taylor} as follows.
Suppose the current timestamp is $t$.
For $v_t$, we take the velocity readings which are currently available and transform them to $v_t$ by using a dynamic model - depending on the agent type, we adopt different dynamic models for this task, which shall be introduced shortly.
For $v_{t+1}, v_{t+2}, ...$,
we estimate their values by applying an integral function based on the interaction-aware accelerations as follows.
\[
v_{t+i}:= \int_t^{t+i}{\mathcal{A}^{\mathcal{S}}}dt,
\]
where $i = 1, 2, ..., L$.
Next, we introduce the dynamic models for transforming the velocity readings which are based on a vehicle-centric coordinating system to those based on a global coordinating system such that they could be plotted in the Equation~\ref{taylor}. Depending on the agent type, we introduce two dynamic models, namely the Vehicle Dynamic Model (VDM) and the Pedestrian Dynamic Model (PDM). As the names imply, the former is for the case where vehicles are agents and the latter is for pedestrians are agents.
\subsubsection{Vehicle Dynamic Model (VDM)}
By following \cite{pepy2006path},
we implement the vehicle dynamic model as a classical bicycle model \cite{taheri1992investigation}.
Specifically, suppose $s := (x, y, \theta, v_x, v_y, r)$ is the current reading involving velocities, where $x$ and $y$ are the coordinates, $\theta$ is the orientation, $v_x$ and $v_y$ are velocities,
and $r$ is the yaw rate.
The bicycle model is written as follows.
\begin{align*}
\dot{x} &= v_x\cdot \cos{\theta} -v_y \cdot \sin{\theta},\\
\dot{y} &= v_x\cdot \sin{\theta} + v_y \cdot \cos{\theta},\\
\dot{\theta} &= r.
\end{align*}
$\dot{x}$ and $\dot{y}$ are the transformed velocities along the $x$ and $y$ directions, respectively.
For more details, the readers are referred to \cite{pepy2006path}.
\subsubsection{Pedestrian Dynamic Model (PDM)}
Models for predicting pedestrian dynamic have been explored largely \cite{kooij2014analysis,zhou2012understanding,scovanner2009learning},
and any of these models could be applied here. \\
To simplify this layer, we regard all agents as mass points and their motion behaviors are described in the basic kinematic motion equations $\dot{x} = v_x$ and $\dot{y} = v_y$.
\subsubsection{Operator Representation}
At timestamp $t$, the motion layer in an operator formula is written as,
\begin{align*}
\textbf{Motion}: \mathcal{A}_{t}^{\mathcal{S}} \longmapsto \mathcal{T}_{t},
\end{align*}
where $\mathcal{T}_{t}$ is an interaction-aware trajectory from $t+1$ to $t+L$.
\subsection{Filter Layer}\label{Filter}
In the filter layer, we establish a model
based on the Kalman filter
to estimate the dynamic trajectories $\mathcal{S}_{t}$ based on the \emph{interaction-aware trajectories} $\mathcal{T}_{t}$ used as observations.
\subsubsection{Filter Model}
To fit the Kalman filter as described in Section \ref{Kalman},
we let the dynamic trajectories $\mathcal{S}_t$ be the states, the \emph{interaction-aware trajectories} $\mathcal{T}_t$ be the observations and the dynamic accelerations $\mathcal{U}_{t}$ be the controls in a linear model.
As a result,
the equation of the linear dynamic model could be written as follows.
\begin{align}
\mathcal{S}_t &= \mathcal{F}\cdot \mathcal{S}_{t-1} + \mathcal{B}\cdot \mathcal{U}_{t-1} + \omega_t, \label{kf1}\\
\mathcal{T}_t &= \mathcal{S}_{t} + \eta_t \label{kf2}
\end{align}
where $\mathcal{F}$ is the state transition matrix,
$\mathcal{B}$ is the control matrix,
$\omega_t \sim \mathcal{N}\big(0, \mathcal{Q}_t\big)$ are the time-varying process noises and
$\eta_t \sim \mathcal{N}\big(0,\mathcal{R}_t\big)$ are the measurement noises.
The time-varying covariances $\mathcal{Q}_t$ and $\mathcal{R}_t$ will be learned by time-varying noise models
which consist of LSTM neural networks and will be introduced later.
Note that here we assume the observation matrix $\mathcal{H}$ is an identity matrix for simplicity.
\subsubsection{Specifications of the Layer}
Notice that we can always assume $N$ agents (dynamic obstacles) in the multi-agent system. At timestamp $t$, the state $\mathcal{S}_t$ and the observation $\mathcal{T}_t$ of our Equation \ref{kf1} and \ref{kf2} could be written as follows.
\[
\mathcal{S}_t :=
\begin{bmatrix}
\mathcal{S}_t^1\\
\vdots \\
\mathcal{S}_t^N\\
\end{bmatrix}_{(2\cdot N \cdot L )\times1} \text{and\hspace*{1em}}
\mathcal{T}_t:=
\begin{bmatrix}
\mathcal{T}_t^1\\
\vdots \\
\mathcal{T}_t^N \\
\end{bmatrix}_{(2\cdot N \cdot L )\times1},
\]
where the state $\mathcal{S}_t^i$ includes positions $p_k^i$ from GPS and velocities $v_k^i$ from the wheel odometer and the observation $\mathcal{T}_t^i$ includes the predicted positions $\bar{p}_k^i$ and the predicted velocities $\bar{v}_k^i$, where $t + 1 \leq k \leq t + L$.
Specifically, we have the following.
\[
\mathcal{S}_t^i :=
\begin{bmatrix}
p_{t+1}^i\\
v_{t+1}^i\\
\vdots\\
p_{t+L}^i\\
v_{t+L}^i\\
\end{bmatrix}_{(2\cdot L)\times1} \text{and\hspace*{1em}}
\mathcal{T}_t^i :=
\begin{bmatrix}
\bar{p}_{t+1}^i\\
\bar{v}_{t+1}^i\\
\vdots\\
\bar{p}_{t+L}^i\\
\bar{v}_{t+L}^i\\
\end{bmatrix}_{(2\cdot L)\times1},
\]
where the subscript $L$ is the predicted time horizon.
Next, we define the state transition matrix $\mathcal{F}$ and the control matrix $\mathcal{B}$ in our model.
Firstly, we define two matrix blocks $M_1$ and $M_2$ as follows.
\[
M_1 :=
\begin{bmatrix}
1 & \Delta t & & & \\
& 1 & & & \\
& & \ddots & & \\
& & & 1 & \Delta t\\
& & & & 1
\end{bmatrix}_{(2\cdot L)\times (2 \cdot L)}
\]
and
\[
M_2 :=
\begin{bmatrix}
\frac{1}{2}\Delta t^2 & & \\
\Delta t & & \\
& \ddots & \\
& & \frac{1}{2}\Delta t^2 \\
& & \Delta t
\end{bmatrix}_{(2\cdot L)\times L}
\]
where $\Delta t$ is the time difference between two adjacent traffic environment observations.
Then, $\mathcal{F}$ and $\mathcal{B}$ are block diagonal matrices that are defined as follows.
\[
\mathcal{F} := diag\big(\underbrace{M_1, \dots,M_1}_{N}\big), \text{ and }\mathcal{B} := diag\big(\underbrace{M_2, \dots,M_2}_{N}\big).
\]
\subsubsection{Prediction and Updated Steps}
The prediction step of the Kalman filter is defined as,
\begin{align*}
\mathcal{S}_t^- &= \mathcal{F}\cdot \hat{\mathcal{S}}_{t-1} + \mathcal{B} \cdot \mathcal{U}_{t-1},\\
\mathcal{P}^-_t &= \mathcal{F}\cdot \hat{\mathcal{P}}_{t-1}\cdot \mathcal{F}^T + \mathcal{Q}_t,
\end{align*}
and the update step is as,
\begin{align*}
\mathcal{K}_t &= \mathcal{P}^-_t\cdot (\mathcal{P}^-_t +\mathcal{R}_t)^{-1},\\
\hat{\mathcal{S}}_t & = \mathcal{S}^-_t + \mathcal{K}_t\cdot(\mathcal{T}_t - \mathcal{S}^-_t),\\
\hat{\mathcal{P}}_t &= (\mathcal{I}-\mathcal{K}_t) \cdot \mathcal{P}^-_t,
\end{align*}
where $\mathcal{Q}_t$ and $\mathcal{R}_t$ are the outputs of the time-varying noise models that we introduce next.
\subsubsection{Time-varying Noise Model}\label{tvn}
Since our desired filter model is time-varying,
we assume both the process noises and the measurement noises to follow a zero-mean Gaussian noise model with its covariances formulated as $\mathcal{Q}_t:= LSTM_Q\big(\mathcal{S}_{t_0:t}^-\big)$ and $\mathcal{R}_t:=LSTM_R\big(\mathcal{T}_{t_0:t}\big)$, respectively.
\subsubsection{Operator Representation}
At timestamp $t$, the filter layer in an operator formula is written as,
\begin{align*}
\textbf{Filter}_{\{\mathcal{W}, b\}}: \big\{\hat{\mathcal{S}}_{t-1}, \mathcal{T}_{t}, \mathcal{U}_{t-1} \big\} \longmapsto \hat{\mathcal{S}}_{t},
\end{align*}
where $\hat{\mathcal{S}}_{t}$ is the predicted trajectory from $t+1$ to $t+L$.
\subsection{Loss Function}
The loss function of architecture IaKNN (\textbf{Interaction}$_{\{\mathcal{W}, b\}}$, \textbf{Motion}, and \textbf{Filter}$_{\{\mathcal{W}, b\}}$) is defined as the sum of displacement error of prior estimation $\hat{\mathcal{S}}$ of dynamic trajectories and ground truth $G$ over all time steps and agents, as follows.
\[
\mathcal{L}_{\{\mathcal{W}, b\}}:= \frac{1}{(L'+1)\cdot N} \cdot \sum_{i=1}^N \sum_{t=t_0}^{t_0+L'} ||\hat{\mathcal{S}}_t^i - G^i_t||^2,
\]
where $G^i_t$ means the ground truth of the future trajectory of $i$-th agent at the start timestamp $t$. Noth that $L'$ is the observation time horizon as defined in the above.
\section{Preliminaries}\label{Prelim}
\section{Kalman Filter}\label{Kalman}
In this part, we provide some background of the Kalman filter (KF)~\cite{bishop2001introduction}
which shall be used as a building block in our model introduced in this paper.
KF is an optimal state estimator in the mean square error (MSE) sense with a linear (dynamic) model and Gaussian noise assumptions. Suppose the state, control, and observation of the linear model are $s_t$, $u_t$ and $z_t$, respectively. The model could be expressed with a process equation and a measurement equation as follows.
\begin{align*}
s_t &= \mathcal{F} \cdot s_{t-1} + \mathcal{B} \cdot u_{t-1} + \omega, \\
z_t &= \mathcal{H} \cdot s_t + \eta,
\end{align*}
where $\mathcal{F}$ is a dynamic matrix, $\mathcal{B}$ is a control matrix, $\mathcal{H}$ is an observation matrix, which are all known.
Moreover, $\omega \sim \mathcal{N}(0, \mathcal{Q})$ is the process noise
and $\eta \sim \mathcal{N}(0,\mathcal{R})$ is the measurement noise
based on the noise covariance matrices $\mathcal{Q}$ and $\mathcal{R}$, respectively.
The process of KF is as follows.
It iterates between a prediction phase and an update phase for each of the observations. In the prediction phase, the current state $s^-_t$ and the error covariance matrix $\mathcal{P}^-_t$ are estimated as follows.
\begin{align*}
s_t^- &= \mathcal{F} \cdot \hat{s}_{t-1} + \mathcal{B} \cdot u_{t-1},\\
\mathcal{P}^-_t &= \mathcal{F} \cdot \hat{\mathcal{P}}_{t-1}\cdot \mathcal{F}^T + \mathcal{Q}.
\end{align*}
In the update phase, once the current observation $z_t$ is received,
the Kalman gain $\mathcal{K}_t$, the prior estimation $\hat{s}_t$ and the error covariance matrix $\hat{\mathcal{P}}_t$ are calculated as follows.
\begin{align*}
\mathcal{K}_t &= \mathcal{P}^-_t \cdot \mathcal{H}^T \cdot (\mathcal{H} \cdot \mathcal{P}^-_t \cdot \mathcal{H}^T + \mathcal{R})^{-1},\\
\hat{s}_t & = s^-_t + \mathcal{K}_t \cdot(z_t - \mathcal{H} \cdot s^-_t),\\
\hat{\mathcal{P}}_t &= ( I - \mathcal{K}_t \cdot \mathcal{H}) \cdot \mathcal{P}^-_t.
\end{align*}
For a comprehensive review of KF, the readers could refer to standard references \cite{bishop2001introduction}.
\section{Problem Statement}\label{problemstatement}
We assume there are $N$ vehicles in the multi-agent system (traffic scene).
For each vehicle, we collect at each timestamp $t$ its position $p_t$, velocity $v_t$, acceleration $a_t$, width $w_t$, length $l_t$, and relative distances $\{d_t^j\}_{j=1}^{N-1}$ from other agents.
We call the observations of all vehicles at timestamp $t$ the environmental observation at timestamp $t$ and denote them by $o_t$.
Given the $h$-length past environmental observations $\mathcal{O}_t := \{o_{t-h+1}, o_{t-h+2},\cdots, o_{t}\}$,
the problem is to predict for each vehicle its future $L$-length trajectories.
\section{Related Work}
\subsubsection{State Estimation in Robotics}
State estimation is one of the common techniques in robotics to estimate the state of a robot from various measurements which involve nosies. Classic approaches of state estimation can be found in \cite{barfoot2017state}. Nowadays, artificial intelligence approaches have been explored largely for state estimation in robotics. For example, Coskun et al. train a triple-LSTM neural network architecture to learn the kinematic motion model, process noise, and measurement noise in order to estimate human pose in a camera image \cite{coskun2017long}. Haarnoja et al. propose a discriminative approach for state estimation in which they train a neural network to learn features from highly complex observations and then filter the learned features to output underlying states \cite{haarnoja2016backprop}.
Our IaKNN model has a different goal from those models,
i.e., IaKNN aims to predict the trajectories of vehicles.
\subsubsection{Data-driven Approach in Trajectory Prediction}
Trajectory prediction for a smart vehicle,
which is an important task for autonomous driving,
has been largely studied~\cite{lefevre2014survey}.
Among those methods for this task,
the data-driven ones are dominating.
For example, Alahi et al. propose a deep learning model to predict the motion dynamics of pedestrians in a crowded scene in which they build a fully connected layer called social pooling to learn the social tensor based on pedestrians \cite{alahi2016social}.
Gupta et al. propose a GAN-based encoder-decoder framework for trajectory prediction with a pooling mechanism to aggregate information across people \cite{gupta2018social}.
In \cite{deo2018convolutional},
the authors extract a social tensor with a convolutional social pooling layer and then feed the social tensor to a maneuver-based motion model for trajectory prediction.
Kuefler et al. \cite{kuefler2017imitating} and Bhattacharyya et al. \cite{bhattacharyya2018multi} use imitation learning approach to learn human drivers' behaviors for trajectory prediction. The learned policies are able to generate the future driving trajectories that match those of human drivers better and can also interact with neighboring vehicles in a more stable manner over long horizons.
Ma et al. propose an LSTM-based two-layers model TrafficPredict for heterogeneous traffic-agents in an urban environment \cite{ma2018trafficpredict}.
Our IaKNN model differs from these models in two aspects.
First, IaKNN captures the interactive effects in a form of accelerations which could then be feed to kinematics models and thus it enjoys the merits of both the classic Physics models and the data-driven process (of capturing the interactive effects).
Second, IaKNN employs the Kalman filter for optimizing the state estimation, where LSTM neural networks are used for learning the time-varying process and measurement noises that are used in the Kalman model, and this is the first of its kind for trajectory prediction.
\section{Traffic Datasets}\label{trafficdataset}
There are four datasets,
namely Cityscapes~\cite{cordts2016cityscapes}, KITTI~\cite{geiger2013vision}, ApolloScape~\cite{huang2018apolloscape}, and NGSIM~\cite{colyar2007us},
which are publicly available and involve traffic scenes.
The first three were collected from the first person perspective and have been widely used for single-agent systems in robotics.
The fourth one, NGSIM, was collected on the southbound US101 road and the eastbound I-80 road with a software application called NG-VIDEO
which transcribes vehicles' trajectories from an overhead video.
In this work, we use NGSIM since among the four datasets, NGSIM is only one that is suitable for the study of a multi-agent system which we target in this paper.
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{iaknn.pdf}
\caption{Illustration of the \textbf{IaKNN} Model: In the diagram, at timestamp $t$, the environmental observation $\mathcal{O}_{t}$ flows into the interaction layer which generates the \emph{interaction-aware acceleration} $\mathcal{A}_{t}^{\mathcal{S}}$. Then, we calculate the \emph{interaction-aware trajectory} of vehicles $\mathcal{T}_t$ w.r.t Vehicle Dynamic Model (VDM) in motion layer. In the end, time-varying multi-agent Kalman neural network runs over the predicted time horizon $L$ to fuse dynamic trajectory $\mathcal{S}_t$ and \emph{interaction-aware trajectory} $\mathcal{T}_t$. Particularly, the time-varying process and measurement noises in the filter layer are set by zero-mean Gaussian noises with covariance formulated in a gated-structure neural network.}
\vspace{-2mm}
\label{figure1}
\end{figure*}
|
{
"timestamp": "2019-03-01T02:11:02",
"yymm": "1902",
"arxiv_id": "1902.10928",
"language": "en",
"url": "https://arxiv.org/abs/1902.10928"
}
|
\section{Introduction}
The nature of the vector charmoniumlike state $Y(4260)$ has remained controversial
since its discovery in the initial-state radiation process
$e^+e^-\to \gamma_{ISR} J/\psi\pi^+\pi^-$~\cite{Aubert:2005rm}. There is no room for
the $Y(4260)$ in the charmonium spectrum predicted in the naive quark model~\cite{Godfrey}, and the $Y(4260)$ does not show
strong couplings to ground-state open-charm decay modes~\cite{Pakhlova:2009jv},
which is unexpected for conventional vector $c\bar{c}$ states above
the $D\bar{D}$ threshold. Such peculiar properties have initiated a
lot of theoretical and experimental studies, see Refs.~\cite{Chen:2016qju,Hosaka:2016pey,Lebed:2016hpi,Esposito:2016noz,Guo:2017jvc,Ali:2017jda,Olsen:2017bmm,Karliner:2017qhf,Yuan:2018inv,Kou:2018nap,Cerri:2018ypt} for recent reviews. On the theoretical side, models have been proposed to interpret the
$Y(4260)$ as a hybrid
state~\cite{Zhu:2005hp,Close:2005iz,Kalashnikova:2008qr}, an excited
charmonium~\cite{LlanesEstrada:2005hz,Li:2009zu,Shah:2012js}, a baryonium~\cite{Qiao:2005av},
a hadrocharmonium~\cite{Dubynskiy:2008mq,Li:2013ssa}, a tetraquark
state~\cite{Maiani:2005pe,Ali:2017wsf,Wang:2018ntv}, a hadronic molecule of
$\bar{D}D_1(2420)$~\cite{Ding:2008gr,Wang:2013cya,Li:2013yla,Cleven:2013mka} or
$\omega\chi_{c0}$~\cite{Dai:2012pb}, or an interference effect~\cite{Chen:2010nv,Chen:2017uof}. On the experimental side, resonant structures with a Breit--Wigner mass ranging from $4.21$ to $4.26\GeV$
have been observed and analyzed in different channels such as
$e^+e^-\to J/\psi\pi^+\pi^-$~\cite{Aubert:2005rm,Ablikim:2016qzw},
$h_c\pi^+\pi^-$~\cite{BESIII:2016adj},
$\omega\chi_{c0}$~\cite{Ablikim:2015uix},
$X(3872)\gamma$~\cite{Ablikim:2013dyn}, $\psi'\pi^+\pi^-$~\cite{Ablikim:2017oaf},
and $D^0 D^{\ast -}\pi^++\mathrm{c.c.}$~\cite{Ablikim:2018vxx}.
The signals from all of these channels could be from the $Y(4260)$. The last one is the first observation in an open-charm channel, and the final state $D\bar D^*\pi$ is as expected from the $D\bar D_1$ hadronic molecular model~\cite{Cleven:2013mka,Qin:2016spb}.
In this work, we will study the possible light-quark components of
the $Y(4260)$ to help reveal its internal structure. We will focus
on the $\pi\pi$ invariant mass spectrum of the reaction
$e^+e^-\to Y(4260) \to J/\psi \pi\pi$, which is one of the most accurately measured channels and is the discovery channel of the $Y(4260)$. In this process, the dipion invariant mass
reaches above the $K\bar{K}$ threshold, and thus allows us to extract the
information of the light-quark SU(3) flavor-singlet and flavor-octet components.
The ratio of the cross sections ${\sigma(e^+e^- \to J/\psi K^+ K^-)}/{\sigma(e^+e^- \to J/\psi \pi^+\pi^-)}$
is relevant to the strange-quark component, and will also be taken into account. If the $Y(4260)$
contains no light quarks (as in the hybrid state or the charmonium
scenarios), the light-quark source provided by the $Y(4260)$ has
to be in the form of an SU(3) singlet state. Thus the determination
of the contributions from different SU(3) eigenstate components is
instructive to clarify the structure of the $Y(4260)$, especially in
the case if a nonzero SU(3) octet component is found to be
indispensable to reproduce the experimental data.
The conservation of parity and $C$-parity constrains the dipion system in $e^+e^-\to Y(4260) \to J/\psi \pi\pi$
to be in even partial waves. The dipion invariant mass $m_{\pi\pi}$ goes up to more than $1.1\GeV$. In this energy region, there are strong coupled-channel final-state interactions (FSIs) in the $S$-wave, which include the scalar resonances $f_0(500)$ and $f_0(980)$ and can be taken into account
model-independently using dispersion theory. Based on unitarity and
analyticity, the modified Omn\`es representation
is used in this study, where the left-hand-cut contributions are approximated
by the sum of the $Z_c(3900)$-exchange mechanism and the triangle diagrams $Y(4260) \rightarrow \bar{D}D_1(2420)\rightarrow \bar{D}D^\ast\pi (\bar{D}D_s^\ast K )\rightarrow J/\psi\pi\pi(J/\psi K\bar{K})$~\cite{Cleven:2013mka,Wang:2013hga,Albaladejo:2015lob}.\footnote{We also need
to take account of the $ Y(4260) \to J/\psi
K \bar K $ process in the coupled-channel FSI.} At low energies, the amplitude should agree with the leading chiral results,
so the
subtraction terms in the dispersion relations can be determined by matching to the chiral contact terms.
For the leading contact couplings for $Y(4260)J/\psi\pi\pi$ and $Y(4260)J/\psi K\bar K$, we construct the chiral Lagrangians in the spirit of
the chiral effective field theory ($\chi$EFT) and the heavy-quark nonrelativistic
expansion~\cite{Mannel}. The parameters are then fixed from fitting to the BESIII data.
A diagrammatic representation of all contributions is given in Fig.~\ref{fig.FeynmanDiagram}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{FeynmanDiagram}
\caption{Feynman diagrams considered for $e^+ e^- \to
Y(4260) \to J/\psi \pi \pi $. (a1) and (a2) denote
the contributions of the chiral contact $Y\psi\Phi\Phi$ terms. (b1)
and (b2) correspond to the contributions of the $Z_c$-exchange
terms. (c1) and (c2) denote the triangle diagrams. The crossed diagrams of (b1), (c1), (b2), and (c2) are not shown
explicitly. The gray blob denotes the effects of FSI.
}\label{fig.FeynmanDiagram}
\end{figure}
This paper is organized as follows. In Sec.~\ref{theor}, we describe
the theoretical framework and elaborate on the calculation of the
amplitudes as well as the dispersive treatment of the FSI. In Sec.~\ref{pheno}, we present the fit results
and discuss the light-quark components of the $Y(4260)$ and its
structure. A brief summary is given in Sec.~\ref{conclu}.
\section{Theoretical framework}\label{theor}
\subsection{Lagrangians}
In general, the $Y(4260)$ can be decomposed into
SU(3) singlet and octet components of light quarks,
\be
\label{eq.YComponents} |Y(4260)\rangle=a|V_1\rangle+b|V_8\rangle\,,
\ee
where $|V_1\rangle \equiv V_1^{\text{light}}\otimes
V^{\text{heavy}}=
\frac{1}{\sqrt{3}}(u\bar{u}+d\bar{d}+s\bar{s})\otimes
V^{\text{heavy}}$ and $|V_8\rangle \equiv
V_8^{\text{light}}\otimes
V^{\text{heavy}}=\frac{1}{\sqrt{6}}(u\bar{u}+d\bar{d}-2
s\bar{s})\otimes V^{\text{heavy}}$, and the ratio of the component
strengths $r\equiv b/a$ can be determined through fitting to the data.
Expressed in terms of a $3\times3$ matrix in the SU(3) flavor space, it is written as
\begin{equation}
\frac{a}{\sqrt{3}} V_1 \cdot \mathbbm{1}+\frac{b}{\sqrt{6}} V_8\cdot \text{diag} \left( 1, 1, - 2\right) .
\end{equation}
The effective Lagrangian for the $Y(4260)J/\psi\pi\pi$ and
$Y(4260)J/\psi K\bar{K}$ contact couplings, at leading order in the chiral expansion
and respecting the heavy-quark spin symmetry,
reads~\cite{Mannel,Chen2016,Chen:2016mjn}
\begin{equation}\label{LagrangianYpsipipi}
\L_{Y\psi\Phi\Phi} = g_1\bra V_{1}^\alpha J^\dag_\alpha \ket \bra u_\mu
u^\mu\ket +h_1\bra V_{1}^{\alpha} J^\dag_\alpha \ket \bra u_\mu u_\nu\ket
v^\mu v^\nu +g_8\bra J^\dag_\alpha \ket \bra V_{8}^{\alpha} u_\mu u^\mu\ket
+h_8\bra J^\dag_\alpha \ket \bra V_{8}^{\alpha} u_\mu u_\nu\ket v^\mu v^\nu
+\mathrm{H.c.}\,,
\end{equation}
where $\langle\ldots\rangle$ denotes the trace in the SU(3) flavor space, $J= (\psi/\sqrt{3}) \cdot \mathbbm{1}$, and
$v^\mu=(1,\vec{0})$ is the velocity of the heavy quark. The lightest pseudoscalar mesons,
being the pseudo-Goldstone bosons from the spontaneous breaking of chiral symmetry, can
be filled nonlinearly into
\begin{equation}
u_\mu = i \left( u^\dagger\partial_\mu u\, -\, u \partial_\mu
u^\dagger\right) \,, \qquad
u = \exp \Big( \frac{i\Phi}{\sqrt{2}F} \Big)\,,
\end{equation}
with the Goldstone fields
\begin{align}
\Phi &=
\begin{pmatrix}
{\frac{1}{\sqrt{2}}\pi ^0 +\frac{1}{\sqrt{6}}\eta _8 } & {\pi^+ } & {K^+ } \\
{\pi^- } & {-\frac{1}{\sqrt{2}}\pi ^0 +\frac{1}{\sqrt{6}}\eta _8} & {K^0 } \\
{ K^-} & {\bar{K}^0 } & {-\frac{2}{\sqrt{6}}\eta_8 } \\
\end{pmatrix} . \label{eq:u-phi-def}
\end{align}
Here $F$ is the pion decay constant in the chiral limit, and we take the physical value $92.1\MeV$ for it.
We need to define the $Z_c Y(4260)\pi$ and the $Z_c J/\psi
\pi$ interacting Lagrangians to calculate the contribution of the
intermediate $Z_c$ states, namely $Y(4260) \to Z_c\pi \to J/\psi \pi\pi$.
Note that there is no hint so far for the existence of a hidden-charm strange partner of the $Z_c$ state~\cite{Shen:2014gdm}.
We thus parametrize the $Z_c$ states in a matrix as
\begin{equation} \label{eq:Z-field}
Z^i_{c}=
\left( {\begin{array}{*{3}c}
\frac{1}{\sqrt{2}}Z^{0i}_{c} & Z^{+i}_{c} & 0 \\
Z^{-i}_{c} & -\frac{1}{\sqrt{2}}Z^{0i}_{c} & 0 \\
0 & 0 & 0
\end{array}} \right)\,.
\end{equation}
The leading-order Lagrangians are~\cite{Guo2011}
\begin{align}
\L_{Z_c Y\pi} = C_{Z_c Y\pi} Y^i \bra {Z^i_{c}}^\dagger u_\mu \ket v^\mu
+\mathrm{H.c.} \,, \notag\\
\L_{Z_c\psi\pi} = C_{Z_c \psi\pi} \psi^i \bra {Z^i_{c}}^\dagger u_\mu \ket
v^\mu +\mathrm{H.c.} \,, \label{LagrangianZc}
\end{align}
which give the $S$-wave pionic vertices proportional to the pion energy.
Note that the SU(3) singlet and octet
components of the $Y(4260)$ are not distinguishable in the $Z_c Y(4260)\pi$ interaction, as the strange-quark component is irrelevant here.
In order to calculate the triangle diagrams $Y(4260) \rightarrow \bar{D}D_1(2420)\rightarrow \bar{D}D^\ast\pi (\bar{D}D_s^\ast K )\rightarrow J/\psi\pi\pi(J/\psi K\bar{K})$,\footnote{Here and in the following, $\bar D D_1$ always means the negative $C$-parity combination of $\bar D D_1$ and $ D \bar D_1$.}
we need the Lagrangians for the coupling of the $Y(4260)$ to $\bar{D}D_1$ as well as the couplings of the $D_1$ to $D^\ast\pi$ and $D_s^\ast K$~\cite{Colangelo:2005gb,Wang:2013cya,Guo:2013nza},
\begin{align}\nonumber
\L_{YD_1D}&=\frac{y}{\sqrt{2}}Y^i\left(\bar{D}_a^\dag
D_{1a}^{i\dag}-\bar{D}_{1a}^{i\dag} D_a^\dag\right)+{\rm H.c.} \,, \\
\L_{D_1D^\ast P}&=i\frac{h^\prime}{F}
\big[3D_{1a}^i(\partial^i\partial^j\Phi_{ab})D^{*
j\dag}_{b}-D_{1a}^i(\partial^{j}\partial^j\Phi_{ab})D_{b}^{* i\dag}+ ... \big]+{\rm H.c.} \,,
\label{LagrangianD1}
\end{align}
where $P$ denotes the pseudoscalar meson $\pi$ or $K$.
We also need the Lagrangian for the $J/\psi D^\ast D \pi$ and $J/\psi D_s^\ast D K$ vertices, which at leading order in heavy-meson chiral perturbation theory
is~\cite{Mehen2013}
\begin{equation}\label{LagrangianJpsiDstarDpi}
\L_{\psi D^\ast D P}= \frac{g_{\psi P}}{2} \bra \psi\bar{H}_a^\dagger H_b^\dagger \ket u_{ab}^0\,,
\end{equation}
where the charm mesons are collected in
$H_a=\vec{V}_a \cdot \boldsymbol{\sigma}+P_a$ with
$P_a(V_a)=(D^{(*)0},D^{(*)+},D_s^{(*)+})$, and
$\bar{H}_a=- \bar{\vec{V}}_a \cdot \boldsymbol{\sigma}+\bar{P}_a$
with
$\bar{P}_a(\bar{V}_a)=(\bar{D}^{(*)0},D^{(*)-},D_s^{(*)-})$~\cite{Mehen2008}.
The gauge-invariant $\gamma^\ast(\mu)$ and $Y(4260) (\nu)$ two-point coupling is given by
\be
iV_{\gamma^{\ast\mu}Y^\nu}=2i(g^{\mu\nu}p^2-p^\mu
p^\nu)c_\gamma \,,
\ee
where $p$ is the momentum of the virtual photon $\gamma^\ast.$
\subsection{Amplitudes of \boldmath{$ Y(4260) \to J/\psi PP $} processes}
First we consider the decay amplitude of $ Y(4260)(p_a) \to
J/\psi(p_b) P(p_c)P(p_d) $, which is described in terms of the
Mandelstam variables
\begin{align}
s &= (p_c+p_d)^2 , \qquad
t_P=(p_a-p_c)^2\,, \qquad u_P=(p_a-p_d)^2\,,\nn\\
3s_{0P}&\equiv s+t_P+u_P=
M_{Y}^2+M_{\psi}^2+2m_P^2 \,.
\end{align}
The variables $t_P$ and $u_P$ can
be expressed in terms of $s$ and the scattering angle $\theta$
according to
\begin{align}
t_P &= \frac{1}{2} \left[3s_{0P}-s+\kappa_P(s)\cos\theta \right]\,,&
u_P &= \frac{1}{2} \left[3s_{0P}-s-\kappa_P(s)\cos\theta \right]\,, \nn\\
\kappa_P(s) &\equiv \sigma_P
\lambda^{1/2}\big(M_{Y}^2,M_{\psi}^2,s\big) \,, & \sigma_P &\equiv
\sqrt{1-\frac{4m_P^2}{s}} \,, \label{eq:tu}
\end{align}
where $\theta$ is defined as the angle between the positive
pseudoscalar meson and the $Y(4260)$ in the rest frame of the $PP$ system, and
$\lambda(a,b,c)=a^2+b^2+c^2-2(ab+ac+bc)$ is the K\"all\'en triangle function. We define $\vec{q}$ as the
3-momentum of final $J/\psi$ in the rest frame of the $Y(4260)$ with
\be \label{eq:q} |\vec{q}|=\frac{1}{2M_{Y}}
\lambda^{1/2}\big(M_{Y}^2,M_{\psi}^2,s\big) \,. \ee
For the $Y(4260) \to J/\psi \pi^+ \pi^-$ process, since the
crossed-channel exchanged $Z_c$ and $DD^\ast$ can be on-shell, the left-hand cut (l.h.c.)
produced intersects and overlaps with the right-hand cut (r.h.c.).
Implementing the modified Omn\`es solution method to obtain the
amplitude including FSI relies on the ability to separate the
amplitude into two parts having either l.h.c.\ or r.h.c.\ only. A way of separating the two has been proposed in Ref.~\cite{Moussallam:2013una}, using the
spectral representation of the resonance propagator as well as a
consistent application of the $i\epsilon$ prescription for the
energy variables.\footnote{As discussed in Ref.~\cite{Schmid:1967ojm}, the l.h.c.\ is in fact in the unphysical Riemann sheet. The proper $i\epsilon$ helps to locate the l.h.c.\ in the right position so that it does not overlap with the r.h.c.\ in the physical Riemann sheet.} Similarly we use the spectral
representations of the $Z_c$ propagator and the $D_1$ propagator~\cite{Moussallam:2013una},
\begin{align}\label{eq.SpectralPropagator}
\widetilde{BW}_R(x)=\frac{1}{\pi}\int_{x^{\text{thr}}_R}^\infty \diff x^\prime
\frac{\textrm{Im} [BW_R(x^\prime)]}{x^\prime-x}\,,
\end{align}
where
$BW_R(x^\prime)=(M_{R}^2-x^\prime-iM_{R}\Gamma_{R}(x^\prime))^{-1}$, and $R$ denotes $Z_c$ or $D_1$.
The off-shell-width effects of
the broad intermediate resonances could play a role in the process discussed~\cite{Cleven:2013mka,Qin:2016spb}, and
we construct the energy-dependent widths
for the broad vector resonances. Taking into account that the $Z_c J/\psi \pi$ vertex is in an $S$-wave and proportional to the energy of the pion, and the $D_1 \to D^\ast\pi$ decays in a $D$-wave, the energy-dependent widths of $Z_c$ and $D_1$ read
\begin{align} \label{widthi}
\Gamma_{Z_c}(s)&=\Gamma_{Z_c} \frac{E_{\psi\pi}^2(s)}{E_{\psi\pi}^2(M_{Z_c}^2)} \frac{k_{\psi\pi}(s)M_{Z_c}}{k_{\psi\pi}(M_{Z_c}^2)\sqrt{s}}\,, \nonumber\\
\Gamma_{D_1}(s)&=\Gamma_{D_1} \frac{k_{D^\ast\pi}^5(s)M_{D_1}}{k_{D^\ast\pi}^5(M_{D_1}^2)\sqrt{s}}\,,
\end{align}
where
$k_{QP}(s)=\lambda^{{1}/{2}}(M_Q^2,m_P^2,s)/(2\sqrt{s})$ is the magnitude of the three-vector momentum of the pion, and $E_{QP}(s)=\sqrt{m_\pi^2+k_{QP}^2(s)}.$
The thresholds in Eq.~\eqref{eq.SpectralPropagator} are $x_{D_1}^{\text{thr}}=(M_D+m_\pi)^2$ and $x_{Z_c}^{\text{thr}}=(M_\psi+m_\pi)^2$, respectively.\footnote{In this paper we aim at describing the dipion invariant mass spectrum. The $Z_c$ enters only through providing parts of the l.h.c.. In this case, we can neglect the subtlety due to the closeness of the $Z_c$ mass to the $D\bar D^*$ threshold in the spectral function. On the contrary, if we want to fit to the $Z_c$ line shape, such an effect has to be taken into account properly, see Refs.~\cite{Hanhart:2015cua,Albaladejo:2015lob,Guo:2016bjq,Pilloni:2016obd}.} Notice that the integration convolves with other parts of the amplitude. Now the $Z_c$-exchange amplitude reads
\begin{align}\label{eq.MZc}
\hat{M}^{Z_c,\pi}(s,\cos\theta)&=\frac{2}{F^2}\sqrt{M_{Y}M_{\psi}}M_{Z_c}C_{Y\psi}p_c^0
p_d^0\bigg(\widetilde{BW}_{Z_c}(t)+\widetilde{BW}_{Z_c}(u)\bigg)\nonumber\\
&=\sum_{l=0}^\infty \hat{M}_l^{Z_c,\pi}(s)P_l(\cos\theta)\,,
\end{align}
where $C_{Y\psi}^{Z_c}\equiv C_{Z_c Y\pi} C_{Z_c \psi\pi}$ is the product
of the coupling constants for the exchange of the $Z_c$. The amplitude has been partial-wave decomposed, and
$P_l(\cos\theta)$ are the standard Legendre polynomials. Parity and
$C$-parity conservation (or isospin conservation combined with Bose symmetry) require the pion pair to be in even angular momentum partial waves. We only take into account the $S$-
and $D$-wave components in this study, neglecting the effects of
higher partial waves. Explicitly, the projections of $S$- and $D$-waves
of the $Z_c$-exchange amplitude read
\begin{align}\label{eq.M0Zc}
\hat{M}_0^{Z_c,\pi}(s)&=-\frac{2 \sqrt{M_Y M_\psi}M_{Z_c}}{\pi F^2
\kappa_\pi(s)}C_{Y\psi}\int_{x_{Z_c}^{\text{thr}}}^\infty \diff x^\prime
\frac{M_{Z_c}\Gamma_{Z_c}(x^\prime)}{(x^\prime-M_{Z_c}^2)^2+M_{Z_c}^2
{\Gamma_{Z_c}^2(x^\prime)}}\Big\{
\big(s+|\vec{q}|^2\big)Q_0(y(s,x^\prime))\nonumber\\
&\quad -|\vec{q}|^2\sigma_\pi^2\big[y^2(s,x^\prime)
Q_0(y(s,x^\prime))-y(s,x^\prime)\big] \Big\},
\end{align}
and
\begin{align}\label{eq.M2Zc}
\hat{M}_2^{Z_c,\pi}(s)&=-\frac{5 \sqrt{M_Y M_\psi}M_{Z_c}}{\pi F^2
\kappa_\pi(s)}C_{Y\psi}\int_{x_{Z_c}^{\text{thr}}}^\infty \diff x^\prime
\frac{M_{Z_c}\Gamma_{Z_c}(x^\prime)}{(x^\prime-M_{Z_c}^2)^2+M_{Z_c}^2
{\Gamma_{Z_c}^2(x^\prime)}}\Big\{ \big[s+|\vec{q}|^2-|\vec{q}|^2\sigma_\pi^2
y^2(s,x^\prime)\big] \nonumber\\&\quad
\times\big[(3y^2(s,x^\prime)-1)Q_0(y(s,x^\prime))-3y(s,x^\prime)\big]
\Big\}\,,
\end{align}
respectively, where $y(s,x^\prime)\equiv {(3s_0-s-2x^\prime)}/{\kappa_\pi(s)}$,
and $Q_0(y)$ is the Legendre function of the second kind,
\begin{equation}\label{eq.Q0}
Q_0(y)=\frac{1}{2}\int_{-1}^1 \frac{\diff z}{y-z}P_0(z)
= \frac{1}{2}\log \frac{y+1}{y-1}\,.
\end{equation}
Notice that the analytic continuation of $Q_0(y)$ should be taken into account since the
$Z_c$ can be on-shell in the physical region. There are two finite
branch points in $Q_0(y(s,x^\prime))$,
\be
s_{\pm}(x^\prime)=\frac{1}{2x^\prime}\Big\{(M_Y^2+M_\psi^2)(m_\pi^2+x^\prime)-M_Y^2 M_\psi^2-(x^\prime-m_\pi^2)^2\pm \lambda^{{1}/{2}}(M_Y^2,x^\prime,m_\pi^2)\lambda^{{1}/{2}}(M_\psi^2,x^\prime,m_\pi^2) \Big\}\,.
\ee
In the range of $s_- < s <s_+$, the
argument of the logarithm in Eq.~\eqref{eq.Q0} becomes negative, and the continuation reads~\cite{Barton:1961ms,Bronzan:1963mby,Kambor:1995yc}
\begin{equation}\label{eq.Q0Continuation}
Q_0(y)
= \frac{1}{2}\log \Big|\frac{y+1}{y-1}\Big|+i\frac{\pi}{2}\,.
\end{equation}
Now we briefly discuss the calculation of the triangle diagrams.
We only keep the terms proportional to
$\bm{\epsilon}_{Y}\cdot \bm{\epsilon}_{\psi}$, and omit the remaining terms proportional to contractions of
momenta with the polarization vectors, which are suppressed in the heavy-quark nonrelativistic expansion~\cite{Chen:2016mjn}.
Explicitly, the partial-wave projections of the triangle amplitude for the $ Y(4260) \to J/\psi \pi\pi (J/\psi K \bar{K}) $ process read
\begin{align}\label{eq.MlLoop}
\hat{M}_l^{\text{loop},\pi(K)}(s)&=\frac{2l+1}{2}\frac{ \sqrt{M_Y M_\psi}M_{D_1}M_{D}M_{D_{(s)}^\ast}}{4\pi F^2
}C_{Y\psi}^{\text{loop}}\int_{-1}^1 \diff \cos\theta P_l(\cos\theta) \int_{x_{D_1}^{\text{thr}}}^\infty \diff x^\prime \text{Im}[BW_{D_1}(x')]
\nonumber\\
&\times \int \frac{\diff^d l}{(2\pi)^d}\bigg\{\frac{i |\vec{p_d}|^2 p_c^0}{(l^2-x^\prime+i\epsilon)\big[(p_a-l)^2-M_D^2+i\epsilon\big]\big[(l-p_d)^2-M_{D_{(s)}^\ast}^2+i\epsilon\big]}
\nonumber\\& \qquad +\frac{i |\vec{p_c}|^2p_d^0}{(l^2-x^\prime+i\epsilon)\big[(p_a-l)^2-M_D^2+i\epsilon\big]\big[(l-p_c)^2-M_{D_{(s)}^\ast}^2+i\epsilon\big]}\bigg\}\,,
\end{align}
where $C_{Y\psi}^{\text{loop}}\equiv y h^\prime g_{\psi P}$ is the product of the coupling constants for the triangle diagrams.
For the chiral contact terms, using the Lagrangians in
Eq.~\eqref{LagrangianYpsipipi}, we have
\begin{align}
M^{\chi,\pi}(s,\cos\theta)&=-\frac{4}{F^2}\sqrt{M_{Y}M_{\psi}}\bigg[\Big(g_1+\frac{g_8}{\sqrt{2}}\Big)p_c\cdot
p_d +\Big(h_1+\frac{h_8}{\sqrt{2}}\Big)p_c^0 p_d^0 \bigg]\,, \notag\\
\label{eq.ContactPi+KRaw}
M^{\chi,K}(s,\cos\theta)&=-\frac{4}{F^2}\sqrt{M_{Y}M_{\psi}}\bigg[\Big(g_1-\frac{g_8}{2\sqrt{2}}\Big)p_c\cdot
p_d +\Big(h_1-\frac{h_8}{2\sqrt{2}}\Big)p_c^0 p_d^0 \bigg]\,.
\end{align}
The projections of the $S$- and $D$-waves of the chiral contact terms are
given by
\begin{align}
M_0^{\chi,\pi}(s)&=-\frac{2}{F^2}\sqrt{M_{Y}M_{\psi}}
\bigg\{\Big(g_1+\frac{g_8}{\sqrt{2}}\Big) \left(s-2m_\pi^2 \right)
+\frac{1}{2}\Big(h_1+\frac{h_8}{\sqrt{2}}\Big) \bigg[s+\vec{q}^2\Big(1
-\frac{\sigma_\pi^2}{3} \Big)\bigg]\bigg\}\,, \notag\\
M_0^{\chi,K}(s)&=-\frac{2}{F^2}\sqrt{M_{Y}M_{\psi}}
\bigg\{\Big(g_1-\frac{g_8}{2\sqrt{2}}\Big) \left(s-2m_K^2 \right)
+\frac{1}{2}\Big(h_1-\frac{h_8}{2\sqrt{2}}\Big) \bigg[s+\vec{q}^2\Big(1
-\frac{\sigma_K^2}{3} \Big)\bigg]\bigg\}\,, \notag\\
\label{eq.M0+2Pi+Kchiral}
M_2^{\chi,\pi}(s)&=\frac{2}{3
F^2}\sqrt{M_{Y}M_{\psi}}\,\Big(h_1+\frac{h_8}{\sqrt{2}}\Big)
|\vec{q}|^2\sigma_\pi^2\,.
\end{align}
For the $D$-wave, where the $\pi\pi$ scattering is almost elastic
in the energy range considered here, we only
give the amplitude of the process involving pions.
\subsection{Final-state interactions with a dispersive approach, Omn\`es solution }
There are strong FSI in the $\pi\pi$ system in particular in the
isospin-0 $S$-wave, which can be taken into account
model-independently using dispersion theory. Since the invariant
mass of the pion pair reaches above the $K\bar{K}$ threshold, we
will consider the coupled-channel ($\pi\pi$ and $K\bar
K$) FSI for the dominant $S$-wave component, while for the $D$-wave
only the single-channel ($\pi\pi$) FSI will be considered.
For $Y(4260) \to J/\psi \pi^+ \pi^-$, the
partial-wave expansion of the amplitude including FSI reads
\be
\M^\text{full}(s,\cos\theta) = \sum_{l=0}^{\infty}
\left[M_l^\pi(s)+\hat{M}_l^\pi(s)\right] P_l(\cos\theta)\,,
\label{eq.PartialWaveFullAmplitude}\ee
where $M_l^\pi(s)$ contains
the r.h.c.\ part and accounts for the $s$-channel rescattering,
and the ``hat function'' $\hat{M}_l^\pi(s)$ represents the l.h.c., contributed by the crossed-channel pole terms or the open-flavor loop effects.
In this study, we approximate the l.h.c.\ by the sum of the $Z_c$-exchange diagram and the triangle diagrams,
i.e., $\hat{M}_l^\pi(s)=\hat{M}_l^{Z_c,\pi}(s)+\hat{M}_l^{\text{loop},\pi}(s)$. The method of
approximating the l.h.c.\ in dispersion relations by including the most relevant resonance
exchanges (in the case of no loops) has been applied previously e.g.\ in
Refs.~\cite{Moussallam-gamma,KubisPlenter,ZHGuo,Kang,Dai:2014lza,Dai:2014zta,Dai:2016ytz,Chen2016}.
For the $S$-wave, we will take into account the two-channel
rescattering effects. The functions $\hat{M}_l(s)$ do not have
a r.h.c., so the two-channel unitarity condition leads to the discontinuity of the production amplitudes as
\begin{equation}\label{eq.unitarity2channel}
\textrm{disc}\, \vec{M}_0(s)=2i T_0^{0\ast}(s)\Sigma(s)
\left[\vec{M}_0(s)+\hat{\vec{M}}_0(s)\right] ,
\end{equation}
where the two-dimensional vectors $\vec{M}_0(s)$ and
$\hat{\vec{M}}_0(s)$ stand for the r.h.c.\ and the l.h.c.\
parts of both the $\pi\pi$ and the $K\bar{K}$ final states,
respectively,
\begin{equation}
\vec{M}_0(s)=\left( {\begin{array}{*{2}c}
{M^\pi_0(s)} \\
{\frac{2}{\sqrt{3}}M^K_0(s)} \\\end{array}} \right), \hspace{0.5cm}\hat{\vec{M}}_0(s)=\left( {\begin{array}{*{2}c}
{\hat{M}^\pi_0(s)} \\
{\frac{2}{\sqrt{3}}\hat{M}^K_0(s)} \\
\end{array}} \right).
\end{equation}
The two-dimensional matrices $T_0^0(s)$ and $\Sigma(s)$ are given by
\begin{equation}\label{eq.T00}
T_0^0(s)=
\left( {\begin{array}{*{2}c}
\frac{\eta_0^0(s)e^{2i\delta_0^0(s)}-1}{2i\sigma_\pi(s)} & |g_0^0(s)|e^{i\psi_0^0(s)} \\
|g_0^0(s)|e^{i\psi_0^0(s)} & \frac{\eta_0^0(s)e^{2i\left(\psi_0^0(s)-\delta_0^0(s)\right)}-1}{2i\sigma_K(s)} \\
\end{array}} \right),
\end{equation}
and $\Sigma(s)\equiv \text{diag}
\big(\sigma_\pi(s)\theta(s-4m_\pi^2),\sigma_K(s)\theta(s-4m_K^2)\big)$.
Three input functions enter the $T_0^0(s)$ matrix: the
$\pi\pi$ $S$-wave isoscalar phase shift $\delta_0^0(s)$, and the modulus and phase of the $\pi\pi \to
K\bar{K}$ $S$-wave amplitude $g_0^0(s)=|g_0^0(s)|e^{i\psi_0^0(s)}$. To estimate the uncertainty due to the
dispersive input for the $\pi\pi/K\bar{K}$ rescattering, we will use two different $T_0^0(s)$ matrices, the
Dai--Pennington (DP)~\cite{Dai:2014lza,Dai:2014zta,Dai:2016ytz} and the Bern/Orsay (BO)~\cite{Leutwyler2012,Moussallam2004} parametrizations, and compare the results.
Note that the inelasticity parameter $\eta_0^0(s)$ in Eq.~\eqref{eq.T00} is
related to the modulus $|g_0^0(s)|$ by
\begin{equation}
\eta_0^0(s)=\sqrt{1-4\sigma_\pi(s)\sigma_K(s)|g_0^0(s)|^2\theta(s-4m_K^2)}\,.
\end{equation}
These inputs are used up to $\sqrt{s_0}=1.3\GeV$, below the onset of
further inelasticities from the $4\pi$ intermediate states, where the $f_0(1370)$ and $f_0(1500)$ resonances become important that couple strongly to
$4\pi$~\cite{Tanabashi:2018oca,Ropertz:2018stk}. Above $s_0$, the phases
$\delta_0^0(s)$ and $\psi_0^0$ are guided smoothly to 2$\pi$ by
means of~\cite{Moussallam2000}
\begin{equation}
\delta(s)=2\pi+(\delta(s_0)-2\pi)\frac{2}{1+({s}/{s_0})^{3/2}}\,.
\end{equation}
\noindent The solution of the inhomogeneous coupled-channel unitarity condition in
Eq.~\eqref{eq.unitarity2channel} is given by
\begin{equation}\label{OmnesSolution2channel}
\vec{M}_0(s)=\Omega(s)\bigg\{\vec{P}^{n-1}(s)+\frac{s^n}{\pi}\int_{4m_\pi^2}^\infty
\frac{\diff
x}{x^n}\frac{\Omega^{-1}(x)T(x)\Sigma(x)\hat{\vec{M}}_0(x)}{x-s}\bigg\}
\,,
\end{equation}
where $\Omega(s)$ satisfies the homogeneous coupled-channel
unitarity relation
\begin{equation}\label{eq.unitarity2channelhomo}
\textrm{Im}\, \Omega(s)=T_0^{0\ast}(s)\Sigma(s) \Omega(s),
\hspace{1cm} \Omega(0)=\mathbbm{1} \,,
\end{equation}
and its numerical results have been computed, e.g., in
Refs.~\cite{Leutwyler90,Moussallam2000,Hoferichter:2012wf,Daub}.
For the $D$-wave, the single-channel FSI will
be taken into account. In the elastic $\pi\pi$ rescattering region,
the partial-wave unitarity condition reads
\begin{equation}\label{eq.unitarity1channel}
\textrm{Im}\, M_2(s)= \left[M_2(s)+\hat{M}_2(s)\right]
\sin\delta_2^0(s) e^{-i\delta_2^0(s)}\,,
\end{equation}
where the phase of the isoscalar $D$-wave amplitude
$\delta_2^0$ coincides with
the $\pi\pi$ elastic phase shift, as required by
Watson's theorem~\cite{Watson1,Watson2}.
The modified Omn\`es solution of Eq.~\eqref{eq.unitarity1channel}
can be obtained as~\cite{Leutwyler96,Chen2016}
\be\label{OmnesSolution1channel}
M_2(s)=\Omega_2^0(s)\bigg\{P_2^{n-1}(s)+\frac{s^n}{\pi}\int_{4m_\pi^2}^\infty
\frac{\diff x}{x^n} \frac{\hat
M_2(x)\sin\delta_2^0(x)}{|\Omega_2^0(x)|(x-s)}\bigg\} \,, \ee
where the polynomial $P_2^{n-1}(s)$ is a subtraction function, and the
Omn\`es function is defined as~\cite{Omnes}
\begin{equation}\label{Omnesrepresentation}
\Omega_2^0(s)=\exp
\bigg\{\frac{s}{\pi}\int^\infty_{4m_\pi^2}\frac{\diff x}{x}
\frac{\delta_2^0(x)}{x-s}\bigg\}\,.
\end{equation}
We will use the result given by the Madrid--Krak\'ow group~\cite{Pelaez} for $\delta_2^0(s)$, which is smoothly continued to $\pi$ for $s\to\infty$.
In order to determine the necessary number of subtractions that
guarantees the convergence of the dispersive integrals in
Eqs.~\eqref{OmnesSolution2channel} and \eqref{OmnesSolution1channel}, we need to
investigate the high-energy behavior of the integrands. First, it is
known that for a phase shift $\delta_l^I(s)$ approaching $k\,\pi$ at
high energies, the corresponding single-channel Omn\`es function
falls asymptotically as $s^{-k}$. As a consequence, we have $\Omega_{2}^0(s) \sim 1/s$ at large $s$. Furthermore, the
coupled-channel Omn\`es function $\Omega_l^I(s)$ is found to fall
asymptotically as $1/s$ for large $s$~\cite{Moussallam2000}, provided the asymptotic condition $\sum
\delta_l^I(s) \geq 2\pi$ for $s\to\infty$, where $\sum
\delta_l^I(s)$ is the sum of the eigenphase shifts. Second, we have
checked that in the intermediate energy region of $1\GeV^2 \lesssim s
\ll M_{Y(4260)}^2$, the inhomogeneity contributed by the
$Z_c$-exchange and the triangle diagrams grows at most linearly in $s$. So we conclude
that in the dispersive representations of
Eqs.~\eqref{OmnesSolution2channel} and \eqref{OmnesSolution1channel}, three subtractions for each of them are sufficient
to make the dispersive integrals convergent. On the other hand, at
low energies the amplitudes $\vec{M}_0(s)$ and $M_2(s)$ should match to those from $\chi$EFT. Namely, in the limit of
switching off the FSI at $s=0$, $\Omega(0)=\mathbbm{1}$ and $\Omega_2^0(0)=1$,
the subtraction terms should agree well with the low-energy chiral amplitudes
given in Eq.~\eqref{eq.M0+2Pi+Kchiral}.
Therefore, for the
$S$-wave, the integral equation takes the form
\begin{equation}\label{M02channel}
\vec{M}_0(s)=\Omega(s)\bigg\{\vec{M}_0^{\chi}(s)+\frac{s^3}{\pi}\int_{4m_\pi^2}^\infty
\frac{\diff
x}{x^3}\frac{\Omega^{-1}(x)T(x)\Sigma(x)\hat{\vec{M}}_0(x)}{x-s}\bigg\}
\,,
\end{equation}
where $ \vec{M}^{\chi}_0(s)=\big(
M_0^{\chi,\pi}(s),
2/\sqrt{3}\,M_0^{\chi,K}(s)
\big)^{T}$, while
for the $D$-wave, it can be written as
\be\label{M21channel}
M_2^\pi(s)=\Omega_2^0(s)\bigg\{M_2^{\chi,\pi}(s)+\frac{s^3}{\pi}\int_{4m_\pi^2}^\infty
\frac{\diff x}{x^3} \frac{\hat
M_2^\pi(x)\sin\delta_2^0(x)}{|\Omega_2^0(x)|(x-s)}\bigg\} \,.
\ee
The amplitude for $Y(4260) \to J/\psi \pi^+\pi^-$
can be expressed in terms of the ingredients discussed above as
\begin{equation}
M^{\text{decay}}(s,\cos\theta) =
M_0^\pi(s)+\hat{M}_0^\pi(s)+\left[M_2^\pi(s)+\hat{M}_2^\pi(s)\right]
P_2(\cos\theta)\,.\label{eq.DecayAmplitude}
\end{equation}
The polarization-averaged modulus-square of the $e^+e^- \to Y(4260)
\to J/\psi \pi^+\pi^-$ amplitude can be written as
\begin{equation}
|\bar{M}(E^2,s,\cos\theta)|^2 = \frac{4\pi\alpha
c_\gamma^2|M^{\text{decay}}(s,\cos\theta)|^2}{3|E^2-M_Y^2+iM_Y\Gamma_Y|^2
M_\psi^2}\left[ 8 M_\psi^2 E^2+(s-E^2-M_\psi^2)^2
\right],\label{eq.eetoJpsipipiAmplitudeSquar}
\end{equation}
where $E$ is the center-of-mass energy of the
$e^+e^-$ collisions, and we set the $\gamma^\ast Y(4260)$ coupling
constant $c_\gamma$ to 1 since it can be absorbed into the overall
normalization when we fit to the event distributions. Here we use the energy-independent width for the
$Y(4260)$, and the values of the $Y(4260)$ mass and width are taken as $4222\MeV$ and $44.1\MeV$, respectively, which are the central values of the BESIII fit in Ref.~\cite{Ablikim:2016qzw}. We also have tried to allow the
mass and width to float freely, and found
that the fit quality changes only slightly.
At last, the $\pi\pi$ invariant mass distribution of $e^+e^- \to J/\psi \pi^+\pi^-$ reads
\begin{equation}
\frac{\diff\sigma}{\diff m_{\pi\pi}} =\int_{-1}^1
\frac{|\bar{M}(E^2,s,\cos\theta)|^2
|\vec{k_3^\ast}||\vec{k_5}|}{128\pi^3 |\vec{k_1}|E^2}\diff
\cos\theta\,,\label{eq.pipimassdistribution}
\end{equation}
where $\vec{k_1}$ and $\vec{k_5}$ denote the 3-momenta of $e^\pm$ and $J/\psi$ in the
center-of-mass frame, respectively, and $\vec{k_3^\ast}$ is the 3-momenta of $\pi^\pm$ in the rest frame of the
$\pi\pi$ system. They are given as
\begin{equation}
|\vec{k_1}|=\frac{E}{2}\,, \quad
|\vec{k_3^\ast}|=\frac{1}{2}\sqrt{s-4m_\pi^2}\,, \quad
|\vec{k_5}|=\frac{1}{2E} \lambda^{1/2}\big(E^2,s,M_\psi^2\big) \,.
\end{equation}
For $e^+e^- \to Y(4260) \to J/\psi K^+ K^-$, the relevant Feynman diagrams can be obtained by
replacing all external pions by kaons in
Fig.~\ref{fig.FeynmanDiagram} (for (c1), the exchanged $D^*$ needs to be replaced by $D_s^*$ ), but without diagram~(b1) due to the
absence of the $Z_c \psi K$ vertex.
Most ingredients of the amplitude
of $e^+e^- \to Y(4260) \to J/\psi K^+ K^-$ have
been given in the above.
\section{Phenomenological discussion}\label{pheno}
\subsection{Characteristics of singlet and octet contributions}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Mpipi_hiovergi}
\caption{The shapes of the $\pi\pi$ invariant mass
spectra contributed from the singlet (left) and octet (right)
chiral contact terms using the DP (top) or the BO (bottom) parametrizations. The black solid, magenta dash-dot-dotted, red dot-dashed,
blue dashed, and green dotted lines correspond to the contributions
with $h_i/g_i$ ($i=1,8$) fixed at 0.1, 0.3, 1, 3, and 10, respectively. For the normalizations we set the highest point to be 1 for each group. }\label{fig.Mpipi_hiovergi}
\end{figure}
The two pions in the final state must come from light-flavor sources. It is instructive to discuss what would be expected for the dipion invariant mass distributions produced from pure SU(3) flavor singlet and octet sources, which are proportional to $(\bar u u + \bar d d + \bar s s)/\sqrt{3}$ and $(\bar u u + \bar d d - 2\bar s s)/\sqrt{6}$, respectively, without considering the left-hand-cut contribution. It is well known that the nonstrange and strange scalar pion form factors, $\langle 0 |(\bar u u+\bar d d)|\pi^+\pi^-\rangle$ and $\langle 0 |s\bar s|\pi^+\pi^-\rangle$, behave very differently. The former has a broad bump around $0.5\GeV$, and has a narrow dip at around $1\GeV$, while the latter has a narrow peak at around $1\GeV$. The narrow structures are manifestations of the scalar meson $f_0(980)$, which couples differently to the nonstrange and strange sources~\cite{Daub:2012mu,Daub}. It is therefore natural to expect that the SU(3) singlet and octet pion scalar form factors should also be dramatically different.
To demonstrate the characteristic structures in the dipion mass spectrum from the singlet and octet sources for the current problem, we need to take into account the energy dependence in the chiral contact terms. Their contributions are separately shown with varying $h_i/g_i$ in Fig.~\ref{fig.Mpipi_hiovergi}.
We consider a large range for the ratio $h_i/g_i$ ($i=1,8$). The black solid, magenta dash-dot-dotted, red dot-dashed,
blue dashed, and green dotted curves in the figure correspond to the ratio taking values of 0.1, 0.3, 1, 3, and 10, respectively. For an easy comparison, the maxima of the curves in each plot are normalized to 1.
One observes that the basic characteristic structures of both the singlet and octet spectra are stable against the variation of $h_i/g_i$: the singlet spectra display a broad bump below $1\GeV$, and around $1\GeV$ there is a dip for $h_1/g_1\lesssim 1$; the octet spectra have little contribution below $0.9\GeV$, and show a sharp peak around $1\GeV$, corresponding to the $f_0(980)$.
It is also worthwhile to notice that both of them have different behaviors from both the nonstrange and the strange pion scalar form factors.
Therefore, one expects that precise measurements of the dipion invariant mass distributions can provide valuable information about the light-quark content of the source, considering the $J/\psi$ to be a SU(3) flavor singlet.
\subsection{Fitting to the BESIII data}
In this work we perform fits taking into account the experimental data sets of the $\pi\pi$
invariant mass distributions of $e^+e^- \to J/\psi
\pi^+\pi^-$ and the ratios of the cross sections ${\sigma(e^+e^- \to J/\psi
K^+ K^-)}/{\sigma(e^+e^- \to J/\psi \pi^+\pi^-)}$ measured at two energy points
$E=4.23\GeV$ and $E=4.26\GeV$ by the BESIII
Collaboration~\cite{Collaboration:2017njt,Ablikim:2018epj}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Mpipi}
\caption{Fit results of the $\pi\pi$ invariant mass spectra in
$e^+e^- \to J/\psi \pi^+\pi^-$ for Fits~Ia (top left),
Ib (top right), IIa (bottom left), and IIb (bottom right).
The borders of the bands represent our best fit results using two different $T_0^0(s)$ matrices.
The background-subtracted and efficiency-corrected experimental data are taken from Ref.~\cite{Collaboration:2017njt}.
}\label{fig.Mpipi}
\end{figure}
As in Refs.~\cite{Collaboration:2017njt,Pilloni:2016obd}, we regard the measurements at $E=4.23\GeV$ and $E=4.26\GeV$ as independent, and thus the coupling constants are allowed to be
different in the fits of these two data sets.
For the normalization factor for each dataset,
we choose to absorb it into the coupling constants.
There are six free parameters in our fits: $g_{1,8}$, $h_{1,8}$, $C_{Y\psi}^{Z_c}$, and $C_{Y\psi}^{\text{loop}}$. The parameters $g_1$ and $h_1$
correspond to the low-energy constants in the $Y\psi\Phi\Phi$ Lagrangian in Eq.~\eqref{LagrangianYpsipipi} for the SU(3) singlet component of the $Y(4260)$, $g_8$ and $h_8$ are the corresponding parameters for the SU(3) octet component. $C_{Y\psi}^{Z_c}$ and $C_{Y\psi}^{\text{loop}}$ are related to the
$Z_c$-exchange contribution\footnote{
The parameter $C_{Y\psi}^{Z_c}$, as a product of the $YZ_c\pi$ and $Z_c\psi\pi$ couplings, is related to the partial widths of the $Y\to Z_c\pi$ and $Z_c\to J/\psi\pi$. In principle, it can be determined from a thorough analysis of the $Z_c$ and $Y$ line shapes; such an analysis that takes into account the $\pi\pi$ FSI is not available yet. Thus, here we make a compromise by focusing on the $\pi\pi$ distribution and taking $C_{Y\psi}^{Z_c}$ as a free parameter.} and triangle-diagram contribution, respectively.
To illustrate the effect of the SU(3) octet
component, we perform two fits for each data set (Fits~Ia
and Ib for $E=4.23\GeV$, and Fits~IIa and IIb for
$E=4.26\GeV$). To be specific, in Fits~Ia and IIa
we only consider the SU(3) singlet component, the
$Z_c$-exchange terms, and the triangle diagrams, while in Fits~Ib and IIb,
the SU(3) octet components are taken into account in addition.
The coupled-channel FSI is considered in all the fits.
\begin{table}
\caption{\label{table-ratioes} Experimental and theoretical
values for the cross sections ratios ${\sigma(e^+e^- \to J/\psi K^+ K^-)}/{\sigma(e^+e^- \to J/\psi\pi^+\pi^-)}\times 10^{2}$. The experimental data are taken
from Ref.~\cite{Ablikim:2018epj}. The theoretical results are obtained with two different $T_0^0(s)$ matrices (DP vs.\ BO).}
\renewcommand{\arraystretch}{1.2}
\begin{center}
\begin{tabular}{l|ccccc}
\toprule
&Experiment & Fit~Ia, DP & Fit~Ib, DP& Fit~Ia, BO & Fit~Ib, BO \\
\hline
$\frac{{\sigma( J/\psi
K^+ K^-)}}{{\sigma( J/\psi
\pi^+\pi^-)}} \times 10^{2}, E=4.23\GeV$ & $6.44\pm 1.15$ & $7.82\pm 0.83$& $7.75\pm 1.10$& $5.88\pm 0.82$& $2.83\pm 1.05$ \\
\toprule
&Experiment & Fit~IIa, DP & Fit~IIb, DP& Fit~IIa, BO & Fit~IIb, BO \\
\hline
$\frac{{\sigma( J/\psi
K^+ K^-)}}{{\sigma( J/\psi
\pi^+\pi^-)}} \times 10^{2} , E=4.26\GeV$ & $4.99\pm 1.10$ & $4.46\pm 0.82$& $4.67\pm 0.98$& $5.37\pm 1.03$& $5.38\pm 0.82$ \\
\botrule
\end{tabular}
\end{center}
\renewcommand{\arraystretch}{1.0}
\end{table}
\begin{table}
\caption{\label{tablepar1} Fit parameters from the best fits of
the $\pi\pi$ mass spectrum in $e^+e^- \to J/\psi \pi^+\pi^-$ and the ratios
${\sigma(e^+e^- \to J/\psi K^+ K^-)}/{\sigma(e^+e^- \to J/\psi \pi^+\pi^-)}$ at
$E=4.23\GeV$ (Fit~Ia and Ib) and $E=4.26\GeV$
(Fit~IIa, IIb, IIc, and IId), respectively, using the DP $T$-matrix parametrization.}
\renewcommand{\arraystretch}{1.2}
\begin{center}
\begin{tabular}{l|cccccc}
\toprule
& Fit~Ia, DP
& Fit~Ib, DP
& Fit~IIa, DP
& Fit~IIb, DP
& Fit~IIc, DP
& Fit~IId, DP\\
\hline
$g_1~[\text{GeV}^{-1}]$ & $ -0.29\pm 0.04$ & $ 1.87\pm 0.13$ & $ 0.21\pm 0.04$ & $ -0.99\pm 0.11$ & $ 0.52\pm 0.02$& $ 0.20\pm 0.08$\\
$h_1~[\text{GeV}^{-1}]$ & $ -0.29\pm 0.02$ & $ -0.31\pm 0.06$ & $ -0.32\pm 0.02$ & $ 0.03\pm 0.04$ & $ 0.02\pm 0.01$& $ 0.09\pm 0.04$ \\
$g_8~[\text{GeV}^{-1}]$ & 0 (\text{fixed}) & $1.25\pm 0.11$ & 0 (\text{fixed}) & $ -1.18\pm 0.03$ & 0 (\text{fixed})& $ 1.01\pm 0.10$ \\
$h_8~[\text{GeV}^{-1}]$ & 0 (\text{fixed}) & $-1.96\pm 0.10$ & 0 (\text{fixed}) & $ 1.70\pm 0.18$ & 0 (\text{fixed})& $ -1.28\pm 0.08$\\
$C_{Y\Psi}^{Z_c}\times 10^{2}$ & $ 0.7\pm 0.6$ & $ 2.0\pm 0.8$ & $ 4.6\pm 0.3$ & $ 6.9\pm 0.3$ & 0 (\text{fixed})& 0 (\text{fixed})\\
$C_{Y\Psi}^{\text{loop}}~[\text{GeV}^{-3}]$ & $ 4.5\pm 1.0$ & $ 38.8\pm 2.5$ & $ 12.5\pm 0.8$ & $ -19.4\pm 2.1$ & 0 (\text{fixed})& 0 (\text{fixed})\\
\hline
${\chi^2}/{\rm d.o.f.}$ & $\frac{405.1}{(44-4)}=10.13$ & $\frac{102.1}{(44-6)}=2.69$ & $\frac{182.7}{(46-4)}=4.35$ & $\frac{63.9}{(46-6)}=1.60$ & $\frac{428.9}{(46-2)}=9.75$ & $\frac{148.2}{(46-4)}=3.53$ \\
\botrule
\end{tabular}
\end{center}
\renewcommand{\arraystretch}{1.0}
\end{table}
\begin{table}
\caption{\label{tablepar2} Fit parameters from the best fits of
the $\pi\pi$ mass spectrum in $e^+e^- \to J/\psi \pi^+\pi^-$ and the ratios
${\sigma(e^+e^- \to J/\psi K^+ K^-)}/{\sigma(e^+e^- \to J/\psi \pi^+\pi^-)}$ at
$E=4.23\GeV$ (Fit~Ia and Ib) and $E=4.26\GeV$ (Fit~IIa, IIb, IIc, and IId), respectively,
using the BO $T$-matrix.}
\renewcommand{\arraystretch}{1.2}
\begin{center}
\begin{tabular}{l|cccccc}
\toprule
& Fit~Ia, BO
& Fit~Ib, BO
& Fit~IIa, BO
& Fit~IIb, BO
& Fit~IIc, BO
& Fit~IId, BO\\
\hline
$g_1~[\text{GeV}^{-1}]$ & $ -0.20\pm 0.04$ & $ 1.34\pm 0.08$ & $ 0.30\pm 0.04$ & $ -1.24\pm 0.05$& $ 0.57\pm 0.02$& $ 0.32\pm 0.11$ \\
$h_1~[\text{GeV}^{-1}]$ & $ -0.32\pm 0.02$ & $ -0.07\pm 0.03$ & $ -0.35\pm 0.01$ & $ 0.02\pm 0.03$& $ -0.02\pm 0.01$& $ -0.01\pm 0.06$ \\
$g_8~[\text{GeV}^{-1}]$ & 0 (\text{fixed}) & $1.65\pm 0.15$ & 0 (\text{fixed}) & $ -1.31\pm 0.05$& 0 (\text{fixed})& $ 0.85\pm 0.12$ \\
$h_8~[\text{GeV}^{-1}]$ & 0 (\text{fixed}) & $-2.37\pm 0.02$ & 0 (\text{fixed}) & $ 2.03\pm 0.06$& 0 (\text{fixed})& $ -1.14\pm 0.11$ \\
$C_{Y\Psi}^{Z_c} \times 10^{2}$ & $ 6.3\pm 0.6$ & $ 3.4\pm 0.7$& $ 6.5\pm 0.2$ & $ 8.0\pm 0.2$ & 0 (\text{fixed})& 0 (\text{fixed})\\
$C_{Y\Psi}^{\text{loop}} ~[\text{GeV}^{-3}]$ & $ 8.0\pm 0.8$ & $ 40.9\pm 3.6$ & $8.7\pm 1.0$ & $-34.0\pm 1.9$ & 0 (\text{fixed})& 0 (\text{fixed})\\
\hline
${\chi^2}/{\rm d.o.f.}$ & $\frac{308.7}{(44-4)}=7.72$ & $\frac{121.4}{(44-6)}=3.19$ & $\frac{170.4}{(46-4)}=4.06$ & $\frac{94.3}{(46-6)}=2.36$ & $\frac{446.5}{(46-2)}=10.15$& $\frac{176.7}{(46-4)}=4.21$ \\
\botrule
\end{tabular}
\end{center}
\renewcommand{\arraystretch}{1.0}
\end{table}
The uncertainty due to the
dispersive input for the $\pi\pi/K\bar{K}$ rescattering is estimated by comparing the fits with the two different $T_0^0(s)$ matrices (DP~\cite{Dai:2014lza,Dai:2014zta,Dai:2016ytz} vs.\ BO~\cite{Leutwyler2012,Moussallam2004}).
In Fig.~\ref{fig.Mpipi}, the best fit results of the $\pi\pi$ mass spectrum in $e^+e^- \to J/\psi \pi^+\pi^-$
are shown, where the borders of the bands represent the fit results using these two different $T_0^0(s)$ matrix parametrizations. The fit results of the ratios of the cross sections ${\sigma(e^+e^- \to J/\psi K^+ K^-)}/{\sigma(e^+e^- \to J/\psi \pi^+\pi^-)}$ are given in Table~\ref{table-ratioes}. The fitted parameters as well as the $\chi^2/\text{d.o.f.}$
are shown in Tables~\ref{tablepar1} and~\ref{tablepar2} for the DP and BO parametrizations, respectively.
As can be seen from Fig.~\ref{fig.Mpipi} as well as Tables~\ref{tablepar1} and~\ref{tablepar2},
the fit quality to the data set at $E=4.23\GeV$ is
worse than that at $E=4.26\GeV$, in particular in the region close to the lower kinematical boundary and for the highest data point.
Notice that by using the inputs from known scattering observables in the dispersion relations, the effects of resonances in the considered partial waves, i.e.\ the $f_0(500)$, $f_0(980)$, and $f_2(1270)$, are included automatically. Since the dataset at $E=4.26\GeV$ has a larger phase space to reveal the
nontrivial structure and the fits are better, we discuss the fit results of
this data set in more details.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Mpipi_4260_NoLefthandcuts}
\caption{Fit results of the $\pi\pi$ invariant mass spectra in
$e^+e^- \to J/\psi \pi^+\pi^-$ for Fits~IIc (left) and IId (right). The borders of the bands represent our best fit results using two different $T_0^0(s)$ matrices. The background-subtracted and efficiency-corrected experimental data are taken from Ref.~\cite{Collaboration:2017njt}. }\label{fig.Mpipi_4260_NoLefthandcuts}
\end{figure}
It is interesting to compare Fits~IIa and IIb. In Fit~IIa, the
SU(3) octet chiral contact terms are not included. The experimental
data, especially the broad peak in the region lower than $0.6\GeV$, cannot be described well.
In contrast, in Fit~IIb, including the SU(3) octet chiral
contact terms, the fit quality is improved significantly. A similar improvement is also observed comparing Fits~Ib and Ia.
We also perform two further Fits~IIc and IId for the $E=4.26\GeV$ dataset, considering only the contact terms and switching off the left-hand cuts: in Fit~IIc we only retain the SU(3) singlet component, while in Fit~IId, both the SU(3) singlet and octet components are taken into account. The result is shown in Fig.~\ref{fig.Mpipi_4260_NoLefthandcuts}, and the fit couplings are also listed in Table~\ref{tablepar1}. Comparing Fits~IIc and IId, one also finds that adding the SU(3) octet component increases the fit quality significantly.
It is instructive
to analyze the ratio of the parameters for the SU(3) octet component relative to those for
the SU(3) singlet component.
Using the results of Fit~IIb as shown in Tables~\ref{tablepar1} and~\ref{tablepar2}, we have $g_8/g_1=1.2 \pm 0.2$ and $h_8/h_1=57\pm 76$ in the DP parametrization and $g_8/g_1=1.1 \pm 0.1$ and $h_8/h_1=102\pm 152$ in the BO one,
which agree well with each other within errors. Note that $h_8/h_1$ is not as stable as $g_8/g_1$: the reason is that $h_1$ is small in most fits. In the $\bar{D}D_1$
hadronic molecule scenario of $Y(4260)$, one has
\be
|Y(4260)\rangle=\frac{1}{2}\big[|D_1^0 \bar{D}^0\rangle+|D_1^+
D^-\rangle\big]+\mathrm{c.c.}\,,
\ee
from which the light-quark component reads $|u \bar{u}+d\bar{d}\rangle/\sqrt{2} = (\sqrt{2}
V_1^{\text{light}} +V_8^{\text{light}})/\sqrt{3}$, where the definitions of
the singlet and octet components $V_1^{\text{light}}$
and $V_8^{\text{light}}$ have been given below
Eq.~\eqref{eq.YComponents}. They thus give the ratio of $1/\sqrt{2}$.
Certainly our results (values of $g_8/g_1$) differ significantly from the result of the pure $\bar{D}D_1$
hadronic molecule scenario. In addition to the
$\bar{D}D_1$ hadronic molecule, the $Y(4260)$ may contain other
SU(3) singlet sources, e.g., from $|c\bar{c}\rangle$ or a hybrid. Assuming in
the transition $Y\to\psi\Phi\Phi$ the strengths of the light-quark
components from the $\bar{D}D_1$ hadronic molecule and the other SU(3)
singlet source are $\alpha$ and $\beta$, respectively, namely,\footnote{Notice that any isoscalar pair of nonstrange charm and anticharm mesons has the same SU(3) structure.}
\be
\frac{\alpha}{\sqrt{3}}\Big(\sqrt{2}V_1^{\text{light}}
+V_8^{\text{light}}\Big)+\beta\, V_1^{\text{light}}\,,
\ee
we can
estimate the ratio of $\beta/\alpha=-0.30\pm 0.05$ based on our
results of $g_8/g_1$. Thus we conclude that there is a large
light-quark SU(3) octet component in the $Y(4260)$, and
scenarios of a hybrid or conventional charmonium are disfavored
since the light quarks have to be produced in the SU(3)
singlet state in such states. Also our study shows that the $\bar D D_1$ component of the $Y(4260)$ may not be completely dominant.
This is not unnatural, as the $Y(4260)$ mass, being around $4.22\GeV$, is about $70\MeV$ below the $\bar D D_1$ threshold.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Moduli}
\caption{The moduli of the $S$- (left) and $D$-wave (right)
amplitudes for $e^+e^- \to J/\psi \pi^+\pi^-$ in
Fit~IIb, using the DP (top) or the BO (bottom) parametrizations. The red solid lines represent our best fit results,
while the blue dot-dashed, darker green dashed,
and magenta dotted lines correspond to the contributions
from the chiral contact terms, $Z_c$-exchange, and the triangle diagrams, respectively.
}
\label{fig.Moduli}
\end{figure}
In Fig.~\ref{fig.Moduli}, we plot the moduli of the $S$- and
$D$-wave amplitudes from the chiral contact terms, the $Z_c$-exchange terms, and the triangle diagrams for Fit~IIb.
An interesting feature is that the $D$-wave contribution is comparable
to the $S$-wave contribution in almost the whole phase space. Such a large $D$-wave
contribution in the $Y\psi \Phi\Phi$ transition again indicates that the $Y(4260)$ cannot be a conventional charmonium state, for which the $\pi\pi$ $S$-wave should be dominant.
Notice that in the $\bar D D_1$ hadronic molecule interpretation~\cite{Cleven:2013mka,Lu:2017yhl}, the $\pi\pi$ $D$-wave emerges naturally since the $D_1$ decays dominantly into $D$-wave $D^*\pi$.
Also one observes that the contributions from the chiral contact terms and the l.h.c.\ contributions are of the same order. Amongst the l.h.c.\ contributions, both the $Z_c$ term and the triangle diagrams appear far from negligible. A better distinction of the effects of the
$Z_c$ and the open-charm loops requires a detailed analysis of the $J/\psi\pi$ distribution and is beyond the scope of the present paper.
\section{Conclusions}
\label{conclu}
We have used dispersion theory to study the processes
$e^+e^-\to Y(4260) \to J/\psi \pi\pi(K\bar{K})$. In particular, we have analyzed the roles of the
light-quark SU(3) singlet state and SU(3) octet state in these
transitions. The strong FSI, especially the coupled-channel ($\pi\pi$ and $K\bar{K}$) FSI in
the $S$-wave, has been considered in a model-independent way, and
the leading chiral amplitude acts as the subtraction function in the
modified Omn\`es solution.
Through fitting to the data of the $\pi\pi$
invariant mass spectra of $e^+e^-\to Y(4260) \to J/\psi \pi\pi$
and the ratios of the cross sections ${\sigma(e^+e^- \to J/\psi K^+ K^-)}/{\sigma(e^+e^- \to J/\psi \pi^+\pi^-)}$,
we find that the light-quark SU(3) octet state plays a
significant role in the $Y(4260)J/\psi\Phi\Phi$ transition, which
indicates that the $Y(4260)$ contains a large light-quark component.
Thus we conclude that the $Y(4260)$ is in all likelihood neither a hybrid nor a
conventional charmonium state. Furthermore, through an analysis of the ratio
of the light-quark SU(3) octet and singlet
components, we show that the $Y(4260)$ does not behave like a pure $\bar D D_1$ hadronic
molecule. We also find that there is a large $D$-wave component in the $\pi\pi$ invariant spectrum of the $Y(4260)$.
We close this manuscript by anticipating a combined analysis of both the $Y(4260)$ and $Z_c(3900)$ data. Such an analysis is a necessary step toward revealing the nature of both states, as there is evidence that the $Z_c(3900)$ events in the $J/\psi\pi\pi$ are only produced when the latter is constrained in the $Y(4260)$ region~\cite{Abazov:2018cyu}.
\section*{Acknowledgments}
We acknowledge Rong-Gang Ping for helpful discussions on the experimental analyses and for kindly providing us with the efficiency-corrected data, and Christoph Hanhart for
a careful reading of the manuscript and valuable comments and suggestions.
This work is supported in part by the Fundamental Research Funds
for the Central Universities under Grants No.~531107051122 and No.~06500077, by the National Natural Science Foundation of China (NSFC) under Grants No.~11805059, No.~11747601, and No.~11835015, by NSFC and Deutsche Forschungsgemeinschaft (DFG) through
funds provided to the Sino--German Collaborative Research Center ``Symmetries and the
Emergence of Structure in QCD'' (NSFC Grant No.~11621131001,
DFG Grant No.~TRR110), by the Thousand Talents Plan for Young
Professionals, by the CAS Key Research Program of Frontier Sciences
(Grant No.~QYZDB-SSW-SYS013), by the CAS Key Research Program (Grant No.~XDPB09), and by the CAS Center for Excellence in Particle Physics (CCEPP).
|
{
"timestamp": "2019-04-19T02:08:36",
"yymm": "1902",
"arxiv_id": "1902.10957",
"language": "en",
"url": "https://arxiv.org/abs/1902.10957"
}
|
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\begin{document}
\maketitle
\vspace*{-5mm}
\begin{abstract}
{\small This paper studies high-order evaluation complexity for partially separable convexly-constrained optimization involving non-Lipschitzian group sparsity terms in a nonconvex objective function. We propose a partially separable adaptive regularization algorithm using a $p$-th order Taylor model and show that the algorithm can produce an
$(\epsilon,\delta)$-approximate $q$-th-order stationary point
at most
$O(\epsilon^{-(p+1)/(p-q+1)})$ evaluations of the objective function and its first $p$
derivatives (whenever they exist).
Our model uses the underlying rotational symmetry of the Euclidean norm function to build a
Lipschitzian approximation for the non-Lipschitzian group sparsity terms, which are defined by the group $\ell_2$-$\ell_a$ norm with $a\in (0,1)$. The new
result shows that the partially-separable structure and non-Lipschitzian group sparsity terms in the objective
function may not affect the worst-case evaluation complexity order. }
\end{abstract}
{\small
\textbf{Keywords:} complexity theory,
nonlinear optimization,
non-Lipschitz functions,
partially-separable problems, group sparsity, isotropic model.
}
{\small
\textbf{AMS subject classifications:}, 90C30, 90C46, 65K05
}
{\footnotesize \numsection{Introduction}}
Both applied mathematicians and computer scientists have, in recent years,
made significant contributions to the fast-growing field of worst-case
complexity analysis for nonconvex optimization (see \cite{CartGoulToin18a} for
a partial yet substantial bibliography). The purpose of this paper is to
extend the available general theory in two distinct directions. The first is
to cover the case where the problem involving non-Lipschitzian group sparsity terms.
The second is to show that the ubiquitous
partially-separable structure (of which standard sparsity is a special case)
can be exploited without affecting the complexity bounds.
We consider the partially-separable convexly constrained nonlinear optimization
problem:
\begin{equation}
\label{problem}
\min_{x \in \calF} f(x)
= \sum_{i \in \calN} f_i(U_ix) + \sum_{i \in \calH} \|U_ix-b_i\|^a
\eqdef \sum_{i \in \calN \cup \calH} f_i(U_ix)
\end{equation}
where
$\calN\cup\calH \eqdef \calM$,
$\calN\cap\calH = \emptyset$,
$f_i$ is a continuously $p$ times differentiable function from $\Re^{n_i}$
into $\Re$ for $i\in \calN$, and $f_i(x)=\|U_ix-b_i\|^a$ for $i\in \calH$,
$a \in (0,1)$,
$\|\cdot\|$ is the Euclidean norm,
$U_i\in \Re^{n_i \times n}$ with $n_i \leq n$,
and $b_i\in \Re^{n_i}$.
Without loss of generality, we assume that, for each $i \in \calM$, $U_i$ has
full row rank and $\|U_i\| = 1$, and that the ranges of the $U_i^T$ for $i\in
\calN$ span $\Re^n$ so that the intersection
of the nullspaces of the $U_i$ is reduced to the origin.
We also assume that the ranges of the $U_i^T$ (for $i
\in \calH$) are orthogonal, that is
\beqn{Ui-ortho-H}
U_iU_j^T = 0 \tim{for} i\neq j,\, i,j\in \calH.
\eeqn
Without loss of generality, we furthermore assume that the rows of $U_i$ are
orthonormal for $i \in \calH$. Our final assumption, as in
\cite{ChenToinWang17}, is that the feasible set $\calF\subseteq \Re^n$ is
non-empty closed and convex, and that it is ``kernel-centered'' in that
is, if $P_\calX[\cdot]$ is the orthogonal projection onto the convex set
$\calX$ and $^{\dagger}$ denotes the Moore-Penrose generalized inverse,
then
\beqn{kernel-centered}
U_i^\dagger b_i + P_{\ker(U_i)}[\calF] \subseteq \calF
\tim{whenever} b_i \in U_i\calF, \quad i\in \calH.
\eeqn
These assumptions do not restrict our study for applications. For example, consider
the row sparse problem in multivariate regression \cite{huang2009,huang2010benefit,obozinski2011}
\begin{equation}\label{Prob4}
\displaystyle \min_{X\in R^{\nu\times \gamma}} \, \|HX-B\|^2_F +\lambda \|X\|_{\ell_a/\ell_2},
\end{equation}
where $H\in \Re^{\kappa\times \nu}, B \in \Re^{\kappa\times \gamma}$,$
\|\cdot\|_F$
is the Frobenius norm of a matrix,
\[
\|HX-B\|_F^2
= \sum_{j=1}^{\gamma}\sum^\kappa_{i=1} (\sum_{\ell=1}^\nu H_{i\ell} X_{\ell, j}-B_{ij})^2
\quad {\rm and} \quad
\|X\|_{\ell_a/\ell_2}=\sum^\nu_{i=1}\big(\sum^\gamma_{j=1} X_{ij}^2\big)^{\frac{a}{2}}.
\]
Let $n=\nu\gamma$, $\calF=\Re^n, b_i=0$,
$x=(x_{11}, x_{12},\ldots, x_{\nu\gamma})^T\in \Re^n$ and set $U_i\in \Re^{\nu\times n}$ for $i\in \calN=\{1,\ldots, \gamma\}$
be the projection whose entries are 0, or 1 such that $U_ix$ be the
$i$th column of $X$ and $U_i\in \Re^{\gamma\times n}$ for $i\in \calH=\{1,\ldots, \nu\}$ be the
projection whose entries are 0, or 1 such that $U_ix$ be the $i$th row of $X$. Then problem \req{Prob4} can be written in the form of
(\ref{problem}). It is easy to see that the $\{U_i^T\}_{i \in \calN}$ span $\Re^n$. Hence, all assumptions mentioned above hold
for problem \req{Prob4}.
Problem (\ref{problem}) encompasses the non-overlapping group sparse
optimization problems. Let $G_1, \ldots, G_m$ be subsets of $\{1,\ldots, n\}$
representing known groupings of the decision variable with size $n_1,\ldots,
n_m$ and $G_i\cap G_j=\emptyset, i\neq j$. In this case, problem
\req{problem} reduces to
\begin{equation}\label{Prob2}
\min_{x \in \calF} {\displaystyle f_1(x) + \lambda \sum^m_{i=1}\|U_ix\|^a},
\end{equation}
where $f_1:\Re^n\to \Re_+$ is a smooth loss function, $\lambda >0$ is a positive number and
$U_i \in \Re^{n_i\times n}$ is defined in the following way
\[
(U_i)_{kj}=\left\{\begin{array}{ll}
1 & \, {\rm if} \, j\in G_i\\
0 & \, {\rm otherwise}
\end{array} \quad \quad {\rm for} \quad k=1,\ldots, n_i.
\right.
\]
Thus $U_ix=x_{G_i}$ is the $i$th group variable vector in $\Re^{n_i}$ with components
$x_j, j\in G_i.$ If $\calF=\{x \, | \, \alpha_i \le x_i\le \beta_i, i=1,\ldots,n \}$
with $\alpha_i <0 <\beta_i$, then all assumptions mentioned above with $U_1=I\in \Re^{n\times n}$,
for $1\in \calN$ hold for problem \req{Prob2}.
In problem (\ref{problem}), the decision variables have a group structure so
that components in the same group tend to vanish simultaneously. Group
sparse optimization problems have been extensively studied in recent years due
to numerous applications. In machine learning and statistics, when the
explanatory variables have high correlative nature or can be naturally
grouped, it is important to study variable selection at the group sparsity
setting \cite{breheny2015,huang2009,huang2010benefit,Lee2016,MaHuang,Yuan}.
In compressed sensing, group sparsity is refereed to as block sparsity and has
been efficiently used to recovery signals with special block structures
\cite{Ahsen,Eldar,juditsky2012,Lee2012,Lv}. In spherical harmonic
representations of random fields on the sphere, group Lasso penalty grouped
the coefficients of homogeneous harmonic polynomials of the same degree is
rotationally invariant while Lasso penalty ($G_i=\{i\}$) is not
\cite{gia2018}.
Problem (\ref{problem}) with $a\in (0,1)$ and $n_i=1, i\in \calH$ has been
studied in \cite{Bian_Chen_SIOPT,Bian-Chen-Ye,CXY,CNY,CGWY,Chen_Rob}. Chen,
Toint and Wang \cite{ChenToinWang17} show that an adaptive regularization
algorithm using a $p$-th order Taylor model for $p$ \emph{odd} needs in
general at most $O(\epsilon^{-(p+1)/p})$ evaluations of the objective function
and its derivatives (at points where they are defined) to produce an
$\epsilon$-approximate first order critical point. Since this complexity bound
is identical in order to that already known for convexly constrained
Lipschitzian minimization, the result in \cite{ChenToinWang17} shows that
introducing non-Lipschitzian singularities in the objective function may not
affect the worst-case evaluation complexity order.
The unconstrained optimization of smooth partially-separable was first considered in Griewank and Toint \cite{GrieToin82a}, studied by many researchers
\cite{GoldWang93,Gay96,ChenDengZhan98,MareRichTaka14,ConnGoulSartToin96a,ConnGoulToin00}
and extensively used in the popular
{\sf CUTEst} testing environment \cite{
GoulOrbaToin15b} as well as in the AMPL \cite{FourGayKern87}, {\sf LANCELOT}
\cite{ConnGoulToin92} and {\sf FILTRANE} \cite{GoulToin07b}
packages.
In problem (\ref{problem}), all these ``element
functions'' $f_i$ depend on $U_ix \in \Re^{n_i}$ rather than on $x$, which is most useful when $n_i \ll n$. Letting
$$x_i=U_ix \in \Re^{n_i} \,\, {\rm for} \,\, i\in \calM \quad {\rm and} \quad f_\calI(x)= \sum_{i\in
\calI}f_i(x) \, \, \mbox{for any} \,\, \calI \subseteq \calM,$$
we define
\[
f_\calN(x) \eqdef \sum_{i \in \calN} f_i(U_ix) = \sum_{i \in \calN} f_i(x_i)
\quad \, {\rm and} \quad\,
f_\calH(x) \eqdef \sum_{i \in \calH} f_i(U_ix) = \sum_{i \in \calH} f_i(x_i).
\]
The $p$-th degree Taylor series
\beqn{taylor}
T_{f_\calN,p}(x,s) = f_\calN(x) + \sum_{j=1}^p \frac{1}{j!}\nabla_x^jf_\calN(x)[s]^j,
\tim{where}
\nabla_x^jf_\calN(x)[s]^j
= \sum_{i \in \calN} \nabla_{x_i}^jf_i(x_i)[U_is]^j,
\eeqn
indicates that, for each $j$, only the $|\calN|$ tensors
$\{\nabla_{x_i}^jf_i(x_i)\}_{i\in \calN}$ of dimension $n_i^j$ needs to be computed
and stored. Exploiting derivative tensors of order larger than 2
--- and thus using the high-order Taylor series \req{taylor} as a local model
of $f_{\calN}(x+s)$ in the neighbourhood of $x$ --- may therefore be practically
feasible in our setting since $n_i^j$ is typically orders of magnitude
smaller than $n$. The same comment applies to
$f_\calH(x)$ whenever $\|U_ix-b_i\|\neq 0.$
\comment
Interestingly, the use of high-order Taylor models for optimization was
recently investigated by Birgin \emph{et al.} \cite{BirgGardMartSantToin17}
in the context of adaptive regularization algorithms for unconstrained
problems. Their proposal belongs to this emerging class of methods pioneered
by Griewank \cite{Grie81}, Nesterov and Polyak \cite{NestPoly06} and Cartis,
Gould and Toint \cite{CartGoulToin11d} for the unconstrained
case and by these last authors in \cite{CartGoulToin12b} for the convexly
constrained case of interest here. Such methods are distinguished by their
excellent evaluation complexity, in that they need at most
$O(\epsilon^{-(p+1)/p})$ evaluations of the objective function and their
derivatives to produce an $\epsilon$-approximate first-order critical point,
compared to the $O(\epsilon^{-2})$ evaluations which might be necessary for
the steepest descent and Newton's methods (see
\cite{CartGoulToin10a} for details). However, most adaptive regularization
methods rely on a non-separable regularization term in the model of the
objective function, making exploitation of structure difficult\footnote{The
only exception we are aware of is the unpublished note
\cite{GoulHoggReesScot16} in which a $p$-th order Taylor model is coupled
with a regularization term involving the (totally separable) $r$-th power of
the $r$ norm ($r \geq 1$).}. We note that complexity issues for
non-Lipschitzian problems have already been investigated
\cite{CartGoulToin16,GrapNest17,Mart17}, but the Lipschitz assumption on the
derivatives is then replaced by a (weaker) H\"older condition. Our ambition
here is to assume considerably less, since our purpose is to cover severe
singularities as present in cusps and norms of fractional index, for which
H\"older conditions fail.
}
The main contribution of this paper is twofold.
\begin{itemize}
\item We propose a partially separable adaptive regularization algorithm with
a $p$-th order Taylor model which uses the underlying rotational symmetry
of the Euclidean norm function for $f_\calH$ and the first $p$ derivatives
(whenever they exist) of the ``element functions'' $f_i$, for $i \in \calM$.
\item We show that the algorithm can produce an
$(\epsilon,\delta)$-approximate $q$-th-order critical point of problem
\req{problem} at most $O(\epsilon^{-(p+1)/(p-q+1)})$ evaluations of the
objective function and its first $p$ derivatives for any $q \in \ii{p}$.
\end{itemize}
Our results extend worst-case evaluation complexity bounds for smooth
nonconvex optimization in \cite{CartGoulToin18b,CartGoulToin18a} which
do not use the structure of partially separable functions and do not
consider the Lipschitzian singularity. Moreover, our results subsume
the results for non-Lipschitz nonconvex optimization in \cite{ChenToinWang17}
which only consider the complexity with $q=1$ and $n_i=1$ for $i\in \calH$.
This paper is organized as follows. In Section 2, we define an $(\epsilon,
\delta)$ $q$-order necessary optimality conditions for local minimizers of
problem \req{problem}. A Lipschitz continuous model to approximate $f$ is
proposed in Section~3. We then propose the partially separable adaptive
regularization algorithm using the $p$-th order Taylor model in Section 4. In
Section 5, we show that the algorithm produces an
$(\epsilon,\delta)$-approximate $q$-th-order critical point at most
$O(\epsilon^{-(p+1)/(p-q+1)})$ evaluations of $f$ and its first $p$
derivatives.
\comment
The main purpose of the present paper is to establish that first-order worst-case
evaluation complexity for nonconvex minimization subject to convex constraints
is not affected by the introduction of the non-Lipschitzian singularities in
the objective function \req{problem}. This requires several intermediate
steps. The first is to derive, in Section~\ref{optimality-s}, new first-order
necessary optimality conditions that take the non-Lipschitzian nature of
\req{problem} into account. These conditions motivate the introduction of a new
'two-sided' symmetric model of the singularities which is then exploited in the proposed
algorithm. Because the new necessary conditions involve the gradient of a partial
objective with a number of singular terms itself depending on the approximate
solution (see Theorem~2.1 below), this prevents the aggregation of all terms
in \req{problem} in a single abstracted objective function. As a consequence,
complexity bounds must be derived while preserving the additive
partially-separable structure of the objective function. Our second step is
therefore to show, in Section~\ref{psarp-s}, that first-order worst-case
complexity bounds are not affected by the use of partially-separable
structure. In Section~\ref{complexity-s}, we then specialize our analysis to
a wide class of kernel-centered feasible sets and show that complexity bounds
are again unaffected by the presence of the considered non-Lipschitzian singularities. The
final step is to show in Section~\ref{true-model-s} that (weaker) complexity
results may still be obtained if one considers feasible sets which are not
kernel-centered. All these results are discussed in Section~\ref{discuss-s} and
some conclusions are presented in Section~\ref{concl-s}.
We end this section by introducing notations used in the next four sections.
\noindent
{\bf Notations.}
For a
symmetric tensor $S$ of order $p$,
$S[v]^p$ is the result of applying $S$ to $p$ copies of the vector $v$ and
\beqn{Tnorm}
\|S\|_{[p]} \eqdef \max_{\|v\|=1} | S [v]^p |
\eeqn
is the associated induced norm for such tensors. If $S_1$ and $S_2$ are tensors,
$S_1\otimes S_2$ is their tensor product and $S_1^{k\otimes}$ is the product
of $S_1$ $k$ times with itself. For any set $\calX$, $|\calX|$ denotes its
cardinality.
Because the notion of partial separability hinges on geometric interpretation
of the problem, it is useful to introduce the various subspaces of interest
for our analysis. We will extensively use the following definitions.
As will become clear in Section~\ref{optimality-s}, we will need to
identify
\beqn{Cxeps-def}
\calC(x,\epsilon) \eqdef \{ i \in \calH \mid \|U_ix-b_i\| \leq \epsilon \}
\tim{and}
\calA(x,\epsilon) \eqdef \calH \setminus \calC(x,\epsilon),
\eeqn
the collection of hard elements which are close to singularity for a given
$x$ and its complement (the ``active'' elements), and
\beqn{Rxeps-def}
\calR(x,\epsilon)
\eqdef \bigcap_{i\in\calC(x,\epsilon)} \ker(U_i)
= \left[\spanset_{i\in\calC(x,\epsilon)}(U_i^T)\right]^\perp
\eeqn
the subspace in which those nearly singular elements are invariant.
(When $\calC(x,\epsilon)=\emptyset$, we set $\calR(x,\epsilon)=\Re^n$.)
For convenience, if $\epsilon=0$, we denote $\calC(x)\eqdef \calC(x,0)$,
$\calA(x) \eqdef \calA(x,0)$, $\calR(x)\eqdef \calR(x,0)$ and
$\calW(x)\eqdef \calW(x,0)$.
From these definitions, we have
\beqn{dd}
U_id = 0, \tim{for} i\in \calC(x), \,\,\, d\in \calR(x).
\eeqn
Also denote by
\beqn{Rii-def}
\calR_{\{i\}} \eqdef \spanset(U_i^T)
\eeqn
and observe that \req{Ui-ortho-H} implies that the $\calR_{\{i\}}$ are
orthogonal for $i \in \calH$. Hence $\calR_{\{i\}}$ is also
the subspace in which all singular elements are invariant but the $i$-th.
We also denote the ``working'' collection of elements not close to singularity
by
\beqn{Wxeps-def}
\calW(x,\epsilon) \eqdef \calN \cup \calA(x,\epsilon).
\eeqn
If $\{x_k\}$ is a sequence of iterates
in $\Re^n$, we also use the shorthands
\beqn{CRW-short}
\calC_k = \calC(x_k,\epsilon),
\ms
\calA_k = \calA(x_k,\epsilon),
\ms
\calR_k = \calR(x_k,\epsilon)
\tim{and}
\calW_k = \calW(x_k,\epsilon).
\eeqn
We will make frequent use of
\beqn{fWk-def}
f_{\calW_k}(x) \eqdef \sum_{i \in \calW_k}f_i(x),
\eeqn
which is objective function ``reduced'' to the elements ``away from singularity''
at $x_k$.
For some $x, s\in \Re^n$, we often use the notations $
r_i = U_ix - b_i
$ and $s_i=Us$.
\numsection{Necessary optimality conditions}\label{optimality-s}
At variance with the theory developed in \cite{ChenToinWang17}, which solely
covers convergence to $\epsilon$-approximate first-order stationary points, we
now consider arbitrary orders of optimality. To this aim, we follow
\cite{CartGoulToin18b} and define, for a sufficiently smooth function
$h:\Re^n\rightarrow\Re$ and a convex set $\calF\subseteq\Re^n$, the vector $x$ to be an
$(\epsilon,\delta)$-approximate $q$-th-order stationary point ($\epsilon>0, \delta>0$, $q\in\ii{p}$)
of $\min_{x\in \calF} h(x)$ if, for some $\delta \in (0,1]$
\beqn{optimality-smooth}
\phi_{h,q}^\delta(x) \leq \epsilon \chi_q(\delta)
\eeqn
where
\beqn{phi-def}
\phi_{h,q}^\delta(x)
\eqdef h(x)-\min_{\stackrel{x+d\in \calF}{\|d\|\leq\delta}}T_{h,q}(x,d),
\eeqn
and
\beqn{chi-def}
\chi_q(\delta) \eqdef \sum_{\ell=1}^q \frac{\delta^\ell}{\ell!}.
\eeqn
In other words, we declare $x$ to be an $(\epsilon,\delta)$-approximate
$q$-th-order stationary point if the scaled maximal decrease that can be
obtained on the $q$-th order Taylor series for $h$ in a neighbourhood of $x$ of
radius $\delta$ is at most $\epsilon$. We refer the reader to
\cite{CartGoulToin18b} for a detailed motivation and discussion of this
measure. For our present purpose, it is enough to observe that
$\phi_{h,q}^\delta(x)$ is a continuous function of $x$ and $\delta$ for any $q$. Moreover,
for $q=1$ and $q=2$, $\delta$ can be chosen equal to one and $\phi_{h,1}^1(x)$
and $\phi_{h,2}^1(x)$ are easy to compute. In the unconstrained case,
\[
\phi_{h,1}^1(x) = \|\nabla_x^1h(x)\|
\]
and computing $\phi_{h,2}^1$ reduces to solving the standard trust-region problem
\[
\phi_{h,2}^1(x) = \left|\min_{\|d\|\leq 1} \nabla_x^1h(x)[d] + \half
\nabla_x^2h(x)[d]^2 \right|.
\]
In the constrained case,
\[
\phi_{h,1}^1(x) = \left| \min_{\mystack{x+d \in \calF}{\|d\|\leq
1}}\nabla_x^1h(x)[d]\right|,
\]
which is the optimality measure used in \cite{CartGoulToin12b} or
\cite{ChenToinWang17} among others. However, given
the potential difficulty of solving the global optimization problem in
\req{phi-def} for $q>2$, our approach remains, for now, conceptual for such
high optimality orders.
We now claim that we can extend the definition \req{optimality-smooth} to
cover problem \req{problem} as well. The key observation is that, by the
definition of $\calW(x,\epsilon)$ and $\calR(x,\epsilon)$,
\beqn{f=fWonReps}
f_{\calW(x,\epsilon)}(x)=f_{\calW(x,\epsilon)}(x+d)\le f(x+d)
\leq f_{\calW(x,\epsilon)}(x+d)+\epsilon^a|\calH| \tim{for all} d \in \calR(x,\epsilon).
\eeqn
Note now that $f_{\calW(x,\epsilon)}$ is smooth around $x$ because it only contains
elements which are away from singularity, and hence that
$T_{f_{\calW(x,\epsilon)},p}(x,s)$ is well-defined. We may therefore define
$x$ to be an $(\epsilon,\delta)$-approximate $q$-th-order stationary point
for \req{problem} if, for some $\delta \in (0,1]$
\beqn{optimality}
\psi_{f,q}^{\epsilon,\delta}(x) \leq \epsilon \chi_q(\delta),
\eeqn
where we define
\beqn{psi-def}
\psi_{f,q}^{\epsilon,\delta}(x)\eqdef
f(x)-\min_{\mystack{x+d\in\calF}
{\|d\|\leq\delta,\, d\in\calR(x,\epsilon)}} T_{f_{\calW(x,\epsilon)},q}(x,d).
\eeqn
By the definition of $\calW(x,\epsilon)$, we have $f_{\calW(x,\epsilon)}(x)\le f(x)$ and thus
\[f_{\calW(x,\epsilon)}(x)-\min_{\mystack{x+d\in\calF}
{\|d\|\leq\delta,\, d\in\calR(x,\epsilon)}} T_{f_{\calW(x,\epsilon)},q}(x,d) \le \psi_{f,q}^{\epsilon,\delta}(x).
\]
Taking $\epsilon=0$, $x$ is $q$-th-order stationary point if
$\psi_{f_{\calW(x)},q}^{0,\delta} =0$, as we now prove.
\begin{theorem}
If $x_*$ is a local minimizer of (\ref{problem}), then there is $\delta \in (0,1]$ such that
\begin{equation}\label{Op1}
\psi_{f,q}^{0,\delta}(x_*)=0.
\end{equation}
\end{theorem}
\proof{
Suppose first that $\calR(x_*) = \{0\}$ (which happens if there exists $x_* \in \calF$ such that $f_\calH(x_*)=0$ and
$\spanset_{i\in \calH}\{U_i^T\} = \Re^n$). Then \req{Op1}
holds vacuously with any $\delta\in (0,1]$. Now suppose that $\calR(x_*) \neq \{0\}$.
Let
\[
\delta_1=\min\left[1, \min_{i\in \calA (x_*)} \|U_ix_*-b_i\|\right] \in (0,1].
\]
Since $x_*$ is a local minimizer of \req{problem}, there
exists $\delta_2>0$ such that
\begin{equation*}
\begin{split}
f(x_*) & = \bigmin_{\mystack{x_*+d \in \calF}{\|d\|\leq\delta_2}}f_{\calN}(x_*+d)+\sum_{i\in\calH}\|U_i(x_*+d)-b_i\|^a\\
& \leq \bigmin_{\mystack{x_*+d \in \calF}{\|d\|\leq\delta_2,\,d \in \calR(x_*)}}f_{\calN}(x_*+d)+\sum_{i\in\calH}\|U_i(x_*+d)-b_i\|^a\\
& = \bigmin_{\mystack{x_*+d \in \calF}{\|d\|\leq\delta_2,\,d \in \calR(x_*)}}f_{\calN}(x_*+d)+\sum_{i\in\calA(x_*)}\|U_i(x_*+d)-b_i\|^a\\
& = \bigmin_{\mystack{x_*+d \in \calF}{\|d\|\leq\delta_2,\,d \in \calR(x_*)}}f_{\calW(x_*)}(x_*+d),
\end{split}
\end{equation*}
where we used (\ref{dd}) and (\ref{Wxeps-def}) to derive the last two equalities, respectively.
Now we consider the reduced problem
\beqn{reduced}
\min_{\mystack{x_*+d\in \calF}{\|d\|\leq\delta_2,\, d \in\calR(x_*)}}f_{\calW(x_*)}(x_*+d).
\eeqn
Since we have that
\[
f_{\calW(x_*)}(x_*)
=f_{\calN}(x_*)+\sum_{i\in\calA(x_*)}\|U_i x_*-b_i\|^a
=f_{\calN}(x_*)+\sum_{i\in\calH}\|U_i x_*-b_i\|^a
=f(x_*),
\]
we obtain that
\[
f_{\calW(x_*)}(x_*)
\leq \min_{\mystack{x_*+d \in \calF}{\|d\|\leq\delta_2,\,d \in \calR(x_*)}}f_{\calW(x_*)}(x_*+d)
\]
and $x_*$ is a local minimizer of problem
\req{reduced}.
Note that for any $x_*+d$ in the ball $B(x^*, \delta_3)$ with $\delta_3<\delta_1$, we have
\[
\|U_i(x_*+d)-b_i\|
\geq \|U_ix_*-b_i\|-\|U_id\|
\geq \delta_1-\|U_i\|\|d\|
= \delta_1-\delta_3 > 0, \ms i\in \calA(x_*).
\]
Hence $f_{\calW(x_*)}(x_*+d)$ is $q$-times continuously differentiable, and has
Lipschitz continuous derivatives of orders 1 to $q$ in $B(x^*, \delta_3)$. By
Theorem 3.1 in \cite{CartGoulToin17c}, there is a $\delta \in \big(0,
\min[\delta_2, \delta_3]\big]$, such that
\[
\psi_{f_{\calW(x_*)},q}^{0,\delta}(x_*)
=f_{\calW(x_*)}(x_*)-\min_{\mystack{x_*+d\in \calF}{\|d\|\leq\delta,\ d\in \calR(x_*)}} T_{f_{\calW(x_*)},q}(x_*,d)=0
\]
This, together with
$f(x_*)=f_{\calW(x_*)}(x_*)$,
gives the desired result \req{Op1}.
}
We call $x_*$ is
a $q$-th-order stationary point of \req{problem} if there is $\delta \in (0,1]$ such that
(\ref{Op1}) holds.
\lthm{epsilon_q}{
For each $k$, let $x_k$ be an $(\epsilon_k, \delta_k)$-approximate
$q$-th-order stationary point of \req{problem} with
$1 \geq \delta_k\geq \bar{\delta}>0$ and $\epsilon_k\rightarrow 0$.
Then any cluster point of $\{x_k\}$ is
a $q$-th-order stationary point of \req{problem}.
}
\proof{
Let $x_*$ be a cluster point of $\{x_k\}$. Without loss of generality, we
assume that $x_* = \lim_{k\rightarrow\infty}x_k$.
From $0< \chi_q(\delta)\le 2$ and $\psi_{f,q}^{\epsilon, \delta}(x)\ge 0$, we
have from \req{optimality} that
$\lim_{k\rightarrow\infty}\psi_{f,q}^{\epsilon_k, \delta_k}(x_k)=0$. We now
need to prove that $\psi_{f,q}^{0, \bar{\delta}}(x_*)=0$.
If $\calR(x_*)=\{0\}$, \req{Op1} holds vacuously with any $\delta >0$, and
hence $x_*$ is a $q$th-order-necessary minimizer of \req{problem}.
Suppose now that $\calR(x_*) \neq \{0\}$.
We first claim that there exists a $k_*\geq 0$ such that
\beqn{CkCstar}
\calC(x_k,\epsilon_k ) \subseteq \calC(x_*)
\tim{ for } k \geq k_*.
\eeqn
To prove this inclusion, we choose $k_*$
sufficiently large to ensure that
\beqn{kstar-def}
\|x_k-x_*\| + \epsilon_k < \min_{j \in \calA(x_*)}\|U_jx_*-b_j\|,
\tim{ for } k \geq k_*.
\eeqn
Such a $k_*$ must exist, since the right-hand side of this inequality is strictly positive
by definition of $\calA(x_*)$.
For an arbitrary $k \geq k_*$ and an index $i \in
\calC(x_k,\epsilon_k )$, using the definition of $\calC(x,\epsilon)$, the
identity $\|U_i\| = 1$ and \req{kstar-def}, we obtain that
\[
\|U_ix_*-b_i\|
\leq \|U_i(x_*-x_k)\| + \|U_ix_k -b_i\|
\leq \|x_* - x_k\| + \epsilon_k
< \min_{j \in \calA(x_*)}\|U_jx_*-b_i\|.
\]
This implies that $\|U_ix_*-b_i\|= 0$ and $i \in \calC(x_*)$. Hence \req{CkCstar} holds.
By the definition of $\calR(x,\epsilon)$ and $\calW(x, \epsilon)$,
\req{CkCstar} implies that, for all $k$,
\beqn{RkRstar}
\calR(x_*) \subseteq \calR(x_k,\epsilon_k )
\tim{ and }
\calW(x_*) \subseteq \calW(x_k,\epsilon_k ).
\eeqn
For any fixed $k\ge k_*$,
consider now the following three minimization problems:
\beqn{prob-a}
\hspace*{6mm}(A,k) \ms \left\{
\begin{array}{cl}
\min_d & \quad T_{f_{\calW(x_k,\epsilon_k)},q}(x_k,d) \\
\mbox{s.t.} & \quad x_k+d\in\calF,\,d\in\calR(x_k,\epsilon_k), \,\|d\|\le \delta_k,
\end{array}
\right.
\eeqn
\beqn{prob-b}
(B,k) \ms \left\{
\begin{array}{cl}
\min_d & \quad T_{f_{\calW(x_k,\epsilon_k)},q}(x_k, d) \\
\mbox{s.t.} & \quad x_k+d\in\calF,\, d\in\calR(x_*), \, \|d\|\le \delta_k,
\end{array}
\right.
\eeqn
and
\beqn{prob-c}
(C,k) \ms \left\{
\begin{array}{cl}
\min_d & \quad T_{f_{\calW(x_*)},q}(x_k, d) \\
\mbox{s.t.} & \quad x_k+d\in\calF, \, d\in\calR(x_*), \, \|d\|\le \delta_k.
\end{array}
\right.
\eeqn
Since $d=0$ is a feasible point of these three problems, their minimum values,
which we respectively denote by $\vartheta_{A,k}$, $\vartheta_{B,k}$ and
$\vartheta_{C,k}$, are all smaller than $f(x_k)$. Moreover, it follows
from the first part of \req{RkRstar} that, for each $k$,
\beqn{vartBA}
\vartheta_{B,k} \geq \vartheta_{A,k}.
\eeqn
It also follows from \req{CkCstar} and \req{Ui-ortho-H} that
\[
T_{f_{\calW(x_k,\epsilon_k)},q}(x_k, d)=
T_{f_{\calW(x_*)},q}(x_k, d) - f_{\calW(x_*)}(x_k) + f_{\calW(x_k,\epsilon_k)}(x_k)
\leq T_{f_{\calW(x_*)},q}(x_k, d) + |\calH|\epsilon_k^a
\]
for all $d\in \calR(x_*)$, and thus \req{vartBA} becomes
\beqn{vartABC}
\vartheta_{A,k} \leq \vartheta_{B,k} \le \vartheta_{C,k}+|\calH|\epsilon_k^a
\tim{ for all}k\ge k_*.
\eeqn
The assumption that $x_k$ is an $(\epsilon_k, \delta_k)$-approximate
$q$th-order necessary minimizer of \req{problem} implies that
\beqn{vart2}
0 \leq f(x_k)-\vartheta_{C,k}-|\calH|\epsilon_k^a
\leq f(x_k)-\vartheta_{A,k}\le\epsilon_k \chi_q(\delta_k),
\tim{for all} k \geq k_*.
\eeqn
Now (\ref{RkRstar}) implies that
$T_{f_{ \calW(x_*)},q}(x_k,d)\leq T_{f_{\calW(x_k)},q}(x_k,d)\leq T_{f,q}(x_k,d)$. Hence
\[
f(x_k)-\vartheta_{C,k}=f(x_k)-\min_{\mystack{x_k+d\in
\calF}{\|d\|\le \delta_k, \ d\in \calR(x_*)}}T_{f_{ \calW(x_*)},q}(x_k,d)
\ge\phi_{f,q}^{0,\delta_k}(x_k).
\]
As a consequence, \req{vart2} implies that
\beqn{epsilonc}
\phi_{f,q}^{0,\delta_k}(x_k)
\leq \epsilon_k \chi_q(\delta_k)+|\calH|\epsilon_k^a.
\eeqn
In addition, the feasible sets of the three problems \req{prob-a}-\req{prob-c}
are convex, and the objectives functions are polynomials with degree $q$.
By the perturbation theory for optimization problems
\cite[Theorem~3.2.8]{ConnGoulToin00}, we can claim that
\beqn{chiCk}
\lim_{k \rightarrow \infty} \vartheta_{C,k}
=\min_{\mystack{x_*+d\in \calF}{\|d\|\le \delta_*, \ d\in \calR(x_*)}}T_{f_{ \calW(x_*)},q}(x_*,d),
\eeqn
where $\delta_*=\liminf_{k \rightarrow \infty}\delta_k\geq \bar{\delta}$.
This implies that letting $k\to \infty$ in \req{epsilonc} gives
\[
\psi_{f,q}^{0, \bar{\delta}}(x_*)
= f(x_*)
-\min_{\mystack{x_*+d\in \calF}{\|d\|\leq \bar{\delta},\,d\in\calR(x_*)}} T_{f_{ \calW(x_*)},q}(x_*,d)=0.
\]
}
We conclude this section by an important observation. The optimality measure
\req{optimality} may give the impression (in particular in its use of
$\calR(x,\epsilon)$) that the ``singular'' and ``smooth''
parts of the problem are merely separated, and that one could possibly apply the
existing theory for smooth problems to the latter. Unfortunately, this
is not true, because the ``separation'' implied by \req{optimality} does
depend on $\epsilon$, and one therefore need to show that the complexity of
minimizing the ``non-singular'' part does not explode (in particular with the
unbounded growth of the Lispchitz constant) when $\epsilon$ tends to
zero. Designing an suitable algorithm and proving an associated complexity
result comparable to what is known for smooth problems is the main challenge
in what follows.
\numsection{A Lipschitz continuous model of $f_{\calW_k}(x+s)$}\label{model-s}
Our minimization algorithm, described in the next section, involves the
approximate minimization of a \emph{model} $m(x_k,s)$ of $f_{\calW_k}$ in the
intersection of a neighbourhood of $x_k$ with $\calR_k$. This model,
depending on function and derivatives values computed at $x_k$, should be
able to predict values and derivatives of $f$ at some neighbouring point
$x_k+s$ reasonably accurately. This is potentially difficult if the current
point happens to be near a singularity.
Before describing our proposal, we need to state a useful technical result.
\llem{dersnorm-l}{Let $a$ be a positive number and $r\neq 0$.
Define, for a positive integer $j$,
\beqn{factor-def}
\pi(a-j) \eqdef a\prod_{i=1}^{j-1}(a-i).
\eeqn
Then, if $\nabla_\cdot^j\big\|r\big\|^a$ is the value
of the $j$-th derivative tensor of the function $\|\cdot\|^a$ with
respect to its argument, evaluated at $r$, we have that,
\beqn{ders}
\nabla_\cdot^j \|r\|^a
= \bigsum_{i=1}^j \phi_{i,j}
\|r\|^{a-2i} \, r^{(2i-j) \otimes} \otimes I^{(j-i)\otimes}
\eeqn
for some scalars $\{\phi_{i,j}\}_{i=1}^j$ such that
$\sum_{i=1}^j \phi_{i,j} = \pi(a-j)$ , and that
\beqn{dersnorm}
\big\|\,\nabla_\cdot^j \|r\|^a \,\big\|_{[j]} = |\pi(a-j)|\,\|r\|^{a-j}.
\eeqn
Moreover, if $\beta_1, \beta_2$ are positive reals and $\|r\|=1$, then
\beqn{difdersnorm}
\left\|\,\nabla_\cdot^j \|\beta_1 r\|^a - \nabla_\cdot^j \|\beta_2 r\|^a\,\right\|_{[j]}
= |\pi(a-j)|\,\left|\beta_1^{a-j} - \beta_2^{a-j}\right|.
\eeqn
}
\proof{See appendix. }
Consider now the elements $f_i$ for $i\in \calN$. Each such element is $p$ times
continuously differentiable and, if we assume that its $p$-th derivative
tensor $\nabla_x^pf_i$ is globally Lipschitz continuous with constant $L_i
\geq 0$ in the sense that, for all $x_i,y_i \in \Re^{n_i}$
\beqn{tensor-Lip-fi}
\|\nabla_{x_i}^pf_i(x_i) - \nabla_{x_i}^p f_i(y_i)\|_{[p]} \leq L_i \|x_i-y_i\|,
\eeqn
then it can be shown (see \cite[Lemma~2.1]{CartGoulToin18b}) that
\beqn{f-Lip-1}
f_i(x_i+s_i) = T_{f_i,p}(x_i,s_i) + \frac{1}{(p+1)!} \tau_i L_i \|s_i\|^{p+1}
\tim{ with }| \tau_i | \leq 1.
\eeqn
Because $\tau_i L_i$ in \req{f-Lip-1} is usually unknown in
practice, it may not be possible to use \req{f-Lip-1} directly to model $f_i$
in a neighbourhood of $x$. However, we may replace this term with an
adaptive parameter $\sigma_i$, which yields the following
$(p+1)$-rst order model for the $i$-th ``nice'' element
\beqn{miN-def}
m_i(x_i,s_i)=T_{f_i,p}(x_i,s_i)+\frac{1}{(p+1)!}\ \sigma_i\|s_i\|^{p+1},
\ms (i \in \calN).
\eeqn
Using the associated Taylor's expansion
would indeed ignore the non-Lipschitzian singularity occurring for $r_i=0$
and this would restrict the validity of the model to a possibly very small
neighbourhood of $x_k$ whenever $r_i$ is small for some $i \in
\calA(x_k,\epsilon)$. Our proposal is to use the underlying rotational
symmetry of the Euclidean norm function to build a better Lipschtzian
model. Suppose that $r_i \neq 0 \neq r_i+s_i$ and let
\beqn{uiui+-def}
u_i = \frac{r_i}{\|r_i\|},
\ms
r_i^+ = r_i(x+s) = r_i+s_i
\tim{ and }
u_i^+ = \frac{r_i^+}{\|r_i^+\|}.
\eeqn
Moreover, let $R_i$ be the rotation in the $(u_i,u_i^+)$ plane\footnote{If
$u_i=u_i^+$, $R_i= I$. If $n_i=1$ and $r_ir_i^+<0$, this rotation is just
the mapping from $\Re_+$ to $\Re_-$, defined by a simple sign change, as in
the two-sided model of \cite{ChenToinWang17}.} such that
\beqn{roti-def}
R_iu_i^+ = u_i.
\eeqn
We observe that, given the isotropic nature of the Euclidean norm, the value
$\|r_i\|^a$ and of its derivatives with respect to $s_i$ can be deduced
from those $\| \, \|r_i\|u_i^+ \,\|^a$. More precisely, for any $d\in\Re^{n_i}$,
\beqn{rotation}
\big\|\, \|r_i\|u_i^+ \, \big\|^a = \|r_i\|^a
\tim{and}
\nabla_\cdot^\ell\big\| \,\|r_i\|u_i^+\,\big\|^a[d]^\ell
=\nabla_\cdot^\ell\|r_i\|^a[R_id]^\ell.
\eeqn
For example, when $j=1$, one
verifies that
\[
\begin{array}{ll}
\nabla_\cdot^1\|r_i\|^a[R_id]
& = a \|r_i\|^{a-2}r_i^TR_id \\*[1.5ex]
& = a \|r_i\|^{a-2} \|r_i\|( R_i^Tu_i)^Td \\*[1.5ex]
& = a \big\|\,\|r_i\| u_i^+\,\big\|^{a-2} (\|r_i\|u_i^+)^Td\\*[1.5ex]
& = \nabla_\cdot^1\big\| \,\|r_i\|u_i^+\,\big\|^a [d].
\end{array}
\]
We may then choose to compute the Taylor's expansion for the function $\|\cdot\|^a$
around $\|r_i\|u_i^+$, that is
\[
\begin{array}{ll}
\|r_i^+\|^a
& = \big\|\,\|r_i^+\|u_i^+\,\big\|^a \\*[1.5ex]
& =
\big\|\,\|r_i\|u_i^+\,\big\|^a+\bigsum_{\ell=1}^\infty\bigfrac{1}{\ell!}
\nabla_\cdot^\ell\big\|\|r_i\|u_i^+\big\|^a
\big[(\|r_i^+\|-\|r_i\|)u_i^+\big]^\ell\\*[1.5ex]
& = \|r_i\|^a
+\bigsum_{\ell=1}^\infty\bigfrac{(\|r_i^+\|-\|r_i\|)^\ell}{\ell!}
\nabla_\cdot^\ell\big\|r_i\big\|^a\big[R_iu_i^+\big]^\ell\\*[1.5ex]
& = \|r_i\|^a
+\bigsum_{\ell=1}^\infty
\bigfrac{(\|r_i^+\|-\|r_i\|)^\ell}{\ell!}
\nabla_\cdot^\ell\big\|r_i\big\|^a[u_i]^\ell.
\end{array}
\]
Using the expression \req{ders} applied to $\ell$ copies of the unit vector
$u_i$ and the fact that $r_i^{j\otimes}[u_i]^j= (r_i^Tu_i)^j=\|r_i\|^j$ for all
$j\in\Na$, we now deduce that, for $\zeta_i= \|r_i+s_i\|-\|r_i\|\geq -\|r_i\|$,
\beqn{Taylorsi}
\|r_i+s_i\|^a
= \|r_i\|^a + \bigsum_{\ell=1}^\infty
\bigfrac{\pi(a-\ell)}{\ell!}\zeta_i^\ell
\|r_i\|^{a-\ell},
\eeqn
which is nothing else than the Taylor's expansion of $\|r_i+\zeta_iu_i\|^a$
(or, equivalently, of $\big\| \|r_i\|u_i^+ + \zeta_iu_i^+\big\|^a$)
expressed as a function of the scalar variable $\zeta_i \geq -\|r_i\|$. As can be
expected from the isotropic nature of the Euclidean norm, the value of
$\|r_i^+\|^a$ (and of its derivatives after a suitable rotation) only
depend(s) on the distance of $r_i^+$ to the singularity at zero.
Thus, limiting the development \req{Taylorsi} to degree $p$ (as in
\req{miN-def}), it is natural to define
\beqn{TiH-def}
\mu(\|r_i\|, \zeta_i)
\eqdef \|r_i\|^a
+\bigsum_{\ell=1}^p\bigfrac{\pi(a-\ell)}{\ell!}\zeta_i^\ell
\|r_i\|^{a-\ell},\\*[1.5ex]
\ms
(i \in \calA(x,\epsilon)),
\eeqn
which is a unidimensional model of $\|r_i+\zeta_iu_i\|^a$ based of the
residual value $\|r_i\|$.
Note that $\mu(\|r_i\|,\zeta_i)$ is Lipschitz continuous as a function of
$\zeta_i$ as long as $\|r_i\|$ remains uniformly bounded
away from zero, with a Lipschitz constant depending on the lower bound. We
then define the isotropic model
\beqn{miH-def}
m_i(x_i,s_i) \eqdef \mu(\|r_i\|,\zeta_i) = \mu_i(\|r_i\|,\|r_i+s_i\|-\|r_i\|)
\tim{ for } i\in \calA(x,\epsilon),
\eeqn
so that, abusing notation slightly,
\[
m_i(x_i,s_i) = T_{m_i,p}(x_i,s_i) \eqdef \|r_i\|^a
+\bigsum_{\ell=1}^p\bigfrac{\pi(a-\ell)}{\ell!}\zeta_i^\ell
\|r_i\|^{a-\ell},\\*[1.5ex]
\ms
(i \in \calA(x,\epsilon)).
\]
\noindent
We now state some useful properties of this model.
\llem{miprop-l}{Suppose that $p$ is odd and that $\calA(x) \neq \emptyset$ for
some $x \in \calF$.
Then, for $i \in \calA(x)$,
\beqn{overestH}
m_i(x_i,s_i) \geq \|r_i+s_i\|^a,
\eeqn
and, whenever $\|r_i+s_i\|\leq \|r_i(x)\|$,
\beqn{dersmi-odd}
\nabla_\zeta^\ell \mu(\|r_i\|,\zeta_i))
\geq \nabla_\zeta^\ell \mu(\|r_i\|,0)
= \pi(a-\ell)\|r_i\|^{a-\ell}
\ms
(\ell \mbox{~odd}),
\eeqn
and
\beqn{dersmi-even}
\nabla_\zeta^\ell \mu(\|r_i|,\zeta_i)
\leq \nabla_\zeta^\ell\mu(\|r_i\|,0)
= \pi(a-\ell)\|r_i\|^{a-\ell}
\ms
(\ell \mbox{~even}).
\eeqn
As a consequence, $m_i(x_i,\zeta_i)$ is a concave function of $\zeta_i$ on
the interval $[-\|r_i\|,0]$.
}
\proof{Let $i\in \calA(x)$.
From the mean-value theorem and \req{Taylorsi}, we have that, for some $\nu
\in (0,1)$,
\beqn{mvmi}
\|r_i+s_i\|^a
= \|r_i\|^a + \bigsum_{\ell=1}^p
\bigfrac{\pi(a-\ell)}{\ell!}\zeta_i^\ell
\|r_i\|^{a-\ell} + \bigfrac{\pi(a-p-1)}{(p+1)!} \zeta_i^{p+1}\|r_i+\nu
\zeta_i u_i\|^{a-p-1}.
\eeqn
Since $p$ is odd, we obtain that $\pi(a-p-1)<0$ and $\zeta_i^{p+1} \geq 0$.
Thus \req{overestH} directly results from \req{mvmi}, \req{TiH-def} and \req{miH-def}.
Now \req{TiH-def} and \req{miH-def} together imply that
\beqn{derzeta}
\nabla_\zeta^\ell \mu(\|r_i\|,\zeta_i)
= \nabla_\zeta^\ell \mu(\|r_i\|,0)
+ \sum_{j=\ell+1}^p\frac{\pi(a-j)}{(j-\ell)!}\zeta_i^{j-\ell}\|r_i\|^{a-j}
\eeqn
for $\zeta_i = \|r_i(x)+s_i\|-\|r_i(x)\| \leq 0$.
Suppose first that $\ell$ is odd. Then we have that $\pi(a-j)$ is negative for
even $j$, that is exactly when $\zeta_i^{j-\ell}$ is non-positive. Hence
every term in the sum of the right-hand side of \req{derzeta} is non-negative
and \req{dersmi-odd} follows. Suppose now that $\ell$ is even. Then $\pi(a-j)$
is negative for odd $j$, which is exactly when $\zeta_i^{j-\ell}$ is
non-negative. Hence every term in the sum of the right-hand side of
\req{derzeta} is non-positive and \req{dersmi-even} follows. The last
conclusion of the lemma is then deduced by considering $\ell=2$ in
\req{dersmi-even} and observing that
$\pi(a-\ell)\|r_i(x)\|^{a-\ell}=a(a-1)\|r_i(x)\|^{a-2}<0$.
}
\noindent
Thus the isotropic model $m_i(x_i,s_i)$ overestimates the true function
$\|r_i(x)+s_i\|^a$ and correctly reflects its concavity in the direction of
its singularity. But $m_i(x_i,s_i)= \mu(\|r_i\|,\zeta_i)$ is now Lipschitz
continuous as a function of $s_i$, while $\|r_i(x)+s_i\|^a$ is not.
Combining \req{miN-def} and \req{miH-def} now allows us to define a model for
the complete $f$ on $\calR(x,\epsilon)$ as
\beqn{mfull-def}
m(x,s)
\eqdef \sum_{i\in \calW(x,\epsilon)} m_i(x_i,s_i).
\eeqn
Since $r_i(x)\neq 0$ for $i \in \calA(x,\epsilon)$, this model is in turn
well defined.
We conclude this section by observing that writing the problem in the
partially-separable form \req{problem} is the key to expose the singular parts
of the objective function, which then allows exploiting their rotational
symmetry.
\numsection{The adaptive regularization algorithm}\label{algo-s}
Having defined a model of $f$, we may then use this model within a
regularization minimization method inspired from the AR$p$ algorithm in
\cite{CartGoulToin18b}. In such an algorithm, the step from an iterate $x_k$
is obtained by attempting to (approximately) minimize the model (\req{mfull-def}
in our case). If an $(\epsilon,\delta)$-approximate $q$-th-order-necessary
minimizer is sought, this minimization is terminated as soon as the step $s_k$
is long enough, in that
\beqn{longs}
\|s_k\|\geq \varpi \epsilon^{\frac{1}{p-q+1}}
\eeqn
for some constant $\varpi \in (0,1]$,
or as soon as the trial point $x_k+s_k$ is an
approximate $q$-th-order-necessary minimizer \emph{of the model}, in the sense
that
\beqn{term-m}
\psi_{m,q}^{\epsilon,\delta_k}(x_k,s_k)
\leq \min\left[\frac{\theta\|s_k\|^{p-q+1}}{(p-q+1)!},
\, a \min_{i\in \calA(x_k+s_k, \epsilon)}\|r_i(x_k+s_k)\| \right]\chi_q(\delta_k)
\eeqn
for some $\theta, \delta_k \in (0,1]$, where $\psi_{m,q}^{\epsilon,\delta_k}(x_k+s_k)$
is the optimality measure \req{psi-def} computed for the model $m(x,s)$,
that is
\beqn{psim-def}
\psi_{m,q}^{\epsilon,\delta}(x,s)
\eqdef m(x,s)-\min_{\mystack{x+s+d \in \calF}{\|d\|\leq\delta, d\in \calR(x,\epsilon)}}T_{m,q}(x,s+d).
\eeqn
In view of the optimality condition \req{optimality}, we also require that, if
$\|r_i(x+s)\| \leq \epsilon$ occurs for some $i \in \calH$ in the course of
the model minimization, the value of $r_i(x+s)$ is fixed, implying that the
remaining minimization is carried out on $\calR(x_k+s,\epsilon)$. As a
consequence, the dimension of $\calR(x_k+s,\epsilon)$ (and thus of $\calR_k$) is
monotonically non-increasing during the step computation and across all
iterations. It was shown in \cite[Lemma~2.5]{CartGoulToin18b} that, unless
$x_k$ is an $(\epsilon,1)$-approximate $p$-th-order-necessary minimizer (which
is obviously enough for the whole algorithm to terminate), a step satisfying
\req{term-m} can always be found. The fact that this condition must hold on a
subspace of potentially diminishing dimension clearly does not affect the
result, and indicates that \req{term-m} is well-defined. This model
minimization is in principle simpler than the original problem because the
general nonlinear $f_i$ have been replaced by locally accurate polynomial
approximations and also because the model is now Lipschitz continuous, albeit
still non-smooth. Importantly, the model minimization \emph{does not involve
any evaluation of the objective function} or its derivatives, and model
evaluations within this calculation therefore do not affect the overall
evaluation complexity of the algorithm.
We now introduce some useful notation for describing our algorithm. Define
\[
x_{i,k} \eqdef U_ix_k,
\ms r_{i,k} \eqdef U_ix_k-b_i,
\ms s_{i,k} \eqdef U_is_k,
\ms u_{i,k} \eqdef \frac{r_{i,k}}{\|r_{i,k}\|}
\]
and
\[
\calA_k^+ \eqdef \calA(x_k+s_k,\epsilon),
\ms
\calR_k^+ \eqdef \calR(x_k+s_k,\epsilon)
\tim{and}
\calW_k^+ \eqdef \calW(x_k+s_k,\epsilon).
\]
Also let
\[
\Delta f_{i,k} \eqdef f_i(x_{i,k}) - f_i(x_{i,k}+s_{i,k}),
\ms
\Delta f_k
\eqdef f_{\calW_k^+}(x_k) - f_{\calW_k^+}(x_k+s_k)
= \sum_{i \in \calW_k^+} \Delta f_{i,k},
\]
\[
\Delta m_{i,k} \eqdef m_i(x_{i,k},0) - m_i(x_{i,k},s_{i,k}),
\ms
\Delta m_k
= \sum_{i \in \calW_k^+} \Delta m_{i,k},
\]
and
\beqn{deltaT-def}
\begin{array}{lcl}
\Delta T_k
& \eqdef & T_{f_{\calW_k^+},p}(x_k,0) - T_{f_{\calW_k^+},p}(x_k,s_k) \\*[2ex]
& = & [T_{f_\calN,p}(x_k,0) - T_{f_\calN,p}(x_k,s_k)]
+ \big[m_{\calA_k^+}(x_k,0) - m_{\calA_k^+}(x_k,s_k)\big]\\*[2ex]
& = & \Delta m_k+\bigfrac{1}{(p+1)!}\bigsum_{i \in \calN}\sigma_{i,k}\|s_{i,k}\|^{p+1}.
\end{array}
\eeqn
Our partially-separable adaptive regularization degree-$p$ algorithm PSAR$p$ is then
given by Algorithm~\ref{psarp} \vpageref{psarp}.
\algo{psarp}{Partially-Separable Adaptive Regularization (PSAR$p$)}{
\vspace*{-3mm}
\begin{description}
\item[Step 0: Initialization:] $x_0\in\calF$ and $\{\sigma_{i,0}\}_{i\in\calN}
>0$ are given as well as the accuracy $\epsilon \in (0,1]$ and constants $0
<\gamma_0 < 1 < \gamma_1 \leq \gamma_2$, $\eta \in (0,1)$, $\theta \geq0$, $\delta_{-1}=1$,
$\sigma_{\min} \in (0, \min_{i\in\calN}\sigma_{i,0}]$ and $\kap{big}>1$. Set $k =0$.
\item[Step 1: Termination:] Evaluate $f(x_k)$ and
$\{\nabla_x^j f_{\calW_k}(x_k)\}_{j=1}^q$. If
\beqn{optimk}
\psi_{f,q}^{\epsilon,\delta_{k-1}}(x_k) \leq \epsilon \chi_q(\delta_{k-1})
\eeqn
return $x_\epsilon=x_k$ and terminate. Otherwise evaluate
$\{\nabla_x^j f_{\calW_k}(x_k)\}_{j=q+1}^p$.
\item[Step 2: Step computation:] Attempt to compute a
step $s_k\in \calR_k$ such that $x_k+s_k \in \calF$, $m(x_k,s_k)< m(x_k,0)$
and either \req{longs} holds or \req{term-m} holds for some $\delta_k\in (0,1]$.
If no such step exists, return $x_\epsilon=x_k$ and terminate.
\item[Step 3: Step acceptance:] Compute
\beqn{rhok-def}
\rho_k = \bigfrac{\Delta f_k}{\Delta T_k}
\eeqn
and set $x_{k+1} = x_k$ if $\rho_k < \eta$, or $x_{k+1} = x_k+s_k$ if $\rho_k \geq \eta$.
\item[Step 4: Update the ``nice'' regularization parameters:] For
$i \in \calN$, if
\beqn{up-cond}
f_i(x_{i,k}+s_{i,k}) > m_i(x_{i,k},s_{i,k})
\eeqn
set
\beqn{sig-incr}
\sigma_{i,k+1} \in [ \gamma_1 \sigma_{i,k}, \gamma_2 \sigma_{i,k} ].
\eeqn
Otherwise, if either
\beqn{muchover-neg}
\rho_k \geq \eta
\tim{ and }
\Delta f_{i,k} \leq 0
\tim{ and }
\Delta f_{i,k} < \Delta m_{i,k} - \kap{big}|\Delta f_k|
\eeqn
or
\beqn{muchover-pos}
\rho_k \geq \eta
\tim{ and }
\Delta f_{i,k} > 0
\tim{ and }
\Delta f_{i,k} > \Delta m_{i,k} + \kap{big}|\Delta f_k|
\eeqn
then set
\beqn{sig-decr}
\sigma_{i,k+1} \in [\max[\sigma_{\min}, \gamma_0\sigma_{i,k}], \sigma_{i,k} ],
\eeqn
else set
\beqn{sig-keep}
\sigma_{i,k+1} = \sigma_{i,k}.
\eeqn
Increment $k$ by one and go to Step~1.
\end{description}
}
Note that an $x_0 \in \calF$ can always be computed by projecting an
infeasible starting point onto $\calF$. The motivation for the second and
third parts of \req{muchover-neg} and \req{muchover-pos} is to identify cases
where the isotropic model $m_i$ overestimates the element function $f_i$ to an
excessive extent, leaving some room for reducing the regularization and hence
allowing longer steps. The requirement that $\rho_k \geq \eta$ in both
\req{muchover-neg} and \req{muchover-pos} is intended to prevent a situation
where a particular regularization parameter is increased and another decreased
at a given unsuccessful iteration, followed by the opposite situation at the
next iteration, potentially leading to cycling.
It is worthwhile to note the differences between the PSAR$p$ algorithm and
the algorithm discussed in \cite{ChenToinWang17}.
The first and most important is that the new algorithm is intended to find an
$(\epsilon,\delta)$-approximate $q$-th-order necessary minimizer for problem
\req{problem}, rather than a first-order critical point. This is made possible
by using the $q$-th-order termination criterion \req{optimk} instead of a
criterion only involving the first-order model decrease, and by simultaneously
using the step termination criteria \req{longs} and \req{term-m} which again
replace a simpler version again based solely on first-order information.
The second is that the PSAR$p$ algorithm applies to the more general problem
\req{problem}, in particular using the isotropic model \req{TiH-def} to allow
$n_i>1$ for $i\in \calH$.
As alluded to above and discussed in \cite{CartGoulToin18a} and
\cite{BellGuriMoriToin18}, the potential termination of the algorithm in
Step~2 can only happen whenever $q>2$ and $x_k=x_\epsilon$ is an
$(\epsilon,1)$-approximate $p$-th-order-necessary minimizer within $\calR_k$,
which, together with \req{optimality}, implies that the same property holds
for problem \req{problem}. This is a significantly stronger optimality
condition than what is required by \req{optimk}. Also note that the
potentially costly calculation of \req{term-m} may be avoided if \req{longs}
holds.
Let the index set of the ``successful'' and ``unsuccessful'' iterations be
given by
\[
\calS \eqdef \{ k \geq 0 \mid \rho_k \geq \eta \}
\tim{and}
\calU \eqdef \{ k \geq 0 \mid \rho_k < \eta \}.
\]
Also define
\[
\calS_k \eqdef \calS \cap \iibe{0}{k}
\tim{ and }
\calU_k \eqdef \iibe{0}{k} \setminus \calS_k.
\]
We then state a bound on $|\calU_k|$ as a function of $|\calS_k|$. This is a
standard result for non-partially-separable problems (see
\cite[Theorem~2.4]{BirgGardMartSantToin17} for instance), but needs careful
handling of the model's overestimation properties to apply to our present
context.
\llem{a-succ-unsucc}{
Suppose that AS.2 and AS.3 hold and that $\sigma_{i,k}\leq \sigma_{\max}$ for
all $i \in \calM$ and all $k\geq0$. Then, for all $k \geq 0$,
\[
k \leq \kappa^a |\calS_k| + \kappa^b,
\]
where
\[
\kappa^a \eqdef 1 + \frac{|\calN|\,|\log \gamma_0|}{\log \gamma_1}
\tim{ and }
\kappa^b \eqdef \frac{|\calN|}{\log \gamma_1}
\log\left(\frac{\sigma_{\max}}{\sigma_{\min}}\right).
\]
}
\proof{See \cite[Lemma~4.11]{ChenToinWang17}. The proof hinges on \req{overestH}.}
\numsection{Evaluation complexity analysis}\label{complexity-s}
We are now ready for a formal analysis of the evaluation complexity of the
PSAR$p$ algorithm for problem \req{problem}, under the following assumptions.
\ass{AS.1}{ The feasible set $\calF$ is closed, convex, non-empty and
kernel-centered (in the sense of \req{kernel-centered}).}
\vspace*{-4mm}
\ass{AS.2}{ Each element function $f_i$ ($i \in \calN$) is $p$ times
continuously differentiable in an open set containing $\calF$, where $p$ is
odd whenever $\calH \neq \emptyset$.}
\vspace*{-4mm}
\ass{AS.3}{The $p$-th derivative of each $f_i$ ($i \in \calN$) is
Lipschitz continuous on $\calF$ with associated Lipschitz constant $L_i$ (in
the sense of \req{tensor-Lip-fi}).
}
\vspace*{-4mm}
\ass{AS.4}{There exists a constant $f_{\rm low}$ such that $f_\calN(x) \geq f_{\rm
low}$ for all $x \in \calF$.
}
\vspace*{-4mm}
\ass{AS.5}{If $\calH\neq\emptyset$, there exists a constant
$\kappa_\calN \geq 0$ such that
$ \|\nabla_x^j f_i(U_ix)\| \leq \kappa_\calN $ for all
$x \in \calV$, $i \in \calN$ and $j \in \ii{p}$, where
\beqn{calV-def}
\calV \eqdef \left\{ x \in \calF \mid \tim{there exists} i \in \calH
\tim{with}\|r_i(x)\| \leq \frac{a}{16}\right\}.
\eeqn
}
\noindent
Note that AS.4 is necessary for problem \req{problem} to be well-defined.
Also note that, because of AS.2, AS.5 automatically holds if $\calF$ is
bounded or if the iterates $\{x_k\}$ remain in a bounded set. It is possible
to weaken AS.2 and AS.3 by replacing $\calF$ with the level set $\calL = \{ x
\in \calF \mid f(x) \leq f(x_0) \}$ without affecting the results
below. Finally observe that $\calV$ need not to be bounded, in particular if
$\spanset_{i\in \calH}(U_i)$ is a proper subspace of $\Re^n$. AS.5 is, of
course, unnecessary if $\calF$ or $\calV$ are bounded or the iterates remain
in a bounded set. The motivation for the particular choice of $\sfrac{1}{16}a$
in \req{calV-def} will become clear in Lemma~\ref{away-l} below.
We first recall a result providing useful bounds.
\llem{psnorm-l}{
There exist a constant $\varsigma>0$ such that,
for all $s \in \Re^m$ and all $v \geq 1$,
\beqn{s-sumsi}
\varsigma^v \|s\|^v
\leq \sum_{i\in \calN} \|s_i\|^v
\leq |\calN| \,\|s\|^v.
\eeqn
}
\proof{See \cite[Lemma~4.1]{ChenToinWang17}.}
\noindent
This lemma states that $\sum_{i\in\calN}\|\cdot\|$ is a norm on $\Re^n$ whose
equivalence constants with respect to the Euclidean one are $\varsigma$ and
$|\calN|$.
Our next step is to specify under which conditions the standard
$\epsilon$-independent overestimation and derivative accuracy bounds typical of
the Lipschitz case (see \cite[Lemma~2.1]{CartGoulToin18b} for instance) can be
obtained for the elements functions of \req{problem}. We define, for a given
$k\geq 0$ and a given constant $\phi>0$ independent of $\epsilon$,
\beqn{Ok-def}
\calO_{k,\phi} \eqdef \{ i \in \calA_k^+
\mid \min[\, \|r_{i,k}\|, \,\|r_{i,k}+s_{i,k}\|\,] \geq \phi \}.
\eeqn
Observe that if, for some $i\in\calH$ and $b_i\not\in U_i\calF$, then
both $\|r_{i,k}\|$ and $\|r_{i,k}+s_{i,k}\|$ are bounded away from zero,
so $i \in \calO_{k,\phi}$ for all $k$ and all $\phi$ such that $\phi \leq
\min_{x\in\calF}\|U_ix-b_i\|$. Thus
we assume, without loss of generality, that
\beqn{binUF}
b_i\in U_i\calF \tim{ for all } i\in\calH.
\eeqn
We then obtain the following crucial error bounds.
\llem{Lip-th}{
Suppose that AS.2 and AS.3 hold. Then, for $k \geq 0$ and $L_{\max} \eqdef
\max_{i\in\calN}L_i$,
\beqn{fi-modi-N}
f_i(x_{i,k}+s_{i,k})
= m_i(x_{i,k},s_{i,k}) +
\frac{1}{(p+1)!}\Big[ \tau_{i,k} L_{\max} - \sigma_{i,k} \Big]\|s_{i,k}\|^{p+1}
\tim{ with }| \tau_{i,k} | \leq 1,
\eeqn
for all $i \in \calN$.
If, in addition, $\phi >0$ is given and independent of
$\epsilon$, then there exists a constant $L(\phi)$ independent of $\epsilon$
such that, for $\ell \in \ii{p}$,
\beqn{gf-gmod}
\| \nabla_x^\ell f_{\calN\cup\calO_{k,\phi}}(x_k+s_k)
- \nabla_s^\ell T_{f_{\calN\cup\calO_{k,\phi}},p}(x_k,s_k) \|
\leq \frac{L(\phi)}{(p-\ell+1)!} \|s_k\|^{p-\ell+1}.
\eeqn
}
\proof{
First note that, if $f_i$ has a Lipschitz continuous $p$-th derivative as a
function of $x_i=U_ix$, then \req{taylor} shows that it also has a Lipschitz
continuous $p$-th derivative as a function of $x$. It is therefore enough to
consider the element functions as functions of $x_i$.
Observe now that, for each $k$ and $i \in \calN$, AS.2 and AS.3 ensure that
\req{fi-modi-N} and the inequality
\beqn{LipN-ders}
\| \nabla_{x_i}^\ell f_i(x_{i,k}+s_{i,k})-\nabla_{s_i}^\ell T_{f_i,p}(x_{i,k},s_{i,k}) \|
\leq \frac{L_i}{(p-\ell+1)!} \|s_{i,k}\|^{p-\ell+1}
\eeqn
immediately follow from the known bounds for $p$ times continuously
differentiable functions with Lipschitz continuous $p$-th derivative (see
\cite[Lemma~2.1]{CartGoulToin18b}). Consider now $i \in \calO_{k,\phi}$ for
some $k$ and some fixed $\phi>0$, implying that
$\min[\|r_{i,k}\|,\|r_{i,k}+s_{i,k}\|] \geq \phi >0$. Then
\beqn{dersf+}
\nabla_\cdot^\ell\|r_{i,k}+s_{i,k}\|^a[d]^\ell
=\nabla_\cdot^\ell\big\| \|r_{i,k}+s_{i,k}\|u_{i,k}^+\big\|^a[d]^\ell
= \nabla_\cdot^\ell\big\| \|r_{i,k}+s_{i,k}\|u_{i,k}\big\|^a[R_{i,k}d]^\ell
\eeqn
where $R_{i,k}$ is the rotation such that $R_{i,k}u_{i,k}^+=u_{i,k}$. We
also have from \req{rotation} with $x$ replaced by $x_k+s_k$ that
\beqn{dersT+}
\nabla_{s_i}^\ell T_{m_i,p}(x_{i,k},s_{i,k})[d]^\ell
= \nabla_\cdot^\ell \big\| \|r_{i,k}\|u_{i,k} \big\|^a[R_{i,k}d]^\ell.
\eeqn
Taking the difference between \req{dersf+} and \req{dersT+}, we obtain,
successively using the definition of the tensor norm, the fact that
$R_{i,k}$ is orthonormal and \req{difdersnorm} in Lemma~\ref{dersnorm-l},
that
\[
\begin{array}{l}
\big\|\nabla_\cdot^\ell\|r_{i,k}+s_{i,k}\|^a
-\nabla_{s_i}^\ell T_{m_i,p}(x_{i,k},s_{i,k})\big\|_{[\ell]}\\*[1.5ex]
\hspace*{30mm} = \bigmax_{\|d\|=1}\left|\nabla_\cdot^\ell\|r_{i,k}+s_{i,k}\|^a[d]^\ell
-\nabla_{s_i}^\ell T_{m_i,p}(x_{i,k},s_{i,k})[d]^\ell\right|\\*[1.5ex]
\hspace*{30mm} = \bigmax_{\|d\|=1}
\left|\nabla_\cdot^\ell\big\| \|r_{i,k}+s_{i,k}\|u_{i,k}\big\|^a[R_{i,k}d]^\ell
-\nabla_\cdot^\ell \big\| \|r_{i,k}\|u_{i,k} \big\|^a[R_{i,k}d]^\ell\right|\\*[1.5ex]
\hspace*{30mm} =
\left\|\,\nabla_\cdot^\ell\big\| \|r_{i,k}+s_{i,k}\|u_{i,k}\big\|^a
-\nabla_\cdot^\ell \big\| \|r_{i,k}\|u_{i,k} \big\|^a\,\right\|_{[\ell]}\\*[1.5ex]
\hspace*{30mm} =
|\pi(a-\ell)| \left| \|r_{i,k}+s_{i,k}\|^{a-\ell}-\|r_{i,k}\|^{a-\ell}\right|.
\end{array}
\]
Now the univariate function $\nu(t)\eqdef t^a$ is (more than) $p+1$ times
continuously differentiable with bounded $(p+1)$-rst derivative on the
interval $[t_1,t_2]$ and thus, from Lemma~\ref{Lip-th}, we have that
\[
\pi(a-\ell)\left|t_1^{a-\ell}-t_2^{a-\ell}\right|
= \left|\frac{d^\ell\nu}{dt^\ell}(t_1)-\frac{d^\ell\nu}{dt^\ell}(t_2)\right|
\leq \frac{L_\nu}{(p-\ell+1)!}|t_1-t_2|^{p-\ell+1},
\]
where $L_\nu$ is the upper bound on the $(p+1)$-rst derivative of $\nu(t)$ on
interval $[t_1,t_2]$, that is
$
L_\nu = |\pi(a-p-1)|\min[t_1,t_2]^{a-p-1}.
$
As a consequence, we obtain that
\[
\big\|\nabla_\cdot^\ell\|r_{i,k}+s_{i,k}\|^a
-\nabla_{s_i}^\ell T_{m_i,p}(x_{i,k},s_{i,k})\big\|_{[\ell]}
\leq
\bigfrac{L(\phi)}{(p-\ell+1)!}\big|\|r_{i,k}+s_{i,k}\|-\|r_{i,k}\|\big|^{p-\ell+1},
\]
where we use the fact that $\min[\|r_{i,k}\|,\|r_{i,k}+s_{i,k}\|] \geq \phi$
to define
\[
L(\phi) = \max\big|\pi(a-p-1)|\phi^{a-p-1}, L_{\max}\big].
\]
We then observe that
$\|s_{i,k}\|=\|r_{i,k}+s_{i,k}-r_{i,k}\|\geq
\big|\|r_{i,k}+s_{i,k}\|-\|r_{i,k}\|\big|$
which finally yields that
\[
\big\|\nabla_\cdot^\ell\|r_{i,k}+s_{i,k}\|^a
-\nabla_{s_i}^\ell T_{m_i,p}(x_{i,k},s_{i,k})\big\|_{[\ell]}
\leq \bigfrac{L(\phi)}{(p-\ell+1)!} \|s_{i,k}\|^{p-\ell+1}.
\]
Combining this last inequality with \req{LipN-ders} and the fact that
$\nabla_{x_i}^\ell\|r_{i,k}+s_{i,k}\|^a=\nabla_\cdot^\ell\|r_{i,k}+s_{i,k}\|^a$
then ensures that \req{gf-gmod} holds.
}
\noindent
Observe that the Lipschitz constant $L$ is independent of $\phi$ whenever
$\calH = \emptyset$. Our model definition also implies the following bound.
\llem{mdecr}{
For all $k\geq 0$ before termination, $s_k \neq 0$, \req{rhok-def} is
well-defined and
\beqn{Dphi}
\Delta T_k \geq \frac{\sigma_{\min}\varsigma^{p+1}}{(p+1)!}\, \|s_k\|^{p+1}.
\eeqn
}
\proof{
We immediately deduce that
\beqn{Dphii}
\Delta T_k \geq \frac{\sigma_{\min}}{(p+1)!}\, \sum_{i \in \calN}\|s_{i,k}\|^{p+1}
\eeqn
from \req{deltaT-def}, the observation that, at
successful iterations, the algorithm enforces $\Delta m_k > 0$ and
\req{sig-decr}. As a consequence, $s_k\neq 0$. Hence at least one
$\|s_{i,k}\|$ is strictly positive because of \req{s-sumsi}, and \req{Dphii}
therefore implies that \req{rhok-def} is well-defined. The inequality
\req{Dphi} then follows from Lemma~\ref{psnorm-l}.
}
Following a now well-oiled track in convergence proofs for regularization
methods, we derive an upper bound on the regularization parameters.
\llem{sigmax}{
\cite[Lemma~4.6]{ChenToinWang17}
Suppose that AS.2 and AS.3 hold. Then, for all $i \in \calN$ and all $k \geq 0$,
\beqn{sigma-max}
\sigma_{i,k} \in [\sigma_{\min}, \sigma_{\max}],
\eeqn
where $\sigma_{\max} \eqdef \gamma_2 L_{\max}$.
}
\proof{
Assume that, for some $i \in \calN$ and $k \geq 0$, $\sigma_{i,k} \geq L_i$.
Then \req{fi-modi-N} gives that \req{up-cond} must fail, ensuring
\req{sigma-max} because of the mechanism of the algorithm.
}
\noindent
It is important to note that $\sigma_{\max}$ is independent of
$\epsilon$. We now verify that the trial step produced by Step~2 of the
PSAR$p$ Algorithm either essentially fixes the residuals $r_i$ to zero (their
value being then fixed for the rest of the calculation), or is long enough
(i.e.\ \req{longs} holds), or maintains these residuals safely away from zero
in the sense that their norm exceeds an $\epsilon$-independent constant.
\llem{away-l}{
Suppose that AS.1, AS.2, AS.3 and AS.5 hold, that $\calH \neq
\emptyset$ and that \req{longs} fails. Let
\beqn{omega-def}
\omega \eqdef
\min\left[ \frac{a}{16},
\left(\frac{a}{12|\calN|\left(\kappa_\calN
+\frac{\sigma_{\max}}{(p-q+1)!}\right)}\right)^{\frac{1}{1-a}}\right].
\eeqn
Then, if, for some $i \in \calH$,
\beqn{a-assxik}
\|r_{i,k}\| < \omega,
\eeqn
we have that
\beqn{gap}
\|r_{i,k}+ s_{i,k}\| \leq \epsilon
\tim{ or }
\|r_{i,k}+ s_{i,k}\| \geq \omega.
\eeqn
}
\proof{The conclusion is obvious if $i \in \calC_k^+= \calH\setminus\calA_k^+$.
Consider now $i \in \calA_k^+$ and suppose, for the purpose of deriving a
contradiction, that
\beqn{absurd}
\|r_{i,k}+s_{i,k}\| \in (\epsilon, \omega) \tim{ for some } i \in \calA_k^+,
\eeqn
and immediately note that
the failure of \req{longs} and the orthonormality of the rows of $U_i$ imply that
\beqn{skleq1}
\|s_{i,k}\| \le \|s_k\|< \varpi \epsilon^{\frac{1}{p-q+1}} \leq 1
\eeqn
and also that \req{term-m} must hold.
As a consequence, for some $\delta_k \in (0,1]$,
\beqn{psib-1a}
a \|r_{i,k}+s_{i,k}\| \chi_q(\delta_k)
\geq \psi_{m,q}^{\epsilon,\delta_k}(x_k,s_k).
\eeqn
Consider now the vector
\beqn{dk-def}
d_k = -\min\big[\delta_k ,\|r_{i,k}+s_{i,k}\| \big] v_{i,k}^+
\tim{with} v_{i,k}^+ = U_i^\dagger u_{i,k}^+\eqdef U_i^\dagger
\frac{r_{i,k}+s_{i,k}}{\|r_{i,k}+s_{i,k}\|}.
\eeqn
We now verify that $d_k$ is admissible for problem \req{psim-def}. Clearly
$\|d_k\| = \delta_k$ because the rows of $U_i$ orthonormal.
We also see that \req{Ui-ortho-H} and \req{Rii-def} imply that, since $i \in
\calA_k^+$,
\beqn{dkinRk+}
d_k \in \calR_{\{i\}} \subseteq \calR_k^+.
\eeqn
Moreover, we have that
\beqn{ininter}
x_k+s_k+d_k \in [\![ x_k+s_k, x_k+s_k- U_i^\dagger(r_{i,k}+s_{i,k})]\!],
\eeqn
where $[\![v,w]\!]$ denotes the line segment joining the vectors $v$ and $w$.
But
\[
\begin{array}{ll}
x_k+s_k - U_i^\dagger(r_{i,k}+s_{i,k})
& = x_k+s_k - U_i^\dagger U_i(x_k+s_k) + U_i^\dagger b_i \\
& = (I- U_i^\dagger U_i)(x_k+s_k) + U_i^\dagger b_i \\
& = P_{\ker(U_i)} [x_k+s_k] + U_i^\dagger b_i\\
& \in \calF,
\end{array}
\]
where we have used \req{kernel-centered} to deduce the last inclusion.
Since $\calF$ is convex and $x_k+s_k\in \calF$, we deduce from
\req{ininter} that $x_k+s_k +d_k \in \calF$. As a consequence, $d_k$ is
admissible for problem \req{psim-def} and hence, using \req{psib-1a},
\beqn{psib-1}
a \|r_{i,k}+s_{i,k}\| \chi_q(\delta_k)
\geq \psi_{m,q}^{\epsilon,\delta_k}(x_k,s_k)
\geq \max\left[0,m(x_k,s_k)-T_{m,q}(x_k,s_k-d_k)\right].
\eeqn
Moreover \req{dkinRk+} and \req{miH-def} imply that
\beqn{psib-2}
\begin{array}{ll}
m(x_k,s_k)-T_{m,q}(x_k,s_k-d_k)&\\*[1.5ex]
&\hspace*{-4cm} =
m_\calN(x_k,s_k)-T_{m_\calN,q}(x_k,s_k-d_k)
+ m_i(x_{i,k},s_{i,k})- m_i(x_{i,k},s_{i,k}-U_id_k)\\*[1.5ex]
&\hspace*{-4cm} \geq
-\left|m_\calN(x_k,s_k)-T_{m_\calN,q}(x_k,s_k-d_k)\right|
+ m_i(x_{i,k},s_{i,k})- m_i(x_{i,k},s_{i,k}-U_id_k).
\end{array}
\eeqn
We start by considering the first term in the right-hand side of this
inequality. Observe now that \req{a-assxik} ensures that $x_k\in \calV$ (as
defined in \req{calV-def}). Hence AS.5, \req{a-assxik} and \req{skleq1}
together imply that, for each $i\in \calN$,
\beqn{psib-3}
\begin{array}{l}
|m_i(x_k,s_k)-T_{m_i,q}(x_k,s_k-d_k)|\\*[2ex]
\hspace*{10mm}
\leq \left|\bigsum_{\ell=1}^q \frac{1}{\ell!} \nabla_x^\ell
T_{m_i,q}(x_{i,k},s_{i,k})[-U_id_k]^\ell\right|\\*[2.5ex]
\hspace*{10mm}
= \left|\bigsum_{\ell=1}^q \frac{1}{\ell!}\left(
\sum_{t=\ell}^p \frac{1}{(t-\ell)!}\nabla_x^t f_i(x_{i,k})[s_{i,k}]^{t-\ell}
+ \bigfrac{\sigma_{i,k}}{(p+1)!}\big\|\nabla_\cdot^\ell\|s_{i,k}\|^{p+1}
\big\| \right)[-U_id_k]^\ell\right|.
\end{array}
\eeqn
Using now the identity $\|U_id_k\| = \|d_k\|= \delta_k$ and the fact that
\[
\sum_{t=\ell}^p \frac{1}{(t-\ell)!} \leq 1+\chi_{p-\ell}(1) < 3,
\]
we obtain from \req{psib-3}, the triangle inequality and \req{skleq1} that
\beqn{psib-4}
|m_i(x_k,s_k)-T_{m_i,q}(x_k,s_k-d_k)|
< \bigsum_{\ell=1}^q\frac{1}{\ell!} \left(3 \|\nabla_x^t f_i(x_{i,k})\|_{[t]}
+ \bigfrac{\sigma_{i,k}}{(p+1)!}\big\|\nabla_\cdot^\ell\|s_{i,k}\|^{p+1}\big\|\right)\delta_k^\ell.
\eeqn
But we have from Lemma~\ref{dersnorm-l} and \req{skleq1} that, for
$\ell\in \ii{q}$,
\beqn{psib-5}
\big\|\nabla_\cdot^\ell\|s_{i,k}\|^{p+1}\big\|
= |\pi(p-\ell+1)|\,\|s_{i,k}\|^{p+1-\ell}
\leq \frac{(p+1)!}{(p-q+1)!},
\eeqn
and therefore that
\beqn{psib-6}
\begin{array}{ll}
\left|m_\calN(x_k,s_k)-T_{m_\calN,q}(x_k,s_k-\delta_k \|r_{i,k}+s_{i,k}\|v_{i,k}^+)\right|
& < 3|\calN|\left(\kappa_\calN+\frac{1}{(p-q+1)!}\sigma_{\max}\right)\chi_q(\delta_k)\\*[1.5ex]
& \leq \quarter a \omega^{a-1}\chi_q(\delta_k),
\end{array}
\eeqn
where we have used \req{omega-def} to derive the last inequality.
Let us now consider the second term in the right-hand side of \req{psib-2}.
Applying Lemma~\ref{miprop-l}, we obtain that
$\mu(\|r_{i,k}+s_{i,k}\|,\cdot)$ is concave between $0$ and $-\|r_{i,k}+s_{i,k}\|$
and $\mu(\|r_{i,k}\|,\cdot)$ is concave between $0$ and $-\|r_{i,k}\|$.
Therefore, because \req{a-assxik}, we may deduce that
\[
\begin{array}{ll}
m_i(x_{i,k},s_{i,k})- m_i(x_{i,k},s_{i,k}-U_id_k)
& = \mu(\|r_{i,k}+s_{i,k}\|,0) - \mu(\|r_{i,k}+s_{i,k}\|,\|U_id_k\|)\\*[1.5ex]
&\geq \nabla_\zeta^1\mu(\|r_{i,k}+s_{i,k}\|,0)\|U_id_k\|\\*[1.5ex]
&\geq \nabla_\zeta^1\mu(\|r_{i,k}\|,\|r_{i,k}+s_{i,k}\|-\|r_{i,k}\|)\|U_id_k\|\\*[1.5ex]
&\geq a\|r_{i,k}\|^{a-1}\delta_k\\*[1.5ex]
&\geq \half a \omega^{a-1}\chi_q(\delta_k),
\end{array}
\]
where the second and third inequalities result from \req{dersmi-odd}.
Combining now this inequality with \req{psib-1}, \req{psib-2} and \req{psib-6}, we deduce
that
\[
a\|r_{i,k}+s_{i,k}\|\chi_q(\delta_k)
> \half a \omega^{a-1}\chi_q(\delta_k) - \quarter a \omega^{a-1}\chi_q(\delta_k)
= \quarter a \omega^{a-1} \chi_q(\delta_k).
\]
Finally, we obtain using \req{absurd} that
\[
\omega > \quarter \omega^{a-1},
\]
which impossible in view of \req{omega-def}. Hence \req{absurd} cannot hold
and the proof is complete.
}
\noindent
This last result is crucial in that it shows that there is a ``forbidden''
interval $(\epsilon,\omega)$ for the residual's norms $\|r_i(x_k+s_k)\|$,
where $\omega$ only depends on the problem and is independent of $\epsilon$. This in turn
allows to partition the successful iterates into subsets, distinguishing
iterates which ``fix'' a residual to a near zero value, iterates with long
steps and iterates with possibly short steps in regions where the considered
objective function's $p$-th derivative tensor is safely bounded independently
of $\epsilon$. Our analysis now follows the broad outline of
\cite{ChenToinWang17} while simplifying some arguments. Focusing on the case
where $\calH \neq \emptyset$, we first isolate the set of successful iterations which
``deactivate'' a residual, that is
\[
\calS_\epsilon \eqdef \{ k \in \calS \mid
\|r_{i,k}+s_{i,k}\| \leq \epsilon
\tim{and}
\|r_{i,k}\| > \epsilon
\tim{for some} i \in \calH\},
\]
and notice that, by construction
\beqn{Seps-bound}
|\calS_\epsilon| \leq |\calH|.
\eeqn
We next define the $\epsilon$-independent constant
\[
\alpha = \threequarters \omega
\]
and
\beqn{Ss-def}
\calS_{\|s\|}
\eqdef \{ k \in \calS \mid \|s_k\| \geq \quarter \omega \}.
\eeqn
Moreover, for an iteration $k\in \calS\setminus (\calS_\epsilon \cup
\calS_{\|s\|})$, we verify that $\calA_k$ can be partitioned into
\[
\begin{array}{ll}
\calI_{\heartsuit,k} \eqdef
& \{ i \in \calA_k \mid \|r_{i,k}\| \in [\alpha,+\infty) \tim{and}
\|r_{i,k}+s_{i,k}\| \in [\alpha,+\infty)\} \\
\calI_{\diamondsuit,k} \eqdef
& \{ i \in \calA_k \mid \big( \|r_{i,k}\| \in [\omega,+\infty) \tim{and}
\|r_{i,k}+s_{i,k}\| \in (\epsilon,\alpha) \big)\\
& \hspace*{13mm} \tim{ or }
\big( \|r_{i,k}\| \in (\epsilon,\alpha) \tim{and}
\|r_{i,k}+s_{i,k}\| \in [\omega,\infty) \big)
\} \\
\calI_{\clubsuit,k} \eqdef
& \{ i \in \calA_k \mid \|r_{i,k}\| \in (\epsilon,\omega) \tim{and}
\|r_{i,k}+s_{i,k}\| \in (\epsilon,\omega)\}.
\end{array}
\]
Morever, Lemma~\ref{away-l} shows that $\calI_{\clubsuit,k}$ is always empty
and one additionally has that, if $i\in\calI_{\diamondsuit,k}$, then
\[
\|s_k\|
\geq \|s_{i,k}\|
\geq \big|\|r_{i,k}+s_{i,k}\|-\|r_{i,k}\|\big|
\geq \omega - \alpha
= \quarter \omega,
\]
implies that $k \in \calS_{\|s\|}$. Hence $\calI_{\diamondsuit,k}$ is also
empty and
\beqn{Bk-heart}
\calA_k = \calI_{\heartsuit,k}
\tim{ for } k\in \calS\setminus (\calS_\epsilon\cup\calS_{\|s\|}) \eqdef \calS_\heartsuit.
\eeqn
The next important result shows that steps at iteration belonging to
$\calS_\heartsuit$ are long enough, because they are taken over region where a
good $\epsilon$-independent Lipschitz bounds holds. Indeed, if
$\calH\neq\emptyset$ and assuming that $\epsilon\leq \alpha$, we have, for $k
\in \calS_\heartsuit$, that $\calA_k^+ = \calA_k$ and thus that
$\calW_k^+=\calW_k$ and $\calR_k^+=\calR_k$. Moreover, the definition of
$I_{\heartsuit,k} = \calA_k$ ensures that
$
\calA_k \subseteq \calO_{k,\alpha}
$
and thus that Lemma~\ref{Lip-th} (and in particular \req{gf-gmod}) guarantees
that $f_{\calW_k}$ satisfies standard derivative error bounds for functions
with Lipschitz continuous $p$-th derivative (with corresponding Lipschitz
constant $L(\alpha)$). We may therefore apply known results for such functions
to $f_{\calW_k}$. The following lemma is extracted from \cite{CartGoulToin18b}, by
specializing Lemma~3.3 in that reference to optimization of $f_{\calW_k}$ over
$\calR_k$ for functions with Lipschitz continuous $p$-th derivative
(i.e. $\beta=1$ in \cite{CartGoulToin18b}).
\llem{longs-Lip-l}{
Suppose that AS.1 -- AS.3 and AS.5 hold, that
\beqn{epsalpha}
\epsilon \leq \alpha \tim{ if } \calH \neq \emptyset
\eeqn
and consider $k \in \calS_\heartsuit$ such that the PSAR$p$ Algorithm does
not terminate at iteration $k+1$. Then
\beqn{longs-Lip}
\|s_k\| \geq \kappa_\heartsuit \epsilon^{\frac{1}{p-q+1}}
\tim{ with }
\kappa_\heartsuit
\eqdef \left(\frac{(p-q+1)!}{L(\alpha)+\theta +\sigma_{\max}}\right)^{\frac{1}{p-q+1}}.
\eeqn
}
\noindent
We may finally establish our final evaluation complexity bound by combining
our results so far.
\lthm{compl-succ-l}{Suppose that AS.1--AS.5 and \req{epsalpha} hold.
Then the PSAR$p$ Algorithm requires at most
\beqn{max-succ-its}
\left\lfloor \kappa_\calS(f(x_0)-f_{\rm low}) \epsilon^{-\frac{p+1}{p-q+1}}\right\rfloor+ |\calH|
\eeqn
successful iterations and at most
\beqn{max-evals}
\left\lfloor \left \lfloor
\kappa_S ( f(x_0)- f_{\rm low})
\left( \epsilon^{-\frac{p+1}{p-q+1}} \right)
\right \rfloor
\left(1+\frac{|\log\gamma_1|}{\log\gamma_2}\right)+
\frac{1}{\log\gamma_2}\log\left(\frac{\sigma_{\max}}{\sigma_0}\right)\right\rfloor
+ |\calH| +1
\eeqn
evaluations of $f$ and its $p$ first derivatives to return an
$(\epsilon,\delta)$-approximate $q$-th-order-necessary minimizer for problem
\req{problem}, where
\beqn{kappaS-def}
\kappa_S \eqdef \frac{(p+1)!}{\eta\sigma_{\min}\varsigma^{p+1}}
\left(\frac{(p-q+1)!}
{L(\alpha)+\theta +\sigma_{\max}}\right)^{-\frac{1}{p-q+1}}.
\eeqn
}
\proof{
Consider $k\in \calS$ before termination.
Because the iteration is successful, we obtain from AS.4, Step~3
of the algorithm and Lemma~\ref{mdecr} that
\beqn{cs-1}
f(x_0)-f_{\rm low}
\geq f(x_0)-f(x_{k+1})
= \sum_{j\in \calS_k}\Delta f_k
\geq \eta \sum_{j\in \calS_k} \Delta T_k
\geq \frac{\eta\sigma_{\min}\varsigma^{p+1}}{(p+1)!}
\sum_{j\in \calS_k}\|s_k\|^{p+1}.
\eeqn
Defining now
\[
\calS_{\epsilon,k} \eqdef \calS_\epsilon \cap \iibe{0}{k},
\ms
\calS_{\|s\|,k} \eqdef \calS_{\|s\|} \cap \iibe{0}{k}
\tim{and}
\calS_{\heartsuit,k} \eqdef \calS_\heartsuit \cap \iibe{0}{k},
\]
we verify that $\calS_{\|s\|,k}$ and $\calS_{\heartsuit,k}$ form a partition
of $\calS_k\setminus \calS_{\epsilon,k}$.
As a consequence, we have that
\[
\begin{array}{ll}
f(x_0)-f_{\rm low}
& \geq \bigfrac{\eta\sigma_{\min}\varsigma^{p+1}}{(p+1)!}
\left\{
|\calS_{\|s\|,k}|\bigmin_{j \in \calS_{\|s\|,k}}\|s_k\|^{p+1}
+ |\calS_{\heartsuit,k}|\bigmin_{j \in \calS_{\heartsuit,k}}\|s_k\|^{p+1}
\right\}\\
& \geq \bigfrac{\eta\sigma_{\min}\varsigma^{p+1}}{(p+1)!}
\left\{
|\calS_{\|s\|,k}|(\quarter \omega)^{p+1}
+ |\calS_{\heartsuit,k}|\left(\kappa_\heartsuit \epsilon^{\frac{1}{p-q+1}}\right)^{p+1}
\right\}\\
& \geq \bigfrac{\eta\sigma_{\min}\varsigma^{p+1}}{(p+1)!}
\left\{|\calS_{\|s\|,k}|+|\calS_{\heartsuit,k}|\right\}
\min\left[(\quarter \omega)^{p+1},\left(\kappa_\heartsuit \epsilon^{\frac{1}{p-q+1}}\right)^{p+1}\right]\\
& \geq \bigfrac{\eta\sigma_{\min}\varsigma^{p+1}}{(p+1)!}
|\calS_k\setminus\calS_{\epsilon,k}|\,\kappa_\heartsuit^{\frac{1}{p-q+1}}\,\epsilon^{\frac{1}{p-q+1}},
\end{array}
\]
where we have used \req{cs-1}, Lemma~\ref{psnorm-l}, \req{Ss-def} and
\req{longs-Lip} to deduce the second inequality, and the assumption that
(without loss of generality in view of \req{longs-Lip})
$\kappa_\heartsuit\leq \quarter \omega$ to deduce the last. The above
inequality yields that
\[
|\calS_k|
= |\calS_k\setminus\calS_{\epsilon,k}| + |\calS_{\epsilon,k}|
\leq \kappa_S (f(x_0)-f_{\rm low})\epsilon^{-\frac{p+1}{p-q+1}}+|\calS_{\epsilon,k}|,
\]
where $\kappa_S$ is given by \req{kappaS-def}.
Since $|\calS_{\epsilon,k}|\leq |\calS_\epsilon| \leq |\calH|$, we finally
deduce that the bound \req{max-succ-its} holds. The bound \req{max-evals}
then follows by applying Lemma~\ref{a-succ-unsucc} and observing that $f$ and its first
$p$ derivatives are evaluated at most once per iteration, plus once at termination.
We conclude our development by recalling that the above result is valid for
$\calH = \emptyset$, in which case the problem is a smooth
convexly-constrained partially-separable problem. Note that the
norm-equivalence constant $\varsigma$ occurs in \req{kappaS-def}, which
indicate that the underlying geometry of the problem's invariants subspaces
$\ker(U_i)$ may have a significant impact on complexity.
\numsection{Conclusions}\label{concl-s}
We have shown that an $(\epsilon,\delta)$-approximate
$q$-th-order critical point of partially-separable convexly-constrained
optimization with non-Lipschitzian singularities can be found at most
$O(\epsilon^{-(p+1)/(p-q+1)})$ evaluations of the objective function and its
first $p$ derivatives for any $q \in \{1,2,\ldots, p\}$ whenever the smooth
element functions $f_i$, $i\in \calN$ of the objective function are $p$ times
differentiable. This worst-case complexity is obtained via our Algorithm 4.1
(PSAR$_p$) with an $p$-th order Taylor model which uses the underlying
rotational symmetry of the Euclidean norm function for $f_\calH$ and the first
$p$ derivatives (whenever they exist) of the ``element functions'' $f_i$, for
$i\in \calM$.
Several observations are of interest. A first one is that the results remain
valid if Lipschitz continuity is not assumed on the whole of the feasible set,
but restricted to the segments of the ``path of iterates'', that is
$\cup_k[\![x_k,x_{k+1}]\!]$. While this might in general be difficult to ensure a
priori, there may be case where problem structure could help. A second
observation is that convexity of the feasible set is only used on the segments
$\cup_{i,k} [\![x_{i,k}, U_i^\dagger b_i]\!]$. Again this might be exploitable in
some cases. The third observation is that, in line with \cite{CartGoulToin18b},
it is possible to replace the Lipschitz continuity assumption by a weaker
H\"{o}lder continuity.
While it may be possible to handle non-kernel-centered
feasible sets (maybe along the lines of the discussion in
\cite{ChenToinWang17}), this remains open at this stage. Another interesting
perspective is a more generic exploitation of geometric symmetries inherent to
optimization problems: our treatment here focuses on a specific case of
rotational symmetry, but this should not, one hopes, be limitative.
\noindent
\section*{\footnotesize Acknowledgements}
\vspace*{-3mm}
{\footnotesize
Xiaojun Chen would like to thank Hong Kong Research Grant Council for grant
PolyU153000/17p. Philippe Toint would like to thank the Hong Kong Polytechnic
University for its support while this research was being conducted.
|
{
"timestamp": "2019-03-01T02:02:47",
"yymm": "1902",
"arxiv_id": "1902.10767",
"language": "en",
"url": "https://arxiv.org/abs/1902.10767"
}
|
\section{Introduction}
\label{sec:intro}
Long-range Rydberg molecules (LRRMs), first predicted nearly two decades ago \cite{Greene2000} and subsequently observed almost a decade later \cite{Bendkowsky}, are a stunning highlight of the nearly century-long study of the interactions between Rydberg atoms and neutral systems. In 1934 Edoardo Amaldi and Emilio Segr\'{e}\footnote{Who, along with Oscar D'Agostino, Ettore Majorana, Bruno Pontecorvo, and Franco Rasetti were known as the ``{Via Panisperna} Boys'', a research group led by Enrico Fermi and most known for their foundational work in nuclear physics.} observed that Rydberg atoms could be excited even when immersed in a dense gas of other atoms (``perturbers''), although at an energy shifted from that of an isolated Rydberg atom \cite{Segre}. Curiously, depending on the species of perturber, these line shifts could be either blue or red detuned from the atomic line. This behavior qualitatively contradicted the classical model, which treats the surrounding gas of polarizable atoms as a dielectric material and predicts only a red shift. Fermi resolved this mystery by developing a quantum scattering theory which introduced foundational concepts like the scattering length and zero-range pseudopotential \cite{Fermi}. His model recognized that a ground state atom's polarization potential extends over only a fraction of the Rydberg electron's enormous de Broglie wavelength, and thus it only adds a scattering phase shift to the Rydberg wave function\footnote{This standard ``origin story'' of Rydberg molecules fails to acknowledge simultaneous independent measurements performed in Rostock by F\"{u}chtbauer and coworkers \cite{Fuchtbauer1,Fuchtbauer2,Fuchtbauer3}, as was graciously pointed out to me by T. Stielow and S. Scheel from that same university}. The resulting energy shift is equivalent to that provided by a delta function potential located at the perturber and proportional to the electron-atom scattering length. This simple concept became the basis for many subsequent studies of the interactions between Rydberg atoms and neutral systems, especially after Omont generalized it to incorporate energy-dependent scattering lengths and arbitrarily high partial waves \cite{Omont}. With this pseudopotential researchers studied such diverse phenomena as collisional broadening, Rydberg quenching, $l$-changing collisions, and charge transfer \cite{LebedevFabrikant,LebFabrikant2,Fabrikant1986,Masnou-Seeuws}. Many properties of these phenomena exhibited oscillatory behavior, reflecting oscillations in the Rydberg wave function through the delta function potential.
From a different direction -- the study of excited molecular states -- hints of this same oscillatory behavior were also found, such as in the Born-Oppenheimer potential energy curves (PECs) of some excited heteronuclear dimers \cite{DuGreene87,DePrunele2,DuGreene89}. Although the energy regime of these molecular states is unlike the regime where the Fermi pseudopotential was first designed, the pseudopotential still semi-quantitatively reproduced PECs calculated through more sophisticated approaches \cite{DuGreene87,DePrunele}. Thus, by the 1980s, theorists were aware that the Fermi pseudopotential accurately described Rydberg-neutral interactions, and that excited (but not yet ``Rydberg'') molecules existed and possessed some oscillatory features. The key concepts underlying long-range Rydberg molecules (LRRMs) were therefore at hand, but it was not until the realization of Bose-Einstein condensates (BEC) in the mid-1990s that it became possible to fully forge the link between these concepts: \textit{Rydberg molecules} consisting of a Rydberg atom loosely bound to a distant perturber\footnote{ These molecules are distinct from \textit{Rydberg states of molecules} (such as H$_2$ \cite{Herzberg, Jungen}), which are ``typical'' molecules bound by traditional chemical bonds with small internuclear distances, but which have a highly excited electron. These are the molecular analogue of a Rydberg atom.}. The condensate's high density -- so that two atoms could be found at the right proximity for photoassociation -- and extremely cold temperature -- so that thermal motion would not immediately destroy the fragile molecular bond -- could provide the right conditions. This led Greene and coworkers in 2000 \cite{Greene2000} to make a key leap: they took the Fermi pseudopotential seriously as the foundation for Born-Oppenheimer PECs for a Rydberg and a ground state atom, and showed that this led to a new type of chemical bond. They calculated molecular spectra and found that these molecules were stable at internuclear distances of hundreds of nanometers. Furthermore, in some configurations the molecules would exhibit surprisingly non-trivial properties due to the high Coulomb degeneracy. Most notably, they possess large \textit{permanent electric dipole moments} and electronic wave functions resembling the trilobite fossil of antiquity\footnote{Trilobites were marine anthropods having distinctively ridged exoskeletons with three lobes down the length of the body. They declined into extinction about 250 million years ago, leaving fossils of thousands of different species worldwide.}. Soon after, a second molecular species with a wave function shaped like a butterfly were predicted \cite{HamiltonGreeneSadeghpour,KhuskivadzeJPB}. These zoomorphic colloquialisms have persisted.
The number of atomic species successfully laser cooled to ultracold temperatures grew rapidly in the last two decades, as did the ability of experimentalists to excite Rydberg atoms in such an environment \cite{BECRb, BECNa, BECCa, BECSr,HydrogenBECPRLs,HydrogenBECPRLs2,Cr,dysprosium,erbium}. It was thus only a matter of time before these molecules, if the predictions were correct, would be observed. Indeed, within a decade of Greene \textit{et al}'s prediction, the group of Tilman Pfau observed vibrational spectra of LRRMs in ultracold rubidium \cite{Bendkowsky}. This confirmed the basic veracity of the theory and sparked considerable interest in the larger community. Within a few years evidence for all the predicted molecular states in a variety of atomic species had been gathered and was accompanied by renewed theoretical interest in these exotic molecules \cite{BoothTrilobite,Butterfly}.
\subsection{Outline and related work}
This tutorial describes the current state of and future prospects for this theory. It focuses on a pedagogical description that can serve as a foundation for future exploration, and is intertwined with a summary of the experimental impetus for these theoretical developments. Although this tutorial is primarily a review of previously published material, in several places original calculations are presented. Section \ref{sec:inter} reviews the theory of electron collisions and spectroscopy in the context of Rydberg atoms and electron-atom scattering phase shifts. This section is tailored for researchers unfamiliar with Rydberg atoms, and can be skipped by the expert reader with the caveat that much of the notation used in later sections is only defined here. Section \ref{sec:primer} describes the features common to nearly all LRRMs and provides the theoretical ``skeleton'' for the following sections, which focus on three particularly interesting aspects of the molecular structure. First, section \ref{sec:polyintro} elucidates the structure and experimental signatures of polyatomic LRRMs. Next, Section \ref{sec:spinintro} incorporates all relevant spin degrees of freedom. These modify the molecular states due to the Rydberg fine structure, the hyperfine structure of the perturber, and the relativistic spin-orbit splitting of the electron-atom scattering. These are necessary for a theoretical description of similar accuracy to what is now experimentally attainable. Finally, section \ref{sec:fieldstudies} reviews how these molecules interact with and can be controlled by electric or magnetic fields. These can be applied externally in the laboratory or generated by the dipole moments of other LRRMs. Section \ref{sec:conclusions} concludes with speculations for the future.
Four other publications have similar aims as this tutorial. Ref. \cite{marcassareview} overlaps section \ref{sec:primer} and also describes a second class of highly excited molecules called Rydberg macrodimers \cite{Boisseau,Farooqi,DeiglmayrRydRyd,RaithelRydRyd}. These are formed by two Rydberg atoms bound weakly together by long-range multipole interactions and have bond lengths exceeding one micron\footnote{ Note that, unless otherwise specified, we always mean the ``trilobite''-type of Rydberg molecule bound by the Fermi pseudopotential rather than these two other similarly named molecules.}. Ref. \cite{CsReview} reviews experiments on both types of molecules in Cs, and Ref. \cite{RydbergRev} reviews experiments in the high density regime, complementing the perspective given in Sec. \ref{sec:polyintro}. Ref. \cite{Hosseinreview} also reviews both types of molecules, and it discusses some of the spin effects reviewed here. Several other relevant reviews are either highly specific (Ref. \cite{Gaj2016} describes experimental aspects of field control) or quite generic (Ref. \cite{DunningRev} covers Rydberg states of alkaline-earth atoms and Ref. \cite{GallagherPillet} discusses Rydberg interactions). This tutorial is designed to complement these reviews by tailoring the discussion to a more pedagogical focus. We use atomic units throughout except when specified.
\section{Physics background: high energy Rydberg atoms and low energy collisions}
\label{sec:inter}
At first glance, the principal components of LRRMs appear rather paradoxical. An atom in an ultracold gas, cooled to just a few hundred nanokelvin, absorbs several electron volts of energy from a laser beam. The highly excited electronic wave function swells, spanning several thousands of angstroms in diameter. The electron, storing nearly all of its energy in the Coulomb potential between it and the distant positively charged ionic core of the atom, has very low velocity and hence a large de Broglie wavelength. As such its interaction with a perturber -- whose spatial extent is dwarfed by the Rydberg wavelength -- can be described by the Fermi pseudopotential. In this way the highly energetic and spatially extended Rydberg atom and the ultra-low-energy collision between an electron and a point-like perturber conspire to form LRRMs. Of particular advantage to the theorist is that this three-body system contains only two-body interactions. The simplest, between the ionic core and the perturber, is given by the potential
\begin{equation}
\label{eq:polarizationotential}
V_\text{ion-atom}=-\frac{\alpha}{2R^4},
\end{equation}
where $\alpha$ is the perturber's polarizability and $R=|\vec R|$ is the internuclear distance. The first half of this section reviews the second, and strongest, interaction: the Coulomb attraction between the electron and the ionic core. The second half discusses the electron-perturber interaction, in particular its determination via Fermi's model by scattering phase shifts. This section relies on scattering and quantum defect theory covered in greater detail in e.g. Refs. \cite{OrangeRMP, FanoRau,FriedrichBook}.
\subsection{Rydberg atoms}
\label{sec:interryd}
We study first Rydberg atoms, notorious for their exaggerated properties such as large size, long lifetime, and powerful long-range interactions \cite{GallagherBook,Stebbings}. None of these properties can be characterized accurately without knowing the Rydberg spectrum of that particular atom. This spectrum can either be regular and predictable, as in the alkali atoms, or highly complex due to mixing between intertwined Rydberg series, as in the alkaline earth atoms. The powerful theoretical tools of multichannel quantum defect theory (MQDT) and eigenchannel $R$-matrix theory can be used to disentangle and interpret the spectra of the outer valence electron(s) \cite{ OrangeRMP,FanoRau, FanoJOSA, AymarReview1984, AymarTelmini1991,CookeCromer,Seaton}. In this section we will use quantum defect theory to understand both classes of spectra. A solid grasp of the spectrum of one-electron Rydberg states of alkali atoms is essential to understanding any aspect of Rydberg physics. We also provide a glimpse into the rich physics of two-electron Rydberg spectra, hinting at the diverse range of behavior exhibited by atoms beyond the first column of the periodic table.
\subsubsection{Spectra of alkali atoms.}
\label{subsec:alkaliryd}
Alkali atoms are ubiquitous in ultracold laboratories due to the conceptual and experimental simplicity of their sole valence electron. This electron, when excited to a Rydberg state, only interacts with the other electrons over a very small region of its total volume. It is shielded from the full attraction of the atomic nucleus by the other, more tightly bound, electrons. It is thus an excellent approximation to include the shielding, polarization, and exchange effects of the deeply bound electrons within a model potential $V(r)$, which approaches the Coulomb potential as $r$ increases and the effects of the shell electrons fade away.
The Rydberg wave function $\Psi_{nlm}(\vec r)$ is an eigenfunction of the time-independent Schr\"{o}dinger equation,
\begin{align}
\label{eq:ryd1}
E\Psi_{nlm}(\vec r)&=\left(-\frac{1}{2}\nabla_r^2 + V(r)\right)\Psi_{nlm}(\vec r);\\
\Psi_{nlm}(\vec r) &= \frac{u_{nl}(r)}{r}Y_{lm}(\theta,\phi)\label{eq:ryd2}.
\end{align}
We use spherical coordinates since $V(r)$ is spherically symmetric. This results in a separable wave function where $Y_{lm}(\theta,\phi)$ is a spherical harmonic. The quantum numbers $l$ and $m$ give the orbital angular momentum and its projection on the $z$ axis, respectively. The principal quantum number $n$ is related to the energy $E$, and fixes the number of radial nodes of a bound state of Eq. \ref{eq:ryd1} to be $n_r = n-l-1$. Together with $l$ and $m$ it defines a complete set of quantum numbers and uniquely defines the eigenenergies $E_{nl}$. All that remains is to solve the radial Schr\"{o}dinger equation,
\begin{equation}
\label{rydbergham}
0=\left(-\frac{1}{2}\frac{d^2}{dr^2}+\frac{l(l+1)}{2r^2}+V_l(r)-E_{nl}\right)u_{nl}(r).
\end{equation}
We have specialized here to an $l$-dependent model potential $V_l(r)$\footnote{The dependence of this potential operator on $l$ makes it non-local, but this creates no problems in our treatment.}. Many parameterizations of this potential exist. We have used that of Ref. \cite{Marinescu},
\begin{align}
\label{eq:modelpotential}
V_l(r) &= -\frac{Z_l(r)}{r} - \frac{\alpha_c}{2r^4}\left[1 - e^{-(r/r_c)^6}\right],\\
Z_l(r) &= 1 + (Z - 1)e^{-a_1r}-r(a_3 + a_4r)e^{-a_2r},
\end{align}
where $Z$ is the nuclear charge, $a_1$, $a_2$, $a_3$, and $a_4$ are fit parameters, $\alpha_c$ is the static dipole polarizability of the positive ion, and $r_c$ cuts off the unphysical behavior of the $r^{-4}$ potential at the origin. Ref. \cite{Marinescu} determined these parameters by fitting calculated $E_{nl}$ from Eq. \ref{rydbergham} to experimentally obtained atomic energy levels. Once $V_l(r)$ is determined one can solve Eq. \ref{rydbergham} numerically to find the energies of higher lying Rydberg states. However, this quickly grows tedious as the energy levels become densely spaced while the wave functions grow spatially diffuse. It also provides little to no information about the general properties of the spectrum. An analytic theory is thus necessary.
Quantum defect theory exploits one central idea: the highly excited Rydberg electron traverses a large domain of space, defined by $r\ge r_0$, where $V_l(r)$ is synonymous with the Coulomb potential. In this region of space the two linearly independent solutions $f_l(r)$ and $g_l(r)$ of Eq. \ref{rydbergham} are given analytically in terms of confluent hypergeometric functions\footnote{The properties of these functions are rather complicated, but they are unnecessary for the present discussion.} \cite{OrangeRMP,Seaton,StrinatiGreene}. For positive energy $E = \frac{k^2}{2}$ their asymptotic behavior at large $r$ is
\begin{align}
\label{eq:fposenco}
f_l(r)&\to (2/\pi k)^{1/2}\sin\left(kr+\frac{1}{k}\ln r + \eta_l\right),\\
\label{eq:gposenco}
g_l(r)&\to -(2/\pi k)^{1/2}\cos\left(kr+\frac{1}{k}\ln r + \eta_l\right).
\end{align}
These functions are energy normalized and include the Coulomb phase $\eta_l$ \cite{OrangeRMP}. Since they are linearly independent, any valid wave function for $r>r_0$ must be a linear combination of these two functions since the non-Coulombic potential is restricted to $r<r_0$. This superposition is written
\begin{equation}
\label{eq:positiveenergysol}
u_{nl}(r) = \mathcal{N}\left[f_l(r) - \tan\delta_l g_l(r)\right],r\ge r_0,
\end{equation}
where $\mathcal{N}$ is a normalization constant and $\delta_l$, as in standard scattering theory, is the phase shift for the $l$th partial wave. This phase reflects the mixing of solutions caused by the non-Coulomb part of $V_l(r)$, since $f_l(r)$ is the physical solution (obeying the proper boundary conditions) for the pure Coulomb potential. The ``quantum defect'' is related to this phase through $\mu_l=\delta_l/\pi$. It is readily determined by computing\footnote{Using, for example, Numerov's algorithm.} the solution $F(r)$, $0\le r\le r_0$, of Eq. \ref{rydbergham} subject to the boundary condition $F(0)=0$. $F(r)$ is obtained using a small-scale numerical calculation since this region of space is much smaller than the span of the actual Rydberg wave function, and additionally it can be computed at a small, but arbitrary, positive energy. Since we are interested in highly excited Rydberg states at energies satisfying $1\gg|E|$, we expect over this small $r$ range that $V(r)$, with its massive Coulomb forces, dominates the total energy: $|V(r<r_0)|\gg |E|$. As a result $F(r)$ is insensitive to the exact value, and even the sign, of the energy for small $r$. Matching $F(r)$ along with its derivative $F'(r)$\footnote{Throughout this tutorial, a primed function represents its derivative with respect to its full argument, $f'(x) = \frac{df(x)}{dx}$.} to the asymptotic solutions at $r_0$ determines $\mu_l$:
\begin{equation}
\label{eq:defnmu}
\tan\pi\mu_l = \left.\frac{F'(r)f_l(r) - F(r)f_l'(r)}{F'(r)g_l(r)-F(r)g_l'(r)}\right|_{r=r_0}.
\end{equation}
Eqs. \ref{eq:fposenco}-\ref{eq:defnmu} have set the form of the wave function and, by requiring continuity at $r_0$, applied one of its two boundary conditions. Furthermore, since $F(r)$ is nearly independent of $E$ and the energy-normalized solutions used to construct the wave function are ``nearly analytic'' in energy \cite{Seaton}, $\mu_l$ is a very smooth function of energy. We have thus combined a small numerical calculation over the region of complicated non-Coulomb potential with the analytic Coulomb functions valid outside of this range to obtain a parameter, the quantum defect $\mu_l$, which is essentially constant for all Rydberg states of interest (those having $n\gtrapprox10$). The next step is to determine the quantized Rydberg eigenspectrum in terms of $\mu_l$ by analytically imposing the second boundary condition: the bound-state wave function must be normalizable.
We first need the asymptotic wave function (Eq. \ref{eq:positiveenergysol}) at negative energy, $E = -\frac{k^2}{2} = \frac{(i\kappa)^2}{2}$. The regular and irregular functions can be obtained by careful analytic continuation $k\to i\kappa$ of the exact solutions\footnote{There are considerable technical details involved due to a small non-analyticity in the solutions at $E = 0$ and the desire for real solutions both below and above threshold. Refs. \cite{OrangeRMP,Seaton,StrinatiGreene} present several different approaches using the properties of confluent hypergeometric functions or WKB theory arguments. It is for this reason that one cannot simply analytically continue the asymptotic solutions in Eqs. \ref{eq:fposenco} and \ref{eq:gposenco}, although one can get the essence by considering how the $kr$ and $\frac{1}{k}\ln r$ terms lead to terms of the form $e^{\pm \kappa r}$ and $r^{\pm \nu}$ in Eqs. \ref{eq:fnegenco} and \ref{eq:gnegenco}.}, giving
\begin{align}
\label{eq:fnegenco}
f_l&\to \left(\pi\kappa\right)^{-1/2}\left(\sin\beta D^{-1}r^{-\nu}e^{\kappa r} - \cos\beta Dr^\nu e^{-\kappa r}\right),\\
\label{eq:gnegenco}
g_l&\to -\left(\pi\kappa\right)^{-1/2}\left(\cos\beta D^{-1}r^{-\nu}e^{\kappa r} +\sin\beta Dr^\nu e^{-\kappa r}\right),
\end{align}
where $\nu = \kappa^{-1}$ and $\beta = \pi(\nu -l)$. $D$ is a parameter\footnote{The explicit form of $D$, which depends on $\nu$ and $l$, is not needed in any of what follows}. The asymptotic function $u_{nl}(r)$ follows by inserting these expressions into Eq. \ref{eq:positiveenergysol}:
\begin{align}
u_{nl}(r)
\to\frac{\mathcal{N'}}{\cos\pi\mu_l}&\Bigg[\sin(\beta + \pi\mu_l)D^{-1}r^{-\nu} e^{\kappa r}\\&-\cos(\beta+\pi\mu_l)Dr^\nu e^{-\kappa r}\Bigg].\nonumber
\end{align}
To ensure that $u_{nl}(r)$ is normalizable the diverging $e^{\kappa r}$ term of this solution must be totally eliminated. This is possible if its coefficient, $\sin(\beta + \pi\mu_l)$, vanishes. Thus,
\begin{equation}
\pi(\nu - l+\mu_l) = N\pi,\,\,N\in\mathcal{Z}.
\end{equation}
We define $n=N+l$, and hence,
\begin{equation}
\label{eq:quantumdefect1}
E_{nl} = -\frac{\kappa^2}{2} = -\frac{1}{2\nu^2} = -\frac{1}{2(n-\mu_l)^2}.
\end{equation}
This quantization condition is the famed Rydberg formula defining the infinite number of bound state energies of Eq. \ref{rydbergham}. To improve this formula one can add small correction terms to the quantum defect to compensate for the weak energy dependence ignored in this derivation (see Eq. \ref{eq:quantumdefects}). We obtain a simple analytic expression for the corresponding eigenfunctions by noticing that Eq. \ref{eq:positiveenergysol} can now be written
\begin{equation}
\label{eq:newsuperposition}
u_{nl}(r) = -f_l(r)\cos\beta - g_l(r)\sin\beta.
\end{equation}
For non-zero quantum defects the coefficient of $g_l(r)$ does not vanish, and hence this linear combination diverges as $r\to 0$. However, for our application -- \textit{long-range} Rydberg molecules -- we never need the exact Rydberg wave function at such small distances\footnote{And if this is ever required, then a numerical solution of Eq. \ref{rydbergham} is straightforward using Numerov's algorithm since the eigenenergy is already determined.}. The superposition in Eq. \ref{eq:newsuperposition} is related to the Whittaker function\footnote{WhittakerW$[\dots]$ in Mathematica.} \cite{OrangeRMP,WhitWats} through
\begin{equation}
\label{fdefradial}
u_{nl}(r) = \frac{W_{\nu,l+1/2}\left(\frac{2r}{\nu}\right)}{\sqrt{\nu^2(\Gamma(\nu+l+1)\Gamma(\nu-l)}}.
\end{equation}
This wave function is normalized so that $1 = \int_{r_0}^\infty |u_{nl}(r)|^2\dd{r}$, i.e. we ignore the tiny contribution from $0\le r \le r_0$ that is present in the exact wave function.\footnote{The expression in Ref. \cite{OrangeRMP} is energy-normalized. To go between normalization conventions one can multiply the energy-normalized functions by $(\nu^3+\frac{d\mu_l}{d\varepsilon})^{-1/2}$. We have used the approximate conversion factor $\nu^{-3/2}$ since the quantum defects are essentially constant in energy. }
The key point of this discussion is that, since the interaction of a Rydberg electron with the ionic core extends over such a small range, the key differences between its spectrum and that of hydrogen are encapsulated by a few essentially energy-independent quantum defects. These can be determined numerically, using Eq. \ref{eq:defnmu}, or fit to measured energies. As a result, Rydberg wave functions and energy levels are excellently described entirely analytically using Eqs. \ref{fdefradial} and \ref{eq:quantumdefect1}, respectively.
\begin{table*}[ht]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|| c|| c c| c|| c c| c|| c c||}
\hline
{\bf Li} & $\mu(0)$ & $\mu'(0)$ &{\bf Na} & $\mu(0)$ & $\mu'(0)$ &{\bf K} & $\mu(0)$ & $\mu'(0)$ \\
\hline \hline
$s_{1/2}$& 0.3995101 & 0.0290 & $s_{1/2}$& 1.347964 & 0.060673 & $s_{1/2}$& 2.1801985 & 0.13558 \\
$p_{1/2}$ & 0.0471780 & -0.024 & $p_{1/2}$& 0.855380 & 0.11363 & $p_{1/2}$& 1.713892 & 0.233294 \\
$p_{3/2}$ & 0.0471665 & -0.024 & $p_{3/2}$& 0.854565 & 0.114195 &$p_{3/2}$& 1.710848 & 0.235437 \\
$d_{3/2}$ & 0.002129 & -0.01491 & $d_{3/2}$& 0.015543 & -0.08535 & $d_{3/2}$& 0.2769700 & -1.024911 \\
$d_{5/2}$ & 0.002129 & -0.01491& $d_{5/2}$& 0.015543 & -0.08535 & $d_{5/2}$& 0.2771580 & -1.025635 \\
$f_{5/2}$ & -0.000077 &0.021856 & $f_{5/2}$& 0.0001453 & 0.017312 & $f_{5/2}$& 0.010098 & -0.100224 \\
$f_{7/2}$ & -0.000077 &0.021856 & $f_{7/2}$& 0.0001453 & 0.017312 & $f_{7/2}$& 0.010098 & -0.100224 \\
\hline
{\bf Rb} & $\mu(0)$ & $\mu'(0)$ &{\bf Cs} & $\mu(0)$ & $\mu'(0)$ & {\bf Sr} & $\mu(0)$ & $\mu'(0)$\\
\hline \hline
$s_{1/2}$& 3.1311804 & 0.1784 & $s_{1/2}$& 4.049325 & 0.2462 & $5sns^3$S$_1$ &3.371 & 0.5\\
$p_{1/2}$ & 2.6548849 & 0.2900 & $p_{1/2}$& 3.591556 & 0.3714 & $5snp^3$P$_{2(1)}$ & 2.8719 (2.8824) & 0.446(0.407) \\
$p_{3/2}$ & 2.6416737 & 0.2950 & $p_{3/2}$& 3.559058 & 0.374 & $5snp^3P_0$ & 2.8866 & 0.44\\
$d_{3/2}$ & 1.34809171 & -0.60286& $d_{3/2}$& 2.475365 & 0.5554 & $5snd^3D_{3(2)}$ &2.612(2.662) & $-41.4(-15.4)$\\
$d_{5/2}$ & 1.34646572 & -0.59600& $d_{5/2}$& 2.466210 & 0.067 & $5snd^3D_1$ & 2.673 & -5.4\\
$f_{5/2}$ & 0.0165192 & -0.085 & $f_{5/2}$& 0.033392 & -0.191 & $5snf^3F_{4(3)}$ & 0.120(0.120) & -2.4(-2.2)\\
$f_{7/2}$ & 0.0165437 & -0.086 & $f_{7/2}$& 0.033537 & -0.191 & $5snf^3F_2$ & 0.120 & -2.2 \\
\hline \hline
& $\alpha$ (a.u.) & $\alpha_c$ (a.u.)& & $\alpha$ (a.u.) & $\alpha_c$ (a.u.) & & $\alpha$ (a.u.) & $\alpha_c$ (a.u.)\\
\hline
{\bf Li} & 164.9$^{a},164.2^b$ & 0.1923 \protect\cite{Marinescu} & {\bf Na} & 165.9$^a$,162.7$^b$ & 0.9448 \protect\cite{Marinescu} & {\bf K} & 307.5$^a$,290.6$^b$ & 5.3310\protect\cite{Marinescu}\\
{\bf Rb} & 319.2$^{b}$ & 9.12 \protect\cite{Gallagher2016a}& {\bf Cs} &401$^b$ & 15.544 \cite{GallagherCS} & {\bf Sr}&186$^c$ & 86 \protect\cite{SrPlusPol} \\
\hline
\end{tabular}}
\caption{Quantum defect parameters (used in Eq. \ref{eq:quantumdefects}) $\mu$, $\mu'$ and polarizabilities $\alpha$, $\alpha_c$ for the alkalis and strontium.
The quantum defects are from Ref. \protect\cite{GoyLi} and Ref. \protect\cite{NiemaxLi} (Li), Ref. \protect\cite{NiemaxLi} (Na and K), Ref. \protect\cite{LiRb,JamilRb} (Rb), Ref. \protect\cite{GoyCs,MoreCs} (Cs), and Refs. \protect\cite{SrQD1,SrQD2,SrQD3,SrQD4,DingSpec} (Sr). The different polarizabilities are calculated$^a$ using the model potential or measured in $^b$Ref. \protect\cite{updatedPols,Miffre2006,CsPolHigh,Polarizabilities} and$^c$ Ref. \protect\cite{SrPol}. Due to the multichannel physics discussed in the text the quantum defects for Sr are not guaranteed to be energy-independent over the entire Rydberg series, and should be confirmed for a given Rydberg state. }
\label{tab:datatable2}
\end{table*}
A more realistic treatment of the Rydberg atom must include relativistic fine structure effects. One of these, the $p^4$ correction to the kinetic energy, is automatically included in the model potential since it is fit to empirical energies which intrinsically include this shift. More complicated is the spin-orbit coupling between the electron's orbital and spin angular momenta, which can be included with an additional model potential of the form \cite{GreeneAymar},
\begin{equation}
V_{so}^{slj}(r) = \frac{(g-1)\alpha_\text{FS}^2}{2}\frac{1}{r}\frac{dV_l(r)}{dr}\left[\tilde{V_l}(r)\right]^{-2}\vec s_1\cdot\vec l,
\end{equation}
where $\tilde{V_l}(r) = 1 - \frac{(g-1)\alpha_{FS}^2}{2}V_l(r)$, $\alpha_\text{FS}$ is the fine-structure constant, $\vec s_1$ is the electron spin operator, and $g$ is the electron $g$-factor. This potential couples the orbital and spin angular momenta, but it is diagonal in the total angular momentum $\vec j = \vec s_1+ \vec l$ and its projection $m_j$. We can generalize the Rydberg formula, Eq. \ref{eq:quantumdefect1}, to include the spin-orbit splitting by making it $j$-dependent:
\begin{equation}
\label{eq:Rydbergformula}
E_{n(s_1l)jm} = -\frac{1}{2(n-\mu_{(s_1l)j}(n))^2}.
\end{equation}
To better match observed energy levels we use quantum defects with a linear energy dependence,
\begin{equation}
\label{eq:quantumdefects}
\mu_{(s_1l_1)j}(n) = \mu_{(s_1l_1)j}(0) + \frac{\mu'_{(s_1l_1)j}(0)}{\left[n - \mu_{(s_1l_1)j}(0)\right]^2}.
\end{equation}
Table \ref{tab:datatable2} displays these quantum defect parameters for low-$l$ Rydberg levels. For higher angular momenta, the quantum defects are determined almost entirely by the core polarization $\alpha_c$. Their values,
\begin{align}
\mu_l(n) &= \left(\frac{\alpha_c[3n^2 - l(l+1)]/4}{n^2(l-1/2)l(l+1/2)(l+1)(l+3/2)}\right)\label{eq:coredefects},
\end{align}
are given by calculating perturbatively the first-order energy shift of the polarization potential.
The fine structure splitting for these nonpenetrating high-$l$ ($l>3$) states is given by the formula obtained for hydrogen using the Dirac equation,
\begin{equation}
\label{eq:finestrucsplitting}
\Delta E_{n(s_1l)jm} = -\frac{\alpha^2}{2n^3}\left(\frac{1}{j+1/2} - \frac{3}{4n}\right).
\end{equation}
Eqs. \ref{eq:quantumdefects}- \ref{eq:finestrucsplitting} thus fully specify the Rydberg spectrum of an alkali atom.
To complete this discussion we give the spin-dependent electronic wave function,
\begin{equation}
\label{eq:jdepefuncs}
\Psi _{n(ls_1)jm_j}(\vec r) = \sum_{m,m_1}C_{lm,s_1m_1}^{jm_j}\frac{u_{nlj}(r)}{r}Y_{lm}(\hat r)\chi_{m_1}^{s_1},
\end{equation}
where $\chi_{m_1}^{s_1}$ is the Rydberg electron's spin state. $u_{nlj}(r)$ is equivalent to $u_{nl}(r)$ but with $j$-dependent $\nu$.
\subsubsection{The multichannel Rydberg spectra of two-electron atoms.}
\label{MQDTrydbergs}
The previous section showed that a set of nearly energy-independent quantum defects defines the Rydberg spectrum of alkali atoms. We now introduce the basic concepts of multichannel Rydberg systems by considering the spectra of atoms with two valence electrons. Interest in such atomic species dates back decades \cite{OrangeRMP,AymarReview1984,GreeneAymar}, and has recently resurged in theoretical studies of Rydberg interactions \cite{Vaillant,LRinteractionsFrancisBooth,Eiles2015} and in ultracold Rydberg spectroscopy of atoms such as Sr, Ho, and Yb \cite{DingSpec,SrMeas2,holmium,YtRyd}. The discussion here is intended to spark increased interest in the possibilities of these atoms in the context of LRRMs and to present the reader with a more complete picture of the richness and intricacy of Rydberg atoms.
The non-relativistic Hamiltonian for the two valence electrons is
\begin{equation}
\label{twoelec1}
H_{2e} = -\frac{1}{2}\nabla_{r_c}^2 - \frac{1}{2}\nabla_r^2 + V_{l_c}(r_c) + V_{l}(r) + \frac{1}{|\vec r_c - \vec r|}.
\end{equation}
We have labeled the position operators of the two electrons $\vec r_c$ and $\vec r$ to clarify the following essential point: we focus on energy regimes far below the double ionization threshold, and hence the two-electron wave function $\Psi(\vec r_c,\vec r)$ vanishes when $r_c>r_0$. The size of $r_0$ is set by the spatial extent of the excited core states we wish to include, but typically is a few tens of atomic units. The model potential $V_{l_i}(r_i)$ is similar to Eq. \ref{eq:modelpotential}, except that it is modified to represent a doubly, rather than singly, charged positive ion\footnote{Ref. \cite{OrangeRMP} gives some explicit expressions and fit parameters for the alkaline-earth atoms.}.
To efficiently describe the six-dimensional $\Psi(\vec r_c,\vec r)$ we define a set of \textit{channel functions}, $\Phi_i(\omega)$, where $\omega$ refers to all coordinates except for $r$, and $i$ is a set of quantum numbers defining each channel. By definition, $\Phi_i(\omega)$ is an eigenfunction of a smaller Hamiltonian, $H'$, involving only the coordinates $\omega$. Since $H'$ is spherically symmetric, it shares eigenstates with $\vec l_c^2$, $\vec l^2$, $\vec L^2$, and $L_z$, where $L$ is the total orbital angular momentum. The eigenvalue equation for the channel functions $\Phi_{i=l_clLM,\epsilon_{l_c}}(\omega)$ is
\begin{align}
\nonumber\left(\left[-\frac{\nabla_c^2}{2} + V_{l_c}(r_c)\right] + \frac{\vec l^2}{2r^2}\right)&\Phi_{l_clLM,\epsilon_{l_c}}(\omega) \\= \left(\epsilon_{l_{c}} + \frac{l(l+1)}{2r^2}\right)&\Phi_{l_clLM,\epsilon_{l_c}}(\omega),
\end{align}
where $\epsilon_{l_c}$ is the eigenenergy of the inner electron whose Hamiltonian is contained in the square brackets. These channel functions define a complete set of basis functions to expand the full wave function into:
\begin{equation}
\label{channelexp}
\Psi_{i'}(\vec r_c,\vec r) = \mathcal{A}\sum_ir^{-1}\Phi_i(\omega)F_{ii'}(r).
\end{equation}
$\mathcal{A}$ denotes the antisymmetrization operator and $F_{ii'}(r)$ is the $i'$th linearly independent radial wave function in the $i$th channel. A matrix of radial solutions $F_{ii'}$ is necessary because, after imposing boundary conditions at the origin but before imposing boundary conditions at infinity, an $N$-channel Schr\"{o}dinger equation has $N$ independent solutions in each channel. These unknown radial functions are found by projecting $H_{2e}\Psi(\vec r_c,\vec r)$ onto the channel functions $\Phi_i(\omega)$ to obtain the \textit{coupled-channel equations},
\begin{align}
\label{eq:outersolns}
\nonumber&\sum_{j=1}^N\Bigg[\left(-\frac{1}{2}\frac{d^2}{dr^2} +\frac{l_{i}(l_{i}+1)}{2r^2}+V_{l_i}(r)+\frac{1}{2\nu_i^2}\right)\delta_{ij}\\& \,\,\,+\int\frac{\Phi_i^*(\omega)\Phi_j(\omega)\dd{\omega}}{|\vec r_c-\vec r|}\Bigg] F_{ji'}(r) = 0,\,\,r\le r_0
\end{align}
where we have truncated to $N$ channels. Exchange effects recrease rapidly for $r >r_0$ and are neglected in these equations.
The channel-dependent quantum number $\nu_i$ defines the energy of the outer electron via $(E - \epsilon_{i})=-\frac{1}{2\nu_i^2}$, where $E$ is the total energy.
Now that the channel structure of the wave function is laid out, we can generalize the single channel equations. For $r>r_0$, $V_{l_i}(r) = -\frac{2}{r}$, and using the multipole expansion of $1/|\vec r_c - \vec r|$ we find that the coupling term in the second line of Eq. \ref{eq:outersolns} is, to first order, a diagonal Coulomb potential $\frac{1}{r}\delta_{ij}$\footnote{Higher multipolar coupling terms can be ignored for now provided $r_0$ is not too small, and they can treated perturbatively later if necessary.}. Thus, the coupled channel equations decouple into radial Coulomb-Schr\"{o}dinger equations asymptotically. Following Eq. \ref{eq:positiveenergysol}, we express $F_{ii'}(r)$ as a linear combination of $f_i(r)$ and $g_i(r)$, the two linearly independent solutions in channel $i$. The multichannel generalization of Eq. \ref{eq:positiveenergysol} is \cite{OrangeRMP}
\begin{align}
\label{mqdtlongrange}
\Psi _{i^{\prime }}&=\mathcal{A}\sum_{i}\Phi _{i}(\omega )\left[f_i(r)\delta_{ii'}-g_i(r)K_{ii'}(r)\right],r>r_0,
\end{align}
where $\underline{K}$, the \textit{reaction matrix}, is related to the phase shift matrix through the equation $\underline{K} = \tan\underline{\delta}$ \footnote{We could also use the scattering matrix $\underline{S} = e^{2i\underline{\delta}}$ to set up this derivation. By converting this exponential function into trigonometric form we can identify the relationship between the scattering and reaction matrices,
\begin{equation}
\underline{K} = \frac{i(\underline{I} - \underline{S})}{\underline{I} + \underline{S}}. \nonumber
\end{equation}
We prefer the $K$-matrix formalism because all arithmetic is explicitly real.}.
The form of this equation provides physical intuition for the mathematical statement above about the number of linearly independent solutions. As the Rydberg electron, in channel $i$, careens into the ion and interacts with the inner electron, it swaps angular momentum and energy with this electron and exits the interaction region in channel $i'$ through the matrix element $K_{ii'}$.
As in the single-channel case, we impose long-range boundary conditions by eliminating the exponential growth of $f_i$ and $g_i$ as $r\to\infty$. Using Eqs. \ref{eq:fnegenco} and \ref{eq:gnegenco} reveals the relevant exponential terms, $f_i\to D\sin\beta_ie^{\kappa_ir}r^{-\nu_i}$ and $g_i\to-D\cos\beta_ie^{\kappa_ir}r^{-\nu_i}$. As before, $\underline\beta = \pi(\underline\nu - \underline l)$. We utilize the flexibility to choose any superposition of linearly independent solutions $\Psi_{i'}$ to form a wave function $\Psi = \sum_{i'}\Psi_{i'}B_i'$. At large $r$,
\begin{equation}
\label{sdada}
\Psi\to De^{\underline{\kappa}r}r^{-\underline{\nu}}(\sin\underline\beta + \cos\underline\beta \underline K)\vec B,
\end{equation}
where all matrices except for $\underline{K}$ are diagonal.
This expression must vanish, and so $(\sin\underline\beta + \cos\underline\beta \underline K)\vec B=0$. This system of equations has a non-trivial solution if the determinant vanishes,
\begin{equation}
\label{deteqn}
\det(\tan\underline\beta + \underline K) = 0.
\end{equation} This defines a relationship between the $\nu_i$ which, in combination with energy conservation fully determines the energies. After Eq. \ref{deteqn} is solved we can set $\underline K\vec B= -\tan\underline\beta\cdot\vec B$, and hence
\begin{align}
\label{mqdtlongrange}
\Psi &=\mathcal{A}\sum_{i}\Phi _{i}(\omega )\left[f_i(r)\cos\pi\beta_{i}+\sin\pi\beta_{i}g_i(r)\right]\frac{B_{i}}{\cos\beta_{i}}.
\end{align}
This linear combination of $f_i$ and $g_i$ is just the channel $u_{nl}(r)$ function, and so:
\begin{align}
\label{MQDTwf}
\Psi(\vec r_c,\vec r) &= \sum_i\frac{1}{r}\Phi_i(\omega)u_{n_il_i}(r)\frac{-B_i}{\cos\beta_i},\,\,r>r_0
\end{align}
This wave function, due to channel coupling from the electron-electron interaction, is a mixture of channel functions weighted by the coefficients $-B_i/\cos\beta_i$.
\subsubsection{Determination of the $K$ matrix and Lu-Fano plots.}
Eq. \ref{deteqn} and \ref{MQDTwf} show that we can obtain the energies and wave functions of multichannel Rydberg states from the $K$-matrix through a similar, but algebraically more involved, process as in the single channel case. We now turn to the practical matter of how to obtain $\underline{K}$, focussing on a semi-empirical method which also illustrates some important concepts of these multichannel Rydberg states. Ref. \cite{OrangeRMP} explains the nearly \textit{ab initio} determination of $\underline{K}$ using the $R$-matrix method, which is also briefly summarized in the context of electron-atom scattering in the following section.
Of critical importance is the representation that diagonalizes or approximately diagonalizes the Hamiltonian also diagonalizes $\underline K$. In the previous discussion we constructed channels using the $LS$-coupling scheme: the orbital ($l_c$ and $l$) and spin ($s_c$ and $s$) angular momenta of the two electrons are coupled separately to form $L$ and $S$. These are subsequently coupled to form the total angular momentum $J$, and the channel functions are $\ket{(l_cl)L(s_cs)S]JM_J}$. $H$ is approximately diagonal in this coupling scheme since non-relativistic effects are so far ignored, and therefore so is $\underline{K}$, $K_{ii^{\prime
}}^{(LS)}=\delta _{ii^{\prime }}\tan \pi \mu _{i}$.
We illustrate this with an example: the $J=0$ Rydberg states of silicon, which has the ground state configuration Ne $3s^2 3p^2$. For each parity there are two relevant $LS$-coupled channels: $^3P^e(d)$, $^3P^e(s)$, $^1S^o(p)$, and $^3P^o(p)$, where $(l)$ labels the Rydberg electron's angular momentum and $^{2S+1}L^\pi$ is the standard term symbol. An approximately energy-independent $K$-matrix is extracted from measured energy levels for these four configurations \cite{BrownGinterGinter,BGG2,BGG3}. These quantum defects are nearly constant in energy over several low-lying excited states, confirming the basic principle of quantum defect theory.
This approach is disrupted by the spin-orbit splitting of $\Delta E=35.7$meV between the $j_c = 1/2$ and $j_c = 3/2$ states of the Si$^+$ ion, where $j_c=l_c+s_c$. When the Rydberg electron is near the core this splitting is dominated by the strong electrostatic and exchange interactions, and since the long-range potential far from the core is still a purely diagonal Coulomb potential one might naively think that this splitting has essentially no effect on the Rydberg spectrum. However, energy conservation requires that the total energy $E$ of the system be partitioned between the two electrons, and hence the channel quantum numbers are defined relative to these two different thresholds $\epsilon_{3/2}$ or $\epsilon_{1/2}$:
\begin{equation}
E = \epsilon_{3/2} - \frac{1}{2\nu_{3/2}^2} = \epsilon_{1/2} - \frac{1}{2\nu_{1/2}^2}.
\end{equation}
Clearly, the kinetic energy available to the Rydberg electron depends very sensitively on the state of the inner electron, which in turn causes the electron to accumulate phase at very different rates depending on the $j_c$ state of the core. $LS$ coupling is fundamentally unable to include this non-perturbative effect as $j_c$ is not a good quantum number in this coupling scheme, and so we must use a different set of quantum numbers for the long-range behavior of the Rydberg electron. The $jj$-coupling scheme
represented by the ket $|[(l_{c}\frac{1}{2})j_{c}(l\frac{1}{2}%
)J_e]jM_{J}\rangle $ can accomplish this since it explicitly labels the $j_c$ and $j=l+s$.
This is an example of a very generic problem in atomic and molecular physics: we have a physical system obeying different symmetries, and therefore described by different sets of quantum numbers, in different regions of space. It can be tackled using the powerful technique of a frame transformation \cite{ChrisFra,LeeLu,OrangeRMP}. In the present case, it is only once $r\sim1000a_0$ that the channel radial wave functions begin to dephase. We can still use $LS$-coupling at small $r$ to obtain a $K$-matrix, and then in the region where both coupling schemes are roughly equivalent we can simply project the wave function in the $LS$-coupling scheme onto the $jj$-coupling scheme. In the present case, this projection is effected by a \textquotedblleft geometric\textquotedblright\ orthogonal frame
transformation matrix $U_{ij}$. This recoupling matrix rotates the $K$-matrix from the $LS$ coupling scheme to the $jj$-coupled representation \cite{LeeLu} and is given by standard angular momentum algebra \cite{Varsh}:
\begin{align*}
U_{ij} &= \sqrt{\lfloor j_c \rfloor\lfloor j_e \rfloor\lfloor L\rfloor\lfloor S\rfloor}\left\{\begin{array}{ccc} l_e &s_e &j_e\\l_c & s_c & j_c\\ L & S & J\end{array}\right\},
\end{align*}
where $\lfloor x\rfloor = (2x+1)$ and $\{\dots\}$ is a Wigner 9J Symbol. The $jj$-coupled $K$ matrix is obtained via $%
K_{ii^{\prime }}^{(jj)}=\sum_{jj'}U_{ij}K_{jj^{\prime }}^{(LS)}U_{j^{\prime }i^{\prime
}}^{\dagger }$.
\begin{figure}[t]
\begin{centering}
{\normalsize \includegraphics[width = \columnwidth]{siliconLuFano-min.png}
\vspace{-20pt}
}
\caption{Lu-Fano plots for a) $J=0$, b) $J=1$, c) $J=2$, and d) $J=3$
symmetries. Blue points are $l\approx 0$ odd parity; red are $l \approx 1$ even parity, and green are $l \approx 2$ odd parity.
Intersections of the solid curves (Eq. \ref{deteqn}) with the diagonal lines (Eq. \ref{qnumrel}; only a few representative ones are shown) give the positions of bound states (points). This figure is modified from Ref. \cite{Eiles2015}. }
\label{lufanoJs}
\end{centering}
\end{figure}
We thus transform the $LS$-coupled $K$-matrix obtained from experimental energy levels into $jj$-coupling, and then solve Eq. \ref{deteqn} to obtain the Rydberg series leading to each ionization threshold. These series are labeled by the principal quantum numbers in each channel, which are related by energy conservation,
\begin{equation}
\label{qnumrel}
\nu_{1/2}(\nu_{3/2}) =\left(\nu_{3/2}^{-2}-2\Delta E\right)^{-1/2}.
\end{equation}
In a system with two thresholds a Lu-Fano plot, shown in Fig. \ref{lufanoJs}, graphically illustrates the behavior of the quantum defects. The solutions of Eq. \ref{deteqn} are colored curves, while Eq. \ref{qnumrel} determines the black lines. We show only a few representative ones in Fig. \ref{lufanoJs}a. At the intersections of these curves lie bound states \cite{GreeneKim,FrancisGreene}. Since the only relevant information contained in the quantum defect is its non-integer part, we collapse all energy levels onto a single curve by plotting $\nu_{3/2}$ and $\nu_{1/2}$ modulo one.
The Lu-Fano plots contain a great deal of information about the channel couplings and behavior of this Rydberg system. In the $J=0$ case exemplified here, we see that the even parity Rydberg series (where the Rydberg electron is either in an $s$ or a $d$ state) are essentially two uncoupled Rydberg series, since the bound states lie on straight lines. This means that the quantum defect in one channel is independent of the other, and these series are effectively single-channel. The odd parity curves, on the other hand, are not flat; a pronounced avoided crossing reveals strong channel interactions. For bound states around this avoided crossing the mixing coefficients in Eq. \ref{MQDTwf} will be significant, leading to wave functions which mix angular momentum as well as levels of radial excitation, since states with very different principal quantum numbers mix. These channels mix strongly because an energy level (if they could be treated independently) in one channel is nearly degenerate with one in a channel corresponding to the other threshold. Away from this avoided crossing, the curves are approximately flat: these are energetically isolated Rydberg states that are predominantly single-channel. With these tools for multichannel systems in hand: the graphical analysis provided by the Lu-Fano plot, the powerful set of approximations contained in the frame transformation, and the multichannel spectrum and wave functions determined by Eqs. \ref{deteqn} and \ref{MQDTwf}, one can determine the rich spectrum of multichannel Rydberg systems.
\subsection{Electron-atom scattering phase shifts}
\label{sec:interphases}
We now study the scattering of a very low energy electron from a neutral atom. Through the partial wave decomposition this process is described by a collection of phase shifts. At low energy only a few partial waves are relevant, and the Fermi pseudopotential described in the following section utilizes this simplicity to parametrize the interaction of a Rydberg electron with an atom in terms of just $s$ and $p$-wave phase shifts. Since these phases directly determine the properties of LRRMs, it is paramount that they be computed accurately. This section outlines this calculation and discusses the properties of these phase shifts in the alkali atoms relevant to LRRMs.
\subsubsection{Details of the calculation.}
We keep the basic philosophy undergirding the previous section: the multidimensional coordinate space can be partitioned into two regions. In a small volume around the atomic core the system's dynamics are complicated due to the strong interactions between the scattered electron and the atomic electrons. We use the $R$-matrix method to compute the logarithmic derivative of the wave function on the surface of this volume \cite{OrangeRMP}. The phase shift is extracted upon matching this to the correct long-range solutions. We give only an abridged discussion of this calculation, and the reader may consult Refs. \cite{OrangeRMP, TaranaCurikLi,EilesHetero,tennyson,PanStaraceGreene1} for more details. Here we describe only the relatively simple (but most relevant to LRRM) scenario of alkali atom-electron scattering.
The first step is to compute single-electron wave functions satisfying Eq. \ref{rydbergham} and which vanish at $r=0$ and $r=r_0$. For moderately large $r_0$, $r_0\approx40a_0$, the first few eigenstates are the physical atomic states. Because of the hard-wall boundary condition at $r_0$, the rest of the spectrum consists of positive energy solutions that, while not corresponding to any physical states, give a complete set of states to represent continuum scattering states. With these ``closed'' functions we can accurately describe the total wave function within the $R$-matrix volume. We also calculate two ``open'' functions which are non-zero at $r_0$; these describe the part of the wave function corresponding to the scattering electron, which is non-zero at the surface of the $R$-matrix volume.
A two-electron basis $y_k$, satisfying the proper symmetry of the state under consideration, is constructed from these one-electron functions. The Hamiltonian is identical to Eq. \ref{twoelec1}, except $V_{l_i}(r_i)$ are the model potentials for a singly-charged ion defined in Eq. \ref{eq:modelpotential}. For the light alkali atoms we ignore fine structure, and hence for $s$ and $p$-wave scattering we must compute four scattering phase shifts for the symmetries $^1S$, $^1P$, $^3S$, and $^3P$. Relativistic effects become important in the heavier atoms Rb and Cs, splitting $J=L+S$ levels into a fine structure. These phase shifts have $J$ labels: $^1S_0$, $^1P_1$, $^3S_1$, and $^3P_{0,1,2}$. The $^1S$ symmetry is particularly important as each alkali atom has a bound anion of this symmetry, and in order for the computed ground state of $H_{2e}$ to reproduce the correct electron affinity a dielectronic polarization potential,
\begin{equation}
\label{eq:dielpolarizability}
V_{pol}=-\frac{\alpha}{r_1^2r_2^2}(1 - e^{-(r_1/r_c)^3})(1 - e^{-(r_2/r_c)^3})P_1(\hat r_1,\hat r_2),
\end{equation}
must be added to $H_{2e}$ \cite{Laughlin}. $V_{pol}$ describes how one electron influences the other by polarizating the positively charged core. $P_l$ is a Legendre polynomial, and $r_c$ is a fitting parameter. Without Eq. \ref{eq:dielpolarizability} the model Hamiltonian overpredicts the electron affinity. This can have a strong influence on the phase shifts, particularly the resonant $P$-wave shifts \cite{ChrisJJ}, and must be included.
The logarithmic derivative $b$ for a given scattering energy $E$ can be obtained via a variational calculation using the trial wave function $\Psi = \sum_ky_kC_k$ \cite{OrangeRMP,ChrisRmat1}. This requires solving a generalized eigenvalue equation, $\underline\Gamma \vec C = \underline{\Lambda}\vec Cb$, where
\begin{equation}
\Gamma_{kl} = 2(EO_{kl} - (H_{2e})_{kl}-L_{kl}),
\end{equation}
and $\Lambda_{kl} = \int y_ky_l\delta(r - r_0)\dd{V}$. $O_{kl}$ and $\Lambda_{kl}$ are volume and surface overlap matrix elements, respectively, and $(H_{2e})_{kl}$ is a matrix element of the Hamiltonian. $L_{kl}$ is a matrix element of the Bloch operator, $\frac{1}{2r}\delta(r - r_0)\frac{\partial}{\partial r}r$. Even though many basis states are involved in constructing these matrices, the overlap matrix $\underline{\Lambda}$ is singular because most basis functions have no surface amplitude. Only as many eigenvalues as there are open channels are non-zero; in particular for elastic low-energy scattering we have only one open channel, the atomic ground state. Once $-b = F'(r)/F(r)$ is obtained, the phase shifts are extracted immediately from Eq. \ref{eq:defnmu} with the proper long range solutions $f_l$ and $g_l$.
Typically one expects that the long-range solutions for an electron in the field of a neutral object correspond to a free electron,
\begin{equation}
\label{asymptoticsolns}
f_l(r) = \sqrt{\frac{2}{\pi k}}krj_{l}(kr),\,\,g_l(r) = -\sqrt{\frac{2}{\pi k}}kry_{l}(kr),
\end{equation}
where $j_n(x)$ and $y_n(x)$ are the spherical Bessel and Neumann functions, respectively. However, if the off-diagonal coupling elements in Eq. \ref{eq:outersolns} are still significant at $r_0$ then it is not yet valid to match to these diagonal solutions. The coupled channel equations can be adiabatically diagonalized, decoupling the channels at long-range but introducing a polarization potential \cite{WatanabeGreene}. The quantum defect theory has been generalized to this $r^{-4}$ potential \cite{WatanabeGreene} and can be used to analytically match the functions. We opt instead to numerically propagate wave functions in the polarization potential from $r_0'$ inward to $r_0$. At $r_0'$ the polarization potential is vanishingly small and the functions in Eq. \ref{asymptoticsolns} provide the initial conditions. Once obtained, the phase shifts define the energy-dependent $s$-wave scattering length and $p$-wave scattering volume,
\begin{align}
\label{asdef}
a_s[k(R)]&=\left(-\frac{\tan\delta_s[k(R)]}{k(R)}\right)\\
\label{apdef}
a_p^3[k(R)] &= \left(-\frac{\tan\delta_p[k(R)]}{[k(R)]^3}\right),
\end{align}
respectively.
\subsubsection{Phase shifts and scattering lengths.}
\begin{figure}[b]
\includegraphics[width= 1\columnwidth]{phases_tutorial-min.png}
\caption{\label{fig:phaseshifts} Alkali atom phase shifts for $^1P$ (green, dot-dashed), $^3P_J$ (red, dashed), $^3S$ (black, solid), and $^1S$ (blue, dotted), ignoring the spin-orbit splitting of the $^3P$ states. This figure is modified from Ref. \cite{EilesHetero}.}
\end{figure}
Fig. \ref{fig:phaseshifts} shows electron-atom phase shifts for the lighter alkali atoms calculated with this approach \cite{EilesHetero}. They share many similar features between species.
The $^3S$ phase shift is positive near zero energy, signalling a negative zero-energy scattering length necessary for Rydberg molecule formation. The point where it changes sign identifies the location of a Ramsauer-Townsend zero. The $^3P$ phase shift exhibits a shape resonance, as the scattering electron is temporarily trapped behind the centrifugal barrier. This shape resonance leads to a divergence in the scattering volume; as a result the $p$-wave interaction includes a far larger contribution than expected based purely on the Wigner threshold law\cite{BahrimThumm,BahrimThumm2,Wigner1948}.
The $^1S$ and $^1P$ phase shifts are comparitively featureless. Since the $^1S$ symmetries support an anionic bound state, the positive zero-energy scattering length of this symmetry is unsurprising.
No calculation is capable of converging results at zero energy, so an effective range expansion is employed to extrapolate to zero energy. The energy dependence of the $s$-wave scattering length for a long-range polarization potential is well described by the effective range formula \cite{OmalleyRosenbergSpruch}
\begin{align}
a(k) &\approx a(0) + \frac{\alpha \pi}{3}k+\frac{4}{3}a(0)k^2\ln(1.23\sqrt{\alpha}k)\\&\nonumber+\left(\frac{R_e}{2} + \frac{\sqrt{\alpha}\pi}{3} - \frac{\sqrt{\alpha^3}\pi}{3a(0)^2}\right)a(0)^2k^2 \\&- \frac{\pi}{3}\alpha k^3\left(a(0)^2 + \frac{7\alpha}{117}\right)+\cdots\nonumber,
\end{align}
which has two adjustable parameters, the zero-energy scattering length $a(0)$ and an effective range parameter $R_e$. The first two terms in this expression, linear in $k$, have been used occasionally in the Rydberg molecule community to approximate the full phase shifts. We recommend against this rather crude procedure as this linear approximation rapidly and strongly deviates from the actual values. The scattering volume diverges as $k\to 0$. This is irrelevant in Rydberg molecules as $k\to 0$ implies an infinitely extended wave function, rather than the physical Rydberg wave function. For numerical stability we simply extrapolate the scattering volume to some finite value as $k\to 0$; the PECs calculated below are independent of the specific extrapolation.
\begin{center}
\begin{table}[t]
\begin{tabular}{||c|| c| c || }
\hline
& | & | \\
&$a_s^T(a_0)$&$a_s^S(a_0)$ \\
\hline \hline
Li& \specialcell{$-7.12^a,-7.43^b$,\\$-5.66^d,-6.7^c$} & \specialcell{$3.04^a,2.99^b$,\\$3.65^d,3.2^c$} \\
\hline
Na & $-6.19^a,-5.9^d,-5.7^c$ & $4.03^a,4.2^d,4.2^c$ \\
\hline
K & $-15^d,-15.4^g,-14.6^c$ & $0.55^d,0.57^g,0.63^c$ \\
\hline
Rb & \specialcell{$-16.1^i,-16.9^{g,c},-13^j$,\\$-19.48^l,-14\pm 0.5^m$} & $0.627^{i,c},2.03^g$ \\
\hline
Cs & \specialcell{$-21.7^i,-22.7^g$,\\$-17^j,-21.8\pm 0.2^k$} & \specialcell{$-1.33^i,-2.40^g$,\\$-3.5\pm 0.4^k$} \\
\hline
Sr & $-18^{n,\dagger},-13.2\pm 0.1^o$ & \\
\hline
\end{tabular}
\caption{A summary of theoretical and experimental values (extracted from molecular spectroscopy) of the zero-energy scattering lengths for the triplet (T) and singlet (S) symmetries of the alkali atoms as well as for the ground state of Sr. These values are from: a) \cite{LiNaNorcross}, b) \cite{TaranaCurikLi}, c) \cite{EilesHetero}, d)\cite{Karule}, e) \cite{JohnstonBurrow}, f) \cite{BFKNa}, g) \cite{Fabrikant1986}, h) \cite{Moores1976}, i) \cite{RbCsFr}, j) \cite{BahrimThumm}, k) \cite{Sass},l) \cite{quantumreflection}, m) \cite{AndersonPRL}, n) \cite{BartschatSadeghpour}, o) \cite{DeSalvo2015}.
}
\label{tab:phases}
\end{table}
\end{center}
\begin{figure}[tbp]
{\normalsize
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig12-eps-converted-to-min.pdf}
\end{center}
}
\caption{Scattering phase shifts for Cs (a) and Rb (b), extracted from Ref. \cite{KhuskivadzePRA}. In panel a the unshifted phases are shown as faint curves; the thick curves were shifted slightly to better reflect experimentally observed resonance positions. This figure is taken from Ref. \cite{EilesSpin}. }
\label{fig:phases}
\end{figure}
The zero-energy scattering lengths presented in Table \ref{tab:phases} are crucially important for LRRMs, as they set the overall strength of the molecular bond. The discrepancies in these values, which differ by 10-20\% between reference, are presumably caused by differences in the model Hamiltonian or the level of accuracy in determining its spectrum, approximate long-range potentials, or the zero-energy extrapolation. Electron-atom scattering lengths are extremely difficult to measure, and so one promising application of the vibrational spectroscopy of LRRMs is to extract their values from the spectrum \cite{CsReview,MacLennan}.
We did not calculate \textit{relativistic} phase shifts for the heavier alkali atoms, Rb and Cs. Refs \cite{KhuskivadzePRA,BahrimThumm} are the standard references for their phase shifts, reproduced in Fig. \ref{fig:phases}. In most of this tutorial we neglect the spin-orbit splitting of the $p$-wave interaction \footnote{In Rb this is qualitatively acceptable, with a few notable acceptions; on the other hand the PECs of Cs molecules are not even qualitatively correct without including this splitting.}. The non-relativistic phase shifts of Ref. \cite{BahrimThumm} are used for these calculations, and we have verified that the theoretical approach described agrees with these values. When spin-orbit effects are included, as in Sec. \ref{sec:spinintro}, we use phase shifts from Ref. \cite{KhuskivadzePRA} with a slight modification: the Cs $^3P_J$ phases are shifted by $\sim1$ meV to align the resonance positions with experimental values \cite{pwaveresonance1,pwaveresonance2}. Since no direct experimental measurements of the Rb resonance positions yet exist, we did not modify these phase shifts.
Low-energy phase shifts for other atomic species are unfortunately very uncommon in the literature. To the best of our knowledge the Ca, Sr, and Mg phase shifts published in Ref. \cite{BartschatSadeghpour} are the only sufficiently high-resolution calculations of low-energy phase shifts available. Refs. \cite{Saha,scattNoble,Schwartz} provide zero-energy scattering lengths for He, the noble gas atoms, and H, respectively, but without energy dependent phase shifts or higher partial waves these have limited quantitative utility in the context of LRRMs. In addition to higher resolution calculations for these species, scattering length calculations for more complex atoms -- such as the lanthanide species recently in vogue in ultracold experiments -- would be a highly desirable goal for theory.
\section{A Rydberg molecule primer}
\label{sec:primer}
The previous section described the spectrum of a Rydberg atom and the phase shifts describing low energy electron-atom collisions. We now unite these two concepts, using the Fermi pseudopotential, to describe the Born-Oppenheimer PECs of Rydberg molecules. By presenting only the simplest description of these molecules, focusing on alkali atoms and neglecting electronic and nuclear spin, we draft a blueprint which will then allow us to systematically introduce further complexity in later sections.
\subsection{Fermi pseudopotential}
\label{subsec:fermi}
In principle, the properties of LRRMs could be calculated using standard techniques from quantum chemistry. One could solve the Schr\"{o}dinger equation for an electron in the modified Coulomb field of the atomic nucleus along with the polarization potential of the perturber to obtain the Born-Oppenheimer PECs. In practice this approach is excessively difficult. As the previous section revealed, the electron-atom interaction is sensitive not just to the long-range polarization potential but also to the detailed electron-electron interactions, exchange, and correlation that occur by the perturber. Treating these at an \textit{ab initio} level is extremely challenging. Additionally, the inherent two-center nature of the problem makes a full calculation extremely imposing due to the lack of symmetry. Finally, the overwhelming number of Rydberg states accessible at these high energies and the vast spatial dimensions these wave functions occupy quickly discourage attempts to converge numerical calculations.
In contrast, the Fermi pseudopotential almost immediately provides accurate results. Its predictions have been verified in a multitude of experimental contexts and in comparison with alternative theoretical methods of increasing complexity \cite{DuGreene87,DuGreene89,DePrunele,KhuskivadzeJPB,KhuskivadzePRA,Lebedev,TaranaCurik, GadeaDickinson,JCPabinitioRydMol}. Rather than re-deriving the Fermi pseudopotential here, we instead survey the literature surrounding this topic. A derivation of the $s$-wave pseudopotential most closely tied to the Rydberg context can be found in Fermi's original paper \cite{Fermi}\footnote{A more recent presentation is found in \cite{RydbergRev}.}. Fermi's approach relied on the nature of the zero-energy wave function and its relationship to the scattering length; Omont generalized this by expanding the Rydberg wave function into plane waves near the perturber \cite{Omont}. Independently, Huang and Yang formulated an equivalent pseudopotential for hard sphere scattering in the context of many-body physics\cite{HuangYang}. Their pseudopotential was valid for all partial waves, but unfortunately contained an algebraic mistake for $l>0$ that created substantial confusion in the community once researchers began studying $p$-wave scattering in detail. Before this discrepancy was fully resolved several groups found alternative derivations, and the results and methodologies of these papers may be useful for the Rydberg molecule community \cite{Deutsch,Derevianko,Idziaszek}. Several of these approaches contain explicitly a regularization operator in the pseudopotential. This is necessary for exact calculations using the three-dimensional delta function operator, due to its highly singular nature, but since we never encounter irregular wave functions in the perturbative calculations employed for LRRMs we can ignore this operator.
Omont formulates the pseudopotential as
\begin{align}
\label{omontgeneralization}
V_\text{fermi}(\vec r, \vec R) &= 2\pi\sum_{l=0}^\infty(2l+1)\delta^3(\vec r - \vec R)\\&\times\left(-\frac{\tan\delta_l[k(R)]}{k(R)}\right)P_l\left(\frac{\cev{\nabla}\cdot\vec\nabla}{[k(R)]^2}\right).\nonumber
\end{align}
With the origin at the Rydberg core, $\vec r$ and $\vec R$ are the position operators of the electron and perturber, respectively. The backwards vector symbol on the gradient operator implies that it acts on the bra in a matrix element, thus making the operator Hermitian. In this plane wave approximation $k(R)$ is the semiclassical momentum of the Rydberg electron, $k(R) = \sqrt{2\left(\frac{1}{R} +E\right)}$, where $E$ is the electron's energy. In principle $k(R)$ should be determined self-consistently by iteratively recalculating the electronic eigenenergies until they become stable, but so far the small errors implied by this semiclassical momentum have not demanded this more complicated approach. This definition of $k(R)$ implies two ambiguities: which electronic energy should be chosen when calculating matrix elements $\bra{i}V\ket{j}$ when the electronic state energies $E_i$ and $E_j$ differ, and what happens in the classically forbidden region where $k(R)<0$? In all of our calculations we set $E = -\frac{1}{2n_H^2}$, where $n_H$ is the principal quantum number of the nearest hydrogenic manifold, to eliminate these ambiguities at the expense of neglecting the small effect of quantum defects on $k(R)$. This also fixes the classical turning point $R_\text{out} = 2n_H^2$ for all electronic states. For $R>R_\text{out}$ we either set $k(R>R_\text{out}) = k(R_\text{out})$ or $k(R>R_\text{out}) = -\sqrt{2\left|\frac{1}{R} +E\right|}$ and smoothly interpolate the phase shift from positive to negative $k(R)$. Although both of these approaches are unphysical, they result in smooth PECs\footnote{ Care should be taken on this point when quantitatively comparing PECs from different references. In particular, the behavior near zero energy leads to irrelevant kinks near the classically turning point in some references, and the choice of total electron energy frequently varies.}-- a primarily aesthetic choice since the classical turning point lies outside of the potential wells which support bound states, and thus has little effect on the spectrum.
It is convenient to recast Eq. \ref{omontgeneralization} to include only $s$ and $p$ partial waves; this will pave the way for a more concise notation for later sections as well. The index $\xi$ differentiates the four terms: $\partial_\xi = 1$, $\partial_r$, $\frac{1}{r}\partial_\theta$, and $\frac{1}{r\sin\theta}\partial_\phi$ for $\xi=1$, $2$, $3$, and $4$, respectively. It additionally denotes
\begin{align}
a_\xi[k(R)] &= a_s[k(R)],\xi=1\\
&=3a_p^3[k(R)],\xi>1,\nonumber
\end{align}
where the scattering length/volume were defined in Eqs. \ref{asdef} and \ref{apdef}. We can rewrite Eq. \ref{omontgeneralization} for $s$ and $p$ partial waves as
\begin{equation}
\label{fermicompact}
V_\text{fermi}(\vec R,\vec r) = 2\pi\sum_{\xi = 1}^4a_\xi[k(R)]\cev{\partial_\xi} \delta^3(\vec r-\vec R)\vec{\partial_\xi},
\end{equation}
where the individual potential terms define $V_\xi(\vec R,\vec r) = a_\xi[k(R)]\cev{\partial_\xi} \delta^3(\vec r-\vec R)\vec{\partial_\xi}$.
\subsection{Rydberg molecule potential energy curves}
Within the standard Born-Oppenheimer framework, the potential energy curves (PECs) for Rydberg molecules (unless otherwise stated, these are assumed to be dimers) are the eigenenergies of the electronic Hamiltonian depending parametrically on the internuclear coordinate $\vec R$,
\begin{equation}
H\Psi(\vec r;\vec R) = E(\vec R)\Psi(\vec r;\vec R).
\end{equation}
The most rudimentary electronic Hamiltonian consists of the Rydberg electron's Hamiltonian $H_0$ and the Fermi pseudopotential,
\begin{equation}
H = H_0 + V_\text{fermi}(\vec R,\vec r).
\end{equation}
The Fermi pseudopotential is valid if the neutral atom has a strictly perturbative effect on the Rydberg wave function. It is therefore sufficient to use perturbation theory to compute the PECs as well. The zeroth order states are the Rydberg wave functions satisfying $H_0\ket{nlm} = -\frac{1}{2(n-\mu_l)^2}\ket{nlm}$. These electronic states are shifted by the Fermi pseudopotential, giving rise to the PECs. The nuclear Hamiltonian, $H_\text{nuc}(\vec R)=-\frac{1}{2M}\nabla_R^2 + E(\vec R)$, is solved afterwards to find the vibrational spectrum of the molecule with reduced mass $M$.
In general, a diatomic molecule only possesses cylindrical symmetry, and as such the only conserved quantum number is the projection $\Omega$ of the total angular momentum onto the internuclear axis, which we set parallel to $\hat z$. Molecular states of cylindrical symmetry are classified by their $\Omega$ value ($\Omega=0$ is a $\Sigma$ state, $|\Omega|=1$ is a $\Pi$ state, etc.). The $V_1$ and $V_2$ operators are only non-zero if $\bkt{\vec r=r\hat z}{nlm}\ne 0$, which is only true for $m=0$ states\footnote{$|Y_{lm}(0,\phi)|^2=\frac{2l+1}{4\pi}\delta_{m,0}$.}. The resulting molecules are therefore classified as $\Sigma$ states. The $V_3$ and $V_4$ operators change, through their angular derivatives, $Y_{l|m|=1}(\theta,\phi)$ functions into $Y_{l0}(\theta,\phi)$ functions, and therefore correspond to $\Pi$ symmetry. Since we have only included $s$ and $p$ wave pseudopotentials we only have $\Sigma$ and $\Pi$ symmetries.
\subsubsection{Low-l Rydberg molecules.}
We first calculate the PECs associated with electronic Rydberg states having low angular momentum $l\le l_\text{min}$, where typically $l_\text{min}=2$. Since these states have finite quantum defects they are energetically isolated, and so the molecular PECs associated with these non-degenerate levels are computed by evaluating $\langle nlm| V_\text{fermi}(\vec r,\vec R)|nlm\rangle$ at each value of $R$\footnote{This breaks down for atoms with a $p$-wave shape resonance which can couple many electronic states together, and later we will develop a more sophisticated method.}. The resulting $\Sigma$ and $\Pi$ PECs,
\begin{align}
\label{lowlNone}
E^\Sigma_{l_\text{min}}(R)& =2\pi\left(\frac{2l+1}{4\pi}\right) \Bigg(a_s[k(R)][u_{nl}(R)/R]^2\\&+3a_p^3[k(R)]\left|\frac{du_{nl}(R)/R}{dR}\right|^2\Bigg), \nonumber\\
E^\Pi_{l_\text{min}}(R) &= 6\pi a_p^3[k(R)]\left(\frac{(2l+1)(l+1)l}{8\pi}\right)\left[\frac{u_{nl}(R)}{R^2}\right]^2,
\end{align}
depend only on $R=|\vec R|$.
There are two degenerate $\Pi$ PECs, and they are far weaker than the $\Sigma$ PECs because of an additional $R^{-2}$ factor. Crucially, these PECs depend on the product of two terms. The first, the scattering length/volume, determines their overall strength and repulsive or attractive nature. The second is the radial probability density ($s$-wave) and its gradient ($p$-wave), which cause the PECs to oscillate as a function of $R$. For negative scattering lengths the perturber is therefore trapped in the lobes of the Rydberg wave function, sketched in Fig. \ref{fig:ndfunc} for a Rydberg $nD$ state. This linking of the oscillations in the atomic wave function with oscillations in the PECs is one of the distinguishing features of LRRMs in marked contrast to covalent or ionic bonds. Eq. \ref{lowlNone} reveals that the depth of these potentials increases with $l$ since the electronic density can focus along the internuclear axis at higher $l$. As such, we anticipate that high-$l$ Rydberg states can form very deeply bound molecules due to this probability enhancement.
\begin{figure}[t]
\begin{centering}
\begin{center}
\includegraphics[width= 0.9\columnwidth]{dwavetutorial-min.png}
\end{center}
\end{centering}
\caption{ A probability isosurface plot of a Rydberg $nD$ state. Potential wells are located in the lobes of the wave function, which can trap the perturber (red sphere). }
\label{fig:ndfunc}
\end{figure}
\subsubsection{High-$l$ Rydberg molecules: ``trilobites'' and ``butterflies''.}
As $l$ increases the quantum defects rapidly shrink (see Table \ref{tab:datatable2}). The strength of the Fermi pseudopotential dwarfs the energy splitting between states, and they can be treated as exactly degenerate, just as in the hydrogen atom. Eq. \ref{lowlNone}, derived using non-degenerate perturbation theory, is therefore clearly inadequate for $l>l_\text{min}$. We must use \textit{degenerate} perturbation theory to compute PECs in this high-$l$ subspace. The perturbed eigenstates will be superpositions of the many degenerate unperturbed states, which can therefore combine very effectively into a perturbed wave function that extremizes the potential and no longer resembles the unperturbed states \footnote{For this reason one should always keep in mind the maxim that ``degenerate perturbation theory is non-perturbative.''}.
The degenerate subspace includes all Rydberg wave functions $\ket{nlm}$ having $l>l_\text{min}$ and identical $n$. We illustrate here the diagonalization of the potential within this subspace using a single $V_\xi$. For brevity, we define a shorthand\footnote{When only $\phi_{nlm}(\vec r)$ is used without an index, it is assumed that $\xi=1$ and this is the standard hydrogenic wave function.} for the wave function and spherical gradient components:
\begin{equation}
\label{estatedef}
\phi^\xi_{nlm}(\vec r) =\partial_\xi\phi_{nlm}(\vec r).
\end{equation}
The matrix elements of $V_\xi(\vec R,\vec r)$ are thus proportional to
\begin{align}
\label{matelforshow}
\bra{nlm}\cev{\partial_\xi} \delta(\vec r-\vec R)\vec{\partial_\xi}\ket{nl'm'}&= \left[\phi_{nlm}^\xi(\vec R)\right]^*\phi_{nl'm'}^\xi(\vec R)
\end{align}
This defines a rank 1 {\it separable matrix} and so, despite its large ($\sim n^2$) dimension in this representation it has only one non-zero eigenvalue,
\begin{equation}
\label{evalpert}
E(\vec R) = a_\xi[k(R)]\sum_{l,m}\left[\phi^\xi_{nlm}(\vec R)\right]^*\phi^\xi_{nlm}(\vec R).
\end{equation}
The summation is over $|m|\le l$ and $l_\text{min}<l<n-1$. The corresponding (un-normalized) perturbed wave function is
\begin{align}
\label{eqeigenstatesgeneral}
\Psi_{\xi}(\vec R,\vec r) &= \sum_{l,m}\left[\phi_{nlm}^\xi(\vec R)\right]^*\phi_{nlm}(\vec r).
\end{align}
These formulas betray a recurring pattern: repeatedly we have to sum a product of Rydberg wave functions or their derivatives. It is therefore very useful to study the following object, which we call the \textit{trilobite overlap}, in depth:
\begin{equation}
\label{eqtomdef}
\+{\Upsilon_{pq,n}^{\pmb\alpha\pmb\beta}}= \sum_{l,m}\left[\phi^\alpha_{nlm}(\vec R_p)\right]^*\phi^\beta_{nlm}(\vec R_q).
\end{equation}
The full generality of this formula will be useful throughout this tutorial. With this notation we express Eq. \ref{evalpert} as $E(\vec R) = a_\xi[k(R)]\+{\Upsilon_{RR,n}^{\pmb\xi\pmb\xi}}$ and Eq. \ref{eqeigenstatesgeneral} as $\Psi_\xi(\vec R,\vec r) =\+{\Upsilon_{Rr,n}^{\pmb\xi\pmb1}}$, with the prescription that a subscript $R$ or $r$ implies $\vec R_p=\vec R$ or $\vec R_p=\vec r$, respectively. The generic subscripts ``$p$'' and ``$q$'' imply that $\+{\Upsilon_{pq,n}^{\pmb\alpha\pmb\beta}}$ is evaluated at $\vec R_p$ and $\vec R_q$, two specific points in space. The trilobite overlap's structure is thus reminiscent of a Green's function or a two-point correlation function.
Surprisingly, the trilobite overlap can be analytically summed provided $l_\text{min}=0$, i.e. neglecting quantum defects\footnote{This introduces only small errors which can be subtracted later if necessary}. This was accomplished by Chibisov and coworkers \cite{ChibisovPRL2,Chibisov2000} and used occasionally in their study of LRRMs \cite{KhuskivadzePRA}. We feel that the utility of this summation has not been appreciated in the following literature, and therefore provide here a sketch of the derivation and key results\footnote{The author recently became aware of even more under-appreciated articles in the mathematical chemistry literature which seem to have also been unknown to Chibisov and coworkers: Refs. \cite{Blinder1993,Bartell1996} present these same formulas several years before Ref. \cite{ChibisovPRL2}.}.
All of these summations can be derived from the unnormalized trilobite state,
\begin{align}
\+{\Upsilon_{Rr,n}^{11}}=\sum_{l = 0}^{n-1}\sum_{m = -l}^{m=l}[\phi_{nlm}(\vec R)]^*\phi_{nlm}(\vec r).
\label{sumformgoal}
\end{align}
This expression is akin to the Coulomb Green's function, defined as a summation (which extends into an integral over continuum states) over the complete set of orthogonal Coulomb functions:
\begin{equation}
G(\vec r, \vec R,E) = \sum_{nlm}\frac{\phi^*_{nlm}(\vec r)\phi_{nlm}(\vec R)}{E-E_n}.
\end{equation}
We can neglect all continuum states and even all discrete states except for one degenerate manifold by evaluating the Green's function at a bound state energy:
\begin{equation}
\label{gff1}
G(\vec r, \vec R, E \to E_n) \approx \frac{1}{E-E_n}\sum_{lm}\phi^*_{nlm}(\vec r)\phi_{nlm}(\vec R).
\end{equation}
This formula shows that we can evaluate this sum provided that we can obtain the Green's function by another means and properly cancel out the divergent $\frac{1}{E-E_n}$ term. Hostler and Pratt \cite{HostlerPratt} derived a closed form expression for the Coulomb Green's function,
\begin{align}
\label{eq:hostlerprattgreenfunction}
G(\vec r,\vec R,E) &= \frac{\Gamma(1 - \nu)}{2\pi|\vec r - \vec R|}\left(\pd{(x/\nu)}-\pd{(y/\nu)}\right) \\&\times W_{\nu,1/2}(x/\nu)M_{\nu,1/2}(y/\nu),\nonumber
\end{align}
in terms of the variables $(x,y) = r + R \pm |\vec r - \vec R|$ and Whittaker functions $M_{\nu,1/2}(\tau)$ and $W_{\nu,1/2}(\tau)$. These are related by
\begin{align}
\label{eq:irregularwhittaker}
&\Gamma(1 - \nu)M_{\nu,1/2}(\tau)\\ &= (-1)^{1 + \nu}\frac{\Gamma(1 - \nu)}{\Gamma(1 + \nu)}W_{\nu,1/2}(\tau) + (-1)^\nu W_{-\nu,1/2}(-\tau).\nonumber
\end{align}
As $\nu$ approaches an integer $n$, as occurs at a bound state, $\Gamma(1 - \nu)$ diverges as
\begin{equation}
\Gamma(1 - \nu)|_{\nu\to n }=\frac{(-1)^n}{n^3\Gamma(n)}\frac{1}{E - E_n},
\end{equation}
where $E = -(2\nu^2)^{-1}$ and $E_n = -(2n^2)^{-1}$. Now, by matching Eqs. \ref{gff1} and \ref{eq:hostlerprattgreenfunction} as $E \to E_n$, we have
\begin{align}
\label{eq:matchingGF}&\frac{1}{E-E_n}\sum_{lm}\phi^*_{nlm}(\vec r)\phi_{nlm}(\vec R)\\ &=\nonumber\frac{1}{E - E_n}\Bigg[\frac{(-1)^n}{n^3\Gamma(n)}\frac{1}{2\pi|\vec r - \vec R|}\left(\pd{(x/\nu)}-\pd{(y/\nu)}\right)\\&\left.\,\,\,\,\nonumber\times W_{\nu,1/2}(x/\nu)\frac{ (-1)^{1 + \nu}}{\Gamma(1 + \nu)}W_{\nu,1/2}(y/\nu)\right|_{\nu=n}\Bigg].
\end{align}
Since both sides of this equation diverge identically as $1/(E-E_n)$, the summation on the left is equivalent to the bracketed term on the right.
We insert the standard hydrogen radial wave function $u_{nl}(r)$ using Eq. \ref{fdefradial}, and evaluating the derivatives Eq. \ref{eq:matchingGF} simplifies to
\begin{align}
\label{chibisovtrilobite}
\+{\Upsilon_{Rr,n}^{11}}&= \frac{u_{n0}'(t_-)u_{n0}(t_+) - u_{n0}(t_-)u_{n0}'(t_+)}{4\pi\Delta t},
\end{align}
where $t_{\pm} =\frac{1}{2}\left(R+r\pm\sqrt{R^2 + r^2 -2Rr\cos\gamma}\right)$, $\Delta t = t_+ - t_-$, and $\gamma$ is the angle between $\vec R$ and $\vec r$. Differentiation of Eq. \ref{chibisovtrilobite} with respect to $R$, $\theta$, or $\phi$ generates the three types of butterfly orbitals $\+{\Upsilon_{Rr,n}^{\pmb\xi 1}}$:
\begin{align}
\label{chibisovRbutterfly}
\+{\Upsilon_{Rr,n}^{21}} &=\frac{(r\cos\gamma-R)\mathcal{F}(t_+,t_-)}{8 \pi\Delta t^3}\\&\nonumber+\frac{u_{n0}(t_+)u_{n0}''(t_-)-u_{n0}(t_-)u_{n0}''(t_+)}{8 \pi\Delta t}\\
\label{chibisovthetabutterfly}
\+{\Upsilon_{Rr,n}^{31}}&= \cos\theta_R\cos\varphi_R\Upsilon_x\\&\nonumber + \cos\theta_R\sin\varphi_R\Upsilon_y-\sin\theta_R\Upsilon_z \\
\label{chibisovphibutterfly}
\+{\Upsilon_{Rr,n}^{41}}&= -\sin\varphi_R\Upsilon_x + \cos\varphi_R\Upsilon_y,\end{align}
where
\begin{align}
\label{eq:symproperties}
\begin{pmatrix}\Upsilon_x\\\Upsilon_y\\\Upsilon_z\end{pmatrix} = \frac{r\mathcal{F}(t_+,t_-)}{8\pi(\Delta t)^3}\begin{pmatrix}\sin\theta_r\cos\varphi_r\\\sin\theta_r\sin\varphi_r\\\cos\theta_r\end{pmatrix}
\end{align}
and
\begin{align}\mathcal{F}(t_+,t_-)&=- 2(\Delta t) u_{n0}'(t_+) u_{n0}'(t_-)\\&-u_{n0}(t_-)[2 u_{n0}'(t_+) - (\Delta t) u_{n0}''(t_+)]\nonumber \\ &+ u_{n0}(t_+)[ 2u_{n0}'(t_-) +(\Delta t) u_{n0}''(t_-)].\nonumber
\end{align}
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=\textwidth]{1orbital_new-min.pdf}
\vspace{-200pt}
\caption{Trilobite and butterfly states for $n=30$ and $R=1232$, plotted as isosurfaces and as a density plot. For the isosurfaces, the deep blue color is a cut of $|\Psi|^2 = X$ and a full rotation about $\hat z$ is plotted. The grayer surfaces are for $X/10$ and $X/100$, respectively, and are only plotted for half a rotation about $\hat z$ to reveal the inner structure. The density plot for $\rho|\Psi(\rho,z)|^2$ is shown in cylindrical coordinates aligned parallel to the isosurfaces. The scale factor $\rho$ emphasizes the nodal structure, but introduces an additional zero along $\hat z$. The angular butterfly lacks the $\sin\phi$ or $\cos\phi$ modulation seen in Eq. \ref{eq:symproperties}. }
\label{fig:trilobitebasis}
\end{center}
\end{figure*}
\noindent
The diagonal elements $\+{\Upsilon_{pp,n}^{\pmb\xi\pmb\xi}}$ are obtained by carefully evaluating equations (\ref{chibisovtrilobite} - \ref{chibisovphibutterfly}) as $\vec R_p$ approaches $\vec R_q$ using L'Hopital's rule\footnote{We eliminate second and third derivatives using the radial Schr\"{o}dinger equation}:
\begin{align}
\label{diagonalTrilo}
\+{\Upsilon_{RR,n}^{11}}&= \frac{(2n^2 - R)\left(u_{n0}(R)/n\right)^2+ Ru_{n0}'(R)^2}{4\pi R}\\
\label{diagonalRbutterfly}
\+{\Upsilon_{RR,n}^{22}}&= \+{\Upsilon_{RR,n}^{33}}-\frac{u_{n0}(R)}{12\pi R^3}\left[3Ru'_{n0}(R)+2u_{n0}(R)\right]\\
\label{diagonalAngbutterfly}
\+{\Upsilon_{RR,n}^{33}}&=\+{\Upsilon_{RR,n}^{44}}\\&\nonumber
= \frac{4\pi R(2n^2 - R)\+{\Upsilon_{RR,n}^{11}} - n^2u_{n0}'(R)u_{n0}(R) }{12\pi n^2R^2}.
\end{align}
This analysis confirms that the two $\Pi$ butterflies are degenerate. Surprisingly, although both terms in the numerator of Eq. \ref{diagonalAngbutterfly} oscillate with $R$, $\+{\Upsilon_{RR,n}^{33}}$ does not (see Fig. \ref{fig:Rbmanifold}). This is particularly intriguing in light of the summation representation of $\+{\Upsilon_{RR,n}^{33}}$,
\begin{equation}
\+{\Upsilon_{RR,n}^{33}}=\sum_{l=0}^{n-1}\left[\frac{u_{nl}(R)}{R^2}\right]^2\frac{(2l+1)(l+1)(l)}{8\pi},
\end{equation}
which shows that the coefficients $c_l = (2l+1)(l+1)l$ guarantee perfect cancellation of all oscillations in the radial wave functions.
These eigenstates are shown in Fig. \ref{fig:trilobitebasis}. Their appearance in cylindrical coordinates (plotted as ``shadows'' in Fig. \ref{fig:trilobitebasis} and more explicitly as surface plots in the insets of Fig. \ref{fig:Rbmanifold}) calls to mind a trilobite fossil or a butterfly with spread wings, respectively\footnote{Whether this is an epiphany or an apophany is up to the reader.}. Their distinctive nodal patterns have attracted interest due to their deep connections to periodic orbit theory and to a near separability of the diatomic Hamiltonian in elliptic coordinates \cite{Granger,Lambert}. One underappreciated point is the extent to which these wave functions localize the wave function near the perturber. The trilobite representats the delta function in this truncated degenerate subspace, and as seen in Fig. \ref{fig:trilobitebasis} its density is focused into a small region around the perturber. The radial butterfly is also localized around the perturber, but as it maximizes the gradient in the $z$ direction it has a node directly on the perturber. As $\Sigma$ states, the trilobite and $R$-butterfly orbitals are invariant under rotation around the $z$ axis, and Eqs. \ref{chibisovtrilobite} and \ref{chibisovRbutterfly} are correspondingly independent of $\phi_r$. The angular butterfly molecules have a node along the internuclear axis in accordance with their $\Pi$ symmetry, as reflected by the $\sin\varphi_r$ and $\cos\varphi_r$ modulation factor in Eq. \ref{eq:symproperties}. This factor is dropped in Fig. \ref{fig:trilobitebasis} to facillitate the visualization. Further details of the symmetry properties of these butterfly states, as pertaining to the symmetries of polyatomic molecules, are discussed in Ref. \cite{JPBdens}.
\subsubsection{Beyond perturbation theory.}
Thus far we have computed the molecular states separately for low-$l$ and high-$l$ Rydberg atoms. The accuracy of the resulting PECs (Eqs. \ref{lowlNone} and \ref{evalpert}), computed in first order perturbation theory, is compromised for several reasons:
\begin{itemize}
\item The matrix elements of $V_\xi$ were calculated and diagonalized separately, ignoring any coupling between trilobite and butterfly states.
\item The coupling between the trilobite/butterfly states and low-$l$ Rydberg states, which can become large depending on the strength of the perturbation compared to the energy gap due to the quantum defects, was ignored.
\item The $p$-wave shape resonance creates an unphysical divergence in the PECs that catastrophically reduces their accuracy. This must be remedied by including couplings between different $n$ manifolds adjacent to the manifold of interest. The resulting level repulsion constrains this divergence and gives sensible results \cite{HamiltonGreeneSadeghpour}.
\end{itemize}
To address these problems we expand the exact wave function into the complete and orthonormal basis of Rydberg wave functions, truncating only in the number of $n$-manifolds we include:
\begin{align}
\tilde\Psi(\vec r; \vec R) = \sum_{n=n_1}^{n=n_2}\sum_{l=0}^{l=n-1}&\sum_{m=-l}^{m=l}c_{nlm}(\vec R)\phi_{nlm}^1(\vec r).\label{exp1}
\end{align}
The expansion coefficients $c_{nlm}$ are determined variationally by diagonalizing the Hamiltonian, $\bra{n'l'm'}H_0 + V_\text{fermi}\ket{nlm}$. This is the standard approach, and we will refer to it as the ``Rydberg basis'' method. Typically an expansion such as this converges provided the number of basis states is not truncated too low. However, its accuracy in this context is a matter of some controversy since the delta function potential is not formally convergent \cite{Fey}. Care must be taken in choosing the number of $n$ manifolds due to this non-convergent behavior, and will be discussed more in Section \ref{sec:spinintro}.
This approach requires the diagonalization of a $\mathcal{M}n^2$-dimensional matrix, where $\mathcal{M}$ is the number of $n$ manifolds included. As Eq. \ref{matelforshow} reveals, there is a huge redundancy as the matrix of a given $V_\xi$, of dimension $n^2$, has only a single eigenstate given in terms of the $l=0$ radial wave function only. The evaluation of so many high-$l$ basis states in order to diagonalize $H$ in the full Rydberg basis seems particularly wasteful.
In this tutorial we develop an alternative method inspired by Ref. \cite{Rost2006}, which considered the trilobite states, rather than the Rydberg basis, as the fundamental unperturbed basis. In the context of polyatomic molecules Ref. \cite{JPBdens} included coupling terms between the butterfly and trilobite states. Here we fully generalize this concept to include all couplings between different $l$ states and $n$ manifolds. This is, to our knowledge, the first time this approach has been presented. We refer to this approach as the ``trilobite basis'' method as its core idea is that we can replace Eq. \ref{exp1} with a new trial wave function which collapses all of the redundant degenerate high$-l$ states into just four states per $n$ manifold: one trilobite and three butterfly ($\xi = 2,3,4$) states. This trial wave function contains these, along with the few non-degenerate low-$l$ states:
\begin{align}
\label{trialwavetrilo}
\Psi(\vec r; \vec R) = \sum_{n=n_1}^{n=n_2}\Bigg[\sum_{l=0}^{l=l_\text{min}}&\sum_{m=-l}^{m=l}c_{nlm}(\vec R)\phi_{nlm}^1(\vec r)\\&+\sum_{\xi=1}^4\mathcal{C}_{n\xi}(\vec R)\+{\Upsilon_{Rr,n}^{\pmb\xi1}}\Bigg].\nonumber
\end{align}
We solve for the coefficients $c_{nlm}(\vec R)$ and $\mathcal{C}_{n\xi}(\vec R)$ by projecting $H\Psi(\vec r;\vec R)$ onto each basis function. Since the trilobite states are not orthogonal,
\begin{equation}
\int\left[\+{\Upsilon_{pr,n}^{\pmb\alpha1}}\right]^*\+{\Upsilon_{qr,n'}^{\pmb\beta1}}\ddn{3}{r} = \+{\Upsilon_{pq,n}^{\pmb\alpha\pmb\beta}}\delta_{nn'},
\end{equation}
this results in a generalized eigenvalue equation $\underline{H}\vec c = E(\vec R)\underline{\Lambda}\vec c$. The Hamiltonian matrix has a block structure. The first block,
\begin{align}
\label{matelementdimer1}
&\bra{\+{\Upsilon_{Rr,n}^{\pmb\alpha 1}}}H_0+V_\text{fermi}\ket{\+{\Upsilon_{Rr,n'}^{\pmb\beta 1}} }\\&=-\frac{1}{2n^2}\+{\Upsilon_{RR,n}^{\pmb\alpha\pmb\beta}}\delta_{nn'}+ 2\pi\sum_{\xi=1}^4a_\xi\+{\Upsilon_{RR,n}^{\pmb\alpha\pmb\xi}}\+{\Upsilon_{RR,n'}^{\pmb\xi\pmb\beta}}, \nonumber
\end{align}
couples trilobite and butterfly states together. The next block couples the non-degenerate low-$l$ states
\begin{align}
&\bra{\phi_{nlm}^1}H_0 + V_\text{fermi}\ket{\phi_{n'l'm'}^1} \\&= -\frac{\delta_{nn'}\delta_{ll'}\delta_{mm'}}{2(n-\mu_l)^2} + 2\pi \sum_{\xi=1}^4a_\xi\phi_{nlm}^\xi(\vec R)^*\phi_{n'l'm'}^\xi(\vec R).\nonumber
\end{align}
Finally, we have an off-diagonal block coupling trilobite and butterfly states to the low-$l$ functions,
\begin{align}
&\bra{\+{\Upsilon_{Rr,n}^{\pmb\alpha 1}}}H_0+V_\text{fermi}\ket{\phi_{n'l'm'}^{1} }\nonumber\\&
= 2\pi \sum_{\xi=1}^4a_\xi\+{\Upsilon_{RR,n}^{\pmb\alpha \pmb\xi}}\phi_{n'l'm'}^\xi(\vec R).
\label{offdiagcouptrilo}
\end{align}
The overlap matrix $\underline{\Lambda}$ has a trilobite block given by $\+{\Upsilon_{RR,n}^{\pmb\alpha\pmb\beta}}\delta_{nn'}$, a purely diagonal low-$l$ block, $\delta_{nn'}\delta_{ll'}\delta_{mm'}$, and no off-diagonal blocks. In the calculations presented below which use this approach, we set $n_1=n-1$, $n_2=n+1$ and $l_{<} = 3$ to account for the non-negligible $\mu_f$.
\subsection{Alternative approaches}
Before we examine the PECs, we briefly mention several alternative approaches. The earliest such approach, predating LRRMs and developed in the context of collisional broadening, is the Borodin and Kazansky (BK) model \cite{BKmodel}. This yields PECs that agree in shape and magnitude with the Fermi model, but lack the oscillatory nature from the electronic density. They are determined purely by the phase shifts:
\begin{equation}
\label{BKmodel}
E^{BK}_l(R) = -\frac{1}{2n^2}+\frac{1}{2}\left(n-\frac{\delta_l[k(R)]}{\pi}\right)^{-2}.
\end{equation}
This approximation provides a useful comparison when attempting to understand the convergence challenges of the delta function potential \cite{Fey,EilesSpin}, since these approximate PECs do not diverge when $\delta_p$ rises by $\pi$. This confirms that the $p$-wave shape resonance is converged adequately by level repulsion when multiple $n$ manifolds are included.
Immediately following the first trilobite prediction, Green's function techniques were developed by Greene and coworkers\cite{HamiltonThesis,Crowell} and, essentially simultaneously, by Fabrikant and coworkers \cite{KhuskivadzeJPB,KhuskivadzePRA}. These methods differ in implementation but are built around a similar logic. The basic idea is that the electron, outside of the non-Coulombic region near the Rydberg core or the polarization potential region surrounding the perturber, experiences only a Coulomb potential. It is thus described by hydrogenic wave functions. Although the wave function differs near the Rydberg core and the perturber, its exact form there is irrelevant since (taking the perspective of quantum defect theory as discussed in Sec. \ref{sec:inter}) the wave function outside of this region is determined only by the quantum defects and scattering phase shifts. Thus, the non-Coulomb regions impart new boundary conditions on the wave function, and these can be included readily once the Green's function is known \cite{HostlerPratt}. The method of Ref. \cite{KhuskivadzePRA} is particularly sophisticated and includes the Rydberg fine structure, singlet and triplet scattering, and even the relativistic $^3P_J$ splitting of the electron-atom phase shifts \cite{KhuskivadzePRA}. For many years, this was the only approach which properly included the fine structure of the phase shifts (Sec. \ref{sec:spinintro} describes a different approach). Tarana and Curik \cite{TaranaCurik} developed an $R$-matrix program which solves directly the two-electron interaction near the perturber before matching to the long-range Coulomb functions. This technique improves slightly upon Ref. \cite{KhuskivadzePRA} since it is available for higher partial waves and does not require input of the scattering phases from an external calculation. It appears to be quite accurate, particularly at small internuclear distances where the Fermi model is unsuitable. Unfortunately so far only fairly low-lying ($n\le 20$) molecular PECs for H$_2$ have been computed with this model, and it would be very useful to extend this line of research to verify the performance of these methods in the alkali atoms.
\subsection{Potential energy curves}
\begin{figure*}[h]
\begin{center}
\includegraphics[width=\textwidth]{1potential_key-min.pdf
\caption{The Rb$_2$ PECs obtained by diagonalizing the Fermi pseudopotential (black) or using Eq. \ref{BKmodel} (red,dashed). The location of the $p$-wave divergence is shown as a blue line. Different important regions are highlighted: (a) The trilobite PEC, and the distinctive wave function (plotted in cylindrical coordinates); (b) the $nS$ PEC and its intersection with the butterfly potential; (c) the $nD$ PEC and its intersection with the butterfly potential; (d) the potential wells that support butterfly states and a butterfly wave function in cylindrical coordinates; (e) a the singlet trilobite PEC; (f) the singlet butterfly PEC. }
\label{fig:Rbmanifold}
\end{center}
\end{figure*}
Figure \ref{fig:Rbmanifold} shows the potential energy landscape between the $n=29$ and $n=30$ manifolds for a Rb$_2$ Rydberg molecule. The regularity of the Rydberg spectrum implies that this same picture is repeated largely unchanged between every two Rydberg levels, according to a collection of scaling laws. The Rydberg level splittings decrease as $n^{-3}$. Away from the $p$-wave shape resonance, the potential wells associated with low-$l$ states get shallower as $n^{-6}$, while the trilobite potential wells, due to the mixing between high-$l$ states, decrease as $n^{-3}$. In contrast, the position of the $p$-wave shape resonance is approximately independent of $n$ while the range of the PECs grows as $n^2$; as a result the relative importance of the $p$-wave shape resonance decreases at higher $n$. Any properties associated with the perturber, such as its zero-energy scattering length or hyperfine splitting, are $n$-independent\footnote{One slight nuance is that the energy-dependence of the scattering length varies with $n$ due to the $n$-dependence of the semiclassical $k(R)$.}. We now examine Fig. \ref{fig:Rbmanifold} in detail as it shows most of the key features of this unusual class of molecules, and we will expand upon this picture throughout this tutorial.
The main features are four nearly flat potentials, corresponding to the three non-degenerate low-$l$ states and the manifold of unperturbed states at the hydrogen energy $-1/n^2$, two degenerate smooth $\Pi$-butterfly potentials, and four oscillatory potentials: the triplet and singlet trilobite potentials (labeled (a) and (e), respectively), and the triplet and singlet $\Sigma$ radial butterfly potentials ((d) and (f), respectively). The BK model (dashed red) confirms the accuracy of the Fermi pseudopotential. The insets hold density plots in cylindrical coordinates of the trilobite and butterfly states. The Rydberg core is marked with a blue dot, and the perturber lies under the ``twin peaks'' of probability density. It is clear from the asymmetric bunching of electron density that these have non-zero dipole moments.
The triplet trilobite potential curve, marked (a), is several GHz deep and possesses a global minimum due to the change in sign of the phase shift (see Fig. \ref{fig:phases}). The singlet trilobite potential curve, highlighted (e), is basically monotonically increasing because the singlet phase shift exhibits no such Ramsaeur-Townsend minimum. The trilobite curve is shown in more detail in Fig. \ref{fig:Rbzooms}a. The triplet butterfly potential, which also has $\Sigma$ symmetry, dives downward through the trilobite potential and the low-$l$ potentials, creating a series of sharp avoided crossings. Its interaction with the $n=29$ manifold is clearly critical to constrain the divergent scattering volume (the location of this divergence is highlighted as the blue line). A series of butterfly wells are formed at relatively short internuclear distances. The $\Pi$ angular butterfly curves do not couple to any other potentials, and are non-oscillatory as predicted by Eq. \ref{diagonalAngbutterfly}. The butterfly potential wells are highlighted in Fig. \ref{fig:Rbzooms}b.
Although these states are the most theoretically appealing due to their remarkable wave functions, deep potentials, and large dipole moments, they are experimentally the most challenging to produce. Since they are composed of high-$l$ basis states, dipole selection rules prohibit excitation from the ground state without a three-photon process. It was not until 2015 that a trilobite molecule with a kilo-Debye dipole moment was observed in Cs \cite{BoothTrilobite}. Cesium has a unique advantage over Rb: its $s$-wave quantum defect is very close to an integer ($\mu_s = 4.05$). As such, the trilobite state intersects and couples to the $nS$ potential curve, allowing a two-photon pathway through this admixture. In $n=30$ Rb these states are separated by $>25$GHz, making this coupling extremely weak \cite{PfauSci}. However, following this same logic, formation of butterfly molecules should be possible because the butterfly potential wells are energetically close to the $(n+2)P$ potential curve, and hence have some mixing. Indeed, Rb butterfly states were observed in 2016, photoassociated via single photon excitation \cite{Butterfly}.
\begin{figure}[b]
\begin{center}
\includegraphics[width=\columnwidth]{multipanel1-min.pdf}\\
\includegraphics[width=\columnwidth]{multipanel3-min.pdf}\\
\includegraphics[width=\columnwidth]{multipanel2-min.pdf}
\caption{Details of the adiabatic potential curves. (a) The triplet trilobite potential curve; (b) the triplet butterfly potential wells; (c) singlet and triplet $nS$ potential curves. The blue vertical line is located at the $p$-wave shape resonance. The dashed red curves show the BK model, while the dashed blue curve in (c) shows the $nS$ potential curve without the influence of the $p$-wave potential. }
\label{fig:Rbzooms}
\end{center}
\end{figure}
In contrast, all of the low-$l$ molecules have been studied extensively since they can be directly coupled to the ground state via single or double photon excitation. Fig. \ref{fig:Rbzooms}c highlights the $nS$ PECs, which are prototypical for the other low-$l$ states in this spin-independent picture. The deep(shallow) curve corresponds to triplet(singlet) scattering. Although the singlet scattering length is positive, small $p$-wave contributions cause the potential wells to fall below the asymptotic energy. The first LRRMs observed were bound in the outermost well of the triplet potential \cite{Bendkowsky}. The dashed blue PEC in Fig. \ref{fig:Rbzooms}c neglects $p$-wave contributions, and shows that the outermost potential well is determined entirely by $s$-wave scattering. The butterfly PEC plunges through the $s$-wave potential at about $R=1000$ and, due to this avoided crossing, strongly affects the vibrational states, particularly the excited ones not localized in the outermost potential well. The $p$-wave shape resonance leads to a sharp drop in the potential curve, and the lack of an inner barrier seems to suggest that vibrational states will rapidly decay or even be destroyed. However, it was observed and pointed out in Ref. \cite{quantumreflection} that these states can still exist due to quantum reflection: at exactly the bound state energies the molecular wave function exponentially decays in the plateau region to the left of the shape resonance, reflecting the strange quantum mechanical principle that a potential drop can function similarly to a barrier.
This narrow $p$-wave crossing also calls into question the accuracy of the adiabatic Born-Oppenheimer approximation since its applicability depends not only on the difference between nuclear and electronic masses but also on the energy separation between PECs. Ref. \cite{UltracoldChem} studied how these sharp avoided crossings lead to novel chemical pathways and non-adiabatic processes like $l$-changing collisions. Sr does not have a $p$-wave shape resonance, and hence its PECs more closely resemble the dashed blue curve in Fig. \ref{fig:Rbzooms}c. Strontium LRRMs \cite{DeSalvo2015} are therefore useful to compare with Rb in order to investigate the role of this $p$-wave resonance on the decay channels and lifetimes of these molecules \cite{Camargo2016,WhalenLifetimes,Rblifetimes}.
The other low-$l$ states, asymptotically associated with Rydberg $nP$ and $nD$ states, have very similar potential curves as the $nS$ molecules just discussed. In Rb, $nP$ \cite{Niederprum} and $nD$ \cite{AndersonPRL,MacLennan,PfauKurz} molecules have been observed, and in Cs $nP$ molecules have been studied\cite{Sass}. Since these Rydberg states have fine structure, which non-trivially couples to the perturber's hyperfine structure, the observed spectra require the full spin-dependent calculation described in Sec. \ref{sec:spinintro} and discuss these states further there.
Although we do not study it in detail, there is one final step after obtaining these PECs before the properties of LRRMs are known: the nuclear Schr\"{o}dinger equation must be solved using these adiabatic PECs. This yields the molecular spectrum, and shows that the vibrational states are typically split by several tens of MHz (for $20\le n \le 40$) and have rotational splittings, inversely proportional to their bond length, on the order of kHz which are typically unresolved. The lifetimes of these states are comparable to those of Rydberg atoms, although somewhat shortened due to additional decay routes provided by the molecular structure.
\subsection{Electric dipole moments}
\begin{figure}[t]
{\normalsize
\begin{centering}
\begin{center}
\includegraphics[scale =0.08]{fig8-min.png}
\end{center}
\end{centering}
}
\caption{ Hydrogenic dipole, quadrupole, and octupole moments for $n =23$, as determined by Eq. \ref{eq:dipolehydrogen}. The trilobite (blue,dashed) and $\Sigma$ butterfly state (red,solid) oscillate as a function of $R$, while the $\Pi$ butterfly state (black,dot-dashed) does not. This figure is taken from Ref. \cite{EilesSpin}. }
\label{fig:multipoles}
\end{figure}
The state mixing induced by the perturber creates permanent electric dipole moments exceeding hundreds of Debye. Because of the coupling between the polar trilobite/butterfly states and the low-$l$ states, even these exhibit dipole moments of a few Debye \cite{PfauSci}. These dipole moments have sparked interest in the application of these molecules in dipolar gases and ultracold chemistry. Here we calculate arbitrary multipole moments for the trilobite/butterfly states, $
d_{\alpha}^{k,q} =\bra{\+{\Upsilon_{Rr,n}^{\pmb\xi\pmb1}}}T^k_q\ket{\+{\Upsilon_{Rr,n}^{\pmb\xi\pmb1}}}$. The multipole moments from classical electrostatics \cite{Jackson} are promoted to quantum-mechanical operators:
\begin{equation}
T^k_q = -r^kC_{kq}(\hat r),\,\,\,C_{kq}(\hat r) = \sqrt{\frac{4\pi}{2k+1}}Y_{kq}(\hat r).\nonumber
\end{equation}
Here $k$ and $q$ label tensor operator components; $T^1_0$ is the usual dipole operator.
A straightforward calculation provides
\begin{align}
\label{eq:dipolehydrogen}\langle \xi| T_q^k|\xi\rangle &= \sum_{l,l'}\frac{\left[\phi_{nlm}^\xi(R)\right]^*\phi_{nl'm'}^\xi(R)}{\+{\Upsilon_{RR,n}^{\pmb\alpha\pmb\alpha}}} R_{nl}^{nl'}(k)\\& \times C_{l'\Omega,kq}^{l\Omega}(-1)^{k-l'}\sqrt{(2l'+1)}\begin{pmatrix}l & l' & k\\ 0 & 0 & 0 \end{pmatrix}.\nonumber
\end{align}
The radial matrix element
\begin{equation}
\label{eq:radmatelforfield}
R_{nl}^{nl'}(k) = \int_0^\infty u_{nl}(r)u_{nl'}(r)r^k\dd{r},
\end{equation}
can be evaluated analytically \cite{radmatel}. These multipole moments scale as $n^{2k}$, and are displayed in Fig. \ref{fig:multipoles} up to the octupole moments. At small $R$ the dipole moments of both butterfly symmetries become negative.
This section introduced the foundational concepts and properties of Rydberg molecules: oscillatory PECs, extremely large bond lengths and, in the trilobite and butterfly cases, highly localized wave functions with exotic nodal structure and large permanent electric dipole moments. All of the resoundingly succesful experimental observations of these molecules, with the exception of satellite peak observations at thermal temperatures\cite{Crowell,NiemaxHet}, have occurred recently -- within the last decade. In the following sections we discuss the theoretical progress made in response to this experimental success.
\section{Polyatomic Rydberg molecules}
\label{sec:polyintro}
Rydberg molecules can be formed over the whole range of principle quantum number $n$. For small $n$ they were photoassociated directly from bound Rb-Rb molecules \cite{Carollo1,Carollo2}. As $n$ increases, or equivalently as the atomic density $\rho$ increases, the average number of perturbers $N\sim \rho n^6$ within a Rydberg volume grows rapidly. As $N$ increases, so does the probability that polyatomic molecules -- trimers, tetramers, and so on -- can form. For $N\gg1$, the individual molecular lines smear into a mean field energy shift linear in density and proportional to the zero-energy scattering length, and we return to the scenario originally studied by Amaldi, Segre and Fermi \cite{PfauBEC,RydbergRev}. Between the two extremes of $N = 1$ and $N\gg 1$ resides a range of fascinating phenomena involving polyatomic LRRMs, and this section investigates their structure and properties. The electronic Hamiltonian of Eq. \ref{fermicompact} is expanded\footnote{Atom-atom van der Waals interactions are negligible at the Rydberg-scale internuclear distances we consider here.} to include $N$ perturber atoms located at $\vec R_i=(R_i,\theta_i,\varphi_i)$:
\begin{align}
\label{ham}
H_N({\vec r};\{\vec R_i\})
& = H_0 + 2\pi \sum_{i=1}^N\sum_{\xi = 1}^4V_\xi(\vec R_i,\vec r).
\end{align}
Since the number of spatial degrees of freedom grows rapidly in a polyatomic molecule, calculation of the potential energy surfaces becomes computationally challenging and visualization becomes nearly impossible. In the following examples illustrating the generic structure of polymers we will study only the breathing mode, i.e. the cut through the potential surfaces where all atoms share a common distance $R$ from the Rydberg core. Although this gives only a small glimpse of the full physical picture, these cuts illustrate most of the interesting physics determining the molecular structure. A more complete analysis can be found in Refs. \cite{FeyKurz,FeyTrimer}, which study the potential surfaces of triatomic LRRMs and reveal the role of stretching modes.
Several experiments have observed polyatomic LRRMs of the simplest type: an $nS$ Rydberg excitation. Soon after the first observation of diatomic states, additional spectral lines much deeper than the deepest dimer state were seen \cite{quantumreflection}. These states were identified as trimers for the following reasons. First, the $nS$ state is non-degenerate, and thus we can sum the PEC from Eq. \ref{lowlNone} over all perturbers. Second, since the $nS$ state is spherically isotropic and along the breathing mode all $R_i = R$, there is no dependence on $i$ in Eq. \ref{lowlNone}, and hence the polyatomic PEC is $N E_{l_\text{min}=0}^\Sigma(R)$. Since the outermost potential well is approximately harmonic, this scaling implies that the vibrational states of a ($N+1$)-mer are $N$ times deeper than the dimer state, and are thus clearly identified in an experimental spectrum. In later experiments in both Rb and Sr \cite{MolSpec,Schlag,Whalen2} even more polymer (trimer, tetramer, and pentamer) lines were identified. Excited vibrational states were also assigned in this fashion in the measurements of Ref. \cite{Whalen2}, which span four orders of magnitude in spectral intensity. At higher $n$ or $\rho$ these vibrational lines blend together into an overall shift with a distinctive lineshape. Some aspects of this lineshape in Rb were attributed to the $p$-wave shape resonance \cite{Schlag}, while other results were interpreted via quantum many-body calculations adapted to study Rydberg ``polarons'' \cite{WhalenPoly,Demler,Whalen2}. These observations are fundamentally reliant upon the accuracy of the potential energy surfaces defining the molecular structure, which is our focus here.
\begin{figure*}[t]
\begin{centering}
\includegraphics[width = \textwidth]{trimer_1-min.pdf}
\caption{{Breathing mode PECs, relative to the $n=30$ manifold, for a Rb$_3$ trimer in a collinear geometry. Only triplet states are shown. The insets highlight the trilobite and butterfly potential wells. The faint gray curves show the dimer PECs. The diagram in the bottom right depicts schematically the dimer orbitals, in pink and blue, stretching to each perturber. The vertical blue line denotes the $p$-wave resonance position.}}
\label{fig:poly1}
\end{centering}
\end{figure*}
\begin{figure*}[t]
\begin{centering}
\includegraphics[width = \textwidth]{trimer_2-min.pdf}
\caption{{Breathing mode PECs for an $n=30$ Rb$_3$ trimer in two different geometries. Only triplet states are shown. The top panels highlight the trilobite region, while the bottom panels highlight the butterfly potential wells; the left panels are for a right-angle geometry while the right panels are for an angle $\pi/10$. The faint gray curves running through all four figures show the dimer PECs. The red curve neglects butterfly couplings. }}
\label{fig:polymore}
\end{centering}
\end{figure*}
We develop a generic description of polyatomic LRRMs \footnote{Just as Rydberg molecules (of the H$_2$ variety), ultra-long-range Rydberg molecules, and Rydberg-Rydberg macrodimers can be easily confused due to their similar appellations, so it is with polyatomic Rydberg molecules. Several references have investigated a different type of polyatomic molecule formed by replacing the perturber with a polar dimer such as KRb \cite{Rittenhouse2010,Rittenhouse2011, Mayle2012,Rosario2015}, or even with several polar perturbers \cite{polarperturberpoly}.} using the trilobite orbital method. We enlarge the basis set used in the trial wave function in Eq. \ref{trialwavetrilo} to include trilobite and butterfly states associated with each new perturber:
\begin{align}
\label{trialwavepoly}
\Psi(\vec r; \vec R) = \sum_{n=n_1}^{n=n_2}\Bigg[\sum_{l=0}^{l=l_\text{min}}&\sum_{m=-l}^{m=l}c_{nlm}(\vec R)\phi_{nlm}^1(\vec r)\\&+\sum_{i=1}^N\sum_{\xi=1}^4\mathcal{C}_{ni\xi}\+{\Upsilon_{ir,n}^{\pmb\xi1}}\Bigg].\nonumber
\end{align}
The basis size, $\mathcal{M} (4N+(l_\text{min}+1)^2)$, grows linearly with $N$. The Hamiltonian matrix elements derived for the dimer readily generalize. The block of high$-l$ trilobite states now contains additional overlap terms between trilobite states associated with each perturber,
\begin{align}
\label{matelementpolymer1}
&\bra{\+{\Upsilon_{pr,n}^{\pmb\alpha 1}}} H_N({\vec r};\{\vec R_i\}) \ket{\+{\Upsilon_{qr,n'}^{\pmb\beta 1}} }\\&=-\frac{1}{2n^2}\+{\Upsilon_{pq,n}^{\pmb\alpha\pmb\beta}}\delta_{nn'}+ 2\pi \sum_{i=1}^N\sum_{\xi = 1}^4a_\xi\+{\Upsilon_{pi,n}^{\pmb\alpha\pmb\xi}}\+{\Upsilon_{iq,n'}^{\pmb\xi\pmb\beta}}.\nonumber
\end{align}
The quantum defect block is unchanged except for an additional summation over perturbers,
\begin{align}
&\bra{\phi_{nlm}^1}H_N({\vec r};\{\vec R_i\}) \ket{\phi_{n'l'm'}^1} \\&= -\frac{\delta_{nn'}\delta_{ll'}}{2(n-\mu_l)^2} + 2\pi \sum_{i=1}^N\sum_{\xi = 1}^4a_\xi\phi_{nlm}^\xi(R_i)^*\phi_{n'l'm'}^\xi(R_i),\nonumber
\end{align}
and the off-diagonal coupling blocks are
\begin{align}
\label{endmat}
&\bra{\+{\Upsilon_{pr,n}^{\pmb\alpha 1}}}H_N({\vec r};\{\vec R_i\}) \ket{\phi_{n'l'm'}^{1} }\\&
= 2\pi \sum_{i=1}^N\sum_{\xi = 1}^4a_\xi\+{\Upsilon_{p i,n}^{\pmb\alpha \pmb\xi}}\phi_{n'l'm'}^\xi(R_i).\nonumber
\end{align}
The overlap matrix has now a larger trilobite block given by $\+{\Upsilon_{pq,n}^{\pmb\alpha\pmb\beta}}\delta_{nn'}$, but the rest is unchanged. This matrix is typically one to two orders of magnitude smaller than that required when using the Rydberg basis, a major increase in computational efficiency. Furthermore, it reveals the essential structure of the trilobite states of these molecules that is difficult to extract from full basis.
This structure is most easily observed by considering just the trilobite states within a single $n$-manifold. Eqs. \ref{matelementpolymer1}-\ref{endmat} then simplify, and the PECs are the $N$ eigenvalues of $\+{\Upsilon_{pq,n}^{11}}$, an $N\times N$ matrix\footnote{This is strictly true only if the scattering length is equal for all perturbers; in fact, the PECs are eigenvalues of a generalized matrix equation $\sum_ia(R_i)\+{\Upsilon_{pi,n}^{11}}\+{\Upsilon_{iq,n}^{11}}\vec c=E\+{\Upsilon_{pq,n}^{11}}\vec c$}. The average value of these eigenvalues for the polyatomic system equals the single eigenvalue of the dimer. The off-diagonal elements of $\+{\Upsilon_{pq,n}^{\pmb1\pmb1}}$,$\alpha\ne\beta$ correspond to the overlap between trilobite states extending from the Rydberg core to different perturbers, and their size determines the splittings between the polyatomic eigenvalues and their average value. If these off-diagonal elements vanish, the eigenvalues are simply $N$ copies of the dimer eigenvalue $\+{\Upsilon_{RR,n}^{11}}$. The formation of polyatomic states is therefore predicated on the overlap between these trilobite basis states, lending this approach some physical meaning beyond its mathematical effectiveness. The angular dependence of the trilobite wave functions likewise contributes considerably to the energy landscape\cite{FeyTrimer}.
In Figs. \ref{fig:poly1} and \ref{fig:polymore} we show three sets of breathing mode PECs for a triatomic molecule. This calculation includes three hydrogenic manifolds and, as a result of the computational gains granted by the trilobite overlap method, are the first calculations showing the trimer butterfly potential curves. In the collinear configuration shown in Fig. \ref{fig:poly1} the high level of symmetry leads to large overlap elements, and hence large splittings in the trimer potential curves. These have gerade and ungerade symmetry, and are especially notable in the angular butterfly curves which, due to their initial degeneracy in the diatomic case, couple strongly and are shifted wildly from the dimer limit. At large $R$ the overlap vanishes and the PECs converge to the dimer curve.
Fig. \ref{fig:polymore} provides more detail about the effect of the molecular geometry. In the 90$^o$ configuration, the trilobite-like states overlap much less than in the collinear configuration, and the potential curves exhibit much smaller splittings. An angular arrangement at a very small angle, $\pi/10$, is shown on the right. The overlaps now increase at larger $R$, unlike the other two configurations, and moreover the trilobite curves to the left of the butterfly crossing no longer oscillate about the dimer potential. The trilobite potentials without any $p$-wave contributions (red) behave as expected, oscillating about the dimer PEC. This shows that the coupling between butterfly and trilobite states is quite sensitive to the geometry; in this configuration the pronounced avoided crossing between states shifts the trilobite potentials on either side of the crossing so that they are higher (left) and lower (on the right) than the dimer value. These strong couplings are also manifest in the butterfly potentials in the lower right panel, which evince exagerrated, yet slow, oscillations.
\begin{figure*}[ht]
\begin{center}
\subfigure[]{\label{hoodoo2}\includegraphics[scale = 0.4035]{exampfig1-eps-converted-to-min.pdf}}
\subfigure[]{\label{hoodoo2}\includegraphics[scale = 0.4]{fig2b-eps-converted-to-min.pdf}}\subfigure[]{\label{hoodoo3}\includegraphics[scale = 0.4]{fig2c-eps-converted-to-min.pdf}}
\end{center}
\begin{center}
\caption{(a) PECs for an octagonal molecular geometry, color-coded by their irrep. The symmetry operations for $C_{8v}$ symmetry are shown in the inset. (b) ``Hoodoo'' symmetry adapted trilobite orbitals for evaluated at $R = 840$ $a_0$. The probability amplitude $\sqrt{r^2|\psi(x,y,0)|^2}$ is plotted in the $xy$ plane. (c) The electron probability corresponding to the one-dimensional irrep $B_1$ (top) and one of the doubly-degenerate $E_1$ irreps (bottom) are plotted. This figure is modified from Ref. \cite{JPBdens}. }
\end{center}
\label{mainhoodoofigure1}
\end{figure*}
The methodology developed in the present section applies to an arbitrary arrangement of perturbers, and shows that the spectroscopic signatures of polyatomic formation in non-isotropic Rydberg states will not be so clear as in $nS$ Rydberg states. Some additional understanding of the dependence of these potential curves on the geometry is gained by characterizing the symmetry group of the molecule. The molecular symmetry group is a
subgroup of the complete nuclear permutation inversion group of
the molecule~\cite{Bunker, Bunker2}, which commutes with the molecular
Hamiltonian in free space. Therefore, the eigenstates of such a
Hamiltonian can be classified in terms of the irreducible
representations (irreps) of the given molecular symmetry group, called {\it symmetry-adapted orbitals} (SAOs). Given a molecular symmetry
group, it is possible to calculate the SAOs associated
with each irrep of the group using the projection operator method \cite{Bunker}. The projection operator also gives the coefficients $\mathcal{A}_p^{(\alpha,j)}$ for the SAO $\mathcal{G}^{(\alpha,j)}(\vec r)$ corresponding to the $\alpha^\text{th}$ orbital and $j^\text{th}$ irrep:
\begin{equation}
\label{sao}
\mathcal{G}^{(\alpha,j)}(\vec r)=\sum_{p = 1}^N\+{\Upsilon_{pr,n}^{\pmb\alpha\pmb1}}\mathcal{A}_p^{(\alpha,j)}.
\end{equation}
The prescription for calculating the projection operator depends on the orbital in question, and becomes quite involved for the angular butterfly states since they mix together under symmetry operations. Ref. \cite{JPBdens} describes this calculation in full detail; here we report only the particularly elegant expression for the PEC for the trilobite molecule of the $j^\text{th}$ irrep:
\begin{equation}
\label{saoswave}
E^{(j)} = 2\pi a_s(k)\sum_{p,q=1}^N\mathcal{A}_p^{j}\+{\Upsilon_{pq}^{11}}\mathcal{A}_q^{j}.
\end{equation}
The PECs of an octagonal arrangement of atoms, characterized by their irreps, and symmetry adapted trilobite orbitals are displayed in Fig. \ref{mainhoodoofigure1} . Panel a) shows the PECs, color-coded by their symmetry irrep. This shows how the coupling strengths between trilobite and butterfly potentials depend on the irrep and how PECs of different irreps have real crossings. They symmetry irreps also determine the degeneracy remaining in the system: the $E_x$ irreps are all doubly degenerate, and hence only five rather than the anticipated eight PECs are visible. Panels b and c show representative symmetry adapted orbitals as density plots at the internuclear distance $R = 840$ $a_0$. These explicitly exhibit the allowed symmetries and the interference between trilobite orbitals in their beautiful nodal patterns.
\begin{figure}[t]
\begin{center}
\includegraphics[width= \columnwidth]{trimer-min.png}\\
\includegraphics[width= \columnwidth]{trimer2-min.png}
\caption{Top: PECs showing the Borromean collinear trimer state in Na$_3$. Bottom: $^1P$ and borromean trimer curves in Na$_3$ (left) and K$_3$ (right). The energy scale is relative to the $n=30$ Rydberg energy.}
\label{fig:trimers}
\end{center}
\end{figure}
The large oscillations in the breathing mode potentials in polyatomic molecules appear strongest in these symmetric configurations, and more stable LRRMs can thus be engineered by exploiting these features. They can also lead to exotic Borromean trimer\footnote{Borromean rings are three interlaced rings which, although any two of them are not attached together, cannot as a trio be separated.} states, originally predicted by Ref. \cite{Rost2009} to occur in neon. Ne has a small positive scattering length, and thus has weakly repulsive trilobite PECs which cannot support bound dimer states. However, when a third atom is introduced to form a collinear trimer with the Rydberg atom in the middle, the large oscillatory gerade/ungerade splitting creates deeper potential wells which can support bound states \cite{Rost2009}. Sodium is a more experimentally viable species to realize this scenario. It has a larger singlet scattering length and a nearly integer ($\sim$0.86) $nP$ quantum defect which brings this state energetically close to the $(n-1)$ hydrogenic manifold. It therefore couples to the ungerade trilobite trimer as shown in in Fig. \ref{fig:trimers}a. The gerade trimer remains uncoupled due to its opposite parity. This Borromean trimer could be excited via the $nP$ admixture, circumventing the need to excite a high-$l$ state in the original proposal. Fig. \ref{fig:trimers}b,c show that this same phenomena occurs in the singlet butterfly curves, which have potential wells nearly an order of magnitude deeper in the trimer configuration (black) than in the dimer (dashed blue).
What can we learn in general as $N$ tends towards larger values? The fact that the trilobite polymer PECs average to diatomic PEC reveals that the trilobite interaction is profoundly non-additive. The sophisticated quantum many-body treatment of Refs. \cite{WhalenPoly,Whalen2} has performed excellently in predicting the observed line shapes, but as it relies on the additive scaling of the $nS$ vibrational states it is not clear how to interpret spectra from states with higher $l$. In particular, the dramatic localization of the trilobite state about the location of the perturber causes the overlap matrix elements to be particularly large along the breathing mode, lending some physical import to the breathing mode potentials studied here. The lack of large overlap elements causes the spectrum in a random gas, even for fairly large $N$, to resemble that of the dimer \cite{EilesHyd}. As $N$ increases further the trilobite states fill the Rydberg volume and begin to have significant overlap. One now expects the presence of so many perturbers to destabilize the trilobite, leading to only delocalized polyatomic wave functions heavily perturbed by the dense gas. However, a second process competes with this in a truly random disordered environment. As explained in Ref. \cite{RostLuukko}, as $N$ increases so does the probability that two or more perturbers are in close proximity until it is essentially guaranteed\footnote{This is analogous to the ``birthday paradox:'' there is a 50\% probability that two people in a room of twenty-three will share a birthday.}. When a cluster of just two nearby atoms forms, the Hamiltonian matrix contains a $2\times 2$ sub-block that, to first order, has identical elements. This sub-block approximately decouples from the rest of the matrix, and has one vanishing eigenvalue and one at twice the dimer energy. A cluster of $N$ atoms thus behaves like a perturber with $N$ times the scattering length, which attracts the wave function more strongly and resurrects the trilobite molecule \cite{RostLuukko}.
\section{Spin and relativistic effects}
\label{sec:spinintro}
In all calculations until now we ignored spin-dependent terms in the electronic Hamiltonian, and thus the triplet and singlet phase shifts were studied independently. However, the energy scale of the resulting PECs and vibrational spectrum is similar to the size of the fine and hyperfine structure. Thus, an accurate description of Rydberg molecules must include the relevant spin-dependent interactions. This section discusses each of these new interactions separately, focusing mostly on the fine structure of the $p$-wave scattering phase shifts, as these have either been ignored or treated only approximately in most calculations until recently \cite{EilesSpin}. The trilobite overlap approach used in the rest of the tutorial is not adopted here, as these spin-dependent interactions are simpler to describe in the Rydberg basis.
\subsection{Construction of the Hamiltonian}
\label{sec:spinham}
The fine structure of the Rydberg atom, caused by the spin-orbit splitting, was discussed in Sec. \ref{sec:inter}. As depicted in the second panel of Fig. \ref{fig:cartoonPRA}, it can be added to our calculations by simply extending the Rydberg basis $\ket{nlm}$ to the spin-dependent basis $\ket{n(ls_1)jm_j}$. This almost doubles the number of low-$l$ potentials shown previously, but has very little effect on the trilobite and butterfly potentials due to the negligible fine structure of the high$-l$ states.
The hyperfine structure of the perturber\footnote{The hyperfine splitting depends on the electron's wave function amplitude at the nucleus \cite{AllHF}. Since the Rydberg wave function has very little overlap with the ionic core the hyperfine splitting of Rydberg states decays rapidly with $n$ and can be safely ignored. \cite{AllHF}. Likewise, since the trilobite wave function has such a large overlap with the perturber, one might expect that this contributes an additional energy shift. We neglect this as well as the trilobite amplitude at the perturber is several orders of magnitude smaller than the amplitude of the perturber's valence electron, but this effect could merit further quantitative exploration. Finally, we ignore the dependence on the hyperfine state in the electron phase shifts. This is relevant near threshold where the electron's energy depends on the hyperfine state, and could therefore modify slightly the PECs near the classical turning point.} couples the nuclear spin $i$ to the perturber's electronic spin $s_2$ (See Fig. \ref{fig:cartoonPRA}). This was first included in the theoretical treatment of Ref. \cite{AndersonPRA}, which showed that it mixes singlet and triplet symmetries. These states were measured in subsequent spectroscopy of $nD$ and $nP$ Rb states \cite{AndersonPRL,SingTripMix,MacLennan}. Measurements in Cs have also revealed this hyperfine-induced mixing \cite{Sass}. One curious effect of the hyperfine structure is its interplay with other energy splittings, particularly the fine structure. As the hyperfine splitting is $n$-independent while the fine structure splitting changes with $n$, degeneracies between molecular states can be engineered by changing principal quantum number. This was utilized in Ref. \cite{Niederprum} to induce spin-flips in the perturber, and in Ref. \cite{PfauRaithel} to excite trilobite molecules in Rb due to a ``spin-bridge'' when the hyperfine splitting becomes comparable to the splitting between the trilobite state and the $(n-3)S$ state.
The hyperfine Hamiltonian is $ H_{HF}=A\vec i\cdot \vec s_2$, where the values of $A$ for several isotopes are given in Table \ref{tab:datatableHF}. Since this Hamiltonian commutes with the Rydberg Hamiltonian $H_0$ we choose an uncoupled basis $\ket{n(ls_1)jm_j}\times\ket{s_2m_2;im_i}$, and use $\alpha = \{n,l,s_1,j,m_j\}$ to describe the Rydberg quantum numbers. The matrix elements of $ H_{HF}$ are
\begin{align}
& \bra{\alpha im_i,s_2m_2}A\vec I\cdot\vec S_2 \ket{\alpha' i'm_i',s_2'm_2'}\\&= \frac{A}{2}\delta_{\alpha\alpha'}\sum_{FM_F}C_{s_2m_2,im_i}^{FM_F} C_{s_2m_2',im_i'}^{FM_F}\nonumber\\&\times\left[(F(F+1)-i(i+1)-s_2(s_2+1)\right].\nonumber
\end{align}
\begin{figure}[t]
{\normalsize
\begin{centering}
\begin{center}
\includegraphics[width=\columnwidth]{diagram-min.pdf}
\vspace{-110pt}
\end{center}
\end{centering}
}
\caption{The relevant coordinates and angular momenta. The internuclear axis lies parallel to the body-frame $z$ axis passing through the ionic core (black) and the perturber (red). The black (red) orbits represent the semiclassical orbits of the Rydberg (perturber's) electron. The Rydberg electron is located at $\vec r$ relative to the core and at $\vec X = \vec r - \vec R$ relative to the perturber. The spin of the Rydberg electron, $\vec s_1$, couples to its orbital angular momentum relative to the core, $\vec l$, to give a total angular momentum $\vec j$. The spin of the perturber's outer electron, $\vec s_2$ (cyan) interacts with the perturber's nuclear spin, $\vec i$ to form $\vec F$. The electron-atom scattering potential depends on the total electronic spin, $\vec S = \vec s_1+ \vec s_2$, coupled to the orbital angular momentum $\vec L$ relative to the perturber to form total angular momentum $\vec J$. }
\label{fig:cartoonPRA}
\end{figure}
\noindent These terms, along with the Fermi pseudopotential, give the relativistic Hamiltonian
\begin{equation}
\label{eq:Hamiltonian}
H(\vec r;\vec R) = H_{Ryd}(\vec r) + V^J_\text{fermi}(\vec R,\vec r) + H_{HF}.
\end{equation}
$ V_\text{fermi}^J(\vec R,\vec r)$ is the $J$-dependent Fermi pseudopotential which we derive now. The spin orbit coupling in the scattering interaction couples the orbital angular momentum $L$ to the total spin $S$ and results in split $^3P_J$ scattering states. Ref. \cite{EilesSpin} presents two complementary derivations of the spin-dependent pseudopotential. The first involves a recoupling of the tensorial operators in the Fermi pseudopotential using Wigner-Racah angular momenta algebra. The second, expanded here, reformulates the pseudopotential so that it is diagonal in the representation $\ket{(LS)J\Omega}$, just like the scattering Hamiltonian. Since the operator $\vec L^2$ does not commute with the Rydberg Hamiltonian, which has an operator $\vec l^2$ defining the angular momentum about the Rydberg core rather than about the perturber, we must develop a frame-transformation like that encountered in Sec. \ref{sec:inter}. While that one transformed between $LS$ and $jj$ coupling schemes, here we derive one to transform $l$ to $L$.
\begin{table}
\begin{center}
\begin{tabular}{| c | c | c |}
\hline \hline
Atom & $i$ & A (MHz)\\
$^{6}$Li &$1$ & $152.137$\\
$^7$Li & ${3}/{2}$ & $ 401.752 $\\
$^{23}$Na& ${3}/{2}$ & 885.813\\
$^{39}$K& ${3}/{2}$& 230.86\\
$^{41}$K& ${3}/{2}$ & 127.01\\
$^{85}$Rb & $5/{2}$ & 1011.9 \\
$^{87}$Rb & ${3}/{2}$ & 3417\\
$^{133}$Cs &${7}/{2}$ & 2298\\
\hline
\end{tabular}
\end{center}
\caption{Hyperfine constants $A$ and nuclear spin $i$ for the common isotopes of these alkali metals. \protect\cite{Beckmann1974,RbHF,AllHF,CsHF}.}
\label{tab:datatableHF}
\end{table}
Our first goal is to write the Fermi pseudopotential so that it is diagonal in the proper representation, i.e. it has the form $\sum_J\ket{(LS)J\Omega}A_{LSJ}\bra{(LS)J\Omega}$, where $A_{LSJ}$ is a scattering length function depending on $L$, $S$, and now $J$. Thus far we have used a pseudopotential that leaves $S$ and $L$ uncoupled,
\begin{align}
\label{der1}
V_\text{fermi} &= \sum_{L'',S}\mathcal{A}(L''S,k) \cev\nabla {}^{L'' }\delta^3(\vec X)\cdot \vec\nabla^{L''}\\&\times\nonumber\sum_{M_{S}}\ket{SM_S}\bra{SM_S},
\end{align}
where $\vec X = \vec r - \vec R$ and $\mathcal{A}(LS,k) = (2L+1)2\pi a_{LS}(k)$. We apply the projection operator $\sum_{LM_L}\ket{LM_L}\bra{LM_L}$ to both sides of Eq. \ref{der1}:
\begin{align}
V_\text{fermi}&=\sum_{L'',S,{M_{S}}}\sum_{\substack{LM_L\\L'M_L'}}\mathcal{A}(L''S,k)\ket{SM_S}\bra{SM_S}\\&\times \ket{LM_L}\bra{LM_L}\cev\nabla {}^{L'' }\delta^3(\vec X)\cdot \vec\nabla^{L''}\ket{L'M_L'}\bra{L'M_L'}\nonumber
\end{align}
We next perform the inner integration over angular coordinates $\hat X$
\begin{align}
\label{angint11}
&\bra{LM_L}\cev\nabla {}^{L'' }\delta(\vec X)\cdot \vec\nabla^{L''}\ket{L'M_L'}\\&= \int Y^*_{LM_L}(\hat X)\cev\nabla{}^{L''}\cdot \delta^3(\vec X)\vec\nabla^{L''}Y_{L'M_L'}(\hat X)d\hat X.\nonumber
\end{align}
Since this integration is over the solid angle $\hat X$, the final operator still retains a derivative with respect to $X$. With the benefit of foresight, we know that the basis states will eventually be written as a power series in $X$ (see Eq. \ref{firstorder}). The action of the derivative operator on this power series is
\begin{align}
\label{der2}
\partial_X^{L''}\left.\sum_La_LX^L\right|_{X=0} &= \left.\sum_L\frac{a_LL!X^{L-L''}}{(L-L'')!}\right|_{X = 0}\\
&= a_LL!\delta_{LL''}.\nonumber.
\end{align}
Similarly, the conjugate of this acting on the bra will give a Kronecker delta $\delta_{L'L}$. This product of these two Kronecker deltas gives $\delta_{L'L''}$, i.e. the operator is diagonal in $L$. Thus, we integrate Eq. \ref{angint11} separately for $L=0$,
\begin{align}
\int Y^*_{00}(\hat X) \delta^3(\vec X)Y_{00}(\hat X)d\hat X&=\frac{\delta(X)}{X^2}\left|Y_{00}(0,0)\right|^2,
\end{align}
and for $L=1$,
\begin{align}
&\int Y^*_{1M_L}(\hat X)\cev\nabla\cdot \delta^3(\vec X)\vec\nabla Y_{1M_L'}(\hat X)d\hat X\\&=\frac{\delta(X)}{X^2}\Bigg(\partial_X'\partial_X Y_{1M_L}(0,0)Y_{1M_L'}(0,0) \nonumber\\&+\frac{1}{X^2}\frac{(2L+1)(L+1)L}{8\pi}\delta_{M_L,M_L'}\delta_{|M_L|,1} \Bigg).\nonumber
\end{align}
Here $\partial_X'\partial_X$ is the radially-dependent term of the dot product of the two gradient operators, where $\partial_X'$ acts to the left. For $L=1$, inspection of Eq. \ref{der2} shows that the effect of each of these derivatives is equivalent to multiplying by $X^{-1}$. We therefore replace $\partial_X'\partial_X\to X^{-2}$ factor to obtain a compact form for the pseudopotential:
\begin{align}
V_\text{fermi}&=2\pi\sum_{LM_L,M_L'}\sum_{SM_S}\ket{LM_L,SM_S} \frac{(2L+1)^2}{4\pi}\\&\mathcal{A}(LS,k)\frac{\delta(X)}{X^{2(L+1)}}\delta_{M_L,M_L'}\bra{LM_L',SM_S}.
\end{align}
\begin{figure}[t]
\begin{center}
{\normalsize
\begin{centering}
\includegraphics[scale =0.09]{fig3-min.png}
\end{centering}
}
\end{center}
\caption{PECs of Rb$_2$, $\Omega = 1/2$. The results of Ref. \cite{KhuskivadzePRA} (red crosses) are overlaid in red. This figure is taken from Ref. \cite{EilesSpin}. }
\label{fig:RbSpinFig}
\end{figure}
\noindent We now couple $L$ and $S$, let the scattering length factor depend on $J$, and sum over $M_L'$, giving
\begin{align*}
V_\text{fermi}&= 2\pi\sum_{LM_L}\sum_{SM_S}\sum_{J\Omega,J'\Omega'}\frac{(2L+1)^2\delta(X)}{4\pi X^{2(L+1)}}\mathcal{A}(LSJ,k)\\\,\,\,\,&\times\ket{(LS)J\Omega} C_{LM_L,SM_S}^{J\Omega} C_{LM_L',SM_S}^{J'\Omega'}\bra{(LS)J'\Omega'}
\end{align*}
The sum over $M_L$ and $M_S$ gives
\begin{equation}
\sum_{M_L,M_S}C_{LM_L,SM_S}^{J\Omega} C_{LM_L',SM_S}^{J'\Omega'}=\delta_{JJ'}\delta_{\Omega\Omega'},
\end{equation}
which imposes the triangularity condition between $L$, $S$, and $J$. Finally, in terms of the collective quantum number for the scattering interaction, $\ket\beta=\ket{(LS)J\Omega}$, we have
\begin{align}
\label{finalprojectionform}
V_\text{fermi}&=\sum_{\beta}\ket{\beta}\frac{(2L+1)^2}{2}\mathcal{A}(LSJ,k)\frac{\delta(X)}{X^{2(L+1)}} \bra{\beta}.
\end{align}
The quantum numbers $\beta$ are incompatible with $\alpha$, which characterize the Rydberg eigenstates. We expand the Rydberg wave function of Eq. (\ref{eq:jdepefuncs}) to first order about the position of the perturber:
\begin{align}
\label{firstorder}
&\psi_{nlm}(\vec r) \approx\left[\phi_{nlm}(\vec R) + \vec\nabla\left(\phi_{nlm}(\vec R)\right)\cdot\vec X\right].
\end{align}
We define the following $Q$-functions,
\begin{align}
\label{eqn:Qdef}
Q_{LM_L}^{nl}(R) &=\delta_{m,M_L} \left[\vec \nabla^L\left(\phi_{nlm}(R)\right)\right]_{M_L}^L.\\
\label{eqn:Qfuncs}
Q_{00}^{nl}(R)&= \frac{u_{nl}(R)}{R}\sqrt{\frac{2l+1}{4\pi}},\\
Q_{10}^{nl}(R) &= \sqrt{\frac{2l+1}{4\pi}}\partial_R\left(\frac{u_{nl}(R)}{R}\right),\\
Q_{1\pm 1}^{nl}(R)&=\frac{u_{nl}(R)}{R^2}\sqrt{\frac{(2l+1)(l+1)l}{8\pi}},l>0.
\end{align}
\begin{figure*}[t]
\begin{center}
{\normalsize
\begin{centering}
\vspace{-10pt}
\includegraphics[width=0.9\textwidth]{Cspic-min.pdf}
\vspace{-20pt}
\end{centering}
}
\end{center}
\caption{PECs of Cs$_2$ for the projection $\Omega = 1/2$ are plotted in black. Several characteristic electronic wave functions are shown. This figure is modified from Ref. \cite{EilesSpin}. }
\label{fig:CsSpinFig}
\end{figure*}
\noindent After using the spherical tensor representation of $\vec\nabla\phi_{nlm}(\vec R)$ given by the $Q$ functions and expressing $\vec X$ in terms of spherical harmonics $Y_{LM}(\hat X)$ centered at the perturber, it becomes clear that this expansion mediates the transformation from spherical harmonics relative to the Rydberg atom, $Y_{lm}(\hat r)$, to $S$ and $P$ partial waves relative to the perturber, $Y_{LM}(\hat X)$:
\begin{align}
\label{taylorexpform}
\psi_{nlm}(\vec r)&\approx \sum_{L=0}^1X^Lf_L Q_{LM_L}^{nlj}(R)Y_{LM_L}(\hat X),
\end{align}
where $f_L = \sqrt{\frac{4\pi}{(2L+1)}}$.
We can now form matrix elements of Eq. \ref{finalprojectionform} in this representation, requiring only a trivial integration over the radial coordinate $X$. These are compactly expressed by first constructing the matrix representation of Eq. (\ref{finalprojectionform}) in the $\ket{\beta}$ basis
\begin{align}
\label{eqn:diagpotential}
U_{\beta,\beta'} &= \delta_{\beta,\beta'}\frac{(2L+1)^2} {2} a(SLJ,k).
\end{align}
The transformation of this diagonal matrix into one in the $\ket{\alpha s_2m_2}$ basis is mediated by a frame-transformation matrix $\mathcal{A}$, which transforms between the $\ket{\alpha s_2m_2}$ and $\ket{\beta}$ representations, analogous to what is done in multiple scattering theory \cite{DillDehmerJCP}. $\mathcal{A}$ is readily deduced from the prior steps of the derivation:
\begin{align}
\mathcal{A}_{\alpha s_2m_2,\beta} & =\sum_{M_L=-L}^{M_L=L}C_{LM_L,SM_S} ^{JM_J}f_{L} Q_{LM_L}^{nlj}(R) . \nonumber
\end{align}
The final scattering matrix in the Rydberg basis consists of a block matrix
\begin{align}
\label{eqn:scattpotential}
&V_{ii'}=\sum_{jj'}{\mathcal A_{ij}}{U_{jj}}{\mathcal A_{ji'}^\dagger}
\end{align}
for every $n$ and $l$. The mixing of $M_L,M_L'$ implied by Eq. \ref{eqn:scattpotential} is critical for an accurate physical description of this splitting, since the total spin vector $\vec{S}$ and total
orbital $\vec{L}$ precess during each
$P$-wave collision. This was recognized and incorporated in the Green's function calculation of Ref. \cite{KhuskivadzePRA}, but all subsequent work has neglected this detail. We expect that the much simpler description developed here will correct this oversight. This mixing of $M_L$ projections invalidates the use of $\Sigma$ and $\Pi$ symmetry labels to categorize the $^3P_J$ PECs. Incidentally, the Clebsch-Gordan coefficients vanish for $M_L = 0$ for the $^3P_1$ state, so that it remains a $\Pi$ state. Inclusion of the hyperfine interaction eliminates even this symmetry.
\subsection{Spin-dependent potential energy curves and dipole moments}
To confirm the validity of this approach and to allow for a direct comparison with the spin-independent potentials of Fig. \ref{fig:Rbmanifold}, Rb$_2$ PECs are shown in Fig. \ref{fig:RbSpinFig}. The calculations of Ref. \cite{KhuskivadzePRA}, which ignored hyperfine and fine structure and thus cannot be compared directly, are overlayed. The main features of Ref. \cite{KhuskivadzePRA} are reproduced excellently, validating the accuracy of our $^3P_J$ pseudopotential. The hyperfine structure adds significant complexity, increasing the multiplicity of the trilobite and butterfly states and splitting the low-$l$ states by several GHz.
Fig. \ref{fig:CsSpinFig} shows PECs for Cs$_2$, which reveal the impact of this relativistic splitting in this molecule. The positions of the $^3P_J$ shape resonances and their energy dependences strongly modify the butterfly potential wells. The $^3P_0$ resonance in cesium occurs at such a low electronic energy that the associated PECs cross the low-l states at very large internuclear distances, affecting the vibrational states to a greater degree than in Rb. The much larger $^3P_J$ splittings in Cs greatly spread the butterfly wells, limiting the density of avoided crossings. Some of these butterfly states are plotted in the insets. In the bottom right an unusual butterfly, situated in the deepest $^3P_0$ well at a large $\sim 1250a_0$, is shown. Its size and overall shape resemble a trilobite due to the similar bond length, but the specifics of its nodal character, particularly the node at the perturber, reveal its butterfly nature. Although in Cs$_2$ this state is challenging to excite since it contains very little low-$l$ character, Ref. \cite{EilesHetero} showed that in the \textit{heteronuclear} molecule NaCs the outer wells of this potential intersect the $nP$ Rydberg state and thus admix $P$ character into the butterfly. The top left inset shows a butterfly with mixed $\Pi$ and $\Sigma$ symmetry, and hence it has peaks of electron density both near the perturber and on the opposite side of the Rydberg core. The middle inset shows the mixed trilobite -$(n+4)S$ state, whose $S$ character is accessible experimentally\cite{TallantCS,BoothTrilobite}. The improved description of the nearly-degenerate high-$l$ manifold with the very close $(n+4)s$ state given here lends a more complete theoretical description of this state that should encourage further exploration of the trilobite state in Cs.
\begin{figure}[t]
\begin{center}
{\normalsize
\begin{centering}
\includegraphics[width=\columnwidth]{fig6-min.png}
\end{centering}
}
\end{center}
\caption{ PECs of Cs$_2$, $\Omega = 1/2$. The results using the $\{29,30,31\}$ basis (dot-dashed, blue), the $\{28,29,30,31\}$ basis (solid, black) and the $\{27,28,29,30,31\}$ basis (dashed,red) are plotted. Each panel displays a different regime, showing that at long-range the calculation is quite well converged with either basis, but the short-range butterfly curves in particular vary severely with the basis size. This figure is taken from Ref. \cite{EilesSpin}. }
\label{fig:convergencecomparison}
\end{figure}
Fig. \ref{fig:convergencecomparison} serves two purposes. First, it demonstrates the dependence of the PECs on basis size. Three different basis sets ($\{n_H-q,...,n_H,n_H+1\}$, with $q = 3,2,1$) are used. At large $R$ the inclusion of additional manifolds {\it below} the level of interest does not contribute to the non-convergent increase in well depth seen by \cite{Fey}, but at short range these additional manifolds have a strong effect on the potential wells, particularly the butterfly wells, repulsing them upwards. Setting $q=2$ agrees well with Ref. \cite{KhuskivadzePRA} and with the BK model. Nevertheless, the large variation in butterfly PECs with basis size reveals the convergence difficulties caused by the shape resonance.
Second, Fig. \ref{fig:convergencecomparison} highlights some special features of this spin coupling. In the top panel, showing the trilobite potentials, we see that the hyperfine coupling leads to three trilobite potentials \cite{AndersonPRA}, which intersect and mix with the $(n+4)S$ state. In the middle panel we see that many butterfly potentials are associated with each scattering phase shift, depending on the other spin quantum numbers. It has recently been argued that the multiplicity of such potential curves leads to singlet, doublet, and triplet vibrational lines observed in Rb \cite{SpinFey}. Finally, the bottom panel shows how the hyperfine splitting mixes the singlet and triplet potential curves of Fig. \ref{fig:Rbmanifold}, resulting in one deep pure triplet state and a shallow mixed state.
\begin{figure}[b]
{\normalsize
\begin{center}
\includegraphics[width=0.45\textwidth]{fig9-eps-converted-to-min.pdf}
\end{center}
}
\vspace{-20pt}
\caption{ Analytic permanent electric dipole moments (black, dashed), permanent electric dipole moments ignoring the $^3P_J$ splitting (black, solid, labeled $^3P$), and permanent electric dipole moments from the full spin model for electronic states dominated by $^3P_0$ scattering (blue,solid), and $^3P_1$ scattering (red,solid), are plotted. The red squares are placed at the observed bond lengths and permanent electric dipole moments \cite{Butterfly}. The $^3P_0$ and $^3P_1$ permanent electric dipole moments correspond to states of mixed $M_L$, although the mixing is quite weak for $^3P_1$ scattering and the analytic and exact results agree more closely. The $^3P_2$ case is not shown, for simplicity. This figure is taken from Ref. \cite{EilesSpin}. }
\label{fig:dipoles}
\end{figure}
We conclude by showing an experimental signature of this fine structure in the butterfly molecule dipole moments, shown as a function of bond length in Fig. \ref{fig:dipoles}. The dashed black lines are the predictions of Eq. \ref{eq:dipolehydrogen}, which assume no coupling between butterfly states and zero quantum defects. The solid black line neglects the $^3P_J$ splitting. The red points are measured \cite{Butterfly}. The blue and red curves are calculated from the $^3P_0$ and $^3P_1$ electronic eigenstates. The $^3P_0$ dipole moments are weaker than the $J$-independent ones since the spin orbit splitting mixes $M_L$. The $M_L = 0$ butterfly molecules focus the electronic wave function near the perturber, while $|M_L| = 1$ states maximize the wave function closer to the Rydberg core; this is reflected in their dipole moments (positive for $M_L=0$, negative for $|M_L| = 1$), and can be seen in the exemplary wave functions sketched in Fig. \ref{fig:fieldbutterflies}. Quantitative agreement is seen between the experimental values and the $J$-dependent calculation, which are both systematically smaller (by $\sim$25\%) than the $J$-independent calculation. This is evidence that even though the relatively small e-Rb $^3P_J$ scattering splittings do not dramatically shift the PECs, these splittings do have significant impact on observables such as the dipole moments.
\section{Interactions with external fields}
\label{sec:fieldstudies}
The past three sections described the properties of LRRMs in increasing detail. Throughout a major theme was how these theoretical descriptions were paired with experimental developments, resulting in a fruitful cooperation between these twin pillars of physics which led to the observation and characterization of these molecules. We now turn to a more practical matter which was one of the initial motivations for studying these molecules: their large permanent dipole moments, which make them remarkably sensitive to external control. We focus first on weak fields which primarily address the rotational structure of the molecules; this discussion follows Refs. \cite{Butterfly,EilesPendular} and demonstrates the pendular nature of the rotational spectrum of these molecules\footnote{Dipolar molecules librate around the field axis analogously to a pendulum, and have an evenly spaced level structure like the quantum harmonic oscillator \cite{Friedrich}.}. Following this we study how such pendular LRRMs interact \cite{EilesPendular}. Finally, we develop and present a semi-perturbative treatment of external fields which are strong enough to shape the vibrational and electronic structure of the LRRMs, and apply this to some exemplary trimer configurations. These final calculations are presented here for the first time. Note that in the following we neglect the spin terms used in the previous section.
\subsection{Pendular spectrum in a weak electric field}
We first consider an external electric field $F<1$V/cm which is too weak to modify the electronic structure of the Rydberg molecule. The field-free PECs discussed previously are therefore still applicable, as are the vibrational states which only couple to the external field via the electronic PECs. The field-free rotational spectrum is set by the rotational constant $B_\nu = \frac{1}{m\langle R^2\rangle}$ in the rigid rotor approximation, where $m$ is the molecule's reduced mass and $\langle R^2\rangle$ is the vibrationally-averaged squared bond length. Due to the large size of LRRMs this constant is usually only some hundreds of kHz, smaller than the energy shift of the dipole-field coupling $-\vec{d}\cdot \vec{F}$, where $\vec{d}$ is the dipole moment. We therefore want to determine the effect of this field on the rotational spectrum. In the absence of external fields, polar molecules rotate freely with random orientations. Setting the quantization axis parallel to the electric field, the rotational Hamiltonian in an electric field is
\begin{equation}
\label{eq:rotationalham}
H_\text{rot}=B_\nu\hat{N}^2-dF\cos\theta.
\end{equation}
The rotational angular momentum operator is $\hat{N}$. We define a dimensionless parameter $\omega=\frac{dF}{N_\nu}$. The large dipole moments and small rotational constants of LRRMs conspire together to make this parameter very large, $\omega\sim 10^2-10^3$, about four orders of magnitude larger than in typical heteronuclear molecules at the same field strength ~\cite{KRb}. Trilobite or butterfly molecules are therefore ideal candidates to realize \text{pendular molecules}, since in this high $\omega$ limit the eigenstates of Eq. \ref{eq:rotationalham} resemble harmonic oscillator states, which librate like a pendulum about the electric field axis ~\cite{Rost,Friedrich}. These pendular states are obtained by diagonalizing $H_\text{rot}$ in the basis of spherical harmonics $Y_{NM_{N}}(\theta,\phi)$, $N=0,1,...$. $H_\text{rot}$ is diagonal in $M_N$ since the quantization axis is parallel to the electric field. The eigenstates $|\tilde NM_N\rangle$ are thus characterized by $M_N$ and their librational quantum number, $\tilde N$.
\begin{figure}[b]
{\normalsize
\begin{center}
\includegraphics[width=\columnwidth]{Fig3a-eps-converted-to-min.pdf}
\includegraphics[width=\columnwidth]{Fig3b-eps-converted-to-min.pdf}
\end{center}
}
\caption{(a) Stark spectrum of the $n=24$ pendular states, showing the energy shift $\Delta$ as a function of the applied electric field for $M_{N}=0$. The red points correspond to the two-dimensional harmonic oscillator approximation. (b) Orientation $(x=1)$ (orange) and alignment $(x=2)$ (blue) of the lowest pendular state [orange line in (a)]. This figure is taken from Ref. \cite{EilesPendular}.
}
\label{fig:pendular}
\end{figure}
Fig. \ref{fig:pendular}a shows the resulting Stark spectrum for $n=24$ butterfly molecules. At zero field, the typical $N(N+1)$ spacing of rotational states can be seen; this adiabatically switches into the equally-spaced energy levels of a harmonic oscillator. The ground state, $\Psi_{00}(\theta,\phi)$ highlighted in orange, is the most aligned. Fig. \ref{fig:pendular}b shows the orientation $\langle \cos{\theta} \rangle$ and alignment $\langle \cos^2{\theta} \rangle$ of this state. The almost perfect alignment at $F=1$V/cm is unmatched in previous efforts with traditional molecules.
Greater insight into the character of these pendular states is given in the limit $\omega\to\infty$. Using the explicit form for $\hat N^2$ in spherical coordinates, the Schr\"{o}dinger equation defined by $H_\text{rot}$ is
\begin{align}
&0=\\&\left(\frac{\partial^2}{\partial\theta^2} +\cot\theta\frac{\partial}{\partial\theta} + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}+\omega\cos\theta + W\right)\Psi(\theta,\phi),\nonumber
\end{align}
where $W = E/B_\nu$. This equation maps onto the 2D harmonic oscillator by setting $\xi = 2\alpha\tan(\theta/2)$, where $\alpha = \sqrt{\omega/2}$. A separable solution in $\xi$ and $\phi$ is then obtained, where $\Psi(\xi,\phi) = U(\xi)\frac{1}{\sqrt{2\pi}}e^{i|M_N|\phi}$. $U(\xi)$ is then given by:
\begin{align}
0&= \left(1 + \frac{\xi^2}{4\alpha}\right)^2\left[\frac{d^2}{d\xi^2} + \frac{1}{\xi}\frac{d}{d\xi} - \frac{M_N^2}{\xi^2}\right] U(\xi)\\&\nonumber+\frac{WU(\xi)}{\alpha}+ \frac{\omega}{\alpha}\frac{4 - \xi^2/\alpha}{4 + \xi^2/\alpha}U(\xi).
\end{align}
Since $\alpha\gg 1$ in the pendular regime we discard all terms of order $1/\alpha$ to obtain the Schr\"{o}dinger equation of a harmonic oscillator
\begin{equation}
\left[\left(\frac{d^2}{d\xi^2} + \frac{1}{\xi}\frac{d}{d\xi} - \frac{M_N^2}{\xi^2}\right) + \beta - \xi^2\right]U(\xi) = 0,
\end{equation}
where $\alpha\cdot\beta = W + \omega$. The energies of the pendular states are
\begin{equation}
E = B_\nu(\sqrt{2\omega}(2\tilde N + |M_N| + 1)-\omega),
\end{equation}
and the pendular states are
\begin{equation}
\Psi_{\tilde N,M_N}(\xi)=(-1)^{M_N}\mathcal{N}e^{-\frac{\xi^2}{2}}\xi^{M_N}L_{\tilde N}^{M_N}(\xi^2)e^{i|M_N|\phi},
\end{equation}
where $L_{N}^M(x)$ is a Laguerre polynomial and $\mathcal{N}=\sqrt{\frac{\tilde N!}{\pi\Gamma(\tilde N+M_N+1)}}$. The red points in Fig. \ref{fig:pendular} were computed using this approximation. This Stark map has been experimentally mapped out in butterfly molecules in Ref. \cite{Butterfly}, providing clear evidence of the polar nature of these molecules. The theory discussed in this section allows for the extraction of dipole moments as well as bond lengths from such measurements, which was how the dipole moments presented in Fig. \ref{fig:dipoles} were measured \cite{Butterfly}. In Fig. \ref{fig:fieldbutterflies} the $r$ and $\theta$-type butterfly molecules pendular molecules are shown using the same plotting procedure as in Fig. \ref{fig:trilobitebasis}. This figure illustrates better than Fig. \ref{fig:trilobitebasis} the reason for the ``butterfly'' moniker since the shorter internuclear distance exaggerates the ``wings''. Another interesting facet of these molecules is that, due to the extended electron cloud beyond the perturber, the dipole moments can actually exceed $R$, the classical limit \cite{Butterfly}.
\begin{figure}[t]
{\normalsize
\begin{center}
\includegraphics[scale =0.12]{butterfly_tutorial_nn-min.jpg}
{\vspace{-0pt}\scalebox{1}[-1]{\includegraphics[scale =0.12]{butterfly_tutorial_mm-min.jpg}}}
\end{center}
}
\caption{Top) A $\Sigma$ butterfly molecule shown using isosurfaces as in Fig. \ref{fig:trilobitebasis}. The small black sphere represents the Rydberg core; the perturber is placed in between the small blue lobes below it. The red arrow denotes the direction of the dipole moment. Bottom) A $\Pi$ butterfly molecule, plotted in the same way. The electron's wave function is now localized opposite the perturber, causing the dipole moment to point in the opposite direction as above. }
\label{fig:fieldbutterflies}
\end{figure}
\subsection{Intermolecular interactions}
The large multipole moments of LRRMs also imply that they can interact at long range. This is very similar to the interactions between isolated Rydberg atoms giving rise to the famous Rydberg blockade, which prevents two Rydberg atoms from both being excited if their internuclear separation is less than the blockade radius. This is the distance at which their mutual interaction exceeds the linewidth of the laser, and hence shifts the state with two excitations out of resonance \cite{DipoleBlockade}. At larger internuclear separations the Rydberg-Rydberg interaction can form attractive potential wells at long range which support the Rydberg macrodimer states \cite{Boisseau,DeiglmayrRydRyd}.
\begin{figure}[t]
{\normalsize
\begin{center}
\includegraphics[width=\columnwidth]{1dipole-min.pdf}
\end{center}
}
\caption{ The interaction potential $V(x,y)$ between two aligned pendular butterfly states with $n=24$. Blue(red) regions are attractive(repulsive). The inner white region (outer white line) demark contours at $|V(x,y)|= 0.5(0.1)$MHz. Lines of force are overlayed in black arrows. The length of a side of the figure is $2\times 10^5$ a.u. On the right the basic geometry is sketched, showing the electric field direction, the two butterfly molecules, and defining the intermolecular distance $R$ and interaction angle $\theta$. This figure is modified from Ref. \cite{EilesPendular}. }
\label{fig:potentials}
\end{figure}
These same effects also apply in LRRMs. Whereas in atoms the interactions are typically isotropic van der Waals potentials \footnote{It is possible in special circumstances to realize anisotropic dipole-dipole interactions with Rydberg atoms, such as in the presence of an external field \cite{Marcassa} or near a F\"{o}rster resonance \cite{SaffmanBlockade}.}, in LRRMs these permanent dipole moments are present automatically due to the molecular structure. Their internal molecular structure also provides vibrational and rotational degrees of freedom that can be accessed to tune the interactions. It is simplest and most experimentally interesting to explore these interactions between the pendular molecules discussed above, since the direction of the applied electric field provides an easily accessible experimental knob.
The potential between two pendular molecules, aligned by an electric field at an angle $\theta$ relative to the axis connecting their center of masses, is given by the two-center multipolar expansion of the Coulomb force \cite{MargenauInt,Stone,vanderAvoird1980},
\begin{align}
\label{eqn:interaction}
\hat V&=\sum_{L_A,L_B}q(L_A)q(L_B)\frac{f_{L_A,L_B}^{n}}{R^{L+1}}\sum_{m_A,m_B,m}\begin{pmatrix}L_A & L_B &L\\m_A & m_B & m\end{pmatrix}\nonumber\\ &\times D_{m_A0}^{L_A}(\theta_A,\phi_A)^*D_{m_B0}^{L_B}(\theta_B,\phi_B)^*D_{m0}^{L}(\theta,0)^*.
\end{align}
Here $q(L)$ is the molecular multipole moment of order $L$ averaged over the vibrational molecular wave function and rescaled by the principle quantum number $n$, $q(L)=Q_0^{L}/n^{2L}$, and $L = L_A+L_B$. Several Wigner D-matrices are used: $D_{m_X0}^{L_X}(\theta_X,\phi_X)^*$ rotates the multipole operator between the lab and molecule frame, and $D_{m0}^L(\theta,0)$ rotates between the lab (field) axis and the intermolecular axis. We also have defined
\begin{align}
f_{L_A,L_B}^{n}&= (-1)^{L_A}n^{2L}\left[\frac{(2L+1)!}{(2L_A)!(2L_B)!}\right]^{1/2}.
\end{align}
Eq. \ref{eqn:interaction} is valid provided the electron clouds do not overlap \cite{LeRoy}. This same potential diagonalized within a basis of Rydberg states to compute the Born-Oppenheimer PECs for Rydberg macrodimers \cite{Boisseau,Farooqi, CsReview,DeiglmayrRydRyd,RaithelRydRyd}. In our case this basis becomes prohibitively large due to the additional molecular configurations in the basis. However, we expect the dominant term to be that of the dipole-dipole interaction which is obtained immediately in first-order perturbation theory. We thus proceed perturbatively, using the pendular ground state calculated previously for the molecular state (see the diagram in Fig. \ref{fig:potentials} for the geometry). We include terms up to $L=4$ in first order perturbation theory and up to $L= 2$ in second order perturbation theory. This gives the potential as a power series in $1/R$ accurate to order $R^{-6}$. The van der Waals term is proportional to $n^{11}/R^6$, familiar from Rydberg scaling laws, and contains induction $(C_{6i})$ and dispersion $(C_{6d})$ contributions \cite{vanderAvoird1980}. In total, the intermolecular potential is
\begin{align}
\label{eqn:interactionresults}
&V(R,\theta)= -\frac{2C_3d^2n^4}{R^3}P_2(x)-\frac{8n^8}{R^5}P_4(x)\left(C_{5a}do-C_{5b}q^2\right) \nonumber\\&-2\frac{4d^4n^{11}}{R^6}\left(C_{6i}^a[P_2(x)]^2+C_{6i}^b\frac{(xy)^2}{4}\right)\\
&-\frac{4d^4n^{11}}{R^6}\left(C_{6d}^a[P_2(x)]^2+\frac{C_{6d}^c}{4}y^4+C_{6d}^b\frac{(xy)^2}{4}\right),\nonumber
\end{align}
where $x=\cos\theta,y=\sin\theta$, and where all coefficients $C$ are positive \cite{EilesPendular}. Using the large $\omega$ harmonic oscillator approximation applicable for the pendular states the coefficients $C_3$, $C_{5a}$, and $C_{5b}$ are
\begin{align}
C_3&= 1 + \frac{3}{2\omega} - \sqrt{\frac{2}{\omega}}\\C_{5a} &= 1 + \frac{14}{\omega}-\frac{7}{\sqrt{2\omega}}\\C_{5b} &=\frac{3}{8\omega}(21 - 6\sqrt{2\omega} + 2\omega).
\end{align}
The coefficients of the second-order terms are complicated and can be found in Ref. \cite{EilesPendular}. This potential surface is displayed for $n=24$ in Fig. \ref{fig:potentials}. It has the characteristic anisotropic shape of the dipole-dipole interaction, and even for the relatively low $n=24$ has 100kHz size shifts at distances exceeding $10\mu$m. At this scale the presence of higher-order potentials is only seen for $\theta$ near the ``magic angle'' where $P_2(x_\text{magic})=0$. In Ref. \cite{EilesPendular} this anisotropy was used to propose an anisotropic blockade effect between LRRMs that could be used to create a crystalline formation of LRRMs with tunable separations.
\subsection{Influence of fields on electronic potential energy curves}
\label{sec:strongfields}
We now consider electric and magnetic fields which can modify the electronic structure of the Rydberg atom, and hence change the PECs. These fields break the cylindrical symmetry present in dimers and add a second preferred direction to the system; the resulting potential energy surfaces depend on both $R$ and $\theta$, the internuclear distance and its angle relative to the field axis, respectively. This turns a rotational degree of freedom into a vibrational one and makes it possible for the fields to align LRRMs. This has been observed in Rb $nD$ states in a magnetic field Ref. \cite{PfauKurz} Ref. \cite{KurzSchmelcherPRA} investigated these potential energy surfaces and alignment in an electric field, while Refs. \cite{Lesanovsky,KurzSchmelcherJPB,Hummel,Hummel2} concentrate on magnetic fields effects. These studies showed that the strong coupling to external fields can prove destructive to LRRMs, as the Zeeman splitting can eliminate the trilobite states \cite{Lesanovsky}. By applying both an electric and a magnetic field new control possibilities open up \cite{KurzSchmelcherJPB}, making it possible to control the hybridization of Rydberg orbitals \cite{GajKrupp}. Ref. \cite{Rosario2016} presents an exploratory investigation into the effect of an electric field on trimer molecules.
In the following we approach this problem by extending the trilobite overlap formalism to include the effect of external fields. Since this approach hinged on the degeneracy of high-$l$ Rydberg states, external fields are clearly problematic since they destroy this atomic degeneracy. The Zeeman Hamiltonian splits equally all of the Rydberg $m_l$ levels, while the Stark Hamiltonian adds a linear shift to each high-$l$ state. Nevertheless, we find that it this method remains surprisingly accurate so long as the field influences remain largely perturbative, i.e. we can use the eigenstates of the Fermi pseudopotential as the zeroth-order states to calculate the energy shifts of the external fields.
We first write down the field operators, focussing on the simple scenario of a single applied field\footnote{The formalism we develop is general to any form of the field operators, but this offers an illustrative case.}. We set the quantization axis parallel to the field, and obtain
\begin{equation}
H_B = \frac{B}{2}\hat L_z + \frac{B^2}{8}r^2\sin^2\theta
\end{equation}
for the magnetic field and
\begin{equation}
H_F = Fr\cos\theta.
\end{equation}
for the electric field. In the generic case with $N$ perturbers the Hamiltonian is $H = H_N({\vec r};\{\vec R_i\}) + H_B + H_F$; the matrix elements of $H_N$ are computed as in Sec. \ref{sec:polyintro} and the new matrix elements are written below. The matrix elements of the linear Zeeman term are straightforward to calculate:
\begin{align}
\bra{\+{\Upsilon_{p r,n}^{\pmb\alpha\pmb 1}}}\frac{B}{2}\hat L_z\ket{\+{\Upsilon_{qr,n'}^{\pmb\beta \pmb1}}}&=\frac{B\delta_{nn'}}{2}\sum_{lm}m\phi_{nlm}^\alpha(\vec R_p)\phi_{nlm}^\beta(\vec R_q)\\
\bra{\+{\Upsilon_{p r,n}^{\pmb\alpha \pmb1}}}\frac{B}{2}\hat L_z\ket{\phi_{n'l'm'}^1}&=\frac{B\delta_{nn'}}{2}\phi^\alpha_{nlm}(\vec R_p)m\\
\bra{\phi_{nlm}^1}\frac{B}{2}\hat L_z\ket{\phi_{n'l'm'}^1}&=\frac{B}{2}\delta_{nn'}\delta_{ll'}\delta_{mm'}m.
\end{align}
The spatial dependence of the Stark operator and the quadratic $B$-field operator complicates the computation of their matrix elements. Since these operators separate in spherical coordinates we can write them in general as $H_\text{field} = a(r)b(\theta)$. Their matrix elements, $a_{nl,n'l'}$ and $b_{lm,l'm'}$ are found in standard references \cite{radmatel,Varsh}, and so
\begin{align}
&\bra{\+{\Upsilon_{pr,n}^{\pmb\alpha 1}}} H_\text{field}\ket{\+{\Upsilon_{qr,n'}^{\pmb\beta 1}} }\\&=\sum_{lm}\sum_{l'm'}\left[\phi_{n'l'm'}^\beta(\vec R_q)\right]^* a_{nl,n'l'}b_{lm,l'm'}\phi_{nlm}^\alpha(\vec R_p)\nonumber\\
&\bra{\+{\Upsilon_{p r,n}^{\pmb\alpha \pmb1}}}H_\text{field}\ket{\phi_{n'l'm'}^1}\\&=\sum_{lm}\left[\phi_{n'l'm'}^\beta(\vec R_q)\right]^* a_{nl,n'l'}b_{lm,l'm'}\nonumber\\
&\bra{\phi_{nlm}^1}H_\text{field}\ket{\phi_{n'l'm'}^1}= a_{nl,n'l'}b_{lm,l'm'}.
\end{align}
The matrices $b_{lm,l'm'}$ are often sparse due to angular momentum selection rules so that these summations can be efficiently evaluated. To improve the accuracy of this approximation and reduce the inaccuracies created by the symmetry-breaking of the external fields, it is advantageous to increase $l_\text{min}$ (we set $l_\text{min}=5$) to help compensate for the degeneracy-breaking of the fields.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.2\columnwidth]{1butterflyfields-min.pdf
\hspace{-50pt}
\caption{Radial cuts in the $n=30$ Rb$_2$ butterfly potential energy surfaces in the presence of a) a magnetic field parallel to the internuclear axis, b) an electric field perpendicular to the internuclear axis, c) an electric field parallel to the internuclear axis, d) an electric field anti-parallel to the internuclear axis. Several different fields strengths are plotted. }
\label{fig:butterflyfields}
\end{center}
\end{figure}
\begin{figure*}[h]
\begin{center}
\includegraphics[width=\textwidth]{1EfieldTrimer1-min.pdf
\caption{Rb$_3$ collinear trimer breathing mode PECs in the presence of an electric field aligned with the internuclear axis. The curved lines were calculated with the approximate trilobite basis approach, while the dots for the 250V/cm calculation were computed using exact diagonalization. The top panel shows the trilobite region. The bottom panel shows the full range of PECs. The vertical blue line marks the shape resonance position.{This figure and B field figure changed to remove arrows}}
\label{fig:efieldtrilobites}
\end{center}
\end{figure*}
\begin{figure*}[h]
\begin{center}
\includegraphics[width=\textwidth]{1BfieldTrimer1-min.pdf
\vspace{-50pt}
\caption{Rb$_3$ collinear trimer breathing mode PECs in the presence of an magnetic field aligned perpendicular to the internuclear axis. The curved lines were calculated with the approximate trilobite basis approach, while the dots for the 100G calculation were computed using exact diagonalization. The top panel shows the trilobite region. The bottom panel shows an enlarged view, but the butterfly region is not shown here as it is barely affected as in the electric field case. }
\label{fig:bfieldtrilobites}
\end{center}
\end{figure*}
In Fig. \ref{fig:butterflyfields} we present, for the first time, radial cuts through the potential energy surfaces for butterfly molecules in the presence of external fields of varying strength. We chose three different geometries. In Fig. \ref{fig:butterflyfields}a we show that the magnetic field has an almost completely negligible effect on the butterfly potentials in this geometry. This is because the diamagnetic term is very small at these field strengths, and so when the magnetic field and internuclear axes are parallel only non-$\Sigma$ states are shifted by the Zeeman effect. Thus the $P$ Rydberg levels and $\Pi$ butterfly curves evince linear shifts in $B$, but the $\Sigma$ butterfly wells are unchanged. The shift in the molecular lines is in accordance with the expected scale of the Zeeman splitting $B/2$, which on the order of 1GHz for the highest $B$ field considered here.
The other three panels show the behavior of the PECs when exposed to an electric field. We see in Fig. \ref{fig:butterflyfields}b that the perpendicular configuration, through coupling between $\Sigma$ and $\Pi$ molecular states, does reveal some field dependence through pronounced avoided crossings despite having to first order a vanishing Stark shift in this geometry. These avoided crossings lead to potential wells in the $\Pi$ PECs due to the oscillations in the $\Sigma$ PECs. In Fig. \ref{fig:butterflyfields} c and d the linear Stark shift $d\cdot F$ is quite obvious, as is the change in dipole direction between $\Sigma$ and $\Pi$ states, which shift in opposite directions. Based on the energy shifts witnessed in these figures it is clear that the electric field can provide angular alignment through its effect on the electronic states.
Our results for the dimer states are in good agreement with previous studies \cite{KurzSchmelcherPRA,KurzSchmelcherJPB}, and so we turn to field effects on triatomic molecules. These help elucidate some of the qualitative effects of different external fields, and have not been studied in detail before \cite{Rosario2016}. We show breathing mode PECs for a collinear Rydberg trimer in the presence of a parallel electric field (Fig. \ref{fig:efieldtrilobites}); this is an interesting study because the trimer, unlike the dimer, has no dipole moment and thus should respond very differently to the external field. We show the same potentials in Fig. \ref{fig:bfieldtrilobites}, but now in the presence of a perpendicular magnetic field. In each case we also compare with the exact diagonalization for a single field strength, $250$V/cm for the electric field and $100$G for the magnetic field. Even for these high field strengths the agreement is excellent, proving the robustness of the trilobite overlap method. Due to the much lower numerical demands of the trilobite overlap method all PECs could be calculated in less time than was required to compute the few points using the atomic basis, and this allowed us to study butterfly states which could not be converged in earlier, more heavily truncated, calculations \cite{KurzSchmelcherJPB,Rosario2016}.
The bottom panel of Fig. \ref{fig:efieldtrilobites} gives the global picture of the PECs. Since it has no dipole moment the trimer is unaffected by the electric field over nearly the whole range of $R$. Curiously, the trilobite states respond strongly to the field, as shown in the enlarged top panel view. Here we see yet again the importance of degeneracy in determining energy shifts. For $R<1100$, the PECs are essentially unchanged from the trimer curves seen in Fig. \ref{fig:poly1}. These PECs correspond to non-degenerate gerade and ungerade states. However, as the coupling between this states drops off at large $R$ these states collapse onto the dimer potential curve as shown in Sec. \ref{sec:polyintro}. The electric field now takes advantage of this degeneracy to mix the gerade/ungerade trimer states into upstream/downstream trilobites pointing against/along the electric field. These states now respond to the electric field essentially as the dimer does \cite{KurzSchmelcherPRA}. The size of this splitting is on par with the expected energy shift of a dipole, $dF$.
The collinear trimer also responds very little to the external magnetic field except near the trilobite wells, and so in Fig. \ref{fig:bfieldtrilobites} we focus on this region. Here, as was also observed in dimers \cite{KurzSchmelcherJPB}, the magnetic field magnifies the coupling between trilobite and butterfly curves, which leads to very large avoided crossings at around $R = 1100$ which strongly destabilize the long-range trilobite wells. This remains dominant here, and in fact one can see that in this orientation the long-range trimer states remain basically degenerate, and thus must not be coupled by the magnetic field.
Our analysis of these molecules is intended to illustrate how the trilobite overlap method, although approximate in the presence of fields, results in very accurate calculations and has significant technical advantages over the standard atomic basis approach. Furthermore, we discussed some of the structural implications of these fields, but certainly there is far more to study. As the fields provide angular confinement this section has important implications for the controllability and structure of these molecules in laboratory environments and in future applications.
\section{Conclusions}
\label{sec:conclusions}
The study of long-range Rydberg molecules has matured into a vibrant field since its origin almost two decades ago. This has occurred due to theoretical efforts to widen the scope of this topic and especially to the tremendous experimental accomplishments of the last decade. This tutorial has focused on fundamental aspects of these molecules. Section \ref{sec:inter} detailed their ``ingredients:'' the highly excited bound states of the Rydberg atom and the very low-energy scattering states of the perturber-electron complex. These concepts are united in the Fermi pseudopotential, which then determines the structure of the Born-Oppenheimer PECs. The fundamental characteristics of these potentials were described in Section \ref{sec:primer}. Experimental effort has demanded a deeper theoretical understanding of these molecules, particularly in the three areas described in Sections \ref{sec:polyintro}-\ref{sec:fieldstudies}. We have therefore built on the foundation of Sec. \ref{sec:primer} to include multiple perturbers, the perturber's fine and hyperfine structure, and external fields.
What is next for trilobites? Given the unpredictable paths that research takes through nature's mysteries, it is dangerous to speculate too much on what the future of this field might resemble. However, there are several paths based on the major themes of this tutorial that seem likely to even the most cautious prognosticator.
\begin{itemize}
\item Although the qualitative success of the theory described here is well documented, it still cannot calculate vibrational bound states with an accuracy matching that of modern experimental spectra. This is especially evident in regimes complicated by the $p$-wave shape resonance. The origin of this difficulty is still unclear. One major source of error is that these calculations are only as accurate as the scattering phase shifts used as input. Small errors in the position of the shape resonance and the value of the zero-energy scattering lengths correlate directly with shifts in the PECs. Several groups have attempted to fit their experimental spectra to zero-energy scattering lengths or even to fully energy-dependent phase shift calculations \cite{Sass,DeSalvo2015,MacLennan}, but the diverging values that they have obtained suggest that other uncertainties cloud this approach. Many of these certainly stem from the Fermi pseudopotential, given its problematic $p$-wave divergence and the subtle pathologies of the three-dimensional delta function potential. These issues must be understood before these fits can be expected to yield consistent results.
\item This desire for higher accuracy should motivate the development and use of more accurate techniques, for example the Green's Function methods described in Sec. \ref{sec:primer}. At present, the three generalizations -- polymers, spin, and fields -- discussed here are still dependent on the Fermi model, so whatever alternative techniques are developed must also be able to incorporate these effects.
\item There is still a need for a theoretically rigorous calculation of rovibrational line strengths. This has been explored previously \cite{Markson,OttGroup}, but these studies still lack conclusive assigments of vibrational states, particularly excited ones. The understanding of the mean field lineshapes in dense environments is an impressive success \cite{Schlag, Whalen2}, but this theory must be generalized to non-$S$ state LRRMs.
\item Due to the many degrees of freedom of the polyatomic molecules presented here, detailed information about the spectra of trimers or tetramers in trilobite-like states is still absent even from a theoretical perspective. Since these polymers will have many metastable configurations their spectra will be very complicated and probably only observable as broadened dimer lines, as was observed in Cs $nD$ trimers \cite{FeyNew}. Since nearly all experimental exploration of polyatomic molecules has thus far been restricted only to isotropic Rydberg $S$ states, there is very little known about polyatomic signatures in anisotropic Rydberg states.
\item Studies in structured environments -- such as a Rydberg atom in an optical lattice or near engineered mesoscopic structures like a tube or a torus -- could take advantage of the enhanced sensitivity to the environment's symmetry seen in polyatomic systems. Non-adiabatic effects and conical intersections stemming from symmetry breaking, as in the Jahn-Teller effect, could be studied. LRRMs can span multiple lattice sites and hence could be used to probe lattice disorder.
\item The spin-dependent approach of Sec. \ref{sec:spinintro} has not yet been generalized so that it can be unified with the trilobite overlap method. This should be fairly straightforward to accomplish, and it will have a significant impact on the practicality of including spin into a study of polyatomic molecules. As the present moment the enlarged Hilbert space including the hyperfine structure of each perturber rapidly increases the basis size needed for diagonalization \cite{FeyNew,Hummel}, and the massive reduction in basis size within the trilobite overlap approach would dramatically reduce the computational effort.
\item Building off of the interactions between aligned butterfly molecules described in Sec. \ref{sec:fieldstudies}, one could explore the consequences of the inherently mixed nature of a system of both Rydberg atoms and LRRMs immersed in a sea of bosonic (or fermionic) atoms. This would be a new regime for studying impurity physics with strong interactions in the ultracold. Likewise, it could even be possible to form more exotic systems consisting of Rydberg atoms bound to LRRMs, an extreme version of the Rydberg-polar perturber molecules predicted in Refs. \cite{Rittenhouse2010,Rittenhouse2011,Mayle2012,Rosario2015,polarperturberpoly} . Mixtures of Rydberg atoms with dipolar molecules like those studied in Refs. \cite{Kuznetsova1,Kuznetsova2,Kuznetsova3} could be realized at exaggerated scales in such a system. Finally, the intermolecular interactions could even lead to the formation of tetramers, bound states between two LRRMs. This would be a fascinating unification of LRRMs and macrodimers.
\item The decay mechanism of these molecules is still not fully understood, and could be influenced by a variety of non-adiabatic processes.
\end{itemize}
We conclude with a rapid summary of recent developments that have arisen as researchers have begun to study LRRMs not for their intrinsic interest but for their utility as microscopic tools. This reveals the versatility of these molecules in many new regimes. For example, LRRMs formed in an optical lattice can non-destructively probe the Mott transition \cite{NiederpruemMott}. They could be used to probe density distributions in ultracold mixtures \cite{EilesHetero}. Recently it was proposed that a Rydberg-molecule dressed state could act as a tunable optical Feshbach resonance, and this has already been successfully implemented \cite{RosarioFR,OttFR}. LRRMs can serve as microscopic ``colliders'' to study atom-ion \cite{NewPfau} and electron-atom \cite{CsReview,MacLennan} scattering. Rydberg atoms can interact with many perturbers in a BEC and can either induce condensate losses or imprint a phase on the condensate\cite{Wuester1,Karpiuk2017,PfauBEC,TiwariWuester}. In a mixed system with two different species of atoms they can probe local density distributions \cite{EilesHetero}{, or they can study the effects of fermionic or bosonic statistics by probing the pair correlation function\cite{WhalenFermions}}. They are the launching point for studies of ultracold ion-atom hybrid systems \cite{IonColdMeinert,IonRydBlockade}. Formation of LRRMs has been linked to oscillations in photon retrieval measurements in Rydberg electromagnetically-induced transparency experiments \cite{RydbergEIT}. Finally, some exotic predictions have been made supporting even more unusual types of LRRMs. One such type is a ``ghost'' molecule formed when ultrafast electric and magnetic field pulses are used to shape a Rydberg wave function so that it is identical to the trilobite wave function, despite the absence of a perturber atom. It then looks as if the Rydberg atom is bound to a point in empty space \cite{EilesGhost}. A second type occurs in a totally different context. Theoretical evidence supports the existence and stability of ``Rydberg nuclear molecules'' composed of Rydberg-like states of halo nuclei \cite{NucRydMols}. This last point illustrates the remarkable convergence of diverse physical ideas, and allows us to conclude this tutorial on a a symmetric note. Just as we began with Fermi's pioneering work, we conclude unexpectedly with a connection to nuclear physics, another field where Fermi -- who was instrumental in creating the first nuclear reactor -- also played an oversized role.
\ack {I am greatly indebted to Chris Greene for his invaluable support and guidance through the zoo of Rydberg physics, and for his keen physical insight and intuition into all of these intriguing topics. I thank F. Robicheaux for encouraging me to take on the project of turning my dissertation into this tutorial and for many valuable ideas and advice. I studied several aspects of Rydberg physics in close collaboration with Jesus Perez-Rios, whose enthusiastic help was greatly appreciated. I also thank Christian Fey and Frederic Hummel for many helpful comments and enlightening discussions about polyatomic molecules. At the MPI-PKS I have benefited from many discussions with A. Eisfeld, J.M. Rost, and P. Giannakeas. This work is supported in part by the National Science Foundation under Grants PHY-1306905 and PHY-1404419, and by the Max Planck Society.}
\bibliographystyle{iopart-num}
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\section{Introduction}
The \texttt{iopart-num} Bib\TeX{} style is intended for use in
preparing manuscripts for Institute of Physics Publishing journals,
including Journal of Physics. It provides numeric citation with
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\section{General instructions}
To use the \texttt{iopart-num} style, include the command
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|
{
"timestamp": "2019-03-01T02:04:19",
"yymm": "1902",
"arxiv_id": "1902.10803",
"language": "en",
"url": "https://arxiv.org/abs/1902.10803"
}
|
\section{Introduction and Motivation}
A \emph{cube category} is a category whose objects are (usually) finite-dimensional cubes, and whose morphisms are mappings of some sort between these cubes.
There are many different cube categories~\cite{bezem_et_al:LIPIcs:2014:4628, Ed-Morehouse-Varieties-of-Cubical-Sets, oriended-cube, Antolini2002, Clive-Newstead-cubical-sets}, and they are used to encode higher categorical structures.
Homotopy type theory~\cite{hottbook} is a variation of Martin-L\"of's intensional type theory.
The characteristic and novel view adapted in Homotopy type theory is that types carry the structure of higher categories, or, to be precise, higher groupoids (i.e.\ all morphisms are invertible).
This view supports Voevodsky's \emph{univalence principle} which should seen as a central concept of homotopy type theory.
The first model of such a type theory, given by Voevodsky~\cite{voevodsky_univalentFoundationsProjectNSF} (see also the presentation by Kapulkin and Lumsdaine~\cite{kapulkin2012simplicial}), uses \emph{simplicial sets}.
However, it is still an open question how simplicial sets can be used to build a \emph{constructive} model of type theory with univalent universes.
Instead, this has been achieved by Bezem, Coquand, and Huber~\cite{bezem_et_al:LIPIcs:2014:4628} using \emph{cubical sets}.
Starting from there, cubes have gathered a lot of attention in the type theory community, leading to various \emph{cubical type theories} which have univalence not as an axiom but as a built-in derivable principle~\cite{cubicaltt, CartesianCubicalTypeTheory, awodey2018cubical, PittsAM:aximct-jv}.
Many different cube categories have been considered in this context.
The important cube category used by Bezem, Coquand, and Huber~\cite{bezem_et_al:LIPIcs:2014:4628} (from now on referred to as the \emph{BCH cube category}) uses finite sets of variable names as objects, and a morphism from a set $I$ to a set $J$ is a function $f : I \to J \cup \{0,1\}$ which is ``injective on the left part'', i.e.\ $f(i_1) = f(i_2) = j$ with $j : J$ implies $i_1 = i_2$.
One goal of this paper is to develop several alternative presentations of this category, mainly using graph morphisms.
We have two main motivations to do this.
The first is that, as we hope, our alternative and intuitive (but equivalent) definitions enable new views on the category and facilitate the discovery of further observations.
The second motivation is that a minor change in the definition will allow us to construct a new cube category, the \emph{twisted cubes} from the title. We will come back to this in a moment.
The idea of \emph{directed type theory}~\cite{AndreasNuytsThesis,riehl2017type,north2018towards} is to generalise other type theories by replacing \emph{(higher) groupoids} by general \emph{(higher) categories}.
In a nutshell, this means that ``equality'' (or whatever takes the place of equality) is not necessarily invertible.
This happens naturally in the universe, since not every function is invertible.
Defining directed type theories is a very active topic of current research.
Our long-term goal is to make the connection with cubical type theories and create some sort of \emph{directed cubical type theory}.
\begin{wrapfigure}[5]{r}[0pt]{0pt}
\begin{tikzpicture}[x=1.5cm,y=1.25cm,baseline=(current bounding box.center)]
\node (L) at (0,0) {$00$};
\node (M) at (0,1) {$01$};
\node (N) at (1,0) {$10$};
\node (P) at (1,1) {$11$};
\draw[->,midarrow] (L) to node {} (M);
\draw[->,midarrow] (L) to node {} (N);
\draw[->,midarrow, dashed] (M) to node {} (P);
\draw[->,midarrow] (N) to node {} (P);
\end{tikzpicture}
\end{wrapfigure}
The standard way to create models (of both higher categories and type theories) using simplicial or cubical index categories is to take presheaves and equip them with certain \emph{Kan-filling conditions}.
These filling conditions encode composition of morphisms as well as associativity and all higher coherence laws that one needs.
A typical such Kan-filling condition says for the $2$-cube says that, given the ``partial square'' of three solid edges on the right, one can always find the dashed edge (together with an actual filler for the square).
\begin{wrapfigure}[5]{l}[0pt]{0pt}
\begin{tikzpicture}[x=1.5cm,y=1.25cm,baseline=(current bounding box.center)]
\node (L) at (0,0) {$x$};
\node (M) at (0,1) {$y$};
\node (N) at (1,0) {$x$};
\node (P) at (1,1) {$x$};
\draw[->,midarrow] (L) to node [left] {$\scriptstyle p$} (M);
\draw[->,midarrow] (L) to node [above] {$\scriptstyle \id_x$} (N);
\draw[->,midarrow, dashed] (M) to node {} (P);
\draw[->,midarrow] (N) to node [right] {$\scriptstyle \id_x$} (P);
\end{tikzpicture}
\end{wrapfigure}
\noindent
When we want to model directed type theory, an issue with the BCH cube category and other cube categories is that invertibility of morphisms is built-in.
Consider the partial square on the left, where two of the three solid edges are identities and the third is an actual non-trivial morphism (or equality) $p$ from $x$ to $y$.
Using the Kan filling operation described above, we get a morphism from $y$ to $x$, which is something that should not necessarily exist when considering a model of directed type theory.
\begin{wrapfigure}[5]{r}[0pt]{0pt}
\begin{tikzpicture}[x=1.5cm,y=1.25cm,baseline=(current bounding box.center)]
\node (L) at (0,0) {$00$};
\node (M) at (0,1) {$01$};
\node (N) at (1,0) {$10$};
\node (P) at (1,1) {$11$};
\draw[<-,midarrow] (M) to node {} (L);
\draw[->,midarrow] (L) to node {} (N);
\draw[->, dashed,midarrow] (M) to node {} (P);
\draw[->,midarrow] (N) to node {} (P);
\end{tikzpicture}
\end{wrapfigure}
To remedy this, we ``twist'' the left-most edge of the $2$-dimensional cube, as shown on the right, to ensure that the construction from before becomes impossible.
Using our graph morphisms that we develop for the BCH cube category, it becomes very easy to define this twisting for cubes of all dimensions.
\Cref{fig:3-cube-tw} and \Cref{fig:4-cube-tw} show the twisted $3$- and $4$-dimensional cube, from two different perspectives.
The construction can be roughly described as follows:
Naturally, the faces of a twisted $n$-cube have to be twisted $(n - 1)$-cubes.
Looking at \Cref{fig:3-cube-tw}, we see that the twisted $3$-cube can be constructed by taking a twisted $2$-cube and ``thickening'' it.
Thickening means that we multiply it with the interval (the $1$-cube), i.e.\ take the ``cylinder object'', in order to create a new dimension; then, we reverse all the edges in the ``domain'' copy of the $2$-cube that we started with.
This construction works for all $n \geqslant 1$, and of course, the twisted $0$-cube is simply a point.
\begin{figure}[ht]
\begin{center}
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\node (N\x\y\z) at (\xcord, \ycord) {$\x\y\z$};
}}}
\draw[->,midarrow] (N000) to node {} (N100);
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\draw[->,midarrow] (N110) to node {} (N111);
\draw[->,midarrow] (N001) to node {} (N101);
\draw[-,color=white,line width=6pt] (N010) to node {} (N110);
\draw[->,midarrow] (N010) to node {} (N110);
\draw[-,color=white,line width=6pt] (N100) to node {} (N110);
\draw[->,midarrow] (N100) to node {} (N110);
\fill [black,opacity=0.1] (N001) rectangle (N111);
\end{tikzpicture}
\hspace*{1cm}
\begin{tikzpicture}[x=.6cm,y=.6cm,baseline=(current bounding box.center)]
\foreach \x in {0,1}{
\foreach \y in {0,1}{
\foreach \z in {0,1}{
\node (N\x\y\z) at (3*\x*\z-1.5*\z+2*\x,3*\y*\z-1.5*\z+2*\y) {$\x\y\z$};
}}}
\draw[->,midarrow] (N000) to node {} (N100);
\draw[->,midarrow] (N011) to node {} (N111);
\draw[<-,midarrow] (N010) to node {} (N000);
\draw[<-,midarrow] (N011) to node {} (N001);
\draw[->,midarrow] (N101) to node {} (N111);
\draw[->,midarrow] (N000) to node {} (N001);
\draw[<-,midarrow] (N011) to node {} (N010);
\draw[<-,midarrow] (N101) to node {} (N100);
\draw[->,midarrow] (N110) to node {} (N111);
\draw[->,midarrow] (N001) to node {} (N101);
\draw[->,midarrow] (N010) to node {} (N110);
\draw[->,midarrow] (N100) to node {} (N110);
\end{tikzpicture}
\caption{The $3$-dimensional twisted cube, perspective view and projected. In both cases, the lid (i.e.\ the last face which can be recovered by filling) is marked. In the right picture, this face is the big square. The lid should be seen as the composite of the other faces.} \label{fig:3-cube-tw}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[x=0.7cm,y=0.7cm]
\foreach \x in {0,1}{
\foreach \y in {0,1}{
\foreach \z in {0,1}{
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\pgfmathsetmacro\zycof{0.4}
\pgfmathsetmacro\wcof{2}
\pgfmathsetmacro\xsign{2*\x - 1}
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\node (M\x\y\z\w) at (4.4 * \x + 2.2 * \z - 1.1 * \w
,4.4 * \y + 1.1 * \z + 2.2 * \w)
{\mytiny{$\x\y\z\w$}};
\node (N\x\y\z\w) at (\wmult * \xsum + 13, \wmult * \ysum + 4)
{\mytiny{$\x\y\z\w$}};
}}}}
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\draw[->,midarrow] (M0\x\y\z) -- (M1\x\y\z);
\draw[->,midmidarrow] (N0\x\y\z) -- (N1\x\y\z);
\pgfmathsetmacro\trg{\x}
\pgfmathtruncatemacro\src{1 - \trg}
\draw[->,midarrow] (M\x\src\y\z) -- (M\x\trg\y\z);
\draw[->,midmidarrow] (N\x\src\y\z) -- (N\x\trg\y\z);
\pgfmathtruncatemacro\trg{0.5*((2*\x - 1) * (2*\y - 1) + 1)}
\pgfmathtruncatemacro\src{1 - \trg}
\draw[->,midarrow] (M\x\y\src\z) -- (M\x\y\trg\z);
\draw[->,midmidarrow] (N\x\y\src\z) -- (N\x\y\trg\z);
\pgfmathtruncatemacro\trg{0.5*((2*\x - 1) * (2*\y - 1) * (2*\z - 1) + 1)}
\pgfmathtruncatemacro\src{1 - \trg}
\draw[->,midarrow] (M\x\y\z\src) -- (M\x\y\z\trg);
\draw[->,midmidarrow] (N\x\y\z\src) -- (N\x\y\z\trg);
}}}
\fill [black,opacity=0.1] (-1.1, 2.2) -- (3.3, 2.2) -- (5.5, 3.3) --
(5.5, 7.7) -- (1.1, 7.7) -- (-1.1, 6.6) -- cycle;
\end{tikzpicture}
\end{center}
\caption{The $4$-dimensional twisted cube, perspective view and projected. The lid is shadowed on the left. It is the biggest cube on the right.} \label{fig:4-cube-tw}
\end{figure}
Twisted cubes do not only remove the discussed source of invertibility,
but they also make the composition of morphisms somewhat more natural.
The filling of a ``standard'' square can be interpreted as saying that the composition of two edges equals the composition of the other two edges.
However, there is no clear actual composite.
In contrast, in the twisted square, the lid should be seen as the single composite of the three other edges.
The right half of \Cref{fig:3-cube-tw} shows the projection of the twisted $3$-cube, and the biggest square (001-101-111-011) is the lid.
As for the square, this lid should be seen as the composite of the other (here five) faces.
Intuitively, one starts with the small inner square, composes it with the top and the bottom squares, and extends it to the left and the right.
\Cref{fig:4-cube-tw} shows the similar situation for the $4$-dimensional twisted cube where one starts with the inner $3$-cube, then extends to the front and the back, the top and the bottom, and the left and the right.
The ``twisting'' pattern also appears in the \emph{twisted arrow category}~\cite{TwistedArrowCategory}, also known as the \emph{category of factorisations}~\cite{CategoryOfFactorizations}.
However, it is unclear how to generalise this idea to more than squares; it has been developed to solve a different problem.
In the main body of the paper, we first introduce the framework of graph morphisms for standard (non-twisted) cubes.
We consider the properties of meet/join and dimension preservation of graph morphisms, and conclude that both of these are suitable refinements to ensure that the category of graph morphisms matches the BCH category. The proof of this is the main result of \Cref{sec:standardcubes}.
We use this development to introduce and examine \emph{twisted cubes} in \Cref{sec:twistedCubes}.
We will see that they have many characteristic properties that standard cubes are lacking.
Some of them, such as a Hamiltonian path through the cube and the fact that vertices are totally ordered, as well as the observation that that injective maps from smaller cubes always correspond to faces, are more familiar from simplicial structures but not from cubical ones.
Another interesting feature, neither familiar from cubical nor from simplicial structures, is that surjective maps are unique (i.e.\ there is only one way to degenerate a twisted cube).
These and other observations allow us to define a further representation of the category of twisted cubes which does not make use of graphs.
\mysubparagraph{Setting}
We use a standard version of Martin-L\"of's dependent type theory as our meta-language.
We assume function extensionality, but we do not require other axioms or features since we mostly work with finite sets, which are extremely well-behaved by default (in particular, it does not matter for us whether UIP/Axiom K is assumed or not).
\mysubparagraph{Summary of Contributions}
Our main contributions are as follows:
\begin{itemize}
\item We give several alternative but equivalent presentations of the BCH cube category.
\item We introduce \emph{twisted cubes}, a variation of the BCH cube category which allows for filling conditions without built-in invertibility.
\item We show several results about twisted cubes.
These include a connection to simplices and the perhaps surprising property that surjective maps (and thus degeneracies) are unique.
\end{itemize}
\section{A Standard Cube Category} \label{sec:standardcubes}
In this section, we discuss various representations of the cube category $\shcube$.
This category was used by Bezem, Coquand, and Huber to present a constructive model of univalence~\cite{bezem_et_al:LIPIcs:2014:4628}.
In \Cref{sec:twistedCubes}, we will see how minimal modifications lead to a category of twisted cubes.
Keeping in mind that we use type theory as the language in which the results are presented (i.e.\ as our meta-theory), we use the following notations:
$\N$ are the natural numbers, including $0$.
For $n : \N$, the set $\finset n$ is the finite set with elements $\{0, 1, \ldots, n-1\}$.
In particular, $\finset 2$ is the set of booleans.
As usual, ${\finset n}^{\finset m}$ is simply the function set $\finset m \to \finset n$.
We denote elements of ${\finset 2}^{\finset n}$ by binary sequences as in $0 \cdot 1 \cdot 1 \cdot 0$.
This means a function $f$ is denoted by $f(0) \cdot f(1) \cdot f(2)\ldots f(n-1)$.
If there is no risk of confusion, we omit the $\cdot$ and simply use juxtaposition as in $0110$.
In several situations, we want to consider a type of functions into a coproduct which is injective ``on the \emph{left} part of the codomain''.
To make this precise, we introduce a notation:
\begin{definition}[$\tolinj$] \label{def:inj-left}
Assume $A$, $B$, and $C$ are given types.
For a function $f : A \to (B+C)$, we say that $f$ is \emph{injective on the left part} if
\begin{equation}
\linj(f) \defeq \Pi(x,y : A, z : B). (f(x) = \inl(z)) \to (f(y) = \inl(z)) \to x = y.
\end{equation}
We write the type of functions which are injective on the left part as
\begin{equation}
(A \tolinj B+C) \defeq \Sigma (f: A \to (B+C)). \linj(f).
\end{equation}
\end{definition}
The following simple but useful (and well-known) result will be necessary.
It could be formulated in higher generality, but a version which is sufficient for us is this:
\begin{lemma} \label{lem:inj-partial-sym}
Given $m,n : \N$, injective partial functions from $\finset m$ to $\finset n$ are in bijection with injective partial functions from $\finset n$ to $\finset m$.
In other words, we have an equivalence
\begin{equation}
\left(\finset m \tolinj \finset n + \finset 1\right) \simeq \left(\finset n \tolinj \finset m + \finset 1\right).
\end{equation}
\end{lemma}
\begin{proof}
The equivalence can be constructed directly. Given an $f : \finset m \tolinj \finset n + \finset 1$, we have to construct a function
$g : \finset n \tolinj \finset m + \finset 1$.
For $i : \finset n$, we can decide whether there is a $k$ such that $f(k) = \inl(i)$.
If so, then this $k$ is unique due to injectivity, and we set $g(i) \defeq \inl(k)$; otherwise, we set $g(i) \defeq \inr(0)$.
Checking that this is an equivalence is routine.
\end{proof}
The presentation of the cube category in question that we start with is the one given by Bezem, Coquand, and Huber~\cite{bezem_et_al:LIPIcs:2014:4628} (which is the same as in Huber's PhD thesis~\cite{simon:thesis}).
Since it is sufficient for our purposes, we use a skeletal variation: our objects are not finite sets but rather natural numbers.
\begin{definition}[category $\shcube$~\cite{bezem_et_al:LIPIcs:2014:4628,simon:thesis}] \label{def:shcube}
The category $\shcube$ has natural numbers as objects and, for $m, n : \N$, a morphism in $\shcube(m,n)$ is a function $f : \finset m \to \finset n + \finset 2$ which is injective on the $\finset n$-part.
In type-theoretic notation:
\begin{align}
& \obj{\shcube} \defeq \N & \mor{\shcube}(m,n) \defeq \finset m \tolinj \finset n + \finset 2
\end{align}
Composition $g \circ f$ is defined to be the set-theoretic composition $(g + \mathsf{id}_2) \circ f$.
\end{definition}
\noindent
What we will need is the opposite of this category, $\shcubeop$.
While the above definition is short and abstract, a description closed to the intuitive idea of cubes is helpful for our later developments.
Let us consider \emph{graphs} $G = (V, E)$ of nodes (vertices) and edges, where $E$ is a subset of $V \times V$.
A standard way to implement this is to give $E$ the type $E : V \times V \to \Prop$, although we write $(s,t) : E$ for $E(s,t)$ and assume that $E$ is given in the ``total space'' formulation.
Furthermore, in our cases $E$ will always be a \emph{decidable} subset.
$E$ being a subset means that our graphs do \emph{not} have multiple parallel edges, i.e.\ for any pair of vertices, there is at most one edge between them, and it is decidable whether there is an edge between two given vertices.
Given a graph, we construct a new graph as follows.
Note that the ``total space'' of the edges of the new graph is $E + E + V$, but in order to make clear which vertices these new edges connect, we use ``set theory style'' notation:
\begin{definition} \label{def:ord-graph-iter}
Given $G = (V, E)$, the \emph{ordinary iteration} of $G$,
denoted as \\ $\ordgraphiter \; (G) \defeq (\ordgraphiter \; (V), \ordgraphiter \; (E))$ is another graph where
\begin{align}
\ordgraphiter \; (V) \; \defeq \; & \bool \times V
\\
\ordgraphiter \; (E) \; \defeq \; & \{ \;(\;(0, \;s), \;(0, t)\;)\;\; | \;\;(s, t) : E \} \nonumber
\\
\cup \; & \{ \;(\;(1, \;s), \;(1, t)\;)\;\; | \;\;(s, t) : E \}
\\
\cup \; & \{ \;(\;(0, \;v), \;(1, v)\;)\;\; | \;\;v : V \}. \nonumber
\end{align}
\end{definition}
This allows us to define the standard cube as a graph:
\begin{definition} \label{def:ord-cube-graph-rec}
Given $n : \nat$, the standard cube $\cube n$ is defined as follows:
\begin{alignat}{2}
\cube 0 \; \defeq\; (\finset 1, \; \{(0, 0)\}) \qquad\qquad\qquad\qquad\qquad
\cube {n + 1} \;\defeq\; \ordgraphiter \; (\cube n)
\end{alignat}
\end{definition}
Another way of defining $\cube n$, without recursion, is the following.
Here, we give the ``total space'' of edges $\mathsf{edges}(\cube n)$ together with functions $\mathsf{src}, \mathsf{trg} : \mathsf{edges}(\cube n) \to \mathsf{nodes}(\cube n)$:
\begin{definition} \label{def:ord-cube-graph-nonrec}
In the following, our convention is that $\finset {-1}$ is empty (i.e.\ the same as $\finset 0$):
\begin{align}
& \mathsf{nodes}(\cube n) && \defeq && \bool^{\finset n} \label{eq:def-cube-nonrec-nodes} \\
& \mathsf{edges}(\cube n) && \defeq && \bool^{\finset n}
+ \left(\finset n \times \bool^{\finset{n-1}} \right) \label{eq:def-cube-nonrec-edges} \\
& \mathsf{src}(\inl(v)) \quad \defeq \quad \mathsf{trg}(\inl(v)) && \defeq && v \\
& \mathsf{src}(\inr(i, x_0x_1\ldots x_{n-2})) && \defeq && x_0x_1\ldots x_{i-1} 0 x_i \ldots x_{n-2} \\
& \mathsf{trg}(\inr(i, x_0x_1\ldots x_{n-2})) && \defeq && x_0x_1\ldots x_{i-1} 1 x_i \ldots x_{n-2}
\end{align}
\end{definition}
In \Cref{def:ord-cube-graph-nonrec}, the left part ($\bool^{\finset n}$) are the ``identities'' (one for each node),
while the right part ($\finset n \times \bool^{\finset{n-1}}$) represents the non-trivial edges.
\Cref{fig:ord-cube-graph-123} shows drawings for $\cube 0$ to $\cube 3$.
\begin{lemma}
\Cref{def:ord-cube-graph-rec} and \Cref{def:ord-cube-graph-nonrec}
define isomorphic graph structures. \qed
\end{lemma}
This observation allows us to use whichever is more convenient in any given situation.
\begin{figure}[H] \centering
\begin{tikzpicture}[x=1.2cm,y=1.2cm,baseline=(current bounding box.center)]
\node (N) at (0,0.5) {$\epsilon$};
\node (N0) at (2,0.5) {$0$};
\node (N1) at (3,0.5) {$1$};
\draw[->,midarrow] (N0) to node[above,sloped]{\mytiny{$\langle 0, \epsilon \rangle$}} (N1);
\foreach \x in {0,1}{
\foreach \y in {0,1}{
\node (N\x\y) at (\x + 5,\y) {$\x\y$};
}}
\draw[->,midarrow] (N00) to node[above,sloped]{\mytiny{$\langle 0, 0 \rangle$}} (N10);
\draw[->,midarrow] (N01) to node[above,sloped]{\mytiny{$\langle 0, 1 \rangle$}} (N11);
\draw[->,midarrow] (N00) to node[above,sloped]{\mytiny{$\langle 1, 0 \rangle$}} (N01);
\draw[->,midarrow] (N10) to node[above,sloped]{\mytiny{$\langle 1, 1 \rangle$}} (N11);
\foreach \x in {0,1}{
\foreach \y in {0,1}{
\foreach \z in {0,1}{
\node (N\x\y\z) at (1.5*\x*\z-0.75*\z+\x + 9,1.5*\y*\z-0.75*\z+\y) {$\x\y\z$};
}}}
\foreach \x in {0,1}{
\foreach \y in {0,1}{
\draw[->,midarrow] (N0\x\y) to node[above,sloped]{\mytiny{$\langle 0, \x\y \rangle$}} (N1\x\y);
\draw[->,midarrow] (N\x0\y) to node[above,sloped]{\mytiny{$\langle 1, \x\y \rangle$}} (N\x1\y);
\draw[->,midarrow] (N\x\y0) to node[above,sloped]{\mytiny{$\langle 2, \x\y \rangle$}} (N\x\y1);
}}
\end{tikzpicture}
\caption{An illustration of $\cube n$ for $n \leqslant 3$. The labels on the vertices and edges are in accordance with \eqref{eq:def-cube-nonrec-nodes} and \eqref{eq:def-cube-nonrec-edges}. The identity loops are hidden to tidy up the diagrams. This allows us to to unambiguously hide the constructor $\inr{}$ as well. } \label{fig:ord-cube-graph-123}
\end{figure}
A \emph{graph morphism} from $G = (V,E)$ to $G' = (V',E')$ is, as usual, a function between the node types which preserves the edges:
\begin{equation}
\graphhom{(V,E)}{(V',E')} \defeq \Sigma (f : V \to V'). \Pi(v_0, v_1 : V). E(v_0,v_1) \to E'(f(v_0),f(v_1))
\end{equation}
We can now consider the following category:
\begin{definition}[category $\graphcube$] \label{def:ord-graphs-def}
The category $\graphcube$ has natural numbers as objects. \\ A morphism between $m$ and $n$ is a graph morphism from $\cube m$ to $\cube n$, as in:
\begin{align}
& \obj{\graphcube} \defeq \N
& \mor{\graphcube}(m,n) \defeq \graphhom{\cube m}{\cube n}
\end{align}
Composition is composition of graph morphisms.
\end{definition}
The category $\graphcube$ has more morphisms than $\shcubeop$.
One example would be the morphism in $\graphhom{\cube 2}{\cube 1}$ which maps the three nodes $00$, $01$, $10$ all to $0$ and $11$ to $1$.
Another example is the morphism which maps $00$ to $0$, and $01$, $10$, $11$ all to $1$.
Both of these morphisms do not have analogues in $\shcubeop$.
In other words, $\graphcube$ has \emph{connections}.
We do not want these since the category $\shcubeop$ that we are trying to find alternative definitions for does not have them.
In order to remedy this, we refine the definition of the morphisms in $\graphcube$.
Let us formulate the following auxiliary definitions.
\begin{definition}[free preorder of a graph] \label{def:freepreorder}
For a given graph $G = (V,E)$, we write $G^* = (V, E^*)$ for the free preorder generated by it.
$G^*$ has $V$ as objects and, for $v,u : V$, we have $v \leqslant u$ if there is a chain of edges starting in $v$ and ending in $u$.
When taking about nodes in $G$, we borrow the notions of \emph{meet} (product) and \emph{join} (coproduct) from preorders.
If they exist in $G^*$, we write them as $v \meet u$ and $v \join u$.
\end{definition}
It is easy to see that, in the case of $\cube n$, all meets and joins exist and can be calculated directly: From the programming perspective, they correspond to the bitwise operators $'\texttt{\&}'$ and $'\texttt{|}'$.
Thus, when talking about $\cube n$, we can view $\meet$ and $\join$ as actual functions calculating the binary meet and join:
\begin{equation}
\meet, \join : V \times V \to V
\end{equation}
Given a graph morphism $g : \graphhom{\cube m}{\cube n}$, it is easy to define what it means that it preserves binary meets resp.\ joins:
\begin{align}
& \presmeet(g) \defeq \Pi (u,v : \bool^{\finset m}). g(u \meet v) = g(u) \meet g(v) \\
& \presjoin(g) \defeq \Pi (u,v : \bool^{\finset m}). g(u \join v) = g(u) \join g(v)
\end{align}
Note that preserving meets and joins is a property (a ``mere proposition'') of morphisms.
For general morphisms between graphs which might not have all meets or joins, the definition is more subtle but still straightforward;
one can always define the property of \emph{being a meet (join)}
and then say that any vertex which has this property is mapped to one which also has it.
We omit the precise type-theoretic formulation.
The two mentioned examples of morphisms which are ``too much'' in $\graphcube$ do not preserve binary meets resp.\ joins.
\begin{definition}[category $\graphmeet$] \label{def:graphmeet}
The category $\graphmeet$ has $\N$ as objects and, as morphisms, graph morphisms between standard cubes which preserve meets and joins ($\mathsf{c}$ for \emph{continuous}):
\begin{align}
& \obj{\graphmeet} \defeq \N \\
& \mor{\graphmeet}(m,n) \defeq \Sigma (g : \graphhom{\cube m}{\cube n}).
\presmeet(g) \times \presjoin(g)
\end{align}
\end{definition}
This gives us a category which is indeed equivalent (in fact isomorphic) to $\shcubeop$:
\begin{theorem} \label{thm:shcube-meetjoinpres}
The categories $\shcubeop$ and $\graphmeet$ are isomorphic.
The isomorphism on the object part is the identity, i.e.\ the equivalence is given by a family $e$ as in:
\begin{equation}
e : \Pi(m,n : \N). \mor{\shcubeop}(m,n) \simeq \mor{\graphmeet}(m,n).
\end{equation}
\end{theorem}
Before giving a proof, we formulate the following:
\begin{lemma} \label{lem:graphmeet-reduce}
Consider the full subgraph of $\cube n$ which has exactly $(n+1)$ vertices, namely the ``origin'' $00\ldots 0$ and the ``base vectors'' which have exactly one $1$.
We call this subgraph $B_n$, where the $B$ stands for ``base'', and it comes with the inclusion $i : B_n \hookrightarrow \cube n$.
For any $m$, ``forgetting'' the property of preserving the joins and composing with $i$ as in
\begin{equation}
\lambda g. i \circ (\mathsf{proj}_1(g)) : \; \left(\Sigma (g : \graphhom{\cube n}{\cube m}. \presjoin(g)\right) \; \to \; \graphhom{B_n}{\cube m}
\end{equation}
is an equivalence.
Moreover, $g$ preserves meets if and only if $i \circ (\mathsf{proj}_1(g))$ does.
\end{lemma}
\noindent
\begin{minipage}{.85\textwidth}
\begin{proof}
The only binary joins that $B_n$ has are trivial, so every morphism $\graphhom{B_n}{\cube m}$ is join-preserving. Thus, the first claim of the lemma is that every such morphism can be extended in a unique way as shown in the diagram to the right.
Every node of $\cube n$ which is not in $B_n$, i.e.\ every node which is not the origin or a base vector, can be written as a join of base vectors.
Since we need to preserve joins, it is therefore determined where the node has to be sent to.
The map defined in this way preserves all binary joins, and it preserves binary meets if and only if the input does.
\end{proof}
\end{minipage}
~
\begin{minipage}{.12\textwidth}
\begin{tikzpicture}[x=1.5cm,y=-1.5cm,baseline=(current bounding box.center)]
\node (Gn) at (0,0) {$B_n$};
\node (Cm) at (1,0) {$\cube m$};
\node (Cn) at (0,1) {$\cube n$};
\draw[->] (Gn) to node {} (Cm);
\draw[right hook->] (Gn) to node {} (Cn);
\draw[->,dashed] (Cn) to node {} (Cm);
\end{tikzpicture}
\end{minipage}
\begin{proof}[Proof of \Cref{thm:shcube-meetjoinpres}]
We first give the overview of the argument as a chain of equivalences, then we justify each step.
\begin{alignat*}{3}
&&& \mor{\graphmeet}(m,n) \\
\quad & \equiv & \quad & \Sigma (g : \graphhom{\cube m}{\cube n}). \presmeet(g) \times \presjoin(g) \\
\step 1 \quad & \simeq & \quad & \Sigma (g : \graphhom{B_m}{\cube n}). \presmeet(g) \\
\step 2 \quad & \simeq & \quad & \Sigma(z : {\finset 2}^{\finset n}, d : \finset m \tolinj \finset n + \finset 1). \Pi(i:\finset m, j : \finset n). (d(i) = \inl(j)) \to (z(j) = 0) \\
\step 3 \quad & \simeq & \quad & \Sigma(z : {\finset 2}^{\finset n}, e : \finset n \tolinj \finset m + \finset 1). \Pi(i:\finset m, j : \finset n). (e(j) = \inl(i)) \to (z(j) = 0) \\
\step 4 \quad & \simeq & \quad & \Sigma(z : {\finset 2}^{\finset n}, e : \finset n \to (\finset m + \finset 1)). \linj(f) \times \Pi(i:\finset m, j : \finset n). (e(j) = \inl(i)) \to (z(j) = 0) \\
\step 5 \quad & \simeq & \quad & \Sigma \big(\alpha : \Pi(j : \finset n). \Sigma(e : \finset m + \finset 1, z : \finset 2). \Pi(i : \finset m).(e = \inl(i)) \to z = 0\big). \linj(\mathsf{proj}_1 \circ \alpha) \\
\step 6 \quad & \simeq & \quad & \Sigma \big(\alpha : \Pi(j : \finset n). \finset m + \finset 2\big). \linj(\alpha) \\
\quad & \equiv & \quad & \mor{\shcubeop}(m,n)
\end{alignat*}
\noindent
Step 1 holds by \Cref{lem:graphmeet-reduce}.
Let us look at Step 2.
Giving a graph homomorphism between $B_m$ and $\cube n$ corresponds to choosing where the origin is mapped to, and choosing where each (non-trivial) edge of $B_m$ is mapped to.
For the origin, we use the component $z : {\finset 2}^{\finset m}$.
There are $m$ non-trivial edges in $B_m$, and $z$ is an endpoint of $n$ non-trivial edges and one trivial edge in $\cube n$.
This gives us up to $\finset m \to \finset n + \finset 1$ possible functions, but since we only consider meet-preserving morphisms, every function needs to be injective on the left part, leading to $d : \finset m \tolinj \finset n + \finset 1$.
Moreover, if $d(i) = \inl(j)$ for some $i,j$, then the image of the origin must be the \emph{starting point} of the edge in dimension $j$, i.e.\ $z(j) = 0$.
Step 3 is an application of \Cref{lem:inj-partial-sym} (essentially, it swaps the roles of $m$ and $n$).
Step 4 only unfolds the defition $\tolinj$.
In Step 5, the usual distributivity between $\Sigma$ and $\Pi$ (under the propositions-as-types view referred to as the ``axiom of choice'') is used: $z$, $e$, and the unnamed last component can all be seen as (dependent) functions with domain $\finset n$.
The dependent function $\alpha$ combines them into a single dependent function with domain $\finset n$ and a codomain that consists of multiple components which, again, are called $e$, $z$, and unnamed.
Only the component expressing the ``injectivity on the left part''-property cannot be seen as a function in $\finset n$.
In Step 6, we massage the codomain of $\alpha$:
We have $e : \finset m + \finset 1$ and also $z : \finset 2$, but the condition says that $z$ is determined unless $e = \inr(0)$; thus, the type is equivalent to $\finset m + \finset 2$.
We omit the calculation which shows that the constructed equivalence preserves composition of morphisms in the categories.
\end{proof}
In \Cref{sec:twistedCubes}, we will switch from standard cubes to twisted cubes.
The directions of some edges will be reversed.
It is therefore an advantage to formulate a condition similar to the one about meets and joins without referring to the direction of edges.
This is indeed possible:
\begin{definition}[dimension preserving morphisms; category $\graphdim$] \label{def:ord-dim-pres}
Given the standard cube $\cube n$, where we use the non-recursive definition as in \Cref{def:ord-cube-graph-nonrec}, the \emph{dimension} of an edge is defined as follows:
\begin{align}
& \dimens : \mathsf{edges}(\cube n) \to \finset n + \finset 1
&& \dimens (\inl(v)) \qquad\quad\;\;\, \defeq \inr(0) \\
&&& \dimens (\inr(i, x_0\ldots x_{n-2}) \defeq \inl(i)
\end{align}
We say that a morphism $f : \graphhom{\cube m}{\cube n}$ is \emph{dimension-preserving} if $f$ maps edges of the same dimension to edges of the same dimension,
\begin{equation} \label{eq:dim-preserving}
\dimpres(f) \defeq \Pi(e_1,e_2 : \mathsf{edges}(\cube n)). (\dimens(e_1) = \dimens(e_2)) \to (\dimens(f(e_1)) = \dimens(f(e_2))).
\end{equation}
The category $\graphdim$ makes use of these concepts:
\begin{align}
& \mathsf{obj}(\graphdim) \defeq \N
& \mor{\graphdim}(m,n) \defeq \Sigma (g : \graphhom{\cube m}{\cube n}). \dimpres(g)
\end{align}
\end{definition}
As $\presmeet(g)$ and $\presjoin(g)$, preserving the dimension as in \eqref{eq:dim-preserving} is a proposition in the sense of homotopy type theory (has at most one proof).
\begin{remark} \label{rem:dim-inj-derivable}
For a graph morphism $f$ as in the definition above, the following condition says that $f$ is ``injective on dimensions'' (on the non-trivial part):
\begin{align*}
& \diminj(f) \defeq \Pi(e_1,e_2 : \mathsf{edges}(\cube m), j : \finset n) .
\big( \dimens(f(e_1)) = \inl(j) \times \dimens(f(e_2)) = \inl(j) \big) \\
& \hspace*{6cm} \to (\dimens(e_1) = \dimens(e_2)).
\end{align*}
However, note that this follows directly from $\dimpres(f)$: Assume $e_1, e_2$ are edges such that $\dimens(f(e_1))$ and $\dimens(f(e_2))$ are equal and non-trivial.
If $e_1$ and $e_2$ are not ``parallel'' (i.e.\ not in the same dimension), then we can find $e_1'$ in the same dimension as $e_1$ such that $e_1'$ and $e_2$ are adjacent (i.e.\ the endpoint of one is the starting point of the other). It is clear that $f(e_1')$ and $f(e_2)$ cannot go into the same non-trivial direction, since we can only go one step into a given direction before going back.
\end{remark}
The connection to meet- and join-preserving is given by the following result:
\begin{lemma} \label{lem:join-meet-dim}
A morphisms $f : \graphhom{\cube m}{\cube n}$ is join-and-meet-preserving exactly if it is dimension-preserving.
\end{lemma}
\begin{proof}
This follows easily by going via morphisms $\graphhom{B_m}{C_n}$ as in \Cref{lem:graphmeet-reduce}.
The graph $B_m$ has exactly one edge for every non-trivial dimension, and the proof is analogous to the one of \Cref{lem:graphmeet-reduce}.
\end{proof}
This allows us to conclude:
\begin{corollary}[Section summary]
The categories $\shcubeop$, $\graphmeet$, and $\graphdim$ are isomorphic. \qed
\end{corollary}
\section{A Category of Twisted Cubes} \label{sec:twistedCubes}
As discussed in the introduction, we build on our framework of graph morphisms to define a category of \emph{twisted cubes}.
A small change of \Cref{def:ord-graph-iter} gives us these twisted cubes:
\begin{definition} \label{def:tw-graph-iter}
Given a graph $G = (V,E)$, the \emph{twisted iteration} of $G$, \\ denoted as $\twgraphiter \; (G) \defeq (\twgraphiter \; (V), \; \twgraphiter \; (E))$ is the graph defined by
\begin{align}
\twgraphiter \; (V) \; \defeq \; & \bool \times V \label{eq:nodes-2n}
\\
\twgraphiter \; (E) \; \defeq \; & \{ \;(\;(0, \;t), \;(0, s)\;)\;\; | \;\;(s, t) : E \}
\\
\cup \; & \{ \;(\;(1, \;s), \;(1, t)\;)\;\; | \;\;(s, t) : E \}
\\
\cup \; & \{ \;(\;(0, \;v), \;(1, v)\;)\;\; | \;\;v : V \}. \label{eq:twisted-edges-from-nodes}
\end{align}
\end{definition}
We then define:
\begin{definition} \label{def:tw-cube-graph-rec}
Given $n : \nat$, the twisted cube $\twcube n$ is defined as follows:
\begin{alignat}{2}
\twcube 0 \; \defeq\; (\finset 1, \; \{(0, 0)\}) \qquad\qquad\qquad\qquad\qquad
\twcube {n + 1} \;\defeq\; \twgraphiter \; (\twcube n)
\end{alignat}
\end{definition}
Alternatively, we can tweak \Cref{def:ord-cube-graph-rec} to get a non-recursive definition. As before, the convention is that $\finset {-1}$ is empty.
\begin{definition} \label{def:tw-cube-graph-nonrec}
The non-recursive definition of $\twcube n$ is as follows:
\begin{align}
& \mathsf{nodes}(\twcube n) && \defeq && \bool^{\finset n} \\
& \mathsf{edges}(\twcube n) && \defeq && \bool^{\finset n}
+ \left(\finset n \times \bool^{\finset{n-1}} \right) \label{eq:def-tw-cube-nonrec-edges} \\
& \mathsf{src}(\inl(v)) \quad \defeq \quad \mathsf{trg}(\inl(v)) && \defeq && v \\
& \mathsf{src}(\inr(i, x_0x_1\ldots x_{n-2})) && \defeq && x_0x_1\ldots x_{i-1} \cdot b \cdot x_i \ldots x_{n-2} \\
& \mathsf{trg}(\inr(i, x_0x_1\ldots x_{n-2})) && \defeq && x_0x_1\ldots x_{i-1} \cdot (1 - b) \cdot x_i \ldots x_{n-2}
\end{align}
where $b = 1$ if the total number of zeros in $x_0x_1\ldots x_{i-1}$ is odd, and $b = 0$ otherwise.
\end{definition}
This means that an edge is reversed (compared to the standard cubes discussed before) exactly if the number of zeros in dimensions that come \emph{before} the edge is odd (note that the condition talks about $x_{i-1}$, not $x_{n-2}$).
The twisted cubes of dimension up to $3$ are illustrated in \Cref{fig:twisted-cube-graph-0123}; see also \Cref{fig:3-cube-tw,fig:4-cube-tw} in the introduction.
\begin{lemma}
\Cref{def:tw-cube-graph-rec} and \Cref{def:tw-cube-graph-nonrec} define isomorphic graph structures. \qed
\end{lemma}
\begin{figure}[H]
\centering
\begin{tikzpicture}[x=1.2cm,y=1.2cm,baseline=(current bounding box.center)]
\node (N) at (0,0.5) {$\epsilon$};
\node (N0) at (2,0.5) {$0$};
\node (N1) at (3,0.5) {$1$};
\draw[->,midarrow] (N0) to node[above,sloped]{\mytiny{$\langle 0, \epsilon \rangle$}} (N1);
\foreach \x in {0,1}{
\foreach \y in {0,1}{
\node (N\x\y) at (\x + 5,\y) {$\x\y$};
}}
\draw[->,midarrow] (N00) to node[above,sloped]{\mytiny{$\langle 0, 0 \rangle$}} (N10);
\draw[->,midarrow] (N01) to node[above,sloped]{\mytiny{$\langle 0, 1 \rangle$}} (N11);
\draw[->,midarrow] (N01) to node[above,sloped]{\mytiny{$\langle 1, 0 \rangle$}} (N00);
\draw[->,midarrow] (N10) to node[above,sloped]{\mytiny{$\langle 1, 1 \rangle$}} (N11);
\foreach \x in {0,1}{
\foreach \y in {0,1}{
\foreach \z in {0,1}{
\node (N\x\y\z) at (1.5*\x*\z-0.75*\z+\x + 9,1.5*\y*\z-0.75*\z+\y) {$\x\y\z$};
}}}
\foreach \x in {0,1}{
\foreach \y in {0,1}{
\draw[->,midarrow] (N0\x\y) to node[above,sloped]{\mytiny{$\langle 0, \x\y \rangle$}} (N1\x\y);
\pgfmathsetmacro\trg{\x}
\pgfmathtruncatemacro\src{1 - \trg}
\draw[->,midarrow] (N\x\src\y) to node[above,sloped]{\mytiny{$\langle 1, \x\y \rangle$}} (N\x\trg\y);
\pgfmathtruncatemacro\trg{0.5*((2*\x - 1) * (2*\y - 1) + 1)}
\pgfmathtruncatemacro\src{1 - \trg}
\draw[->,midarrow] (N\x\y\src) to node[above,sloped]{\mytiny{$\langle 2, \x\y \rangle$}} (N\x\y\trg);
}}
\end{tikzpicture}
\caption{An illustration of $\twcube n$ where $n \leqslant 3$.} \label{fig:twisted-cube-graph-0123}
\end{figure}
$\twcube n$ has an interesting property that the standard cube
$\cube n$ does not have: Its free preorder $\twcube n^*$ is isomorphic
to the total order on $2^n$ elements. This observation was originally
suggested by Paolo Capriotti and Jakob von Raumer in a discussion with
the first author of this paper. Note that this observation should not
be misunderstood to mean that $\twcube n$ itself is uninteresting.
Its edges give it a unique structure, as visualised in
\Cref{fig:rainbows}.
The idea behind this result is that $\twgraphiter$ preserves the
property of having a preorder that is total. To elaborate on this, if
$G^*$ is a total order, then $(\twgraphiter \; G)^*$ consists of two
copies of $G^*$, where the first copy is turned around.
One of the edges added in \eqref{eq:twisted-edges-from-nodes} links the largest node in the first copy to the smallest node in second copy, thus every element of the second copy is larger than all the elements of the first.
\begin{theorem} \label{thm:total-order}
For all $n$, the preorder $\twcube n^*$ is isomorphic to the total order $({\finset 2}^{\finset n}, <)$.
\end{theorem}
Note that \Cref{thm:total-order} is a property which one usually expects for simplicial structures, but not for cubical ones.
Another related observation is that we can find a path from the smallest vertex to the largest vertex of $\twcube n$ which respects the direction of the edges, and which visits each vertex exactly once. Recall that such a path is called a \emph{Hamiltonian path}.
We record this:
\begin{theorem} \label{thm:hamiltonian}
For all $n$, there is exactly one Hamiltonian path through $\twcube {n+1}$.
This path contains exactly one edge in the first dimension (i.e.\ the one which is added when going from $\twcube n$ to $\twcube {n+1}$).
Moreover, this single edge in the new dimension connects the Hamiltonian paths through the two copies of $\twcube n$ of which $\twcube {n+1}$ consists as by definition, cf.~\eqref{eq:nodes-2n}.
\end{theorem}
\begin{proof} [Proof of \Cref{thm:total-order} and \Cref{thm:hamiltonian}]
As before, we denote elements of ${\finset 2}^{\finset n}$ as sequences such as $00101$ (``big endian'') or, for clarity, by $0 \cdot 0 \cdot 1 \cdot 0 \cdot 1$.
We use the endofunction $\rev$ on $\finset 2^{\finset n}$, which simply replaces each $0$ in a sequence by a $1$ and vice versa; i.e.\ it sends the number $i$ to $2^n - 1 - i$ (note that $\rev$ does not reverse the sequence, but the ordering on ${\finset 2}^{\finset n}$).
Let us define endofunctions $f_n$ and $g_n$ on ${\finset 2}^{\finset n}$, by induction on $n$. Note that, at this point, we do not talk about graph morphisms but only about functions between sets.
The base cases of the induction are uniquely determined. We define $f$ and $g$ by
\begin{align}
& f_{n+1} (0 \cdot \vec x) \defeq 0 \cdot f_n(\rev(\vec x)) & & g_{n+1} (0 \cdot \vec x) \defeq 0 \cdot \rev(g_n(\vec x)) \label{eq:def-f-0} \\
& f_{n+1} (1 \cdot \vec x) \defeq 1 \cdot f_n(\vec x) & & g_{n+1} (1 \cdot \vec x) \defeq 1 \cdot g_n(\vec x). \label{eq:def-f-1}
\end{align}
It is easy to calculate that, by induction, $f$ and $g$ are inverse to each other.
We want to show that they extend to morphisms between preorders,
\begin{align}
& \hat f_n : ({\finset 2}^{\finset n}, <) \to \twcube n^*
& \hat g_n : \twcube n^* \to ({\finset 2}^{\finset n}, <).
\end{align}
To construct $\hat f_n$ \emph{and} the Hamiltonian path through the cube, it suffices to show:
for $x,y : {\finset 2}^{\finset n}$ with $x+1=y$, we have an edge $f_n(x) \to f_n(y).$
We do induction on $n$. For $n = 0$, this is vacuously true (such $x,y$ do not exist).
For $n = n' + 1$, there are multiple cases:
\begin{itemize}
\item case $x = 0 \cdot x'$ and $y = 0 \cdot y'$: Then, the assumption gives us $x'+1=y'$ and we have to find an edge $0 \cdot f_n(\rev(x')) \to 0 \cdot f_n(\rev(y'))$.
Looking at \Cref{def:tw-graph-iter}, we can get this if we have
$f_n(\rev(y')) \to f_n(\rev(x'))$.
This holds by induction, since $\rev$ reverses the order which gives us $\rev(y')+1 = \rev(x')$.
\item case $x = 1 \cdot x'$ and $y = 1 \cdot y'$: Similar to the previous case, but nothing gets reversed.
\item $x = 0 \cdot x'$ and $y = 1 \cdot y'$: In this case, we have $x = 0111\ldots$ and $y = 1000\ldots$.
We need to find an edge $0 \cdot f(\rev(111\ldots)) \to 1 \cdot f(000\ldots)$, which simplifies to $0 \cdot f(000\ldots) \to 1 \cdot f(000\ldots)$.
This edge is directly given in \eqref{eq:twisted-edges-from-nodes}.
\item The last case contradicts the assumption $x+1 = y$.
\end{itemize}
This shows that there is a Hamiltonian path, and it is given by $\hat f_n$.
The definition of $f$ as in (\ref{eq:def-f-0},\ref{eq:def-f-1}) also shows that $f_{n+1}$ consists of two copies of $f_n$, implying the last claim of \Cref{thm:hamiltonian}.
In order to prove \Cref{thm:total-order}, we need to construct $\hat g_n$.
It is enough to show that, for an edge from $u$ to $v$ in $\twcube n$, we have $g(u) \leqslant g(v)$.
This follows by straightforward induction, going through the edges in \Cref{def:tw-graph-iter}.
But \Cref{thm:total-order} implies that there is at most one Hamiltonian path.
\end{proof}
\begin{figure}
\begin{center}
\begin{tikzpicture}[x=0.95cm,y=1.0cm]
\def85{85}
\node (N0) at (0 + 0, 0) {$\epsilon$};
\node () at (0 + 0, 0.5) {$\epsilon$};
\node (N0) at (1 + 0, 0) {$0$};
\node (N1) at (1 + 1, 0) {$1$};
\node () at (1 + 0, 1) {$0$};
\node () at (1 + 1, 1) {$1$};
\node (N01) at (3 + 0, 0) {$01$};
\node (N00) at (3 + 1, 0) {$00$};
\node (N10) at (3 + 2, 0) {$10$};
\node (N11) at (3 + 3, 0) {$11$};
\foreach \x in {0,1}{
\foreach \y in {0,1}{
\node () at (3 + 2*\x + \y, 2) {$\x\y$};
}}
\node (N011) at (7 + 0, 0) {$011$};
\node (N010) at (7 + 1, 0) {$010$};
\node (N000) at (7 + 2, 0) {$000$};
\node (N001) at (7 + 3, 0) {$001$};
\node (N101) at (7 + 4, 0) {$101$};
\node (N100) at (7 + 5, 0) {$100$};
\node (N110) at (7 + 6, 0) {$110$};
\node (N111) at (7 + 7, 0) {$111$};
\foreach \x in {0,1}{
\foreach \y in {0,1}{
\foreach \z in {0,1}{
\node () at (7 + 4*\x + 2*\y + \z, 3) {$\x\y\z$};
}}}
\draw[->,midmidarrow] (N0) to [bend left=85]
node[below]{\mytiny{$\langle 0, \epsilon \rangle$}} (N1);
\draw[->,midmidarrow] (N00) to [bend left=85]
node[above]{\mytiny{$\langle 0, 0 \rangle$}} (N10);
\draw[->,midmidarrow] (N01) to [bend left=85]
node[above]{\mytiny{$\langle 0, 1 \rangle$}} (N11);
\draw[->,midmidarrow] (N01) to [bend left=85]
node[below]{\mytiny{$\langle 1, 0 \rangle$}} (N00);
\draw[->,midmidarrow] (N10) to [bend left=85]
node[below]{\mytiny{$\langle 1, 1 \rangle$}} (N11);
\draw[->,midmidarrow] (N000) to [bend left=85]
node[above]{\mytiny{$\langle 0, 00 \rangle$}} (N100);
\draw[->,midmidarrow] (N001) to [bend left=85]
node[above]{\mytiny{$\langle 0, 01 \rangle$}} (N101);
\draw[->,midmidarrow] (N010) to [bend left=85]
node[above]{\mytiny{$\langle 0, 10 \rangle$}} (N110);
\draw[->,midmidarrow] (N011) to [bend left=85]
node[above]{\mytiny{$\langle 0, 11 \rangle$}} (N111);
\draw[->,midmidarrow] (N010) to [bend left=85]
node[above]{\mytiny{$\langle 1, 00 \rangle$}} (N000);
\draw[->,midmidarrow] (N011) to [bend left=85]
node[above]{\mytiny{$\langle 1, 01 \rangle$}} (N001);
\draw[->,midmidarrow] (N100) to [bend left=85]
node[above]{\mytiny{$\langle 1, 10 \rangle$}} (N110);
\draw[->,midmidarrow] (N101) to [bend left=85]
node[above]{\mytiny{$\langle 1, 11 \rangle$}} (N111);
\draw[->,midmidarrow] (N000) to [bend left=85]
node[below]{\mytiny{$\langle 2, 00 \rangle$}} (N001);
\draw[->,midmidarrow] (N011) to [bend left=85]
node[below]{\mytiny{$\langle 2, 01 \rangle$}} (N010);
\draw[->,midmidarrow] (N101) to [bend left=85]
node[below]{\mytiny{$\langle 2, 10 \rangle$}} (N100);
\draw[->,midmidarrow] (N110) to [bend left=85]
node[below]{\mytiny{$\langle 2, 11 \rangle$}} (N111);
\end{tikzpicture}
\end{center}
\caption{Linear drawings of the twisted cubes $\twcube 0$, $\twcube 1$, $\twcube 2$, and $\twcube 3$, demonstrating that the underlying preorders are total orders. The binary sequences on top are the values of $g_n$ from the proof of \Cref{thm:total-order}.} \label{fig:rainbows}
\end{figure}
\begin{remark}
Note that every vertex $v$ in $\twcube n$ is an endpoint of $n$ non-trivial edges.
The number of zeros in the binary representation in the ``order number'' of $v$ (i.e.\ the value $g_n(v)$ in the proof of \Cref{thm:total-order}) equals the number of \emph{outgoing} edges.
\Cref{fig:rainbows} shows this.
\end{remark}
Analogously to \Cref{def:ord-graphs-def}, we can now define the category of twisted graph morphisms:
\begin{definition}[category $\twcubecat$] \label{def:tw-graphs-def}
The category $\twcubecat$ has natural numbers as objects, and morphisms from $m$ to $n$ are graph morphisms between twisted cubes:
\begin{align}
& \obj{\twcubecat} \defeq \N
& \mor{\twcubecat}(m,n) \defeq \graphhom{\twcube m}{\twcube n}
\end{align}
\end{definition}
It is easy to see that the category $\twcubecat$ has a version of connections.
Since we are looking for a ``twisted analogue'' of $\shcubeop$, we need to refine it further.
In \Cref{sec:standardcubes}, we have discussed the restriction to (meet and join)-preserving morphisms, and to dimension-preserving morphisms.
It follows directly from \Cref{thm:total-order} that every morphism in $\twcubecat$ preserves all binary meets and joins, so this condition becomes trivial; it does not avoid connections.
However, preserving dimensions is still a non-trivial condition which does avoid connections.
The definition of equation \eqref{eq:dim-preserving} still works.
\begin{definition}[category $\twgraphdim$]
The category $\twgraphdim$ has dimension-preserving maps between twisted cubes as morphisms:
\begin{align}
& \mathsf{obj}(\twgraphdim) \defeq \N
& \mor{\twgraphdim}(m,n) \defeq \Sigma (g : \graphhom{\twcube m}{\twcube n}). \dimpres(g)
\end{align}
\end{definition}
Note that the explanation of \Cref{rem:dim-inj-derivable} holds for the twisted cube category as well.
A consequence of \Cref{thm:total-order} is that morphisms in $\twgraphdim$ cannot ``swap dimensions''.
But an even stronger result holds, namely that surjective morphisms are unique:
\begin{theorem} \label{lem:surj-dim-unique}
There is exactly one surjective morphism in $\twgraphdim(m,n)$
for $m \geqslant n$. \\ (Clearly, there is none if $m < n$.)
\end{theorem}
\begin{proof}
The key to the proof is
\Cref{thm:hamiltonian}.
Clearly, the Hamiltonian path in $\twcube m$ goes through all vertices.
Due to surjectivity, its image has to go through all vertices of $\twcube n$.
In other words, the $\twcube m$-Hamiltonian path has to be mapped to the $\twcube n$-Hamiltonian path.
Since the graph morphisms that we consider preserve the dimension, the only edge in the $\twcube m$-path which can be mapped to the single edge in the first dimension in the $\twcube n$-path is just this single edge in the first dimension in the $\twcube m$-path; i.e.\ the middle edge has to be mapped to the middle edge.
From here, it follows by induction that there can only be at most one surjective graph morphism.
What is left to show is that there actually is a surjective graph morphism if $m \geqslant n$.
It is enough to construct a surjective graph morphism $f : \twgraphdim(n+1,n)$, from where we get any other by $(m-n)$-fold composition ($0$-fold composition is the identity).
Such a graph morphism is given by
\begin{align}
f(x_0 \ldots x_{n-1} x_n) \defeq (x_0 \ldots x_{n-1}).
\end{align}
Since the directions of the edges do not depend on the very last dimension, this works (cf.~\Cref{def:tw-cube-graph-nonrec}).
\end{proof}
An important consequence of the above result is that there is a unique way to degenerate a twisted cube.
We do not go into this here (but see the conclusions at the end of the paper).
Here, we go into a different direction.
Let us write $\intv$ (``interval'') for the finite set $\{0,1,\star\}$.
Of course, $\intv$ is isomorphic to $\finset 3$, but referring to the last element as $\star$ helps the intuition, we hope.
\begin{definition}
A \emph{face} of the twisted $n$-cube $\twcube n$ is a function $f : \finset n \to \intv$.
The \emph{dimension} of a face, written $\mathsf{dim}(f)$, equals the number of times $f$ takes $\star$ as value (i.e.\ the size of $f^{-1}(\star)$).
The type of faces of dimension $k$ is written as $\mathsf{faces}(n, k)$.
\end{definition}
The face $f : \finset n \to \intv$ represents the full subgraph of $\twcube n$ of vertices on which $f$ ``matches'' (a vertex $x_0x_1\ldots{}x_{n-1}$ is matched if, for every $i$, we have $f(i) = x_i$ or $f(i) = \star$).
\begin{lemma} \label{lem:image-is-face}
The image of $f : \mor{\twgraphdim}(m,n)$ is a face.
\end{lemma}
\begin{proof}
This follows from the property of preserving the dimension.
\end{proof}
\begin{lemma} \label{lem:faces-only-inj}
The $m$-faces are the only injective maps $\twgraphdim(m,n)$:
\begin{equation}
\mathsf{faces}(n, m) \, \simeq \, \Sigma(f : \twgraphdim(m,n)).\isinj(f).
\end{equation}
\end{lemma}
\begin{proof}
Every face gives rise to a canonical injective dimension-preserving morphism, as dictated by the inclusion of the full subgraph that the face represents into $\twcube n$.
The fact that these are the only ones follows from \Cref{thm:total-order} (we cannot ``swap dimensions'') and \Cref{lem:image-is-face}.
\end{proof}
As with \Cref{thm:total-order} before, \Cref{lem:faces-only-inj} is a result which is usually found in simplicial structures, but not in cubical ones.
In any case, we now easily get:
\begin{lemma}[factorisation of dimension preserving morphisms] \label{lem:factoring-dim-pres}
Given a morphism $f : \twgraphdim(m,n)$, there is exactly one way to write it as the composition $f = \mathsf{inj}(f) \circ \mathsf{surj}(f)$ of a surjective dimension preserving graph morphism followed by an injective one.
This means that the map
\begin{align}
& \hspace*{-.3cm} \big(\Sigma (k : \N) . \left(\Sigma (h : \twgraphdim(k,n)). \isinj(h)\right) \times \left(\Sigma (g : \twgraphdim(m,k)). \issurj(g)\right) \big) \to \twgraphdim(m,n)
\label{eq:factorisation}\\
& \hspace*{-.3cm} (k,(h,i),(g,s)) \mapsto h \circ g
\end{align}
is an equivalence.
Moreover, morphisms $\twgraphdim(m,n)$ are in 1-to-1 correspondence with faces of $\twcube n$ of dimension $\leqslant m$.
\end{lemma}
\begin{proof}
A consequence of \Cref{lem:image-is-face} is that the factorisation on the level of sets of vertices works.
The second claim follows from the first:
In \eqref{eq:factorisation}, the $k$ and the surjective map are uniquely determined (i.e.\ contractible components) by \Cref{lem:surj-dim-unique}.
By \Cref{lem:faces-only-inj}, injective maps correspond to faces.
\end{proof}
\begin{remark}
A consequence of \Cref{lem:factoring-dim-pres} and the proof of \Cref{lem:surj-dim-unique} is that all the non-empty fibres of a dimension-preserving morphism between twisted cubes have the same size. The reverse is the case as well: a morphism between twisted graphs where all non-empty fibres have the same size is dimension-preserving.
\end{remark}
Finally, let us record an alternative representation of the category $\twgraphdim$ which does not go via graph morphisms.
\begin{definition}[trinary notation: category $\twternary$]
The category $\twternary$ has natural numbers as objects, and a morphism from $m$ to $n$ is a function $\finset n \to \intv$ which takes $\star$ at most $m$ times as image:
\begin{align}
& \obj{\twternary} \defeq \N
& \mor{\twternary}(m,n) \defeq \Sigma (f : \finset n \to \intv). f^{-1}(\star) \leqslant m
\end{align}
The identity morphisms are the functions that are constantly $\star$.
To define the composition of $f : \twternary(k,m)$ and $g : \twternary(m,n)$,
we need to define a function $g \circ f : \finset n \to \intv$ (which is $\star$ at most $k$ times).
We define $(g \circ f)(i)$ by recursion on $i$, simultaneously with the values $i'$ and $b_i$, as follows:
\begin{align}
(g \circ f)(i) \defeq
\begin{cases}
g(i) & \text{if } g(i) \in \{0,1\} \\
(f(i')) \; \mathsf{xor} \; b_i & \text{if } g(i) = \star \text{ and } f(i') \in \{0,1\} \\
\star & \text{if } g(i) = \star \text{ and } f(i') = \star
\end{cases}
\end{align}
where \begin{itemize}
\item $i'$ is the number of occurrences of $\star$ in the sequence $g(0), g(1), \ldots, g(i-1)$;
\item $b_i$ is $1$ if the number of zeros in the sequence $(g \circ f)(0), (g \circ f)(1), \ldots, (g \circ f)(i-1)$ is odd, and $0$ if it is even.
\end{itemize}
\end{definition}
Note that a morphism in $\mor{\twternary}(m,n)$ can be represented as
a sequence such as $01\slimstar0\slimstar10$ of length $n$ which
contains the symbol $\star$ at most $m$ times, which is why we
refer to it as \emph{ternary notation}.
\begin{remark} \label{rm:semi-twisted-cubes}
There is a category of twisted semi-cubes, denoted by $\twternary^+$,
which is exactly the same as $\twternary$ except that the number of
$\star$ in the sequence must be exactly $m$, i.e.\
``$\leqslant$'' is changed to ``$=$'' in the definition of
$\mor{\twternary}(m,n)$. This category is equivalent to the
sub-category of $\twgraphdim$, denoted as $\twgraphdim^+$,
which consists of \emph{injective} dimension-preserving graph homomorphism.
Note that this injectivity condition is equivalent to replacing $(\finset 1, \finset 1)$ by $(\finset 1, \finset 0)$ in the base case of \Cref{def:tw-cube-graph-rec}.
If we remove the expression (\emph{xor $b_i$}) in the definition of morphisms of $\twternary^+$, then the category becomes
equivalent to the category of standard cubes but without degeneracies and swapping dimensions. In other words, the
expression (\emph{xor $b_i$}) characterises ``twisted-ness''.
\end{remark}
\begin{theorem}
The categories $\twgraphdim$, and $\twternary$ are isomorphic, with the object part being the identity.
In particular, we have:
\begin{equation} \label{eq:tw-cubes-cat-iso}
\mor{\twgraphdim}(m,n) \simeq \mor{\twternary}(m,n)
\end{equation}
\end{theorem}
\begin{proof}
As the following chain of equivalences:
\begin{alignat*}{3}
&&& \mor{\twgraphdim}(m,n) \\
[\Cref{lem:factoring-dim-pres}] \quad & \simeq & \quad & \Sigma (k : \N) . \left(\Sigma (h : \twgraphdim(k,n)). \isinj(h)\right) \times \left(\Sigma (g : \twgraphdim(m,k)). \issurj(g)\right) \\
[\Cref{lem:surj-dim-unique}] \quad & \simeq & \quad & \Sigma (k : \N) . \left(\Sigma (h : \twgraphdim(k,n)). \isinj(h)\right) \times (k \leqslant m) \\
[\Cref{lem:faces-only-inj}] \quad & \simeq & \quad & \Sigma (k : \N) . \; \mathsf{faces}(n, k) \times (k \leqslant m) \\
[simplification] \quad & \simeq & \quad & \Sigma (f : \finset n \to \intv). f^{-1}(\star) \leqslant m \\
\quad & \equiv & \quad & \mor{\twternary}(m,n)
\end{alignat*}
When transported along this isomorphism, the composition of $\twgraphdim$ gets mapped to the composition of $\twternary$, as required.
\end{proof}
\section{Conclusions and Future Work}
In this paper, we have introduced and proved multiple results about \emph{twisted cube categories}.
In future work, we plan to examine them further.
They carry a monoidal structure, defined by $(x_0\ldots x_m) \otimes (y_0\ldots y_n) = (x_0\ldots x_m y_0' \ldots y_n')$ with $y_i' = y_i$ if the number of zeros in $x_0\ldots x_m$ is even, otherwise $y_i' = 1 - y_i$.
Another aspect that we have not discussed in this paper is an algebraic presentation via generators and relations.
Such presentations exist for many different cube categories in the literature.
As far as we are aware, such a definition has not been suggested for the BCH category, but the presentations by Antolini~\cite{Antolini2002} and Newstead~\cite{Clive-Newstead-cubical-sets}
are easy to adapt to that category.
Interestingly, further adapting the generators to the \emph{twisted} setting simplifies them significantly, which mirrors the fact that morphisms between twisted cubes cannot swap dimensions.
Moreover, our \Cref{lem:surj-dim-unique} implies that degeneracies are unique: there is only one single way in which a twisted $n$-cube can be degenerated to get a twisted $(n+1)$-cube.
A consequence is that we do not need to impose relations between different degeneracies.
This, we hope, will help us to develop the higher categorical structures that can be encoded as presheaves on the category of twisted cubes.
Ultimately, our goal is to model some form of \emph{directed cubical type theory} mirroring the model by Bezem, Coquand, and Huber~\cite{bezem_et_al:LIPIcs:2014:4628}.
It seems to be possible to reuse some of their development.
Another direction which we want to explore is to not consider set-valued presheaves, but type-valued presheaves instead.
To facilitate this, we can consider the category of twisted semi-cubes mentioned on~\Cref{rm:semi-twisted-cubes}.
From there, type-valued presheaves can be encoded as Reedy-fibrant diagrams in a known style~\cite{shulman_inversediagrams}.
We can then add a condition reminiscent of Rezk's \emph{Segal-condition}~\cite{rezk2001model} by stating that the projection from twisted semi-cubical types to the sequence of types along the Hamiltonian path is an equivalence.
It seems that this is promising for a construction of composition and higher coherences, although the details remain to be worked out.
\mysubparagraph{Acknowledgements}
We would like to thank Paolo Capriotti and Jakob von Raumer.
Both offered many suggestions during interesting discussions.
In particular, the initial observation on which \Cref{thm:total-order} is based was suggested by them, and the idea of considering graph morphisms was found in one of our many interesting discussions.
We are also grateful to many other people who listened to and commented on the idea of twisted cubes.
|
{
"timestamp": "2019-03-05T02:24:26",
"yymm": "1902",
"arxiv_id": "1902.10820",
"language": "en",
"url": "https://arxiv.org/abs/1902.10820"
}
|
\section{Introduction}
The Internet of Things (IoT) \cite{IoT_Iqbal} provides a concept of connectivity of anything from anywhere at anytime so that the interaction of physical objects connected to the network can be done autonomously.
IoT is closely coupled with sensor technology, because in most of the cases sensors \& actuators are part of a larger IoT network. The use of IoT devices such as laptops, smart-phones, home appliances, industrial systems, ehealth devices, surveillance equipment, precision farming sensors, and other accessories connected to Internet would exceed 45 billion by 2020 \cite{bizanis2016sdn}. These IoT sensors \& actuators may produce large volumes of data. Hence, the need for installing new network access \& core devices will increase. To manage the network devices efficiently, the network hardware resources need to be virtualized.
Virtualization \cite{7387427} is the logical abstraction of the underlying hardware devices within a network, through software implementation. The abstraction decouples the control from hardware, and makes it easier to modify, manage, and upgrade. In recent times, the abstraction has not been limited to hardware only, but rather software embedded into hardware has also been virtualized as independent elements.\par
Traditional networks are usually rigid and fixed. Heterogeneity, scalability, and interoperability has been major challenges due to rapid growth of Internet. Software Defined Networks (SDN) \cite{8187644} and Network Function Virtualization (NFV) \cite{7243304} are the two main solutions to provide virtualization in communication. SDN physically decouples network control plane from the forwarding plane, and centralizes the decision making for physical forwarding devices. It enables the network control to become directly programmable, and the underlying physical infrastructure to be virtually abstracted for applications \& network services. The OpenFlow (OF) \cite{7929710} protocol is the foundation of communication between SDN-enabled devices. Several benefits of SDN include:~(i)~direct network programmability allowing network managers to configure, manage, optimize, and secure network resources dynamically, (ii)~network-wide traffic flow control \& flow installations, (iii)~network intelligence is logically centralized providing a global view of the network, and (iv)~a vendor independent open standard, simplifying network design \& operations. NFV \cite{7073808,Schaffrath:2009:NVA:1592648.1592659,5183468} is the mechanism of abstracting functions, such as firewall, load balancing, path calculation, etc., from dedicated hardware to virtual environment. The key benefits of NFV includes replacing dedicated hardware with commodity servers. It enables to host SDN applications like security functions, load balancing, data collection \& analysis, etc., through deployment of on-demand virtual network functions (VNFs). This enables not only enhanced scalability \& elasticity for deploying vendor independent commodities with reduced cost, but also optimizes computing, memory, storage, and networking capacity of network devices. SDN and NFV are not competing technologies, but are complimentary to each other. The key benefits of both technologies are inter-related. NFV can boost SDN towards virtualizing the SDN controller and other network applications in the cloud. Similarly, SDN with its programmable network connectivity can implement traffic engineering decisions taken by VNFs \cite{7350211}.
Use of SDN along with IoT has been studied to some detail. A number of solutions \cite{7218418,6385039,6324377,6838365} have been proposed to address different IoT optimization challenges by using software defined networking. Similarly, Network Functions \cite{8169853} of IoT devices and ecosystem can also be virtualized to make them more agile, robust, \& cost effective. This will reduce the number of physical devices needed, easily segment networks, and enforce security policies on physical devices.\par
It is important to note that in this work we classify virtualization techniques in IoT from three different aspects: (a) SDN-based IoT refers to IoT solutions which use software defined networking for core communication, (b) NFV for IoT includes solutions which virtualize IoT specific functions in the whole ecosystem, and (c) Software Defined - IoT (SDIoT) refers to techniques which not only makes the network layer virtualized but also includes device and application abstraction, virtual security policy implementation, and virtual device configuration and management, etc.
\textbf{Contributions of this work:} In this paper we provide a comprehensive survey of different virtualization solutions designed specifically for IoT. Following is a list of major contributions.
\begin{enumerate}
\item An overview of general virtualization techniques and their benefit to IoT.
\item Classification of solutions available in literature.
\item In-depth survey of SDN-based solutions with respect to architecture, management, and security in IoT.
\item Detailed analysis of function virtualization techniques in IoT, along with their uses in SDNs for IoT.
\item Details of software defined Internet of Things and virtualization of different elements in ecosystem.
\item Major research directions for virtualization in IoT.
\end{enumerate}
\textbf{Structure of Paper:} Table~\ref{tab:Outline} gives an outline of organization of the paper. Section II gives an overview of different virtualization techniques for SDN, control plane, functions, and devices. IoT and details of its working are given in section III. Section IV elaborates the literature classification and other works similar to this article. Section V, VI, and VII provide detailed survey of solutions for SDN-based IoT, function virtualization in IoT, and software defined IoT, respectively, along with analysis and comparisons. Section VIII gives the future research directions, and conclusion is given in section IX.
\begin{table}[!t]
\centering
\caption{Outline of this article.}
\label{tab:Outline}
\begin{tabularx}{0.99\linewidth}{|>{\hsize=.25\hsize}Z|>{\hsize=1.75\hsize}Y|}\hline
\textbf{Sections} & \textbf{Details} \\\hline
II & Overview of network virtualization techniques\newline
\quad$\bullet$\quad Control plane virtualization \& components\newline
\quad$\bullet$\quad Function virtualization\newline
\quad$\bullet$\quad Device virtualization\\\hline
III & Internet of Things\newline
\quad$\bullet$\quad IoT use case applications\newline
\quad$\bullet$\quad IoT challenges \& SDN benefits\newline
\quad$\bullet$\quad IoT stack and protocols\newline
\quad$\bullet$\quad Sensor networks \& IoT \\\hline
IV & Motivation \& Related works\\\hline
V & Software Defined Network based IoT\newline
\quad Architecture, Security, and Management solutions.\\\hline
VI & Network Function Virtualization for IoT\newline
\quad$\bullet$\quad NFV architectures for IoT solution of NFV for IoT\newline
\quad$\bullet$\quad IoT Management using Virtual Functions\newline
\quad$\bullet$\quad Security in IoT using of NFV\\\hline
VII & Software Defined IoT\newline
\quad$\bullet$\quad Architectural solutions of SDIoT\newline
\quad$\bullet$\quad IoT Management using SD Frameworks\newline
\quad$\bullet$\quad Security solution using SDIoT \\\hline
VIII & Future research directions\\\hline
IX & Conclusion\\\hline
\end{tabularx}
\end{table}
\begin{table}[!t]
\centering
\caption{List of uncommon abbreviations used in this article.}
\label{tab:TableIX}
\begin{tabularx}{0.9\linewidth}{|>{\hsize=.5\hsize}Z|>{\hsize=1.5\hsize}Y|}\hline
\textbf{Abbreviation} & \textbf{Description} \\\hline
API & Application Programming Interface\\\hline
CoAP & Constrained Application Protocol\\\hline
DDS & Data Distribution Service\\\hline
DLT & Distributed Ledger Technology\\\hline
DNP & Distributed Network Protocol\\\hline
DPDK & Data Plane Development Kit\\\hline
DPI & Deep Packet Inspection\\\hline
E/WBI & East/Westbound Interface\\\hline
EP & Entry Point\\\hline
EPC & Evolved Packet Core\\\hline
GMPLS & General Multi-Protocol Label Switching\\\hline
HDFS & Hadoop Distributed File System\\\hline
HLPSL & High Level Protocols Specification Language\\\hline
IPS & Intrusion Prevention System\\\hline
LXC & Linux Container\\\hline
MANO & Management and Orchestration\\\hline
MEC & Mobile Edge Computing\\\hline
MIMO & Multi-input and Multi-output\\\hline
MINA & Multi-network Information Architecture\\\hline
MitM & Man in the Middle\\\hline
MNO & Mobile Network Operator\\\hline
MPC & Mobile Packet Core\\\hline
MQTT & Message Queuing Telemetry Transport\\\hline
MVNO & Mobile Virtual Network Operator\\\hline
NAC & Network Access Control\\\hline
NBI & Northbound Interface\\\hline
NFV & Network Function Virtualization\\\hline
NFVI & Network Function Virtualization Interface\\\hline
NTP & Network Time Protocol\\\hline
NV & Network Virtualization\\\hline
OF & Open Flow\\\hline
ONF & Open Networking Foundation\\\hline
ONOS & Open Network Operating System\\\hline
OVS & Open vSwitch\\\hline
PIGs & Programmable IoT Gateways\\\hline
REST & Representational State Transfer\\\hline
SBI & Southbound Interface\\\hline
SD & Software Defined\\\hline
SDNCH & Software Defined Network Cluster Head\\\hline
SDNi & Software Defined Network Interconnection\\\hline
SDSH & Software Defined Smart Home\\\hline
SFC & Service Function Chaining\\\hline
SMP & Security Management Provider\\\hline
SPF & Sieve, Process, Forward\\\hline
TLS & Transport Layer Security\\\hline
VF & Virtual Function\\\hline
VIM & Virtual Infrastructure Manager\\\hline
VNE & Virtual Network Element\\\hline
VNF & Virtual Network Function\\\hline
VNFM & Virtual Network Function Manager\\\hline
\end{tabularx}
\end{table}
\section{Overview of Network Virtualization Techniques}
Network virtualization \cite{8326277} is the mechanism of combining both software \& hardware resources and network functionality into a logically configured single software-based administrative entity. The term \textit{virtual network} refers to the resulting software network entity. In other words, a successful network virtualization would require platform virtualization along with resource virtualization. This is achieved through the Virtualization Layer, which is an additional abstraction layer between network and storage hardware, and the applications running on it. It can be categorized as either an external virtualization, consisting of many networks into a virtual unit, or internal virtualization serving network-like functionality to software containers on a single network server. In the following subsections we elaborate each element of a virtualized network.
\subsection{Control Plane Virtualization}
Traditionally, a network comprises of hardware devices for connectivity with a dedicated controller built in them. The controller is part of router architecture which instructs switches where to forward packets. Hardware in the physical network devices is managed by the controller. In existing communication network, there is a need for more flexible features from these controllers. An ideal controller can be managed anytime from geographically anywhere in the world. This has opened up the scope for virtualization of the controller, which is implemented through Software Defined Networks (SDNs) \cite{8323847}. The main idea is to separate the control and data plane, i.e. the intelligence of the router/switch is split from the packet-forwarding engine and placed in the control plane. This may be done centrally or in a distributed manner. The SDN controller supports programmability, allowing the underlying infrastructure to be abstracted for applications and network service. Thus, network programmability \cite{7496952} is the process of releasing the network's power in unique ways for more flexible, faster, and intelligent infrastructure that makes the network application-aware. Programmability refers to the ability to enhance network features linking the applications to it and allowing dynamic traffic flow change, providing both network and application-level Quality of Service (QoS).\par
SDN is a network architecture which can be dynamic, manageable, adaptable, cost-effective, appropriate for high bandwidth requirements, and adapts to dynamic nature of today's applications. It is directly programmable, agile, and centrally manageable. It has the ability to prioritize, de-prioritize or even block specific types of packets with a granular state of control while routing packets in a given network. This process may also be referred to as efficient traffic engineering allowing administrator to use less expensive OpenFlow complaint commodity switches. OF is a communications protocol that allows access to the data plane of a network switch.
\subsubsection{SDN Architecture} SDN architecture has three major layers as shown in Figure~\ref{fig:1}. These layers communicate through Application Programing Interfaces (APIs).
\begin{figure}[!t]
\centering
\includegraphics[width=0.95\linewidth]{SDN_Architecture.eps}
\caption{Software Defined Network Architecture.}
\label{fig:1}
\end{figure}
\textbf{ Data Plane:} It is also referred as Forwarding Plane, which includes switches, either hardware or software based, connected through a physical medium and perform a set of elementary operations, such as looking up in a table extracting information about incoming packets. These devices have well-defined instruction sets which are used to take actions (forward to port, drop, forward to controller) for incoming packets. These instructions can also be dynamically configured from control plane.
\textbf{Control Plane: }It is a decoupled entity from data plane and is logically a centralized server, also referred as controller, having a global view of the whole network under its control. Based on its global view, it installs forwarding rules on devices in data plane. Some examples of controllers are POX, NOX, OpenDaylight, Floodlight, etc. \cite{Gude:2008:NTO:1384609.1384625,8447647,6838330,ODL,FL,HassasYeganeh:2012:KFE:2342441.2342446}. These controllers can be centralized or distributed \cite{KARAKUS2017279,OKTIAN2017100} in design.
\textbf{Southbound Interface:} It provides a communication protocol between control plane and data plane. This interface helps controller to program forwarding devices and install flow entries or rules. Some examples of southbound interfaces are \cite{McKeown:2008:OEI:1355734.1355746,doria2010forwarding,Song:2013:PFU:2491185.2491190,pfaff2013open,parniewicz2014design}, but OpenFlow \cite{inferringOFrules_Iqbal} is a widely used protocol in existing SDN implementations.
\textbf{Management Plane:}
Applications designed in management plane can be used to manage and monitor switches in the data plane through the control plane. SDN can be deployed anywhere from enterprise to data centers with the help of management plane, which provides a variety of applications. These applications can be grouped into different categories: Network management and traffic engineering \cite{Jain:2013:BEG:2486001.2486019,6305261,Nakao,Reitblatt:2012:ANU:2342356.2342427,6126682}, Server load balancing \cite{Wang:2011:OSL:1972422.1972438}, Security and network access control \cite{Gember:2012:TSM:2390231.2390233,Gibb2012InitialTO,ETSI,6702549,Nayak:2009:RDA:1592681.1592684,Khurshid:2012:VVN:2342441.2342452,6089085,Jafarian:2012:ORH:2342441.2342467,5689156,5735752,Porras:2012:SEK:2342441.2342466}, Network virtualization \cite{Sharafat:2011:MMV:2018436.2018516,6066002,Gutz:2012:SIS:2342441.2342458,Ferguson:2012:HPS:2342441.2342450,6461196,openstackfoundation}, and Inter-domain routing \cite{Nascimento:2010:QPQ:1851182.1851252,Nascimento:2011:VRS:2002396.2002405,6211892,Caesar:2005:DIR:1251203.1251205,Rothenberg:2012:RRC:2342441.2342445}.
\textbf{Northbound Interface:}
It provides communication between management plane and control plane, where mostly REST API \cite{restapi} is used. There are some controllers (e.g. NOX, PANE, etc.) \cite{Ferguson:2013:PNA:2534169.2486003}, which provide their own northbound APIs and some programming languages (e.g. Frenetic, Procera, etc.) also support them.
\textbf{East/Westbound Interface:}
Scalability and single point of failure are two major challenges in SDN that are resolved by distributed architectures, where multiple controllers work together to attain a global network view. A communication channel is required for these controllers for information sharing. For this purpose East/Westbound Interface (E/WBI) is used, which can interconnect different SDN domains or SDN and traditional network domains. In this context, east refers SDN-to-SDN communication, and west refers to legacy-SDN communication.
\subsubsection{SDN Functionality Details} The decision making is logically decoupled from the network device, and given to the SDN controller in control plane, which is usually a server running relevant SDN software. OpenFlow protocol (OF) is used by the controller to communicate with a physical or virtual switch in data plane through the Southbound Interface (SBI). Forwarding table is created based on the information provided by the control plane and forwarded to the data plane. Network device (switch) uses this table to decide where to send frames or packets. The routing functionality initiates with the switch encapsulating and forwarding the first packet from a flow to an SDN controller, requesting for addition of a flow to flow table of the switches. When the SDN controller adds the new flow for the switch table, the switch then forwards the incoming packet(s) using the correct port. It is also possible that SDN controller may not add a new flow for the switch in the routing table, and instead enforce the policy to drop the packet during a particular flow, permanently or temporarily, due to security purposes, avoiding Denial-of-Services (DoS) attacks, or for traffic management optimization \cite{CISCO}.
SDN creates dynamic and flexible network architecture, to adapt to the changes in networks requiring rapid deployment. Using centralized control and network automation, SDN also adds more benefits, such as the use of API enabled SDN controllers to execute network commands on multiple IoT devices.
\subsection{Function Virtualization}
Function Virtualization is implemented through a Network Function Virtualization (NFV) architecture. Figure~\ref{fig:2} shows a generic NFV architecture, which utilizes IT virtualization technologies to virtualize the complete network node functions into series of building blocks to establish connectivity, and to create communication services among them. Its architecture depends on three main components: Virtual Network Function (VNF) \cite{7890085}, Network Function Virtualization Infrastructure (NFVI), and Network Function Virtualization Management and Orchestration architectural framework (NFV-MANO). NFV implements network functions through a piece of software that is configured under NFVI. These network functions tend to be in the form of VNF, which is responsible for handling specific network operations that run on top of the hardware infrastructure. NFVI consists of both physical and virtual storage, processing, and virtualization software. NFV-MANO architectural framework consists of interfaces and reference points to individual VNFs and NFVI elements.
For example, network function such as firewall, is an instance of plain software, installed inside voluminous switches, storage, and servers, to filter traffic and neutralize vulnerable packets.
Further benefits include, allowing the relocation and initiation of these nodes from geographically different network locations.
\begin{figure}[!t]
\centering
\includegraphics[width=0.95\linewidth]{NFV_logical_Architecture.eps}
\caption{A generic NFV modular structure.}
\label{fig:2}
\end{figure}
\subsection{Device Virtualization}
Device virtualization is the process of virtualizing a switch in the data plane using certain logical abstractions among its components, or only the functionality to be executed on different operating systems. Virtualization, in a computing platform, tends to hide the physical features from the users, and create an abstract computing platform to define unique rules for switches to comply, which may be referred to as VNFs. The software that controls virtualization is called the control program, also referred to as hypervisor \cite{8404837}. Similarly, Sensor virtualization \cite{doi:sensorvirtualization} provides software abstraction of various external IoT objects, and allows applications to easily utilize various IoT resources through open APIs (e.g. Zeroconf\cite{zeroconf}). Zeroconf or similar APIs allows virtual sensor to transparently discover arbitrary sensor device as virtual switches.
It is also able to communicate with different applications using a standard communication interface based on UDP/TCP sockets or even HTTP \cite{5416827}. This way, the applications are not required to deal with sensor specific details.
\section{Internet of Things (IoT)}
Internet of Things~\cite{6722995} is a collection of sensors, actuators, and smart objects, interconnect via the Internet utilizing embedded technology to interact and communicate with the external environment. IoT connectivity and management are two major challenges in its deployment. Usually IoT systems are developed with a specific target and technology.
IoT incorporates everything from a small objects to big machines, appliances to building and industries, body sensors to cloud computing. In essence, it has infiltrated every aspect of our lives. \cite{IoT1} estimates that the potential market value of IoT devices and associated technologies will exceed \$14 trillion in the next 10 years. Similarly, major hardware developers (e.g. Apple, Cisco, Samsung, etc.) have made huge investments in different IoT fields.
\subsection{IoT Use Cases} IoT is playing a significant role in a number of use case applications. Figure~\ref{fig:IoTUsecase2Inscape} shows some examples of IoT ecosystem. The benefits achieved range from small to large scale. Below we briefly introduce some of these use cases, and how they benefit different industries.
\begin{figure}[!t]
\centering
\includegraphics[width=0.95\linewidth]{IoTUsecase2Inscape.eps}
\caption{Example IoT ecosystem. Isolated application specific IoT networks may also communicate with each other over the Internet.}
\label{fig:IoTUsecase2Inscape}
\end{figure}
\subsubsection{Hospitals \& Healthcare} Application of IoT in both hospital premises and e-health systems is not limited to remote monitoring, but also provides a complete automated healthcare ecosystem. A wide range of IoT devices are used in this process, such as, monitoring cameras, connected inhalers, ingestible sensors, smart insulin delivery devices, smart watch \& wearable sensors/data collectors, connected ambulance, etc.
\subsubsection{Intelligent Transportation Systems} There are various uses of IoT applications in this domain. Sensors are used to retrieve information related to available parking spots for efficient parking management solutions. Smart signboard connected to Internet, can disseminate emergency information alongside roads.
Asset tracking allows enterprises to easily locate \& monitor vehicular fleets and other mobile assets. Fleet management helps transport companies reduce investment risks associated to vehicles. It improves efficiency \& productivity, while reducing overall transportation \& management costs. Shipping service uses real time traffic feeds to deliver more packages using efficient algorithms, with lower burden on drivers \& vehicles.
Connected vehicles can better automate many normal driving tasks. Benefits of self-driving cars include accident avoidance, lesser traffic congestion, and other economical efficiencies. Driverless taxis and buses are also a major use case for IoT applications.
Application of IoT technology in transportation eventually reduces traffic congestion, improves safety, mobility, and productivity.
\subsubsection{Industrial Automation \& Supply Chain} Industrial automation uses artificial intelligence with IoT technology, to automate the supply chain process. Supply chain along with asset tracking optimizes logistics, maintains inventory levels, prevent quality issues, and detect theft. Industry 4.0 production lines are greatly influenced by intelligent manufacturing system, such as smart machines (e.g. multiple smart robots used in car assembling works collaboratively) powered by IoT devices. This results in less human errors, increased speed of production process \& quality of the finished products.
\subsubsection{Smart Homes} IoT in such applications provides a complete intelligent ecosystem for connected devices, ranging from lighting control to security and safety. Usually a smart central hub or gateway is used for human interaction, which in turn controls device automation. These devices can be lined to heating systems, lighting control, appliance monitoring and control, utility usage and optimization, security system, support systems for elderly/disabled, etc.
\subsection{IoT Challenges \& SDN} There are many technological challenges for deploying IoT systems so they can function smoothly. These includes security, connectivity, compatibility \& longevity, standards, and intelligent analysis \& actions. IoT networks are usually large, mobile, and dynamically change their topology \& connectivity. They also have heterogeneous devices which support a range of applications. Hence, challenges like IoT device detection, low power consumption, bandwidth, access control, and data encryption become major concerns for large scale deployment.
SDN ensures reliable connectivity at any given time, based on pre-defined policies. SDN supports customized device configuration enabling efficient packet flows \& optimized routing. It is also a vendor independent platform supporting widely used OF protocol, which mitigates the compatibility standardizing challenges. SDN facilitates device-to-device communication without the intervention of base stations. Heterogeneity is a major concern, especially when billions of mobile IoT devices are connected in a network. NFV plays a significant role in connecting and managing heterogeneous IoT elements. Function virtualization and service chaining mechanisms are the core components to mitigate heterogeneity limitation. Combination of SDN \& NFV supports network programmability, which can improve access control \& bandwidth, data encryption, IoT device detection, low power consumption, etc. for large scale deployment of IoT.
\subsection{IoT Stack and Protocols}
IoT is applicable in a diverse range of use cases and industries. Its implementation ranges from embedded standalone devices to real-time and mission critical cloud infrastructures. The layered IoT stack shown in Figure~\ref{fig:3}, presents the standards, technologies, and protocols used in such systems. Application Layer specifies all the shared communication protocols and interface medium used by IoT devices. Network Layer specifies communication path over the network (IP address). Physical/Media Access Control (PHY/MAC) Layer specifies communication path between adjacent nodes and data transfer (MAC address). From an SDN perspective, it is very important to understand the technologies used to build IoT networks. It is important to note that SDN does not only install flows for IP packets, but can also be used for radio resource management, security policies, and channel assignment at physical layer, etc.
\begin{figure}[!t]
\centering
\includegraphics[width=0.95\linewidth]{IoTStack.eps}
\caption{IoT technology stack and protocols.}
\label{fig:3}
\end{figure}
Various applications fall under the umbrella of IoT, that use different technologies as the main communication enabler~\cite{sheng2013survey, withanage2014comparison}. The most commonly used physical layer technologies are:
\textbf{ZigBee (IEEE 802.15.4)}~\cite{kinney2017ieee,zhou2009wireless}:
Specifies the physical layer and media access control for low-rate wireless personal area networks. It has been designed to run on low-power devices enabling M2M communication. It provides low-power consumption and low duty cycle to maximize battery life. ZigBee can also be used in mesh networks, and supports a large number of devices over long distances with many different topologies, connected all together through multiple pathways.
\textbf{WiFi (IEEE 802.11)}~\cite{deng2015ieee}:
Allows local communication between two or more devices using radio waves. It is the most used technology to connect the Internet gateway to devices. WiFi utilizes both 2.4GHz UHF and 5GHz SHF ISM radio bands. WiFi networks operate in the unlicensed 2.4 radio bands, where the access point and the mobile stations share the same channel and communicate in half duplex mode.
\textbf{Bluetooth \& Bluetooth Low Energy (IEEE 802.15.1)}~\cite{nieminen2015rfc,bisdikian2001overview}:
It is used to transfer data over short distances using 2.4 GHz ISM band and frequency hopping, and up to 3 Mbps data rate with 100m as maximum range. The technology is mostly used to connect user phones and small devices with each other.
\textbf{6LoWPAN}~\cite{shelby20116lowpan,kushalnagar2007ipv6}:
It is a networking technology that combined the Internet Protocol (IPv6) with Low-power Wireless Personal Area Networks (LoWPAN), which is one of the most suitable technologies for IoT deployment. It is a good choice for the smaller devices that are limited in processing and transmission capabilities.
\textbf{5G}~\cite{rappaport2013millimeter}:
The fifth-generation wireless is the newest iteration of cellular technology that is based on the IEEE 802.11ac wireless networking standard in order to speed up the transmission data, reduce the latency. Both LTE and MIMO are used as a foundation in 5G network, as well as network slicing.
\subsection{Sensor Networks and IoT}
\textbf{Wireless Sensor Network (WSN)}: It is a distributed and self-organized wireless network that consists of autonomous devices using sensors to observe physical or geographical conditions. According to \cite{niyato2017wireless}, due to the ability to relay messages from one node to another, the area coverage of such networks may differ from a few meters to several kilometers. It is important to note that sensor network and IoT networks are not the same. At best, sensor networks are a subset of IoT ecosystem. They not only differ in deployment, but also in protocols, topologies, use cases, applications, and other technical aspects. A handful of SDN solutions for WSNs have been proposed, but they cannot be directly applied to IoT.
\section{Motivation \& Classification}
In this section we first discuss the existing surveys, and derive the need and motivation for this article. Following it, we present the basic classification and group of literature reviewed.
\subsection{Motivation \& Existing Surveys}
Virtualization, SDN, and IoT have individually attracted attention from the research community. However, there has been very limited effort to review the literature which combines them. Table~\ref{tab:existing_Surveys} lists surveys which have previously been done, and are related to the work in this paper. It is important to note that most of them only target a specific technology. The closest work is \cite{bizanis2016sdn} and \cite{7060643}, which deals with the virtualization in IoT and WSN, respectively. Bizanis~et~al.~\cite{bizanis2016sdn} provide a survey of literature from 2009-2016 and mostly focus on SDN and network virtualization in IoT applications, specific to mobile, cellular and 5G context. It does not cover IoT in-depth nor considers all solutions available. Khan~et~al.~\cite{7060643} focus specifically on WSN and do not collect works on IoT in general.
Some other surveys related to SDN or NFV have also touched IoT in passing. Pan~et~al.~\cite{8089336} focused mainly on IoT application based on future edge cloud and edge computing but the effort is only limited to brief introduction of related challenges and enabling cloud based technologies like SDN and NFV for IoT applications. Akpakwu~et~al.~\cite{8141874} concentrated research on 5G based communication technologies and challenges for IoT applications. But the effort is limited to IoT application usecases for mobile communications. There is only brief introduction of two useful technologies like SDN and NFV to counter IoT management specific issues for future telecommunication system. Cox~et~al.~\cite{8066287} focused research only on SDN state-of-art and challenges with no related solution reviews. Ngu~et~al.~\cite{7582463} presented findings on design of real-time prediction of blood alcohol content using smart-watch sensor data and IoT middleware issues and enabling technologies.
We believe that there is a need to classify and analyze literature, which focuses directly on IoT in terms of different virtualization techniques. Moreover, these virtualization techniques should not be limited to SDNs for IoT, but should also include network function virtualization, network virtualization, and most importantly software defined Internet of Things.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{TaxOfVirIoT1.eps}
\caption{Classification of literature for this article. Literature in three major categories is further grouped into specific types of solutions.}
\label{fig:Organogram}
\end{figure}
\begin{table*}[!h]
\centering
\caption{Summary of related surveys \& contribution of this work.}
\label{tab:existing_Surveys}
\setlength\tabcolsep{3.5pt}
\begin{tabularx}{0.95\linewidth}{|>{\hsize=.6\hsize}Y|>{\hsize=.2\hsize}Z|>{\hsize=1\hsize}Y|>{\hsize=2.2\hsize}Y|}\hline
\textbf{Survey} & \textbf{Year} & \textbf{Main Focus} &\textbf{Details}\\\hline
Pan et~al. \cite{8089336} &
2018 &
IoT applications based on edge, cloud, \& edge computing. &
Brief introduction of challenges \& enabling cloud based technologies for IoT applications: NFV and SDN.\newline[3pt]
Review covers 2009-2016.\\\hline
Akpakwu et~al. \cite{8141874} &
2018 &
5G for IoT: Communication technologies and challenges. &
Limited to IoT application use cases for mobile communications.\newline[3pt]
Briefly introduces SDN \& NFV technologies.\newline[3pt]
Review covers 2002-2017.\\\hline
Cox et~al. \cite{8066287} &
2017 &
SDN advancement survey. &
Discusses SDN state of art \& challenges.\newline[3pt]
Brief discussion on SDN-IoT, NFV, and SDIoT.\newline[3pt]
Review covers 2002-2016.\\\hline
Ngu et~al. \cite{7582463} &
2017 &
IoT Middleware issues and enabling technologies. &
Focuses on middleware with limited discussion on virtualization.\newline[3pt]
Review covers 2003-2016.\\\hline
Bizanis et al. \cite{bizanis2016sdn} &
2016 &
SDN and virtualization for IoT. &
Focuses on SDN and NV in IoT applications, specifically in mobile \& cellular context and limited to 5G \& WSN.\newline[3pt]
Review covers 2009-2016.\\\hline
Khan et al. \cite{7060643} &
2016 &
WSN virtualization. &
Limited to detailed discussion about WSN virtualization, state-of-art, and research issues. IoT is not the main focus.\newline[3pt]
Review covers 2003-2016.\\\hline
This work &
2018 &
IoT virtualization using SDN, NFV, NV, and hybrid SD designs. &
Discusses solutions which are specific to IoT.\newline[3pt]
Literature is covered which utilizes software defined networking (network layer), function virtualization, hypervisors, hybrid NFV and SDN, and software defined Internet of Things.\newline[3pt]
Review covers all literature till 2018.\\\hline
\end{tabularx}
\end{table*}
\subsection{Classification}
In this work we have categorized the IoT virtualization solutions into three main categories, which are then further divided into 3 types of solutions. The main categories, as shown in Figure~\ref{fig:Organogram} are:
\begin{itemize}
\item SDN-based IoT solutions: These solutions only address the virtualization of network layer control (flow management and data transmission).
\item NFV-IoT solution: These solutions are either in combination of SDN or stand alone but focus on individual functions of IoT ecosystem.
\item Software Defined IoT solutions: These are more elaborate and provide broader solutions for IoT.
\end{itemize}
In each category we have grouped the solutions into three types. Some solutions present architectures (with or without implementation), while others are more focused on management of IoT network and devices. The third type are related to security of IoT networks.
\section{Software Defined Network based IoT}
SDN-based IoT is a concept where SDN can facilitate routing efficiency, high data transmission, network management and resource allocation for the IoT devices to meet the growing need of the user demands \cite{Tayyaba:2017:SDN:3102304.3102319}. SDN solutions in IoT environment are expected to resolve traditional network issues \cite{huawei1}, like heterogeneity, interoperability, and scalability among IoT devices, inefficient service deployment (lack of dynamic services), slow adaptation to new services (network upgrade time consumption), and lack of user experience guarantees (minimum bandwidth). To do so, different SDN-based IoT architectures have been proposed in many works until recently.
Commercial solutions such as AR2500 Series \cite{huawei} agile IoT gateways are also available for deployment. In addition to commercial solutions there are numerous proposals and solutions available in academic literature. We classify them into architectural, security, and management solutions. SDN-based IoT architecture deals with clear separation of concern between services provided in the control plane and the data plane. Control plane specifies the management of network traffic and data plane specifies the mechanisms to forward traffic to desired destination. SDN-based IoT management specifies how the applications on top of the Management Layer interacts with the control plane and the coordination among them. It also allows the admin/analyst to define how the control process is to be governed not only by the SDN controller itself but also by human users. SDN-based IoT security specifies different security parameters for access to network, end-point devices, and other control layer elements. It does this by defining security policies for the complete software defined system.
\begin{figure}[!t]
\centering
\includegraphics[width=0.95\linewidth]{IoT_Architecture.eps}
\caption{A generic SDN based architecture for IoT.}
\label{fig:4}
\end{figure}
\subsection{Architecture Solutions}
Works in \cite{IoT_SDN_OFenabled,IoTEcoSysIqbal,salman2015architecture,SIMECA_Iqbal} propose SDN-based cloud platform approaches for IoT network connectivity, \cite{qin2014software,martinezempowering,li2015general,li2016sdn,SDN_IoT_architecture_with_NFV} propose general SDN-based architectures to facilitate the scalability, heterogeneity, and interoperability among IoT devices or nodes, and \cite{6838330} propose SDN-based control plane platform solutions. Figure~\ref{fig:4} depicts the SDN-based IoT architecture. It provides a general overview to show the management plane, control plane, data plane, and perception plane. How the IoT sensors would interact with the data and control plane, is discussed in this section through different research solutions. Table~\ref{tab:TableII} shows these case studies and comparison among different architectures.
\begin{table*}[!t]
\centering
\tiny
\caption{Comparison of different SDN architectures for IoT networks}
\label{tab:TableII}
\setlength\tabcolsep{2.5pt}
\begin{tabularx}{\linewidth}{|>{\hsize=.5\hsize}Y|>{\hsize=0.8\hsize}Y|>{\hsize=1.5\hsize}Y|>{\hsize=.5\hsize}Z|>{\hsize=.6\hsize}Z|>{\hsize=1.65\hsize}Y|>{\hsize=1.45\hsize}Y|}\hline
\textbf {Literature} & \textbf{Objectives} & \textbf{Solutions} & \textbf{Control Plane Arch.} & \textbf{Controller}& \textbf {Benefits} & \textbf {Limitations} \\\hline
Desai et~al. \cite{IoT_SDN_OFenabled} &
Heterogeneity, IoT device to cloud comm. &
OF-enabled management device &
Distributed &
NOX, POX, ODL &
Proposed an OF-enabled management device, which will make network simpler &
Implementation of OF-enabled management device left as future work.\\\hline
Ogrodowczyk et~al. \cite{IoTEcoSysIqbal} &
Scalability, QoS, Reliability, Security &
SDN-based IoT network application working on top of SDN controller.\newline[4pt]
Dynamic creation and management of end-to-end comm. channels among IoT devices and cloud &
Distributed &
Ryu, OpenFlow &
Orchestrated application uses specialized controller for traffic analysis.\newline[4pt]
IoT device categorization, recognition, \& policy enforcement.\newline[4pt]
Real-time data collection, visualization, storage and analysis through automated IoT service deployment. &
Scenario specific solution (smart cities).\newline[4pt]
Each sensor is used by a single tenant only, which limits virtualization at device level.\\\hline
Salman et~al. \cite{salman2015architecture} &
Single solution for multiple challenges: Scalability, Heterogeneity, QoS, Latency, Reliability, Security. &
Centralized SDN control network for IoT with decentralized data management.\newline[4pt]
Layered model: Works in application, control, network, and device layers.\newline[4pt]
Implements SD-gateways in the fog with specialized algorithms. &
Distributed &
SDN controller &
Inter-controller communication.\newline[4pt]
Intelligent fog nodes.\newline[4pt]
SDN controller uses management protocols (i.e. NetConf and Yang, OF-Config \& extended OF).\newline[4pt]
Use of unified application for communication. &
Architecture only. \newline[2pt]
Simulation and implementation is left for future work.\\\hline
Nguyen et~al. \cite{SIMECA_Iqbal} &
Latency, QoS, Overhead, Mobility. &
Services hosted inside edge devices.\newline[4pt]
Packet header translation.\newline[4pt]
Separating end point and routing identity.\newline[4pt]
Lightweight control mechanism. &
Distributed &
SDN controller &
Lightweight solution.\newline[4pt]
Efficient peer-to-peer service abstraction for IoT devices.\newline[4pt]
Reduced signaling \& data overhead.\newline[4pt]
Flexible service deployment \& resource management. &
Controller compatibility with the proposed architecture may become an issue.\\\hline
Qin et~al. \cite{qin2014software} &
Heterogeneity, Interoperability, Scalability , Security, QoS. &
Centralized global view.\newline[4pt]
Heterogeneous devices with various data formats for information modeling.\newline[4pt]
Adaptable network state. &
Centralized &
Layered IoT controller. &
Minimized latency and optimized interoperability \& scalability.\newline[4pt]
Better performance and flow scheduling. &
Limited security and tools for resource provisioning or network control.\\\hline
Li et~al. \cite{li2015general} &
Heterogeneity, Interoperability, Scalability, Availability, Security. &
IoT gateways and SDN switches.\newline[4pt]
Distributed network OS. &
Distributed &
SDN controller &
Distributed OS providing centralized control.\newline[4pt]
Global view of the underlying physical distributed network environment. &
Architecture only. \newline[4pt] No performance evaluation \& implementation available.\\\hline
Li et~al. \cite{li2016sdn} &
Heterogeneity, Interoperability, Latency, Scalability, Reliability, Security. &
SDN gateway/router.\newline
Distributed network OS. &
Distributed &
POX &
Discovering IoT devices from different domains.\newline[2pt]
Real time evaluation for latency in IoT devices \& sensors, using Raspberry Pi. &
No discussion of security mechanisms.\\\hline
Ojo et~al. \cite{SDN_IoT_architecture_with_NFV} &
Heterogeneity, Scalability, Mobility. &
Replacement of traditional gateway with SDN gateway. &
Distributed &
ONOS, ODL &
Improved network efficiency \& agility.\newline[2pt]
SDN-enabled gateway.\newline[2pt]
Intelligent routing protocols \& caching techniques. &
Architecture only.\newline[2pt]
Performance evaluation and implementation left for future work.\\\hline
\end{tabularx}
\end{table*}
Desai \textit{et al.} \cite{IoT_SDN_OFenabled} proposes an architecture where IoT device communication with cloud based processing systems is enabled using SDN. The proposed \emph{management device} structure is designed for a number of different applications, such as smart homes, temperature sensors, etc. It also contains application frameworks for system management, communication drivers, Secure Socket Layer (SSL) and media framework libraries, runtime process, \& virtual machines. SSL is used for data encryption. The respective communications driver, depending on the type of IoT device attempting to establish connection with the OpenFlow-enabled management device, uses appropriate libraries for data encryption and decryption.
The data manger formats the data appropriately for the application layer, and then forwards to the OpenFlow-switch (OF-switch). The OF-switch works in a traditional manner, and consults the forwarding table for packet processing. Once the data reaches the gateway controller, it negotiates with other gateway controllers to determine the destined location where the data should be processed. The destination may be located in the local domain or cloud domain. In case of cloud domain, the data will be sent to the cloud gateway controller from the local gateway controller and is processed, the output is sent to the respective destination based upon negotiations. The output location can be an IoT device which is attached to an OpenFlow-enabled management device. Since the layered architecture is configured in Linux kernel, it can be considered reliable. The authors suggest that implementation or deployment of OpenFlow-enabled management device is expected to be carried out in the future.
Ogrodowczyk \textit{et al.} \cite{IoTEcoSysIqbal} presents an architecture which contains multiple independent IoT ecosystems connected through cloud using SDN infrastructure. The solution is able to generate a global view of all IoT resources using OF Experimenter extensions for auto-detection. Service provisioning is automated by inserting meta-data into the flow information. The solution also proposes a protocol which interfaces between the cloud orchestrator and the OF controller (Ryu). The IoT services are instantiated inside Linus Containers (LCX), which are virtualized isolated Linux systems (containers) controlled by a single kernel. The orchestrated application uses commercial product NoviFlow \cite{NoviFlow}. The Ryu SDN controller \cite{8370397}, in the proposed architecture, is a customized version, rebuilt from scratch in Python.
The solution is evaluated in Poznan Smart City use case. The authors demonstrate the \emph{slicing} of a city into different smart spaces, while connected to a single SDN-based platform. The city wide network is an OF enabled infrastructure integrated with cloud resources, capable of hosting multi-tenant cloud applications for IoT devices. The IoT application was tested with sensors like Libelium \cite{libelium} (i.e. IoT gateway to connect any sensor to any cloud platform), and with the Spirent STC \cite{SpirentSTC} ( i.e. a test emulator to analyze complex traffic pattern). For real-time performance evaluation of a smart city and to scale the entire system, further testing is required to validate the feasibility of using vendor independent sensor devices rather than confined to specific sensors.
Salman \textit{et al.} \cite{salman2015architecture} proposes an architecture, with layered model, for IoT with decentralized data and centralized control. Authors also discuss IoT challenges like scalability, big data, heterogeneity, and security. The proposed four layered model consists of Application, Control, Network, and Device Layer. The architecture uses unique identifiers in device layer that ensures interoperability, security, and quick address. Software Defined Gateways (SD-Gateways), a virtualized abstraction of a common gateway supporting extended OpenFlow protocol to communicate with the SDN controllers, enforces a Genius algorithm \cite{WILLE2015629} as one of the virtual functions on top of it. They are expected to mitigate the IoT challenges, when applied in the Network Layer. Fog nodes (i.e. SD-Gateways) would bridge the communication between IoT devices and the SDN controllers. The authors leave the implementation of SD-Gateways for future work. Control Layer specifies the network orchestration and computation such as collecting the topology data, defining security rules, implementing scheduling algorithms, and computing the forwarding rules with routing algorithms. However, these algorithms have not been addressed in depth in the paper and may possibly be considered as future research directions. Application Layer reveals the use of software functions based on the information provided by the control layer, which is yet to be implemented.
Many works also discuss the migration of traditional IoT network to SDN. Qin \textit{et al.} \cite{qin2014software} discusses MINA (Multi-network Information Architecture), a centralized architecture for heterogeneous nature of IoT. It attempts to address the interoperability challenges with different heterogeneous devices, and exploits various data formats for modeling information. MINA's objective is to minimize latency and optimize interoperability and scalability to improve QoS. A customized Qualnet \cite{QualNet} simulation platform with SDN features based on OpenFlow-like protocol in IP layer is used. It enables effective resource provisioning in IoT multi-networks environment by using Observe-Analyze-Adopt \cite{6838332} loop. It also defines flow scheduling over multi-hop, and heterogeneous ad-hoc paths. It takes advantage of flow matching using heuristic algorithms (i.e. network calculus and genetic algorithm \cite{Seo:2008:GAS:1363686.1364121}) to examine QoS, considering parameters like jitter, delay, and throughput. Its proposed flow scheduling algorithm proves better compared to the existing ones. However, security and availability for sophisticated tools to enable on-the-fly resource provisioning and network control are left for future research work.
Pedro \textit{et al.} \cite{martinezempowering} aims to enhance IoT network by using SDN Controller with an additional IoT Controller. The proposed model tries to integrate SDN and IoT to resolve heterogeneity issue of objects (i.e. IoT devices). The authors analyze the different types of workloads that IoT elements will push to the network, which determines the structure and modularity of IoT Controller. An IoT Controller acts as a functional block, which receives communication interests by the IoT agent installed into the objects, finds the responder in the network graph, uses routing algorithm to calculate the path, builds the forwarding rules based on the nature of protocols holding the object requested, and finally passes such rules to the SDN Controller to be installed on the forwarders (i.e. SDN switches). The advantage of the proposed architecture is that the IoT Controller tends to reduce the workload of the SDN controller but the limitations still may persist, as the nature of routing algorithm is not described. Latency issue to discover objects may also persist, as the author also state that the IoT Controller may sometimes face protocol compatibility issue and hence some rules may need to be handled by the forwarders.
Li \textit{et al.} \cite{li2016sdn} discusses issues like interoperability from the perspective of devices, data, communication protocols, and re-usability of data generated from IoT devices. Moreover, authors suggest resource utilization, openness and interoperability by using a layered architecture which includes Device Layer (responsible for collecting data), Communication Layer (contains SDN enabled switches and gateways), Computing Layer (having SDN Controller), and Service Layer (which provides services). The IoT devices communicates with the SDN gateway/router through sinks, like Raspberry Pi. The gateway/router then forwards the data to the SDN controller. The SDN controller manipulates the data as per the application requirements located at the service layer. This is done by programming the SDN controller. Limitations of this work include sink and sensing devices, which work independently while only responsible for aggregating and caching the data received from IoT devices. Its architecture lacks security mechanisms and routing algorithms both in SDN controller and IoT Gateway.
Nguyen \textit{et al.} \cite{SIMECA_Iqbal} presents a distributed mobile edge-cloud architecture that enables a new network service abstraction called SDN-based IoT Mobile Edge Cloud Architecture (SIMECA). It aims to improve IoT device communication performance, as compared to the Long Term Evolution/Evolved Packet Core (LTE/EPC) architecture. It realizes the abstraction by lightweight control and data planes that significantly reduces signaling and packet header overhead, while supports seamless mobility. Through evaluations with pre-commercial EPC, SIMECA shows promising improvements in data plane overhead, control plane latency, and end-to-end data plane latency, while coordinating large numbers of IoT devices in cellular networks. PhantomNet \cite{PhantomNet} testbed is used for evaluation purpose. Controller information is not available, which may impact the results as different SDN controllers have different features. The proposed system may not be compatible with all SDN controllers due to SDN controller software, interface, and OS compatibility issues. Other issues like heterogeneity, availability, and scalability may also exist from the perspective of physical devices in the network.
Li \textit{et al.} \cite{li2015general} proposes an SDN-based IoT architecture with conceptual virtual functions. It consists of three different layers. Application Layer accommodates IoT servers for various applications and services through APIs, Control Layer accommodates SDN controllers running on distributed OS, and Infrastructure Layer accommodates IoT gateways and SDN switches to enable connections between the SDN controller and IoT devices. It carries different technologies like RFID and sensors using the control plane interface. The benefit of employing distributed OS is that it provides centralized control and global view of underlying physical distributed network environment to process network data forwarding. However, the work presented does not show the performance comparison or real world implementation. The issue of IoT devices is also a concern for this architecture.
Ojo \textit{et al.} \cite{SDN_IoT_architecture_with_NFV} proposes a replacement of traditional IoT gateways, with specialized SDN-enable gateways. These gateways are capable of managing wired \& wireless devices, and claims to be more flexible, efficient, and scalable. Authors also claim that the gateway can perform efficient traffic engineering with intelligent routing protocols and caching techniques across less constrained paths. However, the work is limited in defining intelligent routing algorithms, and performance evaluation or implementation in real time which is considered a future direction.
\textbf{Conclusion:} A number of novel algorithms have been proposed to tackle issue of IoT challenges like heterogeneity, interoperability, latency, security, data manipulation, etc. However, most works only propose the architecture. Real world implementation and experiments are needed to address the performance evaluation. Hence, this is a major research direction for this area. Furthermore, the adaption of existing controllers to IoT is still not completely addressed. Controllers which can seamlessly integrate into access network and can reach devices in the mobile domain, will be necessary to better optimize the IoT ecosystem.
\subsection{Security Solutions}
Traditionally security mechanisms like firewalls, intrusion detection \& prevention system are deployed at the network edge to prevent external attacks. Such mechanisms are no longer enough, considering the dynamic changes in network topology as a result of IoT nodes joining-in and moving-out. As for internal threats, e.g. if an object is corrupted by virus, other uncorrupted objects may also be exposed to threats. Hence, the security parameters for both internal and external threats may need to be reconsidered with the flow of technological advancement.\par
\begin{figure}[!h]
\centering
\includegraphics[width=0.95\linewidth]{SDNIoTSecSol.eps}
\caption{Security solution categorization for SDN-based IoT.}
\label{fig:SDNIoTSecSol}
\end{figure}
The following literature discusses different proposed solutions for SDN-based IoT security issues. Table~\ref{tab:TableIII} shows comparisons among them. We group these works into different categories, as shown in Figure~\ref{fig:SDNIoTSecSol}: communication protocol related vulnerabilities \cite{xu2017defending,sandor2015resilience,hesham2017simplified}, flow-based security issues \cite{bull2016flow,sivanathan2016low}, application layer security issues \cite{sivaraman2015network} \& \cite{HostBasedIntrusion_Iqbal}, architectural security challenges \cite{flauzac2015sdn,gonzalez2016sdn,7785005}, and other attacks and vulnerabilities which expose the network \cite{li2017securing,chakrabarty2015black}.
\textbf{Common Protocol Vulnerabilities.} In an SDN environment, the communication between IoT based devices and servers can be blocked by new flow attacks, that contain a significant amount of unmatched packets injected into routing system. This leads to processing of excessive amount of data packets in both data and control plane, and exhaust either the SDN-enabled switch or the controller or both overloaded with intensive new flows, ultimately cutting off the bridge between IoT devices and IoT servers. To solve this issue, Xu \textit{et al.} \cite{xu2017defending} presents a security framework to defend against such suspicious flow attack for IoT centric OpenFlow switches and SDN controllers. The controller acts as a security middle-ware to filter new-flow vulnerabilities, such as DDoS attack, controller-switch communication flooding, and switch flow table flooding, and uses traditional SDN northbound and southbound interfaces to mitigate them. Both simulation and real-time experiments show feasibility to defend against the cyber attacks although calculation process and result filtering technique still need to be improved to implement in a large-scale scenario.\par
Sandor \textit{et al.} \cite{sandor2015resilience} presents an IoT-based hybrid network framework along with a redundant path switching algorithm using SDN's adjustable routing feature, which would protect against DoS attacks. The architecture is hybrid because it includes SDN switches and non-SDN topology segments that contain both types of Entry Point (EPs) and communication edges. By employing SDN switches, the algorithm (i.e. redundant EP switching logic) executes dynamic switching among different EPs. These SDN switches implementing the forwarding plane of the SDN technology are further controlled by the control plane using OpenFlow protocol. These routing rules may also be received from any external entity (e.g. an application to enforce routing policy). Hence, dynamic traffic switching process takes place between two EPs. The authors undertook experiments to measure the performance of the hybrid architecture which exhibited significant reduction in the effect of DoS attack, hence improving the performance and resilience of the IoT systems.\par
Network access control is a security mechanism which limits the access to authorized devices only. Traditional networks use port-based mechanisms defined by 802.1X~\cite{802.1x_rfc}, for its implementation. Hesham \textit{et al.} \cite{hesham2017simplified}, using SDN technology, presents a novel network access control service for IoT sensor networks and M2M communication by replacing the 802.1X standard based software and hardware. The solution also offers adjustment of available bandwidth and predetermined network access policy for each device, to implement authentication and authorization mechanism. This new device should be able to communicate with the OpenDaylight controller via northbound interface. The entire solution consists of four different steps: authenticate clients, authorize clients, flow installations on SDN controller, and deletion of flows on controllers as soon as clients logs out. The solution also follows two separated policy based databases, termed as the User database and the Policy database. The experiment testbed evaluates the system performance for flow installation delay against a varying number of devices and policies. The primary experiment results show some challenges in flow installation, however, the system is able to successfully authenticate users and register them.
The results may further be improved by using Apache Cassandra which allows thousands of transactions per second and improves scalability, for policy and authentication database. This will be significantly useful when multiple new devices simultaneously connect to the network (i.e. bootstrapping a new subnet).
However, authors also suggested that the authentication and authorization module could have been wrapped-up inside the SDN controller, which would have improved the performance of the system to a great extent, as less flow installation may mean less time consumed to establish device-to-device connectivity. This can be a possible future direction for research community.
\begin{table*}[]
\centering
\caption{Comparison of security solutions using SDN for IoT networks.}
\label{tab:TableIII}
\setlength\tabcolsep{2.5pt}
\begin{tabularx}{\linewidth}{|>{\hsize=.7\hsize}Y|>{\hsize=1.10\hsize}Y|>{\hsize=1.10\hsize}Y|>{\hsize=.7\hsize}Z|>{\hsize=.7\hsize}Z|>{\hsize=2\hsize}Y|>{\hsize=.7\hsize}Z|}\hline
\textbf {Literature} & \textbf{Objectives} & \textbf{Vulnerability} & \textbf{SDN Controller} & \textbf{Switch Type} & \textbf {Implementation \& Evaluation Details} & \textbf{Operational Layer(s)} \\\hline
Xu et~al. \cite{xu2017defending} &
Detection, Mitigation. &
Suspicious flow attack. &
ODL &
OpenvSwitch &
Testbed for attack detection in IoT centric OF switches with OpenDaylight controllers.\newline[4pt]
Novel packet filtering algorithms implemented in Matlab. &
Datalink\\\hline
Sandor et~al. \cite{sandor2015resilience} &
Dynamic switching among redundant entry points. &
DoS attack &
Floodlight &
OpenvSwitch &
SDN-enabled hybrid network infrastructure, using automatic switching, and advanced routing mechanisms. &
Network\\\hline
Hesham et~al. \cite{hesham2017simplified} &
Novel network access control mechanism. &
Unauthorized access to network devices. &
ODL &
Pica8 Switch &
Testbed with in-band topology (merged control \& data plane) to enable connection between clients \& authentication service. &
Datalink\\\hline
Bull et~al. \cite{bull2016flow} &
Detection of anomalous behavior in packet flows. &
TCP flood attack, DoS attack, ICMP based attack on IoT device. &
POX &
OF 1.3 Switch &
Flow monitoring, periodic checking, and flow installation mechanisms to counter TCP flooding and ICMP attacks.\newline[4pt]
Mininet based emulation. &
Datalink\newline[4pt]
Network\\\hline
Sivanathan et~al. \cite{sivanathan2016low} &
Network level monitoring to detect flow-based anonymous packets. &
Self developed two new Python-based emulated attacks. &
SDN controller &
TP-Link SDN-enabled gateway. &
Experimental testbed using C programing. &
Datalink\newline[4pt]
Network\\\hline
Sivaraman et~al. \cite{sivaraman2015network} &
Device Monitoring \& Control. &
Eavesdropping\newline[3pt]
Remote access\newline[3pt]
Privacy\newline[3pt]
Man in the Middle &
Floodlight &
OpenvSwitch &
Prevention mechanisms for suspicious eavesdropping and packet injection attacks in Smart Home appliances. &
Network\\\hline
Nobakht et~al. \cite{HostBasedIntrusion_Iqbal} &
Identify and block attacks. &
Unauthorized access of smart home devices. &
Floodlight &
OF Switch &
Identify suspicious packet flows \& prevent access to Smart Home IoT devices. &
Datalink\\\hline
Flauzac et~al. \cite{flauzac2015sdn} &
Distributed routing.\newline[4pt]
Distributed security rules. &
General security issues. &
Distributed controller &
OF Switch &
Multi-SDN domain access control network architecture\newline[4pt]
Provisioning security for IoT objects (i.e. sensors, smart phone, tablets, etc.). &
Datalink\newline[4pt]
Network\\\hline
Shuhaimi et~al. \cite{7785005} &
Reduced hardware usage.\newline[4pt]
Enhanced security \& privacy. &
3rd party applications\newline[4pt]
Untrusted data\newline[4pt]Privacy &
SDN controller &
OF Switch &
IoT and SDN integrated algorithmic model, to secure attacks from both inside \& outside the domain.&
Datalink\\\hline
Li et~al. \cite{li2017securing} &
Detect Man in the Middle attacks &
TLS vulnerabilities &
Floodlight &
OF 1.3 Switch &
Bloom filters based SDN \& extended OF approach to detect MitM attacks emulated using Mininet. &
Datalink\\\hline
Chakrabarty et~al. \cite{chakrabarty2015black} &
Secure meta-data \& payload within layers.\newline
Privacy\newline
Confidentiality\newline
Integrity\newline
Authentication &
Packet injection\newline[4pt]
Eavesdropping. &
Centralized controller &
OpenvSwitch &
Payload uses novel encryption mechanism.\newline[4pt]
Able to mitigate a wide range of passive and active attacks on IoT net.\newline[4pt]
Uses SDN for routing over multiple topologies.\newline[4pt]
Node sleep and sync. mechanisms. &
Datalink\newline[4pt]
Network\\\hline
\end{tabularx}
\end{table*}
\textbf{Flow-Based Security.} Data flow related challenges of IoT devices and systems have been described by Bull \textit{et al.} \cite{bull2016flow}, where SDN gateways are used in a distributed structure to monitor data traffic and flow characteristics. The authors propose a method to identify and reduce anomalous behavior, claimed from their previous work in \cite{bull2015pre}, add functionality of packet forwarding/blocking, and enhance QoS by the SDN-based IoT gateways. In this approach, to categorize the network state, source and destination flow statistics are collected from the SDN controller. Additionally, the proposed mechanism executes relevant actions (i.e. permit or block traffic flows) to negotiate with the detected anomalous behaviors. The primary results successfully authenticate the approach by showing a small number of attacks being blocked by using this method, although dynamic traffic analysis and hardware based-testbed experiments are reserved for future works.
Sivanathan \textit{et al.} \cite{sivanathan2016low} elaborates the differences between flow-based monitoring approaches and packet-based approaches to prevent vulnerabilities in smart-home IoT devices. Based on the flow-level characterization of IoT traffic, the authors present a system containing SDN-enabled gateway with a cloud-based controller to identify malicious IoT activity in the home network. They propose an analysis engine, Security Management Provider (SMP), that communicates with the SDN controller via northbound APIs to recognize trusted IoT devices at low cost. It requests SDN controller to inspect flows selected by it. The SDN controller then configures home gateway with such rules, referred by the analysis engine, to mirror selected traffic flows towards it. It actively inspects the packet in/out of the IoT device with specific headers and also measures the load of selected flows. Traffic analysis is concluded by stopping the traffic mirroring followed by deletion of pertinent rules inside the home gateway. Traffic flows are managed from the cloud-based software, rather than embedded processing unit of home gateway. Internal and external attacks have been demonstrated in an experimental testbed consisting of real IoT devices to prove that the approach can be effective with minimal cost. However, this method is limited to packet content inspection and plain-text password based attack types. Future research may be carried on flow-level monitoring to mitigate other sophisticated security threats.
\textbf{Application Layer Security Issues.} Usage of SDN in IoT for application specific usecase is very important. This also gives rise to security issue. Sivaraman \textit{et al.} \cite{sivaraman2015network} illustrates that a significant amount of IoT based home appliances such as smart bulbs, motion sensors, smoke alarms, and monitoring/analysis devices, lack basic security functions that may have a negative impact on day-to-day activities. The authors argue that security implementation needs to consider various kind of factors like device capabilities, mode of operation, and manufacturer. They propose a prototype, Security Management Provider (SMP), that can control the access to data on devices, by applying dynamic or fixed content-based policies to identify attacks (e.g. eavesdropping, spoofing, etc.) at the network level. SMP exercises configuration control over the ISP network or home router without being directly on the data path. SMP is invoked via API to provide dynamic/on-demand policy, front-end web interface, static policy via web interface, and OpenFlow capabilities. The solution uses FloodLight controller to configure OpenvSwitch (OVS) and Ruby on Rails as security orchestrator and web-GUI developed in Java script A new module is introduced to the FloodLight controller to implement the API for access control, that works as a wrapper to the FloodLight controller firewall, employing access control policies (based on remote IP). These policies are referred by the external SMP entity for a specific home device. Although, the proposed solution has the potential to block threats at the network level, protecting users' privacy still needs to be addressed in detail with regards to the possible exposure of vital personal data.
Nobakht \textit{et al.} \cite{HostBasedIntrusion_Iqbal} proposes an Intrusion Detection and Mitigation framework (IoT-IDM), providing network-level prevention mechanism against malicious or suspicious ad hoc objects from the external network domain to access Smart Home environment. IoT-IDM users may have enough flexibility to use customized machine learning mechanisms to detect attacks based on learned signature pattern methodology. This framework is realized using SDN technology (i.e. a Java based Floodlight controller) via OpenFlow protocol for remote management purpose and routing efficiency, implemented in real-time using a smart IoT light bulb. However, IoT-IDM works on top of SDN controller, requiring to handle large volume of network traffic. The authors suggest that it is not feasible to use this approach to mitigate the intrusion detection process for all devices, and is applicable to only selected smart home IoT devices.
\textbf{Architectural Security Challenges.} Flauzac \textit{et al.} \cite{flauzac2015sdn} proposes solution which is mainly designed to increase the security of SDN controllers and to solve the scalability issues in multiple IoT-based domains. The work combines wired \& wireless networks, and further extends its solution to ad hoc enabled network and IoT devices like sensors, smart phones, tablets, etc. Each network node acts as a combination of OpenFlow switch and legacy host. Besides, one controller acts as central trusted authority to improve executable security policies while border controllers assist in communication among neighboring IoT domains by establishing communication and exchanging information. However, future work may include the elaboration of management technique of multiple controllers (i.e. security controller, border controller), and inter-SDN controller communication in different layers. It may also include real-time implementation and performance evaluation on how security and border controller may behave and interact among different SDN domains. Security policies may be scrutinized to further enhance access control mechanism.
In a distributed network scenario, Gonzalez \textit{et al.} \cite{gonzalez2016sdn} introduces a proposal that is adequate for an IoT cluster environment, by establishing groups of sensor nodes. OpenFlow and network virtualization technologies have been used for virtual nodes to simulate a distributed cluster based system of 500 devices. Instead of using a traditional approach of the static firewall to block a possible attack, the authors presented an SDN based routing protocol and a dynamic firewall termed as Distributed Smart Firewall that can apply the functionality of an SDN controller. However, the entire framework is not complete as the system is only limited to handle the communications between clusters. Therefore, setting up a dynamic routing protocol along with expanding the simulation to use OpFlex protocol \cite{OpFlexProtocol} functionalities are reserved for future work. Another SDN Controller clustering approach by Shuhaimi \textit{et al.} \cite{7785005}, deals with challenges like availability, heterogeneity, security and privacy in IoT. It also proposes a multi-step novel algorithm, to select SDN Cluster-Head (SDNCH) that works as SDN controller. Its job is not only to manage and control network traffic, but also monitors and prevents the attacks from inside \& outside domains by securing the whole SDNCH domain. It may be considered as a benefit of this proposal, but the work is limited in performance evaluation and implementation. The authors intend to analyze the results from different security attacks such as neighboring attack, black hole, and other related attacks in near future.
\textbf{Miscellaneous Security Challenges.} To detect man in the middle (MitM) attacks in Software-defined IoT-Fog networks, Li \textit{et al.} \cite{li2017securing} proposes a lightweight countermeasure tool. MitM attack is known as one of the common Transport Layer Security (TLS) vulnerabilities \cite{maninthemiddle} and both SDN controller and OpenvSwitch are susceptible to this attack. The authors first demonstrate three different attacks on a simulated environment in Mininet \cite{lantz2010network} using Floodlight controller, and then, by modifying the existing OpenFlow protocol they have proposed a countermeasure to detect these MitM vulnerabilities. The three different attacks are: (i) redirecting flows in the data plane, (ii) exemplifying the attacker's mechanism to collect information from the data plane, and (iii) the attacker's mechanism to infect the controller's view of the network. The most integral part of this tool has been built inside Floodlight controller, so that modules will be loaded automatically during the initialization of the controller. The experiments have shown a significant improvement in performance and detection accuracy of this method, although the number of false positives remains a concern. Passive attacks may also cause damage to the network.
Chakrabarty et al. \cite{chakrabarty2015black} proposes Black SDN to secure SDN-based IoT networks. The Black SDN approach encrypts both payload and packet header at the network layer with the implementation of a single SDN controller that has a global view of the existing network. It also helps to communicate with different resource constrained IoT devices through Black packets. This method can mitigate several passive attacks like inference and traffic analysis attacks and also secures meta data which correlates with each packet or frame of an IoT end-to-end device communication, hence improving payload efficiency. The authors demonstrate the working of Black SDN via simulation using various node states and network topologies, and the achieved results proved effective to defend against many passive attacks. Although, Black SDN provides higher level of security than existing protocols but traffic control may become complicated due to the proposed system's increased communication between the SDN controller and the IoT nodes.
\textbf{Conclusion:} A number of security issues and solutions concerning secure efficient packet routing, monitoring, and corrupt packet prevention and access control mechanisms in different operational layers of SDN based IoT network have been discussed. These prevention mechanisms are mostly developed as an external module to cooperate with the SDN controllers. The research community may focus on possibilities to integrate these modules inside the SDN controllers to achieve enhanced scalability. Efforts may be taken to focus on more real-time evaluation against different threat vectors, which can be helpful in determining the status of the solutions.
\begin{table*}[!t]
\centering
\caption{Comparison of SDN-based solutions for IoT Management.}
\label{tab:TableIV}
\setlength\tabcolsep{2.5pt}
\begin{tabularx}{\linewidth}{|>{\hsize=.6\hsize}Y|>{\hsize=0.9\hsize}Y|>{\hsize=1.4\hsize}Y|>{\hsize=.7\hsize}Z|>{\hsize=.6\hsize}Z|>{\hsize=1.3\hsize}Y|>{\hsize=1.5\hsize}Y|}\hline
\textbf {Literature} & \textbf{Objectives} & \textbf{Solutions} & \textbf{Control~Plane Architecture} & \textbf{Controller}& \textbf {Benefits} & \textbf {Limitations} \\\hline
Hakira et~al. \cite{hakiri2015publish} &
Networking, Mobility, Standardization, Security, QoS. &
IoT architecture combining SDN with message-based publish/subscribe DDS middle-ware. &
Centralized &
SDN controller &
Filtering \& fusion mechanism for efficient traffic engineering. &
Architecture design only.\newline[4pt] No implementation or evaluation.\\\hline
Bera et~al. \cite{bera2016soft} &
Real time working, Flexibility, Simplicity. &
Device \& Topology management. &
Centralized &
Customize controller &
Application-aware service provisioning, \& improved network performance. &
Limited to specific sensor devices.\\\hline
Yiakoumis et~al. \cite{yiakoumis2011slicing} &
Scalability, Reduce latency, Efficient load-balancing. &
Slicing mechanism using Flowvisor for multiple home networks. &
Centralized &
NOX &
Isolating network traffic \& bandwidth.\newline[4pt]
Resource sharing.\newline[4pt]
Cost-effective. &
Architecture lacks compatibility with all applications.\newline[4pt]
Privacy, performance, security, and flexibility may be further improved.\\\hline
Tortonesi et~al. \cite{7543778} &
Network load, Storage \& Cost reduction, D2D communication. &
Information filtering.\newline[4pt]
Prioritization using VoI. &
Centralized &
SDN controller &
Reduced load by information filtering. &
Distributed and disruption tolerant architectures.\newline[4pt]
Efficient information processing functions.\\\hline
Fichera et~al. \cite{fichera2017experimenting} &
Heterogeneity, Scalability. &
Management of data path across IoT, cloud, and edge network. &
Distributed &
ONOS &
Congestion recovery with reliable data delivery. &
Redirection of flows may create delays for time-sensitive mice flows.\\\hline
\end{tabularx}
\end{table*}
\subsection{Management Solutions}
At the existing scale of deployed networks, it is almost impossible to manually configure remote devices. IoT requires that network providers are able to configure and reconfigure devices across the network from a centralized management point. However, this requires the right technology to automate the whole management process. SDN is able to facilitate advanced mechanisms to configure and manage devices (e.g. SDN-enabled switch) across variety of different types of networks. This section discusses different proposed SDN-based IoT management solutions. Table~\ref{tab:TableIV} shows management based comparison of existing SDN-based IoT literature.
Hakiri \textit{et al.} \cite{hakiri2015publish} discusses five key network related challenges of IoT, such as current standardization efforts, mobility management, recurring distributed systems issues, communication protocols, and security \& privacy. They outline an IoT architecture that combines SDN with message-based publish/subscribe Data Distribution Service (DDS) middle-ware to solve variety of issues like networking, mobility, standardization, and QoS (Quality of Service) support. In this framework, smart devices are linked with SDN-based IoT gateways to communicate with SDN forwarding devices. Furthermore, an SDN controller connects to the forwarding devices using southbound APIs allowing asynchronous, anonymous, and many-to-many communication semantics. Within a domain, DDS can provide discovery and communication service between different heterogeneous IoT devices and the controller itself. DDS is utilized in local network whereas SDN is responsible for allowing the connection outside of a local network. A novel SDN-enabled gateway is proposed for smooth handover migration between smart IoT devices in a Wide Area Network (WAN). Future work may be on developing the algorithms for the proposed DDS, defining various communication patterns (i.e. transactional queues for request/response interaction, delivery response, event-based interaction) to publish/subscribe data. Algorithms may also be developed on how to differentiate and prioritize traffic packets.
Bera \textit{et al.} \cite{bera2016soft} proposes leveraging of IoT related application-aware service in Wireless Sensor Network environment. They present an architecture named Soft-WSN that is based on the centralized provisioning of SDN controller. The architecture is divided into three layers: application, control, and infrastructure layer. Application layer generates application specific request to be sent to the SDN controller in the control layer. Control layer has SDN controllers to configure the SDN-enables switches. Control layer has two important entities to assist with policy management. First one is device manager, which deals with device specific control tasks such as scheduling the sensing tasks, sensing delay task, and active-sleep management. Second is topology manager, which deals with network topology control mechanism while focusing on the network connectivity management and forwarding rules. Hence, the topology management system can identify every single node and therefore, it can assist SDN controller to provision according to given configuration policies. The proposed system will be effective for several IoT applications. For example, environment monitoring, traffic monitoring, smart home from both topology and device management perspective. From the experiment results, authors show that Soft-WSN provides better data delivery rate, energy efficiency, and traffic overhead than traditional WSN. However, this method has some compatibility issues with other radio technologies like Bluetooth, and controller placement problem may arise under minimized network delay and the overhead of control messages.
Slicing techniques has always played a key role towards securing and managing a complex network. Network slicing is an effective and powerful virtualization capability. It allows creation of multiple logical networks built on top of a common physical infrastructure. This helps in addressing the efficiency, cost, and flexibility requirement of future networks. Technologies like SDN (through network programmability) and virtualization are the means to realize network slicing. Slices may be optimized in many ways including bandwidth and latency requirements. Usually slices remain isolated from each other in the control and user planes. From the user's perspective they only visualize a single network, regardless of the fact that it may physically be a portion of a layered network. Yiakoumis \textit{et al.} \cite{yiakoumis2011slicing} proposes a prototype where multiple home networks can be sliced and a trustworthy third party can manage whole network using different slicing techniques. Similarly, resources can be shared among multiple service providers to reduce the cost. Authors use FlowVisor \cite{sherwood2009flowvisor} for slicing mechanism in OpenFlow networks, providing bandwidth and traffic isolation. SNMP protocol is used to configure the wireless access-points (such as WiFi, SSID, queues, encryption, etc.), and also to inter-operate with firewalls and NATs in smart home environment (i.e. UDP-in-TCP tunneling). The OpenFlow controller (i.e. NOX) independently controls and manages programmatic control of a slice. It also defines the forwarding logic for a switch (in data plane) to operate. The experiment analyzed Flowvisor which enabled high scalability with low latency, showing efficient load-balancing feedbacks. Future work may include extending OpenFlow protocol to virtualize multiple resources in the proposed scheme: Virtual Device Configuration, Virtual Links, and Virtual Address Space. Moreover, improvement in trade-offs among privacy, performance, security, and flexibility can be future research directions.
SDN technology allows installation and management of communications and computational resources to develop and deploy IoT applications. Sieve, Process, Forward (SPF) by Tortonesi \textit{et al.} \cite{7543778}, is an extended SDN architecture of Open Networking Foundation (ONF). The authors use SPF for information processing, and replacement of data plane with Dissemination plane (i.e. data forwarding plane), and uses a novel SPF-Controller, with Programmable IoT Gateways (PIGs). It uses a solution of data processing (i.e. audio/video analysis, IoT device discovery, tracking \& counting) at the edge of the network rather than in cloud, which reduces high bandwidth usage. SPF architecture has three stakeholders: administrators, service providers, and users. Administrators deploy, run, and operate SPF controllers along with PIGs, allowing the service providers to use it. Service providers develop, deploy, and manage IoT applications. Users may utilize the SPF applications available to them by installing its client app on their smart devices. Moreover, critical information is prioritized by ranking objects (i.e. IoT devices) using Value of Information (VoI) metrics. Future work may include extension of SPF-Controller incorporating interesting functionalities which can extract informations from Twitter, Facebook, Wechat, etc. through mobile devices of customers for data analysis purpose. Moreover, improving information filtering mechanisms of PIGs, utilizing both semantic methodologies and complex event processing, can be done.
A real-time 5G Operating Platform proposed by Fichera \textit{et al.} \cite{fichera2017experimenting}, is able to manage the heterogeneity and scalability of a network. A testbed has been presented in this work for exploiting SDN management capabilities to provide data delivery paths across different network domains under 5G communication. The experiment divides the testbed into IoT-based, cloud-based, and edge networks. To consolidate communication between these environments, an SDN Orchestrator is designed as an application, running on top of an ONOS controller. It is implemented within IoT domains and cloud environment, exploiting network programmability among sensors and Virtualized Functions (VFs), respectively. The real-time 5G operating platform is interlinked to resource infrastructure managers/controllers (i.e. Cloud controller, SDN controller, IoT device manager), lying underneath all the hardware resources (e.g. SDN switch, gateways). Service Orchestrator deals with cloud, SDN, and IoT Orchestrator followed by the respective resource infrastructure manager/controllers.
SDN controller configures routing policies on flow tables for SDN switch, to enable end-to-end IoT device communication. An SDN Orchestrator is able to recover congestion events (e.g. service outages or degradation events) through the traversable path that has been redirected towards those switches or links that rely on constant monitoring of throughput data. Cloud, SDN, and IoT Orchestrator(s) rely on Service Orchestrator. It is responsible for invocation of services through intent-based interfaces and infrastructure service abstractions. Experimented results show that redirected operation took less time, although, packet dropping at congested switch may tend to degrade the real-time assured services of the proposed scheme.
\textbf{Conclusion:} Most of the SDN-based management solutions available deal with data distribution data services, topology management, home network slicing, and resource management. Some of the directions which can be further explored are synchronization \& compatibility of IoT devices. APIs for such services can improve heterogeneity in the IoT ecosystem.
\section{Network Function Virtualization for IoT}
Network Function Virtualization and SDN are complimentary technologies. They do not require or are dependent on each other, but rather improve and facilitate each other's working. NFV provides a collection of virtual applications referred to as Virtual Network Functions (VNFs). These can include processes for deep packet inspection (DPI), routing, security, and traffic management, which can be combined to provide network services specialized for IoT. A hybrid SDN/NFV architecture for IoT, given in Figure~\ref{fig:5}, shows a general interaction of SDN and NFV to provision reliable communication and to facilitate IoT platforms. The architecture is composed of Network Function Virtualization Infrastructure (NFVI), Virtual Network Functions (VNFs), and Management and Orchestration (MANO) plane, leveraging each other to achieve sustainable network virtualization, with uninterrupted network connectivity, and enforcing efficient packet flow rules by the SDN controller. Different components of this architecture are detailed below:
\textbf{Network Function Virtualization Infrastructure}: It consists of all of the networking hardware and software resources required to connect and support carrier network. These resources include operating systems, hypervisors, servers, virtual switches, Virtual Machines (VMs), Virtual Infrastructure Managers (VIMs), and any other virtual and physical assets enabling NFV.
\textbf{Virtual Network Functions}: VNF focuses on network service optimization. It is responsible for managing specific network function that executes on one or multiple VMs. These VMs work on top of physical hardware resources (i.e. switches, router, etc.). Virtual function for routing, firewall, load balancing, Intrusion Prevention System (IPS), etc. defining unified policy for virtualized hardware resources is adopted into a single VNF. In this way, multiple VNFs may be linked together. This linking can form a service chain managed by VNF manager and VIM, respectively.
\textbf{Management and Orchestration Plane}: MANO facilitates connection of services of different modules of NFVI, VNF, and APIs from the Management Plane, and coordinates with the respective subcomponents in MANO plane.
\begin{itemize}
\item{NFV Orchestrator:} NFVO works concurrently with VNFM and VIM, standardizing the functions of virtual networking and enhancing the interoperability of IoT devices. It binds together different functions like service orchestration, coordinating, authorizing, releasing, and engaging NFVI resources, to build an end-to-end resource coordinated service in a dispersed NFV environment.
\item{VNF Manager:} All VNF instances are associated with VNFM. Its operations include initiation, scaling, updating and/or upgrading, and termination of VNFs.
\item{Virtual Infrastructure Manager:} Network hardware resources like IoT gateways, SDNvSwitches, routers, etc., are abstracted through the virtualization layer using VIM. It keeps allocation inventory of virtual and hardware resources, and manages VNF forwarding graphs, security group policies, hardware resources in a multi-domain environment or optimize them for a specific NFVI environment.
\end{itemize}
The rest of the section presents architectural, management, and security solutions of NFV for IoT.
\begin{figure}[!t]
\centering
\includegraphics[width=0.95\linewidth]{NFV_for_IoT_Architecture1.eps}
\caption{A General SDN-IoT Architecture with NFV.}
\label{fig:5}
\end{figure}
\begin{table*}[]
\centering
\caption{Network Function Virtualization Solutions for IoT networks.}
\label{tab:TableV}
\tiny
\setlength\tabcolsep{2.5pt}
\begin{tabularx}{\linewidth}{|>{\hsize=.45\hsize}Y|>{\hsize=1.25\hsize}Y|>{\hsize=1.35\hsize}Y|>{\hsize=.5\hsize}Z|>{\hsize=.75\hsize}Y|>{\hsize=1.7\hsize}Y|>{\hsize=1\hsize}Y|}\hline
\textbf {Literature} & \textbf{Objective(s)} & \textbf{Solution(s)} & \textbf{Control Plane Arch.} & \textbf{Controller \& Switch} & \textbf {Implementation, Evaluation, Benefits} & \textbf {Limitation(s)}\\\hline
Jie Li et~al. \cite{li2015general} \newline[2pt]$\bigcirc$ $\bigtriangleup$ &
Routing, Access control, Security, Traffic control, Virtualization. &
SDN-based IoT framework with NFV implementation. &
Centralized &
SDN controller \& switches with IoT gateways. &
Distributed OS.\newline
Performance, scalability, availability, and security are enhanced due to virtualization. &
Limited to the study of general SDN \& NFV architecture.\\\hline
Ojo et~al. \cite{SDN_IoT_architecture_with_NFV} \newline[4pt] $\bigcirc$ $\bigtriangleup$ &
Interoperability, Device discovery, Scalability, Security, Efficiency and management flexibility.\newline[4pt]
Application specific requirement provisioning. &
An SDN-IoT architecture with NFV implementation. &
- &
SDN controller\newline[4pt]
Virtualized IoT gateways &
Enhanced performance \& management of hardware, software, \& virtual resources.\newline[4pt]
Device discovery with enhanced connectivity. &
Scalability issues still persists due to overloading of data traffic.\\\hline
Batalle et~al. \cite{batalle2013implementation} \newline[4pt]$\bigstar$ &
Efficient routing.\newline[4pt]
Cost effective deployment. &
Resolves CAPEX issues in IoT. &
Centralized &
SDN controller &
Efficient inter-domain routing.\newline[4pt]
Less connected \& deployed devices, hence cost-effective. &
Latency\\\hline
Du et~al. \cite{du2016context} \newline[4pt] $\bigcirc$ $\bigtriangleup$ &
Security \& privacy.\newline[4pt]
Cost effective \& cheaper IoT \& MVNO.\newline[4pt]
Value-added services for MVNOs.\newline[4pt]
Multi-MVNO networks. &
Context-aware processing/forwarding of IoT traffic.\newline[4pt]
Contextual info. recvd. from sensor-layer and application-layer.\newline[4pt]
Bridges gap between IP \& IoT network, using SDN and NFV. &
Centralized &
Central service controller.\newline[4pt]
IoT gateways.\newline[4pt]
MVNO switch. &
IoT framework deployable in current Internet.\newline[4pt]
Cost effective business model for MVNO use in IoT.\newline[4pt]
Programmable MVNO IoT gateways (using Edison \cite{Edison} board).\newline[4pt]
Trailer-slicing for IoT networks. &
Proposed arch. may not become a unified IoT platform.\\\hline
Balon et~al. \cite{balon2012mobile} \newline[2pt] $\bigstar$ $\bigtriangleup$ &
Costs effective.\newline[4pt]
Study cost-benefit analysis of MVNO, MNO, \& security measures.&
MVNO based arch. evolution and economic stakes. &
- &
- &
Business model suggesting sharing of info among operators to reduce cost. &
Limited to business model.\newline[4pt]
No implementation.\\\hline
Vilalta et~al. \cite{vilalta2016end} \newline[4pt] $\bigcirc$ $\bigstar$ &
Low cost IoT.\newline[4pt]
Enhanced scalability \& interoperability. &
An SDN/NFV-enabled edge node, which orchestrates end-to-end SDN IoT services. &
Distributed &
SDN controller\newline[4pt]
IoT gateways\newline[4pt]
OF-switches &
ODL \& OpenStack Nova/Havanna service controller.\newline[4pt]
GMPLS controlled optical network.\newline[4pt]
Multi domain network architecture.\newline[4pt]
Optimized packet response time. &
Not a unified IoT platform.\\\hline
Salman et~al. \cite{salman2015edge} \newline[4pt] $\bigcirc$ $\bigtriangleup$ &
High level management capabilities.\newline[4pt]
Low latency \& Heterogeneity.\newline[4pt]
Mobility using fog computing. &
Edge computing enabling the IoT. &
Distributed &
ODL, Onix \& ONOS controllers\newline[4pt]
SD Fog gateways\newline[4pt]
SD-MEC WHAT IS MEC HERE?\newline[4pt]
OF-switches &
Supports multiple identification and comm. technologies.\newline[4pt]
Multiple SD fog gateways ensure interoperability.\newline[4pt]
Centralization leading to security enhancement to some extent.HOW IS IT CENTRALIZED?\newline[4pt]
Ensures fine-grained flow services using FlowVisor or OpenVirtex. &
Scalability.\newline[4pt]
Infrastructure enhancements exposed to third party causing security vulnerabilities.\\\hline
Maksymuk et~al. \cite{maksymyuk2017iot} \newline[4pt] $\bigcirc$ $\bigstar$ &
Scalability.\newline[4pt]
Efficient interoperability \& traffic engineering. &
Framework for monitor IoT devices in SD 5G networks. &
Centralized &
SDN controllers &
Architecture based on independent IoT system \& shared by multiple MNO.\newline[4pt]
Upgraded MNO parameters to include carrier freq., node velocity, cell ID, etc.\newline[4pt]
use of MQTT to customize monitoring system.\newline[4pt]
Low traffic overhead. &
Not a unified IoT platform.\newline[4pt]
Third party service involvement may cause security threats.\\\hline
Zhang et~al. \cite{Zhang:2016:OPH:2940147.2940155} \newline[4pt] $\bigstar$ &
Efficiency \& Scalability. &
Dynamic manipulation of packets using NFs in docker container. &
Distributed &
SDN controller &
NF-Lib facilitates fast deployment of NFs.\newline[4pt]
Improved scalability. &
Third party library functions may pose to security threats.\\\hline
Massonet et~al. \cite{8114476} \newline[4pt] $\bigtriangleup$ &
Enhance security. &
NFV/SFC approach. WHAT IS SFC? &
Distributed &
SDN controller &
Integrated federated agent in IoT network controller \& gateway.\newline[4pt]
Security VNF within the federated IoT-cloud. &
Limited to architecture design only.\newline[4pt]
Implementation and evaluation left for future work.\\\hline
Al-Shaboti et~al. \cite{8432333} \newline[4pt] $\bigtriangleup$ &
Enhanced security \& latency. &
IPv4 NFV-based ARP server providing security against ARP spoofing \& network scanning. &
Centralized &
Ryu controller &
NFV dispatcher for packet inspection.\newline[4pt]
Secure ARP operations through NFV-based ARP server.\newline[4pt]
Not dependent on mapping between the host \& the port.\newline[4pt]
Both WiFi \& Ethernet port is usable simultaneously.\newline[4pt]
Reduces packet processing delay.&
Focus only ARP attacks.\newline[4pt]
IPv6 for IoT is not considered.\\\hline
\multicolumn{6}{>{\hsize=\dimexpr7\hsize+7\tabcolsep+\arrayrulewidth\relax}X}{$\bigcirc$ $\bigstar$ and $\bigtriangleup$ represent architecture, management and security based solutions respectively.}\\
\end{tabularx}
\end{table*}
\subsection{Architectural solutions of NFV for IoT}
In this sub-section we review architectural solutions proposed in literature for function virtualization in IoT environments. Many of these solutions are hybrid SDN/NFV solutions, which take advantage of each other's capabilities.\par
Li \textit{et al.} \cite{li2015general} proposes one such architecture following a top-down approach. It is divided into application layer (e.g. services like Operation Support System/Business Support System), control layer (i.e. SDN controller with distributed operating system), and infrastructure layer (i.e. IoT switches and gateways). The primary objective is to employ SD \& NFV to meet the IoT challenges, such as heterogeneity, scalability, security, and interoperability. The proposed SDN-based IoT architecture with NFV implementation, can provide a centralized control, and virtualize different IoT services in healthcare, transport, education, etc. The proposal only discusses architectural details of how these services may be realized, and leaves out implementation of methodologies. The authors intend to study the organization and components of each part of SDN/NFV-based IoT framework as a future direction.
Ojo \textit{et al.} \cite{SDN_IoT_architecture_with_NFV} presents an IoT framework based on virtualized elements in an SDN-enabled system. They utilize VNFs for a number of purposes, which are deployed on SDN/NFV edge nodes. By using these edge devices the framework is able to provide services such as, rich user context (location information), low latency, high bandwidth guarantees, and rapid IoT device deployment.
The MANO plane orchestrates control of the network infrastructure and the different network functions through respective managers. It also interacts with the management plane applications to obtain policy and configuration information, and with SDN controllers for communication and network services. The overall architecture is quite similar to the one depicted in Figure~\ref{fig:5}. The SDN elements are logically separate from the NFV layers, and some of the functions of SDN are performed in the NFVI. NFV can also be used to relocate some of the IoT gateway functionality into virtual gateways, which will allow greater scalability, easier mobility management, and faster deployment. Although, theoretically the models proposed in this work are sound, there is no implementation or evaluation available to realize the system. Authors have left it as future direct, which can be taken by the research community to integrate SDN and NFV for IoT.
Du \textit{et al.} \cite{du2016context} focuses on prototyping context-aware forwarding/processing mechanism that can manage IoT traffic depending on contextual information. These contextual information is distributed from both sensor-layer and application-layer to mitigate the challenges of IP-based network and IoT network. These issues are related to scalability, discoverability, security, and reliability, mostly due to computation and battery power limitations. The proposal focuses on software-defined data plane defining novel services for Mobile Virtual Network Operators (MVNOs), which offers network services to customers at low prices by means of obtaining network services from Mobile Network Operators (MNOs), without requiring to have their own wireless network infrastructure. The architecture incorporates programmable MVNO switch on multi-core processors and IoT gateways with Edison \cite{Edison} board. The authors focused at high security and privacy mechanism, performance optimization, and value added service in the IoT-MVNO domain. The MVNO switch collects data from the sensors (e.g. smart watches, wearable glasses) via IoT gateways and sends it to the logical service controller for data processing. Several isolated MVNO networks are associated with different applications to work simultaneously. The architecture uses OpenFlow protocol to communicate between the MVNO switch and IoT application through southbound interface. The MVNO switch is built on FLARE \cite{FLARE} testbed equipped with multi-core network processor. IoT Gateway software ensures trailer slicing on FLARE platform, serving functionalities like IoT device discovery and connectivity, data collection and encapsulation, and context-aware packet forwarding/processing. The simulation output shows significantly high rate of data transmission with low bandwidth and efficient routing. The architecture is also realized as an effective business model for IoT application based on MVNO network to make it highly cost-effective. Future plans include a contextual IoT trailer architecture for a unified IoT platform on top of current Internet protocols.
Balon \textit{et al.} \cite{balon2012mobile} proposes a model for robust security and network performance management. They show a usecase to build a private virtualized MVNO, which can easily be expanded and scaled for high volume traffic and number of user. They also discuss the different components and enablers of MVNO networks and provide a cost-benefit analysis of using MVNO. However, the paper only discusses architecture and market analysis, but does not give details on implementation using the MNO services.
Vilalta \textit{et al.} \cite{vilalta2016end} proposes an SDN-based NFV edge node. The proposed edge node adopts OpenFlow-enabled switch, controlled by edge SDN controller. It also provides storage resources, and computing services via edge cloud/fog controller. The OpenStack Nova handles the NFV framework through the Cloud/Fog Network orchestrator, which has two different orchestrators running below it: (i) Cloud/Fog orchestrator, which deals with the edge cloud \& metro controllers, and (ii) Multi-domain SDN orchestrator, which deals with edge SDN \& DC SDN controllers. This entire orchestration consolidates NFV and SDN together to provide seamless network connectivity between deployed VMs to virtual switch at the edge node or in DC. The IoT gateway acts as the client which requests computing and storage services to the SDN/NFV edge node. Multi-domain SDN orchestrator simulates OpenDayLight and OpenStack Nova to provide end-to-end network services. Eventually, data from IoT gateway flows to the processing resources, which are located in the proposed SDN/NFV edge node. The proposed approach is only limited to the edge nodes and DC. The packet response time is considerably low between the IoT Gateway and edge node VM or core DC VM, which optimized the edge resource usage.
Another similar approach towards edge networking is done by Salman \textit{et al.} \cite{salman2015edge} that presents a fog computing architecture termed as Software-Defined Mobile Edge Computing (SD-MEC) for integrating Mobile Edge Computing (MEC) with IoT, SDN, and NFV. SD-MEC is a four-layer architecture that includes an application layer, a control layer, a device layer and a network layer. All of these layers initiate different tasks for the orchestration of the proposed fog services. In this framework, Software Defined Function (SDF) Gateway plays an essential role. It acts as an inter-operator between the various communication protocols and heterogeneous networks, presenting high management capability, rendered from the SDN features, and also offering heterogeneity abstraction, low latency, and mobility support from the fog devices. Applying the NFV features further facilitates management at network level required in the MEC platforms. However, this work only gives conceptual information regarding Fog architectures and for a specific use case scenario. The real time implementation and performance evaluation to ensure the effectiveness of the architecture proposed, is reserved for future work.
\textbf{Conclusion:} The works presented in this section are mainly architectures only, focusing on scalability of IoT networks and reduction in processing/communication overhead. Implementation and evaluation are two key elements missing from these solutions. Similarly, coupling of SDN and synchronization of different VFs with orchestrator and control layer could lead to improvement in deployment of VNFs in IoT.
\subsection{Management of IoT using NFV}
This sub-section reviews management specific literature for function virtualization in IoT environments. Some of the solutions uses SDN technology besides NFV.
Batalle \textit{et al.} \cite{batalle2013implementation} integrates NFV and SDN to reduce cost in IoT, where centralized controller is responsible for routing which has a global view of the network. This work presents a novel design of a virtualized routing protocol using NFV infrastructure. It simply manages and reduces signaling overhead, particularly when inter-domain routing is required. The NFV implementation for virtualization of the routing function is done over an OpenFlow network. It aims to also reduce the number of connected and deployed devices, hence will reduce the cost as well. Just like OpenFlow, packet is inspected and if required, it is sent to the Floodlight controller which then takes decision after inspecting whether packet belongs to IPv4 or IPv6. Proposed solution is implemented using GEANT \cite{GEANT}, that offers infrastructure to emulate OpenFlow-based SDN solutions. As the amount of communication increases, the proposed solution is able to reduce the number of flow entries by 50\%, which improves performance and scalability. But to improve the robustness of the virtualized function, more evaluation are expected. The experiment leads to a number of open research questions, starting from implementation of dynamic routing protocols in the virtualized host, to different routing policy optimization.\par
Maksymyuk \textit{et al.} \cite{maksymyuk2017iot} adopts IoT-based network monitoring framework to manage the performance of 5G heterogeneous networks under different conditions. In this architecture, Radio Access Network functionalities are virtualized using NFV to simplify load balancing and spectrum allocation. On the other hand, the centralized intelligence of SDN controller is used to implement interference aware spectrum allocation. This allows better load balancing of smaller cells and manages user's mobility. This proposed framework has two main advantages. Firstly, only relevant data will be subscribed by each network operator that can improve the existing monitoring system. It also supports multiple Mobile Network Operators (MNOs). Secondly, the small size of transmittable data block generates less traffic overhead.
Zhang \textit{et al.} \cite{Zhang:2016:OPH:2940147.2940155} proposes an extension to OpenNetVM using Network Function (NF) management module that manages on-demand NFs in lightweight Docker containers. This is to facilitate various service providers, leveraging startup duration and memory consumption of CPUs. OpenNetVM supports flexible and high performance NFV architecture for a smart IoT platform, enabling increased interoperability among NFs. NF management module is an efficient and scalable packet processing architecture that enables dynamic manipulation of packets using service chains. The simulation result shows significantly high rate of throughput for packet transmission leveraging Data Development Kit (DPDK) \cite{DPDK} to improve performance I/O. This creates scope to render complex software based services for deep analysis within the network and data centers. This work may also remove the limitation of managing large volumes of IoT devices to some extent. However, on-demand NF deployment is limited to CPU cores.
\textbf{Conclusion:} In all the efforts mentioned in this subsection, third party services are involved to manage and facilitate the network topology. Usage of SDN controllers may also include management services from vendor specific organizations. Research community may work on developing SDN/NFV-based advanced real-time applications to manage and orchestrate IoT nodes in the context of knowledge-based 5G mobile networks.
\subsection{NFV-based Security solutions for IoT}
This sub-section presents different security solutions, which use NFV to implement security in IoT.
Massonet \textit{et al.} \cite{8114476} proposes an extended federated cloud networking architecture for edge networks and connected IoT device security. The security solution utilizes lightweight virtual functions and Service Function Chaining (SFC). The IoT gateways in the edge networks are responsible for implementing global security policy, by creating a chain of VFs for different purposes, such as, firewall and intrusion detection. They monitor the IoT devices for vulnerabilities and attacks, and isolate the device if it is detected.
SFC is also responsible for flow management within the IoT network and with the federated cloud, which requires the cloud and IoT platforms to have appropriate infrastructure to support it. This is achieved by implementing a \emph{federation agent} at IoT controller or gateway level. The communication itself is done using REST API.
The federated network manager sends configuration information to IoT network Controller, which is then forwarded to gateways for implementation. Finally, the network controller exchanges information with the IoT proxy, which helps manage the data plane using OpenFlow protocol. To secure the IoT-Cloud network slices, a module is implemented inside the IoT network controller. Future work may incorporate enhancing scalability among IoT devices, and algorithms to preserve strong security \& privacy in the edge IoT network.
Al-Shaboti \textit{et al.} \cite{8432333} proposes novel IPv4 address resolution protocol (ARP) server providing NFV security service to defend against ARP spoofing attack, and network scanning. The work also proposes an SDN-based architecture for enforcing network static and dynamic access control of smart home IoT. All ARP requests pass through a virtualized trusted entity called ARP server. It is able to secure all ARP operations, eliminating the ARP broadcast messages, and easily legitimates ARP spoofing through ARP proxy by configuring the ARP server. The work resolves packet processing delay problem using high-speed packet processing technology. Such technologies include deep packet inspection (DPI), multi-core processor, carrier-grade operating system with Linux, and virtualization enabling the sharing of cores between applications. Only NFV-IoT related contribution is focused here. The design architecture includes local components like data plane, NFV dispatcher, and local security services. Security agent, as one of remote components, takes input from user control plane, IoT policy manager, security services, and configures the SDN (Ryu) controller to enforce the corresponding network access control rules. NFV dispatcher receives all mirrored packets relaying from mirror port. Then forwards them to the corresponding security service based on the dispatcher list. Security agents extracts related information to direct security services for each flow. Based on the examination, security decisions/alerts are generated. IPv4 ARP server validation shows that it can protect ARP spoofing, and corresponding data plane deployment kit (DPDK) implementation performs well for the smart home IoT network. Future work will extend incorporating intrusion detection and prevention system into this architecture, and include IPv6 as a key enabler for IoTs.\par
\textbf{Conclusion:} NFV or SDN domains have different elements, applications, orchestration managers, virtual functions, communication APIs, etc. A malicious or compromised element in any of them may have serious effects on the whole system. For example, a malicious VNF by a compromised software vendor, a compromised hypervisor, or MANO component, could harm the entire network domain. If these elements are well secured than integrity, confidentiality, availability, access control, and accountability can be well preserved. Research work on access control \& packet inspection mechanisms, needs further investigation, specially for resource constrained IoT devices.
\section{Software-Defined Internet of Things (SDIoT)}
In this work we classify SDN-based IoT solutions and SDIoT solutions as two separate categories, with different scopes. SDIoT extends the Software-Defined (SD) approach to collect and aggregate data from network devices, sensor platforms, and cloud platforms. It uses sensing applications to provide standard API services for data acquisition, transmission, and processing. The SDN technology provides packet flow configuration for network devices enhancing network connectivity, hence SDN-based IoT is limited to network layer virtualization. NFV implementation extends the network connectivity and security. The basic idea is to virtualize key NFs, and place them on commodity servers. Next step is to connect them via a flexible SD infrastructure managed through a unified orchestration system. For optimization, service provisioning, scalability, performance enhancement, and rapid deployment, the whole IoT ecosystem can be virtualized by SD paradigm. Hence, SDIoT solutions are not limited for a specific layer, but ranges from device up to application.\par
The difference between SDN-based IoT architecture (Figure~\ref{fig:SDN_vs_SD}a) is very subtle but significant as compared to those of SDIoT (Figure~\ref{fig:SDN_vs_SD}b). Control layer is improved by customizing more domain-specific SD-controllers, each executing specific tasks within SDIoT architecture. This reduces the burden on single controller. The control layer is extended not only horizontally, but also vertically. Hence, the function virtualization orchestrator becomes an integrated part of control layer. Protocols/APIs for SDIoT framework varies upon the nature of communication, and the type of IoT devices connected to it. A widely used OpenFlow protocol already exists to communicate between SD-controller and OF-switch, but it needs to extend it's capabilities to communicate with IoT devices beyond virtual switches. Application and management layer communicates with the connectivity layer through the NBI. This layer can also have a management specific framework, which can enforce different policies through the programmable interface for SD-controllers to execute. This framework can also enable different virtual functions at different layers of SDIoT network for groups of different nodes. SD-controllers enforce different policies. Enforcement of these policies is pushed through virtual functions from multiple controllers. For example, SD-Security and SD-Management controller enforce security policies and management related policies, respectively. OpenStack controller orchestrates network slicing. SDN controller configures OF-switch to install data flows, best routing paths, and network control. More controllers can be assigned based on the nature of network management objectives and network performance. Orchestrator is responsible for configuring different SD-controllers on-demand, not only along the horizontal control plane but also vertically. SDIoT controller enforces IoT device specific management rules. It works in collaboration with the orchestrator and other SD-controllers to enhance the communication of perception layer with the control layer via connectivity layer. It eventually enables seamless end-to-end IoT device communication in an SDIoT environment.
The following sub-sections present different architectural, management, and security solutions exploiting different SD-controllers. Table~\ref{tab:TableVI} summarizes \& categorizes SDIoT architecture, management, and security related literature.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{SDN_SDIoTArch1.eps}
\caption{Difference between SDN-based IoT \& SDIoT architectures. Left figure shows the generic SDN based IoT framework, while right side shows a complete software define IoT design.}
\label{fig:SDN_vs_SD}
\end{figure}
\subsection{Architecture Solutions}
This section discusses SDIoT architectural solutions, using multiple controllers for providing different services. Remote configuration of networks and efficient data retrieval has been one of the core challenges of big data analytics for smart cities. Few efforts have been done to use SD and IoT potential to counter these issues.
Din \textit{et al.} \cite{8110222} proposes an SDIoT architecture, which consists of data collection \& management controller. The data passes through Data Processing Layer, Data Management Layer, and Application Layer. The architecture uses multiple SD-controllers/SD-gateways. Data is collected through a novel data collection algorithm, from various IoT-enabled embedded devices. The aggregated data, via various Aggregator Points (i.e. Zone, Local, and Global), is passed on to Data Processing \& Management layers, for real-time data processing and extraction. Since IoT devices generates large volumes of data, the proposed system utilizes Hadoop Distributed File System for data storage \& manipulation purpose. The work contributes by inserting a novel data processing algorithm setting threshold limit values for every data set. The work also uses Information Centric Network \cite{7986623} and Named Data Network \cite{8308426} potentials to fulfill its requirements. The simulation result shows promising aspects. HDFS works significantly well analyzing data with high throughput and less processing time, even though the throughput and processing time may still be improved using cluster based Hadoop system with efficient scheduling mechanisms.
Liu \textit{et al.} \cite{liu2015software} proposes an SDIoT architecture to separate smart urban sensing applications from the existing physical infrastructure, because most of the underlying network element (e.g. sensor nodes) are not SDN-enabled. The control logic of these devices is encapsulated in hardware. The authors divide the entire framework into three layers, i.e. physical infrastructure layer (sensors, smart phones,gateways,etc.), control layer (SD controllers), and application layer (IoT applications). SD controllers are used to manage specific configuration for each hardware resource and provide interface to standard API services for data manipulation. Each of these controllers can be replicated to enhance its robustness and can be physically placed anywhere for resource usage optimization. Basically, data is first aggregated in the sensor platform and passed on to the network infrastructure to calculate the best routing path for end-to-end IoT device data transmission, using SDN-enabled networks. Every sensor platform in SDIoT architecture is facilitated with more than one sensors of similar or different types and shared by many applications. Sensor controller has the global view of the underlying physical infrastructure and capable of activating/deactivating sensors dynamically.The forwarding devices are OpenFlow-enabled and programmable, and the SDN controllers are responsible for scheduling packet flow tables for forwarding devices, and smart traffic steering. Hence, optimizing network resource usage. On the other hand, cloud platform allows urban sensing data to be stored and processed. Cloud controller monitors and maps the underlying server resource pools. Although the architectural design is supported with case studies and qualitative investigations only, it shows promising possibilities to improve network resource utilization as well as dynamic data optimization, processing, and transmission. Future work may focus on controller inter-communication and resource utilization.\par
By applying the fundamental SD features like centralization, virtualization, optimization, another similar approach is taken by Xu \textit{et al.} \cite{xu2016toward}. They proposed an IoT-based software defined Smart Home (SDSH). It supports openness, virtualization, and centralization, integrating the heterogeneous network devices in smart home domain. The entire platform has been divided into three main layers namely controller layer, intelligent hardware layer, and external service layer. The controller acts as a management layer providing compatibility and API support to different smart devices and third party services, respectively. The APIs through the SDN controllers from the control layer handle the communication and interaction between the peripheral IoT devices. These APIs are also responsible for IoT device registration based on their specifications. The architecture also uses virtualization technology to maintain uniform virtual abstraction of hardware computing, storage, and network resources of the whole smart home ecosystem. In addition, it uses virtual network function for access control mechanisms, firewall, load balancing, etc. The literature only reviews key technologies and challenges of SDSH. Although the overall architecture shows promising aspects, simulation or real-time experiment should be carried out in the future to prove effectiveness of the solution.\par
Hu \textit{et al.} \cite{7324414} proposes a dynamic controllable solution for Software Defined Industrial IoT (SDIIoT) with SDN features in it. The solution emphasizes on application specific holistic performance approach of network nodes like field devices, gateways, and sensor cloud in respect to connectivity and interoperability. The proposed architecture has three different network building blocks: IIoT sensor cloud, IIoT gateway, and IIoT field device. The control plane is responsible for configuring these network nodes, and uses different controllers for it. QoS controller enforces QoS policies for the network backbone and field WSN. Network controller handles topology management and data updates. Timeliness is dealt by the data synchronization controller, and security controller enforces security schemes for these network nodes. An additional Data Manger module provides data management services, and control module implements control plane functions. The authors uses Floodlight controller for configuring open virtual switch via IIoT gateway for best routing paths during real-time data transmission, then compare their results with Amazon AWS and sensor cloud server. They show that latency can be reduced by 30\% to 38\%. Moreover, system is reliable and success rate is 100\% because of QoS mechanism in CoAP protocols. Future work can explore and exploit the SDIIoT from big data perspective employing problem specific networking techniques.\par
Wan \textit{et al.} \cite{7467436} proposes a Software-Defined Industrial Internet of Things (SDIIoT) architecture utilizing SDN technology and industrial cloud. The architecture consists of three layers. Physical layer consists of various kinds of hardware devices such as sensor, gateway, switch, router, etc. Control layer manages the physical infrastructure underlying it. It includes SD-controller and SDN controller for processing specific tasks and configure the switches/gateways within its network via SBI. NBI allows applications from the application layer to implement decisions on SD-Controllers based on industry enterprise needs. Application layer also provides different kinds of APIs to monitor equipment fault and usage, and product processing. The proposed SDIIoT architecture also provide three major services: data collection, data transmission, and data processing. Data collection is processed through the application layer APIs, where data sensed is transmitted either via wireless or wired networks. Data processing occurs simultaneously to process one or multiple IoT devices. Decision making is autonomous while the data processing is software defined.
As the system would deal with large scale big data, the SDIIoT service mechanisms require high-quality data process mechanisms/algorithms, which the authors aim to develop in future. The authors also provides security suggestions related to illegal access, vulnerabilities caused due to IoT device mobility and large number of sensor/IoT nodes, which are potential directions for research community.
\textbf{Conclusion:} The contributions in this sub-section include IoT/IIoT concepts with SD features. The solutions mainly focused on incorporating APIs in the application layer to enforce decision rules on SD-Controllers and to exploit network virtualization features, eventually providing global view of IoT nodes beyond virtual switches. Future work may focus on integration of VFs specialized for different controllers and their distributed placement in the network. Moreover, the distribution of different controllers in the networks may improve performance and reduce the communication latency with IoT devices. In this regard, inter controller communication may also require further improvement and standardization.
\begin{table*}[]
\centering
\caption{Software Defined IoT Solutions and their Classification.}
\label{tab:TableVI}
\setlength\tabcolsep{2.5pt}
\begin{tabularx}{\linewidth}{|>{\hsize=0.6\hsize}Y|>{\hsize=0.7\hsize}Y|>{\hsize=1.2\hsize}Y|>{\hsize=0.7\hsize}Z|>{\hsize=1.5\hsize}Y|>{\hsize=1.3\hsize}Y|}\hline
\textbf{Literature \& Classification} & \textbf{Objective(s)} & \textbf{Solution} & \textbf{* Control Plane Architecture} & \textbf{Benefit(s)} & \textbf{Limitation(s) / Future Work} \\\hline
Din et~al. \cite{8110222} \newline[4pt]\textit{Architecture} &
Data sensing, collection, \& processing.\newline[4pt]
Scalability \& availability. &
SDIoT architecture to analyze data of smart cities. &
Distributed &
Uses a Hadoop ecosystem for load balancing.\newline[4pt]
Data collection done using SDN and NDN. &
Complex scheduling algos needed for cluster based Hadoop systems.\\\hline
Liu et~al. \cite{liu2015software} \newline[4pt]\textit{Architecture} &
Sensing \& robustness &
SDIoT architecture for smart urban sensing. &
Distributed &
Dynamic data optimization, processing, and transmission. &
Multiple application configuration persists on shared sensor platform.\\\hline
Xu et~al. \cite{xu2016toward} \newline[4pt]\textit{Architecture} &
Scalability, Mobility, Openness. &
Smart Home IoT device integration with with SDN-based services. &
Centralized &
Virtualization to simplify heterogeneity \& complexity of diff. SDSH protocols. &
Architectural design only. No implementation or evaluation.\\\hline
Hu et~al. \cite{7324414} \newline[4pt]\textit{Architecture} &
Reliability, scalability, security, \& QoS. &
SD-IIoT architecture to manage data exchange and delay. &
Distributed\newline[4pt] \textit{(FloodLight)} &
Application specific approach for node performance, connectivity, \& interoperability.\newline[4pt]
Focus on network controllability: processing, queuing, transmission, and delays. &
Optimization for more than 10 parallel connections not possible.\\\hline
Wan et~al. \cite{7467436} \newline[4pt]\textit{Architecture} &
Reliability, standardization, \& security. &
SD-IIoT architecture for seamless data processing. &
Distributed &
SD-data collection,transmission, \& processing mechanisms.\newline[4pt]
Provides solution for: illegal access, and vulnerabilities caused by IoT device mobility, \& large crowds of IoT nodes. &
Limited evaluation of the proposed solution.\\\hline
Nastic et~al. \cite{6984208} \newline[4pt]\textit{Management} &
Configuration, operation, and access control of cloud system. &
Fleet management system using SDIoT cloud. &
Distributed &
Overall resource usage optimization.\newline[4pt]
Elastic policy based configuration.\newline[4pt]
Cost awareness. &
Limited implementation \& evaluation.\newline[4pt]
Research on run-time SDIoT governance, \& edge network resource usage required.\\\hline
Kathiravelu et~al. \cite{Kathiravelu:2015:CMP:2836127.2836132} \newline[4pt]\textit{Management} &
Sensing, security, \& scalability. &
A middleware solution for context-aware smart buildings using SD WSN. &
Centralized &
Avoids single point of failure.\newline[4pt]
Fast response to dynamic changes. &
Prototype is limited to single building.\\\hline
Wu et~al. \cite{wu2015ubiflow} \newline[4pt]\textit{Management} &
Scalability \& reliability.\newline[4pt]
Mobility. &
Distributed overlay structure to support mobility management, and dynamic flow control. &
Distributed\newline[4pt]\textit{(FloodLight)} &
Mobility management, Handover optimization, and Distributed control. &
Flow-scheduling optimization issues concerning backbone network.\\\hline
Jararweh et~al. \cite{jararweh2015sdiot} \newline[4pt]\textit{Management} & Scalability, Heterogeneity, Agile, \& Inexpensive. &
SD solution for IoT to forward, store, \& secure data. &
Distributed\newline[4pt]\textit{(Multiple SDN controllers)} &
Multiple SD application modules to facilitate IoT network. &
Architectural design only. No implementation or evaluation.\\\hline
Salman et~al. \cite{SDIoTSecurity_Iqbal} \newline[4pt]\textit{Security} &
Security, Privacy, \& Connectivity. &
Security solution for SDIoT utilizing SDN \& NFV technologies. &
Distributed &
Slicing techniques.\newline[4pt]
Cloud based edge computing.\newline[4pt]
Low latency, high throughput \& scalability with location awareness. &
Inter-access controller connectivity challenges.\\\hline
Darabesh et~al. \cite{SDSecurityExpFramework_Iqbal} \newline[4pt]\textit{Security} &
Enhanced security and reduced cost of security cost operations. &
SD-security solution. &
Centralized &
Virtualized SD-security elements: host, switch, and controller.\newline[4pt]
Context-aware security solution.\newline[4pt]
Supports security component configuration. &
Traffic overhead optimization challenges.\\\hline
\multicolumn{6}{>{\hsize=\dimexpr6\hsize+6\tabcolsep+\arrayrulewidth\relax}X}{* Literature either does not mention any controller or assumes generic controller.}\\
\end{tabularx}
\end{table*}
\subsection{Management Solutions}
Managing and configuring a diverse range of IoT devices can be a challenging task. In order to reap benefits of network programmability and efficient resource utilizations a few works have focused on SD-IoT management solutions.
The work done by Nastic \textit{et al.} \cite{6984208} applies SD in IoT, where they try to abstract the IoT resources in cloud by encapsulating them in software defined APIs. The proposed system directly interacts with the underlying physical IoT infrastructure. The main component in the system is the SD-gateway which implements predefined algorithms specified for tracking vehicles utilizing cloud. The objectives are to provision configuration, access, and operation of IoT cloud systems for a unified view. The authors use a vehicle fleet management system as a usecase. The architecture presents fundamental building blocks of SDIoT cloud systems by automating provisioning processes and supporting configuration models, eventually trying to make simple and flexible customization for IoT cloud for operation managers. On the other hand, exchange of raw IoT data in cloud needs a lot of computational resources and bandwidth. The future plan is to consider techniques and mechanisms to support runtime governance of SDIoT systems, enable SDIoT to optimize resource usage of edge networking, and allowing policy based automation of security and data-quality of SDIoT systems.\par
Kathiravelu \textit{et al.} \cite{Kathiravelu:2015:CMP:2836127.2836132} proposes an architecture for Software Defined Building (SDB) \cite{6224230}, using smart clusters. This enables communication among IoT appliances within a multi-building campus. SDB is a platform to enhance the programmability and re-usability of IoT appliances. It also uses Software Defined Sensor Network \cite{6324377} to manage communication mechanisms between sensors \& IoT appliances, and system policy implementation. The prototype CASSOWARY is partially Software Defined because it works on top of traditional SDN environment. It has a two layer architecture. Network layer has control and data plane, whereas, appliance layer manages the integration of smart appliances. The addition of IoT device SD-Controllers to the SDN controller, allows fast response to dynamic changes. Sensors and IoT nodes are connected to the SDN-enabled switches in the data plane. Different controllers deployed are physically distributed in a cluster, which avoids a single point of failure. The message broker in the control plane assists SDN controller to distribute flow information and orchestrate the smart appliances and sensors.
A full scale deployment over real world scenario is complex and authors have left it for future. This may also include energy efficient and access control mechanisms for different smart devices.
Instead of using completely centralized controllers in the IoT based urban mobile networks, Wu \textit{et al.} \cite{wu2015ubiflow} introduces a distributed overlay structure to support ubiquitous mobility management and dynamic flow control where the entire SDIoT network topology is divided into different geographic chunks or clusters. Using a distributed hashing algorithm, each controller is assigned to a single IoT platform to solve the scalability problem. The authors focus on logical centralization of controllers while they are physically placed at different locations. An orchestration controller is used to communicate with local controllers. All controllers are OpenFlow compatible, coordinating with the mobility management of each mobile sensor platform. As the mobile sensor platform finds the gateway managed by one of new local controllers on the move, it sends the event details to the orchestration controller. It then coordinates with the initial/original controller and the local new controller, to provide smooth handover mechanism. The authors emphasize that this process of mobility management supports SDIoT paradigm. Moreover, a unique distributed protocol is used among these controllers independently to handle the single point of failure. However, for backbone network, further provisioning of flow-scheduling optimization is considered as future work.
To address the needs of heterogeneous nature of IoT applications and objects, Jararweh \textit{et al.} \cite{jararweh2015sdiot} proposes a comprehensive SDIoT framework, to provide an improvement over IoT management layer. This model enhances several important aspects like security, storage, and traffic forwarding. It has three main components. First, physical layer deals with all physical devices like sensors, servers, switches/routers and security hardware. Second, control layer is the core of the proposed prototype to manage and coordinate among different SD-controllers, i.e. IoT controllers, SDN controllers, SDStore controllers, and SDSec controllers to abstract the management and control operations from the underlying physical infrastructure. Each of these SD-controllers run specific tasks in the control layer. Third, application layer through NBIs combines fine-grained user applications to facilitate access control and data storage mechanisms. The administrator is able to configure them through the Eastbound Interface and the inter-controller communication may occur using the Westbound APIs. Additional controllers can be added to tackle sophisticated load balancing and inconsistency issues and to deliver fast response time for many requests within the network. The authors in this prototype only exploit the ideas of SDN, SDStore, and SDSec to build the architecture but these SD-controllers' detail functional elements is planned to be developed in the future.
\textbf{Conclusion:} Most of the research contributions mainly focus on extending APIs in the application layer to enforce decision rules on SD-Controllers, SD-gateways, and to exploit network virtualization features. However, in presence of multi-vendor solutions at application, control, and data layers, standardization for communication interfaces (NBI) becomes very critical. Moreover, functionalities specific to IoT devices should also be part of overall management architecture such as mobility and resource management, etc.
\subsection{Security Solutions}
Enforcing policies for security and access control in large scale networks, can be made easier by programmable interfaces. An efficient solution can be defined by adding a dedicated security controller in the SD infrastructure for IoT network. Below we discuss solutions, which have focused on a similar concept.
Salman \textit{et al.} \cite{SDIoTSecurity_Iqbal} discusses the IoT requirements in terms of security and privacy. In addition, an IoT Software Defined security framework is proposed where software defined and virtual function technologies are used for virtualization, and further slicing techniques are used for isolation of the network, blended along with Mobile Edge Computing (MEC). They provide cloud based edge computing services at the user’s proximity, which gives valuable benefits such as location awareness, low latency, augmented reality, high throughput \& scalability, etc. The architecture consists of six layers. There are two types of controllers. Core controller acts as a global network OS, while access controller provides a dynamic control model to support IoT device communication. They are located in the core network and access network, respectively. The devices in the data plane are connected with access points. Each IoT device after registration is provided with initial credentials, and assigned a security level depending on it's capabilities (computation, storage, energy), which affects its authentication. The scheme has Authentication and Authorization Delegation requests passing through different layers based upon IoT device type. IoT devices are also tested using the Automated Validation of Internet Security Protocols and Applications (AVISPA) tool, which uses High Level Protocols Specification Language (HLPSL), a high-level role-based language for security protocol description. Authors have evaluated against only few back-end attack modules to test the security goals. As the overall system is very complex, the evaluation is limited. Future work on inter-controller connectivity and seamless integration of modules may enhance the system.
Darabseh \textit{et al.} \cite{SDSecurityExpFramework_Iqbal} addresses the challenges of providing multiple levels of protection and efficiency in an SD environment. They propose a centralized yet flexible security solution by abstracting the security mechanisms from the hardware to a software layer, providing virtualized testbed environment for Software Defined Security (SDSec) systems, grouped under Software Defined System (SDSys). The system is software defined because the SDSec\_Controller (i.e. software-based POX controller) has the ability to manage diverse data place resources regardless of their vendors. The framework uses Mininet simulator to create a virtual environment for emulating different forms of SDSec policies and test their performance under different scenarios.
The core customized components are SDSec\_Host, SDSec\_Switch, and SDSec\_Controller. The framework uses Catbird \cite{catbird} for policy deployment on hardware assets. It is completely software-based, and unrestricted to hardware, easy to scale, and can adapt it to new changes. It is able to create on-demand new VMs, without the need for manual intervention. Hence, SDSec imports Catbird into its security framework to virtualize the security functions, and to increase the discoverability of the problems and abnormal actions \& activities.
The authors aim to extend the SDSecurity by developing a distributed controller which may reduce the overhead and enhance performance. More security controls inside each SD controller can be further added in future.
\textbf{Conclusion:} The solution presented here mainly focused on preventing DDoS attacks, access \& congestion control mechanisms, and slicing techniques. Future work may include developing APIs that may interact simultaneously through the NBI, SBI, and E/WBI, in order to enforce security decisions \& rules to the SD controllers/gateways. It may also be able to exploit network virtualization features to assign VNFs for preventing different threat vectors. The solutions should be able to defend against a wide range of threats, providing global view of IoT nodes beyond virtual switches.
\section{Future Research Directions}
The fundamental objective of this article is to collect, categorize, and analyze different software defined and virtual function solutions for Internet of Things. From this analysis, we have identified key challenges and possible research directions in this domain. The summaries of these are given at the end of each sub-section in the paper, however, in this section, we elaborate them in greater depth. The success of SDIoT requires improvement in different layers of the overall system, hence we classify them accordingly.
\textbf{Application Layer:} This is the top most layer, and mainly responsible for user/administrator interaction, and other generic application models for enforcing and configuring different policies in lower structure. Some of the core research directions are as follows:
\begin{itemize}
\item In past SDN applications have been written specific to certain functionalities and only for specific controllers. This creates a major bottleneck as there is no standardized application development framework available. Such a framework will be highly beneficial for both research community to build test applications and also for industry in rapid deployment of SD-IoT networks.
\item Similarly existing controllers mostly use REST API for communication with application layer. Unlike SBI there has been little to no effort in standardizing NBI. This effort will certainly be an important step towards widespread adoption and development. An important research element in this regard is to allow diversified application and controller capabilities. As applications and controllers are both specific to different functionalities in SD-IoT framework, hence the standardized NBI must be flexible enough to accommodate different types of communications.
\item Most of the focus with regards to security has been on flow security and attacks on networks. Hence security policy enforcement has been extensively studied. However, security of application modules is also as important as the prior. A compromised application module can mis-configure and severely compromise an whole SD-IoT ecosystem.
\end{itemize}
\textbf{Control Layer:} This is the main focus area of SD-IoT and will require major research and contribution efforts. It is not possible to list all potential directions, hence, we list the major concerns for control layer in SD-IoT.
\begin{itemize}
\item Communication among different elements of control layer is an important aspect. In traditional software defined network, the controllers independently controlled a domain, and those in hierarchy would use proprietary methods of inter-controller communication. However, in SD-IoT there are multiple types of controllers for some domain, which rely on each other for complete working. In addition, the control layer may use multi-vendor solutions, hence a standardized interface is an important research direction. Communication with other domain’s controller may also be investigated for efficient \& optimized communication. \cite{yin2012} did an interesting work in this regard, which may be a starting point.
\item In traditional SDN, single point of failure of control layer is avoided by back-up controller. However, due to diversity in SD-IoT controllers, having a back-up controller of each controller may require investigation for deployment costs and complexity. This also impacts scalability, hence more novel architecture for controller redundancy may provide better solutions.
\item SBI is a major element of control layer. OF has been a defacto interface for network controller to data plane communication. In light of diverse SD controllers, suitability of OF may need reevaluation. SBI which can effectively work for all types of controllers and devices will be another interesting direction. At the same time, the SBI should be able to reach IoT devices beyond vSwitches. OF does not connect hosts, but only allows flow installation on switches. In an IoT networks the mobile devices and AP may also need configuration and other policy enforcements. This requires enhancements to OF or new SBIs which can reach beyond vSwitches.
\item Efficient use of network function virtualization is also a key research direction. Function chaining for various controller processes may enhance the performance, and allow better control in network. As the vertical control layer implements VNFs, their orchestration with the horizontal controllers is also an open research challenge.
\item In addition to other control layer challenges, security of control layers itself, its elements, and communication is extremely important. The security controllers should not only focus on security of data plane and network devices, but should also ensure logical element security. Research in this direction will have a major impact on SD-IoT networks.
\end{itemize}
\textbf{Controller perspectives:} Controller Perspective: SD-IoT will consist of a number of controllers designed for specific operations. This will allow a number of research directions to be explored.
\begin{itemize}
\item Placement of a controller or other control layer elements is a less researched area, mainly due to a single network controller. In SD-IoT networks, the number of controllers and topological structure of IoT devices may require a more close look at the placement in topology for different controllers.
\item IoT networks will compromise of hundreds of devices (if not thousands) in a single SD domain. Hence the scalability of controllers is an important factor. This will include not only scalable architectures, but also languages, thereby capabilities, storage, and processing at controllers. As there are multiple types of controllers, hence, scalability and coupling at a large scale will be very interesting research direction.
\item Synchronization of controllers and their policies will also be an interesting challenge. Furthermore, it will be equally interesting to evaluate the requirements of domain and then utilize only those types of SD controllers which are required. Vertical versus horizontal deployment of controllers and associated VNFs may also present interesting design options.
\item Controller virtualization is an important element in software defined IoT systems. Virtualizing multiple controllers and coordination among them is a challenging task. Similarly, placement of virtualized elements in core or edge network will be an interesting research issue.
\end{itemize}
\textbf{Management Perspective:} The nature and properties of IoT networks has highlighted some newer research challenges, which were not evident in traditional SDNs. In a complete SD-IoT system, these will require significant attention from research community.
\begin{itemize}
\item \emph{Mobility:} In a single SD-IoT domain there may be multiple edge networks, with dozens of diverse mobile IoT devices with high mobility and limited resources. Some solutions have tried to address mobility in SDNs, however, in a hybrid adhoc-infrastructure environment with different physical layer technologies, it will present new research dimensions. Efficient and quick topology discovery in mobile domain, path configuration, hand-over and other scalability challenges should be further investigated.
\item \emph{Device configuration:} The edge and access network in an SD-IoT network will comprise of heterogeneous mobile devices. A major challenge is to configure them according to policy dictated by the application layer. This also requires significant research before a unified framework can be developed.
\item \emph{Virtual functions:} Virtualization of different network functions will be an integral part of SD-IoT ecosystem. Hence, their management in control lance, distribution, virtualization, and integration with other layers \& APIs is a major research area.
\end{itemize}
\textbf{Technology Interaction \& Complexity: } Most of the previous challenges \& directions also deal with complexity of overall architecture, but the research community needs to look at integration of other technologies in the overall ecosystem, such as fog/edge computing, cloud computing, crowd sensing, Blockchain, etc.
\begin{itemize}
\item Crowd sourcing techniques can benefit extensively from SDIoT networks. The functions for task advertisement, auction, bidding, and offloading can be easily implemented through virtual functions, and orchestrated by a crowd sourcing controller placed at the edge node. Such an architecture, can enable rapid deployment of sourcing tasks and collection of data. However, this will certainly require further research in the specific controller design, virtualization of such controllers, and security among other challenges. This will also increase the complexity of overall SDIoT frame work, hence requiring more scalable systems.
\item Blockchain is a relatively new area for IoT, but may prove to be extremely beneficial in financial transactions and other private Blockchain trades. Potential research directions may involve virtualization of complete peers/mines, offloading of complex mathematical functions \& proof of work to other nodes via virtual functions, virtualization of Blockchain ledger, etc. SDIoT may also pave the way for hybrid Blockchains for Internet of Things.
\end{itemize}
\section{Conclusion}
Software defined networks have seen extensive deployment in data centers and core networks, where they have been mostly used for flow optimization and related policies. The recent advancement in Internet of Things has created a keen interest of research community as well as industry to integrate SDN in IoT networks. Similarly, the virtualization in terms of networks, functions, and devices has also seen significant contributions in recent past. In this article, we have reviewed both SDN and virtualization techniques for IoT, and classified them into different types of solutions. SDN is limited to virtualizing the network layer of the stack where the IoT network traffic flow is optimized. Mostly the solutions aim at providing SDN services to resources constrained devices, provide configuration services, or address security threats. Some works have involved function virtualization to implement common network functions in logical domain. An important factor to note is the future of IoT will not only be limited to SDN and isolated virtual functions. The later part of the paper emphasizes on software defined IoT, which is a comprehensive solution, by incorporating controllers for different purposes in the control layer. This also integrates orchestration of virtual functions, as part of the vertical control layer. Additionally, we have presented a number of future research directs in this regard.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2019-03-01T02:09:56",
"yymm": "1902",
"arxiv_id": "1902.10910",
"language": "en",
"url": "https://arxiv.org/abs/1902.10910"
}
|
\section{Introduction}
\subsection{Background}
Zero-knowledge systems are interactive proof systems that the prover (with unlimited computational power) can convince the verifier (with polynomial-time computational power) of the correctness of the claim without leakage of information. By ``without leakage'', we mean that there exists a polynomial-time simulator whose output is indistinguishable from the communication of the interactive proof system. Depending on the definition of the indistinguishability, there are several variants of zero-knowledge: perfect/statistical/computational zero-knowledge, that is, the output of the simulator is identical/statistically close/computational indistinguishable to that of the interactive proof system. It has been a central notion since it was proposed by Goldwasser, Micali, and Rackoff \cite{GMR}. Though it seems impossible, Graph Isomorphism \cite{GMW} and Non Quadratic Residue \cite{GMR} have perfect zero-knowledge systems, for example.
Multi-prover interactive proof systems are introduced in the relation to zero-knowledge, which is also a central notion in computational theory and cryptography. Multi-prover interactive proof systems are interactive proof systems with many provers, who cannot communicate with each other during the protocols \cite{Bab,BGKW,GMR}. Each prover has to reply to the verifier without the information about communications between the other provers and the verifier, and this restricts malicious strategies by the provers. The class of languages that have multi-prover interactive proofs is denoted as $\mathsf{MIP}$. The computational equivalence $\mathsf{MIP=NEXP}$ was proved by Babai, Fortnow and Lund \cite{BFL}, and hence $\mathsf{IP}$, which is the class of languages that have single-prover interactive proofs ($\mathsf{=PSPACE}$ \cite{Sh}), and $\mathsf{MIP}$ are different unless $\mathsf{PSPACE=NEXP}$. Under the statistical or perfect zero-knowledge condition, multi-prover interactive proofs are much stronger than single-prover interactive proofs. Zero-knowledge proofs of NP with a single-prover need the assumption that one-way functions exist \cite{OW}, but multi-prover interactive proofs do not need any computational assumptions. Even multi-prover interactive proofs with perfect zero-knowledge can compute all languages in $\mathsf{MIP}$ \cite{BGKW}.
In quantum complexity theory, quantum analogues of these interactive proof systems have been also deeply studied. Quantum interactive proof system, where the prover and the verifier use quantum communication and the verifier can do polynomial-time quantum computations, were introduced by Watrous \cite{W0}. Quantum statistical zero-knowledge systems were also introduced by Watrous \cite{W1}. An important notion of zero-knowledge systems is the honest zero-knowledge, that is, an interactive proof system is honest zero-knowledge if it leaks no information only to the honest verifier who follows the specified protocol. Restricting the verifier to be honest makes the construction of the simulator easier, but the equivalence between general zero-knowledge and honest zero-knowledge is not obvious. This equivalence was proved by Watrous for statistical zero-knowledge by using a quantum statistical zero-knowledge hard problem \cite{W2}. Kobayashi proved this equivalence for perfect/statistical/computational zero-knowledge by the direct construction of simulators \cite{Ko}. Multi-prover interactive proof systems with entangled provers and the complexity class $\mathsf{MIP^*}$ were introduced by Cleve et al. \cite{CHTW}. $\mathsf {MIP^*}$ is radically different from classical $\mathsf{MIP}$ due to the non-locality. Quantum multi-prover interactive proofs, where the verifier is also quantum, were introduced by Kobayashi and Matsumoto \cite{KM}, and the corresponding class is denoted by $\sf QMIP^*$.
Despite the importance of quantum zero-knowledge and quantum multi-prover interactive proof systems, there is little research about quantum multi-prover interactive proof systems with zero-knowledge. Recently Chiesa et al. \cite{CFGS} proved that $\mathrm{MIP^*}$ protocols with perfect zero-knowledge can compute $\mathsf{NEXP}$, which is the only literature that analyzed the power of $\mathsf{MIP^*}$ with zero-knowledge as far as the author knows.
\subsection{Results}
In this paper, we investigate complexity-theoretical properties of $\rm QMIP^*$ protocols with zero-knowledge, different from $\rm MIP^*$ protocols with zero-knowledge in \cite{CFGS}. The first result is the elimination of the honest condition of quantum multi-prover zero-knowledge systems ($\rm QMZK^*$) without any assumptions.
\begin{theo}[Informal version of Theorem 4]
Any language L computed by honest zero-knowledge quantum multi-prover interactive proof systems can be computed by general zero-knowledge quantum multi-prover interactive proof systems.
\end{theo}
We analyze the public coin protocols of honest verifier quantum multi-prover zero-knowledge systems, which is the analogue of the public coin $\rm{QMIP^*}\ systems$ \cite{KKMV}, since we can convert any honest-verifier $\rm QMZK^*$ protocol into such a restricted protocol. Comparing the systems in \cite{KKMV}, the new difficulty for the multi-prover zero-knowledge case is that the malicious verifier may change messages for each prover even if the honest verifier only sends the same one bit to all provers. The main ingredient in our proof is what we call the GHZ test: it replaces the coin in the public coin protocol of $\sf{QMIP^*}$ \cite{KKMV} by a GHZ state shared by the provers.
We analyze mainly computational quantum zero-knowledge systems, but this result also holds for statistical quantum zero-knowledge systems. In the main text, we add one prover to eliminate the honest condition, but this addition is not necessary. We prove that the honest condition can be eliminated without increasing the number of the provers combining our GHZ test with the rewinding of \cite{Ko,W2} in Appendix.
The second result is the construction of computational quantum zero-knowledge systems for $\sf QMIP^*$ with natural computational assumptions.
\begin{theo}
Let $p$ be any positive integer. If there exists an unconditionally binding and ]computational hiding bit commitment scheme, every language computed by quantum interactive proof systems with $p$ provers has a computational quantum zero-knowledge interactive proof system with $p+1$ provers.
\end{theo}
The main new tool is the Local Hamiltonian based Interactive Protocol (called the LHI protocol in this paper). Local checking of the history of the computation is a very important tool in complexity theory. The most famous example in quantum complexity theory is the Local Hamiltonian problem for $\sf QMA$ \cite{KSV}, which is an analogue of SAT for NP. The LHI protocol is a variation of local checking of the history of the computation for $\rm QMIP^*$ protocols. Recently local checking of the history states of $\rm QMIP^*$ protocols was studied by Ji \cite{Ji1} and Fitzsimons et al.\cite{FJSY}. Their purpose is to construct efficient $\rm MIP^*$ protocols. On the other hand, the verifier in our protocol directly handles quantum history states.
In the LHI protocol, what the verifier does consists of only sending one message to provers and one measurement on the state received from provers by one local Hamiltonian. Then we apply the technique of quantum zero-knowledge proof for $\sf{QMA}$ by Broadbent, Ji, Song and Watrous \cite{BJSW} to the LHI protocols. In their protocol for the Local Hamiltonian problem, the honest prover has to encode the witness of the Local Hamiltonian problem, to send the commitment of the encoding key, and to convince the verifier that the output of the verifier on the encoded witness and the commitment correspond to yes instances by the zero-knowledge protocol for NP. The process of encoding, committing, and convincing the verifier of the correctness of the witness for the Local Hamiltonian problem can be also applied to encode and check the history state in the multi-prover interactive proof.
The protocol also needs the replacement of coins by GHZ states which is used to prove our first result.
Chiesa et al. \cite{CFGS} constructed the perfect zero-knowledge $\mathrm{MIP^*}$ protocol for $\mathsf{NEXP}$. The upper bound of $\mathsf{MIP^*(=QMIP^*}$\cite{RUV}), however, is not obvious and $\mathsf{QMIP^*}$ may not be in $\mathsf{NEXP}$. Hence their result does not imply the existence of zero-knowledge protocols for $\mathsf{QMIP^*}$. On the other hand, our protocol assumes the existence of some computational tools and our protocol is only the computational quantum zero-knowledge system. Chiesa et al.'s protocol, however, is perfect zero-knowledge. Hence our result is incomparable to the result of \cite{CFGS}.
The organization of this paper is as follows. Section 2 is the preliminary section, especially the definition of quantum multi-prover zero-knowledge is here. In Section 3, we prove the elimination of the honest condition. In Section 4, we construct the zero-knowledge protocol for $\sf QMIP^*$. In Section 5, we discuss some open problems. In Appendix, we prove the additional result that any honest $\rm QMZK^*$ protocol can be converted into a general $\rm QMZK^*$ protocol without increasing of the number of provers.
\section{Preliminaries}
We assume that the reader is familiar with basic quantum computation \cite{KSV,NS1} and quantum interactive proof systems \cite{VW}. Our main results heavily use results from \cite{BJSW,KKMV}.
\subsection{Notations}
For a string $x$, $|x|$ denotes the length of $x$. For integers $n,l$, define $[n]=\{1,2,...,n\}$, $[n,l]=\{n,n+1,...,l\}$. In this paper, $p$ denotes the number of provers, and $m$ denotes the number of turns in interactive proof systems. Both are the polynomial size of $|x|$. $poly$ denotes some polynomial in $|x|$. $exp$ denotes some exponential of a polynomial in $|x|$. Usually $c$ is a completeness parameter and $s$ is a soundness parameter in interactive proof systems. A function $\epsilon(|x|)$ is negligible if for any polynomial $g(|x|)$, $\epsilon(|x|)<1/g(|x|)$ holds for any sufficiently large $|x|$. For a vector $|\psi\rangle$, $\||\psi\rangle\|$ denotes the norm of $|\psi\rangle$. For density matrices $\rho,\sigma$, $\|\rho-\sigma\|$ denotes the trace distance between $\rho$ and $\sigma$. $|+\rangle,|-\rangle$ denote $\frac{1}{\sqrt 2}(|0\rangle+|1\rangle),\frac{1}{\sqrt 2}(|0\rangle-|1\rangle)$. The measurement by a binary-outcome POVM $\{M,\mathrm{Id}-M\}$ is often called as the measurement by $M$. In particular, if $M=|\psi\rangle\langle\psi|$ for a state $|\psi\rangle$, the measurement is called as the projection onto $|\psi\rangle$.
\subsection{Universal gate and Clifford Hamiltonians}
The circuit of the verifier consists of tensor products of two Hadamard gates and controlled phase gates, $\{H\otimes H,\Lambda(P)\}$. The set $\{H,\Lambda(P)\}$ is universal, and hence $\{H\otimes H,\Lambda(P)\}$ is also universal. This setting is necessary to apply the result of \cite{BJSW}. They used restricted forms of Hamiltonians, called Clifford Hamiltonians. Clifford Hamiltonians are Hamiltonians realized by Clifford operations, followed by a standard basis measurement
\subsection{$\mathrm{QMIP^*}$ systems}\label{subsectionQMIP*}
A quantum $p$-prover interactive proof system consists of the following data; a verifier with a private register $\mathsf{V}$, $p$ provers with private registers $\mathsf{P}_1,\mathsf{P}_2,...,\mathsf{P}_p$, and message registers $\mathsf{M}_1,\mathsf{M}_2,...,\mathsf{M}_p$. All message registers have $l$ qubits. One of qubits of $\mathsf V$ is the output qubit. At the beginning of the protocol, $(\mathsf{V},\mathsf{M}_1,\mathsf{M}_2,...,\mathsf{M}_p)$ are initialized to $|0\cdots0\rangle$. Provers can set any state $|\psi\rangle$ in $\mathsf{P}_1,\mathsf{P}_2,...,\mathsf{P}_p$. During the protocol, the verifier and the provers apply their operations onto $(\mathsf{V},\mathsf{M}_1,\mathsf{M}_2,...,\mathsf{M}_p$,$\mathsf{P}_1,\mathsf{P}_2,...,\mathsf{P}_p)$ alternatively. The verifier applies the operations onto $(\mathsf{V},\mathsf{M}_1,\mathsf{M}_2,...,\mathsf{M}_p)$. Prover $p'$ applies the operations onto $(\mathsf{M}_{p'},\mathsf{P}_{p'})$. The verifier's operations can be constructed in polynomial time. Prover's operations have no such restrictions. After applying the last operation of the verifier, the verifier measures the output qubit in the computational basis. If the output is $|1\rangle$, then the verifier accepts. Denote this probability as $p_{acc}$.
The $m$-turn verifier for the quantum $p$-prover interactive proof system applies circuits $\{V^1,...,V^{\lceil (m+1)/2 \rceil}\}$ which can be constructed in polynomial time. The verifier uses $V^j$ at the $j$-th verifier's turn. The $m$-turn provers apply operations $\{P_i^1,...,P_i^{\lfloor (m+1)/2 \rfloor}\}_{i=1,...,p}$, where $P_i^j$ is the unitary operator done by the $i$-th prover in the provers' $j$-th turn. Provers' strategy is the tuple of ($|\psi\rangle,\{P_i^1,...,P_i^{\lfloor (m+1)/2 \rfloor}\}_{i=1,...,p}$). $(V,\{P_1,...,P_p\},|\psi\rangle)$ denotes the corresponding interactive proof system, that is, the tuple of the set of circuits of the verifier and the provers' strategy.
\begin{defi}A language $L$ is in $\mathsf{QMIP^*}(p,m,c,s)$ if there is an $m$-turn verifier V, for any $x$,
\begin{enumerate}
\item If $x\in L$, then there are $m$-turn provers $P_1,...,P_p$ and a state $|\psi\rangle$ (called honest provers), $p_{acc}\ge c$.
\item If $x\notin L$, then, for any $m$-turn provers $P_1',...,P_p'$ and any state $|\psi'\rangle$, $p_{acc}\le s$.
\end{enumerate}
\end{defi}
\subsection{Quantum zero-knowledge}
First, we introduce the indistinguishability of states and channels. We note that the definitions of indistinguishability allow the auxiliary quantum state $\sigma$.
\begin{defi}[Indistinguishable states]
Let $S\subseteq\{0,1\}^*$ be a language, and let $r$ be a function bounded by a polynomial. For each $x\in S$, $\rho_x,\zeta_x$ are $r(|x|)$ qubits states. The ensembles $\{\rho_x|x\in S\}$ and $\{\zeta_x|x\in S\}$ are computationally indistinguishable if for any function $s,k$ bounded by a polynomial, for any circuit $Q$, the size of which are $s(|x|)$ and the output of which is $0$ or $1$, and for any $k(|x|)$ qubits state $\sigma$, there is a negligible function $\epsilon$, such that
\begin{equation*}
\|\mathrm{Pr}[Q(\rho_x\otimes\sigma)=1]-\mathrm{Pr}[Q(\zeta_x\otimes\sigma)=1]\|\le\epsilon(|x|).
\end{equation*}
\end{defi}
\begin{defi}[Indistinguishable channels]
Let $S\subseteq\{0,1\}^*$ be a language, and let $q,r$ be functions bounded by a polynomial. For each $x\in S$, $\Psi_x,\Phi_x$ are channels from $q(|x|)$ qubits to $r(|x|)$ qubits for each $x\in S$. The ensembles $\{\Psi_x|x\in S\}$ and $\{\Phi_x|x\in S\}$ are indistinguishable if for any functions $s,k$ bounded by a polynomial, for any circuit $Q$ the size of which is $s(|x|)$, acts on $r(|x|)+k(|x|)$ qubits and outputs $0$ or $1$, and for any $r(|x|)+k(|x|)$ qubits state $\sigma$, there is a negligible function $\epsilon$, such that
\begin{equation*}
\|\mathrm{Pr}[Q(\Psi_x\otimes \mathrm{Id}) (\sigma)=1]-\mathrm{Pr}[Q(\Phi_x\otimes \mathrm{Id})(\sigma)=1]\|\le\epsilon(|x|).
\end{equation*}
\end{defi}
Next, we introduce quantum zero-knowledge proof systems. Let $(V,\{P_1,...,P_p\},|\psi\rangle)$ be a quantum multi-prover interactive proof system. Let $view_{(V,\{P_1,...,P_p\},|\psi\rangle)}(x,j)$ be the state in register $(\mathsf{V},\mathsf{M}_1,...,\mathsf{M}_p)$ at the $j$-th turn of the interactive proof. A malicious verifier $V'$ is a circuit family constructed in polynomial time, which interacts with $\{P_1,...,P_p\}$ instead of $V$. $V'$ has an additional input register $\mathsf W$ and an output register $\mathsf Z$. The interaction of $(V',\{P_1,...,P_p\},|\psi\rangle)$ until the $j$-th turn can be interpreted as a channel $\langle V',\{P_1,...,P_p\},|\psi\rangle \rangle(x,j)$ from $\mathsf W$ to $\mathsf Z$ decided by $(V',x,j)$.
\begin{defi}[Quantum Multi-Prover Zero-Knowledge Systems]
\sloppy A quantum interactive proof $(V,\{P_1,...,P_p\},|\psi\rangle)$ for a language $L$ is computationally zero-knowledge if for any verifier $V'$, there is an ensemble of channels $\{S_{V'}(x,j)\}$ (called a simulator) constructed in polynomial time, and $\{\langle V',\{P_1,...,P_p\},|\psi\rangle \rangle(x,j)|x\in L\}$ and $\{S_{V'}(x,j)|x\in L\}$ are indistinguishable.
Let $\sf {QMZK^*}$$(p,m,c,s)$ be the class decided by quantum multi-prover zero-knowledge systems with $p$ provers, $m$ turns, completeness $c$ and soundness $s$. We define $\mathsf{QMZK^*}:=\cup_{p,m:poly,c-s\ge\frac{1}{poly}}\mathsf{QMZK^*}(p,m,c,s)$.
\end{defi}
\begin{defi}[Honest Verifier Quantum Multi-Prover Zero-Knowledge Systems]
A quantum $p$-prover interactive proof $(V,\{P_1,...,P_p\},|\psi\rangle)$ for a language $L$ is honest verifier zero-knowledge if for the honest verifier $V$, there is an ensemble of states $\{\Psi_{V}(x,j)\}$ constructed in polynomial time and $\{view_{(V,\{P_1,...,P_p\},|\psi\rangle)}(x,j)|x\in L\}$ and $\{\Psi_{V}(x,j)|x\in L\}$ are indistinguishable.
\sloppy Let $\sf {HVQMZK^*}$$(p,m,c,s)$ be the class decided by honest verifier quantum multi-prover zero-knowledge systems with $p$ provers, $m$ turns, completeness $c$ and soundness $s$. We define $\mathsf{HVQMZK^*}:=\cup_{p,m:poly,c-s\ge\frac{1}{poly}}\mathsf{HVQMZK^*}(p,m,c,s)$.
\end{defi}
We note a lemma used to eliminate the honest condition.
\begin{lemm}[\cite{Aa}]
\label{2PO}
Assume a POVM on a state $\rho$ with two outcomes has the outcome 1 with probability $\epsilon$. Denote the state after it outputs 0 by $\rho_0$. Then $\|\rho-\rho_0\|\le\sqrt\epsilon$.
\end{lemm}
\subsection{Cryptographic tools}
Here we introduce a few cryptographic tools used in the $\mathrm{QMZK^*}$ protocol to apply the technique in \cite{BJSW}. Assuming the existence of commitment schemes, computational quantum zero-knowledge proofs for NP \cite{W2} and quantumly secure coin-flipping \cite{DL} exist.
\begin{defi}[commitment schemes]
A quantum computationally commitment scheme for an alphabet $\Gamma$ is an ensemble of functions $\{f_n: \Gamma\times \{0,1\}^{p(n)}\rightarrow\{0,1\}^{q(n)},\ n\in \mathbb{N}\}$ computable in polynomial time such that following holds;
\begin{itemize}
\item Unconditionally binding property. For all $n\in\mathbb{N},a,b\in\Gamma$ and $r,s\in\{0,1\}^{p(n)}$, if $f_n(a,r)=f_n(b,s)$, then $a=b$.
\item Quantum computationally concealing property. For all $a\in\Gamma$ and $n\in\mathbb{N}$, define
\begin{equation}
\rho_{a,n}=\frac{1}{2^{p(n)}}\sum_{r\in\{0,1\}^{p(n)}}|f_n(a,r)\rangle\langle f_n(a,r)|.
\end{equation}
For all $a,b\in\Gamma$, the ensemble $\{\rho_{a,n}|n\in \mathbb{N}\}$ and $\{\rho_{b,n}|n\in \mathbb{N} \}$ are computationally indistinguishable.
\end{itemize}
\end{defi}
\subsubsection{Coin-Flipping}
A coin-flipping protocol is an interactive process that allows two parties to jointly toss random
coins. We only make use of one specific coin-flipping protocol, namely Blum's coin-flipping protocol \cite{Bl} in which an honest prover commits to a random $y\in\{0,1\}$, the honest verifier selects $z\in\{0,1\}$ at random, the prover reveals $y$, and the two participants agree that the random bit generated is $r=y\oplus z$. Damg\r{a}rd and Lunemann \cite{DL} proved that Blum's coin-flipping protocol is quantum-secure, assuming a quantum-secure commitment scheme.
\section{Elimination of the honest condition}
In this section, we eliminate the honest condition of $\rm QMZK^*$ protocols. This needs no computational assumptions. First, we reduce the number of turns. The reduction is almost the same as Kempe et al. \cite{KKMV}. There are no new contents except for the addition of honest zero-knowledge, while it is shown obviously. Hence we only give the statement.
\begin{theo} Let $\sqrt s\le c$. Then, $\sf {HVQMZK^*}$$(p,4m+1,c,s)\subseteq$ $\sf{HVQMZK^*}$$(p,2m+1,\frac{1+c}{2},\frac{1+\sqrt s}{2})$.
\end{theo}
The next proposition is easily derived by applying Theorem 3 to a protocol obtained from repeating the original protocols with threshold value decisions.
\begin{prop}
$\sf {HVQMZK^*}$$(p,m,c,s)\subseteq$ $\sf{HVQMZK^*}$$(p,3,1-\frac{1}{exp},1-\frac{1}{poly})$.
\end{prop}
\begin{proof}[Proof(Sketch)] $\sf {HVQMZK^*}$$(p,m,c,s)\subseteq$ $\sf{HVQMZK^*}$$(p, poly\cdot m,1-\frac{1}{exp},\frac{1}{exp})$ is easily shown by repeating the original $\sf {HVQMZK^*}$$(p,m,c,s)$ protocol $poly$ times, threshold value decisions and applying Chernoff bounds. Apply Theorem 3 to $\sf{HVQMZK^*}$$(p, poly\cdot m,1-\frac{1}{exp},\frac{1}{exp})$.
\end{proof}
The next theorem is our first result. The reason why we add one more prover (prover 0) is to prevent prover $1,\ldots,p$ from acting on register $\mathsf{V}$ (the verifier's private register) by making prover 0 have the register $\sf V$.
\begin{theo}
$\sf{HVQMZK^*}$$(p,3,1-\epsilon,1-\delta)\subseteq \sf{QMZK^*}$$(p+1,2,1-\frac{\epsilon}{2},1-\frac{\delta^2}{8})$.
\end{theo}
\begin{figure}
\hrulefill\\
Let $V_1,V_2$ be the circuit that the verifier of a three-turn protocol for $\mathsf{HVQMZK^*}(p,3,1-\epsilon, 1-\delta)$ applies after receiving message registers $(\mathsf{M}_1,\mathsf{M}_2,...,\mathsf{M}_p)$ from provers $1,...,p$, respectively, at the first turn and the third turn. The two-turn protocol is the following;
\begin{enumerate}
\item Select $b\in\{0,1\}$ uniformly at random. Send $b$ to provers $1,...,p$, and nothing to prover $0$.
\item Do the following tests depending on $b=0,1$:
\begin{enumerate}
\item
($b=0$: Forward test) Receive $\mathsf {V}$ from prover 0 and receive $\mathsf {M}_i$ from prover $i$ for $i=1,...,p$. Apply $V_2$ to $(\mathsf{V},\mathsf{M}_1,\mathsf{M}_2,...,\mathsf{M}_p)$. Accept if the original verifier of the three-turn protocol accepts the state in $(\mathsf{V},\mathsf{M}_1,\mathsf{M}_2,...,\mathsf{M}_p)$. Reject otherwise.\\
\item
($b=1$: Backward test) Receive $\mathsf {V}$ from prover 0 and receive $\mathsf {M}_i$ from prover $i$ for $i=1,...,p$. Apply $(V_1)^\dagger$ to $(\mathsf{V},\mathsf{M}_1,\mathsf{M}_2,...,\mathsf{M}_p)$. Accept if all qubits in $\mathsf{V}$ are $|0\rangle$, otherwise reject.
\end{enumerate}
\end{enumerate}
\hrulefill
\caption{$(p+1)$-prover two-turn honest verifier zero-knowledge protocol constructed from a three-turn protocol for $\mathsf{HVQMZK^*}(p,3,1-\epsilon, 1-\delta)$.}
\label{2turn}
\quad\\
\hrulefill\\
$\mathsf{V},\mathsf{M}_i$ are the same as Figure \ref{2turn}. In addition, we use $\mathsf{G}_0, \mathsf{G}_1, \mathsf{G}_2,...,\mathsf{G}_p$, all of which are 1 qubit registers.
At the start, prover $i$ has $\mathsf{G}_i$.
\begin{enumerate}
\item Select $b'\in\{0,1\}$ uniformly at random. Send $b'$ to provers $1,...,p$, and nothing to prover 0.
\item Do the following tests depending on $b'=0,1$:
\begin{enumerate}
\item
($b'=0$:GHZ test)\\
Prover 0 sends $\mathsf{V,G_0}$. Prover $i$ sends $\mathsf{G}_i$, for $i=1,...,p$. Measure ($\mathsf{G}_0, \mathsf{G}_1, \mathsf{G}_2,...,\mathsf{G}_p$) by the projection onto $\frac{1}{\sqrt2} (|0^{p+1}\rangle+|1^{p+1}\rangle)$. If $\frac{1}{\sqrt2} (|0^{p+1}\rangle+|1^{p+1}\rangle)$ is measured, then accept.
\item
($b'=1$:History test)\\
Prover 0 sends $\mathsf{V,G_0}$. Prover $i$ sends $\mathsf {M}_i$ and $\mathsf{G}_i$, for $i=1,...,p$. Verifier measures $\mathsf{G}_0$ in the computational basis. Let $b$ be the output of this measurement. If $b=0$, the verifier measures ($\mathsf{V},\mathsf{M}_1,...,\mathsf{M}_p)$ as the verifier in Figure \ref{2turn} does at the Forward test. If $b=1$, the verifier measures ($\mathsf{V},\mathsf{M}_1,...,\mathsf{M}_p)$ as the verifier in Figure \ref{2turn} does at the Backward test .
\end{enumerate}
\end{enumerate}
\hrulefill
\caption{General zero-knowledge protocol based on the protocol in Figure \ref{2turn}.}
\label{2gz}
\end{figure}
\begin{proof}
First, we construct a $(p+1)$-prover two-turn protocol from a three-turn protocol for $\sf{HVQMZK^*}$$(p,3,1-\epsilon,1-\delta)$, in which message to provers is the same classical one bit except for prover 0, who receives no messages. The two-turn protocol is described in Figure \ref{2turn}. The main protocol of $\sf{QMZK^*}$$(p+1,2,1-\frac{\epsilon}{2},1-\frac{\delta^2}{8})$ is described in Figure \ref{2gz}. The proof of the correctness of the two-turn protocol in Figure \ref{2turn} is the same as the proof of Lemma 5.4 in \cite{KKMV}. We only discuss the conversion of a two-turn protocol in Figure \ref{2turn} into a general zero-knowledge protocol in Figure \ref{2gz}.\\\\
$\bf Completeness$: The conversion of the two-turn protocol to the general zero-knowledge preserves completeness since the provers only have to do the GHZ test and the history test honestly.\\\\
$\bf Soundness$ : We construct a $\mathrm{HVQMZK^*}$ protocol from a $\mathrm{QMZK^*}$ protocol with a sufficient acceptance probability. Let $U_g^i,U_h^i$ be the unitary operators that prover $i$, for $i=1,\ldots, p$, does for the GHZ test and the history test, respectively. (Prover 0 receives no messages, and hence we can assume prover 0 does only the identity operator.)
Let $\rho'':=\rm{Tr}_{GHZ}$$(\Pi_{GHZ}(\otimes_{i=1}^p U_g^i)\rho(\otimes_{i=1}^p (U_g^i)^\dagger)\Pi_{GHZ})$. Here, $\rho$ is the initial state shared by the provers in the $\mathrm{QMZK^*}$ protocol and $\Pi_{GHZ}$ is a projection onto $\frac{|0^{p+1}\rangle+|1^{p+1}\rangle}{\sqrt2}$ of registers ($\mathsf{G}_0, \mathsf{G}_1, \mathsf{G}_2,...,\mathsf{G}_p$) which is used in the GHZ test, and $\rm Tr_{GHZ}$ is the partial trace operation on registers ($\mathsf{G}_0, \mathsf{G}_1, \mathsf{G}_2,...,\mathsf{G}_p$). Let $\rm Tr$$\rho''=1-\epsilon_1$, where $\epsilon_1$ is the rejection probability of the GHZ test. Let $\frac{\rho''}{1-\epsilon_1}:=\rho'$. We use this state as the initial state of provers. Let $\epsilon_2$ be the rejection probability of the history test.
Now we construct the strategy of the two-turn $\rm{HVQMZK^*}$ protocol from the strategy of $\rm{QMZK^*}$ protocol; The private registers of prover $i$ are $\mathsf{P}_i,\mathsf{G}_i$, for $i=1,...,p$. Initially prover 0 has only register $\mathsf{V}$. The initial state in ($\mathsf{P}_1,...,\mathsf{P}_p,\mathsf{M}_1,...,\mathsf{M}_p$) is $\rho'$. If the verifier sends $b$ to prover $i$ ($i=1,...,p$), prover $i$ sets $|b\rangle$ in register $\mathsf{G}_i$, applies $U^i_h{U^i_g}^\dagger$ and sends $\mathsf{M}_i$ to the verifier. Prover 0 always sends $\mathsf{V}$.
The rejection probability $p_{rej,hv}$ of this $\mathrm{HVMQZK^*}$ protocol is bounded by the rejection probability $p_{rej}$ of the general zero-knowledge protocol as follows. Here, $\Pi_{rej}$ is the projection onto rejected states.
\begin{equation}
\begin{split}
p_{rej,hv}&:=\frac{1}{2}\sum_{b\in\{0,1\}}\mathrm{Tr}[\Pi_{rej}(\otimes_{i=1}^p U_h^i {U_g^i}^\dagger) (|b^p\rangle\langle b^p|\otimes \rho') (\otimes_{i=1}^p U_g^i{U_h^i}^\dagger)]\\
&=\frac{1}{1-\epsilon_1}\mathrm{Tr} [\Pi_{rej} (\otimes_{i=1}^p U_h^i{U_g^i}^\dagger)\Pi_{GHZ}(\otimes_{i=1}^p U_g^i)\rho (\otimes_{i=1}^p {U_g^i}^\dagger)\Pi_{GHZ}(\otimes_{i=1}^p U_g^i{U_h^i}^\dagger)]\\
&\le \sqrt\epsilon_1+\mathrm{Tr} [\Pi_{rej}(\otimes_{i=1}^p U_h^i {U_g^i}^\dagger)(\otimes_{i=1}^p U_g^i)\rho (\otimes_{i=1}^p {U_g^i}^\dagger)(\otimes_{i=1}^p U_g^i{U_h^i}^\dagger)]\\
&=\sqrt\epsilon_1+\mathrm{Tr}[\Pi_{rej}(\otimes_{i=1}^p U_h^i)\rho(\otimes_{i=1}^p {U_h^i}^\dagger)]\\
&=\sqrt\epsilon_1+\epsilon_2\le\sqrt\epsilon_1+\sqrt\epsilon_2\le2\sqrt {2p_{rej}}.
\end{split}
\end{equation}
The first equality follows from the definition of $\rho'=\frac{1}{1-\epsilon_1}\rho''$$=\frac{1}{1-\epsilon_1}\rm{Tr}_{GHZ}$$(\Pi_{GHZ}(\otimes_{i=1}^p U_g^i)\rho(\otimes_{i=1}^p (U_g^i)^\dagger)\Pi_{GHZ})$ and that $\frac{1}{2}(|0^p\rangle\langle0^p|+|1^p\rangle\langle1^p|)$ equals to the $p$-qubits substate
of the $(p+1)$-qubit GHZ state. The first inequality follows since $\|\rho'-\rho\|\le\sqrt\epsilon_1$ holds by regarding $\{\Pi_{GHZ},\mathrm{Id}-\Pi_{GHZ}\}$ as a two-outcome POVM and applying Lemma 1.\\\\
$\bf Zero$ $\bf knowledge$: Before the verifier sends messages, the verifier gets no qubits from provers. Hence it is sufficient to construct a simulator of the malicious verifier for each of the verifier's possible messages.
First, we observe that it is sufficient to prove only the case that the malicious verifier sends the same $b'$ to all provers except for prover 0. Here, we define the action of the honest prover $j$ in Figure 2 as follows;
\begin{enumerate}
\item If the prover $j$ receives $b'=1$, then he/she measures the register $\mathsf{G}_j$ in the computational basis. Let the outcome be $b$. He/she acts as the prover $j$ in Figure 1 and sends $\mathsf{M}_j,\mathsf{G}_j$.
\item If the prover $j$ receives $b'=0$, then he/she sends $\mathsf{G}_j$.
\end{enumerate}
This definition preserves the completeness parameter. Assume that the verifier sends $b'=1$ (history test) to prover $j$ and $b'=0$ (GHZ test) to prover $i$. Prover $i$ sends register $\mathsf{G}_i$ to the verifier, but the qubit in $\mathsf{G}_i$ is the same $|0\rangle$ or $|1\rangle$ of $\mathsf{G}_j$. Hence, if the case that the verifier sends $b'=1$ to all provers can be simulated, the case that prover $i$ receives $b'=0$ can be also simulated since it is sufficient to simulate any verifier who sends $b'=1$ to all provers and aborts $\mathsf{M}_i$.
The case that the verifier sends $b'=0$ to provers $1,...,p$ can be also simulated since it is sufficient to simulate the verifier who sends $b'=1$ to all provers, aborts $\mathsf{M}_i,\mathsf{G}_i$, and prepares $\frac{1}{\sqrt2} (|0^{p+1}\rangle+|1^{p+1}\rangle)$. The case the verifier sends $b'=1$ to all provers is obviously zero-knowledge since the verifier receives only the copies of a uniform random 1 bit and the state that the honest verifier in Figure \ref{2turn} receives.
\end{proof}
Theorem 4 needs an additional prover to eliminate the honest condition, but by the rewinding technique of \cite{Ko,W2}, we can eliminate the honest condition without additional provers. This result essentially follows by combining the proof of Theorem 4 and the rewinding, and our main contribution is described in the proof of Theorem 4. Hence we prove this result in Appendix. To prove this result, we have to restrict the verifier's message to one public classical bit by the technique of \cite{KKMV}. Let $\sf{HVQMZK^*}_{pub,1}$$(k,3,c,s)$ be the class of problems verified by such $\rm HVQMZK$ systems. We omit the proof of the next theorem since it is also the same as \cite{KKMV}.
\begin{theo}Let $\sqrt s\le c$. Then, $\sf {HVQMZK^*}$$(p,3,c,s)\subseteq$ $\sf{HVQMZK^*}_{pub,1}$$(p,3,\frac{1+c}{2},\frac{1+\sqrt s}{2})$.
\end{theo}
We prove the next theorem in Appendix. This result also needs the GHZ test, and rewinding alone is not sufficient to prove this result.
\begin{theo}
$\sf{HVQMZK^*}_{pub,1}$$(p,3,1-\epsilon,1-\delta)\subseteq \sf{QMZK^*}$$(p,3,1-\frac{\epsilon}{2},1-\frac{\delta^2}{8})$.
\end{theo}
\section{LHI protocols and computational quantum zero-knowledge systems}
In this section, we construct the zero-knowledge system for $\sf QMIP^*$. The construction of a zero-knowledge system for $\sf QMIP^*$ consists of the following three steps
\begin{enumerate}
\item Construction of the Local Hamiltonian based Interactive protocol (LHI protocol) corresponding to the $p$-prover $\mathrm{QMIP^*}$ protocol, which extends the protocol for checking the local Hamiltonian problem corresponding to a QMA protocol in \cite{BJSW} to the $\rm QMIP^*$ case.
\item Construction of what we call the LHI+ protocol, which replaces the uniform random queries by the GHZ test.
\item Zero-knowledge protocol for $\sf QMIP^*$ based on the LHI+ protocol.
\end{enumerate}
Our main contribution is step 1 and 2. Step 3 is a direct application of the technique of Broadbent et al. \cite{BJSW}. The analysis of step 3 is almost the same as \cite{BJSW}, and hence we only point out the main differences.
In Section 4.1, we construct the LHI protocol and prove its validity. We only consider three-turn $\mathrm{QMIP^*}$ protocols, as this does not lose generality due to Lemma 4.2 of \cite{KKMV}. In Section 4.2, we provide the LHI+ protocol and prove its validity. In Section 4.3, we overview the quantum zero-knowledge protocol for $\sf{QMA}$ and explain why it works for the LHI+ protocol. In Section 4.4, we construct the final zero-knowledge protocol. We analyze the final protocol in Section 4.5.
\subsection{Construction of the LHI protocol\label{LH2;sec}}
First, we construct a local Hamiltonian based Interactive protocol (LHI protocol) for a three-turn $\mathrm{QMIP^*}$ protocol $\cal{P}$. Intuitively, the LHI protocol checks the history state of the calculation of the interactive proof system in Figure \ref{1com}, which is transformed from the original three-turn protocol $\cal{P}$ with completeness $1-\frac{1}{exp}$ and soundness $1-\frac{1}{poly}$ in Figure \ref{3turn}. We can assume such completeness/soundness errors on $\cal{P}$ due to Lemma 4.2 of \cite{KKMV}. This transformation is done to locally check the communication of the interactions. Let $V_0=U_{t_0}\cdots U_1$ and $V_1=U_T\cdots U_{t_0+2pl+p+1}$ be the circuits which the verifier in $\cal{P}$ uses (see Figure \ref{3turn}). Here, $l$ is the length of the message register $\mathsf{M}_i$ between the $i$-th prover and the verifier in $\cal{P}$, $t_0$ is the number of gates in $V_0$ and $(T-(t_0+2pl+p+1)+1)$ is the number of gates in $V_1$. The reason why the indices of $V_0,V_1$ are not successive is the increase of communication turns by the transformation from Figure \ref{3turn} to Figure \ref{1com}. This transformation is necessary to make prover 0 send message registers $\mathsf{M}_1,...,\mathsf{M}_p$ in the LHI protocol to prevent the malicious attack on message registers by provers $1,...,p$ depending on the verifier's message.
We can assume all $\mathsf{M}_i$ have the same length. The LHI protocol uses $T$ registers $\mathsf{C}_1,...,\mathsf{C}_T$ and $p$ registers $\mathsf{Me}_1,...,\mathsf{Me}_p$ where each of those $(T+p)$ registers consists of a single qubit, in addition to $\mathsf{M}_1,...,\mathsf{M}_p$, and $\mathsf{V}$. Intuitively, $\mathsf{C}_1,...,\mathsf{C}_T$ are the time counters, $\mathsf{Me}_1,...,\mathsf{Me}_p$ are 1 qubit message registers. The LHI protocol is described in Figure \ref{LHp}. Without loss of generality, we can assume $T^5\ll\epsilon^{-1}$ by Lemma 4.2 of \cite{KKMV}.
Though intuitively the LHI protocol corresponds to the protocol in Figure \ref{1com}, we show that if the LHI protocol can be accepted with high probability, then the protocol $\cal{P}$ in Figure \ref{3turn} can be accepted with high probability, since we put Figure \ref{1com} for ease of intuitive understanding LHI protocols, but the correctness of Figure \ref{1com} is logically unnecessary and proving the correctness is only redundant. The next lemma states that if there exist provers who pass the LHI protocol, then there exist provers who pass the original three-turn protocol $\cal P$.
\begin{figure}
\hrulefill\\
\includegraphics[width=0.8\textwidth]{original3turn.pdf}
\caption{Original three-turn protocol $\cal{P}$. In this figure, the number of provers is two.}
\label{3turn}
\hrulefill\\
\end{figure}
\begin{figure}
\hrulefill\\
\includegraphics[width=0.8\textwidth]{1qubitcommunication.pdf}
\caption{Restricting communications to one qubit. In this figure, the length of message register is three. The arrows $\updownarrow$ are SWAP gates. $\mathsf{M'_1}$ is used by honest prover 1 to receive the qubits from the verifier. Note that $\sf{M'_1}$ is introduced to explain the behavior of the honest prover and hence it does not appear in the soundness analysis in the main text. Communications between the verifier and prover 2 are similarly transformed. A,B,A' denote the correspondence to the operators $A,B,A'$ in Eq.(\ref{opr}).}
\label{1com}
\hrulefill
\end{figure}
\begin{lemm}Suppose $T^5\ll\epsilon^{-1}$. If there are provers who pass the protocol in Figure \ref{LHp} with probability $1-\epsilon$, there are provers who pass the original three-turn protocol $\cal{P}$ with probability $1-O(\sqrt[4]{T^5\epsilon})$.
\end{lemm}
\begin{figure}
\hrulefill\\
(A). LHI protocol
\begin{enumerate}
\item The verifier sends $t'\in [2T+n]$ to provers $1,...,p$ and nothing to prover 0.
\item Prover $i$ sends $\mathsf{Me}_i$. Prover 0 sends $\mathsf{V},\mathsf{M}_1,..., \mathsf{M}_p,\mathsf{C}_1,...,\mathsf{C}_T$.
\item The verifier measures these qubits by $H_{t'}$.
\end{enumerate}
Here, $n$ is the number of qubits of $\mathsf{V}$. We can assume $n\le T$.\\
\hrulefill\\
(B). Hamiltonians used in the LHI protocol\\\\
$SWAP _{(k,j),\mathsf{Me}_k}$ is the SWAP operator on the $j$-th qubit of $\mathsf {M}_k$ and $\mathsf{Me}_k$, $CNOT_{\mathsf{C}_{t_0+pl+k},\mathsf{Me}_k}$ is a CNOT operator which is controlled by $\mathsf{C}_{t_0+pl+k}$ and acts on $\mathsf{Me}_k$. $\Pi_{acc}$ is a projection that the verifier in $\cal{P}$ accepts. ${\rm Id}$ is an identity.
The set $\{H_t|t\in [2T+n]\}$ has the following Hamiltonians as described in (a), (b), (b'), (c), (d), (e), and (f).\\\\
In (a), (b), (b'), we define $H_t$ by Eq. (\ref{lh}).
\begin{equation}
\label{lh}
H_t:=|10\rangle\langle10|_{\mathsf{C}_{t-1},\mathsf{C}_{t+1}}\otimes\frac{1}{2}(-|1\rangle\langle0|_{\mathsf{C}_t}\otimes U'_t- |0\rangle\langle1|_{\mathsf{C}_t}\otimes {U'_t}^\dagger+|0\rangle\langle0|_{\mathsf{C}_t}\otimes \mathrm{Id} +|1\rangle\langle1|_{\mathsf{C}_t}\otimes \mathrm{Id}).
\end{equation}
If $t=1$, we replace $|10\rangle\langle10|_{\mathsf{C}_{t-1},\mathsf{C}_{t+1}}$ by $|0\rangle\langle0|_{\mathsf{C}_{t+1}}$ exceptionally since register $\mathsf{C}_0$ does not exist. Similarly we ignore register $\mathsf{C}_0$ in (e).
The unitary operator $U'_t$ in Eq.(\ref{lh}) for (a), (b), (b') is defined as follows;
\begin{itemize}
\item[(a)](verifier's gate step) $t\in[1,t_0]\cup[t_0+2pl+p+1,T]$: $U'_t=U_t$.
\item[(b)](communication step) $t\in[t_0+1,t_0+pl]$ : Let $t=t_0+(k-1)l+j$ ($1\le k\le p,1\le j\le l$). Then $U'_t:=SWAP _{(k,j),\mathsf{Me}_k}$.
\item[(b')](communication step) $t\in[t_0+pl+p+1,t_0+2pl+p]$: Let $t=t_0+pl+p+(k-1)l+j$. Then $U'_t:=SWAP _{(k,j),\mathsf{Me}_k}$.
\end{itemize}
In (c), (d), (e), (f), we define $H_t$ as follows;
\begin{itemize}
\item[(c)](provers' step) $t=t_0+pl+k\ (k=1,...,p)$:\\
\quad$H_{t_0+pl+k}:= |10\rangle\langle10|_{\mathsf{C}_{t_0+pl+k-1},\mathsf{C}_{t_0+pl+k+1}}\otimes|-\rangle\langle-|_{\mathsf{C}_{t_0+pl+k}} CNOT_{\mathsf{C}_{t_0+pl+k},\mathsf{Me}_k}$.
\item[(d)](measurement by the verifier) $t=T+1$: $H_{T+2pl+p+1}:=|1\rangle\langle1|_{\mathsf{C}_T}\otimes({\rm Id}-\Pi_{acc})$.
\item[(e)](consistency of time counters) $t\in[T+2,2T]$: $H_{t}:=|01\rangle\langle01|_{\mathsf{C}_{t-T-1},\mathsf{C}_{t-T}}$.
\item[(f)](initialization of $\mathsf{V}$) $t\in [2T+1,2T+n]$: $H_t:=|0\rangle\langle0|_{\mathsf{C}_1}\otimes(\mathrm{Id}-\Pi_0^{t-2T})$. Here, $\Pi_0^{t-2T}$ is a projection onto $|0\rangle$ of the ($t-2T$)th qubit of $\mathsf{V}$.
\end{itemize}
\hrulefill
\caption{Local Hamiltonian based Interactive Protocol for $\cal{P}$ (in fact, for the $\mathrm{QMIP^*}$ protocol in Figure \ref{1com}). (A) is the protocol. (B) shows the Hamiltonians used in the protocol given in (A). }
\label{LHp}
\end{figure}
\begin{proof}
We construct the $\mathrm{QMIP^*}$ protocol for $\cal{P}$ to compute the original three-turn protocol with acceptance probability at least $1-O(\sqrt[4]{T^5\epsilon})$ based on the protocol in Figure \ref{LHp} with acceptance probability at least $1-\epsilon$. Intuitively, (a),...,(f) in Figure \ref{LHp} check the following;
\begin{description}
\item[(a)]the verifier's gate step
\item[(b) and (b')]communication step
\item[(c)]the provers' step
\item[(d)]the measurement by the verifier
\item[(e)]the consistency of the time counters
\item[(f)]the initialization of $\mathsf V$
\end{description}
Let $|\varphi\rangle$ be the initial state of all the registers in the LHI protocol. Let ${D_{k,j}}$ ($D_{k,j}'$, respectively) be the operation that prover $k$ does in test (b) with query $t=t_0+(k-1)l+j$ ((b') with $t=t_0+pl+p+(k-1)l+j$, respectively). Let $E_k$ be the operation that prover $k$ does in test (c). Denote the Pauli $X$ operator on $\mathsf{Me}_k$ by $X_{\mathsf{Me}_k}$. Denote the projection onto $|0^n\rangle$ of register $\mathsf{V}$ by $\Pi_{0^n}$.
Before we construct the provers' strategy for $\cal{P}$, we define some unitary operators. \sloppy Let $|t\rangle=|1\rangle_{\mathsf{C_1}}|1\rangle_{\mathsf{C_2}}\cdots|1\rangle_{\mathsf{C}_t}|0\rangle_{\mathsf{C_{t+1}}}\cdots|0\rangle_{\mathsf{C}_T}$ be a state in counter registers that corresponds to time $t$. Let
\begin{equation}
\begin{split}
&U_{t_0+(k-1)l+j}:=D_{k,j}^\dagger SWAP _{(k,j),\mathsf{Me}_k}D_{k,j}\ (k\in[1,p],j\in[1,l]),\\
&U_{t_0+pl+p+(k-1)l+j}:={D'_{k,j}}^\dagger SWAP _{(k,j),\mathsf{Me}_k}D'_{k,j}\ (k\in[1,p],j\in[1,l]),\\
&U_{t_0+pl+k}:= {E_k}^\dagger X_{\mathsf{Me}_k}E_k\ (k\in[1,p]),
\end{split}
\end{equation}
and
\begin{equation}
\label{opr}
\begin{split}
&W:=\Sigma_{t\in [0,T]} |t\rangle\langle t| \otimes U_1^\dagger\cdots U_t^\dagger+\Pi_{bad time}\otimes \rm{Id},\\
&A_k:=U_{t_0+(k-1)l+l}\cdots U_{t_0+(k-1)l+1},\\
&B_k:={U}_{t_0+pl+k},\\
&A'_k:={U_{t_0+pl+p+(k-1)l+1}}\cdots {U_{t_0+pl+p+(k-1)l+l}}.
\end{split}
\end{equation}
where $|t\rangle\langle t| \otimes U_1^\dagger\cdots U_t^\dagger$ for $t=0$ is $|0\rangle\langle0|\otimes\rm Id$, and $\Pi_{bad time}$ is a projection onto states rejected by test (e). Intuitively, $A_k,B_k,A'_k$ correspond to A,B,A' in Figure \ref{1com}. Note that $U_t$ for $t\notin [t_0+1,t_0+2pl+p]$ are already defined by the verifier's gates; $V_0=U_{t_0}\cdots U_1$ and $V_1=U_T\cdots U_{t_0+2pl+p+1}$.
Using these operators, we define the strategy. The provers' strategy $(|\overline{\varphi}\rangle,\{P_k\})$ for $\cal{P}$ constructed from the LHI protocol is as follows. (The strategy $(|\overline{\varphi}\rangle,\{P_k\})$ means that the state shared initially by the provers is $|\overline{\varphi}\rangle$, at the first turn prover $k$ sends the substate in $\mathsf{M}_k$ of $|\overline{\varphi}\rangle$ and at the third turn prover $k$ applies $P_k$ on $(\mathsf{P}_k,\mathsf{M}_k,\mathsf{Me}_k)$ and sends back $\mathsf{M}_k$.)
\begin{equation}
\begin{split}
&|\overline{\varphi}\rangle:=\frac{\Pi_{0^n} W |\varphi\rangle}{\|\Pi_{0^n} W |\varphi\rangle\|},\\
&P_k:=A'_kB_kA_k.\\
\end{split}
\label{philine}
\end{equation}
Next, we calculate the acceptance probability of the strategy $(|\overline{\varphi}\rangle,\{P_k\})$.
Now we take $|\varphi\rangle$ as follows;
\begin{equation}
|\varphi\rangle=\Sigma_{t\in[0,T]} |t\rangle|\psi_t\rangle+|\bot\rangle
\label{varphi}
\end{equation}
where $|\psi_t\rangle$ is an (unnormalized) vector, and $|\bot\rangle$ is a state that is rejected by any Hamiltonian of (e)(consistency of the time counters) in Figure 5.
$\||\bot\rangle\|$ is bounded by the following inequality.
\begin{equation}
\begin{split}
\ \||\bot\rangle\|
&=\bigl\|\sum_{t\in[T+2,2T]}|01\rangle\langle01|_{\mathsf{C}_{t-T-1},\mathsf{C}_{t-T}}|\bot\rangle\bigr\|\\
&\le\sum_{t\in[T+2,2T]}\||01\rangle\langle01|_{{\mathsf{C}_{t-T-1},\mathsf{C}_{t-T}}}|\bot\rangle\|\\
&=\sum_{t\in[T+2,2T]}\||01\rangle\langle01|_{{\mathsf{C}_{t-T-1},\mathsf{C}_{t-T}}}|\varphi\rangle\|\\
&\le(T-1)\sqrt{(2T+n)\epsilon}\\
&=O(\sqrt{T^3\epsilon})
\end{split}
\label{boteval}
\end{equation}
The first equality follows since $|\bot\rangle$ is an incorrect time register state. The first inequality is the triangle inequality. The second equality follows by the definition of $|\varphi\rangle$ in (\ref{varphi}). The third inequality follows by the probability of rejection by (e). The last equality follows by the assumption $n\le T$ in Figure \ref{LHp}(A).
We bound the following terms (\ref{eq;vch}), (\ref{eq;com1}), (\ref{eq;com2}), (\ref{eq;pr}), using the assumption that the error probability of the LHI protocol is $\epsilon$ or less.
\begin{align}
&\||\psi_t\rangle-U_t|\psi_{t-1}\rangle\|\ \ (t\in[1,t_0]\cup[t_0+2pl+p+1,T]), \label{eq;vch}\\
&\||\psi_t\rangle-D_{k,j}^\dagger SWAP _{(k,j),\mathsf{Me}_k} D_{k,j}|\psi_{t-1}\rangle\|\ (t\in[t_0+1,t_0+pl],\ t=t_0+(k-1)l+j), \label{eq;com1} \\
&\||\psi_t\rangle-{D'_{k,j}}^\dagger SWAP _{(k,j),\mathsf{Me}_k} D'_{k,j}|\psi_{t-1}\rangle\|\ (t\in[t_0+pl+p+1,t_0+2pl+p],\ t=t_0+pl+p+(k-1)l+j),\label{eq;com2}\\
&\||\psi_t\rangle-E_{k}^\dagger X_{\mathsf{Me}_k} E_{k}|\psi_{t-1}\rangle\|\ \ (t=t_0+pl+k).\label{eq;pr}
\end{align}
We can bound (\ref{eq;vch}) as follows;
\begin{equation}
\label{v-uni}
\begin{split}
\||\psi_{t}\rangle-U_t|\psi_{t-1}\rangle\|
&=\frac{1}{2}(\||\psi_{t}\rangle-U_t|\psi_{t-1}\rangle\|+\||\psi_{t-1}\rangle-U_t^\dagger|\psi_{t}\rangle\|)\\
&\le\frac{\sqrt2}{2}\sqrt{\||\psi_{t}\rangle-U_t|\psi_{t-1}\rangle\|^2+\||\psi_{t-1}\rangle-U_t^\dagger|\psi_{t}\rangle\|^2}\\
&=\frac{\sqrt2}{2}\||t\rangle(|\psi_t\rangle-U_t|\psi_{t-1}\rangle)+|t-1\rangle(|\psi_{t-1}\rangle-U_t^\dagger|\psi_t\rangle)\|\\
&=\frac{\sqrt2}{2}\|H_t|\varphi\rangle\|=O(\sqrt{T\epsilon}).
\end{split}
\end{equation}
The first equality follows from modifying $2\||\psi_{t}\rangle-U_t|\psi_{t-1}\rangle)\|$ to $\||\psi_{t}\rangle-U_t|\psi_{t-1}\rangle)\|$$+\||\psi_{t}\rangle-U_t|\psi_{t-1}\rangle)\|$ and applying $U_t^\dagger$ to the second term. The first inequality is the Cauchy-Schwarz inequality. The second equality follows since $|t\rangle$ and $|t-1\rangle$ are orthogonal vectors. The third equality follows from the definitions of $H_t$ in Figure \ref{LHp}. The final equality follows since the whole rejection probability is at most $\epsilon$ and the verifier selects $H_t$ with probability $\Omega(1/T)$.
We can bound (\ref{eq;com1}) as follows. (\ref{eq;com2}) is bounded similarly;
\begin{equation}
\label{com-uni}
\begin{split}
&\||\psi_{t}\rangle-D_{k,j}^\dagger SWAP _{(k,j),\mathsf{Me}_k} D_{k,j}|\psi_{t-1}\rangle\|\\
&=\frac{1}{2}(\|D_{k,j}|\psi_t\rangle-SWAP _{(k,j),\mathsf{Me}_k} D_{k,j}|\psi_{t-1}\rangle\|+\|SWAP _{(k,j),\mathsf{Me}_k}D_{k,j}|\psi_{t}\rangle-D_{k,j}|\psi_{t-1}\rangle\|)\\
&\le\frac{\sqrt2}{2}\sqrt{\begin{split}&\|D_{k,j}|\psi_t\rangle-SWAP _{(k,j),\mathsf{Me}_k} D_{k,j}|\psi_{t-1}\rangle\|^2+\|SWAP _{(k,j),\mathsf{Me}_k}D_{k,j}|\psi_{t}\rangle-D_{k,j}|\psi_{t-1}\rangle\|^2\end{split}}\\
&=\frac{\sqrt2}{2}\||t\rangle(D_{k,j}|\psi_t\rangle-SWAP _{(k,j),\mathsf{Me}_k} D_{k,j}|\psi_{t-1}\rangle)+|t-1\rangle(D_{k,j}|\psi_{t-1}\rangle-SWAP _{(k,j),\mathsf{Me}_k}D_{k,j}|\psi_t\rangle)\|\\
&=\frac{\sqrt2}{2}\|| H_tD_{k,j}|\varphi\rangle\|=O(\sqrt{T\epsilon}).
\end{split}
\end{equation}
The first equality follows from modifying $\||\psi_{t}\rangle-U_t|\psi_{t-1}\rangle)\|$ to $\frac{1}{2}(\||\psi_{t}\rangle-U_t|\psi_{t-1}\rangle)\|+\||\psi_{t}\rangle-U_t|\psi_{t-1}\rangle)\|)$ and applying $D_{k,j}$ to the first term and $SWAP _{(k,j),\mathsf{Me}_k}D_{k,j}$ to the second term. The first inequality is the Cauchy-Schwarz, the second equality follows since $|t\rangle$ and $|t-1\rangle$ are orthogonal vectors. The third equality follows from the definition of $H_t$ of (b). The final equality follows since the whole rejection probability is $\epsilon$ or less and the verifier selects $H_t$ with probability $\Omega(1/T)$.
We can bound (\ref{eq;pr}) as follows;
\begin{equation}
\label{pr-uni}
\begin{split}
\||\psi_t\rangle-E_k^\dagger X_{\mathsf{Me}_k}E_k|\psi_{t-1}\rangle\|
&=\sqrt{2}\| |-\rangle\langle-|\bigl(|0\rangle_{\mathsf{C}_t} (|\psi_t\rangle-E_k^\dagger X_{\mathsf{Me}_k}E_k|\psi_{t-1}\rangle)\bigr)\|\\
&=\sqrt{2}\||-\rangle\langle-|_{\mathsf{C}_t}\bigl(\sqrt2 |+\rangle_{\mathsf C_t}X_{\mathsf{Me}_k}E_k|\psi_t\rangle -|0\rangle_{\mathsf C_t} (X_{\mathsf{Me}_k}E_k|\psi_{t}\rangle-E_k|\psi_{t-1}\rangle)\bigr)\|\\
&=\sqrt{2}\||-\rangle\langle-|_{\mathsf{C}_t}(|1\rangle_{\mathsf C_t}X_{\mathsf{Me}_k}E_k|\psi_t\rangle+|0\rangle_{\mathsf C_t} E_k|\psi_{t-1}\rangle)\|\\
&=\sqrt{2}\||-\rangle\langle-|_{\mathsf{C}_t} (|t\rangle X_{\mathsf{Me}_k}E_k|\psi_{t}\rangle+|t-1\rangle E_k|\psi_{t-1}\rangle)\|\\
&=\sqrt{2}\||-\rangle\langle-|_{\mathsf{C}_t} \bigl((CNOT_{\mathsf{C}_t,\mathsf{Me}_k}(|t\rangle(E_k|\psi_{t}\rangle+|t-1\rangle E_k|\psi_{t-1}\rangle)\bigr)\|\\
&=\sqrt{2}\|H_tE_k|\varphi\rangle\|=O(\sqrt{T\epsilon}).
\end{split}
\end{equation}
The first equality follows from adding register $\mathsf{C}_t$. The second equality follows from multiplying $|0\rangle_{\mathsf{C}_t} (|\psi_t\rangle-E_k^\dagger X_{\mathsf{Me}_k}E_k|\psi_{t-1}\rangle)$ by $-1$, applying $X_{\mathsf{Me}_k}E_k$ and $\langle-|+\rangle=0$. The third equality follows from $|+\rangle=\frac{|0\rangle+|1\rangle}{\sqrt2}$. The forth equality follows from $|t-1\rangle=|1\cdots1\rangle_{\mathsf{C}_1,\cdots,\mathsf{C}_{t-1}}|0\cdots 0\rangle_{\mathsf{C}_t,\cdots ,\mathsf{C}_{T+2pl+p}}$ and $|t\rangle=|1\cdots 1\rangle_{\mathsf{C}_1,\cdots,\mathsf{C}_t}|0\cdots 0\rangle_{\mathsf{C}_{t+1},\cdots,\mathsf{C}_{T+2pl+p}}$. The fifth equality follows from that $CNOT_{\mathsf{C}_t,\mathsf{Me}_k}$ acts as $X_{\mathsf{Me}_k}$ if and only if register $\mathsf{C}_t$ is $|1\rangle$. The last equality is the definition of $H_t$ of (c) and $E_k$.
The following equations evaluate the norm of $\Pi_{0^n} W |\varphi\rangle$.
\begin{equation}
\label{norm}
\begin{split}
&\!\!\!\! \|\Pi_{0^n}W|\varphi\rangle\|\\
&\ge1-\|({\rm Id}-\Pi_{0^n})W|\varphi\rangle\|\\
&=1-\|({\rm Id}-\Pi_{0^n})(\sum_{t=0}^T |t\rangle\langle t| \otimes U_1^\dagger\cdots U_t^\dagger+\Pi_{bad time}\otimes \mathrm{Id})(\sum_{t=0}^T|t\rangle|\psi_t\rangle+|\bot\rangle)\|\\
&=1-\sum_{t=0}^T\Bigl(\|({\rm Id}-\Pi_{0^n})|t\rangle\bigl(|\psi_0\rangle+\sum_{t'=1}^t (U_1^\dagger\cdots U_{t'}^\dagger|\psi_{t'}\rangle-{U_1}^\dagger\cdots U_{t'-1}^\dagger|\psi_{t'-1}\rangle)\bigr)\|\Bigr)-O(\sqrt{T^3\epsilon})\\
&\ge1-\sum_{t=0}^T\Bigl(\|({\rm Id}-\Pi_{0^n})|\psi_0\rangle\|+\sum_{t'=1}^t \|({\rm Id}-\Pi_{0^n})(U_1^\dagger\cdots U_{t'}^\dagger|\psi_{t'}\rangle-{U_1}^\dagger\cdots U_{t'-1}^\dagger|\psi_{t'-1}\rangle\bigr)\|\Bigr)-O(\sqrt{T^3\epsilon}).\\
\end{split}
\end{equation}
Here, we ignore the term $\sum_{t'=1}^t$ for $t=0$ for the ease of notation. The first inequality follows from the triangle inequality. The first equality follows from the definition of $W$ in (\ref{opr}) and of $|\varphi\rangle$ in (\ref{varphi}). The second equality follows from $U_1^\dagger\cdots U_t^\dagger|\psi_t\rangle=|\psi_0\rangle +\Sigma_{t'=1}^t(U_1^\dagger\cdots U_{t'}^\dagger|\psi_{t'}\rangle-{U_1}^\dagger\cdots U_{t'-1}^\dagger|\psi_{t'-1}\rangle)$ and $\||\bot\rangle\|=O(\sqrt{T^3\epsilon})$ by (\ref{boteval}). The second inequality is the triangle inequality.
The term $\|(\mathrm{Id}-\Pi_{0^n})|\psi_0\rangle\|$ in the last line of (\ref{norm}) is bounded by test (f) in Figure \ref{LHp} as follows;
\begin{equation}
\label{term1}
\|(\mathrm{Id}-\Pi_{0^n})|\psi_0\rangle\|\le\sum_{t\in[2T+1,2T+n]}\|(\mathrm{Id}-\Pi_0^{t-2T})|\psi_0\rangle\| \le O(n\sqrt{T\epsilon}).
\end{equation}
The second inequality follows since each term in the summation of (\ref{term1}) is the rejection probability of each $t$ in test (f) in Figure \ref{LHp}.
The terms $\|(\mathrm{Id}-\Pi_{0^n})(U_T\cdots U_{t'+1}|\psi_{t'}\rangle-U_T\cdots U_{t'}|\psi_{t'-1}\rangle)\|$ in the same line of (\ref{norm}) are bounded as follows;
\begin{equation}
\label{term2}
\|(\mathrm{Id}-\Pi_{0^n})(U_T\cdots U_{t'+1}|\psi_{t'}\rangle-U_T\cdots U_{t'}|\psi_{t'-1}\rangle)\| \le \||\psi_{t'}\rangle-U_{t'}|\psi_{t'-1}\rangle\|\le O(\sqrt{T\epsilon}).
\end{equation}
The first inequality follows by omitting the projection $(\mathrm{Id}-\Pi_{0^n})$ and applying ${U_{t'+1}}^\dagger\cdots {U_T}^\dagger$. The second inequality follows since the term $\||\psi_{t'}\rangle-U_{t'}|\psi_{t'-1}\rangle\|$ equals to one of (\ref{eq;vch},\ref{eq;com1},\ref{eq;com2},\ref{eq;pr}), and (\ref{eq;vch},\ref{eq;com1},\ref{eq;com2},\ref{eq;pr}) are bounded by (\ref{v-uni},\ref{com-uni},\ref{pr-uni}).
From (\ref{norm},\ref{term1},\ref{term2}) and the assumption $n\le T$, we have the following estimation;
\begin{equation}
\|\Pi_{0^n}W|\varphi\rangle\|=1-O(nT\sqrt{T\epsilon})-O(T^2\sqrt{T\epsilon})=1-O(\sqrt{T^5\epsilon}).
\label{linephi}
\end{equation}
From this, the next estimation follows;
\begin{equation}
\label{norm2}
\|(\mathrm{Id}-\Pi_{0^n}) W |\varphi\rangle\|^2=1-\|\Pi_{0^n}W|\varphi\rangle\|^2=O(\sqrt{T^5\epsilon}).
\end{equation}
Finally, we bound the rejection probability $p_{rej}$. Let $\Pi_{rej}$ be the projection onto the rejection of $\cal{P}$. Then,
\begin{equation}
\label{rej}
\begin{split}
\sqrt{p_{rej}}&=\|\Pi_{rej}U_{T}\cdots U_1|\overline{\varphi}\rangle\|\\
&=(1+O(\sqrt{T^5\epsilon}))\|\Pi_{rej}U_T\cdots U_1\Pi_{0^n}W|\varphi\rangle\|\\
&\le(1+O(\sqrt{T^5\epsilon}))(\|\Pi_{rej}U_T\cdots U_1W|\varphi\rangle\|+\|\Pi_{rej}U_T\cdots U_1({\rm Id}-\Pi_{0^n})W|\varphi\rangle\|)\\
&\le(1+O(\sqrt{T^5\epsilon}))\bigl(\|\Pi_{rej}U_T\cdots U_1W|\varphi\rangle\|+O(\sqrt[4]{T^5\epsilon})\bigr)\\
&=\|\Pi_{rej}\sum_{t=0}^T U_T\cdots U_{t+1}|\psi_t\rangle\|+O(\sqrt{T^3\epsilon})+O(\sqrt[4]{T^5\epsilon})\\
&\le\sum_{t=0}^T\Bigl(\|\Pi_{rej}|\psi_T\rangle\|+\sum_{t'=t+1}^T \|\Pi_{rej}U_T\cdots U_{t'+1}|\psi_{t'}\rangle-\Pi_{rej}U_T\cdots U_{t'}|\psi_{t'-1}\rangle\|\Bigr)+O(\sqrt[4]{T^5\epsilon}).\\
\end{split}
\end{equation}
Here, we ignore the term $\sum_{t'=t+1}^T$ for $t=T$ for the ease of notation. The second equality follows from (\ref{linephi}) and the definition of $|\overline\varphi\rangle$ in (\ref{philine}). The first inequality follows from the triangle inequality. The second inequality follows from (\ref{norm2}). The third equality follows from the definition of $W$ in (\ref{opr}), the definition of $|\varphi\rangle$ in (\ref{varphi}), and $\||\bot\rangle\|=O(\sqrt{T^3\epsilon})$ by (\ref{boteval}). The last inequality follows from the triangle inequality.
The term $\|\Pi_{rej}|\psi_T\rangle\|$ in the last line of (\ref{rej}) is bounded by test (d) in Figure \ref{LHp} as follows;
\begin{equation}
\label{term3}
\|\Pi_{rej}|\psi_T\rangle\|\le O(\sqrt{T\epsilon}).
\end{equation}
The terms $\|\Pi_{rej}U_T\cdots U_{t'+1}|\psi_{t'}\rangle-\Pi_{rej}U_T\cdots U_{t'}|\psi_{t'-1}\rangle\|$ in the same line can be bounded by (\ref{v-uni},\ref{com-uni},\ref{pr-uni}) as follows;
\begin{equation}
\label{term4}
\|\Pi_{rej}U_T\cdots U_{t'+1}|\psi_{t'}\rangle-\Pi_{rej}U_T\cdots U_{t'}|\psi_{t'-1}\rangle\| \le \||\psi_{t'}\rangle-U_{t'}|\psi_{t'-1}\rangle\|\le O(\sqrt{T\epsilon}).
\end{equation}
From (\ref{rej},\ref{term3},\ref{term4}), $\sqrt{p_{rej}}\le O(\sqrt[4]{T^5\epsilon})$ follows.
\end{proof}
\subsection{Addition of the GHZ test}
Second, we construct the LHI+ protocol: we add the GHZ test which is an analogue of the GHZ test of Theorem 5 to the LHI protocol, which is described in Figure \ref{int}. The honest verifier of the LHI protocol sends all provers (except prover 0) the same bits, but the malicious verifier may send different bits. To prevent this attack, the provers control which term $H_t$ is measured by the verifier in Figure \ref{LHp}(A). To avoid also that the provers are malicious, we use the GHZ test, which guarantees that the provers really chooses $H_t$ uniformly at random. The reason why we do not use the coin-flipping protocol to decide $t$ is that we do not know any multi-parity coin-flipping protocol among the provers and the verifier. For example, if we use the two-party coin-flipping protocol between each of the provers and the verifier, the malicious verifier could choose different $t$ depending on the provers. The analysis of the LHI+ protocol is almost the same as the proof of Theorem 4.
\begin{lemm}If the completeness/soundness of the protocol in Figure \ref{LHp} are $1-\epsilon/1-\delta$, then the completeness/soundness of the LHI+ protocol in Figure \ref{int} are $1-\frac{\epsilon}{2}$/$1-\frac{\delta^2}{8}$.
\end{lemm}
\begin{proof}
The completeness is obvious. We show the soundness. The analysis is almost the same as the proof of Theorem 5. The only differences are the definition of $\Pi_{GHZ}$ and $\Pi_{rej}$. Let $U_g^i$ be the unitary operator that prover $i$ uses for the GHZ test and $U_h^i$ be the unitary operator that prover $i$ uses for the history test for $i=1,...,p$.
Let $\rho'':=\rm{Tr}_{GHZ}$$(\Pi_{GHZ}(\otimes_{i=1}^p U_g^i)\rho(\otimes_{i=1}^p {U_g^i}^\dagger)\Pi_{GHZ})$. Here, $\rho$ is the initial state of the provers in the LHI+ protocol, $\Pi_{GHZ}$ is the projection onto $(\frac{|0^{p+1}\rangle+|1^{p+1}\rangle}{\sqrt2})^{\otimes u}$ of ($\mathsf{G}_0, \mathsf{G}_1, \mathsf{G}_2,...,\mathsf{G}_p$), and $\rm Tr_{GHZ}$ means the traceout of registers ($\mathsf{G}_0, \mathsf{G}_1, \mathsf{G}_2,...,\mathsf{G}_p$). Denote $\rm Tr$$\rho''=1-\epsilon_1$, where $\epsilon_1$ is the rejection probability of the GHZ test. Let $\frac{\rho''}{1-\epsilon_1}:=\rho'$ be the initial state of provers. Let $\epsilon_2$ be the rejection probability of the history test.
Now we construct the strategy of the LHI protocol from the strategy of the LHI+ protocol. The private registers of prover $i$ are $\mathsf{P}_i,\mathsf{G}_i$, for $i=1,...,p$. Initially prover 0 has only register $\mathsf{V}$. The initial state in ($\mathsf{P}_1,...,\mathsf{P}_p,\mathsf{M}_1,...,\mathsf{M}_p$) is $\rho'$. If the verifier sends $b\in\{0,1\}^u$ to prover $i$ ($i=1,...,p$), prover $i$ sets $|b\rangle$ in register $\mathsf{G}_i$, applies $U^i_h{U^i_g}^\dagger$ and sends $\mathsf{M}_i$ to the verifier. Prover 0 always sends $\mathsf{V}$. The operations by provers are as follows;
\begin{equation}\rho'\rightarrow(\otimes_{i=1}^p U_h^i{U_g^i}^\dagger) ((|b\rangle\langle b|)^{\otimes p}\otimes \rho') (\otimes_{i=1}^p U_g^i{U_h^i}^\dagger).
\end{equation}
The rejection probability of the LHI protocol $p_{rej,lh}$ is bounded by the rejection probability $p_{rej}$ of the LHI+ protocol as follows;
\begin{equation}
\begin{split}
p_{rej,lh}&:=\frac{1}{2^u}\sum_{b\in\{0,1\}^u}\mathrm{Tr}[\Pi_{rej}(\otimes_{i=1}^p U_h^i {U_g^i}^\dagger) ((|b\rangle\langle b|)^{\otimes p}\otimes \rho') (\otimes_{i=1}^p U_g^i{U_h^i}^\dagger)]\\
&=\frac{1}{1-\epsilon_1}\mathrm{Tr} [\Pi_{rej}(\otimes_{i=1}^p U_h^i{U_g^i}^\dagger)\Pi_{GHZ}(\otimes_{i=1}^p U_g^i)\rho (\otimes_{i=1}^p {U_g^i}^\dagger)\Pi_{GHZ}(\otimes_{i=1}^p U_g^i{U_h^i}^\dagger)]\\
&\le \sqrt\epsilon_1+\mathrm{Tr} [\Pi_{rej}(\otimes_{i=1}^p U_h^i{U_g^i}^\dagger)(\otimes_{i=1}^p U_g^i)\rho (\otimes_{i=1}^p {U_g^i}^\dagger)(\otimes_{i=1}^p {U_g^i}{U_h^i}^\dagger)]\\
&= \sqrt\epsilon_1+\mathrm{Tr}[\Pi_{rej}(\otimes_{i=1}^p U_h^i)\rho(\otimes_{i=1}^p {U_h^i}^\dagger)]=\sqrt\epsilon_1+\epsilon_2\le\sqrt\epsilon_1+\sqrt\epsilon_2\le2\sqrt {2p_{rej}}.
\end{split}
\end{equation}
The first equality follows from that $\frac{1}{2}(|0^p\rangle\langle0^p|+|1^p\rangle\langle1^p|)$ equals to the $p$-qubits substate of the GHZ state. The first inequality follows from that $\|\rho'-\rho\|\le\sqrt\epsilon_1$ by Lemma \ref{2PO}.
\end{proof}
\begin{figure}
\hrulefill\\
LHI+ protocol.\\\\
We can assume $2^u=n+2T$ for some integer $u$ without loss of generality. Let $\mathsf{G}_i$ $(i=0,...,p)$ be registers each of which consists of $u$ qubits.
\begin{enumerate}
\item
Select $b\in\{0,1\}$ uniformly at random. Send $b$ to provers $1,...,p$ and nothing to prover 0.
\item
Do the following test depending on $b=0,1$;
\begin{enumerate}
\item[(1)]
($b=0$: GHZ test)\\
(1.1) Prover $i$ ($i=1,...,p$) sends $\mathsf{G}_i$. Prover 0 sends $(\mathsf{V}, \mathsf{M}_1,..., \mathsf{M}_p$, $\mathsf{C}_1,...,\mathsf{C}_T)$ and $\mathsf{G}_0$.\\
(1.2) Measure $\mathsf{G}_0,\mathsf{G}_1,...,\mathsf{G}_p$ by the projection onto $\frac{1}{\sqrt2}(|0^{p+1}\rangle+|1^{p+1}\rangle)^{\otimes u}$. If $(\frac{1}{\sqrt2} (|0^{p+1}\rangle+|1^{p+1}\rangle))^{\otimes u}$ is measured, then accept. Otherwise reject.
\item[(2)]
($b=1$: History test)\\
(2.1) Prover $i$ ($i=1,...,p$) sends $\mathsf{Me}_i$ and $\mathsf{G}_i$. Prover 0 sends $(\mathsf{V}, \mathsf{M}_1,..., \mathsf{M}_p$, $\mathsf{C}_1,...,\mathsf{C}_T)$ and $\mathsf{G}_0$.\\
(2.2) Measure $\mathsf{G}_0$ in the computational basis. Let $t$ be the outcome. Measure by $H_t$ in Figure \ref{LHp}. If the outcome is accepted by the protocol in Figure 5, then accept. Otherwise reject.
\end{enumerate}
\end{enumerate}
\hrulefill
\caption{LHI+ protocol. Note that we add one prover and the GHZ test to the LHI protocol.}
\label{int}
\end{figure}
\subsection{Sketch of the technique of \cite{BJSW} and why it works for our protocol\label{Sketch}}
Finally, we give the zero-knowledge protocol for $\sf QMIP^*$ based on the LHI+ protocol, following the zero-knowledge protocol by Broadbent et al. \cite{BJSW}. In this subsection, we roughly sketch the quantum zero-knowledge protocol by Broadbent et al.\cite{BJSW} and why the technique of \cite{BJSW} works for our protocol.
The result of \cite{BJSW} consists of following ingredients;
\begin{enumerate}
\item The verifier tries to verify a restricted form of the Local Hamiltonian problem, called the Clifford Hamiltonian problem, which is shown to be $\sf{QMA}$ hard. To this end, the verifier only has to measure only Clifford Hamiltonians.
\item The honest prover encodes the witness of the Clifford Hamiltonian problem by a CSS code \cite{NS1}, a quantum one time pad and a permutation. The quantum one time pad and the permutation are the secret key of the encoding.
\item The prover sends the encoded witness and the commitment of the key of the encoding. The verifier measures the encoded witness by one of Hamiltonians and sends the output to the prover. The prover proves that the output of the verifier's measurement corresponds to a yes output of the original Clifford Hamiltonian problem by a zero-knowledge protocol for NP \cite{W2}. This correspondence critically uses the restriction on Hamiltonians and the transversality for Clifford gates which is a characteristic of CSS codes.
\item The malicious verifier's circuits can be replaced by simulators by the assumption of the existence of commitment schemes, and after the replacement of the verifier's circuits, the witness state can be replaced by a state preparable in polynomial time.
\end{enumerate}
If the malicious verifier of the LHI protocol can send only the honest query, the analogues of the above items in our case are as follows;
\begin{enumerate}
\item The verifier of the LHI protocol also measures by only Clifford Hamiltonians.
\item The encoding step of each prover does not depend on the other provers, and hence similar encoding can be done.
\item The corresponding step can be done by one of the provers directly.
\item If the query of the (malicious) verifier is honest and the qubits sent from the provers is honest, this step also can be done directly.
\end{enumerate}
As we cannot assume in general that the malicious verifier of the LHI protocol can send only the honest query, we do not directly use the LHI protocol but the LHI+ protocol, which adds the GHZ test.
\subsection{Final zero-knowledge protocol: the technique of Broadbent et al. \cite{BJSW}\label{Fin;sec}}
We construct the zero-knowledge protocol based on the protocol in Figure \ref{int}. The technique is almost the same as that of Broadbent et al. \cite{BJSW} and the analysis is also almost the same. In this paper, we construct the protocol and explain why the technique of \cite{BJSW} works. The summary of the protocol is given in Figure \ref{Final} and we describe the protocol here.
Let $\mathsf{X}=(\mathsf{X}_1,...,\mathsf{X}_{i_0+p})$ be $(i_0+p)$ registers each of which has 1 qubit, where $i_0$ is the number of qubits of registers ($\mathsf{V},\mathsf{M}_1,...,\mathsf{M}_p,\mathsf{C}_1,...,\mathsf{C}_T$) in Figure \ref{int}. Prover 0 has $(\mathsf{X}_1,...,\mathsf{X}_{i_0})$ that corresponds to ($\mathsf{V},\mathsf{M}_1,...,\mathsf{M}_p,\mathsf{C}_1,...,\mathsf{C}_T$). For $i=1,...,p$, prover $i$ has $\mathsf{X}_{i_0+i}$ that corresponds to register $\mathsf{Me}_i$ in Figure \ref{int}.
\subsubsection{Verifier's message\label{Vmes}}
First, the verifier selects $\overline{b}\in\{0,1\}$ uniformly at random and sends $\overline{b}$ to provers $1,...,p$. Note that $\overline{b}=0$ corresponds to the GHZ test and $\overline{b}=1$ corresponds to the history test. The verifier sends nothing to prover 0.
\subsubsection{Provers' encoding\label{Penc}}
Prover $i$ $(1\le i\le p)$ who received $\overline{b}=0$ sends $\mathsf{G}_i$. Prover $i$ who received $\overline{b}=1$ and prover 0 encode $\mathsf{X}$ in four steps. Let $N$ be the length of a concatenated Steane code that the provers use. Here, a concatenated Steane code is a code such that 1 qubit is encoded by the 7-qubit Steane code several times repeatedly. In \cite{BJSW}, $N$ is taken to be an even power of 7 and bounded by a polynomial in instance size.
\begin{enumerate}
\item For $i=1,...,i_0+p$, $\mathsf{X}_i$ is encoded by the concatenated Steane code to the $N$ qubit state $(\mathsf{Y}_1^i,...,\mathsf{Y}_N^i)$.
\item For each $i$, the provers concatenate additional $N$ qubits to $(\mathsf{Y}_1^i,...,\mathsf{Y}_N^i)$, each of which is chosen from $|0\rangle,|+\rangle$,$|\!\!\circlearrowright\rangle(:=\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle))$ uniformly at random. We call these $N$ qubits as trap qubits. At this point, each $\mathsf{X}_i$ is transfered to $2N$ qubits $(\mathsf{Y}_1^i,...,\mathsf{Y}_{2N}^i)$. The provers store the string $r=r_1\cdots r_{i_0+p}$, where $r_i\in\{0,+,\circlearrowright\}^N$, representing the trap qubits.
\item Select a permutation $\pi\in S_{2N}$ uniformly at random. All provers use the same permutation $\pi$. Permute $2N$ qubits $(\mathsf{Y}_1^i,...,\mathsf{Y}_{2N}^i)$ by $\pi$.
\item Select strings $a=a_1\cdots a_{i_0+p}$, $b=b_1\cdots b_{i_0+p}$, where $a_1,... ,a_{i_0+p},b_1,...,b_{i_0+p}\in\{0,1\}^{2N}$, uniformly at random. Apply quantum one time pad on $(\mathsf{Y}_1^i,...,\mathsf{Y}_{2N}^i)$. Namely, apply $X^{a_i}Z^{b_i}$ on $(\mathsf{Y}_1^i,...,\mathsf{Y}_{2N}^i)$, where for strings $a_i=a^1_i\cdots a^{2N}_i,b^1_i\cdots b^{2N}_i$, $X^{a_i}Z^{b_i}$ means $X^{a^1_i}Z^{b^1_i}\otimes\cdots\otimes X^{a^{2N}_i}Z^{b^{2N}_i}$. Each of provers $0,2,...,p$ sends each encoded qubits and GHZ qubits. Prover 1 sends his encoded qubits, GHZ qubits and the commitment $z=commit(\pi,a,b,s)$. Here, $commit(\pi,a,b,s)$ is a string to commit the string $(\pi,a,b)$ with a random string $s$ using the commitment scheme assumed in Theorem 2.
\end{enumerate}
\begin{figure}
\hrulefill\\
Preparation \\\\
Provers $0,1,...,p$ select and share a tuple $(r,\pi,a,b)$ uniformly at random, where $r=r_1\cdots r_{i_0+p}$ for $r_1,..., r_{i_0+p}\in\{0,+,\circlearrowright\}^N$, $\pi\in S_{2N}$, and $a=a_1\cdots a_{i_0+p}$, $b=b_1\cdots b_{i_0+p}$ for $a_1,... ,a_{i_0+p},b_1,...,b_{i_0+p}\in\{0,1\}^{2N}$.
The provers will use this random string to encode their qubits which correspond to the qubits that the provers in the LHI+ protocol would send. The encoding process is described in Section \ref{Penc}.\\
\hrulefill\\
Protocol\\
Select $\overline{b}=0,1$ uniformly at random. Send $\overline{b}$ to prover $1,...,p$, and nothing to prover 0.
Do the following test depending on $\overline{b}=0,1$:
\begin{enumerate}
\item($\overline{b}=0$: GHZ test)
\begin{enumerate}
\item[(1.1)] Prover $i$ ($i=1,...,p$) sends $\mathsf{G}_i$. Prover 0 encodes $(\mathsf {X}_0,...,\mathsf{X}_{i_0})$
$(=($$\mathsf{V}, \mathsf{M}_1,..., \mathsf{M}_p,\mathsf{C}_1,...,\mathsf{C}_T))$ as described in Section \ref{Penc} and sends these qubits and $\mathsf{G}_0$.
\item[(1.2)] Measure $\mathsf{G}_0,\mathsf{G}_1,...,\mathsf{G}_p$ by the projection onto $\frac{1}{\sqrt2}(|0^{p+1}\rangle+|1^{p+1}\rangle)^{\otimes u}$. If $\frac{1}{\sqrt2} (|0^{p+1}\rangle+|1^{p+1}\rangle)^{\otimes u}$ is measured, then accept. Otherwise reject.
\end{enumerate}
\item($\overline{b}=1$: History test)
\begin{enumerate}
\item[(2.1)] Prover $i$ ($i=1,...,p$) encodes $\mathsf{X}_{i_0+i}(=\mathsf{Me}_i)$ described in Section \ref{Penc} and sends these qubits and $\mathsf{G}_i$. Prover 1 additionally sends $z=commit(\pi,a,b,s)$. Prover 0 encodes $\mathsf {X}_0,...,\mathsf{X}_{i_0}(=(\mathsf{V},\mathsf{M}_1,..., \mathsf{M}_p,\mathsf{C}_1,...,\mathsf{C}_T))$ as described in Section \ref{Penc} and sends these qubits and $\mathsf{G}_0$.
\item[(2.2.1)] The verifier measures $\mathsf{G}_0$ in the computational basis. Let $t$ be the outcome.
\item[(2.2.2)] Prover 1 and the verifier engage in a coin-flipping protocol, choosing a two bit string $v$ uniformly at random. Here, $v$ specifies one of the Clifford gates of $H_t=\sum_v H'_{t,v}$. $H'_{t,v}$ is described in Section \ref{Vmea}.
\item[(2.2.3)] The verifier applies the Clifford operation $C_{t,v}$ transversally to the qubits $(\mathsf{Y}_1^{i_1}$,...,$\mathsf{Y}_{2N}^{i_1})$, $(\mathsf{Y}_1^{i_2},...,\mathsf{Y}_{2N}^{i_2})$,..., $(\mathsf{Y}_1^{i_k},...,\mathsf{Y}_{2N}^{i_k})$ as described in Section \ref{Vmea} and measures all of these qubits in the computational basis, for $(i_1,...,i_k)$ being the indices of the qubits upon which the Hamiltonian term $H_{t,v}$ acts non-trivially. The verifier sends the output to prover 1.
\item[(2.2.4)] Prover 1 checks whether the output sent from the verifier is consistent with the trap qubits and Steane code (described in Section \ref{Pche}). If they are inconsistent, then abort. If they are consistent, prover 1 proves that the output corresponds to a yes output of the LHI+ protocol by a zero-knowledge protocol of NP.
\end{enumerate}
\end{enumerate}
\hrulefill\\
\caption{Summary of Zero-Knowledge Protocol for $\sf QMIP^*$}
\label{Final}
\end{figure}
\subsubsection{Verifier's measurement\label{Vmea}}
If the verifier sends $\overline{b}=0$ (the GHZ test), then he/she measures $\mathsf{G}_0,...,\mathsf{G}_p$ as in the LHI+ protocol.
If the verifier sends $\overline{b}=1$ (the history test), then the verifier measures the state received from provers as follows: $H_t$ is the summation of at most four Clifford gates, that is, $H_t=\sum_v H'_{t,v}$, where $H'_{t,v}$ is a Clifford gate and $v\in\{00,01,10,11\}$ (see Section \ref{CliH} in this paper and Section 2 in \cite{BJSW}). First, the verifier decides $t$ by measuring in the computational basis on $\mathsf{G}_0$ and $v$ by the coin-flipping protocol with prover 1 (i.e., the honest prover 1 commits to random $s_1,s_2 \in\{0,1\}$, the honest verifier selects $s'_1,s'_2\in \{0,1\}$ at random, the prover reveals $s_1,s_2$, and the two participants agree that the random bits are $v_i = s_i \oplus s'_i$ for $i=1,2$). The verifier measures the encoded witness transversally by the projection $H'_{t,v}$. Measuring $\sf G_0$ in the computational basis decides $t$ and coin-flipping decides $v$. Here, the measurement by $H'_{t,v}$ means applying the corresponding Clifford gate $C_{t,v}$ to $(\mathsf{Y}_1^{i_1}$,...,$\mathsf{Y}_{2N}^{i_1})$, $(\mathsf{Y}_1^{i_2},...,\mathsf{Y}_{2N}^{i_2})$,..., $(\mathsf{Y}_1^{i_k},...,\mathsf{Y}_{2N}^{i_k})$ and applying the projection onto computational basis. Here, $i_1,\ldots i_k$ is the indices of qubits on which $H'_{t,v}$ non-trivially acts. The output is a string in $\{0,1\}^{2kN}$. Denote the output $u=u_{i_1}\cdots u_{i_k}$, where $u_{i_1},...,u_{i_k}\in\{0,1\}^{2N}$.
\subsubsection{Provers' Check and Response\label{Pche}}
The verifier sends $k$ strings in $\{0,1\}^{2N}$ to prover 1. There uniquely exist strings $y_i,z_i\in\{0,1\}^N$ such that $\pi (y_i z_i)=u_i$ for each $u_i\in\{0,1\}^N$. Prover 1 continues if the following conditions are satisfied.
\begin{enumerate}
\item For all $i\in\{i_1,\ldots,i_k\}$, $y_i\in D_N$, and at least one $i\in\{i_1,\ldots,i_k\}$, $y_i\in D_N^1$, where $D_N$ is a subset of $\{0,1\}^N$ representing classical codewords of the concatenated Steane code and $D_N^1$ is a subset of $D_N$ corresponding to logical bit 1.
\item $\langle z_{i_1}\cdots z_{i_k}|C_{t,v}^{\otimes N}|r_{i_1}\cdots r_{i_k}\rangle\neq0$.
\end{enumerate}
We define the predicate $R_{t,v}(r,u,\pi)$ which takes the value 1 iff the above two conditions hold. Assume that $R_{t,v}(r,u,\pi)=1$ and prover 1 continues the protocol. For any $a=a_1\cdots a_{i_0+p},b=b_1\cdots b_{i_0+p}$, there uniquely exist $\alpha\in\{\pm i, \pm1\}$ and $c_1,...,c_{i_0+p},d_1,...,d_{i_0+p}\in\{0,1\}^{2N}$ such that the next equation holds and can be computed in polynomial time in $N$.
\begin{equation}
\label{pred}
C_t^{\otimes 2N}(X^{a_1}Z^{b_1}\otimes\cdots\otimes X^{a_{i_0+p}}Z^{b_{i_0+p}})=\alpha(X^{c_1}Z^{d_1}\otimes\cdots\otimes X^{c_{i_0+p}}Z^{d_{i_0+p}}) C_t^{\otimes 2N}
\end{equation}
That is, the following statement is a NP statement: there are a string $s$ and a tuple $(\pi, a,b,r)$ such that $commit(\pi,a,b,s)=z$ and $R_{t,v}(r,u\oplus c_1,...,c_{i_0+p},\pi)=1$, where $c$ is defined by Eq.(\ref{pred}). Prover 1 convinces the verifier of this statement by a zero-knowledge protocol of NP.
\subsection{Analysis}
As we mentioned before, the analysis is almost the same as \cite{BJSW}. Hence we explain only the main difference.
\subsubsection{Clifford gates and Clifford Hamiltonians\label{CliH}}
The zero-knowledge protocol by Broadbent et al. \cite{BJSW} critically uses the condition that all Hamiltonians consist of Clifford gates and projections onto computational basis. Here we prove that the following Hamiltonians are the sums of Clifford Hamiltonian for $U_t=SWAP, CNOT$. $SWAP$ and $CNOT$ can be easily constructed by the product of unitary operators of $\{H\otimes H ,\Lambda(P)\}$. Hence this step is not essentially necessary, but we prove this to simplify the LHI+ protocol. Now $H_t$ is defined as follows;
\begin{equation*}
H_t=|10\rangle_{\mathsf{C}_{t-1},\mathsf{C}_{t+1}}\langle10|_{\mathsf{C}_{t-1},\mathsf{C}_{t+1}}\otimes(|0\rangle\langle0|_{\mathsf{C}_t}\otimes \mathrm{Id}+|1\rangle\langle1|_{\mathsf{C}_t}\otimes \mathrm{Id}-|1\rangle\langle0|_{\mathsf{C}_t}\otimes U_t-|0\rangle\langle1|_{\mathsf{C}_t}\otimes U_t^\dagger)\end{equation*}
$H_t$ acts on $|\rangle_{\mathsf{C}_{t-1},\mathsf{C}_{t+1}}$ as a trivial projection onto computational basis, and hence we consider only other three qubits. For $U_t=SWAP$, $H_t$ is the sum of the projections onto the following vectors;\\\\
\{$|+\rangle(|01\rangle-|10\rangle)$, $|-\rangle|00\rangle$, $|-\rangle|11\rangle$, $|-\rangle(|01\rangle+|10\rangle)$\}.\\\\
Here, $|+\rangle(|01\rangle-|10\rangle)=|+\rangle ((\mathrm{Id} \otimes X) CNOT|-0\rangle)$, and $|-\rangle(|01\rangle+|10\rangle)=|-\rangle ((\mathrm{Id} \otimes X) CNOT|+0\rangle)$ and hence these projections are Clifford Hamiltonians. The control qubits of the CNOT in these operators is the left qubit.
For $U_t=CNOT$, $H_t$ is the sum of the projections onto the following vectors;\\\\
\{$|+1-\rangle$, $|-1+\rangle$, $|-0+\rangle$, $|-0-\rangle$\}.
\subsubsection{Soundness}
In the analysis of Broadbent et al.'s protocol \cite{BJSW}, the prover can prepare the state accepted by the original local Hamiltonian test with high probability by decoding the encoded qubits which can pass their zero-knowledge protocol with high probability. The decoding process is applied to each logical qubit isolatedly. Hence, if the provers in Figure \ref{Final} pass with high probability, then the provers can also pass the LHI protocol with high probability by decoding the qubits in Figure \ref{Final}.
\subsubsection{Zero-Knowledge}
Finally, we discuss zero-knowledge. Similarly to the proof of Theorem 4, we only have to consider the case that the malicious verifier requires provers $1,...,p$ to do the history test, and we can assume that prover $i$ who received $\overline{b}=1$ measures $\mathsf{G}_i$ in the computational basis. In the case of the history test, honest provers send the state that depends on the uniform random variable $t\in[{2^u}]$. The state may depend on $t$, but the analysis of \cite{BJSW} showed that there is only negligible change of the outputs of the simulator if the honest measurement on the state can pass the prover's check with high probability.
\section{Conclusion}
There are obvious open problems: whether there exist the statistical/perfect zero-knowledge systems of $\mathsf{QMIP^*}$. One possible method is the algebraic technique. In quantum complexity theory, the technique of enforcing algebraic structures on the strategy of provers is recently applied to investigate the power of $\mathsf{MIP^*}$ (to prove $\mathsf{NEXP}$ $\subseteq\mathsf{MIP^*}$\cite{IV,NV,Vid}, to construct short proofs for $\sf{QMA}$ with large completeness-soundness gap \cite{NV2}, and to prove $\mathsf{NEXP}$ $\subseteq\mathsf{MIP^*}$ with zero-knowledge \cite{CFGS}, for example). Extensions of such algebraic methods to history states of $\mathrm{QMIP^*}$ protocols may enable perfect zero-knowledge systems for $\mathsf{QMIP^*}$.
The parameters will not be optimal. Though most improvements of parameters will directly follow from improvements of $\mathsf{QMIP^*}$ protocols without zero-knowledge condition, we note an important problem related to zero-knowledge. Parallel repetition is a direct tool to improve completeness/soundness gap. Parallel repetition of zero-knowledge protocols, however, may not preserve zero-knowledge even in single-prover classical zero-knowledge systems (\cite{Gol}, Section 4.5.4). Hence it might be difficult to improve completeness/soundness gap preserving the number of turns by parallel repetition.
Finally, we believe that LHI protocols of interactive proofs would be a powerful tool which makes much previous research for Local Hamiltonian problems applicable to interactive proof systems.
\subsection*{Acknowledgments}
The author is grateful to Prof. Nishimura for useful discussion, careful reading and heavy revision of this paper.
\newpage
|
{
"timestamp": "2019-03-01T02:06:56",
"yymm": "1902",
"arxiv_id": "1902.10851",
"language": "en",
"url": "https://arxiv.org/abs/1902.10851"
}
|
\section{Introduction}\label{intro}
\setcounter{equation}{0}
As a prototype of problems with interface singularities, this paper studies {\em a priori} error estimates of mixed finite element methods for the following interface problem (i.e., the diffusion problem with discontinuous coefficients):
\begin{equation}\label{scalar}
-\nabla\cdot \,(\alpha(x)\nabla\, u) = f
\quad \mbox{in} \,\,\Omega
\end{equation}
with homogeneous Dirichlet boundary conditions (for simplicity)
\begin{equation}\label{bc1}
u = 0 \quad \mbox{on } \partial \O,
\end{equation}
where $\Omega$ is a bounded polygonal domain in $\rm I\kern-.19emR^d$ with $d=2$ or $3$; $f \in L^{2}(\O)$ is a given function; and diffusion coefficient $\alpha(x)$ is positive and piecewise constant with possible large jumps across subdomain boundaries (interfaces):
\[
\alpha(x)=\alpha_i > 0\quad\mbox{in }\,\O_i
\quad\mbox{for }\, i=1,\,...,\,n.
\]
Here, $\{\Omega_i\}_{i=1}^n$ is a partition of the domain $\O$ with $\O_i$ being an open polygonal domain.
It is well known that the solution $u$ of problem (\ref{scalar}) belongs to $H^{1+s}(\O)$ with possibly very small $s> 0$, see for example Kellogg \cite{Kel:75}. But we should also note that even the global regularity is low, when a finite element mesh is given, the singularity or those elements whose solution having a large gradient often only appear bear some points, or along a curve. Thus it is a bad idea to use the global regularity and a global uniform mesh-size to do the a priori error estimate.
In \cite{CHZ:17}, we introduced the idea of robust and local optimal a priori error estimate. The robustness means that the genetic constants appeared in the estimates are independent of the parameters of the equation, the coefficient $\alpha$ in our case. The local optimality means that in the error estimate, the upper bound is optimal with the regularity of each element and local mesh sizes, instead of using a global uniform mesh size and a global regularity.
The local optimal and robust a priori error estimate is very important for the adaptive mesh refinement algorithm. Since that all mesh refinements algorithms are based on the so-called "error equi-distribution" principle \cite{NoVe:12}, that is, each element has an almost equal size of the error measured in an appropriate norm, we need to show this is possible via a priori error estimate. In some sense, if we have a known exact solution $u$ so that the a priori error bound can be computed exactly, we should be able to find an optimal mesh with a fixed number of degrees of freedom that each element has a very similar size of the error. Also, in the robust a posteriori error analysis, we always try to find an equivalence between some intrinsic norm of the error and a computable error estimator, the so call the reliability and efficiency bounds. When constructing the error estimator, it is essential to realize that the best the adaptive numerical method can get is restricted by the robust local a priori estimates with respect each elements. This is especially important for the mixed methods, since there are two unknowns, the flux and the potential, and there are various post-processing methods. It is important to find which is the right quantity and norm to estimate in the a posteriori error estimates.
The proof of local optimal and robust a priori error estimate often contains two parts: one is the {\bf robust best approximation} result (Cea's lemma type of result), which has its own importance; the other is the {\bf robust local approximation properties of the interpolation operator}.
Before we discuss the robust best approximation result and robust local interpolations results for the mixed approximations, we first discuss the corresponding results for the conforming, Crouzeix-Raviart nonconforming, and discontinuous Galerkin results of the interface problem.
For the interface problem (\ref{scalar}), the robust best approximation property is well known and it almost trivial for the $H^1$ conforming approximation:
$$
\|\alpha^{1/2}\nabla (u-u_k^c)\|_0 \leq \inf_{v_k^c \in V_k^c}\|\alpha^{1/2}\nabla (u-v_k^c)\|_0,
$$
where $V_k^c$ is the $k$-th degree $H^1_0$-conforming finite element space, and $u_k^c$ is the corresponding $H^1$ conforming approximation.
On the other hand, the proofs of the robust best approximation for CR nonconforming and discontinuous Galerkin is not easy. In \cite{CHZ:17}, for the Croueix-Raviart nonconforming element approximation, we showed the robust best approximation property (the constant $C$ independent of $\alpha$ and mesh size):
$$
\|\alpha^{1/2}\nabla_h (u-u_1^{nc})\|_0 \leq C\left( \inf_{v_1^{nc} \in V_1^{nc}}\|\alpha^{1/2}\nabla_h (u-v_1^{nc})\|_0 +{\mbox{osc}}_{\alpha,nc} \right),
$$
where $V_1^{nc}$ is the Crouzeix-Raviart non-conforming finite element space, and $u_1^{nc}$ is the corresponding non-conforming approximation, and ${\mbox{osc}}_{\alpha,nc}$ is a robust oscillation term. Also in \cite{CHZ:17}, for the discontinuous Galerkin approximation, we showed the robust best approximation property (the constant $C$ independent of $\alpha$ and mesh size):
$$
|\!|\!| u-u_k^{dg}|\!|\!|_{dg} \leq C\left( \inf_{v_k^{dg} \in D_k}|\!|\!| u-v_k^{dg}|\!|\!|_{dg} +{\mbox{osc}}_{\alpha,dg} \right),
$$
where $D_k$ is the $k$-th degree discontinuous finite element space, and $u_k^{dg}$ is the corresponding discontinuous Galerkin approximation, $|\!|\!| \cdot|\!|\!|_{dg}$ is the $\alpha$-weighted $H^1$ discontinuous Galerkin norm, and ${\mbox{osc}}_{\alpha,dg}$ is a robust oscillation term.
The local approximation properties of the interpolation operators for the DG space and Crouzeix-Raviart is easy to show. For the conforming finite element approximation, there are two types of local interpolations: nodal interpolations which require high regularity of the solution, and the Scott-Zhang or Clement interpolations whose regularity requirement is very low. For the nodal interpolation, it is completely local in each element, but the it need very high regularity to exist, especially in three dimensions. For the Scott-Zhang/Clement interpolations, since they are defined on a local patch, their local robustness depends on a non-realistic assumption, the quasi-monotonicity assumption, see \cite{DrSaWi:96, BeVe:00, CaZh:09, CHZ:17}. Thus, the existence of robust local optimal result for the conforming finite element approximation for the low regularity interface problem is still open.
For the mixed methods, we have two unknowns, one is the flux $\mbox{\boldmath$\sigma$}$, and the other is the potential $u$. For the potential $u$, the discontinuous finite element approximation is used, so the robust local interpolation property is obvious. We use Raviart-Thomas or Brezzi-Douglas-Marini elements to approximate to the flux variable, a robust local interpolation property can be proved by the average Taylor series technique developed in \cite{DuSc:80}. This leaves the main task of proving the robust local optimal error estimates to the proof of the robust best approximation properties of the mixed methods. Unlike the conforming, non-conforming, or DG methods, we have several choices of the norms and the approximations spaces.
Our first robust best approximation property is simple, the weighted $L^2$-norm of the flux error in the equilibrated discrete spaces, see Theorem 3.2 and 3.3.
For the potential $u$, in the standard analysis of the mixed method, the $L^2$ norm is used. It turns out that we have difficulties to have a robust inf-sup condition with the weighted $L^2$ norm for the discrete approximation $u_h$ and a modified $H({\rm div})$ norm. Thus, we use the $\alpha$- and mesh-dependent norms to do the robust analysis. The choice of norm for $u_h$ is a norm similar to the standard discontinuous Galerkin norm, that is, a weighted discrete $H^1$ norm. With this $\alpha$- and mesh-dependent norm analysis, we show robust best approximation result for the potential approximation in the $\alpha$-dependent discrete $H^1$ norm. But since the approximation space for the potential $u$ is not rich enough, the order of approximation of $u$ in the $\alpha$-dependent discrete $H^1$ norm is one or two orders lower than the flux approximation. This order discrepancy suggests that we should not try to do the robust estimate of the $\alpha$ weighted discrete $H^1$-norm of the potential approximation in the a posteriori error analysis, as stated the earlier discussion by Kim \cite{Kim:07}.
For the flux approximation, with the help of $\alpha$- and mesh-dependent analysis, we show the robust best approximation result in the non-equilibrated RT/BDM space with an $\alpha$ and $h$ weighted $H({\rm div})$ norm for the first time. The corresponding robust and local a priori error estimates are also given without order loss even for the BDM approximations.
Finally, since the discrete $H^1$ norm of the potential approximation $u_h$ is often of a lower order than the corresponding flux approximation, we use Stenberg's post-processing to recover a new approximation with a compatible polynomial degree. We show that for the recovered potential approximation, the robust local best approximation result is true and a robust local a priori error estimates of the same order as the flux approximation is obtained. We also prove a new trace inequality of the normal trace. We also point out in the paper that any recovery or post-processing should based on the flux approximation since it is more accurate.
There are many a priori estimates for mixed methods available. The standard analysis can be found in the books and papers \cite{DR:82, BBF:13, RT:91, Ga:14}. In these analysis, $L^2$ or $H({\rm div})$ norms are used for the flux approximation and the $L^2$ norm is used for the potential approximation. No robust analysis is discussed in these papers or books. The mesh-dependent norm analysis can be found in \cite{BrVe:96,LS:06}, also, no robust analysis is discussed. In \cite{Voh:07,Voh:10,Kim:07}, many a priori and a posteriori error results are presented for the mixed methods, some are robust and some are non-robust. No robust and local optimal estimates are discussed for mixed methods before.
The paper is organized as follows. Section 2 describes the mixed finite element methods for the model problem. Various robust best approximations results and robust and local a priori error estimates are presented in Section 3, including the robust best approximation results for the flux in the weighted $L^2$ norm in the discrete equilibrated space and in the weighted $H({\rm div})$ norm in the whole mixed approximations spaces, the robust best approximation result for the potential in weighted discrete $H^1$ norm. In Section 4, we discuss Stenberg's of post-processing and show its robust and local optimal a priori error estimates in each elements. In Section 8, we make some concluding remarks.
\section{Mixed Finite Element Methods}
Introducing the flux
\[
\mbox{\boldmath$\sigma$} = -\alpha(x)\nabla u,
\]
the mixed variational formulation for the problem in (\ref{scalar}) and (\ref{bc1}) is to find
$(\mbox{\boldmath$\sigma$},\,u)\in H({\rm div};\O)\times L^2(\O)$ such that
\begin{equation}\label{mixed}
\left\{\begin{array}{lclll}
(\alpha^{-1}\mbox{\boldmath$\sigma$},\,\mbox{\boldmath$\tau$})-(\nab\cdot \mbox{\boldmath$\tau$},\, u)&=&0 \quad & \forall\,\, \mbox{\boldmath$\tau$} \in H({\rm div};\O),\\[2mm]
(\nab\cdot \mbox{\boldmath$\sigma$}, \,v) &=& (f,\,v)&\forall \,\, v\in L^2(\O).
\end{array}\right.
\end{equation}
Let ${\cal T}=\{K\}$ be a regular triangulation of the domain
$\Omega$ (see, e.g., \cite{Cia:78, BrSc:08}).
Denote by $h_K$ the diameter of the element $K$. Assume that
interfaces $\{\partial\O_i\cap\partial\O_j\,:\, i,j=1,\,...,\,n\}$ do not cut
through any element $K\in{\cal T}$. For any element $K\in{\cal T}$,
denote by $P_k(K)$ the space of polynomials on $K$ with total degree less than or equal to $k$.
Define the discontinuous piecewise polynomial space of degree $k$ by
$$
D_k = \{ v \in L^2(\O)\, :\, v|_K \in P_k \; \forall\, K\in{\cal T}\}.
$$
Define the $H({\rm div})$ conforming Raviart-Thomas (RT) finite element space
and Brezzi-Douglas-Marini (BDM) finite element space of order $k$ by
$$
RT_k = \{ \mbox{\boldmath$\tau$} \in H({\rm div};\O)\, :\, \mbox{\boldmath$\tau$}|_K \in P_k(K)^d + {\bf x} P_k(K) \; \forall\, K\in{\cal T}\}.
$$
and
$$
BDM_k = \{ \mbox{\boldmath$\tau$} \in H({\rm div};\O)\, :\, \mbox{\boldmath$\tau$}|_K \in P_k(K)^d \; \forall\, K\in{\cal T}\}.
$$
For mixed problems, $RT_k\times D_k$ and $BDM_{k+1}\times D_k$ are stable pairs. Thus, we use the notation $\Sigma_k$ to denote $RT_k$ or $BDM_{k+1}$.
The mixed finite element approximation is to find $(\mbox{\boldmath$\sigma$}_h,\,u_h) \in \Sigma_k \times D_k$ such that
\begin{equation}\label{problem_mixed}
\left\{\begin{array}{lclll}
(\alpha^{-1}\mbox{\boldmath$\sigma$}_h,\,\mbox{\boldmath$\tau$}_h)-(\nab\cdot \mbox{\boldmath$\tau$}_h,\, u_h)&=&0
\quad & \forall\,\, \mbox{\boldmath$\tau$}_h \in \Sigma_k,\\[2mm]
(\nab\cdot \mbox{\boldmath$\sigma$}_h,\, v_h) &=& (f,\,v_h)&\forall \,\, v_h\in D_k.
\end{array}\right.
\end{equation}
Difference between (\ref{mixed}) and (\ref{problem_mixed}) yields the following error equation:
\begin{equation}\label{erroreq_mixed}
\left\{\begin{array}{lclll}
(\alpha^{-1}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h),\,\mbox{\boldmath$\tau$}_h)-(\nab\cdot \mbox{\boldmath$\tau$}_h,\, u-u_h)&=&0
\quad & \forall\,\, \mbox{\boldmath$\tau$}_h \in \Sigma_k,\\[2mm]
(\nab\cdot (\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h),\, v_h) &=& 0&\forall \,\, v_h\in D_k.
\end{array}\right.
\end{equation}
\section{Robust and Local Optimal A Priori Error Estimates}
\setcounter{equation}{0}
\subsection{Mixed finite element interpolations and approximation properties}
For a fixed $r>0$, denote by $I^{rt,k}_{h}: H({\rm div};\,\Omega) \cap [H^r(\O)]^d \mapsto RT_k$ the standard $RT$ interpolation operator and $I^{bdm,k}_{h}: H({\rm div};\,\Omega) \cap [H^r(\O)]^d \mapsto BDM_k$ the standard $BDM$ interpolation operator. We have the following local approximation property: for $\mbox{\boldmath$\tau$} \in H^{s_K}(K)$, $s_K >0$,
\begin{eqnarray} \label{rti}
\|\mbox{\boldmath$\tau$} - I^{\Sigma,k}_{h} \mbox{\boldmath$\tau$}\|_{0,K}
&\leq& C h_K^{\min\{k+1,s_K\}} |\mbox{\boldmath$\tau$}|_{\min\{k+1,s_K\},K} \quad\forall\,\, K\in {\cal T}
\end{eqnarray}
with $I^{\Sigma,k}_{h} = I^{rt,k}_{h}$ or $I^{bdm,k}_{h}$. The estimate in (\ref{rti}) is standard for $s_K\geq 1$ and can be proved by the average Taylor series developed in \cite{DuSc:80} and the standard reference element technique with Piola transformation for $0<s_K<1$. We also should notice that the interpolations and approximation properties are completely local.
Denote by $Q^k_{h}: L^2 (\O) \mapsto D_k$ the $L^2$-projection onto $D_k$. The following commutativity property is well-known:
\begin{eqnarray}\label{comm}
\nabla\cdot (I^{rt,k}_{h}\,\mbox{\boldmath$\tau$})&=&Q^k_{h}\,\nabla\cdot\mbox{\boldmath$\tau$} \qquad
\quad\forall\,\,\mbox{\boldmath$\tau$}\inH({\rm div};\,\Omega) \cap H^r(\O)^d \,\mbox{ with }\, r>0, \\[2mm]
\label{comm_bdm}
\nabla\cdot (I^{bdm,k}_{h}\,\mbox{\boldmath$\tau$})&=&Q^{k-1}_{h}\,\nabla\cdot\mbox{\boldmath$\tau$} \qquad
\quad\forall\,\,\mbox{\boldmath$\tau$}\inH({\rm div};\,\Omega) \cap H^r(\O)^d \,\mbox{ with }\, r>0.
\end{eqnarray}
\begin{rem}
The requirement $r>0$ in $H({\rm div};\,\Omega) \cap [H^r(\O)]^d$ is to make sure that the mixed interpolations are well defined. Another choice is $\{\mbox{\boldmath$\tau$}\in L^p(\O)^d\mbox{ and }\nabla\cdot \mbox{\boldmath$\tau$} \in L^2(\O)\}$ for $p>2$ or $W^{1,t}(K)$ for $t>2d/(d+2)$ as in \cite{BBF:13}. We use the Hilbert space based choice since it is more suitable for our analysis.
\end{rem}
\subsection{Robust best approximation in the discrete equilibrated space for the flux}
Define the discrete equilibrated space
$$
\Sigma_k^f = \{\mbox{\boldmath$\tau$}_h \in \Sigma_k : \nabla\cdot \mbox{\boldmath$\tau$}_h =Q^k_{h} f\}.
$$
Note that $\Sigma_k^f = RT_k^f = \{\mbox{\boldmath$\tau$}_h \in RT_k : \nabla\cdot \mbox{\boldmath$\tau$}_h =Q^k_{h} f\}$ for the RT case and $\Sigma_k^f = BDM_{k+1}^f= \{\mbox{\boldmath$\tau$}_h \in BDM_{k+1} : \nabla\cdot \mbox{\boldmath$\tau$}_h =Q^k_{h} f\}$ for the BDM case.
The following theorem is almost standard in the mixed finite element analysis.
\begin{thm}\label{apriori_mixed} (Robust best approximation in the discrete equilibrated space)
Let $(\mbox{\boldmath$\sigma$}, u)$ and $(\mbox{\boldmath$\sigma$}_h,\,u_h) \in \Sigma_k \times D_k$ be the solutions of {\em (\ref{mixed})} and {\em (\ref{problem_mixed})}, respectively, then the following robust best approximation result holds:
\begin{equation}\label{rba_equ}
\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$} -\mbox{\boldmath$\sigma$}_h)\|_{0,\O} \leq \inf_{\mbox{\boldmath$\tau$}_h^f \in \Sigma_k^f} \|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h^f)\|_{0,\O}.
\end{equation}
\end{thm}
\begin{proof}
To establish (\ref{rba_equ}), denote by
\[
{\bf E} = \mbox{\boldmath$\sigma$} -\mbox{\boldmath$\sigma$}_h
\quad\mbox{and}\quad
e = u- u_h
\]
the respective errors of the flux and the solution.
Now, let $\mbox{\boldmath$\tau$}_h^f$ be an arbitrary function in $RT_k^f$, then it follows from the first equation in (\ref{erroreq_mixed}), the fact $\mbox{\boldmath$\sigma$}_h \in \Sigma_k^f$, and the Cauchy-Schwarz inequality that
\begin{eqnarray*}
\|\alpha^{-1/2}{\bf E}\|_{0,\O}^2
&= & (\alpha^{-1}{\bf E},\, \mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h^f) + (\alpha^{-1}{\bf E},\, \mbox{\boldmath$\tau$}_h^f -\mbox{\boldmath$\sigma$}_h)\\
&=&(\alpha^{-1}{\bf E},\, \mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h^f) + (\nab\cdot (\mbox{\boldmath$\tau$}_h^f-\mbox{\boldmath$\sigma$}_h),\,e)\\
&=&(\alpha^{-1}{\bf E},\, \mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h^f)
\leq \|\alpha^{-1/2}{\bf E}\|_{0,\O}\,\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h^f)\|_{0,\O},
\end{eqnarray*}
which implies the result of the theorem.
\end{proof}
\begin{thm}\label{apriori_mixed2} (Robust local a priori error estimates)
Let $(\mbox{\boldmath$\sigma$}, u)$ and $(\mbox{\boldmath$\sigma$}_h,\,u_h) \in \Sigma_k \times D_k$ $(k\geq 0)$ be the solutions of {\em (\ref{mixed})} and {\em (\ref{problem_mixed})}, respectively. Assume that $u\in H^{1+r}(\O)$ with some $r>0$ and that $u|_K\in H ^{1+s_K}(K)$ with an element-wisely defined regularity $s_K>0$ for all $K\in{\cal T}$. Then there exists a constant $C>0$ independent $\alpha$ and $h$ for both the two- and three-dimension such that
\begin{eqnarray}\label{err-bound-L2RT}
\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$} -\mbox{\boldmath$\sigma$}_h)\|_{0} &\leq& C \sum_{K\in{\cal T}} h_K^{\min\{k+1,s_K\}} |\alpha^{1/2}\nabla u|_{\min\{k+1,s_K\},K}, \quad RT_k \mbox{ case} ,
\\[2mm] \label{err-bound-L2BDM}
\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$} -\mbox{\boldmath$\sigma$}_h)\|_{0} &\leq& C \sum_{K\in{\cal T}} h_K^{\min\{k+2,s_K\}} |\alpha^{1/2}\nabla u|_{\min\{k+2,s_K\},K}, \quad BDM_{k+1} \mbox{ case}.
\end{eqnarray}
\end{thm}
\begin{proof}
For the $RT_k \times D_k$ case, the commutativity property in (\ref{comm}) and the second equations in (\ref{mixed}) and (\ref{problem_mixed}) lead to
$$
\nabla\cdot (I_h^{rt,k}\mbox{\boldmath$\sigma$}) = Q^k_{h}\,\nabla\cdot\mbox{\boldmath$\sigma$} = Q^k_{h} f = \nab\cdot
\mbox{\boldmath$\sigma$}_h.
$$
Thus, the result is a direct consequence of the best approximation property in (\ref{rba_equ}) and the local approximation property in (\ref{rti}) by choosing $\mbox{\boldmath$\tau$}_h^f = I_h^{rt,k}\mbox{\boldmath$\sigma$} \in RT_k^f$.
Using the same argument, we can get the result for the $DBM_{k+1} \times D_k$ case.
\end{proof}
\begin{rem}
For those elements with a low regularity $0<s_K<1$, $RT_0$ is enough and there is no need to use BDM or high order RT approximations.
\end{rem}
\begin{rem}
For the case that in each element $K\in {\cal T}$, the diffusion coefficient being a full symmetric positive definite constant matrix $A|_K$ instead of a scalar constant $\alpha_K$, from the proofs, it is clear the above robust best approximation result is also true:
$$
\|A^{-1/2}(\mbox{\boldmath$\sigma$} -\mbox{\boldmath$\sigma$}_h)\|_{0,\O} \leq \inf_{\mbox{\boldmath$\tau$}_h^f \in \Sigma_k^f} \|A^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h^f)\|_{0,\O}.
$$
In each element $K\in{\cal T}$, for the quantity ${\bf q} \in P_k^d$, $A^{-1/2} {\bf q}$ is also in $P_k^d$, and thus $A^{-1/2}I_h^{\Sigma,k} {\bf q} = A^{-1/2}{\bf q}$. Thus for piecewise constant symmetric positive definite constant matrix $A$, we have
$$
\|A^{-1/2}(\mbox{\boldmath$\tau$} - I^{\Sigma,k}_{h} \mbox{\boldmath$\tau$})\|_{0,K}
\leq C h_K^{\min\{k+1,s_K\}} |A^{-1/2}\mbox{\boldmath$\tau$}|_{\min\{k+1,s_K\},K} \quad\forall\,\, K\in {\cal T}.
$$
And we have the robust local a priori error estimatesL
\begin{eqnarray*}
\|A^{-1/2}(\mbox{\boldmath$\sigma$} -\mbox{\boldmath$\sigma$}_h)\|_{0,\O} &\leq& C \sum_{K\in{\cal T}} h_K^{\min\{k+1,s_K\}} |A^{1/2}\nabla u|_{\min\{k+1,s_K\},K}, \quad RT_k \mbox{ case},
\\[2mm]
\|A^{-1/2}(\mbox{\boldmath$\sigma$} -\mbox{\boldmath$\sigma$}_h)\|_{0,\O} &\leq& C\sum_{K\in{\cal T}} h_K^{\min\{k+2,s_K\}} |A^{1/2}\nabla u|_{\min\{k+2,s_K\},K}, \quad BDM_{k+1} \mbox{ case}.
\end{eqnarray*}
The corresponding results for discontinuous Galerkin methods are not proved, since the robustness of the DG method for the diffusion problem depends on the right choice of the weights of the averages and penalty coefficients. For the full tensor case, the right weight is not clear or probably not possible for a full matrix $A$, see \cite{CHZ:17}. For the conforming finite element approximations, due to the lack of the nodal interpolations for the low regularity cases, such robust local optimal estimates is not available. For averaging operators like the Scott-Zhang or Clement interpolations, the robustness with respect to the full tensor $A$ is also impossible since even the famous quasi-monotonicity assumption is not meaningful in the case. For the Crouzeix-Raviart non-conforming finite element approximation, it is possible we can get a similar result by using the relation between the $RT_0$ and Crouzeix-Raviart elements.
\end{rem}
\subsection{Mesh-dependent norm analysis}
In this subsection, we use mesh-dependent norm analysis to derive the robust best approximation properties for the flux and the potential in appropriate norms. Earlier analysis on the mixed methods using mesh-dependent norms can be found in Babu\v{s}ka, Osborn, and Pitk\"{a}ranta \cite{BaOsPi:80}, Braess and Verf\"{u}rth \cite{BrVe:96}, and \cite{CaZh:12}. In the mesh-dependent analysis, we need to restrict ourselves to the scalar case.
First, we discuss the averages of the coefficients on the edge/face $F\in {\cal E}$. For $F = \partial K_F^{+} \cap \partial K_F^{-}\in {\cal E}_{I}$, denote by $\alpha^+_{F}$ and $\alpha^-_{F}$ the restriction of $\alpha$ on the respective $K_F^{+}$ and $K_F^{-}$. Denote the harmonic averages of $\alpha$ on $F \in {\cal E}$ by
\[
\alpha_{F,H} = \left\{\begin{array}{cl}
\displaystyle\frac{\alpha_F^+ \alpha_F^- }{\alpha_F^+ + \alpha_F^-},&\quad F \in {\cal E}_{I},\\[4mm]
\alpha_F^- &\quad F \in {\cal E}_{{_D}}\cup{\cal E}_{{_N}},
\end{array}\right.
\]
which is equivalent to the minimum of $\alpha$:
\begin{equation}\label{a-h}
\displaystyle\frac{1}{2}\min\{\alpha_F^+, \alpha_F^- \}\leq \alpha_{F,H} \leq \min\{\alpha_F^+, \alpha_F^- \} .
\end{equation}
\begin{lem} The bilinear form $(\nabla\cdot \mbox{\boldmath$\tau$}, v)$ for $(\mbox{\boldmath$\tau$},v)\in H({\rm div};\O)\times L^2(\O)$
has the following representation:
\begin{equation} \label{rep}
(\nabla\cdot\mbox{\boldmath$\tau$}, v)
= -\sum_{K\in{\cal T}} (\nabla v,\mbox{\boldmath$\tau$})_{K}
+ \sum_{F\in {\cal E}_{I}} (\mbox{\boldmath$\tau$}\cdot{\bf n}, \jump{v})_F
+ \sum_{F\in {\cal E}_D} (\mbox{\boldmath$\tau$}\cdot{\bf n}, v)_F
\end{equation}
\end{lem}
\begin{proof} The representation (\ref{rep}) is a consequence of integration by parts.
\end{proof}
Define $(\alpha,\,h)$-dependent norms on ${\cal T}$ by
\begin{eqnarray*}
&& \| \mbox{\boldmath$\tau$} \|_{\alpha,h}^2 :=\|\alpha^{-1/2} \mbox{\boldmath$\tau$}\|_{0}^2
+ \displaystyle\sum_{F\in {\cal E} }\frac{h_F}{\alpha_{F,H}} \|\mbox{\boldmath$\tau$} \cdot {\bf n}\|_{0,F}^2,
\quad \forall
\mbox{\boldmath$\tau$} \in \Sigma_k
\\[2mm]
\mbox{and }\,\,&&
|\!|\!| v|\!|\!|_{\alpha, h}^2
=\|\alpha^{1/2} \nabla_h v\|_{0,{\cal T}}^2
+ \displaystyle\sum_{F\in {\cal E}_{I}} \displaystyle\frac{\alpha_{F,H}}{h_F} \|\jump{v}\|_{0,F}^2
+ \displaystyle\sum_{F\in {\cal E}_{D}} \displaystyle\frac{\alpha_{F}}{h_F} \|v\|_{0,F}^2,
\quad \forall v \in D_k.
\end{eqnarray*}
Note that the $|\!|\!|\cdot|\!|\!|_{\alpha,h}$ norm is the standard $\alpha$-weighted DG norm used in the discontinuous Galerkin methods, see \cite{CHZ:17}. For a $v\in H_0^1(\O)$, $|\!|\!| v|\!|\!|_{\alpha, h} = \|\alpha^{1/2}\nabla v\|_{0,\O}$.
\begin{lem} \label{lem_hnorm}
For all $\mbox{\boldmath$\tau$} \in \Sigma_k(K)$, there exists a positive constant $C>0$ independent of $\alpha$ and $h$, such that
\[
\displaystyle\sum_{F\in {\cal E}_K}\frac{h_F}{\alpha_K} \|\mbox{\boldmath$\tau$} \cdot {\bf n}\|_{0,F}^2 \leq C \|\alpha^{-1/2}\mbox{\boldmath$\tau$}\|_{0,K}^2.
\]
\end{lem}
\begin{proof}
The lemma is a simple consequence of the standard scaling argument and the fact that both $RT_k(K)$ and $BDM_{k+1}(K)$ are finite dimensional.
\end{proof}
\begin{thm} \label{thm_hnorm}
The following norm equivalence holds with $C>0$ independent of $\alpha$ and $h$:
\begin{equation} \label{norm_equ}
\|\alpha^{-1/2}\mbox{\boldmath$\tau$}_h\|_0 \leq \|\mbox{\boldmath$\tau$}_h\|_{\alpha,h}
\leq C \|\alpha^{-1/2}\mbox{\boldmath$\tau$}_h\|_0, \quad \forall \mbox{\boldmath$\tau$}_h \in \Sigma_k.
\end{equation}
\end{thm}
\begin{proof}
Since for the harmonic average $\alpha_{F,H}$, we have $1/\alpha_{F,H} = 1/\alpha_{F}^+ +1/\alpha_{F}^-$, by Lemma \ref{lem_hnorm}, we immediately get the robust discrete norm equivalence.
\end{proof}
For $\mbox{\boldmath$\tau$} \in H({\rm div};\O)$, define the following $\alpha$ and $h$ dependent norm:
\begin{equation}
\|\mbox{\boldmath$\tau$}\|_{\alpha,h,H({\rm div})}:= \left(
\|\alpha^{-1/2} \mbox{\boldmath$\tau$}\|_0^2 + \sum_{K\in{\cal T}}h_K^2\|\alpha^{-1/2}\nabla\cdot \mbox{\boldmath$\tau$}\|_{0,K}^2
\right)^{1/2}.
\end{equation}
We also use $\|\mbox{\boldmath$\tau$}\|_{\alpha,h,H({\rm div}),K}$ to denote the norm on a single element $K$.
The following trace inequality can be found in Lemma 2.4 and Remark 2.5 of \cite{CHZ:17}.
\begin{lem} Let $F$ be an edge/face of $K\in{\cal T}$ and ${\bf n}_F$ the unit vector normal to $F$. Assume that $\mbox{\boldmath$\tau$}$ is a given function in $H({\rm div};K)\cap [H^r(K)]^d$, $r>0$ then for any $w_h\in P_k(K)$, we have
\begin{eqnarray}\label{tracecombined}
(\mbox{\boldmath$\tau$}\cdot{\bf n}, w_h)_F
&\leq & C\, h_F^{-1/2}\|w_h\|_{0,F}
\left(\|\mbox{\boldmath$\tau$}\|_{0,K} + h_K\|\nabla\cdot \mbox{\boldmath$\tau$}\|_{0,K}\right).
\end{eqnarray}
\end{lem}
The following two continuity results are true.
\begin{lem}
The following continuity results hold with constants $C_{con,1}>0$ and $C_{con,2}>0$ independent of $\alpha$ and $h$:
\begin{eqnarray} \label{con1}
(\nabla\cdot \mbox{\boldmath$\tau$}_h,v)
&\leq& C_{con,1}\|\alpha^{-1/2}\mbox{\boldmath$\tau$}_h\|_0|\!|\!| v|\!|\!|_{\alpha,h}
, \quad
\forall \mbox{\boldmath$\tau$}_h \in \Sigma_k, \quad v \in H^1_0(\O)
\mbox{ or } v\in D_k,\\[2mm]
\label{cont_hDiv}
(\nabla\cdot \mbox{\boldmath$\tau$},v_h)
&\leq & C_{con,2} \|\mbox{\boldmath$\tau$}\|_{\alpha,h,H({\rm div})}|\!|\!| v_h|\!|\!|_{\alpha,h}, \quad
\forall \mbox{\boldmath$\tau$} \in H({\rm div};\O)\cap [H^r(\O)]^d, \quad v \in D_k.
\end{eqnarray}
\end{lem}
\begin{proof}
The continuity (\ref{con1}) is clear from the representation (\ref{rep}), Cauchy-Schwarz inequality,
the definition of norms $\|\mbox{\boldmath$\tau$}\|_{\alpha,h}$ and $|\!|\!| v|\!|\!|_{\alpha,h}$, and the robust norm equivalent result (\ref{norm_equ}).
To show (\ref{cont_hDiv}), we still start from the representation (\ref{rep}):
$$
(\nabla\cdot\mbox{\boldmath$\tau$}, v_h)
= -\sum_{K\in{\cal T}} (\nabla v_h,\mbox{\boldmath$\tau$})_{K}
+ \sum_{F\in {\cal E}_{I}} (\mbox{\boldmath$\tau$}\cdot{\bf n}, \jump{v_h})_F
+ \sum_{F\in {\cal E}_D} (\mbox{\boldmath$\tau$}\cdot{\bf n}, v_h)_F.
$$
For the term $(\mbox{\boldmath$\tau$}\cdot{\bf n}, \jump{v_h})_F$, where $F\in{\cal E}_I$, by (\ref{tracecombined}),
\begin{eqnarray*}
(\mbox{\boldmath$\tau$}\cdot{\bf n},\,\, \jump{v_h})_F
&\leq & C\, h_F^{-1/2}\|\jump{v_h}\|_{0,F}
\left(\|\mbox{\boldmath$\tau$}\|_{0,K} + h_K\|\nabla\cdot \mbox{\boldmath$\tau$}\|_{0,K}\right),
\end{eqnarray*}
where $K$ is one of the elements having $F$ as an edge/face.
Choosing $K$ to be the element with the smaller $\alpha_K$.
From (\ref{a-h}), the smaller $\alpha_K$ is equivalent to the harmonic average $\alpha_{F,H}$, then
\begin{eqnarray*}
(\mbox{\boldmath$\tau$}\cdot{\bf n},\,\, \jump{v_h})_F
&\leq & C\, \alpha_{F,H}^{1/2}h_F^{-1/2}\|\jump{v_h}\|_{0,F}
\left(\|\alpha^{-1/2}\mbox{\boldmath$\tau$}\|_{0,K} + h_K\|\alpha^{-1/2}\nabla\cdot \mbox{\boldmath$\tau$}\|_{0,K}\right).
\end{eqnarray*}
The term $(\mbox{\boldmath$\tau$}\cdot{\bf n}, v_h)_F$, $F\in{\cal E}_D$, can be handled similarly.
Then by the Cauchy-Schwarz inequality, (\ref{cont_hDiv}) can be easily proved.
\end{proof}
\begin{lem}
The following discrete inf-sup condition
\begin{equation} \label{infsup}
\sup_{\mbox{\boldmath$\tau$}_h \in \Sigma_k} \displaystyle\frac{(\nabla\cdot \mbox{\boldmath$\tau$}_h,v_h)}{ \|\alpha^{-1/2}\mbox{\boldmath$\tau$}_h\|_0}
\geq \beta |\!|\!| v_h |\!|\!|_{\alpha, h} \quad \forall\, v_h \in D_k
\end{equation}
holds with a constant $\beta>0$ independent of $\alpha$ and $h$.
\end{lem}
\begin{proof}
By the robust norm equivalent result (\ref{norm_equ}), we only need to prove the result for $\mbox{\boldmath$\tau$}_h$ in the norm $\|\mbox{\boldmath$\tau$}_h\|_{\alpha,h}$. Since $RT_k \subset BDM_{k+1}$, thus
$$
\sup_{\mbox{\boldmath$\tau$} \in BDM_{k+1}} \displaystyle\frac{(\nabla\cdot \mbox{\boldmath$\tau$}_h,v_h)}{ \|\mbox{\boldmath$\tau$}_h\|_{\alpha,h}}
\geq
\sup_{\mbox{\boldmath$\tau$} \in RT_k} \displaystyle\frac{(\nabla\cdot \mbox{\boldmath$\tau$}_h,v_h)}{ \|\mbox{\boldmath$\tau$}\|_{\alpha,h}}, \quad \forall\, v \in D_k,
$$
we only need to prove the RT version.
Choose a $\tilde{\mbox{\boldmath$\tau$}}_h\in RT_k$ such that
\[
(\tilde{\mbox{\boldmath$\tau$}}_h,\nabla q)_K = -(\alpha \nabla v, \nabla q)_K \quad \forall\, q\in P_{k-1}(K)
\quad\forall\,\, K\in{\cal T}
\]
and that
\begin{equation}\label{n-bc}
\tilde{\mbox{\boldmath$\tau$}}_h \cdot {\bf n} |_F
=\left\{\begin{array}{llll}
\displaystyle\frac{\alpha_{F,H}}{h_F}\jump{v}
& \,\, F\in {\cal E}_{I}, \\[3mm]
\displaystyle\frac{\alpha_F}{ h_F} v & \,\, F\in {\cal E}_D,
\end{array}\right.
\end{equation}
which, together with (\ref{rep}), gives
\begin{equation}\label{5.10}
(\nabla\cdot \tilde{\mbox{\boldmath$\tau$}}_h,v_h)= |\!|\!| v |\!|\!|_{\alpha,h}^2.
\end{equation}
For every $K\in{\cal T}$, by the standard scaling argument, there exists a constant $C>0$ independent of $\alpha$ and the mesh size such that
\[
\|\tilde{\mbox{\boldmath$\tau$}}_h\|_{0,K}^2 \leq C \left (
\|\alpha_K \nabla v \|_{0,K}^2 + h_K \sum_{F\in {\cal E}_K\cap{\cal E}_{I}}
\|\displaystyle\frac{\alpha_{F,H}}{h_F}\jump{v}\|_{0,F}^2
+h_K \sum_{F\in {\cal E}_K\cap{\cal E}_{D}}
\|\displaystyle\frac{\alpha_{F}}{h_F} v\|_{0,F}^2
\right),
\]
which, together with (\ref{a-h}), gives
\[
\|\alpha_K^{-1/2}\tilde{\mbox{\boldmath$\tau$}}_h\|_{0,K}^2 \leq C \left (
\|\alpha_K^{1/2} \nabla v \|_{0,K}^2 +\sum_{F\in {\cal E}_K\cap{\cal E}_{I}} \displaystyle\frac{\alpha_{F,H}}{h_F}
\|\jump{v}\|_{0,F}^2
+ \sum_{F\in {\cal E}_K\cap{\cal E}_{D}}
\displaystyle\frac{\alpha_{F}}{h_F} \|v\|_{0,F}^2
\right),
\]
Hence, there exists a constant $\tilde{C}>0$ independent of $\alpha$ and $h$ such that
\[
\|\tilde{\mbox{\boldmath$\tau$}}_h\|_{\alpha,h} \leq \tilde{C} |\!|\!| v |\!|\!|_{\alpha,h}.
\]
which, together with (\ref{5.10}), leads to the discrete inf-sup condition of the lemma.
\end{proof}
Define the following discrete divergence-free subspace of $\Sigma_k$:
$$
\Sigma_k^0 =\{\mbox{\boldmath$\tau$}_h \in \Sigma_k : \nabla\cdot \mbox{\boldmath$\tau$}_h =0\}.
$$
Its orthogonal complement is
$$
(\Sigma_k^0)^\perp =\{\mbox{\boldmath$\tau$}_h \in \Sigma_k : (\mbox{\boldmath$\tau$}_h, \bf{\rho}_h) =0, \forall \bf{\rho}_h \in \Sigma_k^0\}.
$$
Note that the inf-sup condition (\ref{infsup}) is also equivalent to the following inf-sup condition with $\beta>0$ independent of $\alpha$ and $h$:
\begin{equation} \label{infsup2}
\sup_{v_h\in D_k} \displaystyle\frac{(\nabla\cdot \mbox{\boldmath$\tau$}_h,v_h)}{ |\!|\!| v_h|\!|\!|_{\alpha,h}}
\geq \beta \|\mbox{\boldmath$\tau$}_h\|_{\alpha,h} \geq \beta \|\alpha^{-1/2}\mbox{\boldmath$\tau$}_h\|_0 \quad \forall\, \mbox{\boldmath$\tau$}_h \in (\Sigma_k^0)^\perp.
\end{equation}
The condition (\ref{infsup}) also guarantees that for each $g\in L^2(\O)$, there exists a unique solution $\mbox{\boldmath$\tau$}_h \in (\Sigma_k^0)^\perp$ such that
\begin{equation} \label{orth}
(\nabla\cdot \mbox{\boldmath$\tau$}_h, v_h) = (g, v_h), \quad \forall v_h \in D_k.
\end{equation}
Now let us prove the following robust best approximation property for $|\!|\!| u-u_h|\!|\!|_{\alpha,h}$.
\begin{thm} (Robust best approximation in the weighted discrete $H^1$ norm)
Let $(\mbox{\boldmath$\sigma$}, u)$ and $(\mbox{\boldmath$\sigma$}_h,\,u_h)\in \Sigma_k\times D_k$
be the solutions of
{\em (\ref{mixed})} and {\em (\ref{problem_mixed})}, respectively.
Assume that $u\in H^{1+r}(\O)$ with $r>0$ and that $u|_K\in H ^{1+s_K}(K)$ with element-wisely defined $s_K>0$ for all $K\in{\cal T}$. Then there exists a constant $C>0$ independent of $\alpha$ and $h$ for both the two- and three-dimension such that
\begin{equation} \label{aprioriu}
|\!|\!| u-u_h|\!|\!|_{\alpha,h}
\leq C\left( \inf_{\mbox{\boldmath$\tau$}_h^f\in\Sigma_k^f} \|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h^f)\|_{0,\O}+\inf_{v_h \in D_k}|\!|\!| u -v_h|\!|\!|_{\alpha,h}\right).
\end{equation}
\end{thm}
\begin{proof}
By the inf-sup condition, for each $v_h \in D_k$ we have
\begin{equation}\label{uhvh}
|\!|\!| u_h - v_h|\!|\!|_{\alpha,h} \leq \frac{1}{\beta} \sup_{\mbox{\boldmath$\tau$}_h \in \Sigma_k} \displaystyle\frac{(\nabla\cdot \mbox{\boldmath$\tau$}_h, u_h-v_h)}{\|\alpha^{-1/2}\mbox{\boldmath$\tau$}_h\|_0}.
\end{equation}
By the first equation in the error equations (\ref{erroreq_mixed}),
$$
(\nabla\cdot \mbox{\boldmath$\tau$}_h, u_h-v_h) = (\nabla\cdot \mbox{\boldmath$\tau$}_h, u-v_h) + (\nabla\cdot \mbox{\boldmath$\tau$}_h, u_h-u) =
(\nabla\cdot \mbox{\boldmath$\tau$}_h, u-v_h) - (\alpha^{-1}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h),\,\mbox{\boldmath$\tau$}_h).
$$
Then, by the continuity result \eqref{cont_hDiv} and the Cauchy-Schwarz inequality,
$$
(\nabla\cdot \mbox{\boldmath$\tau$}_h, u_h-v_h) \leq C \|\mbox{\boldmath$\tau$}_h\|_{\alpha,h} |\!|\!| u-v_h|\!|\!|_{\alpha,h} + \|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_0 \|\alpha^{-1/2}\mbox{\boldmath$\tau$}_h\|_{0}.
$$
Thus by \eqref{uhvh} and the equivalence of $ \|\mbox{\boldmath$\tau$}_h\|_{\alpha,h}$ and $\|\alpha^{-1/2}\mbox{\boldmath$\tau$}_h\|_{0}$,
$$
|\!|\!| u_h - v_h|\!|\!|_{\alpha,h} \leq C(|\!|\!| u-v_h|\!|\!|_{\alpha,h}+ \|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_0).
$$
A simple application of the triangle inequality yields
$$
|\!|\!| u-u_h|\!|\!|_{\alpha,h} \leq |\!|\!| u-v_h|\!|\!|_{\alpha,h}+|\!|\!| u_h-v_h|\!|\!|_{\alpha,h}
\leq C\left(\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_0+ |\!|\!| u -v_h|\!|\!|_{\alpha,h}\right).
$$
By the optimal convergence results of $\mbox{\boldmath$\sigma$}_h$, we have the robust best approximation result of the theorem.
\end{proof}
\begin{rem}
Even though we have the robust best approximation result (\ref{aprioriu}), due to the fact that the approximation orders of $\Sigma_k$ and $D_k$ are different for the corresponding norms, the order of convergence for $u-u_h$ in the discrete $H^1$ norm $|\!|\!| \cdot|\!|\!|_{\alpha,h}$ is one or two order lower than the corresponding weighted $L^2$ RT or BDM approximation errors in Theorem \ref{apriori_mixed2}, respectively.
Due to this order difference, in the a posteriori error analysis, we should only construct the error estimator related to $\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_0$.
\end{rem}
Now, let us show the robust best approximation property in $\Sigma_k$.
\begin{thm} (Robust best approximation in the mixed approximation space)
The following robust best approximation properties are true with a constant $C$ independent of $\alpha$ and $h$:
\begin{eqnarray}
\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_0 &\leq& C \inf_{\mbox{\boldmath$\tau$} \in \Sigma_k}\|\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\tau$}_h\|_{\alpha,h,H({\rm div})}, \\[2mm] \label{rbahdiv}
\|\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\sigma$}_h\|_{\alpha,h,H({\rm div})} &\leq& C \inf_{\mbox{\boldmath$\tau$} \in \Sigma_k}\|\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\tau$}_h\|_{\alpha,h,H({\rm div})}.
\end{eqnarray}
\end{thm}
\begin{proof}
For an arbitrary $\mbox{\boldmath$\tau$}_h \in \Sigma_k$, by \eqref{orth}, there exists a unique $\boldsymbol{\zeta}_h \in (\Sigma_k^0)^\perp$, such that
$$
(\nabla\cdot \boldsymbol{\zeta}_h, v_h) = (\nabla\cdot (\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\tau$}_h), v_h), \quad\forall v_h \in D_k,
$$
and
\begin{equation}
\beta \|\alpha^{-1/2}\boldsymbol{\zeta}_h\|_0 \leq \sup_{v_h \in D_k} \displaystyle\frac{(\nabla\cdot \boldsymbol{\zeta}_h, v_h)} {|\!|\!| v_h |\!|\!|_{\alpha,h}}=
\sup_{v_h \in D_k} \displaystyle\frac{(\nabla\cdot (\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\tau$}_h), v_h)} {|\!|\!| v_h |\!|\!|_{\alpha,h}}.
\end{equation}
By the continuity (\ref{cont_hDiv}),
$$
(\nabla\cdot (\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\tau$}_h), v_h) \leq C|\!|\!| v_h |\!|\!|_{\alpha,h} \|\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\tau$}_h\|_{\alpha,h,H({\rm div})}.
$$
Thus,
$$
\|\alpha^{-1/2}\boldsymbol{\zeta}_h\|_0 \leq C \|\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\tau$}_h\|_{\alpha,h,H({\rm div})}.
$$
Setting $\mbox{\boldmath$\tau$}_h^f := \boldsymbol{\zeta}_h + \mbox{\boldmath$\tau$}_h$, it is clear that $\mbox{\boldmath$\tau$}_h^f \in \Sigma_k^f$. Then by the best approximation (\ref{rba_equ}),
$$
\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_0 \leq
\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h^f)\|_0 \leq
\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h)\|_0+\|\alpha^{-1/2}\boldsymbol{\zeta}_h\|_0 \leq C \|\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\tau$}_h\|_{\alpha,h,H({\rm div})}.
$$
On the other hand, since on each element $K\in {\cal T}$,
$$
(\nabla\cdot \boldsymbol{\zeta}_h, v_h)_K = (\nabla\cdot (\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\tau$}_h), v_h)_K, \quad\forall v_h \in P_k(K),
$$
and $\nabla\cdot \boldsymbol{\zeta}_h \in P_k(K)$, we have
$$
\|\nabla\cdot \boldsymbol{\zeta}_h\|_{0,K} \leq \|\nabla\cdot (\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\tau$}_h)\|_{0,K}.
$$
Since $\nabla\cdot(\mbox{\boldmath$\sigma$}_h-\mbox{\boldmath$\tau$}_h^f)=0$, we have
\begin{eqnarray*}
\|\alpha^{-1/2}\nabla\cdot(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_{0,K}&\leq&
\|\alpha^{-1/2}\nabla\cdot(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h^f)\|_{0,K}+\|\alpha^{-1/2}\nabla\cdot(\mbox{\boldmath$\sigma$}_h-\mbox{\boldmath$\tau$}_h^f)\|_{0,K}\\
&=&
\|\alpha^{-1/2}\nabla\cdot(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h^f)\|_{0,K}\\
&\leq&
\|\alpha^{-1/2}\nabla\cdot(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h)\|_{0,K}+
\|\alpha^{-1/2}\nabla\cdot \boldsymbol{\zeta}_h\|_{0,K} \\
&\leq & 2 \|\alpha^{-1/2}\nabla\cdot(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\tau$}_h)\|_{0,K}.
\end{eqnarray*}
With this, the robust best approximation property (\ref{rbahdiv}) in $\|\cdot\|_{\alpha,h,H({\rm div})}$ can proved.
\end{proof}
We classify the elements in the mesh into two sets:
\begin{eqnarray}
{\cal T}_{low} = \{ K\in {\cal T} : 0<s_K<1\} \quad\mbox{and}\quad
{\cal T}_{high} = \{ K\in {\cal T} : 1\leq s_K\}.
\end{eqnarray}
\begin{thm}\label{apriori_mixed3} (Robust local a priori error estimates in weighted $H({\rm div})$ norm)
Let $(\mbox{\boldmath$\sigma$}, u)$ and $(\mbox{\boldmath$\sigma$}_h,\,u_h) \in \Sigma_k \times D_k$ $(k\geq 0)$ be the solutions of {\em (\ref{mixed})} and {\em (\ref{problem_mixed})}, respectively. Assume that $u\in H^{1+r}(\O)$ with some $r>0$ and that $u|_K\in H ^{1+s_K}(K)$ with an element-wisely defined regularity $s_K>0$ for all $K\in{\cal T}$. Then there exists a constant $C>0$ independent $\alpha$ and $h$ for both the two- and three-dimension such that
\begin{eqnarray}\label{err-bound-Div2RT}
\|\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\sigma$}_h\|_{\alpha,h,H({\rm div})} &\leq& C
\sum_{K\in{\cal T}_{low}} \left(h_K^{s_K} |\alpha^{1/2}\nabla u|_{s_K,K} + h_K \|\alpha^{-1/2}f\|_{0,K}\right)\\
&&\quad + C\sum_{K\in{\cal T}_{high}} \left ( h_K^{\min\{k+1,s_K\}} |\alpha^{1/2}\nabla u|_{\min\{k+1,s_K\},K} \right. \\
&&\quad \left. + h_K^{\min\{k+2,s_K\}}\|\alpha^{-1/2}f\|_{\min\{k+1,s_K-1\},K}\right), RT_{k} \mbox{ case}.
\\[2mm] \label{err-bound-Div2BDM}
\|\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\sigma$}_h\|_{\alpha,h,H({\rm div})} &\leq& C
\sum_{K\in{\cal T}_{low}} \left(h_K^{s_K} |\alpha^{1/2}\nabla u|_{s_K,K} + h_K \|\alpha^{-1/2}f\|_{0,K}\right)\\
&&\quad + C\sum_{K\in{\cal T}_{high}} h_K^{\min\{k+2,s_K\}} \left ( |\alpha^{1/2}\nabla u|_{\min\{k+2,s_K\},K} \right. \\
&&\quad \left. + \|\alpha^{-1/2}f\|_{\min\{k+1,s_K-1\},K}\right), BDM_{k+1} \mbox{ case}.
\end{eqnarray}
\end{thm}
\begin{proof}
By the definition of the norm $\|\cdot\|_{\alpha,h,H({\rm div})}$, we only need to discuss the term
$$
h_K\|\alpha^{-1/2}\nabla\cdot (\mbox{\boldmath$\sigma$}- \mbox{\boldmath$\sigma$}_h)\|_{0,K} = h_K\|\alpha^{-1/2}(f-Q^k_h f)\|_{0,K}
$$
for each element $K\in{\cal T}$.
The first case is that the regularity is low in the element $K\in {\cal T}_{low}$, with $0< s_K <1$. In this case, notice that $f \in L^2(K)$, thus
$$
h_K\|\alpha^{-1/2}(f-Q^k_h f)\|_{0,K} \leq h_K\|\alpha^{-1/2}f\|_{0,K}.
$$
Compared to the error $h_K^{s_K} |\alpha^{1/2}\nabla u|_{s_K,K}$ from the weighted $L^2$ approximation, it is of high order.
The other case is that $s_K \geq 1$ in the element $K$. Note that $\alpha_K$ is assumed to be a constant in $K$, thus $f = \nabla\cdot(\alpha_K\nabla u) = \alpha_K \Delta u \in H^{s_K-1}(K)$, thus
$$
h_K\|\alpha^{-1/2}(f-Q^k_h f)\|_{0,K} \leq C h_K^
{\min\{s_K,k+2\}}\|\alpha^{-1/2}f\|_{\min\{s_K-1,k+1\},K}.
$$
Compared with the weighted $L^2$ error, this term is of the same order for the $BDM_{k+1}$ approximation and one order high for the $RT_k$ approximation.
\end{proof}
\begin{rem}
One may want to use the Brezzi's theory directly as in \cite{LS:06} to get the following a priori error estimate
$$
|\!|\!| u - u_h|\!|\!|_{\alpha,h} +\|\mbox{\boldmath$\sigma$} - \mbox{\boldmath$\sigma$}_h\|_{\alpha, h} \leq C\left(\inf_{v\in D_k}|\!|\!| u - v_h|\!|\!|_{\alpha,h} +\inf_{\mbox{\boldmath$\tau$} \in \Sigma_k}\|\mbox{\boldmath$\sigma$} - \mbox{\boldmath$\tau$}_h\|_{\alpha, h}\right ).
$$
This is not right, since for problems with a low regularity, the $L^2$ norm of the trace $\|\mbox{\boldmath$\sigma$}\cdot{\bf n}\|_{0,F}$ is not defined and thus $\|\mbox{\boldmath$\sigma$} \|_{\alpha,h}$ is not well-defined. Also, the result obtained by this is sub-optimal for the flux approximation.
\end{rem}
\begin{rem}
In the standard mixed method analysis, the $L^2$ norm of $u-u_h$ is analyzed and it has the same order convergence as the $RT$ approximation. In the case of the robust local a priori error estimate, we cannot get a robust local estimate for $\|\alpha^{1/2}(u-u_h)\|_0$ since robust an inf-sup condition
$$
\sup_{\mbox{\boldmath$\tau$}_h \in \Sigma_k} \displaystyle\frac{(\nabla\cdot \mbox{\boldmath$\tau$}_h,v_h)}{ \|\mbox{\boldmath$\tau$}_h\|_{\alpha,h,H({\rm div})}}
\geq \beta \| \alpha^{1/2} v_h \|_0 \quad \forall\, v_h \in D_k,
$$
with a constant $\beta$ independent of $h$ and $\alpha$ is not available.
\end{rem}
\section{Stenberg's Post-processing}
Since in the mixed methods, the approximation $u_h$ measured in the weighted discrete $H^1$ energy norm is lower than that of the approximation of the flux, we introduce the Stenberg's post-processing to get a same order approximation.
On each element $K\in {\cal T}$, if $(\mbox{\boldmath$\sigma$}_h,u_h) \in RT_k\times D_k$ ($k\geq 0$) or $(\mbox{\boldmath$\sigma$}_h,u_h) \in BDM_k\times D_{k-1}$ ($k\geq 1$), i.e., the index of the flux approximation space is $k$, we find a $u_{h,K}^* \in P_{k+1}(K)$, such that
\begin{equation}
(\alpha \nabla u_{h,K}^*, \nabla v_h)_K = (f,v_h)_K - (\mbox{\boldmath$\sigma$}_h\cdot{\bf n}, v_h)_{\partial K}, \quad \forall v_h\in P_{k+1}(K)/\rm I\kern-.19emR,
\end{equation}
and
\begin{equation}
\int_K u_{h,K}^* dx = \int_K u_{h} dx.
\end{equation}
We first prove the following trace theorem by using techniques in \cite{BH:01,CaYeZh:11}.
\begin{thm}
For an element $K\in{\cal T}$ with the mesh size $h_K$, we have
\begin{equation} \label{trace}
\|\mbox{\boldmath$\tau$}\cdot{\bf n}\|_{-1/2,\partial K} \leq C(\|\mbox{\boldmath$\tau$}\|_{0,K} + h_K \|\nabla\cdot \mbox{\boldmath$\tau$}\|_{0,K}), \quad \forall \mbox{\boldmath$\tau$} \in H({\rm div};K).
\end{equation}
\end{thm}
\begin{proof}
For any $\mbox{\boldmath$\tau$}\in H({\rm div};K)$ and $v\in H^1(K)$, we have the following identity:
\begin{equation}
\langle v, \mbox{\boldmath$\tau$}\cdot{\bf n} \rangle_{\partial K} =(\mbox{\boldmath$\tau$}, \nabla v)_K + (\nabla\cdot \mbox{\boldmath$\tau$}, v)_K,
\end{equation}
where $\langle v, \mbox{\boldmath$\tau$}\cdot{\bf n}\rangle_{\partial K}$ should be viewed as the duality pair between
$H^{1/2}(\partial K)$ and $H^{-1/2}(\partial K)$.
Thus
$$
\|\mbox{\boldmath$\tau$}\cdot{\bf n}\|_{-1/2,\partial K} = \sup_{v\in H^{1/2}(\partial K)} \displaystyle\frac
{(\mbox{\boldmath$\tau$}, \nabla v)_K + (\nabla\cdot \mbox{\boldmath$\tau$}, v)_K}{\|v\|_{1/2,\partial K}}.
$$
On a reference element $\hat{K}$, given $g\in H^{1/2}(\partial \hat{K})$, consider the following equation
$$
- \Delta z + z =0 \in \hat{K}, \quad z = g \mbox{ on } \partial \hat{K}.
$$
By the elliptic stability theory, we have
$$
\|\nabla z\|_{0,\hat{K}} + \| z\|_{0,\hat{K}} \leq C\|g\|_{1/2, \partial \hat{K}}.
$$
Mapping back to the physical element $K$ we have that given a $g\in H^{1/2}(\partial K)$,
there exits a $w_g\in H^1(K)$ and $w=g$ on $\partial K$, such that
$$
\|\nabla w_g\|_{0, K} + h_K^{-1} \| w_g\|_{0, K} \leq C\|g\|_{1/2, \partial K}.
$$
Thus
$$
\|\mbox{\boldmath$\tau$}\cdot{\bf n}\|_{-1/2,\partial K}
\leq \displaystyle\frac
{(\mbox{\boldmath$\tau$}, \nabla w_g)_K + (\nabla\cdot \mbox{\boldmath$\tau$}, w_g)_K}{\|g\|_{1/2,\partial K}}
\leq
C(\|\mbox{\boldmath$\tau$}\|_{0,K} + h_K \|\nabla\cdot \mbox{\boldmath$\tau$}\|_{0,K}).
$$
The
\end{proof}
\begin{thm} In each element $K\in{\cal T}$, the following robust best approximation property holds:
\begin{equation}
\|\alpha^{1/2}_K\nabla(u-u_{h,K}^*)\|_{0,K} \leq C \left( \inf_{w_h \in P_{k+1}(K)}\|\alpha^{1/2}_K\nabla (u-w_h)\|_{0,K} + \|\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h\|_{\alpha,h, H({\rm div}),K}
\right).
\end{equation}
\end{thm}
\begin{proof}
Let $w_h$ be an arbitrary function in $P_{k+1}(K)$, and $v_h= u_{h,K}^* - w_h$. Let $\overline{v}_h = \int_K v_h dx /|K|$ be the average of $v_h$ on $K$, then $v_h-\overline{v}_h$ belongs to the test space $P_{k+1}(K)/\rm I\kern-.19emR$. Then
\begin{eqnarray*}
\|\alpha^{1/2}_K\nabla(u_{h,K}^* - w_h)\|_{0,K}^2
& = &\|\alpha^{1/2}_K\nabla v_h\|_{0,K}^2 = (\alpha\nabla(u_{h,K}^* - w_h), \nabla v_h)_K\\
& = & (\alpha_K\nabla u_{h,K}^*,\nabla (v_h- \overline{v}_h))_K -(\alpha\nabla w_h, \nabla v_h)_K\\
& = & (f,v_h-\overline{v}_h)_K - (\mbox{\boldmath$\sigma$}_h\cdot{\bf n}, v_h-\overline{v}_h)_{\partial K} -(\alpha_K\nabla w_h, \nabla v_h)_K \\
& = & (\alpha_K \nabla (u-w_h),\nabla v_h)_K +((\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\cdot{\bf n}, v_h-\overline{v}_h)_{\partial K},
\end{eqnarray*}
where we use the fact that
$(\alpha_K \nabla u, \nabla v)_K = (f,v)_K - (\mbox{\boldmath$\sigma$}\cdot{\bf n}, v)_{\partial K}$ is true for any $v \in H^1(K)$.
By the Cauchy-Schwarz inequality, $(\alpha_K \nabla (u-w_h),\nabla v_h)_K \leq \|\alpha_K^{1/2}\nabla (u-w_h)\|_{0,K}\|\alpha_K^{1/2}\nabla v_h\|_{0,K}$. By the definition of the dual norm, the trace inequality \eqref{trace}, and the fact $\|v_h-\overline{v}_h\|_{0,K} \leq C h_K\|\nabla v_h\|_{0,K}$, we have
\begin{eqnarray*}
((\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\cdot{\bf n}, v_h-\overline{v}_h)_{\partial K} &\leq& \|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\cdot{\bf n}\|_{-1/2,\partial K} \|\alpha^{1/2}(v_h-\overline{v}_h)\|_{1/2,\partial K} \\
&\leq & C h_K^{-1}\|\alpha^{1/2}(v_h-\overline{v}_h)\|_{0,K} \|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\cdot{\bf n}\|_{-1/2,\partial K}\\
&\leq & C \|\alpha^{1/2}\nabla v_h\|_{0,K}(\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_{0,K} + h_K\|\alpha^{-1/2}\nabla\cdot(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_{0,K}).
\end{eqnarray*}
Thus
$$
\|\alpha^{1/2}\nabla(u_{h,K}^* - w_h)\|_{0,K} \leq
C(\|\alpha^{1/2}\nabla (u-w_h)\|_{0,K} +
\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_{0,K} + h_K\|\alpha^{-1/2}\nabla\cdot(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_{0,K}).
$$
By the triangle inequality,
\begin{equation}
\|\alpha^{1/2}\nabla(u-u_{h,K}^*)\|_{0,K} \leq \|\alpha^{1/2}\nabla(u-w_h)\|_{0,K}+\|\alpha^{1/2}\nabla(u_{h,K}^* - w_h)\|_{0,K}.
\end{equation}
The theorem is proved.
\end{proof}
By the approximation property of $P_{k+1}(K)$, and the robust local optimal error estimate of $\mbox{\boldmath$\sigma$}_h$, we immediately have the following robust local optimal error estimate for the Stenberg's post-processing.
\begin{thm}
For both the $(\mbox{\boldmath$\sigma$}_h,u_h) \in RT_k\times D_k$ ($k\geq 0$) or $(\mbox{\boldmath$\sigma$}_h,u_h) \in BDM_k\times D_{k-1}$ ($k\geq 1$) case, the Stenberg's recovery $u_{h,K}^* \in P_{k+1}(K)$ has the following robust local a priori error estimate in the low regularity elements $K\in {\cal T}_{low}$ with $0\leq s_K<1$:
\begin{equation}
\|\alpha^{1/2}_K\nabla(u-u_{h,K}^*)\|_{0,K} \leq
C h_K^{s_K} |\alpha^{1/2}\nabla u|_{s_K,K} + h_K \|\alpha^{-1/2}f\|_{0,K}, K \in {\cal T}_{low}.
\end{equation}
For those elements $K\in {\cal T}_{high}$ with $1\leq s_K$, the following robust local a priori error estimate holds:
\begin{eqnarray
\|\alpha^{1/2}_K\nabla(u-u_{h,K}^*)\|_{0,K} &\leq&
C \left( h_K^{\min\{k+1,s_K\}} |\alpha^{1/2}\nabla u|_{\min\{k+1,s_K\},K} \right. \\
&& \left.+ h_K^{\min\{k+2,s_K\}}\|\alpha^{-1/2}f\|_{\min\{k+1,s_K-1\},K} \right), RT_{k}\times D_k \mbox{ case}.
\\[2mm]
\|\alpha^{1/2}_K\nabla(u-u_{h,K}^*)\|_{0,K} &\leq& C h_K^{\min\{k+1,s_K\}} \left (|\alpha^{1/2}\nabla u|_{\min\{k+1,s_K\},K} \right. \\
&&\quad \left. + \|\alpha^{-1/2}f\|_{\min\{k,s_K-1\},K}\right), BDM_{k}\times D_{k
-1} \mbox{ case}.
\end{eqnarray}
\end{thm}
\begin{rem}
There are other post-processings available, such as the one proposed in \cite{AC:95} and analyzed in \cite{Voh:10}. The recovered potential is also mainly from the numerical flux $\mbox{\boldmath$\sigma$}_h$, a similar robust and local optimal a priori error estimate can also be derived.
It is also well known if the mixed method is implemented by hybridization, the Lagrange multiplier is also a better approximation of $u$ than $u_h$, and is a good source for post-processing or solution reconstruction. With careful analysis, it should not be hard to derive robust and local optimal result for the Lagrange multiplier and its post-processed solution under a similar weighted discrete $H^1$ norm.
\end{rem}
\section{Final comments}
In this paper, for elliptic interface problems in two- and three-dimensions with a possible very low regularity, we establish robust and local optimal a priori error estimates for the Raviart-Thomas and Brezzi-Douglas-Marini mixed finite element approximations. For the flux approximation, we show the robust best best approximation in the discrete equilibrated space and the whole mixed approximation space with appropriated norms, an $\alpha$-weighted $L^2$ norm or an $(\alpha,h)$-weighted $H({\rm div})$ norms. We show the robust local optimal error estimates for the flux approximation in these norms. For the potential approximation, we show a robust best approximation result in a weighted discrete $H^1$ norm and show that the convergence order is sub-optimal compared to the flux approximation. We then show that with the flux as the main source of post-processing, the Stenberg's post-processing can recover a potential with the robust local optimal error estimate.
These robust and local optimal a priori estimates provide guidance for constructing robust a posteriori error estimates and adaptive methods for the mixed approximations. For robust a posteriori error for the mixed methods of the interface problem, we should focus on $\|\alpha^{-1/2}(\mbox{\boldmath$\sigma$}-\mbox{\boldmath$\sigma$}_h)\|_0$, like the approaches in \cite{Ain:07,CaZh:10,Kim:07,Voh:10}. The approaches in \cite{BrVe:96,LS:06} are not right since they are all try to put $u_h$ into the estimator. If any post-processing is going to be used to construct the a posteriori error estimator, the main source of information should be the numerical flux $\mbox{\boldmath$\sigma$}_h$, not the numerical potential $u_h$ itself.
|
{
"timestamp": "2019-03-01T02:09:36",
"yymm": "1902",
"arxiv_id": "1902.10901",
"language": "en",
"url": "https://arxiv.org/abs/1902.10901"
}
|
\section{Introduction}
Our understanding of quantum gravity has been dramatically advanced by the AdS/CFT correspondence. In a sense, it provides a precise framework to tackle the gravity path integral by formulating it non-perturbatively in terms of a quantum field theory. As we still grapple with several challenges in black hole physics and cosmology, we require to develop newer tools for calculating various observables and engineer mechanisms to adapt holography to more general settings.
A question of fundamental importance is how can we formulate quantum gravity with some specified boundary conditions and can holography turn out to be useful in this context. A situation where we are posed with this problem appears is in the evolution of a closed universe, wherein the wavefunction of interest is calculated with fixed boundary conditions {\it i.e.~} of an initial state \cite{Hartle:1983ai}. For specified spatial boundary conditions, there has been some progress from studies of the holographic renormalization group \cite{deBoer:1999tgo,deHaro:2000vlm}. Here, attempts were made to interpret the radial cut-off as a Wilsonian cut-off in field theory whereby integrating out bulk geometry corresponds to integrating out high energy modes of the field theory \cite{Faulkner:2010jy,Heemskerk:2010hk}. Identifying the radial cut-off in the bulk with a short-distance cut-off seems to work more naturally in this vein \cite{lecholrg}.
Another procedure for implementing such a regularization is to deform the holographic CFT by some irrelevant operator, the scale associated to which is treated as the cut-off. The important question is then to pinpoint this operator in holographic CFTs. Along these lines, over the last couple of years a novel viewpoint has emerged via the $T{\bar{T}}$ deformation of QFTs. Discovered first in two dimensions, the $T{\bar{T}}$ operator induces a solvable, irrelevant, double-trace deformation of QFTs \cite{Zamolodchikov:2004ce,Smirnov:2016lqw,Cavaglia:2016oda}. The one-parameter family of theories parametrized by the $T{\bar{T}}$ coupling has a number of special properties. Deforming an integrable theory by $T{\bar{T}}$ preserves integrability. The deformed partition function obeys diffusion-like equations \cite{Cardy:2018sdv} and is modular invariant in a unique sense \cite{Datta:2018thy,Aharony:2018bad}. The finite-size spectrum of this theory is exactly the same as Jackiw-Teitelbohm gravity coupled to the undeformed `matter' \cite{Dubovsky:2018bmo}.
The $T{\bar{T}}$ deformation, for large-$c$ CFTs, has been proposed to be holographically dual to AdS$_3$ with a finite radial cut-off with Dirichlet boundary conditions \cite{McGough:2016lol}. Within the pure gravity sector, this geometric notion does reproduce some characteristics of the deformed CFT for a specific sign of the coupling \cite{Kraus:2018xrn}. Some further tests of this cut-off gravity/$T{\bar{T}}$ relation include the finite-size spectrum, signal propagation velocities, stress tensor correlators \cite{Aharony:2018vux,Kraus:2018xrn}, entanglement entropy \cite{Donnelly:2018bef,Chen:2018eqk,Gorbenko:2018oov}. Interestingly, among other observations, \cite{McGough:2016lol} noticed that the flow equation for the trace of the energy momentum tensor in $T\bar{T}$ deformed theory can be re-written as the Hamiltonian constraint in bulk gravity theory on $AdS_3$ spacetime. A different version of $AdS_3$ holography in this context has been put forward in \cite{Giveon:2017myj,Giveon:2017nie}. A Lorenz breaking cousin of this deformation and its holographic interpretation has been proposed in \cite{Guica:2017lia}. An application to de-Sitter holography was studied in \cite{Gorbenko:2018oov}.
The holographic construction with a cut-off has also been generalized to higher dimensions in \cite{Taylor:2018xcy,Hartman:2018tkw} (see also \cite{Chang:2018dge} for a supersymmetric generalization to higher dimensions). The guiding principle behind these works was to define holographic $TT$-deformed theories via a flow equation that originates from the Gauss-Codazzi equation {\it i.e.~} the Hamiltonian constraint. At large $N$, this flow can indeed be seen as coming from the deforming operators that are quadratic in the energy-momentum tensor (see also \cite{Cardy:2018sdv} for generalization in the form of $\det T$). In \cite{Taylor:2018xcy,Hartman:2018tkw} it was shown that such a procedure is consistent with finite cut-off holography through agreements of the quasi-local energy, speed of sound as well as simple correlators within the pure gravity sector. It was also shown in \cite{Shyam:2018sro} that for $d=4$ various results of holographic RG, such as the gradient form of the metric beta functions, are also captured by such irrelevant double trace deformations involving the stress tensor and other suitably defined deforming operators.
In this work, we study the sphere partition functions of $TT$ deformed CFTs in $d\geq 2$. The sphere partition function $Z_{\mathbb{S}^d}$ plays an important role in a wide variety of aspects. The QFT on the sphere is free from IR divergences and for several supersymmetric theories it has been computed exactly by localization techniques \cite{Marino:2011eh,Fuji:2011km,Pestun:2007rz,Pestun:2016zxk}. These have led to many precision tests of holography and a better understanding of RG flows. For CFTs in even dimensions, it captures the anomalies. When anomalies are absent, for instance in 3 dimensions, $F=- \log |Z_{\mathbb{S}^3}|$ serves as an analogue to the central charge in counting the degrees of freedom; $F$ decreases along the RG flow from the UV to the IR \cite{Jafferis:2011zi,Klebanov:2011gs}. Although the $F$-theorem is different in flavour from the even dimensional analogues (the $c$ and $a$ theorems), a unified formulation can be achieved by considering the sphere free energy. This quantity also has other uses. In odd dimensional CFTs, the finite piece of the sphere free energy $F_{d}$ measures the entanglement entropy across a spherical region $\mathbb{S}^{d-2}$ in flat spacetime $\mathbb{R}^{d-1,1}$.
The spectrum of $TT$ deformed CFTs can be computed from the flow equation. The flow equation relies on special factorization properties of the $TT$-operator \cite{Zamolodchikov:2004ce} which are expected to hold true for large $N$ theories in higher dimensions. For the case of the cylinder $\mathbb{S}^{d-1}\times \mathbb{R}^1$, this takes the form of the Burger's equation in hydrodynamics. For the sphere on the other hand, owing to its maximal symmetry, the flow equation can be reduced to an algebraic equation and be solved exactly. This allows us to evaluate the sphere free energy/partition function from the field theory side in a very simple manner. In the cut-off AdS bulk, we calculate the on-shell action with the necessary counterterms \cite{Emparan:1999pm} and observe a precise agreement with the field theory analysis.
Since the holographic flow equation was deduced in \cite{Hartman:2018tkw} from re-writing the bulk Gauss-Codazzi equation in terms of the holographic stress tensor, one may wonder how universal or generic is this flow. Proving this equation starting from the definition of $TT$ operators in higher dimensions and on a curved background in ABJM or $\mathcal{N}=4$ SYM is beyond the scope of this work and remains an important future problem. However, we address the issue of how one can obtain such a universal flow equation from purely field theoretic considerations, namely starting already from a local Callan-Symanzik equation describing a CFT deformed only by an irrelevant $TT$ operator.
The Wheeler-DeWitt equation is a quantum constraint equation in a theory of quantum gravity that encodes the independence of the theory under choice of a foliation of space-time by co-dimension one hypersurfaces. Such a foliation is typically chosen in order to pass to the Hamiltonian formalism, as introduced in \cite{ADM}. The diffeomorphism invariance of the gravitational theory translates into the independence under the choice of foliation, and is thereby encoded in a constraint. In our context, the relevant foliation of the bulk is by successive finite radius cut-off surfaces. The $TT$ flow equation is then mapped to the semiclassical limit of the bulk Wheeler-DeWitt equation. In this work, we solve the mini-superspace Wheeler-DeWitt equation in the WKB approximation and find the Wheeler-DeWitt wavefunction that, indeed, up to holographic counterterms, matches our partition functions.
This paper is organized as follows. In section \ref{sec:holography}, we compute holographic stress-tensor and partition function in Euclidean anti-de Sitter geometries with spherical boundary at finite radial cut-off up to 6 dimensions. In section \ref{sec:field-theory}, we review the $TT$ deformation in field theory and consider the flow equation on the sphere and its solution. This allows us to obtain the sphere partition function and find an exact agreement with the gravity results, at large $N$. We provide field theoretical derivation of the flow equation starting from the local Callan--Symanzik equation and the regularization procedure of the $TT$ operator in section \ref{sec:FlowRegularization}. In Section \ref{sec:Wheeler-DeWitt}, we discuss the (mini-superspace) Wheeler-DeWitt equation and its solution in the WKB approximation that captures our partition functions up to holographic counterterms. Finally, we conclude and pose some open problems in section \ref{sec:conclusions}. Appendix \ref{sec:Gauss-Codazzi} contains a review of the Gauss-Codazzi equation as a flow equation. Appendix \ref{sec:generateR} has some details on the field theory derivation of the flow.
\section{Finite cut-off holography}
\label{sec:holography}
We begin by computing the holographic energy-momentum tensor as well as the sphere partition functions up to $d=6$ dimensions using AdS/CFT with a finite cut-off. In standard holographic computations, at large $N$ and strong 't Hooft coupling, both quantities are related to the bulk gravity action evaluated on Euclidean AdS solution in one higher ($d+1$) dimension\footnote{We ignore additional internal directions in this work.}.
More precisely, we consider regularized gravity action given by the Einstein-Hilbert (EH) and Gibbons-Hawking (GH) terms supplemented by local counterterms
\begin{eqnarray}\label{on-shell-init}
I^{(d+1)}_{on-shell}=-\frac{1}{2\kappa^2}\int_{\mathcal{M}}d^{d+1}x\sqrt{g}\left(R-2\Lambda\right)+\frac{1}{\kappa^2}\int_{\partial \mathcal{M}}d^dx \sqrt{\gamma}K+S_{ct},
\end{eqnarray}
where the counterterm action, up to $d=6$, takes the form \cite{Emparan:1999pm}
\begin{equation}\label{counterterm-action}
S_{ct}=\frac{1}{\kappa^2}\int_{\partial \mathcal{M}}d^dx\sqrt{\gamma}\left[\frac{d-1}{l}c^{(1)}_d+\frac{c^{(2)}_d\,l}{2(d-2)} \tilde{R}+\frac{c^{(3)}_dl^3}{2(d-4)(d-2)^2}\left(\tilde{R}^{ij}\tilde{R}_{ij}-\frac{d}{4(d-1)}\tilde{R}^2\right)\right],
\end{equation}
where $\kappa^2\equiv 8\pi G_N$, $\tilde{R}$ and $\tilde{R}_{ij}$ are the Ricci scalar and Ricci tensor of the cut-off surface. To keep track of different contributions, we introduce $c^{(1)}_d=1$ for $d\ge 2$, $c^{(2)}_d=1$ for $d\ge 3$ and $c^{(3)}_d=1$ is non-zero from $d\ge 5$.
We now consider a Euclidean AdS solution
\begin{equation}
ds^2=\frac{l^2\,dr^2}{l^2+r^2}+r^2d\Omega^2_d\equiv\frac{l^2\,dr^2}{l^2+r^2}+\gamma_{ij}(r,x)dx^idx^j, \label{metric}
\end{equation}
such that for a fixed value of $r=r_c$ the induced metric, $\gamma_{ij}(r_c,x)=r^2_c\gamma^b_{ij}(x)$, describes a sphere with radius $r_c$. Metric $\gamma^b_{ij}(x)$ of the unit sphere will later be identified with the metric of the boundary QFT theory.
The full on-shell action corresponding to this solution can be used to compute the energy-momentum tensor (Brown-York) and the holographic sphere partition function
\begin{eqnarray}
T^d_{ij}[r]\equiv-\frac{2}{\sqrt{\gamma}}\frac{\delta I^{(d+1)}_{on-shell}[r]}{\delta \gamma^{ij}},\qquad\qquad \log Z_{\mathbb{S}^{d}}[r]\equiv -I^{(d+1)}_{on-shell}[r].\label{PFT}
\end{eqnarray}
Note that both quantities explicitly depend on the radius $r$ at which we cut off spacetime. In standard holography, we take $r$ to infinity and the counterterm action yields finite answers (modulo logarithmic divergences that correspond to anomalies). However, in the context of finite cut-off holography, we keep the radial dependence finite -- this can be holographically interpreted as a deformation by a generalization of the $T\bar{T}$ operator to arbitrary dimensions.
In what follows, we evaluate both quantities in \eqref{PFT} from the on-shell action with a finite radial cut-off. From the symmetry of the problem, they are determined by a single function of the radius $\omega[r]$ that we extract from both computations, with exact agreement. We will later demonstrate that this function solves the algebraic flow equation that defines the deformed theory.
\subsection{Holographic stress-tensors}
The holographic stress-tensor \cite{Balasubramanian:1999re,deHaro:2000vlm} is obtained by variation of the on-shell action with respect to the induced metric on the surface of constant $r=r_c$. We first compute the general variation and then show that, as constrained by the spherical symmetry, for our metric \eqref{metric}, energy momentum tensor is proportional to the metric. Performing the standard variations we obtain \cite{Emparan:1999pm,Balasubramanian:1999re,deHaro:2000vlm}\footnote{Note that the signs of the first two terms in \eqref{hol-stress} differ from equation \cite[eq.~(3.3)]{Hartman:2018tkw} since the extrinsic curvature is defined there with an opposite sign. The Gibbons-Hawking term in \cite[eq.~(A.1)]{Hartman:2018tkw} also appears with a minus sign as opposed to our equation \eqref{on-shell-init} where it appears with a plus sign.}
\begin{align}\label{hol-stress}
T_{ij}=-\frac{1}{\kappa^2}&\left\{K_{ij}-K \gamma_{ij}-c^{(1)}_d\frac{d-1}{l}\gamma_{ij}+\frac{c^{(2)}_d\,l}{(d-2)}\tilde{G}_{ij}\right.\nonumber\\
&\quad \left.+\frac{c^{(3)}_dl^3}{(d-4)(d-2)^2}\left[2(\tilde{R}_{ikjl}-\frac{1}{4}\gamma_{ij}\tilde{R}_{kl})\tilde{R}^{kl}-\frac{d}{2(d-1)}\left(\tilde{R}_{ij}-\frac{1}{4}\tilde{R}\gamma_{ij}\right)\tilde{R}\right.\right.\nonumber\\
&\quad -\left.\left.\frac{1}{2(d-1)}\left(\gamma_{ij}\Box \tilde{R}+(d-2)\nabla_i\nabla_j\tilde{R}\right)+\Box \tilde{R}_{ij}\right]\right\},
\end{align}
where $\tilde{G}_{ij}$, $\tilde{R}_{ikjl}$ are the Einstein and Riemann tensors and $\Box$ is the Laplace-Beltrami operator of the induced metric $\gamma_{ij}$. On the field theory side, the above holographic stress tensor will be used to construct the operator (or its expectation value) which deforms the CFT. For future reference, we note the contributions of additional terms for $d\geq 3$ in the counterterm action \eqref{counterterm-action}.
Following the conventions of \cite{Hartman:2018tkw} we denote
\begin{align}\label{C-tensor-def}
{C}_{ij}=\,&\left\{{c^{(2)}_d}\tilde{G}_{ij}+c^{(3)}_db_d\left[2(\tilde{R}_{ikjl}-\frac{1}{4}\gamma_{ij}\tilde{R}_{kl})\tilde{R}^{kl}-\frac{d}{2(d-1)}\left(\tilde{R}_{ij}-\frac{1}{4}\tilde{R}\gamma_{ij}\right)\tilde{R}\right.\right.
\nonumber\\
&\quad -\left.\left.\frac{1}{2(d-1)}\left(\gamma_{ij}\Box \tilde{R}+(d-2)\nabla_i\nabla_j\tilde{R}\right)+\Box \tilde{R}_{ij}\right]\right\}.
\end{align}
with
$b_d=l^2/((d-4)(d-2))$.
Now we evaluate the stress-tensor \eqref{hol-stress} for the metric \eqref{metric} with $r=r_c$, so that the induced metric\footnote{In Section \ref{sec:holography}, $\gamma_{ij}$ refers to this induced metric and we suppress the explicit dependence on $(r_c,x)$. Later, in the field theory part, Section \ref{sec:field-theory}, we will work with boundary $\gamma_{ij}$ related by factor of $r^2_c$. We hope that notation should be clear depending on the context.} becomes $\gamma_{ij}(r_c,x)$. Firstly, the extrinsic curvature terms on the constant $r=r_c$ surface become
\begin{equation}
K_{ij}-K \gamma_{ij}=\frac{d-1}{l}\sqrt{1+\frac{l^2}{r^2_c}} \, \gamma_{ij}.
\end{equation}
The Ricci tensor at $r_c$ is also proportional to the metric $\tilde{R}_{ij}=\frac{d-1}{r^2_c}\gamma_{ij}$ such that the Einstein tensor for the sphere (Einstein manifold) is given by
\begin{equation}
\tilde{G}_{ij}=-\frac{(d-2)(d-1)}{2r^2_c}\gamma_{ij}.
\end{equation}
Then, the contraction of the Riemann tensor with Ricci tensor is also proportional to the metric such that the second line of \eqref{hol-stress} becomes
\begin{equation}
2(\tilde{R}_{ikjl}-\frac{1}{4}\gamma_{ij}\tilde{R}_{kl})\tilde{R}^{kl}-\frac{d}{2(d-1)}\left(\tilde{R}_{ij}-\frac{1}{4}\tilde{R}\gamma_{ij}\right)\tilde{R}=\frac{(d-1)(d-4)(d-2)^2}{8r^4_c}\gamma_{ij}.
\end{equation}
Finally, the last line \eqref{hol-stress} vanishes for the sphere (constant curvature) and we get the holographic energy momentum-tensor \eqref{hol-stress} for the $d$-dimensional sphere at $r=r_c$ in Euclidean $AdS_{d+1}$ \eqref{metric}
\begin{equation} \label{omega-from-AdS}
T^d_{ij}[r_c]=\frac{(d-1)}{\kappa^2 l}\left[c^{(1)}_d+\frac{c^{(2)}_dl^2}{2r^2_c}-\frac{c^{(3)}_dl^4}{8r^4_c}-\sqrt{1+\frac{l^2}{r^2_c}}\right]\gamma_{ij}.
\end{equation}
Indeed, we see that it is proportional to the metric and we define the proportionality function as
\begin{equation}
\omega[r_c]=\frac{(d-1)}{\kappa^2 l}\left[c^{(1)}_d+\frac{c^{(2)}_dl^2}{2r^2_c}-\frac{c^{(3)}_dl^4}{8r^4_c}-\sqrt{1+\frac{l^2}{r^2_c}}\right].
\end{equation}
Clearly, we see that different counterterms in various dimensions contribute with polynomial terms whereas the EH and GH terms yield the square-root part. As we shall show in Section \ref{sec:field-theory}, this function can be obtained by solving the QFT flow equation that becomes an algebraic equation for $\omega[r_c]$.
\subsection{Sphere partition functions}
The next step involves evaluation of the regularized gravity actions and the holographic sphere partition functions with finite cut-off. We evaluate the action \eqref{on-shell-init} in AdS with a cut-off or wall at $r=r_c$ where we also take into account the counterterms \eqref{counterterm-action}.
Our metric \eqref{metric} has a constant negative curvature $R=-d(d+1)/l^2$ and is a solution of the vacuum Einstein equations with negative cosmological constant $\Lambda=-d(d-1)/(2l^2)$. With these ingredients and the formulae of the previous subsection, we can evaluate the on-shell action
\begin{align}
I^{(d+1)}_{\rm on-shell}[r_c]=\frac{dl^{d}S_d}{\kappa^2l}&\left[\int^{r_c}_0\frac{r^d\,dr}{l^{d+1}\sqrt{1+\frac{r^2}{l^2}}}-\left(\frac{r_c}{l}\right)^{d-1}\sqrt{1+\frac{r^2_c}{l^2}}\right.\nonumber\\
&\quad + \frac{r^d_c}{l^d}\left.\left(c^{(1)}_d\frac{(d-1)}{d}+\frac{c^{(2)}_d(d-1)}{2(d-2)} \frac{l^2}{r^2_c}-\frac{c^{(3)}_d(d-1)}{8(d-4)}\frac{l^4}{r^4_c}\right)\right]
\end{align}
The first term comes from the EH action, the second from the GH boundary term and second line from the counterterms. There is an overall factor of the sphere area, $S_d=(2\pi^{\frac{d+1}{2}})/\Gamma\left(\frac{d+1}{2}\right)$.
Moreover, the first two terms {\it i.e.~} the EH and GH terms, can be written under one integral as
\begin{equation}
S_{EH}+S_{GH}=-\frac{d(d-1)S_d}{2\kappa^2l}\int^q_0 \sqrt{l^2q^{d-3}+q^{d-2}}dq,\label{osEHGH}
\end{equation}
where we introduced $q=r^2_c$ and this expression will be important in the Wheeler-DeWitt analysis (Section \ref{sec:Wheeler-DeWitt}).
Performing this integral yields the hypergeometric function and writing the answer in terms of $r_c$ gives the full holographic sphere partition function (up to $d=6$)
\begin{align}
\log Z_{\mathbb{S}^{d}}[r_c] =-\frac{dS_dr^d_c}{\kappa^2l}&\left[-\frac{l}{r_c}\,\,_2F_1\left(-\frac{1}{2},\frac{d-1}{2},\frac{d+1}{2},-\frac{r^2_c}{l^2}\right)\right.
\nonumber\\
& \quad
+\left.c^{(1)}_d\frac{(d-1)}{d}+\frac{c^{(2)}_d(d-1)}{2(d-2)}\frac{l^2}{r^2_c}-\frac{c^{(3)}_d(d-1)}{8(d-4)}\frac{l^4}{r^4_c}\right].\label{SPF}
\end{align}
This is the main result of this section and in Section \ref{sec:Wheeler-DeWitt} we will see how this expression is related to the solution of the Wheeler-DeWitt equation.
The on-shell action \eqref{SPF}, also allows us to extract $\omega[r_c]$. Namely, in general dimensions, the derivative of the sphere partition function with respect to the radius is related to the expectation value of the trace of the energy-momentum tensor. Therefore, we have
\begin{equation}
r_c\partial_{r_c} \log Z_{\mathbb{S}^{d}}[r_c]=-\int d^dx\sqrt{\gamma}\langle T^i_i\rangle=-r^d_cS_d\,d\,\omega[r_c],
\end{equation}
where we used that for the sphere $\vev{T_{ij}}=\omega[r_c] \gamma_{ij}$ and $\vev{T^i_i}=d\,\omega[r_c]$. Differentiating \eqref{SPF} we obtain
\begin{equation}
\omega[r_c]=\frac{(d-1)}{\kappa^2l}\left[c^{(1)}_d+c^{(2)}_d\frac{l^2}{2r^2_c}-c^{(3)}_d\frac{l^4}{8r^4_c}-\sqrt{1+\frac{l^2}{r^2_c}}\right].
\end{equation}
This is precisely the proportionality function derived in the previous subsection. In the next section, we show that it is the solution of the flow equations (with inclusion of anomalies) in all dimensions that we analyze.
\section{Field theory analysis}
\label{sec:field-theory}
\subsection{$T{\bar{T}}$ deformation in general dimensions}
As alluded to in the introduction the $T{\bar{T}}$ operator was initially introduced in 2$d$ by Zamolodchikov \cite{Zamolodchikov:2004ce}. This bi-local operator is defined as the following quadratic combination of the components of the stress-tensor
\begin{align}\label{tt-def}
T{\bar{T}}(z,z') = T_{zz}(z) T_{\bar{z} \bar{z}} (z') - T_{z\bar{z}}(z)T_{z\bar{z}}(z').
\end{align}
This definition is in flat Euclidean space ($z=x+it$). By using symmetries and conservation laws of the stress tensor, it can be shown that the expectation value of this operator is a constant. This fact motivates defining the operator at coincident points. Although there are divergences which do appear upon taking the coincident point limit, it can be shown that these appear as total derivative terms. The operator $T{\bar{T}}$ therefore makes sense unambiguously within an integral. We can then deform a QFT by this operator as follows
\begin{align}\label{action-flow}
\frac{dS(\lambda)}{d\lambda} = \int d^2 x \, T{\bar{T}}(x).
\end{align}
It is crucial to observe that the stress tensor components appearing in the right hand side of the above equation are that of the action $S(\lambda)$ and, therefore, the deformation is in a sense recursive. This leads to modifications of the action/Lagrangian which are generically non-linear in the coupling $\lambda$, see {\it e.g.~} \cite{Cavaglia:2016oda,Bonelli:2018kik}. For deformations of CFTs, the $T{\bar{T}}$ coupling is the only new dimensionful scale of the theory. If a single dimensionful scale is present, the following Ward identity for the effective action holds
\begin{align}\label{eff-Ward-id}
\lambda \frac{dW}{d\lambda} = -\frac{1}{2} \int d^2 x \, \vev{T_i^i} .
\end{align}
Combining the equations \eqref{action-flow} and \eqref{eff-Ward-id} leads to the flow equation
\begin{align}\label{ttb-fac}
\vev{T_i^i} = -2 \lambda \vev{T{\bar{T}}} = -2 \lambda \left(\vev{T_{zz}}\vev{T_{\bar{z} \bar{z}} }- \vev{T_{z\bar{z}}}^2 \right).
\end{align}
If the theory lives on a cylinder $\mathbb{R}\times \mathbb{S}^1$, the second equality of the above equation takes the same form as the inviscid Burger's equation of hydrodynamics \cite{Smirnov:2016lqw,Cavaglia:2016oda}.
The above analysis can be generalized for curved spaces and also to higher dimensions. This has been carried out in \cite{Taylor:2018xcy,Hartman:2018tkw}. The strategy there was to make use of the holographic stress tensor and higher dimensional analogues of \eqref{action-flow} and \eqref{eff-Ward-id} to build the deforming operator.
Although the factorization property \eqref{ttb-fac} is not true in general for curved spaces and $d> 2$, it is still expected to hold for large $N$ theories.
The deforming operator has the following structure
\begin{align}\label{x-def}
X_d = \left(T_{ij}+\frac{\alpha_d}{\lambda^{\frac{d-2}{d}}}C_{ij}\right)^2-\frac{1}{d-1}\left(T^{i}_i+\frac{\alpha_d}{\lambda^{\frac{d-2}{d}}}C^i_i\right)^2+\frac{1}{d }\frac{\alpha_d}{\lambda^{\frac{(d-2)}{d}}}\left(\frac{(d-2)}{2}R+C^i_i\right).
\end{align}
The notation $(B_{ij})^2= B_{ij}B^{ij}$ has been used above.
Here, $\alpha_d$ is a dimensionless parameter depending on the degrees of freedom of the theory -- {\it e.g.~} $\alpha_4 = N/(2^{7/2}\pi)$ for $\mathcal{N}=4$ super-Yang-Mills with an $SU(N)$ gauge group. The last term in \eqref{x-def} vanishes for $d=3, 4$.
The tensor $C_{ij}$ is the contribution to the holographic stress tensor from additional counterterms in $d\geq 3$, equation \eqref{C-tensor-def}. For the field theory on a sphere \eqref{C-tensor-def} become
\begin{equation}
C_{ij}=c^{(2)}_dG_{ij}+c^{(3)}_d\frac{2d\alpha_d\lambda^{\frac{2}{d}}}{d-4}\left[2(R_{ikjl}R^{kl}-\frac{1}{4}\gamma_{ij}R_{kl}R^{kl})-\frac{d}{2(d-1)}\left(RR_{ij}-\frac{1}{4}\gamma_{ij}R^2\right)\right],\label{CijQFT}
\end{equation}
where $c^{(n)}_d$ are defined as in Section \ref{sec:holography} (see below equation \eqref{counterterm-action}) so the first term only appears from $3$ dimensions and the second from $5$ dimensions.\\
For even dimensions, the appropriate anomaly terms are included as a part of the deforming operator. For $d=2$ the factors of $(d-2)$ in front of $R$ and in $\alpha_2$ cancel such that we recover the $T\bar{T}$ flow equation.
The above operator is quadratic in the stress tensor and should be viewed as the large $N$ approximation of a more general operator which could give rise to the dual quantum field theory for cut-off AdS. Specifically, the deforming operators across various dimensions are given by
\begin{align}
X_2 &= \left( T_{ij} \right)^2 - \left( T^i_{i} \right)^2 + \frac{1}{2\lambda} \frac{c}{24\pi} R ,\\
X_3 &= \left( T_{ij} + \frac{\alpha_3}{\lambda^{1\over 3}} G_{ij} \right)^2 - \frac{1}{2}\left( T^i_{i} + \frac{\alpha_3}{\lambda^{1\over 3}} G^i_{i} \right)^2 ,\\
\label{4d-deform}
X_4 &= \left( T_{ij} + \frac{\alpha_4}{\lambda^{1\over 2}} G_{ij} \right)^2 - \frac{1}{3}\left( T^i_{i} + \frac{\alpha_4}{\lambda^{1\over 2}} G^i_{i} \right)^2 ,\\
X_5 &= \left( T_{ij} + \frac{\alpha_5}{\lambda^{3\over 5}} C_{ij} \right)^2 - \frac{1}{4}\left( T^i_{i} + \frac{\alpha_5}{\lambda^{3\over 5}}C^i_{i} \right)^2 +\frac{1}{5\lambda}\frac{\alpha_5}{\lambda^{\frac{3}{5}}} \left(\frac{3}{2}R + C_i^i\right), \\
X_6 &= \left( T_{ij} + \frac{\alpha_6}{\lambda^{2\over 3}} C_{ij} \right)^2 - \frac{1}{5}\left( T^i_{i} + \frac{\alpha_6}{\lambda^{2\over 3}} C^i_{i} \right)^2 + \frac{1}{6\lambda}\frac{\alpha_6}{\lambda^{\frac{2}{3}}} \left(2R +C_i^i\right).
\end{align}
In $2d$, the relation $l^2 = \frac{c\lambda}{3\pi}$ has already been used to obtain the form above from \eqref{x-def}.
Also note that in $4d$, the squares appearing \eqref{4d-deform} can be expanded and the terms corresponding to the anomaly can be manifestly separated
\begin{align}
X_4 &= T_{ij}T^{ij} - \frac{1}{3}( T^i_{i})^2 + 2 \frac{\alpha_4}{\sqrt{\lambda}} \left( G_{ij}T^{ij} - \frac{1}{3} G_i^i T^i_i \right) + \frac{1}{4\lambda}\ \frac{C_T}{8\pi} \left(G_{ij}G^{ij} - \frac{1}{3} \left(G_{i}^i\right)^2\right) .
\end{align}
Here we have used the relation between $\alpha_4$ and the central charge, $C_T=32 \pi \alpha_4^2$ (further details are provided below). This is the expression for the deforming operator in $4d$ which appears in \cite{Hartman:2018tkw}. Similarly, in 6 dimensions, using
\begin{equation}
C_{ij}=G_{ij}+6\lambda \left(\frac{\alpha_6}{\lambda^{\frac{2}{3}}}\right)\left[2\left(R_{ikjl}R^{kl}-\frac{1}{4}\gamma_{ij}R_{kl}R^{kl}\right)-\frac{3}{5}\left(RR_{ij}-\frac{1}{4}\gamma_{ij}R^2\right)\right],
\end{equation}
we can write the operator as
\begin{align}
X_6 =\ & T_{ij}T^{ij}-\frac{1}{5}( T^i_{i})^2 + 2 \frac{\alpha_6}{\lambda^{2/3}} \left( C_{ij}T^{ij} - \frac{1}{5} C_i^i T^i_i \right) \nonumber\\
&+\frac{144\alpha^3_6}{6\lambda}\left[R^{ij}R_{ikjl}R^{kl}-\frac{1}{2}RR_{kl}R^{kl}+\frac{3}{50}R^3\right]+\frac{1}{\lambda^{2/3}}O(R^4).
\end{align}
The term with third order in curvature precisely matches the (negative of) the 6$d$ anomaly \cite{Henningson:1998gx} provided
$
\alpha_6={N}/{24\pi}
$.
Moreover, the terms quartic in curvature can be compactly written as
\begin{equation}
O(R^4)=(C_{ij}-G_{ij})^2-\frac{1}{5}(2R+C^i_i)^2,
\end{equation}
and they come as important part of the operator needed for the correct solution of the flow equation.
The operator \eqref{x-def} was arrived at by using the form of the holographic stress tensor \cite{Hartman:2018tkw} (see also Appendix \ref{sec:Gauss-Codazzi}). In section \ref{sec:FlowRegularization}, we will also provide an independent procedure to derive $X_d$ by using a point-splitting procedure. However, for the rest of this section we assume that this is a
correct flow equation in large $N$ holographic CFTs and employ in a concrete example.
\subsection{The deformation on $\mathbb{S}^d$}
\label{ssec:remarks}
We now consider the $TT$ deformation of a CFT on the unit sphere $\mathbb{S}^d$. Since the sphere is a maximally symmetric space, the stress-tensor expectation values are proportional to the metric\footnote{Here $\gamma_{ij}$ refers to the metric on a unit sphere and all geometric quantities are computed using this metric.} $\vev{T_{ij}}= \omega_d \gamma_{ij}$. We can solve for $\omega_d$ by using the trace equation in higher dimensions
\begin{align}\label{trace-rel}
\vev{T_i^i}= - d {\lambda}\vev{X} .
\end{align}
Inserting the explicit form of the operators, this equation becomes an algebraic equation for $\omega_d$ which can be compactly written as
\begin{eqnarray}\label{flow-alg}
d\,\omega_d&=&d\lambda\left[\frac{d}{d-1}\omega_d^2+\frac{2\alpha_d}{\lambda^{\frac{d-2}{d}}}\frac{1}{d-1}\,C^i_i \omega_d-\frac{1}{d\lambda}\frac{\alpha_d}{\lambda^{\frac{d-2}{d}}}f_d(R)\right].
\end{eqnarray}
where $C_{ij}$ is defined in \eqref{CijQFT} and the last term only depends on the curvature via
\begin{equation}
f_d(R)=\left(\frac{d-2}{2}R+C^i_i\right)+d\lambda\frac{\alpha_d}{\lambda^{\frac{d-2}{d}}}\left(C_{ij}C^{ij}-\frac{1}{d-1}(C^i_i)^2\right).
\end{equation}
The quadratic equation \eqref{flow-alg} can be solved for $\omega_d$ in $d=2,3,4,5,6$ and we get a general formula
\begin{equation}\label{om-general}
\omega_d^{(\pm)}=\frac{d-1}{2d\lambda}\left(1-\frac{2\alpha_d\lambda^{\frac{2}{d}}}{d-1}C^i_i\pm\sqrt{\left(1-\frac{2\alpha_d\lambda^{\frac{2}{d}}}{d-1}C^i_i\right)^2+\frac{4\alpha_d\lambda^{\frac{2}{d}}}{d-1}f_d(R)}\right),
\end{equation}
where the $-$ sign is taken in order to reproduce the anomalies in even dimensions as $\lambda\to0$.
In the ``new" holographic dictionary, the $TT$ coupling, $\lambda$, is expressed by the bulk quantities via the relation \cite{McGough:2016lol,Hartman:2018tkw}
\begin{align}\label{coupling-rel}
\lambda = {4 \pi G_N l \over d r^d_c}.
\end{align}
We note that this relation implies that the $TT$ coupling is dimensionless. This is because there is an additional rescaling by the radius of the sphere, $r_c^d$.
Computing the counterterms and using \eqref{coupling-rel}, in all the examples up to $d\le 6$, the above field theory result \eqref{om-general} agrees with the cut-off AdS computation of the stress tensor \eqref{omega-from-AdS} given $\omega[r_c]=r^{-d}_c\omega_d$ \footnote{This comes form $T^{bulk}_{ij}=r^{2-d}_c T^{bdr}_{ij}$ and our definitions of $\omega$'s.}. We show this explicitly below.
\subsection*{$d=2$}
The case for $d=2$ has been considered earlier in the context of entanglement entropy computations in \cite{Donnelly:2018bef}. We include it here for completeness. For $d=2$ equation \eqref{om-general} is
\begin{align}\label{d2result}
\omega_2^{(\pm)}=\frac{1}{4\lambda}\left(1\pm\sqrt{1+\frac{c\lambda}{3\pi}}\right).
\end{align}
The solution with a $-$ sign in front of the square-root agrees precisely with \eqref{omega-from-AdS}, with the identification \eqref{coupling-rel} for $d=2$ and the usual Brown-Henneaux relation $c = \frac{3l}{2G_N}$. The $\lambda\to 0$ limit of the $-$ branch above reproduces the $2d$ trace anomaly appropriately. The $+$ branch is ruled out since it does not reproduce the trace anomaly in the CFT limit.
\subsection*{$d>2$}
For general $d$, the solution \eqref{om-general} of the flow equation on $\mathbb{S}^d$ \eqref{trace-rel}, is given by (with $\vev{T_{ij}}=\omega_d \gamma_{ij}$)
\begin{align}\label{field-theory-expression}
\hspace{-.3cm}\omega_d ^{(\pm)}
=\frac{d-1}{2d\lambda}\left[1+c_{d}^{(2)}\alpha_d\lambda^{\frac{2}{d}}d(d-2)\left(1-c_{d}^{(3)}\frac{\alpha_d\lambda^{\frac{2}{d}}d(d-2)}{2}\right)\pm\sqrt{1+2d(d-2)\alpha_d\lambda^{\frac{2}{d}}}\right].
\end{align}
There are two branches of the solution since the flow equation yields an algebraic equation quadratic in $\omega_d$.
Now for $3\leq d \leq 6$, the parameter $\alpha_d$ is related to gravitational quantities via the relation\footnote{This can be derived using the relation $a_d r_c^{d-2}=\alpha_d \lambda^{\frac{2-d}{d}}$ and $a_d=\frac{1}{8\pi G_N (d-2)}$ of \cite{Hartman:2018tkw}. Note that \cite{Hartman:2018tkw} works with $l=1$ and therefore powers of $l$ need to be appropriately reinstated.}
\begin{align}\label{alpha-rel}
\alpha_d = \frac{1}{(2d)^{d-2 \over d} (d-2)} \left(l^{d-1} \over 8\pi G_N\right)^{2/d}.
\end{align}
This quantity can be related to the rank of gauge groups of conventional CFT$_d$ duals of $AdS_{d+1}$
as follows
\begin{align}
\alpha_3 = \frac{N_{\rm ABJM}}{6\, 2^{1/3}\pi^{2/3}}, \qquad
\alpha_4 = \frac{ N_{\rm SYM}}{2^{7/2}\pi}, \qquad
\alpha_6 = \frac{N_{(2,0)}}{24\pi},
\end{align}
where, we used the relations for the ratio $l^{d-1}/G_N$ for ABJM, $\mathcal{N}=4$ super-Yang-Mills and the 6$d$ (2,0) theory respectively. Moreover, the following relation between $\alpha_d$, $l$ and $\lambda$ can be verified using \eqref{coupling-rel} and \eqref{alpha-rel}
\begin{align}\label{ads-radius}
\frac{l^2}{r^2_c} = 2d(d-2)\alpha_d\lambda^{2/d}.
\end{align}
Once we use \eqref{ads-radius}, $\omega^{(-)}_d$ is in precise agreement with bulk $\omega[r_c]$ in the bulk stress tensor \eqref{omega-from-AdS}.
The behavior of $\omega_d^{(+)}$ in the $\lambda\to 0$ limit is divergent and, similar to $2d$, this branch is ruled out since this does not reproduce the trace anomaly appropriately in the CFT limit. The situation here should be contrasted with that of the torus partition function, wherein non-perturbative ambiguities exist for the negative values of the coupling \cite{Aharony:2018bad}. In a sense, the CFT trace anomaly provides an additional constraint for partition functions on the sphere.
Finally, we have added appropriate counterterms to obtain the holographic stress tensor and while defining the $TT$ operator. Therefore, the $\lambda \to 0$ limit of the deformed $\mathbb{S}^d$ stress tensor \eqref{field-theory-expression}, for $\omega^{(-)}_d$, is devoid of any divergences even in $d=5,6$.\footnote{These additional counterterms have not been considered in \cite{Hartman:2018tkw}.} Explicitly, $\omega^{(-)}_d$ has the following forms in the undeformed CFT limit
\begin{align}
\omega_{3,5}^{(-)} \approx 0, \qquad \omega_4^{(-)} \approx 12\alpha_4^2= \frac{3N_{\rm SYM}^2}{32\pi^2}, \qquad \omega_6^{(-)} \approx -2880\alpha_6^3 = - \frac{5N_{(2,0)}^3}{24\pi^3}.
\end{align}
These values are perfectly consistent with trace anomalies of the undeformed holographic theory \cite{Henningson:1998gx}.
\section{$TT$ flow equation from the local Callan-Symanzik equation}
\label{sec:FlowRegularization}
The flow equation we have been using so far was derived in \cite{Hartman:2018tkw} starting from the bulk Gauss-Codazzi equation, as explained in appendix \ref{sec:Gauss-Codazzi}, and is taken to be a definition of the dual theory on the boundary. In this section, we shed more light on this flow equation by utilising the Callan-Symanzik (CS) equation for a holographic CFT deformed only by a particular irrelevant operator constructed from the energy momentum tensor\footnote{We would like to stress that, in this section, the Callan-Symanzik equation with only the $TT$ deformation is our starting point and we argue how the full flow equation in curved background emerges from the regularisation procedure of defining the $TT$ operator. We are \textit{not} providing a prescription or an RG scheme that would justify the use of CS with only $TT$. We would like to thank Edgar Shaghoulian for correspondence and clarifications on this point.}. In this section we will work up to $d=5$ and leave the technicalities of $d=6$ as a future problem.
\subsection{$TT$ flow equation vs local CS equation}
The flow equation at large $N$ that serves as the starting point for the analysis presented in the previous section is
\begin{equation}
T^{i}_{i}+\frac{\alpha_{d}}{d\lambda^{\frac{d-2}{2}}}C^{i}_{i}=-d\lambda\left( \left(T_{ij}+\frac{\alpha_{d}}{\lambda^{\frac{d-2}{2}}}C_{ij}\right)^{2}-\frac{1}{d-1}\left(T^{i}_{i}+\frac{\alpha_{d}}{\lambda^{\frac{d-2}{2}}}C^{i}_{i}\right)^{2}\right)-\frac{(d-2)\alpha_{d}}{2\lambda^{\frac{d-2}{2}}}R. \label{rfe}
\end{equation}
This can be made more compact by introducing the `bare' energy momentum tensor
\begin{equation}
\hat{T}^{ij}=T^{ij}+\frac{\alpha_{d}}{\lambda^{\frac{d-2}{2}}}C^{ij},
\end{equation}
and now it reads
\begin{equation}
\hat{T}^{i}_{i}=-d\lambda \left(\hat{T}^{ij}\hat{T}_{ij}-\frac{1}{d-1}(\hat{T}^{i}_{i})^{2}\right)-\frac{(d-2)\alpha_{d}}{d\lambda^{\frac{d-2}{2}}}R.\label{bfe}
\end{equation}
In this section, we aim to obtain the above flow equation from a more intrinsically field theoretic starting point. Namely, the local Callan--Symanzik equation, which expresses the response of the field theory under a local change of scale. This is encoded in the expectation value of the trace of the energy momentum tensor.
First, we notice that on a flat background, the bare flow equation reduces to the one proposed in \cite{Taylor:2018xcy}
\begin{equation}
\hat{T}^{i}_{i}|_{(\gamma_{ij}=\eta_{ij})}=-d\lambda\left(\hat{T}^{ij}\hat{T}_{ij}-\frac{1}{d-1}(\hat{T}^{i}_{i})^{2}\right).
\end{equation}
On such a background, this equation can certainly be seen as coming from the relationship between the energy momentum tensor and the expectation value of a deforming operator\footnote{We assume no other deforming operators are present.}
\begin{equation}
\langle T^{i}_{i}\rangle|_{(\gamma_{ij}=\eta_{ij})}=-d\lambda \langle \mathcal{O}\rangle,
\end{equation}
where $\mathcal{O}(x)$ is the irrelevant operator of interest, and the parameter $\lambda$ is the scale associated to the irrelevant deformation. This relationship is referred to as the local CS equation on flat space.
On curved spaces, this equation generalizes to
\begin{equation}
\langle T^{i}_{i}\rangle=-d\lambda \langle \mathcal{O}(x)\rangle-\mathcal{A}(\gamma). \label{lrg}
\end{equation}
For our purposes, $\mathcal{A}(\gamma)$ is the holographic anomaly which is present in even dimensions.
This equation readily provides the correct flow equation in $d=2$. Here, in the large $c$ limit, we have
\begin{equation}
\langle \mathcal{O}(x)\rangle|_{c\rightarrow \infty}=\limits_{y\rightarrow x}G_{ijkl}(x)\langle T^{ij}(x)T^{kl}(y)\rangle|_{c\rightarrow \infty} =\langle T^{ij} \rangle \langle T_{ij}\rangle -\langle T^{i}_{i}\rangle^{2},
\end{equation}
where $G_{ijkl}=\gamma_{i(k}\gamma_{l)j}-\gamma_{ij}\gamma_{kl}$, and the anomaly takes the form
\begin{equation}
\mathcal{A}(\gamma)=-\frac{c}{24\pi}R(\gamma).
\end{equation}
So, in the end, the two dimensional $T\bar{T}$ deformed flow equation reads
\begin{equation}
T^{i}_{i}=-2\lambda\left( T^{ij} T_{ij} -(T^{i}_{i})^{2} \right) -\frac{c}{24\pi}R(\gamma),
\end{equation}
where the angle brackets are dropped in the large $c$ limit. From this derivation, we see that the coincidence between conformal anomaly and the Ricci scalar was crucially important.
This is no longer the case in $d=3,4,5$. In these dimensions, the anomaly in \eqref{lrg} no longer provides for us the Ricci scalar term in \eqref{rfe}. In fact, in $d=3$ and $d=5$ there is no conformal anomaly whilst in $d=4$ the anomaly is quadratic in the curvature. In order to obtain the Ricci scalar term in the flow equation, it must somehow be `generated' from the definition of $\mathcal{O}(x)$. Furthermore, the anomaly in $d=4$ must somehow also be absorbed into the definition of this operator. These issues are addressed in what follows.
\subsection{From local CS equation to the higher dimensional flow equation}
Our aim, as described in the previous section, is to generate the Ricci scalar term in the equation \eqref{rfe}, from the local CS equation \eqref{lrg}.
In dimensions higher than 2, the deforming operator $\mathcal{O}(x)$ is defined as
\begin{equation}
\mathcal{O}(x)=\limits_{y\rightarrow x}\frac{1}{4} \left(T_{ij}(x)-\frac{1}{d-1}T^{k}_{k}(x)g_{ij}(x)\right)T^{ij}(y).
\end{equation}
It will help to introduce
\begin{equation}
G_{ijkl}(x)=\left(\gamma_{i(k}(x)\gamma_{l)j}(x)-\frac{1}{d-1}\gamma_{ij}(x)\gamma_{kl}(x)\right),
\end{equation}
so that
\begin{equation}
T_{ij}(x)-\frac{1}{d-1}T^{k}_{k}(x)g_{ij}(x)=G_{ijkl}(x)T^{kl}(x).
\end{equation}
From the definition of the energy momentum tensor, we have
\begin{equation}
\langle\mathcal{O}(x)\rangle Z[\gamma]=\limits_{y\rightarrow x}G_{ijkl}(x)\left(\frac{1}{\sqrt{\gamma(x)}}\frac{\delta }{\delta \gamma_{ij}(x)}\left(\frac{1}{\sqrt{\gamma(y)}}\frac{\delta Z[\gamma]}{\delta \gamma_{kl}(y)}\right)\right).
\end{equation}
In order to generate the $R$ term in \eqref{bfe}, we choose more concrete means to implement the coincidence limit. This method is similar to the one of \cite{Ita:2017uvz} although the context is quite different.
We choose to do this through the heat kernel $K(x,y;\epsilon)$, which satisfies the property
\begin{equation}
\limits_{\epsilon\rightarrow 0}K(x,y;\epsilon)=\delta(x,y).
\end{equation}
This property should be thought of as an initial condition for the heat equation
\begin{equation}
\partial_{\epsilon}K(x,y;\epsilon)=(\nabla^{2}_{(x)}+\xi R_{(x)})K(x,y;\epsilon).
\end{equation}
We can now implement the point splitting regularization as follows
\begin{equation}
\limits_{y\rightarrow x}G_{ijkl}\langle T^{ij}(x)T^{kl}(y)\rangle Z[\gamma]=\limits_{\epsilon \rightarrow 0}\int \textrm{d}^{d}yK(x,y;\epsilon) G_{ijkl}(x)\frac{1}{\sqrt{\gamma}(x)}\frac{\delta}{\delta \gamma_{ij}(x)} \left(\frac{1}{\sqrt{\gamma}(y)}\frac{\delta Z[\gamma]}{\delta \gamma_{kl}(y)}\right).
\end{equation}
We also exploit the fact that we can add to the effective action terms involving local functions of the metric
\begin{equation}
Z[\gamma]\rightarrow e^{C[\gamma]}Z[\gamma],
\end{equation}
where $C[\gamma]$ is chosen to be
\begin{equation}
C[\gamma]=\alpha_{0}\left(\epsilon^{\frac{d}{2}-1}\int \textrm{d}^{d}x \sqrt{\gamma}+\frac{(d^{2}-3)\epsilon^{\frac{d}{2}}}{d(d-1)}\int \textrm{d}^{d}x \sqrt{\gamma}R\right).\label{impr}
\end{equation}
Here, $\alpha_{0}$ is a constant given by
\begin{equation}\alpha_{0}=\frac{\alpha_{d}}{\lambda^{\frac{d+2}{2}}}\left(\frac{d-2}{2d^{2}\kappa(d)}\right),
\end{equation}
where
\begin{equation}
\kappa(d)=\frac{(d^{2}-3)(d(d(9d-11)-28)+42)}{12d (d-1)^{2}}.
\end{equation}
With this choice of $\epsilon$ scaling in the improvement term $C[\gamma]$, one can show (as we do in appendix \ref{sec:generateR}) that the deforming operator becomes
\begin{equation}
\langle \mathcal{O}(x) \rangle= \limits_{\epsilon \rightarrow 0}\int \textrm{d}^{d}y K(x,y,\epsilon)G_{ijkl}(x)\langle T^{ij}(x) T^{kl}(y) \rangle+\alpha_{0}R(x)
\end{equation}
which we then subject to the large $N$ limit to obtain
\begin{align}
\langle \mathcal{O}(x)\rangle|_{N\rightarrow \infty}&=
\limits_{\epsilon \rightarrow 0}\int \textrm{d}^{d}y K(x,y;\epsilon)G_{ijkl}(x)\langle T^{ij}(x) \rangle \langle T^{kl}(y)\rangle+\alpha_{0}R(x) \nonumber \\
&=G_{ijkl}(x)\langle T^{ij}(x)\rangle \langle T^{kl}(x)\rangle+\alpha_{0}R(x).
\end{align}
Here we have used the fact that the large $N$ factorized two point function does not suffer any coincidence divergences so the limit can be taken to turn the heat kernel into a delta function, and the $y$ integral can be performed.
We can plug this back into the local CS equation \eqref{lrg}, which now reads, at large $N$
\begin{equation}
T^{i}_{i}=-d\lambda G_{ijkl}T^{ij}T^{kl}-\frac{(d-2)\alpha_{d}}{2\lambda^{\frac{d-2}{2}}}R-\mathcal{A}(\gamma).
\end{equation}
In $d=3$ and $d=5$, the anomaly $\mathcal{A}(\gamma)=0$. Here we immediately obtain \eqref{bfe} provided we make the choice $\hat{T}^{ij}=T^{ij}$.
In $d=4$, the holographic anomaly is given by
\begin{equation}
\mathcal{A}=-\frac{\alpha^{2}_{4}}{\lambda^{2}}\left(G_{ij}G^{ij}-\frac{1}{3}(G^{i}_{i})^{2}\right),
\end{equation}
where $G_{ij}=R_{ij}-\frac{1}{2}R g_{ij}$ is the Einstein tensor, and $a$ is the anomaly coefficient. This can be absorbed into an improvement of the energy momentum tensor, which is subsumed in the definition of the bare energy momentum tensor
\begin{equation}
\hat{T}^{ij}= T^{ij}+\frac{\alpha_{4}}{\lambda}G^{ij}.
\end{equation}
In other words, the equation
\begin{equation*}
T^{i}_{i}=-4\lambda(T^{ij}T_{ij}-\frac{1}{d-1}(T^{i}_{i})^{2})-\frac{\alpha_{4}}{\lambda}R-\mathcal{A}(\gamma)
\end{equation*}
becomes
\begin{equation}
T^{i}_{i}+\frac{\alpha_{4}}{\lambda}G^{i}_{i}=-4\lambda \left(\left(T^{ij}+\frac{\alpha_{4}}{\lambda}G^{ij}\right)\left(T_{ij}+\frac{\alpha_{4}}{\lambda}G_{ij}\right)-\frac{1}{d-1}\left(T^{i}_{i}+\frac{\alpha_{4}}{\lambda}G^{i}_{i}\right)^{2}\right)-\frac{\alpha_{4}}{\lambda}R,
\end{equation}
hence we get \eqref{rfe}.
\subsection{Limitations of this method}
Despite the promise, we find that in $d=4$, this method allows us to readily obtain \eqref{rfe} where as in $d=3,5$, we automatically obtain \eqref{bfe}. The reason for this distinction is that the absorbing the anomaly into the improvement of the energy momentum tensor occurs only in $d=4$. In $d=3$ and $d=5$, the absence of the anomaly leaves us only with the bare flow equation. The inclusion of the counterterms, especially as involved as in $d=6$, should arise from a further improvement of the energy momentum tensor.
In other words, the counterterms are accounted for automatically in $d=4$ whereas must be thought of as an additional input in odd dimensions. Perhaps a different method or scheme would directly give us the renormalized flow equation no matter what dimension we are working in, starting from the local CS equation.
\section{The Wheeler-DeWitt equation}
\label{sec:Wheeler-DeWitt}
In this section, we comment on the role played by the Wheeler-DeWitt equation in deriving the deformed partition function. We shall see that the WKB solution of the (minisuperspace) Wheeler-DeWitt equation perfectly reproduces the bulk and boundary on-shell action without counterterms.
Let us briefly review the Wheeler-DeWitt equation that arises in the minisuperspace approximation (we closely follow \cite{Caputa:2018asc}). The minisuperspace ansatz for the Euclidean asymptotically $AdS$ metric is defined as
\begin{equation}
ds^2=\mathbf{N}^2(r)dr^2+a^2(r)d\Omega^2_{d},
\end{equation}
where $\mathbf{N}(r)$ is the lapse function and $a(r)$ is the scale factor.
We first evaluate the EH and GH actions on this metric and then, in the Euclidean gravity path integral, we redefine the lapse $\mathbf{N}\to \mathbf{N}a^{d-4}$ and introduce a variable\footnote{The main advantage of the $q$ variable here is the canonical kinetic term.} $q=a^2$ such that the action takes the form (see \cite{Caputa:2018asc} and references therein)
\begin{equation}
S_{EH}+S_{GH}=-\frac{d(d-1)S_d}{2\kappa^2}\int dr \left[\frac{q'^2}{4\mathbf{N}}+\mathbf{N}\left(q^{d-3}+l^{-2}q^{d-2}\right)\right],
\end{equation}
where $S_d$ is the sphere area.
To derive the Hamiltonian we compute the canonical momentum conjugate to $q(r)$
\begin{equation}
p=\frac{\partial L}{\partial q'}=-\frac{S_d}{\kappa^2}\frac{d(d-1)}{4\mathbf{N}}q',
\end{equation}
and a Legendre's transform yields
\begin{equation}
H=\mathbf{N}\hat{H}=-\frac{2\kappa^2}{S_d d(d-1)}\mathbf{N}\left[p^2-\left(\frac{d(d-1)S_d}{2\kappa^2l}\right)^2\left(l^2q^{d-3}+q^{d-2}\right)\right].
\end{equation}
Inserting $p=\hbar\frac{d}{dq}$, we derive the Hamiltonian constraint, or the Wheeler-DeWitt equation for the wavefunction $\Psi[q]$ \cite{Caputa:2018asc}
\begin{equation}
\hat{H}\Psi[q]=\left[\hbar^2\frac{d^2}{dq^2}-\left(\frac{d(d-1)S_d}{2\kappa^2l}\right)^2\left(l^2q^{d-3}+q^{d-2}\right)\right]\Psi[q]=0.
\end{equation}
This equation can be solved exactly in terms of special functions for $d=2,3,4$ ({\it e.g.~} in $d=3$ the solution is the Airy function that reproduces the ABJM partition function \cite{Marino:2011eh,Fuji:2011km} with perturbative $1/N$ corrections). However, let us focus just on the semi-classical limit, $G_N\to 0$ (large $N$) but with fixed $q$. In this regime we can use the WKB approximation and the leading order solutions are
\begin{align}
\Psi_{\rm WKB}(q) &\approx \exp \left[ \pm \left(d(d-1)S_d\over 2\kappa^2 l \hbar\right) \int_{0}^{q} \sqrt{ l^2 q^{d-3} + q^{d-2}}\, dq \right].
\end{align}
Performing the integral, we can see that with $q=r^2_c$, the $-$ sign solution in $d$-dimensions becomes\footnote{We also set $\hbar=1$ at the end.}
\begin{align}
\Psi_{\rm WKB}[r_c]
=\exp \left[ \frac{dS_dr^{d-1}_c}{\kappa^2 }\,\,_2F_1\left(-\frac{1}{2},\frac{d-1}{2},\frac{d+1}{2},-\frac{r^2_c}{l^2}\right)\right]
=e^{-\left(I^{\rm on-shell}_{GR}[r_c]-S_{ct}[r_c]\right)}
\end{align}
where we have identified the exponent as the on-shell EH and GH actions (gravity on-shell action without counterterms) evaluated on our Euclidean $AdS$ metric with finite boundary cut-off
\begin{equation}
I^{on-shell}_{GR}[r_c]-S_{ct}[r_c]=S_{EH}[r_c]+S_{GH}[r_c],
\end{equation}
computed in \eqref{osEHGH}. Analogous to \cite{Donnelly:2018bef}, this bare partition function (translated to QFT) can be used to compute entanglement entropy and matched with the Ryu-Takayanagi prescription \cite{Ryu:2006bv} applied to a spacetime with finite cut-off. The details of this computation will be presented elsewhere \cite{CaputaHiranoWIP}.
A few comments are in order at this point. Firstly, this concrete example for the sphere illustrates the known fact that the Wheeler-DeWitt wavefunction should be related to the holographic partition function \cite{deBoer:1999tgo,deBoer:2000cz,Papadimitriou:2004ap}. However, it is the minisuperspace approximation that turns this equation into a powerful tool. Secondly, the counterterms (for the full $TT$ partition function that is obtained from the flow equation) are included by additional canonical transformation as explained, for instance, in \cite{lecholrg}. Thirdly, in the large $N$ limit, it is the WKB solution of the Wheeler-DeWitt equation that can be matched with the on-shell action with finite cut-off. It is therefore an interesting future problem to compare the full solution of the Wheeler-DeWitt equation\footnote{As shown in \cite{Caputa:2018asc}, the full WDW wavefunction is an Airy function in 3$d$. In $2d$, the full solution can be written in terms of the $_1F_1$-hypergeometric function, whilst in 4$d$ it is the parabolic cylinder function or a Hermite polynomial upon variable transformations.} with the CFT deformations at finite $N$.
Finally, let us also recall that the connection between solutions to the radial Wheeler-DeWitt equation and the partition function of the $T\bar{T}$ deformed conformal field theories in two dimensions was first noticed in slightly different guise in \cite{Freidel:2008sh}. The idea there was to define the partition function for the deformed theory through an integral kernel as
\begin{equation}\label{couple}
Z_{\rm QFT}[e]=\int \mathcal{D}f \, e^{\frac{1}{\lambda}\int f^{+}\wedge f^{-}}Z_{\rm CFT}[e+f],
\end{equation}
where $e^{I}_{i}$ is the dyad associated to the metric on the boundary $\gamma_{ij}$. It was then shown that this kernel, when applied to the Weyl Ward identity for the partition function $Z_{\rm CFT}[e]$, resulted in $Z_{QFT}[e]$ satisfying the Wheeler-DeWitt equation. From our discussion above, it follows that this object can be seen as the generating functional for the $T\bar{T}$ deformed theory not including the counter-terms. See \cite{Freidel:2008sh}, \cite{McGough:2016lol} for more details.
It is intriguing to note that \eqref{couple} is very similar to the proposal involving coupling the CFT to Jackiw-Teitelboim gravity in \cite{Dubovsky:2018bmo}. It is also a very interesting open problem to find such integral kernels in higher dimensions.
\section{Conclusions}
\label{sec:conclusions}
In this work we further explored generalized $TT$ deformations in large $N$ CFTs and holography with a finite cut-off. We focused on the deformations defined by the trace of the energy-momentum flow equation in holographic CFTs on the sphere. By computing the energy momentum tensor and sphere partition functions holographically (up to $d=6$), we saw that the crucial information is contained in the proportionality function, $\omega[r_c]$, of the stress tensor, $\vev{T_{ij}}=\omega[r_c]\gamma_{ij}$. In the field theory side, this function solves the (algebraic) $TT$ flow equation provided all the non-trivial ingredients of the holographic dictionary like precise anomalies on $\mathbb{S}^d$ as well as relation between the deformation coupling and the gravity parameters. This program can be generalized to other asymptotic geometries as well as black hole solutions and we leave this for future work.
Since the higher dimensional flow equation originates from the Gauss-Codazzi equation, or the Hamiltonian constraint in gravity, the above results may be seen as a consistency check of AdS/CFT. On the other hand, without the $T\bar{T}$ story, the relation between the radial direction and deformation by irrelevant operators would have remained elusive. This is why there is still a lot to be learned about this new ingredient of holography, especially in higher dimensions. In particular, the definition of $TT$ operators on curved manifolds or purely field theory origin of the flow equation remains challenging. In section \ref{sec:FlowRegularization}, we made some progress on the latter and showed how the field theory flow equation emerges from the regularization procedure in defining the $TT$ operator at large $N$.
We hope that our arguments can be sharpened so that they capture, in arbitrary dimensions, the $1/N$ corrections and additional matter content of the theories. Along these lines, a potentially promising direction to pursue would be to obtain the flow equation for a holographic theory away from large $N$. This is possible by first upgrading the parameter $\lambda$ to a local function of space, ({\it i.e.~} a source) and then to apply the methods of the local renormalization group in the presence of irrelevant operator deformations as was studied recently in \cite{vanRees:2011ir,Schwimmer:2019efk}. Then, setting the parameter to be constant would lead to a flow equation of the kind we are interested in.
The Wheeler-DeWitt equation is ubiquitous in quantum gravity and plays an important role in holographic RG \cite{deBoer:1999tgo,deBoer:2000cz,Papadimitriou:2004ap}. In our example we can see that its mini-superspace version can be employed to reproduce the holographic partition function with a finite cut-off. We may hope that the Wheeler-DeWitt equation can guide us in defining the $TT$ operator and identify its expectation value in the flow equation beyond large $N$. In particular, understanding the relation between the coupling of the $TT$ operator and the cut-off in quantum gravity remains a challenge.
Finally, it is important to explore physical quantities under the $TT$ deformation in various dimensions. In particular, how do correlation functions and transport coefficients ({\it e.g.~} $\eta/s$) get modified. The quasi-normal modes get shifted upon putting a finite cut-off. This should in turn affect the retarded Green's functions. Similarly, an interesting avenue to explore is how thermalization timescales get affected by $TT$. Since the deformation introduces new interaction terms in the Lagrangian and also leads to superluminal signal propagation, one might expect thermalization to occur faster. Last but not least, many recently developed quantum information theoretic quantities in holography correspond to bulk objects that are non-trivially modified by finite cut-off. Non-perturbative comparisons with deformed CFTs, perhaps even beyond the planar limit, may provide important lessons in this directions (see {\it e.g.~} \cite{Akhavan:2018wla,Hashemi:2019xeq}).
\section*{Acknowledgements}
It is a pleasure to thank John Cardy, Thomas Dumitrescu, Monica Guica, Michael Gutperle, Shinji Hirano, Mukund Rangamani, Tadashi Takayanagi, Yunfeng Jiang, Per Kraus, Silviu Pufu and Edgar Shaghoulian for fruitful discussions. Some calculations in this work were performed with the aid of the collection of xAct Mathematica packages. PC is supported by the Simons Foundation through the ``It from Qubit" collaboration and by JSPS Grant-in-Aid for Research Activity start-up 17H06787. SD would like to thank the participants and organizers of CHORD`18 at the KITP for simulating discussions on related topics and UC Berkeley and IIT Kanpur for hospitality during the completion of this work. This research was supported in part by the National Science Foundation under Grant No.~NSF PHY-1748958. VS is supported by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
|
{
"timestamp": "2019-03-20T01:11:32",
"yymm": "1902",
"arxiv_id": "1902.10893",
"language": "en",
"url": "https://arxiv.org/abs/1902.10893"
}
|
\section{Introduction}
When a loan is approved for a person or company, the bank is subject to \emph{credit risk}; the risk that the lender defaults.
To mitigate this risk, a bank will require some form of \emph{security}, which will be collected if the lender defaults.
The bank opens a financial account for the loan, and one or more securities may be connected to it.
It is also possible that a security object is connected to more than one account.
Many-to-many relationships between securities and accounts rarely occur in the private market, but in the corporate market they are not uncommon.
We can model this situation by a bipartite graph.
We have a set $S$ of securities and a set $A$ of accounts.
Each security has a \emph{value} $v_i$ and each account has an \emph{exposure} $e_j$.
If a security $i$ can be used to secure an account $j$, we have an edge from $i$ to $j$.
Let $E$ be the set of edges.
The question is then how much of security $i$'s value should be used to secure account $j$.
Let us use $f_{ij}$ to denote this value.
Clearly, we cannot use a security to more than its value and we do not want to secure an account to more than its amount, i.e.,\footnote{All summations (except if a summation range is explicitly specified) with summation index $i$ are over $i \in S$, all summations with summation index $j$ are over $j \in A$ and all summations with summation indices $i,j$ are over $ij \in E$.}
\begin{align}
\label{v-constraint} \sum_j f_{ij} &\le v_i \qquad &\text{for all securities $i \in S$}\\
\label{e-constraint} \sum_i f_{ij} &\le e_j &\text{for all accounts $j \in A$.}
\end{align}
The unsecured part of the accounts is then $\sum_{j \in A}(e_j - \sum_i f_{ij}) = \sum_j e_j - \sum_{ij} f_{ij}$.
Clearly, we want to make the unsecured part as small as possible, i.e., we want
\begin{equation}\label{max-flow} \sum_{ij} f_{ij} \quad \text{to be maximum.} \end{equation}
In other words, we want a maximum flow from securities to accounts obeying the capacity constraints~\lref{v-constraint} and~\lref{e-constraint}.
The \emph{surplus (unsecured part)} of an account $j$ is equal to $s_j = e_j - \sum_i f_{ij}$ and the \emph{unsecured fraction} or \emph{risk ratio} of an account $j$ is equal to $r_j = s_j/e_j$.
It is desirable that all accounts are secured to the same fraction as much as possible.
Formally, if security $i$ is used for account $j$ ($f_{ij} > 0$) and could be used for account $\ell$ ($i\ell \in E$), then the unsecured fraction of account $\ell$ is at most the unsecured fraction of account $j$ ($r_\ell \le r_j$).
Otherwise, we could divert some of the flow $f_{ij}$ onto the edge $i\ell$ and make the secured fractions more equal.
Formally,
\begin{equation}\label{ratio-balance} \text{if $f_{ij} > 0$ and $i\ell \in E$ then $r_j \ge r_\ell$.} \end{equation}
We have now defined the \emph{ratio-balanced maximum flow problem}: among the maximum flows satisfying the capacity constraints~\lref{v-constraint} and~\lref{e-constraint}, find the one that satisfies the ratio-constraint~\lref{ratio-balance}.
The following example illustrates the concept.
\[\begin{tikzcd}[column sep=10em, row sep=2em]
v_1 = 3 \arrow{r}{3} & e_1 = 4,\ s_1 = 4 - 3 \\
v_2 = 3 \arrow{r}{3} \arrow{ur}{0} & e_2 = 6,\ s_2 = 6 - 4 \\
v_3 = 5 \arrow{r}{4} \arrow{ur}{1} & e_3 = 6,\ s_3 = 6 - 4 \\
\end{tikzcd}\]
In the ratio-balanced maximum flow $f_{11} = 3 $,
$f_{21} = 0 $,
$f_{22} = 3 $,
$f_{32} = 1 $ and
$f_{33} = 4 $, and the ratios are $r_1 = {1}/{4}$ and $r_2 = r_3 = {1}/{3}$.
A related problem is to compute the flow that minimizes the squared 2-norm $\sum_j s_j^2$ of the unsecured parts.
This problem is known as balanced flows~\cite{DPSV08} and can be solved in polynomial time.
The papers~\cite{DPSV08,Darwish-Mehlhorn} suggested to us that ratio-balanced flows can be computed efficiently.
This paper is structured as follows.
In Section~\ref{Alternative Characterization}, we give an alternative characterization for ratio-balanced maximum flows and show that they are the flows minimizing $\sum_j r_j^2 e_j$ subject to the capacity constraints~\lref{v-constraint} and~\lref{e-constraint}.
In Section~\ref{Combinatorial Algorithm} we give a combinatorial algorithm and show that a ratio-balanced flow can be computed by at most $n \log (nM)$ maximum flow computation.
This assumes that all values and exposures are integers bounded by $M$.
In Section~\ref{Quadratic Program} we give a quadratic program for ratio-balanced flows and in Section~\ref{Extensions} we discuss generalizations.
\section{Alternative Characterization}\label{Alternative Characterization}
We call a flow minimizing $\sum_j r_j^2 e_j$ an \emph{MWSR} (\emph{minimum weighted sum of squared risk ratios}) flow.
Let $f$ and $g$ be two flows.
We call $f$ and $g$ \emph{equivalent} if the risk ratios of all accounts with respect to $f$ and $g$ are equal, i.e., for all $j \in A$, $r_j^f = (e_j - \sum_i f_{ij})/e_j = (e_j - \sum_i g_{ij})/e_j = r_j^g$.
\begin{theorem} A flow $f$ is a ratio-balanced maximum flow if and only if it is MWSR.
All ratio-balanced flows are equivalent.
\end{theorem}
\begin{proof} We first show that an MWSR flow is maximum and satisfies the ratio-constraint (4).
Thus an MWSR flow is ratio-balanced.
We then go on to show that any two ratio-balanced flows are equivalent.
\begin{claim} An MWSR flow subject to the capacity constraints is a maximum flow.
\end{claim}
\begin{proof} Assume otherwise and let $f$ be an MWSR flow.
If $f$ is not a maximum flow then there is an augmenting path with respect to it, i.e., a sequence $i_1,j_1,i_2,j_2,\ldots,i_k,j_k$ such that $i_\ell \in S$ and $j_\ell \in A$ for all $\ell$, $\sum_{j \in \delta(i_1)} f_{i_1j} < v_{i_1}$, $\sum_{i \in \delta(j_k)} f_{i j_k} < e_{j_k}$ and $f_{j_\ell i_{\ell + 1}} > 0$ for all $\ell$.
We increase the flow on all edges $(i_\ell, j_\ell)$ by a small amount, decrease the flow on the edges $(j_\ell,i_{\ell + 1})$ by the same amount.
We obtain a flow that obeys the capacity constraints and for which $r_{j_k}$ is smaller.
\end{proof}
\begin{claim} A{n} MWSR flow subject to the capacity constraints satisfies the ratio-constraint~\lref{ratio-balance}.
\end{claim}
\begin{proof}
The derivative of the objective with respect to $f_{ij}$ is equal to
\[ -2 e_j r_j\frac{1}{e_j} = -2 r_j.\]
Therefore decreasing the flow on $(i,j)$ by an infinitesimal amount $\varepsilon$ and increasing the flow on $(i,\ell)$ by the same amount, will change the objective by
\[ (2r_j - 2r_\ell)\varepsilon = 2 (r_j - r_\ell) \varepsilon.\]
If $r_j < r_\ell$, the change would be negative, a contradiction.
\end{proof}
We have shown that a MWSR flow is ratio-balanced.
Now we prove that all ratio-balanced flows are equivalent.
We may assume that every security node can be used for some account.
Otherwise, we may simply remove the security.
Let $f$ and $g$ be two ratio-balanced flows.
Our proof is by induction on the number of nodes in $S$.
If $|S| = 0$ then $f$ and $g$ are equivalent.
Assume $|S| > 0$ and for every graph in which the number of security nodes is less than $|S|$, $f$ and $g$ are equivalent.
For any $j \in A$, let $r^f_j$ and $r^g_j$ be the risk-ratio of node $j$ under $f$ and $g$ respectively.
For any $j \in A$ and $i \in S$, let $f_{ij}$ and $g_{ij}$ be the flow from $i$ to $j$ under $f$ and $g$ respectively.
Without loss of generality we assume that the maximum risk ratio under the flow f is no smaller than the maximum risk ratio under the flow $g$, i.e., $R := \max_j r^f_j \geq \max_j r^g_j $.
If $R = 0$ then $r^f_j = r^g_j = 0$ for all $j \in A$ and $f$ and $g$ are equivalent.
Now assume that $R > 0$.
Let $A' = \{ j; r^f_j = R\}$ be the least secured nodes under $f$.
Let $S'$ be the set of nodes is $S$ which send positive flow to nodes in $A'$ under $f$.
Since $f$ is ratio-balanced, there is no edge from $S \backslash S'$ to nodes in $A'$ and $f_{ij} = 0$ for $i \in S'$ and $j \in A \setminus A'$.
Moreover, since any security node $i \in S'$ is connected to a $j$ such that $r^f_j = R > 0$, $\sum_j f_{ij} = v_i$.
Otherwise, more flow can be sent through $ij$ contradicting $f$ being a maximum flow.
With respect to $f$, the total outflow of the nodes in $S'$ is equal to the total inflow of the nodes in $A'$:
$$ \sum_{i \in S'} v_i = (1 - R) \sum_{j \in A'} e_j.$$
With respect to $g$, the total inflow of the nodes in $A'$ is at most the total outflow of the nodes in $S'$ (there might be flow from $S'$ to $A \setminus A'$):
$$\sum_{j \in A'} (1 - r^g_j) e_j \leq \sum_{i \in S'} v_i.$$
Therefore,
\begin{align*}(1 - R) \sum_{j \in A'} e_j &\geq \sum_{j \in A'} (1 - r^g_j) e_j\\
\intertext{and hence}
\sum_{j \in A'} r^g_j \cdot e_j &\geq \sum_{j \in A'} R \cdot e_j.
\end{align*}
By definition of $R$ we have $R \geq r^g_j$ that for all $j \in A$.
So for every $j \in A'$, $r^g_j = R$ and also
$$\sum_{j \in A'} (1 - r^g_j) e_j = \sum_{i \in S'} v_i,$$
which means that also in $g$, all flow from $S'$ goes into $A'$.
Now remove $A' \cup S'$ from the graph.
The number of security nodes is reduced and according to the induction assumption, $f$ and $g$ are equivalent in the reduced graph.
\end{proof}
\iffalse
Without loss of generality assume that the minimum risk ratio under the flow f is no greater than the minimum risk ratio under the flow g, i.e., $R := \min_j r^f_j \leq \min_j r^g_j $.
Let $A' \subseteq A$ be the nodes which their risk-ratio is equal to $R$ under $f$, which are basically the most secured nodes under $f$.
Let $S'_1$ be the nodes is $S$ which send positive flow to nodes in $A'$ under $f$.
There is no edge from $S'_1$ to nodes in $A \backslash A'$.
Otherwise $f$ would not be a ratio-balanced flow.
Since any security node $i \notin S'_1$ is connected to a $j$ such that $r^f_j > 0$, $\sum_j f_{ij} = v_i$.
Otherwise, more flow can be sent through $ij$ which is contradiction with $f$ being a maximum flow.
Now lets compute the total flow under $f$.
We call this value $F$.
$$F = \sum_i \sum_j f_{ij} = \sum_{i \in S'_1} \sum_j f_{ij} + \sum_{i \in S \backslash S'_1} \sum_j f_{ij} = \sum_{j \in A'} \sum_{i \in S'_1} f_{ij} + \sum_{i \in S \backslash S'_1} v_i = (1 - R) \sum_{j \in A'} e_j + \sum_{i \in S \backslash S'_1} v_i$$
Now lets compute the total flow in $g$.
$$F = \sum_i \sum_j g_{ij} = \sum_{i \in S'_1} \sum_j g_{ij} + \sum_{i \in S \backslash S'_1} \sum_j g_{ij} \leq \sum_{j \in A'} \sum_i g_{ij} + \sum_{i \in S \backslash S'_1} v_i =
\sum_{j \in A'} (1 - r^g_j) . e_j + \sum_{i \in S \backslash S'_1} v_i$$
Therefore:
$$ (1 - R) \sum_{j \in A'} e_j + \sum_{i \in S \backslash S'_1} v_i \leq \sum_{j \in A'} (1 - r^g_j) . e_j + \sum_{i \in S \backslash S'_1} v_i$$
$$ \sum_{j \in A'} R.e_j \geq \sum_{j \in A'} r^g_j . e_j$$
We assumed that $R \leq r^g_j$.
So in order for the inequality to hold, it should be that for every $j \in A': r^g_j = R$.
Let $S'_2$ be the nodes that send positive flow to $A'$ in $g$.
Like $S'_1$, there is no edge from $S'_2$ to nodes in $A \backslash A'$.
Now remove $A' \cup S'_1 \cup S'_2$ from the graph.
$|S|$ is reduced and according to the induction assumption, $f$ and $g$ are identical.
\fi
\section{Combinatorial Algorithm}\label{Combinatorial Algorithm}
\iffalse
\KM{Do we need this Lemma??}
\begin{lemma}\label{characterization} Let $f$ be a MWSR flow and let $R_h < R_{h-1} < \ldots < R_1$ be the different risk ratios of the target nodes.
Let $A_j$ be the set of targets with risk ratio $R_j$ and let $S_j$ be the set of $S$-nodes that send positive flow to a node in $A_j$.
The $S_j$'s are disjoint, there is no flow from $S_j$ to any $A_\ell$ with $\ell > j$ and there is no edge from $S_j$ to any $A_\ell$ with $\ell < j$.
Moreover, the outflow of any node $i \in S_j$, $j < h$ is equal to $v_i$.
If $R_h > 0$, this also holds true for the nodes in $S_h$.
If $R_h = 0$, the inflow into any node in $A_h$ is equal its target value.
Each $R_j$ is a rational number with numerator and denominator bounded by $nM$.
\end{lemma} (We can split this lemma into several lemmas)
\begin{proof}
We know already that the $f$ is ratio-balanced.
First, we prove that the $S_j$'s are disjoint.
Consider a node $i$ in $S$.
Let $A' \subset A$ be the set of nodes that $i$ sends positive flow to.
Based on definition of ratio-balanced flows,
for any nodes $j , \ell \in A'$,
$r_j \leq r_{\ell}$ and $r_{\ell} \leq r_j$.
This implies that $r_j = r_{\ell}$.
So any node in $S$ sends flow to only one of $A_j$'s.
Therefore $S_j$'s are distinct.
This implies that there is no flow from $S_j$ to any $A_{\ell}$ with $\ell > j$.
Second, we prove that there is no edge from $S_j$ to any $A_{\ell}$ with $\ell < j$.
This is obvious based on definition of ratio-balanced flows.
Assume there is an edge from $S_j$ to $A_{\ell}$ and $\ell < j$.
Ratio-balanced definition implies that $r_j \geq r_{\ell}$ which means $j \leq \ell$ and that is a contradiction so no such edge exists.
In order to prove that the outflow of any node $i \in S_j$ is equal to $v_i$ when $r^j > 0$, assume otherwise.
Then there exists $i$ and $j$ such that $i \in S_j$ and its outflow is less than $v_i$.
Consider a node $k \in A_j$ that the edge $ik$ exists.
Node $k$ has not reached its threshold as $r^j > 0$.
Increasing the flow in $ik$ by a small amount would not violate any constraint but decreases the objective's value which is a contradiction.
So the outflow of any node $i \in S_j$ is equal to $v_i$ when $r^j > 0$.
Finally we prove the last claim.
If $r^j = 0$, the claim is obvious.
So assume $r^j > 0$.
All flow into the nodes in $A_j$ comes from the nodes in $S_j$ and the total flow from $S_j$ to $A_j$ is equal to $\sum_{i \in S_j} v_i$.
All nodes in $A_j$ have the same risk-ratio.
This ratio is equal to $1 - \frac{\sum_{i \in S_j} v_i} {\sum_{\ell \in A_j} e_\ell}$.
\end{proof}
\fi
We now give the algorithm for computing a ratio-balanced flow $f$.
The algorithm works in phases.
In each phase, it finds a maximum flow and subsets of $S$ and $A$.
We denote the flow determined in the $k$-th phase by $f^{(k)}$ and the subsets by $S'_k$ and $A'_k$.
The flow $f$ agrees with $f^{(k)}$ on all edges from $S'_k$ to $A'_k$ and has flow zero on all edges from $S'_k$ to $A \setminus \cup_{i \le k} A'_i$.
Let $\lambda$ be a rational number in $[0,1]$ and consider the following flow problem $P_\lambda$.
We add a source node $s$ and an edge $(s,i)$ of capacity $v_i$ for every $i \in S$.
We add a sink node $t$ and an edge $(j,t)$ of capacity $\lambda e_j$ for every $j \in A$.
We set the capacity to $+\infty$ for all edges from $S$ to $A$.
Let $\lambda_1$ be the maximum $\lambda$ such that $P_\lambda$ has a feasible solution.
We discuss below how to find $\lambda_1$.
Consider the residual network with respect to the maximum flow $f^{(1)}$ in $P_{\lambda_1}$ and let $S'_1$ and $A'_1$ be the nodes that cannot be reached from $s$ by a path in the residual network.
Remove $S'_1$ and $A'_1$ from the graph and recurse until either $S$ or $A$ is empty.
\begin{theorem}
The flow $f$ is a maximum ratio-balanced flow.
\end{theorem}
\begin{proof}
Let $S_k$ and $A_k$ be the set of remaining securities and accounts in the beginning of $k$-th step respectively; $S_1 = S$ and $A_1 = A$.
In the $k$-th phase, the flow network has vertices $S \setminus \cup_{i < k} S_i'$ on the $S$-side and vertices $A \setminus \cup_{i < k} A_i'$ on the $A$-side.
Let $\lambda_k$ be the maximum $\lambda$ such that $P_\lambda$ has a feasible solution in the $k$-th phase, let $f^{k)}$ be the maximum flow in the $k$-th phase and let $S'_k \in S_k$ and $A'_k \in A_k$ be the nodes that cannot be reached from $s$ by a path in the residual network with respect to $f^{(k)}$.
Clearly $f^{(k)}_{ij} = 0$ for $i \in S'_k$ and $j \in A_k \setminus A'_k$ and $(i,j)
\not\in E$ for $i \in S_k \setminus S'_k$ and $j \in A'_k$ because of non-reachability in the residual network.
Also for $j \in A_k \setminus A'_k$, the security ratio is larger than $\lambda _k$
because the residual network certifies that we can send more flow.
Which means that if $\ell > k$, $\lambda_{\ell} > \lambda_k$.
So if $f_{ij} > 0$ then there exists $k$ such that $i \in S'_k$ and $j \in A'_k$.
There is no edge from $i$ to $S'_{\ell}$ such that $\ell < k$ and for any $j \in S'_{\ell}$ such that $\ell > k$, $f_{ij} = 0$.
Therefore, $f$ satisfies the ratio-constraint~\lref{ratio-balance}.
Next we prove that $f$ is a maximum flow.
If $\lambda_k < 1$, then for all $i \in S'_k$, $\sum_j f_{ij} = v_i$.
Otherwise $i$ would be reachable from $s$.
Let $h$ be the number of steps.
If $\lambda_h < 1$, then the total flow under $f$ is $\sum_{i \in S} v_i$.
The total flow in a maximum flow cannot exceed this amount.
So, $f$ is a maximum flow.
Now assume that $\lambda_h = 1$.
It means for every $j \in A'_h$, $\sum_i f_{ij} = e_j$.
Hence, the total flow in $f$ is
$$\sum_{i \notin S'_h} v_i + \sum_{j \in A'_h} e_j.$$
Now let the total flow in a maximum flow be $F$.
Then
$$ F = \sum_{i \in S} \sum_{j \in A} F_{ij} = \sum_{i \notin S'_h} \sum_{j \in A} F_{ij} + \sum_{i \in S'_h} \sum_{j \in A} F_{ij} \leq \sum_{i \notin S'_h} v_i + \sum_{j \in A'_h} e_j. $$
\noindent
The last inequality holds because there is no edge from $S'_h$ to $A \backslash A'_h$. So any outflow from $S'_h$ is inflow for $A'_h$.
Therefore, $f$ is a maximum flow.
\end{proof}
We next show how to find $\lambda_i$ efficiently.
For this we assume that the $v_i$ and $e_j$ are integers and use $M$ to the denote their maximum.
For any $j \in A$, let $r^f_j$ be the risk-ratio of node $j$ under $f$.
\begin{lemma}
For every $j \in A$, $r^f_j$ is a rational number with numerator and denominator bounded by $nM$.
\end{lemma}
\begin{proof}
If $r^f_j = 0$, the claim is obvious.
So assume $r^f_j > 0$.
Consider the $k$ such that $j \in A'_k$.
All flow into the nodes in $A'_k$ comes from the nodes in $S'_k$ and the total flow from $S'_k$ to $A'_k$ is equal to $\sum_{i \in S'_k} v_i$.
All nodes in $A'_k$ have the same risk-ratio.
This ratio is equal to $1 - \frac{\sum_{i \in S'_k} v_i} {\sum_{\ell \in A'_k} e_\ell}$.
\end{proof}
\begin{theorem} \cite{Papadimitriou79}
Let $x$ be a fraction, both numerator and denominator of which are bounded by $M$.
Then $x$ can be determined by $O(log(M))$ queries of form "is $x \leq p/q$?", where $p, q \leq 2M$, and $O(log (M))$ arithmetic operations on integers of size not greater than $2M$.
\end{theorem}
Instead of finding $\lambda_i$, we find $1 - \lambda_i$ which is also a fraction with both numerator and denominator bounded by $nM$.
In order to answer each query, we check if $P_{1 - p/q}$ has a feasible solution or not.
If it does, then $\lambda_i \geq 1 - p/q$ which means $1 - \lambda_i \leq p/q$.
Otherwise, $\lambda_i > 1 - p/q$ or $1 - \lambda_i > p/q$.
We need to find at most $n$ $\lambda$-values.
For each one we need to answer $\log (nM)$ queries.
Each query is a maxflow-computation.
\begin{theorem} Let the $v_i$'s and $e_j$'s be integer and let $M$ be their maximum.
A ratio-balanced flow can be computed with $n \log (nM)$ maxflow-computations.
\end{theorem}
For balanced flows (definition given in the introduction), the number of maxflow-computations can be reduced to a single parameterized flow computation~\cite{Darwish-Mehlhorn}.
The same improvement might be possible here.
\section{Solution by Formulation as a Quadratic Program} \label{Quadratic Program}
The task ``minimize $\sum_j e_j r_j^2$ subject to~\lref{v-constraint} and~\lref{e-constraint}'' is a quadratic program.
As such it can be (approximately) solved by any QP-package, e.g., CVXOPT\cite{cvxopt}.
For concreteness, we give the formulation as a standard
QP problem in the notation used in CVXOPT.
\begin{align}
\nonumber \underset{\mathbf{x}}{\operatorname{minimize}} \quad & \frac{1}{2} \mathbf{x}^T P \mathbf{x} + \mathbf{q}^T \mathbf{x} \\
\label{eq:QP} \text{subject to } \quad & G\mathbf{x} \leq \mathbf{h} \\
\nonumber
& A\mathbf{x} = \mathbf{b}
\end{align}
We use $\mathbf{1}$ for the all-ones column vector and $\hat{\mathbf{e}}_j$ for the $j$-th unit vector.
We number the edges of the graph arbitrarily and use $\mathbf{x}$ for the vector of flows.
The matrices $K$ and $V$ connect the flow variables to the securities and accounts:
\[ K_{ij} =
\begin{cases}
1 & \text{if } x_j \text{ is incident to } e_i \\
0 & \text{else}
\end{cases} \qquad
V_{ij} =
\begin{cases}
1 & \text{if } x_j \text{ is incident to } v_i \\
0 & \text{else}
\end{cases}\]
Figure~\ref{ex:large_general} shows an example.
We are now ready to formulate the objective function and the constraints as a QP.
\begin{figure}[t]
\begin{equation*} \footnotesize
\begin{tikzcd}[column sep=6em, row sep=1.5em]
v_1 = 8 \arrow{dr}{x_{2}} \arrow{r}{x_{1}} & e_1 = 12 \\
& e_2 = 8 \\
v_2 = 8 \arrow{r}{x_{5}} \arrow{ur}{x_{4}} \arrow{uur}{x_{3}} & e_3 =16
\end{tikzcd}\quad
\mathbf{x} = \begin{pmatrix}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
x_{5}
\end{pmatrix} \quad
K = \begin{pmatrix}
1 & 0& 1 & 0 &0 \\
0& 1 & 0 & 1 &0 \\
0& 0 & 0 & 0 & 1
\end{pmatrix} \quad
V = \begin{pmatrix}
1 & 1& 0 & 0 &0 \\
0& 0 & 1 & 1 & 1
\end{pmatrix}
\end{equation*}
\caption{\label{ex:large_general} An example with three securities and two accounts.}
\end{figure}
\begin{lemma} In standard form, the exposure-weighted sum-of-squares error function can be written as follows:
\begin{equation*}
\sum_{j } e_j r_j^2
=
\sum_{j }
\frac{\left(e_j - \hat{\mathbf{e}}_j^T K \mathbf{x} \right)^2}{e_j} = \sum_{j } e_j +
\frac{1}{2} \mathbf{x}^T \left( 2 K^T \operatorname{diag} \left( (e_j)^{-1} \right) K \right) \mathbf{x}
- 2\cdot \mathbf{1}^T \mathbf{x},
\end{equation*}
so that $P = 2 K^T \operatorname{diag}\left( (e_j)^{-1} \right) K$ and $\mathbf{q} = -2 \cdot \mathbf{1}$ in Equation \eqref{eq:QP}.
\end{lemma}
\begin{proof}
The goal is to write $\sum_j e_j r_j^2$ in the form $\frac{1}{2} \mathbf{x}^T P \mathbf{x} + \mathbf{q}^T \mathbf{x}$.
To do so, we expand the square by writing
\begin{align*}
\sum_{j} ( e_j)^{-1} \left( e_j - \hat{\mathbf{e}}_j^T K \mathbf{x} \right)^2
&= \sum_{j } ( e_j)^{-1} \left( e_j^2 - 2 e_j^T \hat{\mathbf{e}}_j^T K \mathbf{x} + \mathbf{x}^T K^T \hat{\mathbf{e}}_j \hat{\mathbf{e}}_j^T K \mathbf{x} \right) \\
&= \sum_{j } e_j - \sum_{j =1}^n 2 \hat{\mathbf{e}}_j^T K \mathbf{x} + \mathbf{x}^T K^T \left( \sum_{j } ( e_j)^{-1} \hat{\mathbf{e}}_j \hat{\mathbf{e}}_j^T \right ) K \mathbf{x} \\
&= \underbrace{- 2 \mathbf{1}^T}_{ \mathbf{q}^T} \mathbf{x} + \mathbf{x}^T
\underbrace{K^T \operatorname{diag} \left( ( e_j)^{-1} \right) K}_{\frac{1}{2}P} \mathbf{x} + \sum_{j = 1}^n e_j.
\end{align*}
We
used the fact that $\mathbf{1}^T K = \mathbf{1}^T$ since $\sum_{i} K_{i j} = 1$ for every column $j$.
The diagonal matrix $\operatorname{diag}(e_j^{-1})$ has $e_j^{-1}$ in position $(j, j)$.
\end{proof}
In matrix notation, the constraints are $-I \mathbf{x} \le 0$ and $V \mathbf{x} \le \mathbf{v}$ and $K \mathbf{x} \le \mathbf{e}$.
For the example in Figure~\ref{ex:large_general}, the CVXOPT package solves the QP in 10 milliseconds
and the algorithm uses 5 iterations.
The reported solution is
\begin{equation*}
\mathbf{x} = (x_1, x_2, x_3, x_4, x_5) = \left( 4.88, 3.12, 0.46, 0.43, 7.11 \right),
\end{equation*}
which gives (almost) equal risk ratios
\begin{equation*}
\frac{4.88 + 0.46}{12} \approx \frac{3.12 + 0.43}{8} \approx \frac{7.11}{16} \approx 0.444.
\end{equation*}
\section{Extensions}\label{Extensions}
We discuss some extensions.
\paragraph{Over-Coverage:} Some accounts will be fully covered, meaning that their $r_j$'s will be zero.
Let $A'$ be the set of accounts that are fully covered and let $S'$ be the securities sending flow to them. We restrict the flow problem $P_\lambda$ to these accounts and securities and then proceed as in Section~\ref{Combinatorial Algorithm}. Let $\lambda_1$ be the maximum $\lambda \ge 1$ such that $P_\lambda$ has a feasible solution.
Consider the residual network with respect to the maximum flow $f^{(1)}$ in $P_{\lambda_1}$ and let $S'_1$ and $A'_1$ be the nodes that cannot be reached from $s$ by a path in the residual network.
Remove $S'_1$ and $A'_1$ from the graph and recurse until either $S'$ and $A'$ are empty. In the example below, $\lambda_1 = 1$, $S'_1 = \sset{1}$, $A'_1 = \sset{1}$, $f_{11} = 1$ and $f_{12} = 0$. Next, we have $\lambda_2 = 5$, $S'_2 = \sset{2,3}$, $A'_2 = {2}$, and $f_{22} = 2$ and $f_{32} = 3$.
\begin{center}
\begin{tikzcd}[column sep=8em, row sep=2em]
v_1 = 1 \arrow[swap]{ddr}{0} \arrow{r}{1} & e_1 = 1,\ \lambda_1 = 1 \\
v_2 = 2 \arrow{dr}{2} & \\
v_3 = 3 \arrow{r}{3} & e_2 = 1,\ \lambda_2 = 5
\end{tikzcd}
\end{center}
\paragraph{Limits to a Claim:}
An account might contractually only have claim to parts of the security value.
This is easily modeled by introducing an upper bound on the flow from a security to an account.
The QP-algorithm and the combinatorial algorithm can handle such bounds.
\paragraph{Priorities:}
In the real world, accounts are often arranged by their \emph{priority} to a security object.
If two accounts have prioritized claims to a security object, the account with highest priority (lowest priority number) gets its demand covered first.
Remaining value goes to lower priority accounts.
In the following example, we use
parenthesized superscripts to denote priorities; account 2 has a lower priority (higher priority number) than account 3 on security 2.
In other words, account 3 has ``first rights.''
\begin{equation*}
\begin{tikzcd}[column sep=8em, row sep=2em]
v_1 = 20 \arrow{dr}{x_{2}^{(1)}} \arrow{r}{x_{1}^{(1)}} & e_1 = 20 \\
v_2 = 20 \arrow{r}{x_{3}^{(2)}} \arrow{dr}{x_{4}^{(1)}} & e_2 = 20 \\
& e_3 = 5
\end{tikzcd}
\end{equation*}
Assume we have $P$ different priority classes.
The desired solution is a maximum flow on the edges of priority 1.
Subject to this, it should be a maximum flow on the edges of priority 2, and so on.
Subject to this, it should be a maximum flow on the edges of priority $P$.
Subject to this, the flow should be ratio-balanced.
In the example above, the desired solution is $\mathbf{x} = (17.5, 2.5, 15, 5)$.
The flow on the edges of priority 1 is 25 and the flow on the edges of priority 2 is 15.
Subject to this the flow balances the uncovered fractions of accounts 1 and 2.
Frederic Dorn (Sparebanken Vest) suggested the use of minimum cost flows for modeling the priorities in combination with the objective for a balanced flow.
Let $p=1,2,\ldots, P$ be the priorities, let $E^{{(p)}}$ be the set of edges with priority $p$ and $\epsilon$ a small number.
Consider now the following optimization problem, which is a QP.
\begin{equation} \label{eqn:max_flow_quad} \underset{f}{\operatorname{minimize}} \quad
- \sum_{p=1}^{P} \epsilon^p
\sum_{(i,j) \in E^{(p)}}
f_{ij}
+
\epsilon^{P+1} \sum_{j } e_j r_j^2 \end{equation}
subject to $f \ge 0$ and \lref{v-constraint} and~\lref{e-constraint}.
The first term in the objective sends as much flow as possible through the graph, but prioritizing first priority much stronger than second, the second much stronger than third, and so forth.
The second term states that everything else being equal, a ratio-balanced flow is preferable.
The combinatorial approach of Section~\ref{Combinatorial Algorithm} works too.
We only have to replace the use of a maximum flow algorithm by the use of a minimum cost flow algorithm which minimizes the linear part of the objective in~\lref{eqn:max_flow_quad}.
\newcommand{\htmladdnormallink}[2]{#1}
\bibliographystyle{alpha}
|
{
"timestamp": "2019-03-01T02:17:14",
"yymm": "1902",
"arxiv_id": "1902.11047",
"language": "en",
"url": "https://arxiv.org/abs/1902.11047"
}
|
\section{Introduction}
The aim of this work is to introduce and analyze a notion of \emph{rate-independent} evolution for a
set-valued function $Z: [0,T] \rightrightarrows \Omega$ (with $\Omega$ a bounded domain in $\mathbb{R}^d$),
whose evolution is triggered by that of
another, given set-valued function $F: [0,T]\rightrightarrows \Omega$, in the position of an external force,
through the constraint
\begin{equation}
\label{brittle-constr-intro}
Z(t) \cap F(t) = \emptyset \quad \text{for every } t \in [0,T].
\end{equation}
The evolution of $Z$ is additionally ruled by the competition between
the minimization of the perimeter
and that of volume changes.
\subsection*{Related models: brittle delamination and adhesive contact}
Our study is inspired and motivated by the modeling of delamination between two (elastic)
bodies $O_+$ and $O_- \subset \mathbb{R}^m$, bonded along a prescribed contact surface $\Gamma=\overline{O}_{+}\cap\overline{O}_{-}$ over
time interval $ (0,T)$. Following the approach by \textsc{M.\
Fr\'emond} \cite{Frem88CA,Fre02}, this process can be described in
terms of the temporal evolution of a phase-field type parameter, the
delamination variable $z: (0,T) \times \Gamma \to [0,1]$, which
represents the fraction of fully effective molecular links in the bonding. Therefore, $z(t,x) =1$ ($z(t,x)=0$, respectively) means that the bonding is fully intact (completely broken) at a given time instant $t\in [0,T]$ and in a given material point $x\in \Gamma$. In models for \emph{brittle delamination}, the evolution of $z$ is coupled to that of the (small-strain) displacement variable $u: (0,T) \times O \to \mathbb{R}^m$ (with $O: = O_+ \cup O_- $) through the so-called
\begin{equation}
\label{brittle-constraint-delam}
\text{brittle constraint} \qquad z \JUMP{u} =0 \quad \text{a.e.\ in } (0,T) \times \Gamma.
\end{equation}
In \eqref{brittle-constraint-delam}, $\JUMP{u} = u^{+}|_{\Gamma} - u^{-}|_{\Gamma} $ ($u^+,\, u^-$ denoting the restrictions of $u$ to $O_+,\, O_-$, respectively) is the \emph{jump of $u$} across the interface $\Gamma$. Therefore, \eqref{brittle-constraint-delam} ensures the continuity of
the displacements, i.e.\ $\JUMP{u}=0$, in the (closure of the) set of points where (a portion
of) the bonding is still active, i.e.\ $z>0$. In fact, \eqref{brittle-constraint-delam} allows for displacement jumps only at
points where the bonding is completely broken, namely where $z=0$.
The set $\Gamma \setminus \mathop{\mathrm{supp}}(z)$, where the displacements may jump, can be thus understood as a \emph{crack set}; indeed, brittle delamination can be interpreted as a model for brittle fracture, along a \emph{prescribed} $(m{-}1)$-dimensional surface.
\par
To the best of our knowledge, the analysis of the above described brittle delamination model
has been carried out only in the case this process is treated
as \emph{rate-independent}. Namely, the evolution of the internal
variable, responsible for the dissipation of energy, is governed by a
dissipation potential $\calR$ which is positively homogeneous of degree $1$, i.e.\ fulfilling $\calR(\lambda \dot z) = \lambda \calR(\dot z)$ for all $\lambda\geq 0$. In that case, the process can be mathematically modeled by means of the general concept of \emph{Energetic solution} to a rate-independent system, pioneered in \cite{MieThe99MMRI, MieThe04RIHM} (cf.\ also the parallel notion of \emph{quasistatic evolution} for brittle fracture, \cite{Francfort-Marigo98,DM-Toa2002}). The existence of Energetic solutions to the brittle delamination system was proven in
\cite{RoScZa09QDP} by passing to the limit, as the penalization parameter $k$ blows up, in the Energetic formulation for an approximate system. Therein,
the brittle constraint \eqref{brittle-constraint-delam} was indeed penalized by the
\begin{equation}
\label{adhesive-intro}
\text{adhesive contact term} \qquad \int_{\Gamma} k z|\JUMP{u}|^2 \, \mathrm{d} \mathcal{H}^{m-1}(x), \qquad k \in (0,\infty)
\end{equation}
contributing to the driving energy. For the corresponding rate-independent system, referred to as \emph{adhesive contact} system, the existence of Energetic solution dates back to \cite{KoMiRo006RIAD}.
The
passage from adhesive to brittle was also studied
in \cite{RosTho12ABDM} by extending the existence of
Energetic solutions to the case in which the rate-independent
evolution of the delamination parameter $z$ is coupled to the
\emph{rate-dependent} evolution of the displacement, ruled by a (no
longer quasi-static) momentum balance with viscosity. In order to
overcome the analytical difficulties attached to the coupling of
rate-independent/rate-dependent behavior, in \cite{RosTho12ABDM} a
gradient regularization is advanced for the delamination variable
of $\mathrm{BV}$-type. In fact,
(1)
the constraint $z\in \{0,1\}$ was added to the model, making it closer to Griffith-type model for crack evolution; (2) the term $|\mathrm{D}z|(\Gamma)$ (namely, the total variation of the measure $\mathrm{D}z$) was added to the driving energy functional.
Since $z$ is the characteristic function of the set $Z:= \{ x\in \Gamma\, : \ z(x) >0\}$, the gradient regularization
$
|\mathrm{D}z|(\Gamma)$ coincides with the perimeter $ P(Z,\Gamma) $ of $ Z $ in $ \Gamma. $
\par
The rate-independent evolution of sets $Z: [0,T]\rightrightarrows \Omega$ studied in this paper can be understood as
an abstraction of the delamination process addressed in \cite{RoScZa09QDP, RosTho12ABDM}. Indeed, suppose that $\Omega = \Gamma$, and that the temporal evolution of the set
\begin{equation}
\label{F-delamination}
F(t) := \{ x\in \GC\, : \ \JUMP{u(t,x)}\neq 0 \}
\end{equation}
is \emph{given}. The sets $Z(t)$ correspond to the supports of the delamination variable and, when the set-valued mapping $F:[0,T]\rightrightarrows \Omega$ is given by \eqref{F-delamination}, \eqref{brittle-constr-intro} renders
the brittle constraint \eqref{brittle-constraint-delam}. Therefore, hereafter we shall refer to
\eqref{brittle-constr-intro} as a (generalized) \emph{brittle constraint}, too.
Like in \cite{RosTho12ABDM}, the energy functional driving the evolution of the sets $Z$ will feature their perimeter $P(Z,\Omega)$ in $\Omega$.
Again in analogy with delamination processes, in addition to the brittle case, in which \eqref{brittle-constr-intro} is enforced, we will also address the \emph{adhesive} case, in which \eqref{brittle-constr-intro} is suitably penalized.
\par
Our aim is threefold:
\begin{compactenum}
\item prove the existence of Energetic solutions for both the \emph{adhesive} and the \emph{brittle} evolution of sets;
\item investigate to what extent the fine geometric properties proved in \cite{RosTho12ABDM} for (the supports of) Energetic solutions to the adhesive and brittle delamination systems, carry over to this generalized setting;
\item
gain insight into the connection between this rate-independent evolution of sets, and the well-known mean curvature flow.
\end{compactenum}
\subsection*{Solution concepts and our existence results}
Throughout the paper, we will work under the (additional, w.r.t.\ the delamination case \eqref{F-delamination}) assumption that the `external load' $F: [0,T]\rightrightarrows \Omega$ is monotonically increasing, namely
\begin{equation}
\label{F-increasing}
F(s)\subset F(t) \qquad \text{ for all $0\leq s \leq t \leq T$.}
\end{equation}
Hence, in view of the constraint $Z(t) \subset F^c(t)$ from \eqref{brittle-constr-intro}, it will be natural to enforce for the set function $Z: [0,T] \rightrightarrows \Omega $ the opposite monotonicity property,
viz.\ $Z(t) \subset Z(s)$ for all $0\leq s \leq t \leq T$.
\par
The
adhesive and the brittle processes will be mathematically modeled by
a triple $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})_k$,
$k \in \mathbb{N} \cup \{\infty\}$, with
\begin{compactitem}
\item[-]
$\mathbf{Z} = \{ Z \subset \Omega \, : \ \calL^d(Z)<\infty\}$ the state space;
\item[-] the driving energy functional
\[
\calE_\infty(t,Z):=P(Z,\Omega)+{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,Z) \qquad \text{with }
{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,Z):=
\begin{cases}
0 &\text{ if } Z \cap F(t) =\emptyset,
\\
\infty & \text{ otherwise},
\end{cases}
\]
for the brittle process, while for the adhesive process we will pose
\[
\calE_k(t,Z):=P(Z,\Omega)+{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,Z)\,, \qquad \text{where the term }
{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,Z):= \int_\Omega k f(t) \calX_Z \, \mathrm{d} x, \ k \in (0,\infty),
\]
penalizes the brittle constraint \eqref{brittle-constr-intro}, in that the support of the function $f(t)$ coincides with $F(t)$;
\item[-] the dissipation (quasi-)distance
\[
{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}:\mathbf{Z}\times\mathbf{Z}\to[0,\infty],\quad
{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_1,Z_2):=\left\{
\begin{array}{ll}
a \calL^d(Z_1\setminus Z_2) &\text{if }Z_2\subset Z_1,\\
\infty&\text{otherwise}
\end{array}
\right.
\]
for some $ a>0$,
enforcing that $Z: [0,T]\rightrightarrows \Omega$ is monotonically decreasing.
\end{compactitem}
We will investigate the existence of Energetic solutions to the rate-independent systems $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})_k$,
$k \in \mathbb{N} \cup \{\infty\}$, namely functions $Z: [0,T]\rightrightarrows \Omega$ complying with
\begin{compactitem}
\item[-] the global stability condition
\begin{equation}
\label{stab-intro}
\calE_k(t,Z(t))\leq \calE_k(t,\widetilde Z)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z(t),\widetilde Z) \quad \text{for all }
\widetilde Z \in \mathbf{Z}, \quad \text{for all } t \in [0,T],
\end{equation}
\item[-] the energy-dissipation balance
\begin{equation}
\label{enbal}
\calE_k(t,Z(t))+\mathrm{Var}_{{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}}(Z;[0,t])=\calE_k(0,Z(0))+\int_0^t\partial_t\calE_k(s,Z(s))\,\mathrm{d}t \quad \text{for all } t \in [0,T].
\end{equation}
In \eqref{enbal}, $\mathrm{Var}_{{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}}(Z;[0,t])$ is the total
variation functional induced by the dissipation distance ${\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}$, cf.\
\eqref{totvar} below, and the power functional $\partial_t \calE_k$
has to be suitably understood in the brittle case $k=\infty$, in which
the mapping $t\mapsto \calE_\infty(t,Z)$ ceases to be smooth, see
\eqref{power-infty} below.
\end{compactitem}
As a matter of fact, the derivative-free character of the Energetic concept makes it suitable to formulate evolutions, like ours, set up in spaces lacking a linear or even a metric structure, cf.\ e.g.\ the aforementioned pioneering work \cite{DM-Toa2002} on brittle fractures,
\cite{MaiMie05EREM} for rate-independent processes in general topological spaces, and \cite{BucButLux} for the rate-independent evolution of debonding membranes.
\par
With our first main result, \underline{\bf Theorem \ref{thm:4.2}}, we establish the existence of Energetic solutions for the rate-independent system $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})_k$ in the \emph{adhesive case} $k<\infty$. Its proof will be carried out by proving that (the piecewise constant interpolants $(\piecewiseConstant Z\tau)_\tau$ of) the discrete solutions $(Z_\tau^i)$ of the associated time-incremental minimization scheme, namely
\begin{equation}
\label{tim-intro}
Z_\tau^i \in \mathop{\mathrm{Argmin}_{Z\in\mathbf{Z}}}\big\{\calE_k(\mathsf{t}^i_\tau,Z)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},Z)\big\}
\end{equation}
(where $\Pi_\tau:=\{\mathsf{t}^0_\tau=0<\mathsf{t}^1_\tau<\ldots,<\mathsf{t}^N_\tau=T\}$ is a partition of the time interval $(0,T)$ with step size $\tau$), do converge to an Energetic solution of the adhesive system as $\tau\downarrow 0$. For this, we will pass to the limit in the discrete versions of the stability condition, and of the upper energy-dissipation estimate $\leq$, to obtain their analogues on the time-continuous level.
\par
The situation for the brittle system is more involved, essentially due to the intrinsically \emph{nonsmooth} character of the brittle constraint \eqref{brittle-constr-intro}.
This brings about a nonsmooth time dependence $t\mapsto \calE_\infty(t,Z)$ that has to be handled
with suitable arguments, since most of the techniques for treating the
Energetic formulation of rate-independent systems rely on
the condition that the driving energy functional is at least absolutely continuous w.r.t.\ time, cf.\ also Remark \ref{rmk:smooth} ahead.
In our specific case,
because of \eqref{brittle-constr-intro} we are no longer in a position to show that the discrete solutions to \eqref{tim-intro} satisfy a discrete upper energy estimate. Therefore, it remains an \emph{open problem} to prove the existence of Energetic solutions by passing to the time-continuous limit in scheme \eqref{tim-intro}.
\par Nonetheless,
time incremental minimization yields the existence of limiting curves satisfying the stability condition \eqref{stab-intro}. With a terminology borrowed from the theory of gradient flows \cite{AGS08}, we have chosen to qualify such curves
as \emph{Stable Minimizing Movements}, cf.\ Definition \ref{def:SMM} ahead. Then, \underline{\bf Theorem \ref{thm:4.1}} asserts the existence of Stable Minimizing Movements both in the adhesive case $k\in \mathbb{N}$ and in the brittle case $k=\infty$.
\par
The concept of
Stable Minimizing Movement, though definitely weaker than the Energetic solution notion, seems to be relevant as well. On the one hand, it is
tightly related to time discretization scheme \eqref{tim-intro}, and it is
on the level of \eqref{tim-intro} that we can compare our rate-independent evolution with the mean curvature flow, see Remark \ref{ss:2.3}.
On the other hand, Stable Minimizing Movements enjoy the very same
fine properties as those proved in \cite{RosTho12ABDM} for the
(semi-)stable delamination variables for the
adhesive contact and brittle delamination systems. Namely, the sets
$Z$ fulfill a \emph{lower density estimate}, which prevents outward
cusps, cf.\ Proposition \ref{prop:LDE}. Further geometric properties
of Stable Minimizing Movements are discussed in Section \ref{ss:ulisse}.
\par
As previously mentioned, Thm.\ \ref{thm:4.2} will show that,
for $k\in \mathbb{N}$ Stable Minimizing Movements
enhance to Energetic solutions.
Finally, \underline{\bf Theorem \ref{thm:4.3}} will provide an existence result for the brittle system. Namely,
we will identify a special setting
in which the Energetic solutions $(Z_k)_{k}$ of the adhesive systems $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$,
$k\in\mathbb{N},$ approximate an Energetic solution of the brittle one $(\mathbf{Z},\calE_\infty,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$ as $k\to\infty$. In this case, we will find that the force term
$\int_0^T\partial_t\calE_k(t,Z_k(t))\,\mathrm{d}t$ tends to $0$ as $k\to\infty,$ and thus vanishes for the limit system.
\medskip
\noindent
\paragraph{Plan of the paper.}
In Section \ref{s:2} we fix the setup for the adhesive and brittle evolution of sets, and introduce Stable Minimizing Movements and Energetic solutions for both processes. In Section \ref{s:3} we gain further insight into the time incremental minimization scheme \eqref{tim-intro}; for its solutions we derive a~priori estimates, and discrete versions of the stability condition and of the energy-dissipation inequality.
Section \ref{s:4} contains the statements of all of our existence results, as well as a detailed discussion
on the properties of Stable Minimizing Movements, including some
illustrative numerical experiments in two space
dimensions in the brittle case. Theorems \ref{thm:4.1}, \ref{thm:4.2}, and
\ref{thm:4.3} are then
proved in Section \ref{s:6}.
\medskip
\paragraph{Acknowledgements. }
{\small
We are very grateful for all the discussions we have shared with Alexander Mielke about rate-independent systems, general mathematical questions, and far beyond. We wish him all the best for the next 60 years with a lot of creative thoughts, and the time to enjoy and elaborate on them. We are already looking forward to be part of this process!
\par
Last but not least, the authors also want to acknowledge financial support: M.T.\ acknowledges the support through the DFG within
the project ``\emph{Reliability of efficient approximation schemes for material discontinuities described by functions of bounded variation}''
in the priority programme SPP 1748 "\emph{Reliable Simulation
Techniques in Solid Mechanics. Development of Nonstandard
Discretisation Methods, Mechanical and Mathematical Analysis}". R.R.\ has been
partially supported by a GNAMPA (INDAM) project. U.S.\ acknowledges
support by the Vienna Science and Technology Fund (WWTF)
through Project MA14-009 and by the Austrian Science Fund (FWF)
projects F\,65, P\,27052, and I\,2375.
}
\section{Setup and notions of solutions}
\label{s:2}
First of all, in the upcoming Section \ref{ss:2.1} we fix the setting in which the notion of \emph{Energetic solution} for the adhesive and brittle evolution of sets can be given. As mentioned in the Introduction, along with Energetic solutions we will also address the much weaker concept of \emph{Stable Minimizing Movement}, which originates from the time-incremental minimization schemes for the adhesive and brittle systems. We shall set up these schemes, precisely introduce our solution notions, and compare our notion of evolution to the mean curvature flow, in Section \ref{ss:2.2} ahead.
\subsection{Setup}
\label{ss:2.1}
Throughout this work, $\Omega\subset\mathbb{R}^d$ is a bounded (Lipschitz) domain.
The evolution of sets studied in this paper is driven by a \underline{time-dependent function}
with values in the subsets of $\Omega$, $F: [0,T]\rightrightarrows \Omega$. We will denote by
$F^c(t)=\Omega\backslash F(t)$ the complement of the set $F(t)$ and impose on the mapping $F$
the following conditions, that also involve a function $f: (0,T)\times\Omega \to \mathbb{R}$, suitably related to $F$:
\begin{subequations}
\label{hypF}
\begin{align}
&
\label{hypF1}
F(t) \text{ is open for every } t \in [0,T];
\\
&
\label{hypF2}
F(s) \subset F(t) \quad \text{for every } 0\leq s \leq t\leq T
\\
&
\label{assFsmooth}
\begin{aligned}
&
\exists\, f \in W^{1,1}(0,T;L^1(\Omega)) \quad \text{such that } \text{for a.a.}\, t\in (0,T)
\\
&
f(t,\cdot) \geq 0 \text{ a.e.\ in }\Omega
\,\text{ and }\,
F(t) = \{ x\in \Omega\, : \ f(t,x)>0 \}\,.
\end{aligned}
\end{align}
\end{subequations}
\begin{remark}
\upshape
\label{rmk:smooth}
Condition \eqref{assFsmooth} will ensure, for the energy functional
$\calE_k$ for the \emph{adhesive} system (cf.\ \eqref{en-adh} ahead), that
$\calE_k(\cdot, Z) \in W^{1,1}(0,T)$. This property can be considered
as `standard' within the analysis of rate-independent systems, cf.\
\cite{MieRou-book}. We will rely on it, for instance,
in the proof of the existence of Energetic solutions in the adhesive case, in order to readily conclude the energy-dissipation balance once the energy-dissipation upper estimate and the stability conditions have been verified, cf.\ the proof of Thm.\ \ref{thm:4.2} ahead.
\par
However, even under \eqref{assFsmooth}, for the brittle system
the energy functional $\calE_\infty$
\eqref{en-bri}
fails to be differentiable w.r.t.\ time, due to the intrinsically nonsmooth character of the brittle constraint \eqref{brittle-constr-intro}.
One way
to handle a nonsmooth time-dependence of the driving energy would be to resort to techniques based on the Kurzweil integral, cf.\ \cite{Krejci-Liero} and the references therein.
In the case of our brittle system, though, we will avoid using
these sophisticated tools. Essentially, the (somehow) simplified structure of the energy-dissipation balance will allow us to develop ad-hoc arguments in the proof of the existence of Energetic solutions to the brittle system.
\end{remark}
\par
We now introduce the notation for the \underline{state spaces}
that will enter in our solution concepts for the rate-independent evolution of sets:
\begin{subequations}
\label{defspaces}
\begin{eqnarray}
\mathbf{Z}&:=& \{Z\in \mathscr{L}(\Omega)\, : \,\calL^d(Z)<\infty\} = \mathscr{L}(\Omega) \,,\\
\mathbf{X}&:=&\{ Z\in \mathscr{L}(\Omega) \, : \,P(Z,\Omega)<\infty\}\,,
\end{eqnarray}
\end{subequations}
where $\mathscr{L}(\Omega)$ denotes the $\sigma$-algebra of Lebesgue-measurable sets, while $\calL^d(Z)$ is the Lebesgue measure of the set $Z$ in $\mathbb{R}^d$, and
$P(Z,\Omega)$ its perimeter in $\Omega$.
\par
Since the volume measure $\calL^d$ and the perimeter $P$ are insensitive to null sets, all of our statements will be intended up to null sets. Moreover,
we will identify sets $Z$ from $\mathbf{Z}$ with the functions $z:\Omega
\to \{0,1\}$ such that
$\calX_Z = z$, and indeed
we will often use both notations, even within the same line.
We recall that
\begin{equation}
\label{perZ}
\begin{aligned}
P(Z,\Omega)= \sup\left\{ \int_\Omega z \, \mathrm{div}(\varphi)\, : \ \varphi \in \mathrm{C}_{\mathrm{c}}^1(\Omega;\mathbb{R}^d), \ \|\varphi\|_{L^\infty}\leq 1 \right\} = |\mathrm{D}z|(\Omega),
\end{aligned}
\end{equation}
($\mathrm{C}_{\mathrm{c}}^1(\Omega;\mathbb{R}^d)$ denoting the space of compactly supported $\mathrm{C}^1$-functions on $\Omega$). Hence
$P(Z,\Omega)<\infty$ if and only if $ z = \calX_Z$ has bounded variation on $\Omega$, i.e.\ $z\in \mathrm{SBV}(\Omega;\{0,1\})$, since its distributional derivative $\mathrm{D}z$ has no Cantor part. The state spaces for functions corresponding to $\mathbf{Z}$ and $\mathbf{X}$ are thus
\begin{eqnarray}
\label{defChispaces}
L^1(\Omega)\quad\text{ and }\quad
\mathrm{SBV}(\Omega;\{0,1\}):=\{z:\Omega\to\{0,1\} \text{ is the characteristic function of }Z\in\mathbf{X}\}\,.
\end{eqnarray}
Throughout this paper, we will employ the following notion of convergence of sets:
we will say that a sequence of sets $(Z_{k})_{k}\subset\mathbf{X}$ converges weakly$^{*}$ in $\mathbf{X}$ to a limit set
$Z,$ if for their respective characteristic functions $z_k: = \calX_{Z_k}$ and $z : =\calX_Z$, there
holds weak$^{*}$-convergence in $\mathrm{SBV}(\Omega;\{0,1\}),$ i.e.,
\begin{equation}
\label{sense}
Z_{k}\overset{*}{\rightharpoonup} Z\text{ in }\mathbf{X}\quad\Leftrightarrow\quad
z_k \overset{*}{\rightharpoonup} z \quad \text{ in } \mathrm{SBV}(\Omega;\{0,1\})\,.
\end{equation}
\par
Both for the adhesive and the brittle systems we will consider the \underline{dissipation distance}
\begin{align}
\label{dissip}
{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}:\mathbf{Z}\times\mathbf{Z}\to[0,\infty],\quad
{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_1,Z_2):=\left\{
\begin{array}{ll}
a \calL^d(Z_1{\setminus} Z_2) &\text{if }Z_2\subset Z_1,\\
\infty&\text{otherwise}
\end{array}
\right.
\end{align}
for a constant $ a>0$.
For later use, we will also consider the corresponding dissipation distance (and denote it in the same way) on the space of characteristic functions, namely
\begin{align}
\label{dissip-z}
{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}:
L^{1}(\Omega)\times L^{1}(\Omega) \to[0,\infty],\quad
{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(z_1,z_2):=\left\{
\begin{array}{ll}
a(z_1-z_2)&\text{if }z_2\leq z_1 \quad \text{a.e.\ in } \Omega,\\
\infty&\text{otherwise.}
\end{array}
\right.
\end{align}
Clearly, \eqref{dissip}
makes the evolution of a solution $Z$ to the adhesive/brittle system \emph{unidirectional}, namely $Z: [0,T]\to \mathbf{Z}$ is nonincreasing:
\begin{equation}
\label{monotonicity}
Z(t)\subset Z(s)
\quad \text{if } 0\leq s \leq t \leq T.
\end{equation}
In turn, by this monotonicity property we have that
\begin{equation}
\label{totvar}
\begin{aligned}
\mathrm{Var}_{{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}}(Z;[0,t]) & := \sup\left\{ \sum_{j=1}^{M} {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z(\sigma_{j-1}), Z(\sigma_j))\, : \ 0=\sigma_0< \sigma_{1}<\ldots<\sigma_{M-1}<\sigma_M = t\right\} \\\ & = {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z(0),Z(t))\,.
\end{aligned}
\end{equation}
\par
The evolution of the \emph{adhesive} system will be driven by the \underline{energy functional}
\begin{subequations}
\label{ENADH}
\begin{equation}
\label{en-adh}
\calE_k:[0,T]\times\mathbf{X}\to[0,\infty)\,,\quad
\calE_k(t,Z):=P(Z,\Omega)+{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,Z)\quad\text{for }k\in\mathbb{N}\,,
\end{equation}
where
the functional ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k:[0,T]\times\mathbf{Z} \to [0,\infty]$ penalizes the ``brittle constraint'' \eqref{brittle-constr-intro}, namely
\begin{equation}
\label{Jk}
{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,Z):=\int_\Omega k f(t )z\,\mathrm{d}x
\quad \text{for all } (t,Z)\in[0,T]\times \mathbf{Z},
\qquad\text{for }k\in\mathbb{N}\,,
\end{equation}
where $z\in L^1(\Omega)$ is associated with $Z$ via $z=\calX_Z$.
\end{subequations}
It is immediate to check that $\calE_k$ is differentiable w.r.t.\ time at every $(t,Z) \in [0,T]\times \mathbf{Z}$, with
\begin{equation}
\label{power-k}
\partial_t \calE_k(t,Z) = \int_\Omega k \partial_t f(t,x )z(x)\,\mathrm{d}x
\quad \text{for all } (t,Z)\in[0,T]\times \mathbf{Z}.
\end{equation}
In fact, thanks to \eqref{assFsmooth} we have that $\calE_k(\cdot, Z)\in W^{1,1}(0,T)$ for all $Z\in \mathbf{Z}$.
With slight abuse of notation, we will sometimes write $\calE_k(t,z)$
(with $P(Z,\Omega)$ rewritten in terms of \eqref{perZ}),
in place of $\calE_k(t,Z)$.
\par
The energy functional for the \emph{brittle system} is
\begin{subequations}
\label{ENBRI}
\begin{equation}
\label{en-bri}
\calE_\infty:[0,T]\times\mathbf{X}\to[0,\infty]\,,\quad
\calE_\infty(t,Z):=P(Z,\Omega)+{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,Z)
\end{equation}
with ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty:[0,T]\times\mathbf{Z} \to [0,\infty]$ the indicator functional associated with the constraint
\eqref{brittle-constr-intro}, i.e.
\begin{equation}
\label{Jinfty}
{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,Z):=
\begin{cases}
0 &\text{ if } Z \cap F(t) =\emptyset,
\\
\infty & \text{ otherwise}.
\end{cases}
\end{equation}
\end{subequations}
Since $F^c(t) = \{ x \in \Omega\, : \, f(t,x) =0 \} $,
we have that $Z \subset F^c(t)$ if and only if $z = \calX_{Z}$ fulfills
$z(x) f(t,x) =0$ (a.e.\ in $\Omega$). All in all, we have that
\begin{equation}
\label{J-infty-rephrased}
{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,Z) =
\begin{cases}
0 & \text{ if } z(x) f(t,x) =0 \quad \text{for a.a.}\, x \in \Omega,
\\
\infty & \text{ otherwise.}
\end{cases}
\end{equation}
In fact, also for the brittle system we will sometimes write
${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty$ and
$\calE_\infty$ as functions of $(t,z)$ through the representation $z=\calX_Z$.
In place of the usual power functional $\partial_t
\calE_\infty$, in this case only the left partial time derivative $\partial_t^- \calE_\infty$ is well defined and fulfills
\begin{equation}
\label{power-infty}
\partial_t^-\calE_\infty(t,Z) :=\lim_{h\uparrow 0}
\frac{\calE_\infty(t+h,Z)-\calE_\infty(t,Z)}{h} = \lim_{h\uparrow 0}
\frac{{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t+h,Z)-{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,Z)}{h} =0
\quad \forall\, ( t,Z)\in \mathrm{dom}(\calE_\infty)\,.
\end{equation}
Indeed, $\calE_\infty (t,Z)<\infty$ if and only if ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,Z)=0$, i.e.\ $Z\cap F(t) =\emptyset$. Since $F: [0,T]\rightrightarrows \Omega$ is increasing with respect to time, we then have that $Z\cap F(t+h) =\emptyset$, i.e.\
${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t+h,Z)=0$,
for all $h \in [-t,0)$, which gives \eqref{power-infty}. Let us stress that the monotonicity property
\eqref{hypF2} plays a crucial role in ensuring that $\partial_t^-\calE_\infty$ is well defined.
\subsection{Stable Minimizing Movements and Energetic solutions for the adhesive and brittle systems}
\label{ss:2.2}
\paragraph{\bf Time-incremental minimization.} We consider a partition
$\Pi_\tau:=\{\mathsf{t}^0_\tau=0<\mathsf{t}^1_\tau<\ldots,<\mathsf{t}^N_\tau=T\},$ of the time interval $(0,T)$ with step size
$\tau = \max_{i=1,\ldots, N_\tau}(\mathsf{t}^{i}_\tau {-} \mathsf{t}_\tau^{i-1})$.
Discrete solutions arise from solving the time-incremental minimization problem:
\emph{starting from $Z_\tau^0: = Z_0 $ for a given $Z_0\in \mathbf{X}$, for every $i=1,\ldots, N_\tau$ find}
\begin{equation}
\label{TIM}
Z_\tau^i \in \mathop{\mathrm{Argmin}_{Z\in\mathbf{X}}}\big\{\calE_k(\mathsf{t}^i_\tau,Z)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},Z)\big\}\,.
\end{equation}
Here, $k\in \mathbb{N} \cup\{\infty\}$ is fixed and, for simplicity, we choose to omit the dependence of the discrete solutions
$(Z_\tau^i)_{i=1}^{N_\tau}$
on the parameter $k$.
Time-incremental problem \eqref{TIM} admits a solution
$Z_\tau^i$ for every $i=1,\ldots, N_\tau$
by the \emph{Direct Method}. Indeed,
we inductively suppose that the set of minimizers is nonempty at the previous step $i-1$ and consider
an infimizing sequence $(Z_m)_m$ for the minimum problem at step $i$, with associated functions $(z_m)_m \subset \mathrm{SBV}(\Omega;\{0,1\})$.
Choosing as a competitor in the minimum problem \eqref{TIM}
$Z=\emptyset$ we find that, for every $m\in \mathbb{N}$,
\begin{equation}
\label{est-emptyset}
\begin{aligned}
\calE_k(\mathsf{t}^i_\tau,Z_m)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_\tau^{i-1},Z_m) & \leq
\inf_{Z\in\mathbf{X}}\big\{\calE_k(\mathsf{t}^i_\tau,Z)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},Z)\big\} + \varepsilon_m\\
& \leq \calE_k(\mathsf{t}^i_\tau,\emptyset)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},\emptyset) + \varepsilon_m \\ & = a\calL^d(Z^{i-1}_{\tau}) + \varepsilon_m \stackrel{(1)}{\leq} a\calL^d(Z_0) + \varepsilon_m
\end{aligned}
\end{equation}
with $\varepsilon_m\downarrow 0$ as $m\to\infty$. Note that, for {\footnotesize (1)} we have used that any minimizer at the step $i-1$ fulfills $Z^{i-1}_{\tau}\subset Z_0$ as imposed by ${\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}$.
Since ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k\geq 0$ for every $k\in \mathbb{N}\cup\{\infty\}$,
it is immediate to deduce from estimate \eqref{est-emptyset} that the sequence $(z_m)_m $ is bounded in $ \mathrm{SBV}(\Omega;\{0,1\})$ and, hence, has a weakly-star limit point $\overline{z}$ in $ \mathrm{SBV}(\Omega;\{0,1\})$.
Since $z_m\overset{*}{\rightharpoonup} \overline{z}$ in $ \mathrm{SBV}(\Omega;\{0,1\})$ implies $z_m\to \overline{z}$ in $L^q(\Omega)$ for every $1\leq q <\infty$, we have
\[
\liminf_{m\to\infty} \left( \calE_k(\mathsf{t}^i_\tau,Z_m)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_\tau^{i-1},Z_m) \right)
\geq \calE_k(\mathsf{t}^i_\tau, \overline{Z}) +{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(z_\tau^{i-1}, \overline{Z}) \qquad \text{for } k \in \mathbb{N}\cup\{\infty\},
\]
so that
$ Z^{i}_{\tau}: = \overline{Z} \in \mathop{\mathrm{Argmin}_{Z\in\mathbf{X}}}\big\{\calE_k(\mathsf{t}^i_\tau,Z)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},Z) \big\}$.
\par
We denote by $\piecewiseConstant Z\tau: [0,T]\to \mathbf{X}$ and $\underpiecewiseConstant Z\tau: [0,T]\to \mathbf{X}$
the left-continuous and right-continuous piecewise constant interpolants of the elements
$(Z_\tau^i)_{i=1}^{N_\tau}$, i.e.\
\[
\piecewiseConstant Z\tau(t): = Z_\tau^i \quad \text{for } t \in (\mathsf{t}_\tau^{i-1}, \mathsf{t}_\tau^i],
\qquad
\underpiecewiseConstant Z\tau(t): = Z_\tau^{i-1} \quad \text{for } t \in [\mathsf{t}_\tau^{i-1}, \mathsf{t}_\tau^i) \quad \text{for } i =1,\ldots, N_\tau\
\]
with $\piecewiseConstant Z\tau(0): = Z_\tau^0$ and $\underpiecewiseConstant Z\tau(T) = Z_\tau^{N_\tau}$, while $\piecewiseLinear Z\tau$ is the piecewise linear interpolant
\[
\piecewiseLinear Z\tau: [0,T]\to \mathbf{X}, \qquad \piecewiseLinear Z\tau(t): = \frac{t-\mathsf{t}_\tau^{i-1}}{\tau} Z_\tau^i +
\frac{\mathsf{t}_\tau^{i}-t}{\tau} Z_\tau^{i-1}\quad \text{for } t \in [\mathsf{t}_\tau^{i-1}, \mathsf{t}_\tau^i] \quad \text{for } i =1,\ldots, N_\tau.
\]
We will also work with the (left- and right-continuous) piecewise constant interpolants $\piecewiseConstant \mathsf{t}\tau:[0,T]\to [0,T]$ and $\underline{\mathsf{t}}_\tau :[0,T]\to [0,T]$ associated with the partition $\Pi_\tau$.
\par
Prior to introducing our solution concepts, we qualify the curves arising as limit points (in the sense of \eqref{sense})
of the
interpolants $(\piecewiseConstant Z{\tau_k})_k$ by resorting to a standard terminology for gradient flows, cf.\ \cite{Ambrosio95, AGS08}.
\begin{definition}[Generalized Minimizing Movement]
Let $k\in \mathbb{N} \cup\{\infty\}$.
We call a curve $Z : [0,\widehat T]\to \mathbf{X}$, with $0<\widehat T\leq
T$, a \emph{Generalized Minimizing Movement} starting from $Z_0\in \mathbf{X}$ for the rate-independent system $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$ on the interval $[0,\widehat T]$, and write $Z \in \GMMt 0{\widehat T}{k}$, if
$Z(0)=Z_0$ and
there exists a sequence $\tau_j \downarrow 0$ as $j\to\infty$ such that
\begin{equation}
\label{charact}
\begin{cases}
\exists\, C>0 \ \forall\, j \in \mathbb{N} \ \forall\, t \in [0,\widehat{T}]\,: \quad \calE(t,\piecewiseConstant Z{\tau_j}(t)) \leq C,
\\
\piecewiseConstant z{\tau_j}(t) \overset{*}{\rightharpoonup} z(t) \quad \text{in } \mathrm{SBV}(\Omega;\{0,1\}) \ \text{as } j\to\infty \ \text{ for all } t \in [0,\widehat T],
\end{cases}
\end{equation}
with $z(t)=\calX_{Z(t)}$ and $\piecewiseConstant z{\tau_j}(t) = \calX_{\piecewiseConstant Z{\tau_j}(t)}$
for all $t\in [0,\widehat T]$.
\par
If $\widehat T=T$, we will simply write $\GMM k$ in place of $\GMMt 0{\widehat T}k$.
\end{definition}
Observe that every $Z\in \GMMt 0{\widehat{T}}{k}$ is nonincreasing, i.e.\ \eqref{monotonicity} holds.
Indeed, it is sufficient to observe
that, for $j\in \mathbb{N}$
fixed, $
\piecewiseConstant Z{\tau_j}(t) \subset \piecewiseConstant Z{\tau_j}(s)
$,
since $\piecewiseConstant Z{\tau_j}(s) = Z_{\tau_k}^i$ and $\piecewiseConstant Z{\tau_j}(t) =
Z_{\tau_k}^\ell$ for some $i\leq \ell \in \{0,\ldots,N_j\}$, and
thus $ Z_{\tau_k}^\ell\subset Z_{\tau_k}^i$, as the dissipation
distance fulfills ${\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{\tau_k}^i, Z_{\tau_k}^\ell)<\infty$.
\par
We are now in a position to introduce
the concept of Stable Minimizing Movement.
\begin{definition}[Stable Minimizing Movement]
\label{def:SMM}
Let $k\in \mathbb{N} \cup \{\infty\}$. We say that a curve $Z: [0,\widehat{T}]\to\mathbf{X}$, with $0<\widehat T\leq T$, is a \emph{Stable Minimizing Movement} starting from $Z_0 \in \mathbf{X}$ for the rate-independent system $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$ on the interval $[0,\widehat T]$, and write $Z \in \SMMt 0{\widehat T}k$,
if
\begin{enumerate}
\item
$Z \in \GMMt 0{\widehat T}k$;
\item
$Z$ fulfills the stability condition for all $t\in [0,\widehat{T}]$:
\begin{equation}
\label{stab}
\calE_k(t,Z(t))\leq \calE_k(t,\widetilde Z)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z(t),\widetilde Z) \quad \text{for all } \widetilde Z \in \mathbf{X}.
\end{equation}
\end{enumerate}
We will simply write $\SMM k$ in place of $\SMMt 0{T}k$.
\end{definition}
Requiring the stability condition at all $t\in [0,T]$ clearly implies that the initial datum $Z_0$ will have to be stable at $t=0$, cf.\ \eqref{stab0} ahead.
\par
In the brittle case $k=\infty$, let us straightforwardly derive from the stability condition for $k=\infty$ a result on the life-time of Stable Minimizing Movements.
\begin{lemma}
\label{l:elementary}
Suppose that
$F:[0,T]\rightrightarrows \Omega $ is constant on
some interval $ [t_1,t_2]\subset [0,T]$.
Let $Z\in \SMMt 0{t_1}\infty$ fulfill $\calL^d(Z(t_1))>0$.
Then,
the curve $\widehat{Z}: [0,t_2]\rightrightarrows \Omega$ defined by
\[
\widehat{Z}(t) : = \left\{
\begin{array}{ll}
Z(t) &\text{for } t \in [0,t_1],
\\
Z(t_1) &\text{for } t \in (t_1,t_2]
\end{array}
\right.
\]
is in $\SMMt 0{t_2}\infty$, with
$\calL^d(\widehat{Z}(t))>0$ for all $[0,t_2]$.
\end{lemma}
\par
We now provide a unified definition of Energetic solutions for the adhesive and brittle systems,
where the energy-dissipation balance \eqref{enbal-infty} features the left derivative $\partial_t^- \calE_k$ both for $k \in \mathbb{N}$ and $k =\infty$. Indeed, in the latter case only $\partial_t^- \calE_\infty$ exists. In the former case, it
is sufficient to observe that, since for every $k\in \mathbb{N}$ and $Z\in \mathbf{Z}$ we have that $\calE_k(\cdot, Z) \in W^{1,1}(0,T)$, there holds
\begin{equation}
\label{used-later}
\partial_t^- \calE_k(t,Z) = \lim_{h\uparrow 0}
\frac{\calE_k(t+h,Z)-\calE_k(t,Z)}{h} = \partial_t \calE_k(t,Z) \qquad \text{for a.a.}\, t \in (0,T).
\end{equation}
\begin{definition}[Energetic solution]
Let $k\in \mathbb{N} \cup \{\infty\}$.
We say that a curve $Z: [0,T]\to\mathbf{X}$ is an \emph{Energetic solution} for the rate-independent system $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$ if it satisfies
\begin{enumerate}
\item the monotonicity property \eqref{monotonicity};
\item the stability condition \eqref{stab} for all $t\in [0,T]$;
\item the \color{black} following energy-dissipation balance
for all $t\in [0,T]$
\begin{equation}
\label{enbal-infty}
\calE_k(t,Z(t))+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z(0),Z(t))=\calE_k(0,Z(0))+\int_0^t\partial^-_t\calE_k(s,Z(s))\,\mathrm{d}s \,
\end{equation}
\end{enumerate}
\end{definition}
\begin{remark}[Comparison with mean curvature flows]
\label{ss:2.3}
\upshape
Here we point out some differences of our evolution of sets,
in the \emph{adhesive case},
to the (classical) evolution of sets by a mean-curvature flow.
We perform this comparison in terms of their respective time-discrete schemes. Recall that scheme \eqref{TIM}, from which the discrete solutions to our adhesive system originate,
takes the form
\begin{subequations}
\begin{eqnarray}
\label{ourscheme}
&&Z_{N}^{i}\in\mathrm{argmin}_{\widetilde Z\in\mathbf{Z}}\left\{P(\widetilde Z,\Omega)
+\int_{\widetilde Z}kf\,\mathrm{d}x+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_N^{i-1},\widetilde Z)\right\}
\end{eqnarray}
with $f= f(\mathsf{t}_\tau^i)$.
We highlight that, here, due to
\eqref{dissip} the dissipation potential
accounts for unidirectionality of the
evolution
by enforcing that $Z_{N}^{i} \subset Z_{N}^{i-1}$.
This is a first difference to classical mean curvature flows, which do not take into account a unidirectional evolution.
\par
Thus, for further comparison let us for the moment
disregard unidirectionality and confine the discussion to a \emph{symmetric} dissipation distance, i.e.\
\begin{equation}
\label{D-symmetric}
{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(\widetilde Z,Z)={\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z,\widetilde Z)=\int_{\Omega} a |\tilde z-z|\,\mathrm{d}x\,.
\end{equation}
Since $\tilde z,z$ are the characteristic functions of the finite-perimeter sets $\widetilde Z,Z,$ and thus only take the values $0$ or $1,$ we can equivalently
rewrite our dissipation distance as a \emph{squared} distance, i.e.\
${\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(\widetilde Z,Z)={\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z,\widetilde
Z)=\int_{\Omega} a|\tilde z-z|^{2}\,\mathrm{d}x$. This
makes our scheme \eqref{ourscheme} closer to that for the mean
curvature flow, which is usually related to a quadratic, symmetric distance. More precisely,
with \eqref{D-symmetric} our time-discrete scheme rephrases as
\begin{eqnarray}
\label{general}
Z_{N}^{i}\in\mathrm{argmin}_{\widetilde Z\in\mathbf{Z}}\left\{P(\widetilde Z,\Omega)
+\int_{\widetilde Z}kf\,\mathrm{d}x
+\displaystyle\frac{N}{T}\int_{\Omega} a|\calX_{Z_N^{i-1}}-\calX_{\widetilde Z}|^{2}\,\mathrm{d}x\right\}
\end{eqnarray}
\par
Depending on the values of $f$, minimization problem \eqref{general} may allow for an infinite number of minimizers. This is e.g.\ the case if $f=\mathrm{const}$ in a large open connected set of positive measure, because a minimizer that is translated by a sufficiently small distance is still a minimizer of \eqref{general}.
To make a selection of minimizers that keeps the minimizer pinned, following the classical literature on mean curvature flows, cf.\ e.g.\ \cite{ATW93CDFV,LuckStur95ITDM,Visi97MMCN,Visi98NMCF}, one rather replaces the above quadratic dissipation distance by the following expression $\int_{\Omega}\alpha(-\tfrac{N}{T}\,\mathrm{sdist}(x,\partial Z_N^{i-1}))\tilde Z\,\mathrm{d}x,$
where $\mathrm{sdist}(x,\partial E)=\mathrm{ess\,inf}\{|x-y|,\,y\in\Omega\backslash E\}
-\mathrm{ess\,inf}\{|x-y|,\,y\in E\}$ denotes the signed distance. The
case $\alpha:\mathbb{R}\to\mathbb{R}$ nonconstant, bounded, and monotone is discussed in the works \cite{Visi97MMCN,Visi98NMCF}, while
$\alpha:\mathbb{R}\to\mathbb{R}$ linear is the original and well-established ansatz first proposed in \cite{ATW93CDFV}.
We now combine this specific discrete dissipation distance, multiplied with a prefactor $\varepsilon>0,$
with our choice to obtain the discrete problem
\begin{eqnarray}
\label{Visintin}
Z_{N}^{i}\in\mathrm{argmin}_{\widetilde Z\in \mathbf{Z}}&\bigg\{P(\widetilde Z,\Omega)
+\!\! \displaystyle\int_{\widetilde Z}kf\,\mathrm{d}x
+\!\!\displaystyle\int_{\Omega}\!\! a |\calX_{Z_N^{i-1}}-\calX_{\widetilde Z}|^2 \,\mathrm{d}x
\nonumber\\
&\qquad \quad +\varepsilon \displaystyle\!\!\int_{\Omega}\!\!\alpha\left(-\displaystyle\frac{N}{T}\,\mathrm{sdist}(x,\partial Z_N^{i-1})\right)\tilde Z\,\mathrm{d}x\bigg\}\,.\;
\end{eqnarray}
\end{subequations}
For fixed $\varepsilon>0$, this minimization problem is a particular case of that addressed in \cite{Visi98NMCF}.
Therefore, our time-discrete problem with the symmetric dissipation distance from \eqref{D-symmetric}, i.e.\ \eqref{general}, corresponds to the limit case
$\varepsilon=0$ (so that the term with the signed distance function disappears), in \eqref{Visintin}.
Thus, formally, our scheme \eqref{general} can be understood as a singularly perturbed limit of the flow \eqref{Visintin} set forth in
\cite{ATW93CDFV, Visi98NMCF}.
\end{remark}
\section{The time-discrete problem}
\label{s:3}
In this section we
show that the time-discrete solutions arising from scheme \eqref{TIM} satisfy approximate versions of the stability condition and of the lower energy-dissipation estimates. Instead, as we will see, due to the
unidirectionality constraint on the evolution we will be able to
obtain only a \emph{discrete} energy-dissipation estimate
under the restriction that $k\in\mathbb{N}$, i.e.\ for adhesive systems.
\par
By testing the minimality of $Z^i_{\tau}$ at time step $\mathsf{t}^i_\tau$ (cf.\ scheme \eqref{TIM}), with any $\widetilde Z\in\mathbf{X},$ and by exploiting
that the dissipaton distance ${\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}$ satisfies the triangle inequality, i.e.\
\begin{align*}
\calE_k(\mathsf{t}^i_\tau,Z^i_{\tau})+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},Z^i_{\tau})
\leq\calE_k(\mathsf{t}^i_\tau,\widetilde Z)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},\widetilde Z)
\leq\calE_k(\mathsf{t}^i_\tau,\widetilde Z)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^i_{\tau},\widetilde Z)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},Z^i_{\tau})\,,
\end{align*}
we can show that the time-incremental solutions satisfy the following {\bf stability condition}:
\begin{equation}
\label{discrstab}
\forall\,\widetilde Z\in\mathbf{X}:\quad
\calE_k(\mathsf{t}^i_\tau,Z^i_{\tau})\leq \calE_k(\mathsf{t}^i_\tau,\widetilde Z)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^i_{\tau},\widetilde Z)\,.
\end{equation}
\par
From \eqref{discrstab}, choosing $\widetilde Z=\emptyset$ as a competitor and arguing as for \eqref{est-emptyset}, we deduce
the following uniform a priori bound for the time-incremental
minimizers:
\begin{equation}
\label{unibdstab}
\calE_k(\mathsf{t}^i_\tau,Z^i_{\tau})\leq\calE_k(\mathsf{t}^i_\tau,\emptyset)+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^i_{\tau},\emptyset)\leq{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_0,\emptyset)= a \calL^d(Z_0)\,.
\end{equation}
\par
Moreover, testing stability at time $\mathsf{t}_{i-1}^\tau$ with $Z_i^{\tau}$ we obtain a {\bf discrete
lower energy estimate}:
\begin{equation}
\label{discr-lee}
\begin{split}
\calE_k(\mathsf{t}^{i-1}_\tau,Z^{i-1}_{\tau})
&\leq\calE_k(\mathsf{t}^{i-1}_{\tau},Z^i_{\tau})+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},Z^i_{\tau})\\
&\stackrel{(2)}{=}\left\{
\begin{array}{ll}
\calE_\infty(\mathsf{t}^i_\tau,Z^i_{\tau})+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},Z^i_{\tau})&\text{for }k=\infty\,,\\
\calE_k(\mathsf{t}^i_{\tau},Z^i_{\tau})+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},Z^i_{\tau})-\int_{\mathsf{t}^{i-1}_{\tau}}^{\mathsf{t}^i_\tau}\partial_t\calE_k(t,Z^i_{\tau})\,\mathrm{d}t
&\text{for }k\in\mathbb{N}\,.
\end{array}\right.
\end{split}
\end{equation}
In the brittle case $k=\infty,$
equality {\footnotesize (2)}
is due to the fact that ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(Z^i_\tau,F(\mathsf{t}^{i-1}_\tau))={\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(Z^i_\tau,F(\mathsf{t}^i_\tau))=0$ since
$F(\mathsf{t}^{i-1}_\tau)\subset F(\mathsf{t}^{i}_\tau)$ by monotonicity of $F$.
Instead, in the adhesive case $k\in\mathbb{N}$
we indeed have
\begin{equation}
\calE_k(\mathsf{t}^{i-1}_\tau,Z^i_{\tau})-\calE_k(\mathsf{t}^{i}_\tau,Z^i_{\tau})
=-\int_{\mathsf{t}^{i-1}_\tau}^{\mathsf{t}^i_\tau}\partial_t\calE_k(t,Z^i_{\tau})\,\mathrm{d}t\,.
\end{equation}
All in all, taking into account \eqref{used-later} and the fact
that
$\partial_{t}^{-}\calE_{\infty}(t,Z^{i}_{\tau})=0$ by \eqref{power-infty},
estimate \eqref{discr-lee} can be
rewritten as
\begin{equation}
\calE_k(\mathsf{t}^{i-1}_\tau,Z^{i-1}_{\tau})\leq \calE_k(\mathsf{t}^i_{\tau},Z^i_{\tau})+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},Z^i_{\tau})-\int_{\mathsf{t}^{i-1}_{\tau}}^{\mathsf{t}^i_\tau}\partial_t^{-}\calE_k(t,Z^i_{\tau})\,\mathrm{d}t \qquad \text{for all } k\in\mathbb{N}\cup\{\infty\}\,.
\end{equation}
\par
Finally, in the adhesive case $k\in\mathbb{N}$
one can also obtain a {\bf discrete upper energy-dissipation estimate}
by testing the minimality of $Z^i_{\tau}$ at time $\mathsf{t}^i_\tau$ by $\widetilde Z: = Z^{i-1}_{\tau}$, i.e.\ we get
\begin{equation}
\label{uedek}
\calE_k(\mathsf{t}^i_{\tau},Z^i_{\tau})+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z^{i-1}_{\tau},Z^i_{\tau})\leq \calE_k(\mathsf{t}^i_\tau,Z^{i-1}_{\tau})= \calE_k(\mathsf{t}^{i-1}_\tau,Z^{i-1}_{\tau})
+\int_{\mathsf{t}^{i-1}_{\tau}}^{\mathsf{t}^i_\tau}\partial_t\calE_k(t,Z^{i-1}_{\tau})\,\mathrm{d}t\quad\text{for }k\in\mathbb{N}\,.
\end{equation}
Instead, for $k=\infty$ the constraint
$F(\mathsf{t}^i_\tau) \cap \widetilde Z =\emptyset$ imposed by the
energy contribution
${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(\mathsf{t}_\tau^i,\cdot)$
forbids us to choose $\widetilde Z: = Z^{i-1}_{\tau}$ \color{black} as a competitor in the minimization problem \eqref{TIM}.
In fact, we have
$F(\mathsf{t}^{i-1}_\tau)\subset F(\mathsf{t}^i_\tau)$ and thus
$ Z^{i-1}_{\tau} \subset F^c(\mathsf{t}^{i-1}_\tau)$ need not satisfy the constraint $ Z^{i-1}_{\tau} \subset F^c(\mathsf{t}^{i}_\tau)$.
\par
We are now in a position to deduce from the above observations
(summing up the discrete lower and upper energy inequalities \eqref{discr-lee} and \eqref{uedek} over the index $i$), the following result.
\begin{proposition}
\label{DiscrProps}
Consider the rate-independent systems $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$ for $k\in\mathbb{N}\cup\{\infty\}$ as defined
by \emph{\eqref{hypF}}, \emph{\eqref{dissip}}, and \emph{\eqref{ENADH}} if $k\in \mathbb{N}$, \emph{\eqref{ENBRI}} if $k=\infty$.
Then,
\begin{compactenum}
\item
For every $k\in\mathbb{N}\cup\{\infty\}$ and every $t\in(0,T]$,
the interpolants
$(\piecewiseConstant Z\tau)_\tau$
satisfy the time-discrete
stability condition
\begin{equation}
\label{discr-stab-interp}
\calE_k(\piecewiseConstant \mathsf{t}\tau(t),\piecewiseConstant Z\tau(t)) \leq \calE_k(\piecewiseConstant \mathsf{t}\tau(t),\widetilde Z) +{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(\piecewiseConstant Z\tau(t),\widetilde Z) \quad \text{for all } \widetilde Z \in \mathbf{Z},
\end{equation}
as well as the uniform a priori bound
\begin{equation}
\label{energy-bound-interp}
\exists\, C>0 \ \forall\, k \in \mathbb{N}\cup\{\infty\} \ \forall\,\tau>0\, : \qquad
\sup_{t\in (0,T)}\calE_k(\piecewiseConstant \mathsf{t}\tau(t),\piecewiseConstant Z\tau(t)) \leq C\,.
\end{equation}
\item
For every $k\in\mathbb{N}\cup\{\infty\} $ and every $t\in[0,T]$
there holds the \emph{lower} energy-dissipation estimate
\begin{equation}
\label{lower-discrete}
\begin{split}
\calE_k(0,Z_0)
+\int_{0}^{\piecewiseConstant \mathsf{t}\tau(t)}\partial_t^{-}\calE_k(r,\piecewiseConstant Z{\tau}(r))\,\mathrm{d}r
\leq \calE_k(\piecewiseConstant \mathsf{t}\tau(t), \piecewiseConstant Z\tau(t))+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_0,\piecewiseConstant Z\tau(t))\,.
\end{split}
\end{equation}
\item For every $k\in\mathbb{N}$
and every $t\in [0,T]$ there holds the
\emph{two-sided} energy-dissipation estimate
\begin{equation}
\label{enbd}
\begin{split}
\calE_k(0,Z_0)
+\int_{0}^{\piecewiseConstant \mathsf{t}\tau(t)}\partial_t\calE_k(r,\piecewiseConstant Z{\tau}(r))\,\mathrm{d}r
& \leq
\calE_k(\piecewiseConstant \mathsf{t}\tau(t), \piecewiseConstant Z\tau(t))+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_0,\piecewiseConstant Z\tau(t))
\\
&\leq
\calE_k(0,Z_0)
+\int_{0}^{\piecewiseConstant \mathsf{t}\tau(t)}\partial_t\calE_k(r,\underpiecewiseConstant Z{\tau}(r))\,\mathrm{d}r .
\end{split}
\end{equation}
\end{compactenum}
\end{proposition}
\section{Main results}
\label{s:4}
Our first result ensures that, both for the adhesive and the brittle systems,
the set $\GMM k \neq \emptyset$, and that Generalized Minimizing Movements starting from stable initial data are in fact Stable Minimizing Movements.
\begin{theorem}
\label{thm:4.1}
Let $k\in\mathbb{N}\cup\{\infty\}$.
Let the rate-independent systems $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$ fulfill
\emph{\eqref{hypF}}, \emph{\eqref{dissip}}, and \emph{\eqref{ENADH}}
if $k\in \mathbb{N}$, \emph{\eqref{ENBRI}} if $k=\infty$.
Then,
\begin{enumerate}
\item $\GMM k \neq \emptyset$
for every $Z_0\in \mathbf{X}$,
and for every $Z \in \GMM k $ and for any sequence $(\piecewiseConstant z{\tau_j})_j$ fulfilling
\eqref{charact}
there also holds
\begin{subequations}
\label{refined-convs}
\begin{align}
&
\label{refined-convs-1}
\piecewiseConstant z{\tau_j}(t), \, \underpiecewiseConstant z{\tau_j}(t) \overset{*}{\rightharpoonup} z(t) && \text{ in } \mathrm{SBV}(\Omega;\{0,1\}) \text{ for almost all } t \in (0,T);
\\
&
\label{refined-convs-2}
\piecewiseConstant z{\tau_j}, \, \underpiecewiseConstant z{\tau_j} \overset{*}{\rightharpoonup} z && \text{ in } L^\infty (0,T;\mathrm{SBV}(\Omega;\{0,1\}),
\\
&
\label{refined-convs-3}
\piecewiseConstant z{\tau_j}, \, \underpiecewiseConstant z{\tau_j} \to z && \text{ in } L^q ((0,T){\times} \Omega)
\text{ for every } q \in [1,\infty).
\end{align}
\end{subequations}
\item If in addition
\begin{equation}
\label{stab0}
\calE_k(0,Z_0) \leq \calE_k(0,\widetilde Z) +{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_0,\widetilde Z) \quad \text{for all } \widetilde Z \in \mathbf{X},
\end{equation}
then every Generalized Minimizing Movement for the rate-independent system $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$ is also a Stable Minimizing Movement, i.e.\ $\GMM k =\SMM k$.
\end{enumerate}
\end{theorem}
We postpone the \emph{proof} of Theorem \ref{thm:4.1} to Section \ref{s:6}. Proposition \ref{prop:LDE} below ensures that, both in the adhesive and in the brittle cases, any Stable Minimizing Movement $Z$
enjoys a regularity property,
which prevents the occurrence of outward cusps,
introduced by \textsc{S.\ Campanato} as Property $\frak{a},$
cf.\ e.g.\ \cite{Camp63PHAC,Camp64PUFS},
and also known as \emph{lower density estimate} in e.g.\ \cite{FonFra95RBVQ,AmFuPa05FBVF}. We recall it in the following definition.
\begin{definition}[Property $\frak{a}$]
\label{def-propa}
A set $M\subset\mathbb{R}^d$ has Property $\frak{a}$ if there exists a constant $\frak{a}>0$ such that
\begin{align}
\label{propa}
\forall\,y\in M\;\;\forall\,\rho_\star>0:\quad
\calL^{d}(M\cap B_{\rho_\star}(y))\ge
\frak{a} \rho_\star^{d}\,
\end{align}
\end{definition}
\color{black} In \cite{RosTho12ABDM} we were able to prove the validity of Property $\frak{a}$
for Energetic solutions to adhesive contact and brittle delamination systems, in which $z = \calX_Z$ is a phase-field parameter describing the state of the bonds between two bodies.
This analysis can be extended to the more general context of
our evolution of sets.
More precisely,
we can show that
any set $Z\in \SMM k$
fulfills \eqref{propa}, with constants \emph{uniform} w.r.t.\ the parameter $k$, cf.\ \eqref{LDE} below, and at all points in the support of its characteristic function $z=\calX_Z$,
defined in measure-theoretic way as
\begin{equation}
\label{defsupp}
\mathop{\mathrm{supp}} z :=\bigcap\{A \subset
\mathbb{R}^{d} \, : \,A\text{ closed },\, \calL^{d} (Z{\backslash} A)=0\}.
\end{equation}
However, for this we need to additionally impose that $\Omega$ is \emph{convex}: this is essential for the proof of the uniform relative isoperimetric inequality from \cite[Thm.\ 3.2]{Thom15}, which is in turn a key ingredient for obtaining \eqref{LDE}, cf.\
\cite[Sec.\ 6]{RosTho12ABDM}.
\begin{proposition}
\label{prop:LDE}
Let $k\in\mathbb{N}\cup\{\infty\}$ and the rate-independent systems $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$ fulfill
\emph{\eqref{hypF}}, \emph{\eqref{dissip}}, and \emph{\eqref{ENADH}} if $k\in \mathbb{N}$, \emph{\eqref{ENBRI}} if $k=\infty$.
Suppose in addition that
\begin{equation}
\label{Omega-cvx}
\Omega \text{ is convex.}
\end{equation}
Then, every
$Z\in \SMM k$ satisfies the following lower density estimate:
there are constants $R$ and $\mathfrak{a} = \frak{a}(\Omega,d, a)>0$
depending solely on $\Omega\subset\mathbb{R}^{d},$ space dimension $d,$
and on the parameter $ a,$
such that for every $k\in \mathbb{N} \cup \{\infty\}$ there holds
\begin{align}
\label{LDE}
\forall\,y\in\mathop{\mathrm{supp}} z \quad \forall\,\rho_\star>0:\qquad
\calL^{d}(Z\cap B_{\rho_\star}(y))\ge
\begin{cases}
\mathfrak{a} \rho_\star^{d}&\text{if }\rho_\star< R,\\
\mathfrak{a} R^{d}&\text{if }\rho_\star\geq R\,.
\end{cases}
\end{align}
\end{proposition}
The proof of Proposition \ref{prop:LDE} follows from directly adapting the argument
developed in \cite[Sec.\ 6]{RosTho12ABDM}, to which we refer the reader.
\par
A straightforward consequence of Proposition \ref{prop:LDE} is Corollary \ref{cor:k} below, stating that, the first time $t_*$ at which the complement set $F^c(t_*)$ violates the volume constraint of the lower density estimate \eqref{LDE} is an extinction time for Stable Minimizing Movements in the brittle case.
\begin{corollary}
\label{cor:k}
Let $k=\infty$. Let the rate-independent system $(\mathbf{Z},\calE_\infty,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$ fulfill
(\ref{hypF}), (\ref{dissip}), and (\ref{ENBRI}) if $k=\infty$.
Additionally, assume \eqref{Omega-cvx}.
Suppose that there is
$t_* \in (0,T]$ such that
\begin{equation}
\label{F-tstar-c}
\calL^d(F^c(t_*))<\frak{a}(\Omega)R^d
\end{equation}
for $R$ from \eqref{LDE}.
Then, every $Z\in \SMM \infty$ fulfills $\calL^d(Z(t_*))=0$, and
consequently $\calL^d(Z(t))=0$ for all $t>t_*$.
\end{corollary}
Indeed, from $Z(t_*) \subset F^c(t_*)$ we gather that $\calL^d(Z(t_*))<\frak{a}(\Omega)R^d$, therefore $Z(t_*)$ violates \eqref{LDE}. Then, $ \calL^d(Z(t_*))=0$, which implies that $\calL^d(Z(t))=0$ for all $t>t_*$ due to the monotonicity property $Z(t)\subset Z(t_*)$. Observe that this argument strongly relies on the brittle constraint \eqref{brittle-constr-intro}; in fact, it is not clear how to obtain an analogue of Corollary \ref{cor:k} for the adhesive system.
We shall provide a more detailed discussion of further properties of Stable Minimizing Movements in Section \ref{ss:ulisse}.
\par
In the adhesive case $k\in \mathbb{N}$, Stable Minimizing Movements enhance to Energetic solutions under the very same conditions as for the existence Thm.\ \ref{thm:4.1}.
\begin{theorem}
\label{thm:4.2}
Let $k\in \mathbb{N}$. Let the rate-independent systems $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$ fulfill
(\ref{hypF}), (\ref{dissip}), and (\ref{ENADH}), and let $Z_0\in \mathbf{X}$ comply with the stability condition at $t=0$, cf.\ \eqref{stab0}.
Then, every $Z\in \SMM k $ is an Energetic solution to the rate-independent system $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$.
\end{theorem}
The proof of Thm.\ \ref{thm:4.2}, postponed to Section \ref{s:6}, will be carried out by passing to the limit in the time-discrete scheme \eqref{TIM} in the following steps: it will be shown that any $Z\in \SMM k $ complies with the upper energy-dissipation estimate by passing to the limit in its discrete version \eqref{enbd} via lower semicontinuity arguments; the lower energy estimate will then follow from the stability condition, via a by-now standard technique.
\par
We are not in a position to prove the analogue of Thm.\ \ref{thm:4.2} in the brittle case
$k=\infty$, since the discrete version of the upper energy-dissipation estimate is not at our disposal, and therefore the upper
estimate in the energy-dissipation balance \eqref{enbal-infty} cannot be obtained by passing to the limit in the time-discretization scheme. The existence of Energetic solutions to the brittle system can be proven, though, by taking the limit as $k\to\infty $ in the Energetic formulation at the time-continuous level, provided that the `external force' $F: [0,T]\rightrightarrows \Omega$ additionally fulfills condition
\eqref{power-control} below, and that the initial datum $Z_0$ complies with a suitable compatibility condition, cf.\ \eqref{compatibility} below. This is stated in Theorem \ref{thm:4.3} ahead where,
for completeness, we also give a result on the convergence of Stable Minimizing Movements
for the adhesive system to
Stable Minimizing Movements for the brittle one. The proof of Thm.\ \ref{thm:4.3} shall be also performed in Sec.\ \ref{s:6}.
\begin{theorem}
\label{thm:4.3}
Let the initial data of the adhesive problems $Z_0^{k}$ fulfill \eqref{stab0} for each $k\in\mathbb{N}$ and assume that
\begin{equation}
\label{well-prep1}
Z_{0}^{k}\overset{*}{\rightharpoonup} Z_{0}\text{ in }\mathbf{X}\quad\text{in the sense of \eqref{sense}}\,
\end{equation}
Then,
\begin{enumerate}
\item any sequence $(Z_k)_k$ with $Z_k \in \mathrm{SMM}(\Spz,\calE_k,\calD;Z_0^{k})$ for every $k\in \mathbb{N}$ admits a (not relabeled) subsequence
such that $Z_{k}(t)\overset{*}{\rightharpoonup} Z(t)$ in $\mathbf{X}$ in the sense of \eqref{sense} for all $t\in [0,T]$.
\item
Assume that
\begin{equation}
\label{power-control}
\partial_t f\geq0 \qquad \text{a.e.\ in } [0,T]\times\Omega,
\end{equation}
and that the initial data $(Z_{0}^{k})_{k}$ of the adhesive problems are well-preprared, i.e., in addition to \eqref{well-prep1} there holds
\begin{equation}
\label{well-prep}
\calE_k(0,Z_0^k) \to \calE_\infty (0,Z_0) \qquad \text{as } k \to\infty.
\end{equation}
Further, assume that the limit initial datum $Z_{0}$ satisfies the compatibility condition
\begin{equation}
\label{compatibility}
P(Z_{0},\Omega)={\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{0},\emptyset) = a \calL^d(Z_0) \,.
\end{equation}
Then any sequence $(Z_k)_k$ of Energetic solutions of the adhesive systems $(\mathbf{Z},\calE_k,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$
with $Z_{k}(0)=Z_{0}$ admits a (not relabeled) subsequence converging as $k\to\infty$, in the sense of \eqref{sense}, to an Energetic solution of the brittle system $(\mathbf{Z},\calE_\infty,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}),$ i.e.\ energy-dissipation balance \eqref{enbal-infty} holds true. Moreover, the analogue of the compatibility condition \eqref{compatibility} holds true also
for all $t\in(0,T],$ i.e.,
\begin{equation}
\label{compatibility-t}
P(Z(t),\Omega)={\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z(t),\emptyset)= a \calL^d(Z(t))\,.
\end{equation}
\end{enumerate}
\end{theorem}
A few comments on the above statement are in order:
\begin{enumerate}
\item A close perusal of the proof of Lemma \ref{PropGamma} ahead on the $\Gamma$-convergence of the functionals $(\calE_k)_k$ to $\calE_\infty$ reveals that, under \eqref{well-prep1} there holds
$ P(Z_{0},\Omega) \leq \liminf_{k\to\infty} P(Z_{0}^{k},\Omega)$ and ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty (0,Z_0) \leq \liminf_{k\to\infty} {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(0,Z_0^k)$. Therefore, \eqref{well-prep} is indeed equivalent to requiring that
\begin{equation}
\label{conseq-well-prep}
P(Z_{0}^{k},\Omega)\to P(Z_{0},\Omega) \quad \text{ and } \quad {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(0,Z_0^k) \to {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty (0,Z_0) =0 \qquad \text{as } k \to \infty\,.
\end{equation}
\item
Observe that the `brittle energy-dissipation balance' in fact reads
\begin{equation}
\label{proprio-cosi}
P(Z(t),\Omega) + a \calL^d(Z_0{\setminus}Z(t)) = P(Z_0,\Omega) \qquad \text{for all } t \in (0,T]
\end{equation}
taking into account that ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,Z(t)) = {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(0,Z_0)=0$ and that $\int_0^t \partial_t^-\calE_\infty(s,Z(s)) \, \mathrm{d} s =0$. Combining \eqref{proprio-cosi} with \eqref{compatibility}, we immediately conclude \eqref{compatibility-t}.
The compatibility condition \eqref{compatibility}
may lead to earlier extinction as Example
\ref{ex:4.7} below illustrates. Examples of nontrivial
evolutions complying with \eqref{compatibility} at all times $t\in [0,T]$ are provided with Examples \ref{ex:4.8} ahead.
Finally, we also comment on the outcome of condition
\eqref{compatibility} for the adhesive system later on in Remark \ref{rmk:compat4adh}.
\end{enumerate}
\begin{example}[Extinction of sets under compatibility condition \eqref{compatibility}]
\label{ex:4.7}
\upshape
Let $Z_{0}\subset\Omega$ be a ball of radius $r$ in $\mathbb{R}^{2}$. Compatibility condition
\eqref{compatibility} holds true if ${\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{0},\emptyset)= a\pi r^{2}=P(Z_{0},\Omega)=2\pi r$.
This is satisfied for $r=2/ a$ and for $r=0$. In other words, an Energetic solution can only exist for
the initial ball $Z_{0}$ with $r=2/ a$. As soon as the ball is forced by the brittle constraint to shrink, it is extinguished.
In terms of Stable Minimizing Movements, the ball $Z(t)$ would be extinguished according to \eqref{F-tstar-c}
only if $\calL^{2}(Z(t))<\frak{a}(\Omega)R^{2},$ which may be a
smaller value than enforced by the compatibility condition \eqref{compatibility}.
\par
Yet, the above considerations can be used to provide an example of an Energetic solution for a rate-independent evolution of sets in $\Omega\subset\mathbb{R}^{2}$:
Consider $m$ balls $B_{2/ a}(x_{i}),$ $i=1,\ldots,m,$ with centers $x_{i}\in\Omega$ such that the balls are pairwise disjoint and such that
$\cup_{i=1}^{m}B_{2/ a}(x_{i})\subset\Omega$. Along the time interval $[0,T]$ we choose a partition $t_{0}=0<t_{1}<\ldots<t_{m}=T$ and we define the evolution of the forcing set $F$ through its complement $F^{c}$ by setting $F^{c}(t):=\cup_{i=1}^{m-k}B_{2/ a}(x_{i})$ for all $t\in[t_{k},t_{k+1}),$ for $k=0,1,\ldots,m-1$. In this case, $F^{c}(t)$ itself is a Stable Minimizing Movement, i.e.\ $Z(t)=F^{c}(t),$
which additionally satisfies the compatibility condition \eqref{compatibility} and thus provides an Energetic solution of the system.
\end{example}
We devote the remainder of this section to a collection of remarks, illustrating some more
properties of Stable Minimizing Movements for the \emph{brittle} system.
\subsection{Basic features of Stable Minimizing Movements for the brittle system}
\label{ss:ulisse}
In order to gain further insight into the features of Stable
Minimizing Movements for the brittle system, we shall address a single
step of the Minimizing Movement procedure, starting from some
given initial set $Z^o\subset\Omega$ under the action of the forcing $F$. Along the whole
subsection we assume that $F^c \subset Z^o$. Hence, the {\it
single-step} minimization problem
\[
Z\in \text{\rm argmin} \{P(Z, \Omega)+ a\calL^d(Z^o{\setminus}Z) \ : \
Z \in \mathbf{X}, \
Z \subset F^c \cap Z^o\},
\]
can be equivalently reformulated as
\begin{equation}
Z\in \text{\rm argmin} \{P(Z, \Omega)+ a\calL^d(F^c{\setminus} Z) \ : \
Z \in \mathbf{X}, \
Z \subset F^c\},
\label{c1}
\end{equation}
taking into account that $\calL^d(Z^o{\setminus}Z) = \calL^d(Z^o{\setminus}F^c) + \calL^d(F^c{\setminus}Z)$. Observe that
problem \eqref{c1} has the advantage of being in fact independent of
$Z^o$.
In the following, we comment on
some properties of the minimizers of \eqref{c1}.
\paragraph{\bf Connectedness.}
Even starting from a connected initial set
$Z^o$, connectedness is not necessarily preserved
by Stable Minimizing Movements. Indeed, it is not preserved by solutions of \eqref{c1}.
An example in this direction is given by the incremental
minimizer under the effect of the needle-like forcing $F$ (see Figure \ref{cut}) given by
\begin{equation}
F:=\big(\{\gamma\}\times [-1+\gamma,\infty)\big) \cup \big(\{-\gamma\}\times
(-\infty,1-\gamma]\big) \cup ([-1,1]^2)^c \label{fff}
\end{equation}
for some suitably small $\gamma\in (0,1),$ tuned to $ a$ and
specified later on.
\begin{figure}[h]
\centering
\pgfdeclareimage[width=55mm]{cut.pdf}{cut.pdf}
\pgfuseimage{cut.pdf}
\caption{ An example for nonconnectedness: A needle-like forcing $F$.}
\label{cut}
\end{figure}
Assume by contradiction that $Z$ is connected
and consider the disconnected competitor $Z^{\mathrm{disc}} :=Z\setminus
([-\gamma,\gamma]\times[-1,1])$.
We have that
$$\calL^{2}(F^c {\setminus}Z^{\mathrm{disc}}) = \calL^2( F^c {\setminus}Z)+\calL^2(Z{\setminus}Z^{\mathrm{disc}})
\leq \calL^2( F^c {\setminus}Z) + 4\gamma,$$
where we have estimated $\calL^{2}(Z\backslash
Z^{\mathrm{disc}}) \leq \calL^{2}([-\gamma,\gamma]\times[-1,1])=4\gamma$.
On the one hand, by passing from $Z$ to $Z^{\mathrm{disc}},$ the perimeter drops at
least by twice the distance between the points
$(-\gamma,1-\gamma)$ and $(\gamma,-1+\gamma),$ which is
$2\sqrt{(2\gamma)^{2}+ (2-2\gamma)^2 }$. On the other
hand, by passing
from $Z$ to $Z^{\mathrm{disc}}$, one may gain at most $2\gamma$ in
perimeter at $x = \pm\gamma$. Hence, we find
$$P(Z^{\mathrm{disc}}, \Omega) \leq P(Z, \Omega) - 2 \sqrt{(2\gamma)^2+(2\!-\!2\gamma)^2} + 2\gamma\,.$$
This allows us to conclude that
\begin{align}
P(Z^{\mathrm{disc}},\Omega)+ a \calL^2(F^c {\setminus}Z^{\mathrm{disc}}) &\leq P(Z,\Omega) + a \calL^2( F^c {\setminus}Z)
- 2\sqrt{(2\gamma)^2+(2-2\gamma)^2} + 2\gamma +4 a
\gamma\nonumber\\
&<P(Z, \Omega) + a
\calL^{ 2}(F^c {\setminus}Z)\,, \label{contrad}
\end{align}
where the last strict inequality follows for $\gamma$ small
enough since
$$ - 2\sqrt{(2\gamma)^2+(2-2\gamma)^2} + 2\gamma +4 a
\gamma \to -4.$$
Before closing this discussion, let us point out that the forcing
$F$ from \eqref{fff} does not fulfill the
assumptions \eqref{hypF}. The argument above can however be reproduced
for a suitable smoothing of $F$ as well, at the expense of a somewhat
more involved notation.
\paragraph{\bf Convexity.} In two dimensions, convexity is preserved
by Stable Minimizing Movements. Again, to see this it is sufficient to check the preservation of convexity for the minimizers $Z$ of \eqref{c1},
with $F^c$ convex.
Assume by contradiction that a minimizer $Z$ is not convex and let
$\overline{{\rm co}( Z)}$ be the closed convex hull of $ Z$. Owing
to \cite[Thm.\ 1]{Ferriero-Fusco}, we have that $ P(\overline{{\rm
co}( Z)},\Omega )\leq
P(Z, \Omega)$. As $Z \subset F^c$ implies $\overline{{\rm co}(Z)}\subset \overline{{\rm co}(F^c)}
\equiv F^c $ and we conclude that
$$P({\rm co}( Z),\Omega )+ a \calL^{ 2}(F^c {\setminus} \overline{{\rm co}( Z)})\leq
P( Z, \Omega ) + a\calL^{2}(F^c {\setminus} Z).$$
\noindent
In particular, $\overline{{\rm co}( Z)}$ is a minimizer, too. This implies
that, if the initial datum $Z_0$ for the whole evolutionary process
is convex and the forcing term $F: [0,T] \rightrightarrows \Omega$ is such
that $F^c(t)$ is convex for all $t\in (0,T]$, then any element
$Z\in \SMM \infty$ is
such that
$Z(t)$ is closed and convex for all $t\in[0,T]$.
\par
Note that we cannot apply the same argument in order to ensure that
{\it star-shapedness} with respect to some given point is preserved
along the evolution, for star-shaped rearrangements do not necessarily
decrease the perimeter, cf.\ \cite[Lemma 1.2]{Kawohl}.
\paragraph{\bf Symmetries.} If $Z_0$ and
$F^c(t)$ are balls (radially symmetric), then every $Z\in \SMM \infty$ is a
ball for all $t\in (0,T]$ as well.
This can be checked by induction on
minimizers for problem \eqref{c1}: Assume $Z^o$ and
$F^c$ to be radially symmetric. If $Z$ were not radially
symmetric one would strictly decrease the perimeter by redefining $Z$
to be
a ball with the same volume, included in $ F^c$.
\par
Analogously, other symmetries can be conserved along the
evolution.
For instance, let $Z_0$ be
symmetric with respect to a fixed hyperplane $\pi$ and suppose that the sets
$F^c(t)$ have the same property for all $t\in (0,T]$.
Then, any element
$Z\in \SMM \infty$
is symmetric with respect to
$\pi$ as well. We shall check this again at the level of the time-incremental problem \eqref{c1},
supposing $Z^o$ and $F^c$ to be symmetric with respect to $\pi$.
If $Z$ were not symmetric, one could replace
$Z$ with its Steiner symmetrization $Z^{\mathrm{s}}$ with respect to
$\pi$. This would be admissible, since $F^c$ is
symmetric with respect to $\pi$. Moreover, one would have that
$\calL^d(F^c{\setminus}Z^{\mathrm{s}})=\calL^d(F^c{\setminus}Z)$ and $P(Z^\mathrm{s}, \Omega)
\leq P(Z, \Omega)$. Figure \ref{square} shows some
examples of symmetric minimizers.
\paragraph{\bf Partial $\mathrm{C}^1$ regularity.} Regularity of the evolving set cannot be
expected in general, for it is easy to design forcing sets $F(t)$
resulting in reentrant corners of the solution set (i.e., points $x$ at the
boundary such that the set locally has a cone of amplitude strictly larger than
$\pi$ and vertex at $x$), see
Figure \ref{f3} (right). On the other hand, in two dimensions, the set is smooth out of reentrant corners.
We will check this by considering the case of Cartesian
graphs. Let $ F^c$ be locally the epigraph of the piecewise affine
function $[0,1/ a] \ni x \mapsto \beta|x|$ for $\beta>0$. We will check that the minimizer of \eqref{c1}
is a $\mathrm{C}^1$-set. In order to see this, we show that the minimizer $y \in W^{1,1}(0,1/ a)$
of
\begin{equation}
\calF(y):=\int_{0}^{1/ a} \left(\sqrt{1+(y'(x))^2}
+ a (y(x)-\beta x)\right)\,{\rm d} x
\label{c2}
\end{equation}
under the conditions $y'(0+)=0$ and $ y(x) \geq
\beta x $ for all $x \in (0,1/ a)$
is indeed $\mathrm{C}^1$, see Figure \ref{regularity}.
\begin{figure}[h]
\centering
\pgfdeclareimage[width=95mm]{regularity}{regularity}
\pgfuseimage{regularity}
\caption{The $C^1$ competitor profile.}
\label{regularity}
\end{figure}
Minimum problem \eqref{c2} corresponds to problem \eqref{c1} under
the assumption that the minimizer $Z$ is symmetric w.r.t. the $y$
axis, under mild integrability assumptions. By assuming that the
optimal profile in \eqref{c2} is actually not in contact with the
constraint $\beta|x|$ in some (still unknown) interval $(-\hat x,\hat
x)$ for some $\hat x \in (0,1/a)$, one can consider variations of $\calF$
which are symmetric and compactly supported in $(-\hat x,\hat
x)$ in order to compute the Euler-Lagrange equation
$$\frac{\mathrm{d}}{\mathrm{d}x}\frac{y'(x)}{ \sqrt{1+(y'(x))^2}} = a.$$
We can now solve for $y',$ taking into account
$y'(0+)=0$, and deduce that
$$y'(x) = \frac{ a x}{ \sqrt{1- a^2x^2}}\quad
\text{for all }x\in[0, \hat x)$$
which, by direct integration gives
$$y(x)=y(0)-\frac{1}{ a}\sqrt{1- a^2x^2}.$$
This in particular entails that, independently of the opening
$\beta$, in case of no contact with the constraint $\beta|x|$ the
optimal profile is an arc of a circle with radius $1/a$. Note indeed
that the latter expression makes sense for $|x|\leq |\hat x| \leq 1/a$
only.
In order to determine $y(0)$, we ask $\hat x$ to be a tangency
point between the optimal profile $y(x)$ and the constraint
$\beta|x|$. Indeed, such tangency must occur at some point in
$(0,1/a)$. If this were not the case, one could translate the profile
as $y(x)-k$ for $k>0$ up to tangency, which would contradict
minimality. We can hence assume that $y(\hat x)=\beta\hat x$ and $y'(\hat x)=\beta$
(otherwise this very argument could be repeated at $\hat x$, giving
rise to a contradiction). Moreover, $y(x) = \beta x$ for all $x
\in (\hat x,1/a)$, for the only other options would be to have an arc
of radius $ a$ in $(\hat x,1/ a)$ as well, which would again contradict minimality. This gives
$$\hat x = \frac{\beta}{ a \sqrt{1+\beta^2}} \ \ \text{and} \ \
y(0)=\frac{1}{ a}\sqrt{1+\beta^2}.$$
Note that $\hat x < 1/ a$ for all $\beta>0.$
In particular, the candidate optimal profile is
\begin{equation}
y(x) =
-\frac{1}{ a}\sqrt{1- a^2x^2}+\frac{1}{ a}\sqrt{1+\beta^2}.\label{optimal}
\end{equation}
We now check that the $\mathrm{C}^1$ profile $y$ is optimal by comparing the value of $\calF$ for $y$ with its value for the affine function
$\ell(x) = \beta x;$ the latter corresponds to a nonreentrant
corner (i.e., a
point $x$ at the
boundary such that the complement of the set locally contains a cone of amplitude strictly larger than
$\pi$ and vertex at $x$). Using that $y\equiv \ell$
on $(\hat x,1/ a)$ we have
\begin{align*}
\calF(y) -\calF(\ell)&= \int_0^{\hat x} \sqrt{1
+\frac{ a^2x^2}{1- a^2 x^2}}\,{\rm d}x - \sqrt{1+\beta^2}\hat x + a \int_0^{\hat x} \left(
-\frac{1}{ a}\sqrt{1- a^2x^2}+\frac{1}{ a}\sqrt{1+\beta^2} - \beta x
\right) \, {\rm d} x\\
&= \int_0^{\hat x} \frac{ a^2 x^2}{\sqrt{1- a^2 x^2}} \, {\rm d} x -
\frac{\beta^3}{2 a(1+\beta^2)} <\frac{\beta^3}{3 a(1+\beta^2)^{3/2}} -
\frac{\beta^3}{2 a(1+\beta^2)} <0.
\end{align*}
This in particular shows that a nonreentrant corner at scale $1/a$
is not admissible and that the minimizer is $\mathrm{C}^1$ instead. On the
other hand, the above argument is scale-invariant and nonreentrant
corners are hence excluded at any scale. Note that the optimal profile
$y$ is not $\mathrm{C}^2$, since $y''(\hat x-)>0$.
By combining this analysis with the remark on preservation of
convexity, we can conclude that, in case $F^c$ is convex and piecewise $\mathrm{C}^1$,
the minimizer of problem \eqref{c1} is globally $\mathrm{C}^1$, see also Figures
\ref{f2} (left and right) and \ref{f3} (left). This remark makes
the conclusions of Proposition \ref{prop:LDE} sharper, for in two space dimensions one
can choose $\mathfrak{a}=1/2$.
\paragraph{\bf Regular polygonal forcing.}
In view of the above discussion, the minimizer $Z$ of \eqref{c1} can be
explicitly determined in case $F^c$ is a regular
polygon and $1/ a$ is smaller than half its side. Indeed, the preservation of symmetries implies that the
minimizer $Z$ shares the same symmetries of $F^c$, with rounded
corners of radius $1/ a$, see Figure \ref{square}.
\begin{figure}[h]
\centering
\pgfdeclareimage[width=145mm]{hexagon}{hexagon}
\pgfuseimage{hexagon}
\caption{Solutions of the minimization problem \eqref{c1} with
forcing $F^c$ being an equilateral triangle, a
square, and a regular hexagon, respectively. }
\label{square}
\end{figure}
\begin{examples}[Nontrivial evolutions under compatibility condition \eqref{compatibility}]
\label{ex:4.8}\upshape
Let $B_r$ denote the open ball in $\mathbb{R}^d$ with radius $r\geq0$. By
imposing the compatibility condition
\eqref{compatibility} one has that
$$ P(B_r,\Omega) = a \calL(B_r) \quad \Leftrightarrow \quad
\omega_d r^{d-1} = a \frac{\omega_d}{d}r^d$$
where $\omega_d$ is the surface of the unit sphere in $\mathbb{R}^d$. Hence,
the only ball fulfilling \eqref{compatibility} has radius $d/a$. An
evolution under condition
\eqref{compatibility} and spherical symmetry is necessarily trivial: the ball of radius $d/a$
vanishes as soon as it is forced to evolve. Still, a first
nontrivial evolution example can be obtained by considering a disjoint
collection of balls of radius $d/a$. For instance, the two-dimensional set
\begin{equation}
Z(t) = \displaystyle\cup_{i=1}^{m(t)}B((i,0),2/a) \quad \text{for} \ \ m(t)
=M-\lfloor t \rfloor\label{zzz}
\end{equation}
for some $M\in {\mathbb{N}}$ (and $2/a<1/2$) (with $\lfloor\cdot\rfloor$ the Gauss-bracket), gives an
evolution corresponding to the forcing $F^c(t)=Z(t)$ and fulfills
\eqref{compatibility} for all times, see Figure \ref{palline}.
\begin{figure}[h]
\centering
\pgfdeclareimage[width=125mm]{palline}{palline}
\pgfuseimage{palline}
\caption{The evolution from \eqref{zzz} for $M=5$ and time $t=2$.}
\label{palline}
\end{figure}
In two dimensions, condition \eqref{compatibility} selects the unique minimizer
among each family of rounded polygonal shapes, see Figure
\ref{square}. By considering a collection of
disjoint smoothed polygons fulfilling condition \eqref{compatibility}
and balls of radius $2/a$ (hence fulfilling condition \eqref{compatibility}), one can again design a nontrivial evolution
in the spirit of Figure \ref{palline}.
Let us conclude by showing an example of a nontrivial evolution for a
{\it connected} $Z(t)$ in two dimensions. Consider the smoothed square in the
middle of Figure \ref{square}. Elementary algebra shows that the only
smoothed square fulfilling condition \eqref{compatibility} is
inscribed in the square of side
$(2+\sqrt{\pi})/a$. In particular, the flat portion of each side
measures $\sqrt{\pi}/a\sim 1.77/a$. Let us now consider the union of
discs of radius $1/a$ and the smoothed square, as in Figure
\ref{mickey}. This union can be realized by still fulfilling condition
\eqref{compatibility}, as long as the positioning of each extra disc is such that
the gain in perimeter equates $a$-times the gain in
area. By calling $\alpha$ the angle at the center of the disk which
identifies the arc cut by the side of the smoothed square, the
aforementioned equality reduces to
$$ 2\pi -
\alpha - 4 \sin(\alpha/2)- \sin(\alpha)=0$$
(note that it is independent of $a$), whose unique solution in
$[0,\pi]$ is $\tilde \alpha\sim 2.005$. Note that the cord of the disk
of radius $1/a$
corresponding to $\tilde \alpha$ has length $2\sin(\tilde \alpha/2)/a
\sim 1.687/a$, which is strictly shorter than the flat portion of each
side of the smoothed square.
\begin{figure}[h]
\centering
\pgfdeclareimage[width=155mm]{mickey}{mickey}
\pgfuseimage{mickey}
\caption{An evolution of connected sets fulfilling the
compatibility condition
\eqref{compatibility}, time flows from left to right.}
\label{mickey}
\end{figure}
All configurations in Figure \ref{mickey} hence fulfill
condition \eqref{compatibility}. Moreover, they are stable (with respect
to the forcing corresponding to the interior of their complement), for
they fulfill the interior ball condition with balls of radius
$1/a$. An evolution as depicted in Figure \eqref{mickey} (left to right) can hence be
realized by suitably prescribing the forcing. At all times, such
evolution fulfills \eqref{compatibility}.
\end{examples}.
\subsection{A numerical test}
In order to illustrate the above discussion, we provide some numerical
evidence in a planar setting. We assume $\Omega =
(-3,3)^2$ and consider an initial state $Z^o\subset \Omega$ such that
$F^c \subset Z^o$ where
$$ F=\{(x,y) \in [0,1]\times [-1,1] \ : \
|y| > v(x) \} \cup \{(x,y) \in \mathbb{R}^2 \ : \ x<0 \
\text{or} \ x>1 \},$$ where $v:[0,1] \to (0,1]$ is a given function,
different for each numerical example. In the following,
we seek minimizers $Z$ of the
incremental problem \eqref{c1} of the form
$$
Z=\{(x,y) \in
[0,1]\times [-1,1] \ : \
|y| \leq u(x) \}$$
for some optimal profile $u:[0,1]\to [0,1]$ to be determined
such that $u(x)\leq v(x)$ for all $x\in[-1,1]$,
which is in accordance with the brittle constraint $Z \subset F^c$.
Note that, given the discussion on symmetry
from Subsection \ref{ss:ulisse}, assuming $Z$ to be symmetric with respect
to $\{y=0\}$ is not restrictive, since $F^c$ also is. Moreover, owing to the discussion leading to \eqref{optimal},
it is not restrictive to assume that $[0,1] \times \{0\} \subset Z$
(that is, $Z$ can actually be described by the profile $u$) as long as
the optimal profile $u$ fulfills $$u(x) \geq \sqrt{(a^2 - (x{-}a)^2)^+}+
\sqrt{(a^2-(x{-}1{+}a)^2)^+}\quad \forall x \in [0,1]$$ (that is, if $Z$ contains the two
balls of radius $a$ centered in $(a,0)$ and $(1{-}a,0)$), which happens to be the case for all
computations below.
The problem is discretized in space by partitioning the domain $[0,1]$ of the variable $x$ as $0=x_0<x_1<\dots<x_N=1$ with
$x_i=i/N$ and $N=100$
and by approximating $v$ via its piecewise
affine interpolant on the partition, taking the values
$v(i/N)=:v_i$. In particular, we look for $u$ piecewise affine with
$u(i/N)=:u_i$ minimizing
\begin{align*}
(u_0,\dots,u_N) &\mapsto P(Z,\Omega) + a {\mathcal L}^2(F^c{\setminus} Z) \\
&
= 2\left(u_0+u_N + \sum_{i=1}^N\sqrt{(u_i{-}u_{i-1})^2+N^{-2}} +
a \sum_{i=1}^N N^{-1} |u_i- v_i| \right) \\
&\text{
under the constraints $0\leq u_i\leq v_i$ for $i=1,\dots,N$. }
\end{align*}
This is a
strictly convex minimization problem under convex constraints. In the
following, we solve it by
using the {\tt fmincon} tool of Matlab for different
choices of the function $v$. In all figures, we depict the {\it portions} of
the minimizer
$Z$ (light color) and of the
forcing set $F$ in $[0,1]^2$ (dark color). The reader should
however keep in mind that both forcing and minimizer are actually symmetric along $\{y=0\}$.
Figure \ref{f1} corresponds to the choice $v(x) = (x{-}1/2)^2+1/2$ and
illustrates the effect of changing the parameter
$ a$. A smaller value of $ a$ favors a shorter perimeter at the
expense of a larger distance from $F^c$. Correspondingly, the top adhesion zone, namely the
points where $u\equiv v$, is smaller for smaller $ a$.
\begin{figure}[h]
\centering
\pgfdeclareimage[width=75mm]{f11}{f11}
\pgfdeclareimage[width=75mm]{f12}{f12}
\pgfuseimage{f11}\hspace{-9mm}
\pgfuseimage{f12}
\caption{The effect of changing the parameter $ a$. The two solutions
correspond to $v(x) = (x{-}1/2)^2+1/2$ for $ a=7$ (left) and
$ a=3$ (right). The top adhesion zone is
smaller for smaller $ a$. Note that the parts of the
boundary of $Z$ which are not in contact with $F^c$ are arcs of
circles with radius $1/a$ (recall that $ a$ is different in
the two figures), as predicted in Subsection
\ref{ss:ulisse}.}
\label{f1}
\end{figure}
Let us mention that, in case $ a<2$ one can prove that
$u_0=u_N=0$ which, as mentioned above, may well be not admissible. In order to avoid this pathology,
in all the following simulations, the parameter $ a$ will be always
chosen to be $5$. Correspondingly, in all simulations the optimal
profile $u$ is everywhere well separated from the $y=0$ axis.
Figure \ref{f2} follows by letting $v(x) = 3/4 - \beta(x{-}1/2)^2$
along with two different choices of the parameter $\beta$ and is meant to
illustrate the convexity of the evolution in presence of a convex
forcing $F^{c}$, see Subsection 4.1. In particular, the minimizer $Z$ is
convex. One observes that the optimal profile $u$ detaches from $v$
even in the convex case. This is indeed the case also in Figure
\ref{f2} left, where nonetheless the detachement is not visible due to
the scale.
\begin{figure}[h]
\centering
\pgfdeclareimage[width=75mm]{f21}{f21}
\pgfdeclareimage[width=75mm]{f22}{f22}
\pgfuseimage{f21}\hspace{-9mm}
\pgfuseimage{f22}
\caption{Convex forcing $F^{c}$. The two solutions
correspond to $v(x) = 3/4 - \beta(x{-}1/2)^2$ for $\beta=2$ (left) and
$\beta=1/5$ (right). The minimal set $Z$ is convex. }
\label{f2}
\end{figure}
Figure \ref{f3} corresponds to the choices
$v(x) = 1/2 \pm (|x{-}1/2|-1/4)$ and illustrates
the partial regularity of the solution. Note that nonsmooth
boundary points occur in connection with nonconvex forcings $F^{c}$.
\begin{figure}[h]
\centering
\pgfdeclareimage[width=75mm]{f31}{f31}
\pgfdeclareimage[width=75mm]{f32}{f32}
\pgfuseimage{f31}\hspace{-9mm}
\pgfuseimage{f32}
\caption{Partial $C^1$ regularity. The two solutions
correspond to $v(x) = 3/4 - |x{-}1/2|$ (left) and $v(x) = 1/4
+ |x{-}1/2|$ (right).}
\label{f3}
\end{figure}
Figure \ref{f4} illustrates some special situations. On the left, the solution
for $v(x)=\max\{1-5|x{-}1/2|,1/2\}$. In this case, the optimal
profile is such that $u(1/2)>3/4$. Note that the same holds for any
opening of the cone in $F^c$, whatever small.
On the right, the solution for $v(x)= \lfloor 5x\rfloor/5+1/5$. The profiles $u=v$ touch at the points
$x=1/5, \, 2/5, \, 3/5$, and $4/5$ only. Note that all
nonstraight portions of the boundary of $Z$ are arcs of radius
$1/ a$.
\begin{figure}[h]
\centering
\pgfdeclareimage[width=75mm]{f41}{f41}
\pgfdeclareimage[width=75mm]{f42}{f42}
\pgfuseimage{f41}\hspace{-9mm}
\pgfuseimage{f42}
\caption{Extreme configurations. The solution
for $v(x)=\max\{1-5|x{-}1/2|,1/2\}$ (left) and $v(x)= \lfloor 5x\rfloor/5+1/5$ (right).}
\label{f4}
\end{figure}
\section{Proofs of Theorems \ref{thm:4.1}, \ref{thm:4.2}, and \ref{thm:4.3}}
\label{s:6}
\noindent
We start by the \underline{\bf Proof of Theorem \ref{thm:4.1}:} For $k\in \mathbb{N}\cup \{\infty\}$ fixed, let $(\tau_j)_j$ be a vanishing sequence
and let $(\piecewiseConstant z{\tau_j})_j$ be the characteristic functions of the sets $(\piecewiseConstant Z{\tau_j})_j$. Since the functions
$\piecewiseConstant z{\tau_j}$ are nonincreasing, it is immediate to check that \[
\exists\, C>0 \ \forall\, j \in \mathbb{N} \, : \qquad \| \piecewiseConstant z{\tau_j}\|_{\mathrm{BV}([0,T];L^1(\Omega))} \leq C\,.
\]
Furthermore,
from \eqref{energy-bound-interp} we get that
\[
\exists\, C>0 \ \forall\, j \in \mathbb{N} \, : \qquad \| \piecewiseConstant z{\tau_j}\|_{L^\infty(0,T;\mathrm{SBV}(\Omega;\{0,1\}))} \leq C\,.
\]
We are now in the position to apply a Helly-type compactness result (cf.\ e.g.\ \cite[Thm.\ 3.2]{MaiMie05EREM}), and conclude that
\[
\begin{aligned}
&
\exists\, z \in L^\infty(0,T;\mathrm{SBV}(\Omega;\{0,1\})) \cap \mathrm{BV}([0,T];L^1(\Omega)),
\\
&
\quad \text{with }
z(\cdot, x) \text{ nonincreasing on } [0,T] \ \text{for a.a.}\, x \in \Omega, \text{ such that }
\\
&
\piecewiseConstant z{\tau_j}(t) \overset{*}{\rightharpoonup} z(t) \quad \text{ in } \mathrm{SBV}(\Omega;\{0,1\}) \quad \text{for every } t \in [0,T].
\end{aligned}
\]
Then, the curve $Z: [0,T]\to \mathbf{X}$ defined by
$Z(0)=Z_0$ and
$Z(t): =\{ x\in \Omega\, : \ z(t)=1\}$ for all $t\in (0,T]$ is in $\GMM k$.
\par
In order to prove convergence \eqref{refined-convs-1} for the piecewise constant, right-continuous interpolants $(\underpiecewiseConstant z{\tau_j})_j$ we may argue in this way: by the above Helly argument,
there exists
$\underline z \in L^\infty(0,T;\mathrm{SBV}(\Omega;\{0,1\})) \cap \mathrm{BV}([0,T];L^1(\Omega))$ such that,
up to a further subsequence, $ \underpiecewiseConstant z{\tau_j}(t) \overset{*}{\rightharpoonup} \underline{z}(t) $ in $ \mathrm{SBV}(\Omega;\{0,1\}) $ for all $t\in [0,T]$. Let $J$ be the union of the jump sets of $z$ and $\underline z$: arguing for instance as in the proof of \cite[Thm.\ 4.1]{ThoRouPan},
it can be checked that $z(t) = \underline{z}(t)$ for all $t\in [0,T]\setminus J$.
Convergences \eqref{refined-convs-2} and \eqref{refined-convs-3}
ensue from standard weak and strong compactness arguments.
\par
Let us now pick $Z\in \GMM k $ approximated by a sequence $(\piecewiseConstant Z{\tau_j})_j$ in the sense of
\eqref{charact}. In order to show that $Z$ satisfies the stability condition \eqref{stab}, we will pass to the limit in its discrete
version \eqref{discr-stab-interp}, satisfied by the functions $\piecewiseConstant Z{\tau_j}$, by verifying the so-called \emph{mutual recovery sequence} condition from \cite {MRS06}. Namely, for every fixed $t\in (0,T]$ and every admissible competitor $\widetilde Z \in \mathbf{X}$ for \eqref{stab}, with associated characteristic function $\tilde z$, we will exhibit a sequence $(\tilde z_j)_j\subset \mathrm{SBV}(\Omega;\{0,1\})$ such that
\begin{equation}
\label{MRS}
\limsup_{j\to\infty} \left( \calE_k(t,\tilde{z}_j) {-}\calE_k(t,\piecewiseConstant z{\tau_j}(t)) {+} {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(\piecewiseConstant z{\tau_j}(t),\tilde{z}_j) \right) \leq \calE_k(t,\tilde{z}) {-}\calE_k(t,z(t)) {+} {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}( z(t),\tilde{z}) \,.
\end{equation}
The construction of the sequence $(\tilde z_j)_j$ is slightly adapted from the proof of
\cite[Prop.\ 5.9]{RosTho12ABDM}, which in turn follows the steps of
\cite[Lemma 2.13]{Thom11QEBV}. Hence, in the following lines we shall refer to \cite{RosTho12ABDM} for some details.
First of all, we suppose that
$ {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}( z(t),\tilde{z}) <\infty$, whence $\tilde z \leq z(t) $ a.e.\ in $\Omega$, and, if $k=\infty$, that ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t, \tilde z)=0$, so that
$ \calE_\infty(t,\tilde{z}) <\infty$; otherwise, there is nothing to prove.
Along the foosteps of \cite{RosTho12ABDM}, we set
\begin{equation}
\label{recovery-j}
\tilde{z}_j: = \tilde z \calX_{A_j} +\piecewiseConstant z{\tau_j}(t) (1{-} \calX_{A_j}), \quad \text{where } A_j: = \{ x \in \Omega\, : \ 0\leq \tilde z(x) \leq \piecewiseConstant z{\tau_j}(t,x) \} \,.
\end{equation}
This way, we ensure that
\begin{equation}
\label{1st-props-tildez-k}
\tilde{z}_j \in \{0,1\} \quad\text{and}\quad 0 \leq \tilde{z}_j \leq \piecewiseConstant z{\tau_j}(t) \qquad \text{a.e. in}\ \Omega\,.
\end{equation}
Furthermore, arguing in the very same way as in the proof of \cite[Prop.\ 5.9]{RosTho12ABDM},
where \cite[Thm.\ 3.84]{AmFuPa05FBVF} on the decomposition of $\mathrm{BV}$-functions is applied, we can show that
$\tilde z_j \in \mathrm{SBV}(\Omega;\{0,1\})$. We now split the proof of
\eqref{MRS} in $3$ steps:
\begin{enumerate}
\item Since
\begin{equation}
\label{Lq-conv}
\piecewiseConstant z{\tau_j}(t)\to z(t) \qquad \text{in $L^q(\Omega)$ for all $1\leq q <\infty$},
\end{equation}
and
$\tilde z \leq z(t) $ a.e.\ in $\Omega$, it is immediate to infer that
$\tilde{z}_j \to \tilde z$ a.e.\ in $\Omega$ as $j\to\infty$, which improves to
\begin{equation}
\label{strong-converg}
\tilde{z}_j \to \tilde z \quad \text{in } L^q(\Omega) \quad \text{for all } 1\leq q <\infty,
\end{equation}
as $\tilde{z}_j \in \{0,1\}$ a.e.\ in $\Omega$. Combining \eqref{1st-props-tildez-k} with
\eqref{Lq-conv} and
\eqref{strong-converg} we ultimately conclude that
\begin{equation}
\label{conv-dissip}
\lim_{j\to\infty} {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(\piecewiseConstant z{\tau_j}(t),\tilde{z}_j) = {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}( z(t),\tilde{z}) \,.
\end{equation}
\item For $k\in \mathbb{N}$, we observe that
\begin{subequations}
\label{limsupJ}
\begin{equation}
\label{limsupJk}
\begin{aligned}
\limsup_{j\to\infty} \left( {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,\tilde{z}_j) {-}{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,\piecewiseConstant z{\tau_j}(t))\right) =
\limsup_{j\to\infty} \int_{\Omega} k f(t) (\tilde{z}_j {-} \piecewiseConstant z{\tau_j}(t)) \,\mathrm{d} x
& = \int_{\Omega} k f(t) (\tilde{z}{-} z(t)) \,\mathrm{d} x \\ & = {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,\tilde{z}) {-}{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,z(t))
\end{aligned}
\end{equation}
thanks to \eqref{Lq-conv} and
\eqref{strong-converg}.
For $k=\infty$,
we first of all observe that, since $\sup_{j\in \mathbb{N}} \calE(t,\piecewiseConstant z{\tau_j}(t)) \leq C $ by \eqref{charact},
we have ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty (t,\piecewiseConstant z{\tau_j}(t)) =0$, i.e.\ $f(t,\cdot) \piecewiseConstant z{\tau_j}(t,\cdot)=0$ a.e.\ in $\Omega$, for every $j\in \mathbb{N}$. Then, $0\leq {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,z(t)) \leq \liminf_{j\to\infty} {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty (t,\piecewiseConstant z{\tau_j}(t)) =0$. Furthermore,
since
$ \tilde{z}_j \leq \piecewiseConstant z{\tau_j}(t) $ a.e.\ in $\Omega$ thanks to \eqref{1st-props-tildez-k}, we ultimately conclude that ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty (t,\tilde{z}_j) =0$ for all $j\in \mathbb{N}$. Therefore,
\begin{equation}
\label{limsupJ-infty}
\limsup_{j\to\infty} \left( {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,\tilde{z}_j) {-}{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,\piecewiseConstant z{\tau_j}(t))\right) =
0= {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,\tilde{z}) - {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,z(t))
\end{equation}
\end{subequations}
(recall that we have supposed right from the start that ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,\tilde{z})=0$).
\item Finally, it can be shown that
\begin{equation}
\label{limsup-perims}
\limsup_{j\to\infty} \left( |\mathrm{D} \tilde{z}_j|(\Omega) {-} |\mathrm{D} \piecewiseConstant z{\tau_j}(t)|(\Omega)\right) \leq |\mathrm{D} \tilde{z}|(\Omega) {-} |\mathrm{D} z(t)|(\Omega)
\end{equation}
by repeating the very same arguments as in the proof of \cite[Prop.\ 5.9]{RosTho12ABDM}.
\end{enumerate}
Combining \eqref{conv-dissip}, \eqref{limsupJ}, and \eqref{limsup-perims}, we infer \eqref{MRS} for $k\in \mathbb{N}$ and $k=\infty$,
which concludes the proof of the stability condition \eqref{stab}, and thus of Thm.\ \ref{thm:4.1}.
\mbox{}\hfill\rule{5pt}{5pt}\medskip\par
\paragraph{\bf \underline{Proof of Theorem \ref{thm:4.2}:}} Let $k\in \mathbb{N}$ be fixed,
let $Z\in \SMM k$, and let $(\piecewiseConstant z{\tau_j})_j$ converge to $z=\calX_Z$ as in \eqref{charact}.
We can pass to the limit in the upper energy-dissipation estimate in \eqref{enbd} by observing that
\[
\begin{aligned}
&
\liminf_{j\to\infty} \left(\calE_k(\piecewiseConstant{\mathsf{t}}{\tau_j}(t), \piecewiseConstant Z{\tau_j}(t)) {-}\calE_k(t,z(t))\right) \\ & \geq \lim_{j\to\infty} \left(\calE_k(\piecewiseConstant{\mathsf{t}}{\tau_j}(t), \piecewiseConstant Z{\tau_j}(t)) {-} \calE_k(t, \piecewiseConstant Z{\tau_j}(t))\right) + \liminf_{j\to\infty} \left(\calE_k(t, \piecewiseConstant Z{\tau_j}(t)) {-}\calE_k(t,z(t))\right) =: l_1+l_2 \geq 0,
\end{aligned}
\]
where we have used that
\[
l_1=\lim_{j\to\infty} \int_\Omega k \left( f(\piecewiseConstant{\mathsf{t}}{\tau_j}(t),x) {-} f(t,x)\right)
\piecewiseConstant z{\tau_j}(t,x))
\,\mathrm{d} x \leq k \lim_{j\to\infty} \int_{t}^{\piecewiseConstant{\mathsf{t}}{\tau_j}(t)} \|\partial_t f(s) \|_{L^1(\Omega)} \,\mathrm{d} s =0,
\]
due to \eqref{assFsmooth},
while, by convergence \eqref{charact} we have
\[
l_2 = \liminf_{j\to\infty}\left( |\mathrm{D}\piecewiseConstant z{\tau_j}(t)|(\Omega) {-} |\mathrm{D} z(t)|(\Omega) \right) \geq 0\,.
\]
Thanks to \eqref{charact}, we also have
$ \liminf_{j\to\infty} {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_0,\piecewiseConstant Z{\tau_j}(t)) \geq {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_0, Z(t)) $, and \eqref{refined-convs} gives
\[
\begin{aligned}
\lim_{j \to\infty}
\int_{0}^{\piecewiseConstant \mathsf{t}{\tau_j}(t)}\partial_t\calE_k(r,\underpiecewiseConstant Z{\tau_{j}}(r))\,\mathrm{d}r
& = \lim_{j \to\infty}
\int_{0}^{\piecewiseConstant \mathsf{t}{\tau_j}(t)} \int_\Omega k \partial_t f(r,x) \underpiecewiseConstant z{\tau_{j}}(r,x) \,\mathrm{d} x \,\mathrm{d}r
\\
& = \int_0^t \int_\Omega k\partial_t f(r,x) z(r,x) \,\mathrm{d} x \,\mathrm{d} r =\int_0^t \partial_t \calE_k(r,Z(r)) \,\mathrm{d} r\,.
\end{aligned}
\]
All in all, we conclude the upper energy-dissipation estimate
\[
\calE_k(t,Z(t))+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z(0),Z(t)) \leq \calE_k(0,Z(0))+\int_0^t\partial_t\calE_k(r,Z(r))\,\mathrm{d}r \qquad \text{for all } t \in [0,T].
\]
\par
The lower estimate $\geq $
follows from the stability condition \eqref{stab} via a well-established Riemann-sum technique, cf.\ e.g.\ \cite[Prop.\ 2.1.23]{MieRou-book} for a general result. This concludes the proof that $Z$ is an Energetic solution to the adhesive rate-independent system $(\mathbf{Z},\calE_k, {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$.
\mbox{}\hfill\rule{5pt}{5pt}\medskip\par
\par
Prior to proving Thm.\ \ref{thm:4.3}, we show in the following result that, for every fixed $t\in [0,T]$,
the energies $\calE_k(t,\cdot)$ for the adhesive systems $\Gamma$-converge as $k\to\infty$ to the energy $\calE_\infty(t,\cdot)$ driving the brittle system
\emph{with respect to the weak$^*$-topology of $ \mathrm{SBV}(\Omega;\{0,1\})$} (in the sense of \eqref{sense}).
Namely, we shall prove that
\begin{subequations}
\label{Gamma-conv_E_k}
\begin{align}
&
\label{Ginf}
\text{$\Gamma$-$\liminf$ estimate:} && z_k \overset{*}{\rightharpoonup} z \text{ in } \mathrm{SBV}(\Omega;\{0,1\}) \ \Rightarrow \ \liminf_{k\to\infty}\calE_k(t,z_k) \geq \calE_\infty(t,z),
\\
&
\label{Gsup}
\text{$\Gamma$-$\limsup$ estimate:} && \forall\, \hat{z} \in \mathrm{SBV}(\Omega;\{0,1\}) \ \exists\, (\hat{z}_k)_k\subset
\mathrm{SBV}(\Omega;\{0,1\})\, : \ \hat{z}_k \overset{*}{\rightharpoonup} \hat{z} \text{ in } \mathrm{SBV}(\Omega;\{0,1\}) \\
\nonumber
& &&
\qquad \qquad \text{ and } \limsup_{k\to\infty} \calE_k(t,\hat{z}_k) \leq \calE_\infty(t,\hat{z})\,.
\end{align}
\end{subequations}
\begin{lemma}
\label{PropGamma}
For every $t\in [0,T]$ the functionals $(\calE_k(t,\cdot)_k$ defined by
\emph{\eqref{ENADH}}
$\Gamma$-converge as $k\to\infty$, in the sense of \emph{\eqref{Gamma-conv_E_k}}, to $\calE_\infty(t,\cdot)$ from
(\ref{ENBRI}).
\end{lemma}
\noindent {\it Proof.} To start with, observe that the upper estimate \eqref{Gsup} can be concluded by choosing the constant sequence $(\hat{z}_k: = \hat{z})_k$
as a recovery sequence.
\par To verify the lower estimate \eqref{Ginf}, consider a sequence
$z_k \overset{*}{\rightharpoonup} z$ in $\mathrm{SBV}(\Omega;\{0,1\})$ and
the corresponding sets
of finite perimeter $ Z_k, Z$. We distinguish two cases:
\begin{compactenum}
\item
If $\calL^d( Z\cap F(t))=0,$ then, by the positivity of
${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t, \widetilde Z_k)$
and the lower semicontinuity of the
perimeter w.r.t.\
strong $L^1$-convergence
of characteristic functions,
we find that
\[
\liminf_{k\to\infty}\calE_k(t,z_k)
\geq P(Z,\Omega)=\calE_\infty(t,z).
\]
\item
Assume now that $\calL^d(Z\cap F(t))=c>0$, so that ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,z)=\infty$.
Since $z_k\to z$ strongly in $L^1(\Omega),$ we also have
$f(t) z_k\to f(t) z$ strongly in $L^1(\Omega),$ and hence, for every $\varepsilon>0$ we find an index
$k_\varepsilon\in\mathbb{N}$ such that for all $k\geq k_\varepsilon$ we have that
$\|f(t)z_k-f(t) z\|_{L^1(\Omega)}\leq\varepsilon$. This implies that, for every $\varepsilon\in (0,c)$ there holds
\begin{equation*}
\liminf_{k\to\infty}{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,z_k)
=\liminf_{k\to\infty} \int_{\Omega} k f(t) z_k \, \mathrm{d} x
\geq\liminf_{k\to\infty} k(c-\varepsilon)=\infty,
\end{equation*}
whence again \eqref{Ginf}.
\end{compactenum}
\mbox{}\hfill\rule{5pt}{5pt}\medskip\par
\par
We are now in a position to carry out the \underline{\textbf{proof of Theorem \ref{thm:4.3}:}}
Let us consider a sequence $(Z_k)_k$ with $Z_k\in \SMM k$ for every $k\in \mathbb{N}$ and the associated sequence of characteristic functions $(z_k)_k$.
Since the energy bound \eqref{energy-bound-interp} holds for a constant uniform w.r.t.\ $k\in \mathbb{N}$ as well, and it is inherited by the time-continuous limit,
we infer
\[
\exists\, C>0 \ \forall\, k \in \mathbb{N}\, : \quad \sup_{t\in [0,T]}\calE_k(t,z_k(t)) \leq C, \ \text{ as well as } \ \mathrm{Var}_{{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}}(z_k; [0,T]) \leq C
\]
as the sequence $(z_k)_k$ is nonincreasing in time.
Hence we may repeat the very same compactness arguments as in the proof of Thm.\ \ref{thm:4.1} and conclude that there exist a (not relabeled) subsequence and $z\in L^\infty(0,T;\mathrm{SBV}(\Omega;\{0,1\})) \cap \mathrm{BV}([0,T];L^1(\Omega))$ such that \eqref{sense} holds, as well as
\begin{equation}
\label{convk-L}
z_k\overset{*}{\rightharpoonup} z \text{ in } L^\infty(0,T;\mathrm{SBV}(\Omega;\{0,1\})), \qquad
z_k\to z \text{ in } L^q ((0,T){\times} \Omega)
\text{ for every } q \in [1,\infty).
\end{equation}
Thanks to Lemma \ref{PropGamma} we have
\begin{equation}
\label{liminf_cons}
\liminf_{k\to\infty} \calE_k(t,z_k(t)) \geq \calE_\infty(t,z(t)), \quad \text{in particular } {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,z(t))=0, \quad \text{for all } t \in [0,T].
\end{equation}
\par
The limit passage as $k\to\infty$ in the stability condition \eqref{stab}, for $t\in (0,T]$ fixed, again relies on the mutual recovery sequence condition, i.e.\ on the fact that for every $\tilde z \in \mathrm{SBV}(\Omega;\{0,1\})$ (with
$ {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}( z(t),\tilde{z}) <\infty $ and ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty (t,\tilde z)=0$ to avoid trivial situations) there exists $(\tilde{z}_k)_k \subset \mathrm{SBV}(\Omega;\{0,1\})$ such that
\begin{equation}
\label{MRS-k}
\limsup_{k\to\infty} \left( \calE_k(t,\tilde{z}_k) {-}\calE_k(t,z_k(t)) {+} {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(z_k(t),\tilde{z}_k) \right) \leq \calE_\infty(t,\tilde{z}) {-}\calE_\infty(t,z(t)) {+} {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}( z(t),\tilde{z}) \,.
\end{equation}
To this end, we resort to a construction completely analogous to that in \eqref{recovery-j}
and set
\begin{equation}
\label{recovery-k}
\tilde{z}_k: = \tilde z \calX_{A_k} +z_k(t) (1{-} \calX_{A_k}) \quad \text{with } A_k: = \{ x \in \Omega\, : \ 0\leq \tilde z(x) \leq z_k(t,x) \} \,.
\end{equation}
Exploiting
the convergence properties \eqref{conv-dissip}, \eqref{limsup-perims}, as well as the fact that
\[
\limsup_{k\to\infty} \left( {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,\tilde{z}_k) -{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,z_k(t)) \right) = \limsup_{k\to\infty} \int_\Omega k f(t,x)\left( \tilde{z}_k(t,x) {-} z_k(t,x)\right) \,\mathrm{d} x
\leq 0 = {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty (t,\tilde z)-{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty (t,z(t))
\]
(since $\tilde z_k \leq z_k$ a.e.\ in $\Omega$ by construction), we
obtain \eqref{MRS-k}. This proves that $Z\in \SMM \infty$.
\par
In order to conclude that $Z$ is an Energetic solution to the brittle system $(\mathbf{Z},\calE_\infty,{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F})$ under the additional monotonicity assumption $\partial_t f \geq0$ and compatibilty condition \eqref{compatibility}, we first establish the upper energy-dissipation estimate
\begin{equation}
\label{enbal-infty-upper}
\calE_\infty(t,z(t))+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(z(0),z(t)) \leq \calE_\infty(0,Z_0) \quad \text{for all } t \in [0,T]
\end{equation}
by passing to the limit in \eqref{enbal} as $k\to\infty$.
With this aim, we
observe that
\begin{equation}
\label{convJk}
{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,z_k(t))=\int_\Omega kf(t)z_k(t)\,\mathrm{d}x\to0 \quad\text{as } k\to\infty \quad \text{for all } t \in [0,T].
\end{equation}
To check \eqref{convJk}, first of all we apply the construction of the recovery sequence \eqref{recovery-k}
to $\tilde z:=z(t)$ to find a sequence $(\tilde z_k)_k$, associated with sets $(\widetilde Z_k)_k$.
As observed in the proof of Theorem \ref{thm:4.1} (cf.\ \eqref{limsup-perims}),
the construction ensures
in particular that $\limsup_{k\to\infty}\big(|\mathrm{D} \tilde{z}_k|(\Omega){-} |\mathrm{D} z_k(t)|(\Omega) \big)\leq |\mathrm{D} z(t)|(\Omega)- |\mathrm{D} z(t)|(\Omega) =0$.
Moreover,
$\tilde{z}_k \leq \min\{ z_k(t), \, z(t) \}$ a.e.\ in $\Omega$ and
$\tilde z_k\to z(t)$ in $L^1(\Omega)$, so that
\[
\lim_{k\to\infty}{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(z_k(t),\tilde z_k)=0.
\]
Also observe that ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,\tilde{z}_k)=0$ for all $k\in\mathbb{N},$ because ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,z(t))=0$
by
\eqref{liminf_cons}
and because
$\widetilde Z_k\subset Z(t)$ by construction.
Since ${\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,z_k(t))\geq0$ for all $k\in\mathbb{N}$,
these observations allow us to conclude from the stability of $z_k(t)$ that
\begin{equation*}
0\leq \liminf_{k\to\infty} {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,z_k(t)) \leq\limsup_{k\to\infty}{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(t,z_k(t))
\leq\limsup_{k\to\infty}\big(P(\widetilde Z_k,\Omega)-P(Z_k(t),\Omega)\big)
+\lim_{k\to\infty}{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(z_k(t),\tilde Z_k)=0\,,
\end{equation*}
which gives \eqref{convJk}.
\color{black} We will now show that
\begin{equation}
\label{power2zero}
\int_0^t \partial_t \calE_k(s,z_k(s)) \, \mathrm{d} s = \int_{0}^{t}\int_{\Omega}k\partial_t f(s)z_{k}(s)\,\mathrm{d}x\,\mathrm{d}s \to 0 \quad\text{as } k\to\infty \quad \text{for all } t \in [0,T].
\end{equation}
Indeed,
using the assumption $\partial_t f \geq0$ the energy balance \eqref{enbal} for each $k\in\mathbb{N}$ gives
\begin{equation}
\label{BRAVA}
\begin{aligned}
0&\leq\int_{0}^{t}\int_{\Omega}k\partial_t f(s)z_{k}(s)\,\mathrm{d}x\,\mathrm{d}s
\\ & =P(Z_{k}(t),\Omega)+{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_{k}(t,z_{k}(t))+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{0}^{k},Z_{k}(t))-P(Z_{0}^{k},\Omega) - {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(0,Z_0^k)
\\
&\stackrel{(1)}{\leq} {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{k}(t),\emptyset)
+{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{0}^{k},Z_{k}(t))-P(Z_{0}^{k},\Omega)
\stackrel{(2)}{=}{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{0}^{k},\emptyset)-P(Z_{0}^{k},\Omega)\,,
\end{aligned}
\end{equation}
where {\footnotesize (1)} follows from observing that $ {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_k(0,Z_0^k) \geq 0$ and from choosing $\widetilde{Z} = \emptyset$ in the stability condition \eqref{stab} (which gives $P(Z_{k}(t),\Omega)+{\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_{k}(t,z_{k}(t)) \leq {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{k}(t),\emptyset)$), while {\footnotesize (2)} ensues from the fact that
\[
{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{k}(t),\emptyset) +{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{0}^{k},Z_{k}(t))= a \calL^d(Z_k(t)) + a \calL^d(Z_0^k{\setminus}Z_k(t)) = a \calL^d(Z_0^k) = {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{0}^{k},\emptyset)\,.
\]
We then observe that
\[
\lim_{k\to\infty} \left( {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{0}^{k},\emptyset){-}P(Z_{0}^{k},\Omega) \right)
\stackrel{(3)}{=} {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z_{0},\emptyset)-P(Z_{0},\Omega) \stackrel{(4)}{=}0 \qquad \text{as } k \to \infty\,,
\]
where {\footnotesize (3)} is due to condition \eqref{conseq-well-prep} and {\footnotesize (4)} follows from the compatibility condition \eqref{compatibility}.
Hence we conclude \eqref{power2zero}. Thus,
\eqref{enbal-infty-upper} follows from passing to the limit as $k\to \infty$ in the upper estimate '$\leq$' of \eqref{enbal-infty}: the left-hand side is dealt with by lower semicontinuity arguments, while the limit passage on the right-hand side follows from the well-preparedness \eqref{well-prep}
of the initial data,
and from \eqref{power2zero}.
\par
The lower energy-dissipation estimate,
i.e.\ the converse of inequality \eqref{enbal-infty-upper},
follows from testing the stability condition at $t=0$ with $\widetilde Z = Z(t)$, which gives
\[
\calE_\infty(0,Z_0) \leq {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z(0),Z(t)) + \calE_\infty(0,Z(t)) \leq {\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}(Z(0),Z(t)) + \calE_\infty(t,Z(t)).
\]
Indeed, $\calE_\infty(0,Z(t)) - \calE_\infty(t,Z(t))= {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(0,Z(t)) - {\mathcal J}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}_\infty(t,Z(t))=0$, since $Z(t) \cap F(t) =\emptyset$ and $F(0)\subset F(t) $ imply $Z(t) \cap F(0) =\emptyset$.
\par
We have thus shown that $Z$ complies with the stability condition \eqref{stab} and with the energy-dissipation balance \eqref{enbal-infty}.
This concludes the proof of Theorem \ref{thm:4.3}.
\mbox{}\hfill\rule{5pt}{5pt}\medskip\par
\begin{remark}
\label{rmk:compat4adh}
\upshape
Assume that the external loading $f$ for the \emph{adhesive} system complies with \eqref{power-control}, and that the initial datum $Z_0$ satisfies the compatibility condition \eqref{compatibility}.
From \eqref{BRAVA} we then deduce that
\[
0=\int_{0}^{t}\int_{\Omega}k\partial_t f(s)z_{k}(s)\,\mathrm{d}x\,\mathrm{d}s = \int_0^t \partial_t \calE_k(s,z_k(s)) \, \mathrm{d} s
\]
for every $t\in (0,T]$ and for all $k\in \mathbb{N}$.
Therefore, $\partial_t f z_k =0$ a.e.\ in $\Omega\times (0,T)$, i.e.\ the sets $Z_k$ fulfill
\begin{equation}
\label{weak-constraint}
Z_k(t) \subset ({\mathop{\mathrm{supp}}}(\partial_t f(t)))^c \qquad \text{for a.a.}\, t \in (0,T)\,.
\end{equation}
Taking into account that $ (\mathop{\mathrm{supp}}(f(t)))^c \subset(\mathop{\mathrm{supp}}
(\partial_t f(t)))^c$,
\eqref{weak-constraint} may be interpreted as a weak form of the brittle constraint \eqref{brittle-constr-intro}.
\end{remark}
\bibliographystyle{alpha}
|
{
"timestamp": "2019-03-01T02:01:32",
"yymm": "1902",
"arxiv_id": "1902.10749",
"language": "en",
"url": "https://arxiv.org/abs/1902.10749"
}
|
\section{Introduction}
Machine learning models are often trained on sensitive personal information, e.g.,\ when analyzing healthcare records or social media data. There is hence an increasing awareness and demand for privacy preserving machine learning technology.
This motivated the line of works on {\em private learning}, initiated by \cite{KLNRS11}, {which} provides strong (mathematically proven) privacy protections for the training data. Specifically, these works aim at achieving {\em differential privacy}, a strong notion of privacy that is now increasingly being adopted by both academic researchers and industrial companies. Intuitively, a private learner is a PAC learner that guarantees that every single example has almost no effect on the resulting classifier. Formally, a private learner is a PAC learner that satisfies differential privacy w.r.t.\ its training data. The definition of differential privacy is,
\begin{definition}[\cite{DMNS06}]\label{def:dpIntro}
Let $\mathcal A$ be a randomized algorithm that operates on databases.
Algorithm $\mathcal A$ is $(\varepsilon,\delta)$-{\em differentially private} if for any two databases $S,S'$ that differ on one row, and any event~$T$, we have
$\Pr[\mathcal A(S)\in T]\leq e^{\varepsilon}\cdot \Pr[\mathcal A(S')\in T]+\delta.$
The {notion} is referred to as {\em pure} differential privacy when $\delta=0$, and {\em approximate} differential privacy when $\delta>0$.
\end{definition}
The initial work of \cite{KLNRS11}
showed that any concept class $C$ is privately learnable with sample complexity $O(\log |C|)$ (we omit in the introduction the dependencies on accuracy and privacy parameters). Non-privately, $\Theta(VC(C))$ samples are necessary and sufficient to PAC learn $C$, and much research has been devoted to understanding how large the gap is between the sample complexity of private and non-private PAC learners.
For {\em pure} differential privacy, it is known that a sample complexity of $\Theta(\log|C|)$ is required even for learning some simple concept classes such as one-dimensional thresholds, axis-aligned rectangles, balls, and halfspaces \citep{BBKN12,BNS13,FX14}. That is, generally speaking, learning with pure differential privacy requires sample complexity proportional to log the size of the hypothesis class. For example, in order to learn halfspaces in $\R^d$, one must consider some {\em finite} discretization of the problem, e.g.\ by assuming that input examples come from a finite set $X\subseteq\R^d$. A halfspace over $X$ is represented using $d$ point from $X$, and hence, learning halfspaces over $X$ with pure differential privacy requires sample complexity $\Theta(\log{|X| \choose d})=O(d\log|X|)$. In contrast, learning halfspaces non-privately requires sample complexity $O(d)$. In particular, when the dimension $d$ is constant, learning halfspaces non-privately is achieved with {\em constant} sample complexity, while learning with pure differential privacy requires sample complexity that is proportional to the representation length of domain elements
For {\em approximate} differential privacy, the current understanding is more limited.
Recent results established that the class of one-dimensional thresholds over a domain $X\subseteq\R$ requires sample complexity between $\Omega(\log^*|X|)$ and $2^{O(\log^*|X|)}$ (\cite{BNS13b,BNSV15,Bun16,ALMM18}).
On the one hand, these results establish a separation between what can be learned with or without privacy, as they imply that privately learning one-dimensional thresholds over an infinite domain is impossible.
On the other hand, these results show that, unlike with pure differential privacy, the sample complexity of learning one-dimensional thresholds can be much smaller than $\log|C|=\log|X|$. \cite{BNS13b} also established an upper bound of $\mathop{\rm{poly}}\nolimits(d\cdot2^{\log^*|X|})$ for privately learning the class of axis-aligned rectangles over $X\subseteq\R^d$.
In a nutshell, this concludes our current understanding of the sample complexity of approximate private learning. In particular, before this work, it was not known whether similar upper bounds (that grow slower than $\log|C|$) can be established for ``richer'' concept classes, such as halfspaces, balls, and polynomials.
We answer this question positively, focusing on privately learning halfspaces. The class of halfspaces forms an important primitive in machine learning as learning halfspaces implies learning many other concept classes~(\cite{BL98}). In particular, it is the basis of popular algorithms such as neural nets and kernel machines, as well as various geometric classes (e.g., polynomial threshold functions, polytopes, and $d$-dimensional balls).
\subsection{Our Results}
Our approach for privately learning halfspaces is based on a reduction to the task of privately finding a point in the convex hull of a given input dataset. That is, towards privately learning halfspaces we first design a sample-efficient differentially private algorithm for identifying a point in the convex hull of the given data, and then we show how to use such an algorithm for privately learning halfspaces.
\paragraph{Privately finding a point in the convex hull.}
We initiate the study of privately finding a point in the convex hull of a dataset $S\subseteq X\subseteq\R^d$. Even though this is a very natural problem (with important applications, in particular to learning halfspaces), it has not been considered before in the literature of differential privacy. One might try to solve this problem using the {\em exponential mechanism} of \cite{MT07}, which, given a dataset and a {\em quality function}, privately identifies a point with approximately maximum quality. To that end, one must first settle on a suitable quality function such that if a point $\pt{x}\in X$ has a high quality then this point is guaranteed to be in the convex hull of $S$. Note that the indicator function $q(\pt{x})=1$ if and only if $\pt{x}$ is in the convex hull of $S$ is not a good option, as every point $\pt{x}\in X$ has quality either 0 or 1, and the exponential mechanism only guarantees a solution with {\em approximately} maximum quality (with additive error larger than 1).
Our approach is based on the concept of {\em Tukey-depth} \citep{Tuk75}. Given a dataset $S\subseteq\R^d$, a point $\pt{x}\in\R^d$ has the Tukey-depth at most $\ell$ if there exists a set $A \subseteq S$ of size $\ell$ such that $\pt{x}$ is not in the convex hull of $S\setminus A$. See \cref{def:Tukey} for an equivalent definition that has a geometric flavor. Instantiating the exponential mechanism with the Tukey-depth as the quality function results in a private algorithm for identifying a point in the convex hull of a dataset $S\subseteq X\subseteq\R^d$ with sample complexity $\mathop{\rm{poly}}\nolimits(d,\log|X|)$.\footnote{We remark that the domain $X$ does not necessarily contain a point with a high Tukey-depth, even when the input points come from $X\subseteq\R^d$. Hence, one must first {\em extend} the domain $X$ to make sure that a good solution exists. This results in a private algorithm with sample complexity $O(d^3\log|X|)$.}
We show that this upper bound can be improved to $\mathop{\rm{poly}}\nolimits(d,2^{\log^*|X|})$. Our construction utilizes an algorithm by~\cite{BNS13b} for approximately maximizing (one-dimensional) quasi-concave functions with differential privacy (see Definition~\ref{def:quasiConcave} for quasi-concavity). To that end, we show that it is possible to {find a point with high} Tukey-depth in iterations over the axes, and show that the {appropriate functions are} indeed quasi-concave. This allows us to instantiate the algorithm of \cite{BNS13b} to identify a point in the convex hull of the dataset one coordinate at a time. We obtain the following theorem.
\begin{theorem}[Informal]
Let $ \varepsilon \leq 1$ and $\delta < 1/2$ and let $X \subset\R^d$. There exists an $(\varepsilon,\delta)$-differentially private algorithm that given a dataset $S\in X^m$ identifies (w.h.p.) a point in the convex hull of $S$, provided that
$m=|S|=\mathop{\rm{poly}}\nolimits\left(d,2^{\log^*|X|},\frac{1}{\varepsilon},\log\frac{1}{\delta}\right)$.
\end{theorem}
In fact, our algorithm returns a point with a large Tukey-depth, which is in particular a point in the convex hull of the dataset. This fact will be utilized by our reduction from learning halfspaces, and will allow us to get improved sample complexity bounds on privately learning halfspaces.
\paragraph{A privacy preserving reduction from halfspaces to convex hull.}
Our reduction can be thought of as a generalization of the results by \cite{BNSV15}, who showed that the task of privately learning {\em one-dimensional} thresholds is equivalent to the task of privately solving the {\em interior point problem}. In this problem, given a set of input numbers, the task is to identify a number between the minimal and the maximal input numbers. Indeed, this is exactly the one dimensional version of the convex-hull problem we consider. However, the reduction of \cite{BNSV15} does not apply for halfspaces, and we needed to design a different reduction.
Our reduction is based on the sample and aggregate paradigm:
assume a differentially private algorithm~$\mathcal A$ which gets a (sufficiently large) dataset $D\subset \R^d$ and returns a point in the convex hull of $D$.
This can be used to privately learn halfspaces as follows.
Given an input sample $S$, partition it to sufficiently many subsamples $S_1,\dots,S_k$,
and pick for each $S_i$ an arbitrary halfspace $h_i$ which is consistent with $S_i$. Next, apply $\mathcal A$ to privately find a point in the convex hull of the $h_i$'s (to this end represent each $h_i$ as a point in~$\R^{d+1}$ via its normal vector and bias), and output the halfspace $h$ corresponding to the returned point. It can be shown that if each of the $h_i$'s has a sufficiently low generalization error, which is true if the sample is big enough, then the resulting (privately computed) halfspace also has a low generalization error.
Instantiating this reduction with our algorithm for the convex hull we get the following theorem.
\begin{theorem}[Informal]\label{thm:introHalfspaces}
Let $ \varepsilon \leq 1$ and $\delta < 1/2$ and let $X \subset\R^d$. There exists an $(\varepsilon,\delta)$-differentially private $(\alpha,\beta)$-PAC learner for halfspaces over examples from $X$ with sample complexity
$m=\mathop{\rm{poly}}\nolimits\left(d,2^{\log^*|X|},\frac{1}{\alpha\varepsilon},\log\frac{1}{\beta\delta}\right)$.
\end{theorem}
In particular, for any constant $d$, \cref{thm:introHalfspaces} gives a private learner for halfspaces over $X\subseteq\R^d$ with sample complexity $2^{O(\log^*|X|)}$. Before our work, this was known only for $d=1$.
\paragraph{A lower bound for finding a point in the convex hull.}
Without privacy considerations, finding a point in the convex hull of the data is trivial. Nevertheless, we show that any $(\varepsilon,\delta)$-differentially private algorithm for this task (in $d$ dimensions) must have sample complexity $m=\Omega(\frac{d}{\varepsilon}\log\frac{1}{\delta}+{\log^* {\lvert X\rvert}})$. {In comparison, our algorithm requires sample of size at least
$\tilde{O}(d^{2.5}2^{O(\log^*{\lvert X\rvert})}/\epsilon)$ (ignoring the dependency on $\delta$ and $\beta$).}
Recall that the sample complexity of privately learning a class $C$ is always at most $O(\log|C|)$. Hence, it might be tempting to guess that a sample complexity of $m=O(\log|X|)$ should suffice for privately finding a point in the convex hull of a dataset $S\subseteq X\subseteq\R^d$, even with pure $(\varepsilon,0)$-differential privacy. We show that this is not the case, and that any pure $(\varepsilon,0)$-differentially private algorithm for this task must have sample complexity $m=\Omega(\frac{d}{\varepsilon}\log|X|)$.
\subsection{Other Related Work}\label{sec:otherRelated}
Most related to our work is the work on private learning and its sample and time complexity by \cite{BDMN05, KLNRS11,BDMN05,BBKN12,CH11,BNS13,FX14,BNS13b,BNSV15,BZ16}. As some of these works demonstrate efficiency gaps between private and non-private learning, alternative models have been explored including semi-supervised learning (\cite{BNS15}), learning multiple concepts (\cite{BNS16}), and prediction (\cite{DworkF18}, \cite{BassilyTT18}).
\cite{DV08} showed an efficient (non-private) learner for halfspaces that works in (a variant of) the {\em statistical query (SQ)} model of \cite{Kearns98}. It is known that SQ learners can be transformed to preserve differential privacy \citep{BDMN05}, and the algorithm of \cite{DV08} yields a differentially private efficient learner for halfspaces over examples from $X\subseteq\R^d$ with sample complexity $\mathop{\rm{poly}}\nolimits(d,\log|X|)$.
Another related work is that of \cite{HsuRRU14} who constructed an algorithm for {\em approximately} solving linear programs with differential privacy. While learning halfspaces {\em non-privately} easily reduces to solving linear programs, it is not clear whether the results of \cite{HsuRRU14} imply a {\em private} learner for halfspaces (due to the types of errors they incur).
\section{Preliminaries}
In this section we introduce a tool that enables our constructions, describe the geometric object we use throughout the paper, and present some of their properties.
\paragraph{Notations.}
The input of our algorithm is a multiset $S$ whose elements are taken (possibly with repetition) from a set $X$. We will abuse notation and write that $S \subseteq X$. Databases $S_1$ and $S_2$ are called {\em neighboring} if they differ in exactly one entry. Throughout this paper we use $\varepsilon$ and $\delta$ for the privacy parameters, $\alpha$ for the error parameter, and $\beta$ for the confidence parameter, and $m$ for the sample size.
In this appendix we define differentially private algorithms and the PAC learning model.
\subsection{Preliminaries from Differential Privacy}
Consider a database where each record contains information of an individual. An algorithm is said to preserve differential privacy if a change of a single record of the database (i.e., information of an individual) does not significantly change the output distribution of the algorithm. Intuitively, this means that the information infer about an individual from the output of a differentially-private algorithm is similar to the information that would be inferred had the individual's record been arbitrarily modified or removed. Formally:
\begin{definition}[Differential privacy~\citep{DMNS06,DKMMN06}] \label{def:dp}
A randomized algorithm $\mathcal A$ is $(\varepsilon,\delta)$-differentially private if for all neighboring databases $S_1,S_2\in X^m$, and for all sets $\mathcal{F}$ of outputs,
\begin{eqnarray}
\label{eqn:diffPrivDef}
& \Pr[\mathcal A(S_1) \in \mathcal{F}] \leq \exp(\varepsilon) \cdot \Pr[\mathcal A(S_2) \in \mathcal{F}] + \delta, &
\end{eqnarray}
where the probability is taken over the random coins of $\mathcal A$.
When $\delta=0$ we omit it and say that $\mathcal A$ preserves $\varepsilon$-differential privacy.
\end{definition}
We use the term {\em pure} differential privacy when $\delta=0$ and the term {\em approximate} differential privacy when $\delta>0$, in which case $\delta$ is typically a negligible function of the database size $m$.
We will later present algorithms that access their input database using (several) differentially private algorithms. We will use the following composition theorems.
\begin{theorem}[Basic composition]\label{thm:composition1}
If $\mathcal A_1$ and $\mathcal A_2$ satisfy $(\varepsilon_1,\delta_1)$ and $(\varepsilon_2,\delta_2)$ differential privacy, respectively, then their concatenation $\mathcal A(S)=\langle \mathcal A_1(S),\mathcal A_2(S) \rangle$ satisfies $(\varepsilon_1+\varepsilon_2,\delta_1+\delta_2)$-differential privacy.
\end{theorem}
Moreover, a similar theorem holds for the adaptive case, where an algorithm uses $k$ {\em adaptively chosen} differentially private algorithms (that is, when the choice of the next differentially private algorithm that is used depends on the outputs of the previous differentially private algorithms).
\begin{theorem}[\citep{DKMMN06, DworkLei}]\label{thm:composition3}
An algorithm that adaptively uses $k$ algorithms that preserves $(\varepsilon/k,\delta/k)$-differential privacy (and does not access the database otherwise) ensures $(\varepsilon,\delta)$-differential privacy.
\end{theorem}
Note that the privacy guaranties of the above bound deteriorates linearly with the number of interactions. By bounding the {\em expected} privacy loss in each interaction (as opposed to worst-case), \cite{DRV10} showed the following stronger composition theorem, where privacy deteriorates (roughly) as $\sqrt{k}\varepsilon+k\varepsilon^2$ (rather than $k\varepsilon$).
\begin{theorem}[Advanced composition~\cite{DRV10}, restated]\label{thm:composition2}
Let $0<\varepsilon_0,\delta'\leq1$, and let $\delta_0\in[0,1]$. An algorithm that adaptively uses $k$ algorithms that preserves $(\varepsilon_0,\delta_0)$-differential privacy (and does not access the database otherwise) ensures $(\varepsilon,\delta)$-differential privacy, where $\varepsilon=\sqrt{2k\ln(1/\delta')}\cdot\varepsilon_0+2k\varepsilon_0^2$ and $\delta = k\delta_0+\delta'$.
\end{theorem}
\subsection{Preliminaries from Learning Theory}
We next define the probably approximately correct (PAC) model of~\cite{Valiant84}.
A concept $c:X\rightarrow \{0,1\}$ is a predicate that labels {\em examples} taken from the domain $X$ by either 0 or 1. A \emph{concept class} $C$ over $X$ is a set of concepts (predicates) mapping $X$ to $\{0,1\}$. A learning algorithm is given examples sampled according to an unknown probability distribution $\mathcal D$ over $X$, and labeled according to an unknown {\em target} concept $c\in C$. The learning algorithm is successful when it outputs a hypothesis $h$ that approximates the target concept over samples from $\mathcal D$. More formally:
\begin{definition}
The {\em generalization error} of a hypothesis $h:X\rightarrow\{0,1\}$ is defined as
$${\rm error}_{\mathcal D}(c,h)=\Pr_{x \sim \mathcal D}[h(x)\neq c(x)].$$
If ${\rm error}_{\mathcal D}(c,h)\leq\alpha$ we say that $h$ is {\em $\alpha$-good} for $c$ and $\mathcal D$.
\end{definition}
\begin{definition}[PAC Learning~\citep{Valiant84}]\label{def:PAC}
Algorithm $\mathcal A$ is an {\em $(\alpha,\beta,m)$-PAC learner} for a concept
class $C$ over $X$ using hypothesis class $H$ if for all
concepts $c \in C$, all distributions $\mathcal D$ on $X$,
given an input of $m$ samples $S =(z_1,\ldots,z_m)$, where $z_i=(x_i,c(x_i))$ and each $x_i$
is drawn i.i.d.\ from $\mathcal D$, algorithm $\mathcal A$ outputs a
hypothesis $h\in H$ satisfying
$$\Pr[{\rm error}_{\mathcal D}(c,h) \leq \alpha] \geq 1-\beta,$$
where the probability is taken over the random choice of
the examples in $S$ according to $\mathcal D$ and the random coins of the learner $\mathcal A$.
If $H\subseteq C$ then $\mathcal A$ is called a {\em proper} PAC learner; otherwise, it is called an {\em improper} PAC learner.
\end{definition}
\begin{definition}
For a labeled sample $S=(x_i,y_i)_{i=1}^m$, the {\em empirical error} of $h$ is
$${\rm error}_S(h) = \frac{1}{m} |\{i : h(x_i) \neq y_i\}|.$$
\end{definition}
\subsection{Private Learning}\label{sec:PPAC}
Consider a learning algorithm $\mathcal A$ in the probably approximately correct (PAC) model of~\cite{Valiant84}. We say that $\mathcal A$ is a {\em private} learner if it also satisfies differential privacy w.r.t.\ its training data. Formally,
\begin{definition}[Private PAC Learning~\citep{KLNRS11}]
Let $\mathcal A$ be an algorithm that gets an input $S =(z_1,\ldots,z_m)${, where each $z_i$ is a labeled example}. Algorithm $\mathcal A$ is an {\em $(\varepsilon,\delta)$-differentially private $(\alpha,\beta)$-PAC learner with sample complexity $m$} for a concept
class $C$ over $X$ using hypothesis class $H$ if
\begin{description}
\item{\sc Privacy.} Algorithm $\mathcal A$ is $(\varepsilon,\delta)$-differentially private (as in \cref{def:dp});
\item{\sc Utility.} Algorithm $\mathcal A$ is an {\em $(\alpha,\beta)$-PAC learner} for $C$ with sample complexity $m$ using hypothesis class $H$ {(as in \cref{def:PAC})}.
\end{description}
\end{definition}
Note that the utility requirement in the above definition is an average-case requirement, as the learner is only required to do well on typical samples (i.e., samples drawn i.i.d. from a distribution $\mathcal D$ and correctly labeled by a target concept $c\in C$). In contrast, the privacy requirement is a worst-case requirement, {which} must hold for every pair of neighboring databases (no matter how they were generated, even if they are not consistent with any concept in $C$).
\subsection{A Private Algorithm for Optimizing Quasi-concave Functions -- $\mathcal A_{\rm RecConcave}$}
We next describe properties of an algorithm $\mathcal A_{\rm RecConcave}$ of \cite{BNS16a}. This algorithm
is given a quasi-concave function $Q$ (defined below) and privately finds a point $x$ such that $Q(x)$ is close to its maximum provided that the maximum of $Q(x)$ is large enough
(see~(\ref{eq:largeQ})).
\begin{definition}\label{def:quasiConcave}
A function $Q(\cdot)$ is quasi-concave if $Q(\ell) \geq \min\set{Q(i),Q(j)}$ for every $i < \ell < j$.
\end{definition}
\begin{definition}[Sensitivity] The sensitivity of a function $f : X^m \rightarrow \R$ is the smallest $k$ such that for
every neighboring $D,D' \in X^m$, we have $|f(D)-f(D')
| \leq k$.
\end{definition}
\begin{proposition}[Properties of Algorithm $\mathcal A_{\rm RecConcave}$~\citep{BNS16a}]\label{prop:aRecConcave}
Let $Q:X^*\times\tilde{X}\rightarrow\R$ be a sensitivity-1 function (that is, for every $x \in \tilde{X}$, the function
$Q(\cdot,x)$ has sensitivity $1$).
Denote $\tilde{T}=|\tilde{X}|$ and let $\alpha\leq\frac{1}{2}$ and $\beta,\varepsilon,\delta,r$ be parameters.
There exits an $(\varepsilon,\delta)$-differentially private algorithm, called $\mathcal A_{\rm RecConcave}$, such that the following holds.
If $\mathcal A_{\rm RecConcave}$ is executed on a database $S\in X^*$ such that $Q(S,\cdot)$ is quasi-concave and in addition
\begin{equation}
\label{eq:largeQ}
\max_{i\in\tilde{X}}\{Q(S,i)\} \geq r \geq
8^{\log^* \tilde{T}} \cdot \frac{12 \log^* \tilde{T}}{\alpha\varepsilon}\log\Big(\frac{192(\log^* \tilde{T})^2}{\beta\delta}\Big),
\end{equation}
then with probability at least $1-\beta$ the algorithm outputs an index $j$ s.t.\
$Q(S,j)\geq(1-\alpha)r$.
\end{proposition}
\remove{
We next present an alternative definition of quasi-concave functions that is more convenient to use.
\begin{observation}
Let $i_{\rm max}$ be a value such that $Q(i_{\rm max})$ is maximal.
A function $Q(·)$ is quasi-concave if and only if $Q(i) \leq Q(j)$ for all $i,j\in \tilde X$ such that $i < j < i_{\rm max}$ or $i_{\rm max} < j < i$.
\begin{enumerate}
\item
The function is non-decreasing before $i_{\rm max}$, that is
$Q(i) \leq Q(j)$ for every $i,j\in \tilde{X}$ such that $i < j < i_{\rm max}$, {\em and}
\item
The function is non-increasing after $i_{\rm max}$, that is
$Q(i) \geq Q(j)$ for every $i,j\in \tilde{X}$ such that $i_{\rm max} < i < j$.
\end{enumerate}
\end{observation}
}
\begin{claim}\label{c:minquasiconcave}
Let $\{f_t\}_{t\in\mathcal{T}}$ be a finite family of quasi-concave functions. Then, $f(x)=\min_{t\in \mathcal{T}}f_t(x)$ is also quasi-concave.
\end{claim}
\begin{proof}
Let $i \leq \ell \leq j$. Then,
\begin{align*}
f(\ell) &=\min_{t\in \mathcal{T}}f_t(\ell)\\
&\geq \min_{t\in\mathcal{T}}\{\min\{f_t(i),f_t(j)\}\} \tag{$f_t$ is quasi-concave, $\forall t\in\mathcal{T}$}\\
&= \min\{\min_{t\in \mathcal{T}}f_t(i),\min_{t\in \mathcal{T}} f_t(j)\}\\
&=\min\{f(i),f(j)\}.
\end{align*}
\end{proof}
\subsection{Halfspaces, Convex Hull, and Tukey Depth}\label{def:Tukey}
We next define the geometric objects we use in this paper.
\begin{definition}[Halfspaces and Hyperplanes]
Let $X \subset \R^d$.
For $a_1,\dots,a_d,w\in \R$, let the halfspace
$\operatorname{\rm hs}_{a_1,\dots,a_d,w}:X \rightarrow\{0,1\}$
be defined as $\operatorname{\rm hs}_{a_1,\dots,a_d,w}(x_1,\dots,x_d)=1$ if and only if $\sum_{i=1}^d a_i x_i \geq w$. Define the concept class $\operatorname{\tt HALFSPACE}(X) = \{\operatorname{\rm hs}_{a_1,\dots,a_d,w}\}_{a_1,\dots,a_d,w \in \R}$.
We say that a halfspace $\operatorname{\rm hs}$ contains a point $\pt{x} \in \R^d$ if $\operatorname{\rm hs}(\pt{x})=1$.
The hyperplane $\operatorname{\rm hp}_{a_1,\dots,a_d,w}$ defined by $a_1,\dots,a_d,w$ is the set of all points $\pt{x}=(x_1,\dots,x_d)$ such that $\sum_{i=1}^d a_i x_i = w$.
\end{definition}
\begin{definition}
Let $S\subset \R^d$ be a finite multiset of points. A point $\pt{x} \in \R^d$ is in the convex hull of $S$ if $\pt{x}$ is a convex combination of the elements of $S$, that is, there exists non-negative
numbers $\set{a_{\pt{y}}}_{\pt{y} \in S}$ such that
$\sum_{\pt{y} \in S} a_\pt{y} =1$ and
$\sum_{\pt{y} \in S} a_\pt{y} \pt{y}=\pt{x}$.
\end{definition}
We next define the Tukey median of a point, which is a generalization of a median to $\R^d$.
\begin{definition}[Tukey depth~\citep{Tuk75}]
Let $S\subset \R^d$ be a finite {multiset}
of points. The Tukey depth of a point $\pt{x} \in \R^d$ with respect to $S$, denoted by $\operatorname{\rm td}(\pt{x})$, is the minimum number of points in $S$ contained in a halfspace containing the point $x$,
that is,
$$\operatorname{\rm td}(\pt{x})=\min_{\operatorname{\rm hs} \in \operatorname{\tt HALFSPACE}_{d,T} ,\operatorname{\rm hs}(\pt{x})=1}|\set{\pt{y}\in S:\operatorname{\rm hs}(\pt{y})=1}|.$$
The Tukey median of $S$ is a point maximizing the Tukey depth. A centerpoint is a point of depth at least~$|S|/(d + 1)$.
\end{definition}
\begin{observation}
\label{obs:Tukey}
The Tukey depth of a point is a sensitivity one function of the
multiset $S$.
\end{observation}
\remove{
\knote{this definition is not clear to me} We say that the direction $(a_1,\ldots,a_d)$
is in {\it general position} with respect to $S$
if the numbers $\sum_i a_iy_i$ are distinct
when $(a_1\ldots a_d)$ ranges over all points in $S$. \knote{should this be ``$(y_1\ldots y_d)$ ranges over all points in $S$''??}
\shay{The definition says that no two vectors in $S$ have the same inner product with $(a_1,\ldots, a_d)$. Would it be clearer if stated this way?
I needed this definition in order to fix the sensitivity argument by Amos, but Uri wanted to use an alternative argument which does not require this definition and then we can remove it.}
Note that since $S$ is finite, a standard perturbation argument implies that\knote{this argument holds because the set $S$ is finite, right? if correct, let's mention this.}\shay{added.}
every halfspace $\operatorname{\rm hs}$ can be written as
$\operatorname{\rm hs}_{a_1,\ldots,a_d,w}$ where the direction
$(a_1,\ldots,a_d)$ is in general position with respect to $S$.
Let $\mathsf{GP}(S)\subseteq\mathbb{R}^d$ denote the set of all directions
that are in general position with respect to $S$.
}
\begin{claim}[Tukey depth, alternative definition]\label{c:tukeyalt}
Let $S\subset \R^d$ be a multiset of points.
For a given $a_1,\dots,a_d\in \R$ define the function
\begin{equation}
\label{eq:tdef}
t_{a_1,\dots,a_d}(w)\triangleq\min\set{\left|\{(y_1,\dots,y_d)\in S: \sum_{i=1}^d a_i y_i \geq w\}\right|,\left|\{(y_1,\dots,y_d)\in S: \sum_{i=1}^d a_i y_i \leq w\}\right|}.
\end{equation}
Then,
\begin{equation}
\label{eq:td}
\operatorname{\rm td}(x_1,\dots,x_n)=\min_{(a_1,\dots,a_d)\in\R}t_{a_1,\dots,a_d}\left(\sum_{i=1}^d a_i x_i \right).
\end{equation}
\end{claim}
\begin{claim}[Tukey depth, another alternative definition]\label{c:tukeyalt2}
Let $S\subset \R^d$ be a multiset of points.
The Tukey-depth of a point $\pt{x}$ is the size of the smallest set $A \subseteq S$ such that $\pt{x}$ is not in the convex-hull of $S \setminus A$.
\end{claim}
\begin{claim}[\cite{Yaglom61book,Edelsbrunner87book}]\label{c:centerpoint}
Let $S\subset \R^d$ be a multiset of points.
There exists $\pt{x} \in \R^d$ such that $\operatorname{\rm td}(\pt{x}) \geq |S|/(d + 1)$.
\end{claim}
Thus, a centerpoint always exists and a Tukey median must be a centerpoint. However, not every centerpoint is a Tukey median.
We will use the following regarding the set of points of all points whose Tukey depth is at least $r$.
\begin{fact}[see e.g.~\cite{Xiaohui14Tukey}]\label{fact:counting}
Let $S\subseteq \R^d$ be a multiset of points and $r > 0$.
Define $\mathcal T(r) = \{\pt{x}\in\mathbb{R}^d : \operatorname{\rm td}(\pt{x}) \geq r\}$.
Then $\mathcal T(r)$ is a polytope whose faces are supported by affine subspaces
that are spanned by points from~$S$.
\end{fact}
So, for example the set of all Tukey medians is a polytope and if it is $d$-dimensional then each of its facet is
supported by a hyperplane that passes through $d+1$ points from $S$.
\section{Finding a Point in the Convex Hull}
\label{sec:FindingPoint}
Our goal is to privately find a point in the convex hull of a set of input points (i.e., the database). We will actually achieve a stronger task and find a point whose Tukey depth is at least $|S|/2(d+1)$ (provided that $|S|$ is large enough). Observe that $\pt{x}$ is in the convex hull of $S$ if and only if $\operatorname{\rm td}(\pt{x}) >0$. As we mentioned in the introduction, finding a point whose Tukey depth is high results in a better learning algorithms for halfspaces.
The idea of our algorithm is to find the point $\pt{x}=(x_1,\dots,x_d)$ coordinate after coordinate: we
use $\mathcal A_{\rm RecConcave}$ to find a value $x^*_1$ that can be extended by some $x_2,\ldots,x_d$ so that the depth of $(x_1^*,x_2\ldots,x_d)$ is close to the depth of the Tukey median, then we find a value $x_2^*$ so that there is a point $(x^*_1,x_2^*,x_3\dots,x_d)$ whose depth is close to the depth of the Tukey median, and so forth until we find all coordinates. The parameters in $\mathcal A_{\rm RecConcave}$ are set such that in each step we lose depth of at most $n/2(d+1)^2$ compared to the Tukey median, resulting in a point $(x^*_1,\dots,x^*_d)$ whose depth is at most $n/2(d+1)$ less than the depth of the Tukey median, i.e., its depth is at least $n/2(d+1)$.
\subsection{Defining a Quasi-Concave Function}
To apply the above approach, we need to prove that the functions considered in the algorithm $\mathcal A_{\rm RecConcave}$
are quasi-concave.
\begin{definition}
For every $1 \leq i \leq d$ and every $x^*_1,\dots,x^*_{i-1} \in \R$, define
$$Q_{x^*_1,\dots,x^*_{i-1}}(x_i)\triangleq \max_{x_{i+1},\dots,x_d \in \R} \operatorname{\rm td} (x^*_1,\dots,x^*_{i-1}, x_i,\dots,x_d).$$
\end{definition}
We next prove that $Q_{x^*_1,\dots,x^*_{i-1}}(x_i)$ is quasi-concave. Towards this goal,
we first prove that the function~$t_{a_1,\dots,a_d}(w)$, defined in \cref{eq:tdef}, is quasi-concave.
\begin{claim}
\label{cl:t-concave}
For every $a_1,\dots,a_{d} \in \R$, the function
$t_{a_1,\dots,a_{d}}(w)$ is quasi-concave.
\end{claim}
\begin{proof}
Define
$
f_1(w) = \bigl\lvert\{(y_1,\dots,y_d)\in S: \sum_{i=1}^d a_i y_i \geq w\}\bigr\rvert,$ and
$f_2(w) = \bigl\lvert\{(y_1,\dots,y_d)\in S: \sum_{i=1}^d a_i y_i \leq w\}\bigr\rvert$. Note that $t_{a_1,\dots,a_{d}}(w) = \min\{{f_1(w),f_2(w)}\}$.
These functions count the number of points in $S$ on the two (closed) sides of the hyperplane $\operatorname{\rm hp}_{a_1\ldots a_d,w}$.
The claim follows by Claim~\ref{c:minquasiconcave} since both $f_1,f_2$ are quasi-concave
(in fact, both are monotone).
\end{proof}
We next prove that the restriction of the Tukey depth function to a line is quasi-concave.
This lemma is implied by Fact~\ref{fact:counting} (implying that the set of points whose Tukey depth is at least $r$ is convex). For completeness, we supply a full proof of the claim.
\begin{claim}
\label{cl:tdl-concave}
Fix $\alpha_1,\dots,\alpha_d,\beta_1,\dots,\beta_d \in \R$, and define
$\operatorname{\rm tdl}_{\alpha_1,\dots,\alpha_d,\beta_1,\dots,\beta_d }(t)=\operatorname{\rm td}(\alpha_1 t+\beta_1,\dots,\alpha_d t+\beta_d )$.
The function
$\operatorname{\rm tdl}_{\alpha_1,\dots,\alpha_d,\beta_1,\dots,\beta_d }$ is quasi-concave.
\end{claim}
\begin{proof}
Let $t_0 < t_1 < t_2$ and let $(a_1,\dots,a_d)$ be a direction that minimizes $\operatorname{\rm td}(\alpha_1 t_1+\beta_1,\dots,\alpha_d t_1+\beta_d )$ in~(\ref{eq:td}),
i.e.,
$\operatorname{\rm tdl}_{\alpha_1,\dots,\alpha_d,\beta_1,\dots,\beta_d }(t_1)=\operatorname{\rm td}(\alpha_1 t_1+\beta_1,\dots,\alpha_d t_1+\beta_d )=t_{a_1,\dots,a_d}\left(\sum_{i=1}^d a_i (\alpha_i t_1+\beta_i) \right).$
Consider the function which maps $t$ to $\sum_{i=1}^d a_i (\alpha_i t +\beta_i)= (\sum_{i=1}^d a_i \alpha_i)t+\sum_{i=1}^d a_i \beta_i$. This function is either increasing or decreasing, thus, by Claim~\ref{cl:t-concave},
\begin{eqnarray*}
\operatorname{\rm tdl}_{\alpha_1,\dots,\alpha_d,\beta_1,\dots,\beta_d }(t_1) & = &t_{a_1,\dots,a_d}\left(\sum_{i=1}^d a_i (\alpha_i t_1+\beta_i) \right)\\
& \geq &
\min\set{t_{a_1,\dots,a_d}\left(\sum_{i=1}^d a_i (\alpha_i t_0 +\beta_i) \right),
t_{a_1,\dots,a_d}\left(\sum_{i=1}^d a_i (\alpha_i t_2+\beta_i) \right)} \\
& \geq &
\min\set{\operatorname{\rm td}( \alpha_1 t_0 +\beta_1,\dots, \alpha_d t_0 +\beta_d),
\operatorname{\rm td}( \alpha_1 t_2 +\beta_1,\dots, \alpha_d t_2 +\beta_d)} \\
& = &
\min\set{\operatorname{\rm tdl}_{\alpha_1,\dots,\alpha_d,\beta_1,\dots,\beta_d }(t_0),\operatorname{\rm tdl}_{\alpha_1,\dots,\alpha_d,\beta_1,\dots,\beta_d }(t_1)}.
\end{eqnarray*}
\end{proof}
\begin{lemma}
For every $1 \leq i \leq d$ and every $x^*_1,\dots,x^*_{i-1} \in \R$, the function
$Q_{x^*_1,\dots,x^*_{i-1}}(x_i)$ is a quasi-concave function. Furthermore, $Q_{x^*_1,\dots,x^*_{i-1}}(x_i)$ is a sensitivity $1$ function of the multiset $S$.
\end{lemma}
\begin{proof}
Let $x_i^{0},x_i^1,x_i^2$ such that $x_i^0 < x_i^1 < x_i^2$.
Furthermore, let $x_{i+1}^{0},\dots,x_{d}^{0}$ and $x_{i+1}^{2},\dots,x_{d}^{2}$ be points
maximizing the functions $\operatorname{\rm td} (x^*_1,\dots,x^*_{i-1},x_i^0,\cdot,\dots,\cdot)$ and $\operatorname{\rm td} (x^*_1,\dots,x^*_{i-1},x_i^2,\cdot,\dots,\cdot)$ respectively, that is,
$Q_{x^*_1,\dots,x^*_{i-1}}(x_i^b)=\operatorname{\rm td}(x^*_1,\dots,x^*_{i-1},x_i^b,x_{i+1}^{b},\dots,x_{d}^{b})$ for $b\in\{0,2\}$.
Consider the line $L:\R\rightarrow \R^d$ passing through the points $(x^*_1,\dots,x^*_{i-1},x_i^b,x_{i+1}^{b},\dots,x_{d}^{b})$
for $b\in\{0,2\},$
and scale its parameter such that $L(x_i^b)=(x^*_1,\dots,x^*_{i-1},x_i^b,x_{i+1}^{b},\dots,x_{d}^{b})$.
\remove{
That is, let
$$\alpha_j=\left\{
\begin{array}{ll}
0 & \text{\rm if } 1\leq j \leq i-1 \\
1 & \text{\rm if } j = i \\
(x_j^0-x_j^2)/(x_i^0-x_i^2) & \text{\rm if } i+1 \leq j \leq d
\end{array}
\right. $$
and
$$\beta_j=\left\{
\begin{array}{ll}
x^*_j & \text{\rm if } 1\leq j \leq i-1 \\
0 & \text{\rm if } j = i \\
(x_i^0 x_j^2-x_i^2x_j^0)/(x_i^0-x_i^2) & \text{\rm if } i+1 \leq j \leq d
\end{array}
\right. $$
and define $L(t)=(\alpha_1 t+\beta_1,\dots,\alpha_dt+\beta_d)$.
}
{In particular, for every $x\in \R$ the $i$'th coordinate in $L(x)$ is $x$.}
By Claim~\ref{cl:tdl-concave} and the definition of $Q_{x^*_1,\dots,x^*_{i-1}}$,
$$Q_{x^*_1,\dots,x^*_{i-1}}(x_i^1) \geq \operatorname{\rm td}(L(x_i^1))
\geq \min\set{ \operatorname{\rm td}(L(x_i^0)), \operatorname{\rm td}(L(x_i^2))}
=\set{ Q_{x^*_1,\dots,x^*_{i-1}}(x_i^0), Q_{x^*_1,\dots,x^*_{i-1}}(x_i^2)}.$$
The fact that $Q_{x^*_1,\dots,x^*_{i-1}}$ has sensitivity $1$ is implied by Observation~\ref{obs:Tukey} and the fact that maximum of sensitivity 1 functions is a sensitivity 1 function.
\end{proof}
\subsection{Extending the Domain}
The input to the private algorithm for finding a point in the convex hull is a dataset of points
$S \subseteq X$, where $X$ is a finite set whose size is at most $T$.
We note that the dataset $S$ may contain several copies of the same point (i.e.\ it is a multiset).
By the results of \cite{BNSV15}, the restriction to subsets of a finite set $X$ is essential (even when $d=1$).
Notice that a Tukey median of $S$ might not be a point in $X$.
Furthermore, the proof that the functions $Q_{x^*_1,\dots,x^*_{i-1}}$ are quasi-concave is over the reals.
Therefore, we extend the domain to $\tilde{X}=\prod_{i=1}^d\tilde{X_i}$
such that for every dataset $S$ the functions $Q_{x^*_1,\dots,x^*_{i-1}}$ attain their maximum over the extended domain.
We will not try to optimize the size of $\tilde{X_1}, \ldots, \tilde{X_d}$
as the dependency of the sample complexity of~$\mathcal A_{\rm RecConcave}$ on $|\tilde{X_i}|$ is~$2^{O(\log^* |\tilde{X_i}|)}$.
\begin{claim}
\label{cl:sets}
There exists sets $\tilde{X_1}, \ldots, \tilde{X_d}$ such that
$|\tilde{X_i}| \leq (dT^{d^2(d+1)})^{2^d}$ for $1 \leq i \leq d$ and
for every dataset $S$,
for every $1 \leq i \leq d$, and for every $x^*_1,\dots,x^*_{i-1}\in \tilde{X_1}\times \dots \times \tilde{X_{i-1}}$, there exist
$x^{m}_i,\dots,x^{m}_{d}\in \tilde{X_i} \times \ldots \times \tilde{X_{d}}$ such that
\begin{equation}
\label{eq:ExistsMax}
\max_{x_{i+1},\dots,x_d \in \R} \operatorname{\rm td} (x^*_1,\dots,x^*_{i-1}, x_i,\dots,x_d)
=\operatorname{\rm td} (x^*_1,\dots,x^*_{i-1}, x^m_i,\dots,x^m_d).
\end{equation}
\end{claim}
\begin{proof}
For $1\leq i \leq d$, let $X_i$ be the projection of $X$ to the $i$th coordinate, that is, $$X_i=\set{x: \exists_{x_1,\dots,x_{i-1},x_{i+1},\dots,x_d} (x_1,\dots,x_{i-1},x,x_{i+1},\dots,x_d) \in X }.$$
The construction heavily exploits \cref{fact:counting}.
Let $L$ denote the set of all affine subspaces that are spanned by points in $X_1\times\cdots\times X_d$.
Since each such subspace is spanned by at most $d+1$ points, it follows that~$\lvert L \rvert \leq {T^d \choose d+1}\leq T^{d(d+1)}$.
By \cref{fact:counting}, for every dataset $S$ and every $r>0$,
every vertex of $\mathcal T(r)$ can be written as the intersection of at most~$d$ subspaces in~$L$.
In particular, there exists a Tukey median that is the intersection of at most~$d$ subspaces in~$L$.
We construct the sets in iterations where
we start with $\tilde{X_{j}}=\emptyset$ for every $1 \leq j \leq d$.
In iteration $i$ we do the following: for every $x^*_1,\dots,x^*_{i-1}\in \tilde{X}_1\times \dots \times \tilde{X}_{i-1}$
and for every $d-i$ subspaces in $L$ such that there exists a unique point $x = (x^*_1\ldots x^*_{i-1},x_i\ldots,x_d)$
in the intersection of these $d-i$ subspaces, we add $x_j$ to $\tilde X_j$ for all $j\geq i$.
We next argue that item (ii) in the conclusion of the claim is satisfied:
indeed, by \cref{fact:counting}, this construction contains a vertex of every set of the form
$\mathcal T(r) \cap \{x\in\R^d: x_1 = x^*_1,\ldots x_{i-1}=x^*_{i-1}\}$,
for every $r>0$, $i\leq d$, and every $(x^*_1,\ldots,x^*_{i-1})\in \tilde X_1\times\ldots\times \tilde X_{i-1}$.
In particular, by plugging
\[r= \max_{x_{i+1},\dots,x_d \in \R} \operatorname{\rm td} (x^*_1,\dots,x^*_{i-1}, x_i,\dots,x_d),\]
it contains a point which satisfies \cref{eq:ExistsMax}.
This implies item (ii).
As for item (i), note that the size of $\tilde X_1$ is at most ${\lvert L\rvert \choose d}\leq T^{d^2(d+1)}$.
Similarly, for $i > 1$:
\[\lvert \tilde X_i\rvert\leq {\lvert L\rvert \choose d} + {\lvert L\rvert \choose d-1}\lvert \tilde X_1\rvert + \ldots
+ {\lvert L\rvert \choose d-i}\prod_{j=1}^{i-1}\lvert \tilde X_j\rvert \leq \Bigl( d\cdot {\lvert L\rvert \choose d}\Bigr)^{2^d}\leq (dT^{d^2(d+1)})^{2^d}.\]
\end{proof}
\subsection{The Algorithm}
In \cref{fig:Tukey}, we present an $(\varepsilon,\delta)$-differentially private algorithm $\mathcal A_{\rm FindTukey}$ that with probability at least $1-\beta$ finds a point whose Tukey depth is at least $n/2(d+1)$. The informal description of the algorithm appears in the beginning of \cref{sec:FindingPoint}.
\begin{figure}[thb!]
\begin{center}
\noindent\fbox{
\parbox{.95\columnwidth}{
\begin{center}{ \bf Algorithm $\mathcal A_{\rm FindTukey}$}\end{center}
{\bf Preprocessing:}
\begin{itemize}
\item Construct the sets $\tilde{X}_1,\ldots,\tilde{X}_d$ as in Claim~\ref{cl:sets}. Let $\tilde{T}=\max_{1\leq i \leq d} |\tilde{X_i}|$.
\\$(*$ By Claim~\ref{cl:sets}, $\log^* \tilde{T}=\log^* d +\log^*T +O(1)$. $*)$
\end{itemize}
{\bf Algorithm:}
\begin{itemize}
\item [(i)] Let $\beta,\varepsilon,\delta$ be the utility/privacy parameters, and $S$ be an input database from~$X$.
\item [(ii)]
For $i=1$ to $d$ do:
\begin{itemize}
\item [(a)]
For every $x_i \in \tilde{X}_i$ define $$Q_{x^*_1,\dots,x^*_{i-1}}(x_i)\triangleq \max_{x_{i+1}\in \tilde{X}_{i+1},\dots,x_d \in \tilde{X}_d} \operatorname{\rm td} (x^*_1,\dots,x^*_{i-1}, x_i,\dots,x_d).$$
\item [(b)]
Execute $\mathcal A_{\rm RecConcave}$ on $S$ with the function $Q_{x^*_1,\dots,x^*_{i-1}}$ and parameters $r=\frac{n}{d+1}-\frac{(i-1)n}{d(d+1)}$, $\alpha_0=\frac{1}{2d},\beta_0=\frac{\beta}{d},\varepsilon_0=\frac{\varepsilon}{2\sqrt{2d\ln(2/\delta)}},\delta_0=\frac{\delta}{2d}$. Let $x^*_i$ be its output.
\end{itemize}
\item [(iii)]
Return $x^*_1,\dots,x^*_d$.\\
\end{itemize}
}}
\end{center}
\caption{Algorithm $\mathcal A_{\rm FindTukey}$ for finding a point whose Tukey depth is at least $n/2(d+1)$.\label{fig:Tukey}}
\end{figure}
\begin{theorem}\label{thm:privatecenterpoint}
Let $ \varepsilon \leq 1$ and $\delta < 1/2$ and $X \subset\R^d$ be a set of size at most $T$.
Assume that the input dataset $S\subseteq X$ satisfies
\[|S| =O\Biggl(d^{2.5}\cdot 2^{O(\log^*T +\log^*d)}\frac{\log^{0.5}\bigl(\frac{1}{\delta}\bigr) \log \bigl(\frac{d^2}{\beta\delta}\bigr)}{\varepsilon}\Biggr).\]
Then, $\mathcal A_{\rm FindTukey}$ is an $(\varepsilon,\delta)$-differentially private algorithm
that with probability at least $1-\beta$ returns a point $x^*_1,\dots,x^*_d$ such that
$\operatorname{\rm td}(x^*_1,\dots,x^*_d)\geq\frac{\lvert S\rvert}{2(d+1)}$.
\end{theorem}
\begin{proof}
The proof of the correctness (utility) of $\mathcal A_{\rm FindTukey}$ is proved by induction,
using the correctness of $\mathcal A_{\rm RecConcave}$. The privacy proof follows
from the privacy of $\mathcal A_{\rm RecConcave}$ and using the advanced composition theorem~(\cref{thm:composition2}).
\paragraph{Utility.}
We prove by induction that after step $i$ of the algorithm, with probability at least $1-i\beta/d$,
the returned values $x^*_1,\ldots,x^*_i$ satisfy
$Q_{x^*_1,\dots,x^*_{i-1}}(x^*_i)\geq \frac{\lvert S\rvert}{d+1}(1-\frac{i}{2d})$, i.e., there are $(x_{i+1},\ldots,x_d)\in \tilde{X}_{i+1}\times\dots\times\tilde{X}_{d}$
such that $\operatorname{\rm td}(x^*_1,\dots,x^*_{i},x_{i+1},\dots,x_d)\geq \frac{\lvert S\rvert}{d+1}(1-\frac{i}{2d})$.
The basis is the induction is $i=0$: by Claim~\ref{c:centerpoint} the Tukey median has depth at least~$\lvert S\rvert/(d+1)$
and by Claim~\ref{cl:sets}, the median is in $\tilde{X}_{1}\times \cdots \times\tilde{X}_{d}$.
Thus, with probability $1$ there are $(x_{1},\ldots,x_d)\in \tilde{X}_{1}\times\dots\times\tilde{X}_{d}$
such that $\operatorname{\rm td}(x_{1},\dots,x_d)\geq \frac{\lvert S\rvert}{d+1}$.
Next, by the induction hypothesis for $i-1$,
with probability at least $1-(i-1)\beta/d$ it holds that
\[\max_{x\in\tilde{X_i}}\{Q_{x^*_1,\dots,x^*_{i-1}}(x)\}\geq
\frac{\lvert S\rvert}{d+1}-\frac{(i-1)\lvert S\rvert}{2d(d+1)}=r > \frac{\lvert S\rvert}{2(d+1)}
\geq
8^{\log^* \tilde{T}} \cdot \frac{12 \log^* \tilde{T}}{\alpha_0\varepsilon_0}\log\Big(\frac{192(\log^* T)^2}{\beta_0\delta_0}\Big).
\]
Therefore, by \cref{prop:aRecConcave}, with probability at least
$(1-\beta/d)\bigl(1-(i-1)\beta/d\bigr)\geq 1 - i\beta/d$ Algorithm
$\mathcal A_{\rm RecConcave}$ returns $x^*_i\in \tilde{X}_i$ such that
$$Q_{x^*_1,\dots,x^*_{i-1}}(x^*_i)\geq (1-\alpha)r=\left(1-\frac{1}{2d}\right)\frac{\lvert S\rvert}{d+1}\left(1-\frac{i-1}{2d}\right)>\frac{\lvert S\rvert}{d+1}\left(1-\frac{i}{2d}\right).$$
To conclude, after $d$ steps of the algorithm, $\operatorname{\rm td}(x^*_1,\dots,x^*_d)\geq \frac{\lvert S\rvert}{2(d+1)}$ with probability at least $1-\beta$.
\paragraph{Privacy.}
By \cref{prop:aRecConcave}, each invocation of $\mathcal A_{\rm RecConcave}$ is
$(\varepsilon_0,\delta_0)$-differentially private. $\mathcal A_{\rm FindTukey}$
invokes $\mathcal A_{\rm RecConcave}$ $d$ times.
Thus, by \cref{thm:composition2} (the advanced composition) with $\delta'=\delta/2$,
it follows that $\mathcal A_{\rm FindTukey}$ is $(\frac{\varepsilon}{2}+\frac{\varepsilon^2}{4\ln (2/\delta)},\delta)$ differentially-private,
which implies $(\varepsilon,\delta)$-privacy whenever $\varepsilon \leq 1$ and~$\delta \leq 1/2$.
\end{proof}
\section{Learning Halfspaces Using Convex Hull}
We describe in \cref{fig:reduction} a reduction from learning halfspaces to finding a point in a convex-hull of a {multiset} of points. Furthermore, we show that if
the algorithm we use in the reduction finds a point whose Tukey depth is high (as our algorithm from \cref{sec:FindingPoint} does), then the required sample complexity of the learning algorithm is reduced. As a result, we get
an upper bound of ${\tilde{O}(d^{4.5}2^{\log^*|X| })}$ on the sample complexity of private learning halfspaces (ignoring the privacy and learning parameters).
In comparison, using the exponential mechanism of~\cite{MT07} results in an $(\varepsilon,\delta)$-deferentially private algorithm whose sample complexity is $O(d \log |X|)$, e.g., for the interesting case where $X=[T]^d$ for some $T$, the complexity is $O(d^2 \log T)$.
Our upper bound is better than the sample complexity of the exponential mechanism when $d$ is small compared to $\log |T|$, in particular when $d$ is constant.
\begin{figure}[htb!]
\begin{center}
\noindent\fbox{
\parbox{.95\columnwidth}{
\begin{center}{ \bf Algorithm $\mathcal A_{\rm LearnHalfSpace}$}\end{center}
{\bf Preprocessing:}
\begin{itemize}
\item Fix a set~$H\subseteq\R^{d+1}$ that contains representations of all halfspaces in $\operatorname{\tt HALFSPACE}(X)$, as in Claim~\ref{c:grid}.
\end{itemize}
{\bf Algorithm:}
\begin{enumerate}
\item
Let $\varepsilon,\delta,\alpha,\beta$ be the privacy and utility parameters and let $S$ be a realizable input sample of size~$s$,
where $s$ is as in \Cref{thm:convexhullreduction}.
\item \label{step:partition} Partition $S$ into $m$
equisized subsamples $S_1,\ldots,S_m$, where $m=m(d+1,2|X|^{d+1},\varepsilon,\delta,\beta/2)$ as in \cref{thm:convexhullreduction}.\\
$(*$ Note that each $S_i$ has size $\Theta\bigl(\frac{d\log(\frac{m}{r\alpha}) + \log(2m/\beta)}{r\alpha/m}\bigr)$. $*)$
\item For each $S_i$ pick a consistent halfpace $h_i\in H$ uniformly at random.
\item \label{step:Alg} Apply an $(\varepsilon,\delta)$-differentially private algorithm $\mathcal A$ for finding a point in a convex hull with parameters $\varepsilon,\delta,\frac{\beta}{2}$ on $H_0=(h_1\ldots h_m)$.
\item Output the halfspace $h$ found by $\mathcal A$.
\end{enumerate}
}}
\end{center}
\caption{\label{fig:reduction}A reduction from learning halfspaces to finding a point in a convex hull.}
\end{figure}
We start by showing the existence of a set $H$ that is used by the algorithm. We say that a vector $(a_1,\ldots ,a_d,w)\in\R^{d+1}$ {\it represents} a halfspace $\operatorname{\rm hs} \in \operatorname{\tt HALFSPACE}(X)$
if $\operatorname{\rm hs}(\pt{x})=\operatorname{\rm hs}_{a_1,\ldots, a_d,w}(\pt{x})$ for every $\pt{x} \in X$.
Note that every $\operatorname{\rm hs} \in \operatorname{\tt HALFSPACE}(X)$ has many representations.
\begin{claim}
\label{c:grid}
There exists a set $H \subseteq \R^{d+1}$, where $\lvert H\rvert \leq 2\lvert X \rvert^{d+1}$
which contains one representation of each halfspace $\operatorname{\rm hs}\in\operatorname{\tt HALFSPACE}(X)$.
\end{claim}
\begin{proof}
By standard
bounds from discrete geometry, $\lvert \operatorname{\tt HALFSPACE}(X) \rvert \leq 2\lvert X \rvert^{d+1}$ (see, e.g.~\cite{Gartner94vapnik}).
For each $\operatorname{\rm hs}\in \operatorname{\tt HALFSPACE}(X)$ pick a representation $(a_1\ldots a_d,w)\in \R^{d+1}$.
\end{proof}
\begin{theorem}\label{thm:convexhullreduction}
Assume that Algorithm $\mathcal A$ used in step \ref{step:Alg} of Algorithm $\mathcal A_{\rm LearnHalfSpace}$ is an $(\varepsilon,\delta)$-differentialy private algorithm that finds with probability at least $1-\beta$ a point in a convex hull for a multisets $S \subseteq X \subset \R^d$ whose Tukey depth is at least $r$ provided that
$|S| \geq m(d,|X|,\varepsilon,\delta,\beta)$ for some function $m(\cdot,\cdot,\cdot,\cdot,\cdot)$.
Let $\varepsilon\leq 1, \delta\leq \frac{1}{2}$ and $\alpha,\beta\leq 1$ be the privacy and utility parameters.
Then, $\mathcal A_{\rm LearnHalfSpace}$ is an $(\varepsilon,\delta)$-differentially private $(\alpha,\beta)$-PAC learner with sample complexity $s$ for the class $\operatorname{\tt HALFSPACE}(X)$ for
\[
s=O\Bigl( \frac{m^2\cdot d\log(\frac{m}{r\alpha}) + \log(m/\beta)}{r\alpha} \Bigr)\]
where $m=m(d+1,2|X|^{d+1},\varepsilon,\delta,\beta/2)$.
\end{theorem}
\begin{proof}
We first establish the privacy guarantee of the algorithm
and later argue that it PAC-learns $\operatorname{\tt HALFSPACE}(X)$.
\paragraph{Privacy.}
Let $S^1,S^2$ be two neighboring input samples of size at least $s$, where $s$ is as in the theorem statement.
Let $H_0(S^1),H_0(S^2)$ denote the list of halfspaces that are derived in step~\ref{step:partition} of the algorithm
when it is applied on $S^1,S^2$.
Since the $h_i$'s are constructed from mutually disjoint subsamples
it follows that the datasets $H_0(S_1)$ and $H_0(S_2)$ are neighbors.
The $(\varepsilon,\delta)$-privacy guarantee now follows from the privacy gurantee of $\mathcal A$
since the size of the $H_0(S^i)$'s is $m=m(d+1,2|X|^{d+1},\varepsilon,\delta,\beta/2)$.
\paragraph{Utility.}
We next establish that the algorithm learns $\operatorname{\tt HALFSPACE}(X)$ with confidence $1-\beta$ and error $\alpha$.
Let~$\mathcal D$ denote the target distribution and $c\in\operatorname{\tt HALFSPACE}(X)$ denote the target concept.
Let $S$ denote the input sample of size at least $s$ that is sampled independently from $\mathcal D$ and labeled by $c$.
We first claim that every halfspace $h_i\in H_0$ the probability $h_i$ has error greater than $\frac{r\alpha}{m}$ with respect to the distribution $\mathcal D$ is at most $\beta/2m$.
This follows directly from standard bounds on the (non-private) sample complexity of PAC learning of VC classes,
since the VC dimension of $\operatorname{\tt HALFSPACE}(X)$ is at most $d+1$
and since each $S_i$ has size $\Omega(\frac{d\log(1/\alpha') + \log(1/\beta')}{\alpha'})$
where $\alpha'=\frac{r\alpha}{m}$ and $\beta'=\frac{\beta}{2m}$ (see e.g.\ Theorem 6.8 in~\cite{Shalev14book}).
We next claim that if $h \in \R^{d+1}$ errs on a point $\pt{x}=(x_1,\dots,x_d)$ then at least $r$ halfspaces in $H_0$ err on $\pt{x}$.
By the assumption in the theorem, $\mathcal A$ outputs with probability at least $1-\beta/2$ a halfspace $h$ whose Tukey depth is at least $r$ with respect to $H_0$.
By duality, the set of all halfspaces that err on $\pt{x}$ is
itself a halfspace in $\R^{d+1}$ that contains all points
$(a_1,\dots,a_d,w)\in \R^d$ such that $\mathsf{sign}(\sum_{i=1}^d x_i a_i - w) \neq c(x)$. Denote this halfspace by~$h_{\text{err}}$.
By assumption, $h\in h_{\text{err}}$.
Thus, since the Tukey Depth of $h$ with respect to~$H_0$ is at least $r$,
at least $r$ of the halfspaces in $H_0$ are in $h_{\text{err}}$,
as required.
We are ready to establish the PAC-learning guarantee.
Assume that ${\rm error}_{\mathcal D}(c,h_i) \leq r\alpha/m$ for every $1\leq i \leq m$ and that $\mathcal A$ returns a point whose Tukey rank is at most $r$.
By the argument above and the union bound, this happens with probability at least $1-\beta$.
Let $E_h:X\to\{0,1\}$ denote the indicator the $h$ errs (i.e.\ $E_h(\pt{x})=1$ if and only if $h(\pt{x})\neq c(\pt{x})$).
Similarly, let $E_{h_i}$ denote the indicator that $h_i\in H_0$ errs.
For every $\pt{x}\in X$:
$
E_h(\pt{x}) \leq \frac{1}{r}\sum_{i=1}^m E_{h_i}(\pt{x})
$
(either $E_h(\pt{x})=0$ or $E_h(\pt{x})=1$ and $\sum_{i=1}^m E_{h_i}(\pt{x})\geq r$).
Therefore, by taking expectation over both sides it follows that with probability at least $1-\beta$:\;
${\rm error}_{\mathcal D}(c,h) = \operatorname*{\mathbb{E}}_{\pt{x}\sim\mathcal D}[E_h(\pt{x})] \leq \operatorname*{\mathbb{E}}_{\pt{x}\sim\mathcal D}\Bigl[\frac{1}{r}\sum_{i=1}^m E_{h_i}(\pt{x})\Bigr]=
\frac{1}{r} \sum_{i=1}^m{\rm error}_{\mathcal D}(c,h_i) \leq \frac{m}{r}\cdot(r\alpha/m)= \alpha,$
as required.
\end{proof}
Using $\mathcal A_{\rm FindTukey}$ in step~\ref{step:Alg} of Algorithm $\mathcal A_{\rm LearnHalfSpace}$,
we get the following corollary (which follows from \cref{thm:privatecenterpoint} and \cref{thm:convexhullreduction}).
\begin{corollary}
\label{cor:halfspacelearner}
Let $\varepsilon \leq 1$, $\delta <1/2$, and $X\subseteq \R^d$ be a set.
There exists an $(\varepsilon,\delta)$-differentially private $(\alpha,\beta)$-PAC learner with sample complexity $s$ for $\operatorname{\tt HALFSPACE}(X)$
with
\[
s = \tilde{O}\Bigl(\frac{d^{4.5}2^{O(\log^*|X| +\log^*d)}\log^{1.5}\frac{1}{\delta} \log^2 \frac{1}{\beta}}{\varepsilon\alpha} \Bigr).
\]
\end{corollary}
Corollary~\ref{cor:halfspacelearner} establishes an upper bound on the sample complexity of privately learning halfspaces
whose dependency on the domain size $|X|$ is $2^{O\log^*(|X|)}$.
The crux of the algorithm is a reduction to privately publishing a point with a large Tukey depth with respect to a given input dataset.
A drawback of this approach is that the latter task is likely to be computationally difficult (even without privacy constraints), unless the dimension $d$ is constant
(see \cite{Miller10approx} and references within).
In $\mathcal A_{\rm LearnHalfSpace}$ we can use an algorithm $\mathcal A$ that finds a point in the convex hull (i.e., a point whose Tukey depth is at least $1$).
The resulting learning algorithm require sample complexity of $O(\frac{m^2\cdot d\log(\frac{m}{\alpha}) + \log(m/\beta)}{\alpha})$,
where $m$ is the sample complexity of $\mathcal A$. This may result in a more efficient private learning algorithm for halfspaces as the task of privately finding a point in the convex hull might be easier than the task of privately finding a point with high Tukey degree. Furthermore, in this case, we can use an algorithm that privately finds a hypothesis that is a linear combination with positive coefficients of the hypotheses in $H_0$. This follows from the observation that if all hypotheses in $H_0$ are correct on a point $\pt{x}$, then any linear combination with positive coefficients of the hypotheses in $H_0$ is correct of $\pt{x}$.
\section{A Lower Bound on the Sample Complexity of Privately Finding a Point in the Convex Hull}
In this section we show a lower bound on the sample complexity of privately finding a point in the convex hull of a database $S\subseteq X = [T]^d$.
We show that any $(\varepsilon,\delta)$-differentially private algorithm for this task must have sample complexity $\Omega(\frac{d}{\varepsilon}\log \frac{1}{\delta})$.
Our lower bound actually applies to a possibly simpler task of finding a
non-trivial linear combination of the points in the database.
By \cite{BNSV15}, finding a point in the convex hull (even for $d=1$) requires sample complexity $\Omega(\log^* T)$. Thus, together we get a lower bound on the sample complexity of
$\Omega(\frac{d}{\varepsilon}\log\frac{1}{\delta}+\log^* T)$.
It may be tempting to guess that, even with pure $(\varepsilon,0)$-differential privacy, a sample complexity of $O(\log |X|) = O(d\log T)$ should suffice for solving this task, as the size of the output space is $T^d$, because $S\subseteq [T]^d$, and hence (it seems) that one could privately solve this problem using the exponential mechanism of {\cite{MT07}} with sample complexity that depends logarithmically on the size of the output space. We show that this is not the case, and that any $(\varepsilon,0)$-differentially private algorithm for this task must have sample complexity $\Omega(\frac{d^2}{\varepsilon}\log T)$.
\begin{theorem}\label{thm:lowerBound}
Let $T\geq2$, and $d\geq10$. Let $\mathcal A$ be an $(\varepsilon,\delta)$-differentially private algorithm that takes a database $S\subseteq [T]^d$ of size $m$ and returns, with probability at least $1/2$,
a non-trivial linear combination of the points in $S$. Then,
$$m=\Omega\left(
\min\left\{\frac{d^2}{\varepsilon}\log T,\;
\frac{d}{\varepsilon}\log\frac{1}{\delta}
\right\}\right).$$
\end{theorem}
The proof of Theorem~\ref{thm:lowerBound} builds on the analysis of~\cite{BLR08} for lower bounding the sample complexity of releasing approximated answers for counting queries.
\begin{proof}
Throughout this proof, we use $\operatorname{\rm span}(S)$ to denote the set of all non-trivial linear combinations of the points in $S$.
Let $I=\{\pt{x_1},\pt{x_2},\dots,\pt{x_{d/2}},\pt{x'}\}$ be a multiset of random points, where each point is chosen independently and uniformly from $[T]^d$. Also let $i$ be chosen uniformly from $\{1,2,\dots,d/2\}$. Now define the database $S$ containing $\frac{2m}{d}$ copies of each of $\pt{x_1},\dots,\pt{x_{d/2}}$ and define the database $S'$ containing $\frac{2m}{d}$ copies of each of $\pt{x_1},\dots, \pt{x_{i-1}},\pt{x_{i+1}},\dots \pt{x_{d/2}},\pt{x'}$. Note that $S,S'$ differ in exactly $\frac{2m}{d}$ points.
Observe that the points in $I$ are linearly independent with high probability. To see this note that any set $V$
of size at most $\frac{d}{2}$ spans at most $T^{d/2}$ vectors in $[T]^d$: indeed, without loss of generality
we may assume that $V$ is independent and therefore
can be completed to a {\em basis} $\R^d$ by adding $d-\lvert V\rvert \geq d/2$ unit vectors (since the dimension of $V$ is at most $d/2$ there is at least one unit vector that it does not span, add this vector to $V$ and continue). Without loss of generality, these unit vectors are $\pt{e_1},\dots,\pt{e_{d/2}}$. Thus, every choice from $[T]^{d/2}$ for the last $d/2$ coordinates can be completed in a most one way to a vector spanned by $V$ (because every element of $\R^d$ may be written in a {\em unique} way as a linear combination of elements of the resulting basis). This means that a set of (at most) $d/2$ points spans at most $T^{d/2}$ vectors in $[T]^d$. Hence, by a union bound, the probability that the points in $I$ are not independent is at most $\frac{d}{2}\cdot T^{-d/2}$.
Let $\mathcal B(\pt{b},I)$ be a procedure that operates on a point $\pt{b}$ and a set of points $I$, defined as follows. If $I$ is not linearly independent, or if $\pt{b}\notin\operatorname{\rm span}(I)$, than the procedure outputs $\bot$. Otherwise the procedure returns the point $\pt{x}\in I$ with the largest coefficient when representing $\pt{b}$ as a linear combination of the points in $I$ (ties are broken arbitrarily).
Observe that if $\pt{b}\in\operatorname{\rm span}(S)$ for a subset $S\subseteq I$ and if the points in $I$ are linearly independent then $\mathcal B(\pt{b},I)\in S$. Let $\beta$ denote the probability that $\mathcal A(S)$ fails to return a point in $\operatorname{\rm span}(S)$. As $i$ is uniform on $\{1,2,\dots,d/2\}$ we have
\begin{align*}
\Pr_{I,i,\mathcal A}[\mathcal B( \mathcal A(S), I )=\pt{x_i}]
&\geq \Pr_{I,\mathcal A}\left[
\begin{array}{l}
I \text{ is independent}, \\
\mathcal A(S)\in\operatorname{\rm span}(S)
\end{array}\right]
\cdot \Pr_{I,i,\mathcal A}\left[\mathcal B( \mathcal A(S), I )=\pt{x_i} \left|
\begin{array}{l}
I \text{ is independent}, \\
\mathcal A(S)\in\operatorname{\rm span}(S)
\end{array}
\right.\right]\\
&\geq\left(1-\beta-\frac{d}{2}\cdot T^{-d/2}\right)\cdot \frac{2}{d}
\geq\frac{1}{2d}\;,
\end{align*}
where {the second inequality is implied by the fact that $i$ is chosen with uniform distribution from a set of size $d/2$ and} the last inequality is by asserting that $\beta\leq1/2$, $T\geq2$, and $d\geq10$.
On the other hand observe that if $\mathcal B( \mathcal A(S'), I )=\pt{x_i}$ then
(1) $\mathcal A(S')\in\operatorname{\rm span}(I)$, as otherwise $\mathcal B$ outputs $\bot$,
(2) $I$ is linearly independent, as otherwise $\mathcal B$ outputs $\bot$, and
(3) $\mathcal A(S')\notin\operatorname{\rm span}(I\setminus\{\pt{x_i}\})$, as otherwise the coefficient of $\pt{x_i}$ in $\mathcal A(S')$ is 0, and $\mathcal B$ will not output $\pt{x_i}$.
Let $I_{-i}=I\setminus\{\pt{x_i}\}{=\set{\pt{x_1},\dots,\pt{x_{i-1}},\pt{x_{i+1}},\dots,\pt{x_{d/2},\pt{x'}}}}$ and $\pt{b}\leftarrow\mathcal A(S')$.
We have that
\begin{align*}
\Pr_{I,i,\mathcal A}[\mathcal B( \pt{b}, I )=\pt{x_i}]&\leq\Pr_{I,i,\mathcal A}[\pt{b}\in\operatorname{\rm span}(I) \text{ and } \pt{b}\notin\operatorname{\rm span}(I_{-i}) \text{ and } I \text{ is independent} ]\\%+\Pr[I \text{ is not independent}]\\
&\leq\Pr_{I,i,\mathcal A}[\pt{x_i}\in\operatorname{\rm span}(I_{-i}\cup\{\pt{b}\})]
\, \leq \, T^{-d/2}
\end{align*}
where the
last inequality is because $\pt{x_i}$ is independent of $I_{-i}$ and $\pt{b}$ (recall that we denoted $\pt{b}\leftarrow\mathcal A(S')$, and hence, $\pt{b}$ is a (random) function of $I_{-i}$, which is independent of $\pt{x_i}$).
Therefore, by the privacy guarantees of $\mathcal A$ we get
\begin{align*}
\frac{1}{2d}&\leq\Pr_{I,i,\mathcal A}[\mathcal B( \mathcal A(S), I )=\pt{x_i}]=\sum_{I,i}\Pr[I,i]\cdot\Pr_{\mathcal A}[\mathcal B( \mathcal A(S), I )=\pt{x_i}]\\
&\leq\sum_{I,i}\Pr[I,i]\cdot\left(e^{2\varepsilon m/d}\cdot\Pr_{\mathcal A}[\mathcal B( \mathcal A(S'), I )=\pt{x_i}]+e^{2\varepsilon m/d}\cdot 2\delta m/d\right)\\
&=e^{2\varepsilon m/d}\cdot\Pr_{I,i,\mathcal A}[\mathcal B( \mathcal A(S'), I )=\pt{x_i}]+e^{2\varepsilon m/d}\cdot 2\delta m/d\\
&\leq e^{2\varepsilon m/d}\cdot T^{-d/2}+e^{2\varepsilon m/d}\cdot 2\delta m/d.
\end{align*}
Solving for $m$, this means that
$m=\Omega(
\min\{\frac{d^2}{\varepsilon}\log T,\;
\frac{d}{\varepsilon}\log\frac{1}{\delta}
\}.$
\end{proof}
\bibliographystyle{abbrvnat}
|
{
"timestamp": "2019-03-01T02:00:37",
"yymm": "1902",
"arxiv_id": "1902.10731",
"language": "en",
"url": "https://arxiv.org/abs/1902.10731"
}
|
\section{Introduction}
\label{sec:intro}
Cardiac magnetic resonance (CMR) is the gold-standard technique for assessment of cardiac morphology. Conventional practice is to acquire a stack of breath-hold 2D image sequence in the left ventricular (LV) short axis supplemented by long axis image sequence in prescribed planes to enable reproducible volumetric analysis and diagnostic assessment \cite{alfakih2004assessment}. Disadvantages of this approach for whole-heart segmentation are low through-plane resolution, misalignment between breath-holds and lack of whole-heart coverage. High-resolution 3D image sequences address some of these issues, but also have disadvantages in terms of long acquisition times, relatively low in-plane resolution and lack of clinical availability. However, high-resolution 3D segmentations proved to be crucial for the construction of integrative statistical models of cardiac anatomy and physiology and disease characterization \cite{biffibio, bai2015bi}. For these reasons, a method to reconstruct a 3D high-resolution segmentation from routinely-acquired 2D cines could be highly beneficial - offering high resolution phenotyping robust to artefact in large clinical populations with conventional imaging.
The reconstruction of 3D anatomical structures from a limited number of 2D views has been previously studied via deformable statistical shape models \cite{whitmarsh2011reconstructing}. However, these methods require complex reconstruction procedures and are very computationally-intensive. In recent years, with the advent of learning-based approaches, and in particular of deep learning, a number of alternative strategies have been proposed. The TL-embedding network (TL-net) consists of a 3D convolutional autoencoder (AE) which learns a vector representation of the 3D geometries, whereas a second convolutional neural network attached to the latent space of the AE maps 2D views of the same object to the same vector representation \cite{girdhar2016learning}. More recently, \cite{cerrolaza20183d} proposed a convolutional conditional variational autoencoder (CVAE) architecture for the 3D reconstruction of the fetal skull from 2D ultrasound standard planes of the head. Finally, \cite{biffi2018learning} showed how a convolutional variational autoencoder (VAE) can learn a shape segmentation model of left ventricular (LV) segmentations and how the learned latent space can be exploited to accurately identify healthy and pathological cases and generate realistic segmentations unseen during training.
\begin{figure*}
\begin{center}
\includegraphics[width=0.9\textwidth]{architecture.pdf}
\caption{The proposed conditional variational autoencoder (CVAE) architecture.}
\label{fig:arch}
\end{center}
\end{figure*}
In this work, we present a CVAE architecture that reconstructs a high-resolution 3D segmentation of the LV myocardium from three segmentations of 2D standard cardiac views (one short-axis and two long-axis). Moreover we show how the proposed model naturally produces confidence maps associated to each reconstruction, unlike deterministic models, thanks to its generative properties.
\section{MATERIALS AND METHODS}
\label{sec:method}
\subsection{3D Cardiac Image Acquisition and Segmentation}
A high-spatial resolution 3D balanced steady-state free precession cine MR image sequence was acquired from 1,912 healthy volunteers of the UK Digital Heart Project at Imperial College London using a 1.5-T Philips Achieva system (Best, the Netherlands) \cite{bai2015bi}. Left and right ventricles were imaged in their entirety in a single breath-hold (60 sections, repetition time 3.0 ms, echo time 1.5 ms, flip angle 50$\degree$, field of view $320 \times 320 \times 112 $ mm, matrix $160 \times 95$, reconstructed voxel size $1.2 \times 1.2 \times 2$ mm, 20 cardiac phases, temporal resolution 100 ms, typical breath-hold 20 s). For each subject, a 3D high-resolution segmentation of the LV was automatically obtained using a previously reported technique employing a set of manually annotated atlases \cite{bai2015bi}. In this work, only the end-diastolic (ED) frame was considered.
\subsection{Conditional Variational Autoencoder Architecture}
The outline of the CVAE architecture we propose is shown in Fig. 1. We aim at reconstructing a 3D high-resolution LV segmentation $\bf{Y}$ from $i$ segmentations obtained in as many 2D views $\bf{X}$ $= \{ X_i \; | \; i = 1, 2, 3 \}$. We aim to learn from the training data a conditional generative model $P(\bf{Y}|\bf{X})$ by means of a $d$-dimensional latent distribution $\bf{z}$ and a low-dimensional representation $\bf{x}$ of the views $\bf{X}$. In this work we use a single 2D convolutional neural network (CNN) to encode the 2D views $\bf{X}$ in a low-dimensional representation $\bf{x}$. An alternative encoding strategy was proposed in \cite{cerrolaza20183d}, using a separate branch for each conditional input of the model. However, whilst this latter approach proved efficient when the views suffer from large inconsistencies or variability (e.g., free-hand ultrasound scans), we can notably reduce the model complexity by combining the views $\bf{X}$ as a unique three-channel input as these are consistently acquired in clinical routine.
Directly inferring $P(\bf{Y}|\bf{X})$ is impractical as it would require sampling a large number of $\bf{z}$ values. However, variational inference allows us to approximate $P(\bf{Y}|\bf{X})$ by introducing a high-capacity function $\bf{Q}(\bf{z}|\bf{Y},\bf{X})$ which gives us a distribution over $\bf{z}$ values that are likely to produce $\bf{Y}$. Hence we can learn $P(\bf{Y}|\bf{X})$ by minimizing the following objective:
$$ log(P(\bf{Y}|\bf{X})) - D_{KL}[Q(\bf{z}|\bf{Y},\bf{X}) || P(\bf{z}\given\bf{Y},\bf{X})] = $$
\begin{equation}
E_{\bf{z} \sim \bf{Q}}[log P(\bf{Y} |\bf{z,X})] - D_{KL}[Q(\bf{z}|\bf{Y},\bf{X}) || P(\bf{z}\given\bf{X})]
\end{equation}
where $D_{KL}$ represents the Kullback-Leibler (KL) divergence of two distributions (full mathematical derivation of the equation can be found in \cite{doersch2016tutorial}). The encoding function $\bf{Q}(\bf{z}|\bf{Y},\bf{X})$ can be modelled as a Gaussian distribution parametrized by $\bf{\mu_{z|Y,X}}$ and $\bf{\sigma_{z|Y,X}}$ vectors. These two vectors can be learned by encoding the input 3D segmentation $\bf{Y}$ we want to reconstruct via a 3D CNN to a set of features $\bf{y}$, which are then concatenated together with the lower dimensional representation $\bf{x}$ of the views $\bf{X}$. By concatenating $[\bf{x},$ $\bf{y}]$ with a fully connected neural network to $\bf{\mu_{z|Y,X}}$ and $\bf{\sigma_{z|Y,X}}$ we can thus learn $\bf{Q}(\bf{z}|\bf{Y},\bf{X})$.
If $\bf{Q}(\bf{z}|\bf{Y},\bf{X})$ is modelled by a sufficiently expressive function, then this function will match the real $P(\bf{z} \given \bf{Y},\bf{X})$ and the $D_{KL}[Q(\bf{z}|\bf{Y},\bf{X}) || P(\bf{z}\given\bf{Y},\bf{X})]$ term in (1) will be zero. Therefore optimizing the right side of (1) will correspond to optimizing $P(\bf{Y}|\bf{X})$. In this work, the first term of the right side of (1) is computed as the Dice score (DSC) between $\bf{Y}$ and its reconstruction $\hat{\bf{Y}}$, which is the output of the generative model. The second term in (1) can be computed in a closed form if we assume its prior distribution to be $\mathcal{N}(\bf{0}, \bf{1})$, a $d$-dimensional normal distribution with zero mean and unit-standard deviation, and where $d$ is the number of dimensions of the latent space. Therefore the loss function we optimize becomes $\mathcal{L} = DSC(\bf{Y}, \hat{\bf{Y}}) + \alpha \; D_{KL}[\bf{z}||\mathcal{N}(0,1)]$.
\subsection{Experimental Setup and Network Training}
In this work, we mimicked the two long-axis and the one short-axis views acquired in a routine acquisition with the following steps: (1) we rigidly aligned all the ground truth 3D high-resolution segmentations by performing landmark-based and subsequent intensity-based rigid registration; (2) we kept only the LV myocardium label and we cropped and padded the segmentations to [x = 80, y = 80, z = 80, t = 1] dimension using a bounding box centered at the centre of mass of the LV myocardium; (3) we sampled three orthogonal views passing through the centre of each segmentation (an example is shown in Fig. 1). Thanks to this process we extracted three 2D views showing the same three LV sections consistently for all subjects. In the following experiments, the ground truth 3D high-resolution segmentations and their corresponding 2D views were kept all in the same reference space. Inter-subject pose variability will be addressed in future work, potentially with a simple data augmentation strategy.
The dimension $d$ of the latent space was fixed to 125 as values smaller than 100 provided less accurate results, while above 125 no further improvements were observed. The dimensionality of the low dimensional representation $\bf{x}$ was kept equal to the dimensionality of $\bf{z}$ to guarantee a balanced contribution to the generative model. Simulations for different values of the parameter $\alpha$ in the loss function were performed: low values of $\alpha$ ($\alpha<0.5$) provided better reconstruction results on the training data at the expenses of a strong deviation from normality of the latent space distribution (KL term not converging) causing overfitting. Higher values of $\alpha$ ($\alpha>2$) penalized the reconstruction term in favour of a strictly normal latent space, hence providing poorer reconstruction accuracy. In this work we set $\alpha=1$ as this provided good reconstruction accuracy and convergence of the KL term.
Experiments were performed with different numbers of views $X_i$ as conditions for the proposed model. In particular, referring to the first long-axis view as 1, the second long-axis view as 2 and the short-axis view as 3, we performed the training using either only one view (which we will indicate as CVAE\_1), or a combination or two views (CVAE\_12, CVAE\_23, CVAE\_13), or all the three views (CVAE\_123). We have also studied the feasibility of training a 2D AE to reconstruct the 3 views and used its encoder as a pre-trained conditional encoder (pCVAE\_123). Moreover, the reconstruction capability of the proposed architecture was compared with the one of the TL-net \cite{girdhar2016learning}. Finally, we compared the reconstruction obtained by a VAE with $\bf{z}$=$0$ (VAE\_0) to all our test segmentations, as this represents the best segmentation that the generative model can reconstruct when no information is provided to it. Results obtained with an autoencoder (AE) are also reported since this model yielded better results than different VAEs with distinct $\alpha$ values as it only optimizes the reconstruction accuracy. All the models share the same 3D encoder and decoder architectures.
The dataset was split into training, evaluation and testing sets consisting of 1362, 150 and 400 subjects respectively. Data augmentation included rotation around the three orthogonal axis with rotation angles randomly extracted from a normal distribution $\mathcal{N}(0,6\degree)$ and random closing and opening morphological operations. All the networks were implemented in Tensorflow and training was stopped after 300k iterations, when the total validation loss function had stopped improving (approximately 42 hours per network on an NVIDIA Tesla K80 GPU), using stochastic gradient descent with momentum (Adam optimizer, learning rate = $10^{-4}$) and batch size of 8. During testing, the 3D encoder branch was disabled and the reconstruction were obtained by setting the latent variables $\bf{z}$ $= \bf{0}$.
\begin{center}
\begin{table}[t!]
\resizebox{8.6cm}{!}{
\begin{tabular}{c|c|c|c}
Model & DSC & Hausd. [mm] & MassDiff {[}\%{]} \\
\hline
VAE\_0 & 65.48 $\pm$ 0.38 & 9.32 $\pm$ 0.06 & 35.37 $\pm$ 0.70 \\
\hline
CVAE\_1 & 78.08 $\pm$ 0.33 & 5.29 $\pm$ 0.04 & 3.94 $\pm$ 0.38\\
\hline
CVAE\_23 & 82.90 $\pm$ 0.21 & 4.43 $\pm$ 0.04 & 3.93 $\pm$ 0.19\\
CVAE\_12 & 85.21 $\pm$ 0.20 & 4.46 $\pm$ 0.04 & 3.73 $\pm$ 0.19\\
CVAE\_13 & 83.18 $\pm$ 0.18 & 4.77 $\pm$ 0.04 & 3.69 $\pm$ 0.19\\
\hline
\bf{CVAE\_123} & \bf{87.92 $\pm$ 0.15} & \bf{3.99 $\pm$ 0.03} & \bf{2.70 $\pm$ 0.14}\\
pCVAE\_123& 87.63 $\pm$ 0.16 & 4.04 $\pm$ 0.04 & 3.05 $\pm$ 0.16\\
TL\_net & 82.60 $\pm$ 0.23 & 4.66 $\pm$ 0.04 & 3.85 $\pm$ 0.19 \\
\hline
AE & 90.45 $\pm$ 0.12 & 3.46 $\pm$ 0.03 & 1.50 $\pm$ 0.10 \\
\hline
\end{tabular}
}
\caption{Reconstruction metrics together with their standard error of the mean for all the studied models.}
\label{table}
\end{table}
\end{center}
\section{RESULTS AND DISCUSSION}
\label{sec:results}
\subsection{Accuracy of 3D Reconstruction}
Table 1 shows the reconstruction accuracy in terms of 3D Dice score, 2D slice-by-slice Hausdorff distance and LV mass difference between 3D high-resolution segmentations (ground truth and reconstructed ones) for all the studied architectures. LV mass is an important clinical biomarker, therefore we have estimated for each reconstruction its percentage difference in mass with the ground truth. The results indicate that the reconstruction accuracy decreases when views are removed. From the experiments with two views we can also infer how different views have different importance. In particular, the short-axis view seems to have the smallest impact on the reconstruction accuracy. This could be motivated by the fact that the long-axis views contain more information about the regional changes in curvature of the LV, which strongly influences the Dice Score. The results reported in Table 1 also show how our architecture significantly outperforms the TL-net by a large amount ($p=2.2 \cdot 10^{-16}$), and how the pre-training of the 2D CNN encoder network did not help to achieve better results. Finally, we can observe that the mass difference is systematically overestimated by a small amount that decreases with the number of views provided. We believe this a consequence of using the Dice score in the loss function. On the other hand, models trained using cross entropy in the loss function yielded a systematic underestimation of the mass, often with reconstructions with missing LV apex, as this loss term tends to favour the background instead of the myocardium.
\begin{figure}[t!]
\begin{minipage}[b]{0.9\linewidth}
\centerline{\includegraphics[width=8.5cm]{plots3.pdf}}
\caption{First and third rows, reconstructed segmentation obtained with one and three views (in red, 1v and 3v) overlaid onto the ground truth segmentation (in black, GT) for one random subject. Second and fourth rows, confidence maps for the reconstruction with one and three views - $P(1v)$ and $P(3v)$. First and second columns, long-axis views (LA1 and LA2). Third column, short-axis (SA) view.}
\label{fig:vi}
\end{minipage}
\end{figure}
\subsection{Visualisation and Uncertainty Estimation}
In the first and third rows of Fig.\ref{fig:vi} we report the reconstructed segmentations obtained with one and three views (in red) overlaid onto the ground truth segmentation (in black) for one subject of the testing dataset (with DSC 0.80 and 0.89, respectively). In the second and fourth rows we instead report the confidence maps obtained for the reconstruction with one and three views - $P(1v)$ and $P(3v)$. These maps have been obtained by sampling N times ($N=1,000$) $\bf{z}$ from $\bf{\mathcal{N}(0,1)}$ to reconstruct N segmentations from the same set of views $\bf{X}$. Unlike deterministic architectures (such as the TL-net), by averaging these maps we can compute the probability of each voxel to be labelled as LV myocardium, providing to clinicians a richer and more intuitive interpretation of the reconstruction. It can be seen in Fig. \ref{fig:vi} how the confidence map obtained with only 1 view has greater uncertainty than the one obtained with 3 views, which instead shows lower variability. Moreover, the amount of uncertainty in the $P(1v)$ map for the long-axis view 1 is less than for the other two views, as this view was the one provided to the network as condition. Interestingly, in the reconstruction with one view the areas with more uncertainty correspond to the areas where there is less overlap with the ground truth, i.e. the areas where the network is less accurate in predicting the shape.
\section{CONCLUSIONS}
\label{sec:conclusions}
In this paper we present the first deep conditional generative network for the reconstruction of 3D high-resolution LV segmentations from three segmentations of 2D orthogonal views. The reported results show the potential of this class of models to provide better quantitative cardiac models from sparse data. Future work will focus on using real standard long-axis views (instead of the simulated ones in this work), on reconstructing multiple structures and on extending the proposed framework to pathological datasets, for which acquiring breath-hold sequences is even more challenging.
\section*{\center \small{ACKNOWLEDGMENTS}}
The research was supported by the British Heart Foundation (NH/17/1/32725, RE/13/4/30184); National Institute for Health Research (NIHR) Biomedical Research Centre based at Imperial College Healthcare NHS Trust and Imperial College London; Academy of Medical Sciences Grant (SGL015/1006), and the Medical Research Council, UK.
\newpage
\bibliographystyle{IEEEbib}
|
{
"timestamp": "2019-03-01T02:15:04",
"yymm": "1902",
"arxiv_id": "1902.11000",
"language": "en",
"url": "https://arxiv.org/abs/1902.11000"
}
|
\section{Introduction}
\label{sec:intro}
Synthesizing images from text descriptions (known as {\em Text-to-Image synthesis}) is an important machine learning task, which requires handling ambiguous and incomplete information in natural language descriptions and learning across vision and language modalities.
Approaches based on Generative Adversarial Networks (GANs)~\cite{goodfellow2014generative} have recently achieved promising results on this task~\cite{reed2016generative,reed2016learning,Han16stackgan,Han17stackgan2,xu2017attngan,ma2018gan,hong2018inferring,johnson2018image,zhang2018text}.
Most GAN based methods synthesize the image conditioned only on a global sentence vector, which may miss important fine-grained information at the word level, and prevents the generation of high-quality images. More recently, AttnGAN~\cite{xu2017attngan} is proposed which introduces the attention mechanism \cite{XuBKCCSZB15,YangHGDS16,Dzmitry14,Ashish17} into the GAN framework, thus allows attention-driven, multi-stage refinement for fine-grained text-to-image generation.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.95\linewidth]{general_comparison}
\end{center}
\vspace{-15pt}
\caption{\small Top: AttnGAN~\cite{xu2017attngan} and its grid attention visualization. Middle: our modified implementation of two-step (layout-image) generation proposed in~\cite{hong2018inferring}. Bottom: our Obj-GAN and its object-driven attention visualization. The middle and bottom generations use the same {\it generated} semantic layout, and the only difference is the object-driven attention.}
\vspace{-16pt}
\label{fig:maineg}
\end{figure}
Although images with realistic texture have been synthesized on simple datasets, such as birds~\cite{xu2017attngan,ma2018gan} and flowers~\cite{Han17stackgan2}, most existing approaches do not specifically model objects and their relations in images and thus have difficulties in generating complex scenes such as those in the COCO dataset~\cite{LinMBHPRDZ14}. For example, generating images from a sentence ``several people in their ski gear are in the snow'' requires modeling of different objects (people, ski gear) and their interactions (people on top of ski gear), as well as filling the missing information (\textit{e.g.}, the rocks in the background). In the top row of Fig.~\ref{fig:maineg}, the image generated by AttnGAN does contain scattered texture of people and snow, but the shape of people are distorted and the picture's layout is semantically not meaningful. \cite{hong2018inferring} remedies this problem by first constructing a semantic layout from the text and then synthesizing the image by a deconvolutional image generator. However, the fine-grained word/object-level information is still not explicitly used for generation. Thus, the synthesized images do not contain enough details to make them look realistic (see the middle row of Fig.~\ref{fig:maineg}).
In this study, we aim to generate high-quality complex images with semantically meaningful layout and realistic objects.
To this end, we propose a novel Object-driven Attentive Generative Adversarial Networks (Obj-GAN) that effectively capture and utilize fine-grained word/object-level information for text-to-image synthesis.
The Obj-GAN consists of a pair of object-driven attentive image generator and object-wise discriminator, and a new object-driven attention mechanism. The proposed image generator takes as input the text description and a pre-generated semantic layout and synthesize high-resolution images via multiple-stage coarse-to-fine process. At {\it every} stage, the generator synthesizes the image region within a bounding box by focusing on words that are most relevant to the object in that bounding box, as illustrated in the bottom row of Fig.~\ref{fig:maineg}. More specifically, using a new object-driven attention layer, it uses the class label to query words in the sentences to form a word context vector, as illustrated in Fig.~\ref{fig:attention}, and then synthesizes the image region conditioned on the class label and word context vector. The object-wise discriminator checks every bounding box to make sure that the generated object indeed matches the pre-generated semantic layout. To compute the discrimination losses for all bounding boxes simultaneously and efficiently, our object-wise discriminator is based on a Fast R-CNN~\cite{girshick2015fast}, with a binary cross-entropy loss for each bounding box.
The contribution of this work is three-folded.
(\textit{i}) An Object-driven Attentive Generative Network (Obj-GAN) is proposed for synthesizing complex images from text descriptions. Specifically, two novel components are proposed, including the object-driven attentive generative network and the object-wise discriminator.
(\textit{ii}) Comprehensive evaluation on a large-scale COCO benchmark shows that our Obj-GAN significantly outperforms previous state-of-the-art text-to-image synthesis methods. Detailed ablation study is performed to empirically evaluate the effect of different components in Obj-GAN.
(\textit{iii}) A thorough analysis is performed through visualizing the attention layers of the Obj-GAN, showing insights of how the proposed model generates complex scenes in high quality. Compared with the previous work, our object-driven attention is more robust and interpretable, and significantly improves the object generation quality in complex scenes.
\begin{figure*}[tb]
\begin{center}
\includegraphics[width=1\linewidth]{overall_framework}
\end{center}
\vspace{-16pt}
\caption{\small Obj-GAN completes the text-to-image synthesis in two steps: the layout generation and the image generation. The layout generation contains a bounding box generator and a shape generator. The image generation uses the object-driven attentive image generator.}
\vspace{-18pt}
\label{fig:3steps}
\end{figure*}
\section{Related Work}
\label{sec:related}
Generating photo-realistic images from text descriptions, though challenging, is important to many real-world applications such as art generation and computer-aided design. There has been much research effort for this task through different approaches, such as variational inference~\cite{MansimovPBS15, Gregor15DRAW}, approximate Langevin process~\cite{Reed17parallel}, conditional PixelCNN via maximal likelihood estimation~\cite{Oord16, Reed17parallel}, and conditional generative adversarial networks~\cite{reed2016generative, reed2016learning, Han16stackgan, Han17stackgan2}. Compared with other approaches, Generative Adversarial Networks (GANs)~\cite{goodfellow2014generative} have shown better performance in image generation~\cite{Radford15, DentonCSF15, Salimans2016, Christian2016, pix2pix2017,huang2018turbo}. However, existing GAN based text-to-image synthesis is usually conditioned only on the global sentence vector, which misses important fine-grained information at the word level, and thus lacks the ability to generate high-quality images. \cite{xu2017attngan} uses the traditional grid visual attention mechanism in this task, which enables synthesizing fine-grained details at different image regions by paying attentions to the relevant words in the text description.
To explicitly encode the semantic layout into the generator, \cite{hong2018inferring} proposes to decompose the generation process into two steps, in which it first constructs a semantic layout (bounding boxes and object shapes) from the text and then synthesizes an image conditioned on the layout and text description. \cite{johnson2018image} also proposes such a two-step process to generate images from scene graphs, and their process can be trained end-to-end. In this work, the proposed Obj-GAN follows the two-step generation process as \cite{hong2018inferring}. However, \cite{hong2018inferring} encodes the text into a single global sentence vector, which loses word-level fine-grained information. Moreover, it uses the image-level GAN loss for the discriminator, which is less effective at providing object-wise discrimination signal for generating salient objects. We propose a new object-driven attention mechanism to provide fine-grained information (words in the text description and objects in the layout) for different components, including an attentive seq2seq bounding box generator, an attentive image generator and an object-wise discriminator.
The attention mechanism has recently become a crucial part of vision-language multi-modal intelligence tasks. The traditional grid attention mechanism has been successfully used in modeling multi-level dependencies in image captioning~\cite{XuBKCCSZB15}, image question answering~\cite{YangHGDS16}, text-to-image generation~\cite{xu2017attngan}, unconditional image synthesis~\cite{zhang2018self} and image-to-image translation~\cite{ma2018gan}, image/text retrieval~\cite{Lee2018Stacked}. In 2018, \cite{anderson2017bottom} proposes a bottom-up attention mechanism, which enables attention to be calculated over semantic meaningful regions/objects in the image, for image captioning and visual question-answering. Inspired by these works, we propose Obj-GAN which for the first time develops an object-driven attentive generator plus an object-wise discriminator, thus enables GANs to synthesize high-quality images of complicated scenes.
\section{Object-driven Attentive GAN}
\label{sec:model}
As illustrated in Fig.~\ref{fig:3steps}, the Obj-GAN performs text-to-image synthesis in two steps: generating a semantic layout (class labels, bounding boxes, shapes of salient objects), and then generating the image. In the image generation step, the object-driven attentive generator and object-wise discriminator are designed to enable image generation conditioned on the semantic layout generated in the first step.
The input of Obj-GAN is a sentence with $T_s$ tokens. With a pre-trained bi-LSTM model, we encode its words as word vectors $e \in \mathbb{R}^{D\times T_s}$ and the entire sentence as a global sentence vector $\bar{e}\in \mathbb{R}^{D}$. We provide details of this pre-trained bi-LSTM model and the implementation details of other modules of Obj-GAN in $\S$~\ref{sec:appendix}.
\subsection{Semantic layout generation}
\label{subsec:bbox}
In the first step, the Obj-GAN takes the sentence as input and generates a semantic layout, a sequence of objects specified by their bounding boxes (with class labels) and shapes. As illustrated in Fig.~\ref{fig:3steps}, a box generator first generates a sequence of bounding boxes, and then a shape generator generates their shapes. This part resembles the bounding box generator and shape generator in \cite{hong2018inferring}, and we put our implementation details in $\S$~\ref{sec:appendix}.
\noindent{\bf Box generator.} We train an attentive seq2seq model~\cite{Dzmitry14}, also referring to Fig.~\ref{fig:3steps}, as the box generator:
\begin{equation}\label{eqn:boxgen}
B_{1:T} := [B_1, B_2, \dots, B_T] \sim G_{\text{box}}(e).
\end{equation}
Here, $e$ are the pre-trained bi-LSTM word vectors, $B_t = (l_t, b_t)$ are the class label of the $t$'s object and its bounding box $b = (x, y, w, h) \in \mathbb{R}^4$. In the rest of the paper, we will also call the label-box pair $B_t$ as a bounding box when no confusion arises. Since most of the bounding boxes have corresponding words in the sentence, the attentive seq2seq model captures this correspondence better than the seq2seq model used in \cite{hong2018inferring}.
\noindent{\bf Shape generator.} Given the bounding boxes $B_{1:T}$, the shape generator predicts the shape of each object in its bounding box, \textit{i.e.},
\begin{equation}\label{eqn:hmapgen}
\hat{M}_{1:T} = G_{\text{shape}}(B_{1:T}, z_{1:T}).
\end{equation}
where $z_{t} \sim \mathcal{N}(0,1)$ is a random noise vector. Since the generated shapes not only need to match the location and category information provided by $B_{1:T}$, but also should be aligned with its surrounding context, we build $G_{\text{shape}}$ based on a bi-directional convolutional LSTM, as illustrated in Fig.~\ref{fig:3steps}.
Training of $G_{\text{shape}}$ is based on the GAN framework \cite{hong2018inferring}, in which a perceptual loss is also used to constrain the generated shapes and to stabilize the training.
\begin{figure*}[tb]
\begin{center}
\includegraphics[width=1\linewidth]{generator_pipeline}
\end{center}
\vspace{-16pt}
\caption{\small The object-driven attentive image generator.}
\vspace{-19pt}
\label{fig:ImageGenerator}
\end{figure*}
\subsection{Image generation}
\subsubsection{Attentive multistage image generator}
\label{subsec:imagegenerator}
As shown in Fig.~\ref{fig:ImageGenerator}, the proposed attentive multistage generative network has two generators ($G_0, G_1$). The base generator $G_0$ first generates a low-resolution image $\hat{x}_0$ conditioned on the global sentence vector and the pre-generated semantic layout. The refiner $G_1$ then refines details in different regions by paying attention to most relevant words and pre-generated class labels and generates a higher resolution image $\hat{x}_1$. Specifically,
%
\vspace{-2mm}
\begin{equation*}
\begin{aligned}
h_0 &= F_0(z, \quad \Bar{e}, \quad\text{Enc}(M^0), c^{\text{obj}}, c^{\text{lab}}), \quad \hat{x}_0 = G_0(h_0),\\
h_1 &= F_1(c^{\text{pat}}, h_{0}+\text{Enc}(M^1), c^{\text{obj}}, c^{\text{lab}}), \quad \hat{x}_1 = G_1(h_1),
\end{aligned}
\vspace{-1mm}
\end{equation*}
where (\textit{i}) $z$ is a random vector with standard normal distribution; (\textit{ii}) $\text{Enc}(M^0)$ ( $\text{Enc}(M^1)$ ) is the encoding of low-resolution shapes $M^0$ (higher-resolution shapes $M^1$); (\textit{iii}) $c^{\text{pat}}= F_{\text{attn}}^{\text{grid}}(e,h_0)$ are the patch-wise context vectors from the traditional grid attention, (\textit{iv}) $c^{\text{obj}}=F_{\text{attn}}^{\text{obj}}(e, e^g, l^g, M)$ are the object-wise context vectors from our new object-driven attention, and $c^{\text{lab}}=c^{\text{lab}}(l^g, M)$ are the label context vectors from class labels. We can stack more refiners to the generation process and get higher and higher resolution images. In this paper, we have two refiners ($G_1$ and $G_2$) and finally generate images with resolution $256\times 256$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=1\linewidth]{btup_attn}
\end{center}
\vspace{-16pt}
\caption{\small Object-driven attention.}
\vspace{-18pt}
\label{fig:attention}
\end{figure}
\noindent{\bf Compute context vectors via attention.} Both patch-wise context vectors $c^{\text{pat}}$ and object-wise context vectors $c^{\text{obj}}$ are attention-driven context vectors for specific image regions, and encode information from the words that are most relevant to that image region. Patch-wise context vectors are for uniform-partitioned image patches determined by the uniform down-sampling/up-sampling structure of CNN, but these patches are not semantically meaningful. Object-wise context vectors are for semantically meaningful image regions specified by bounding boxes, but these regions are at different scales and may have overlaps.
Specifically, the patch-wise context vector $c_j^{\text{pat}}$ ( objective-wise context vector $c_t^{\text{obj}}$) is a dynamic representation of word vectors relevant to patch $j$ (bounding box $B_t$), which is calculated by
\vspace{-3mm}
\begin{equation}\label{eq:AttnGAN}
c_j^{\text{pat}} = \sum_{i=1}^{T_s}\beta_{j,i}^{\text{pat}} e_{i}, \quad
c_t^{\text{obj}} = \sum_{i=1}^{T_s}\beta_{t,i}^{\text{obj}} e_{i}.
\vspace{-3mm}
\end{equation}
Here, $\beta_{j,i}^{\text{pat}}$ ( $\beta_{t,i}^{\text{obj}}$ ) indicates the weight the model attends to the $i$'th word when generating patch $j$ (bounding box $B_t$) and is computed by
\vspace{-3mm}
\begin{align}
\beta_{j,i}^{\text{pat}} = \frac{\exp(s_{j,i}^{\text{pat}})}{\sum_{k=1}^{T_s}{\exp(s_{j,k}^{\text{pat}})}}, \; \; \quad \; s_{j,i}^{\text{pat}} = (h_j)^T e_{i}, \label{eq:AttnGANbeta_pat}\\
\beta_{t,i}^{\text{obj}} = \frac{\exp(s_{t,i}^{\text{obj}})}{\sum_{k=1}^{T_s}{\exp(s_{t,k}^{\text{obj}})}}, \; \; \quad \; s_{t,i}^{\text{obj}} = (l_t^g)^T e^g_{i}. \label{eq:AttnGANbeta_inst}
\vspace{-3mm}
\end{align}
For the traditional grid attention, we use the image region feature $h_j$, which is one column in the previous hidden layer $h \in \mathbb{R}^{D^{\text{pat}} \times N^{\text{pat}}}$, to query the pre-trained bi-LSTM word vectors $e$. For the new object-driven attention, we use the GloVe embedding of object class label $l_t^g$ to query the GloVe embedding of the words in the sentence, as illustrated in the lower part of Fig.~\ref{fig:attention}.
\noindent{\bf Feature map concatenation.}
The patch-wise context vector $c_j^{\text{pat}}$ can be directly concatenated with the image feature vector $h_j$ in the previous layer. However, the object-wise context vector $c_t^{\text{obj}}$ cannot, because they are associated with bounding boxes instead of pixels in the hidden feature map. We propose to copy the object-wise context vector $c_t^{\text{obj}}$ to every pixel where the $t$'th object is present, \textit{i.e.}, $M_t \otimes c_t^{\text{obj}}$ where $\otimes$ is the vector outer-product, as illustrated in the upper-right part of Fig.~\ref{fig:attention}. \footnote{This operation can be viewed as an inverse of the pooling operator.}
If there are multiple bounding boxes covering the same pixel, we have to decide whose context vector should be used on this pixel. In this case, we simply do a max-pooling across all the bounding boxes:
\begin{equation}\label{eqn:bthiden}
c^{\text{obj}} = \max_{t~: 1\le t \le T} M_t \otimes c_{t}^{\text{obj}}.
\end{equation}
Then $c^{\text{obj}}$ can be concatenated with the feature map $h$ and patch-wise context vectors $c^{\text{pat}}$ for next-stage generation.
\noindent{\bf Label context vectors.}
Similarly, we distribute the class label information to the entire hidden feature map to get the label context vectors, \textit{i.e.},
\begin{equation}\label{eqn:btlabel}
c^{\text{lab}} = \max_{t~:~1\le t \le T} M_t \otimes e_{t}^{\text{g}}.
\end{equation}
Finally, we concatenate $h$, $c^{\text{pat}}$, $c^{\text{obj}}$ and $c^{\text{lab}}$ and pass the concatenated tensor through one up-sampling layer and several residual layers to generate a higher-resolution image.
\noindent{\bf Grid attention vs. object-driven attention.}
The process to compute the patch-wise context vectors above is the traditional grid attention mechanism used in AttnGAN~\cite{xu2017attngan}. Note that its attention weights $\beta_{j,i}^{\text{pat}}$ and context vector $c_j^{\text{pat}}$ are useful only when the hidden feature $h_j^{\text{pat}}$ in the $G_0$ stage correctly captures the content to be drawn in patch $j$. This essentially assumes that the generation in the $G_0$ stage already captures a rough sketch (semantic layout). This assumption is valid for simple datasets like birds~\cite{xu2017attngan}, but fails for complex datasets like COCO~\cite{LinMBHPRDZ14} where the generated low-resolution image $\hat{x}_0$ typically does {\it not} have a meaningful layout. In this case, the grid attention is even harmful, because patch-wise context vector is attended to a wrong word and thus generate the texture associated with that wrong word. This may be the reason why AttnGAN's generated image contains scattered patches of realistic texture but overall is semantically not meaningful; see Fig.~\ref{fig:maineg} for example. Similar phenomenon is also observed in DeepDream~\cite{mordvintsev2017deep}. On the contrary, in our object-driven attention, the attention weights $\beta_{t,i}^{\text{obj}}$ and context vector $c_t^{\text{obj}}$ rely on the class label $l_t^g$ of the bounding box and are independent of the generation in the $G_0$ stage. Therefore, the object-wise context vectors are always helpful to generate images that are consistent with the pre-generated semantic layout. Another benefit of this design is that the context vector $c_t^{\text{obj}}$ can also be used in the discriminator, as we present in $\S$~\ref{subsec:imageloss}.
\subsubsection{Discriminators}
\label{subsec:imageloss}
We design patch-wise and object-wise discriminators to train the attentive multi-stage generator above. Given a patch from uniformly-partitioned image patches determined by the uniform down-sampling structure of CNN, the patch-wise discriminator is trying to determine whether this patch is realistic or not (unconditional) and whether this patch is consistent with the sentence description or not (conditional). Given a bounding box and the class label of the object within it, the object-wise discriminator is trying to determine whether this region is realistic or not (unconditional) and whether this region is consistent with the sentence description and given class label or not (conditional).
\begin{figure}[tb]
\begin{center}
\includegraphics[width=1\linewidth]{discriminator}
\end{center}
\vspace{-13pt}
\caption{\small Object-wise discriminator.}
\label{fig:discriminator}
\vspace{-10pt}
\end{figure}
\noindent{\bf Patch-wise discriminators.} Given an image-sentence pair $x, \Bar{e}$ ($\Bar{e}$ is the sentence vector), the patch-wise unconditional and text discriminator can be written as
\begin{equation}\label{eq:patchD}
\begin{aligned}
p^{\text{pat,un}}= D_{\text{uncond.}}^{\text{pat}}(\text{Enc}(x)), \quad p^{\text{pat,con}} = D_{\text{text}}^{\text{pat}}(\text{Enc}(x), \Bar{e}),
\end{aligned}
\end{equation}
where $\text{Enc}$ is a convolutional feature extractor that extracts patch-wise features, $D_{\text{uncond.}}$ ( $D_{\text{text}}^{\text{pat}}$ ) determine whether the patch is realistic (consistent with the text description) or not.
\noindent{\bf Shape discriminator.} In a similar manner, we have our patch-wise shape discriminator
\vspace{-2mm}
\begin{equation}\label{eq:patchshape}
\begin{aligned}
p^{\text{pix}} = D^{\text{pix}}(\text{Enc}(x, M)),
\end{aligned}
\end{equation}
where we first concatenate the image $x$ and shapes $M$ in the channel dimension, and then extracts patch-wise features by another convolutional feature extractor $\text{Enc}$. The probabilities $p^{\text{pix}}$ determine whether the patch is consistent with the given shape. Our patch-wise discriminators $D_{\text{uncond.}}^{\text{pat}}$, $D_{\text{text}}^{\text{pat}}$ and $D^{\text{pix}}$ resembles the PatchGAN~\cite{pix2pix2017} for the image-to-image translation task. Compared with the global discriminators in AttnGAN~\cite{xu2017attngan}, the patch-wise discriminators not only reduce the model size and thus enable generating higher resolution images, but also increase the quality of generated images; see Table~\ref{tab:quantative} for experimental evidence.
\noindent{\bf Object-wise discriminators.} Given an image $x$, bounding boxes of objects $B_{1:T}$ and their shapes $M$, we propose the following object-wise discriminators:
\vspace{-2mm}
\begin{equation}\label{eq:instD}
\begin{aligned}
\{h_t^{\text{obj}}\}_{t=1}^T = &\text{FastRCNN}(x, M, B_{1:T}), \\
p_t^{\text{obj,un}} =D_{\text{uncond.}}^{\text{obj}}(h_t^{\text{obj}}),& \quad p_t^{\text{obj,con}} = D^{\text{obj}}(h_t^{\text{obj}}, e_{t}^{\text{g}}, c_t^{\text{obj}}).
\end{aligned}
\end{equation}
Here, we first concatenate the image $x$ and shapes $M$ and extract a region feature vector $h_t^{\text{obj}}$ for each bounding box through a Fast R-CNN model~\cite{girshick2015fast} with an ROI-align layer~\cite{he2017mask}; see Fig.~\ref{fig:discriminator}(a). Then similar to the patch-wise discriminator~\eqref{eq:patchD}, the unconditional (conditional) probabilities $p_t^{\text{obj,un}}$ ( $p_t^{\text{obj,con}}$) determine whether the $t$'th object is realistic (consistent with its class label $e_{t}^{\text{g}}$ and its text context information $c_t^{\text{obj}}$) or not; see Fig.~\ref{fig:discriminator}(b). Here, $e_{t}^{\text{g}}$ is the GloVe embedding of the class label and $c_t^{\text{obj}}$ is its text context information defined in \eqref{eq:AttnGAN}.
All discriminators are trained by the traditional cross entropy loss~\cite{goodfellow2014generative}.
\subsubsection{Loss function for the image generator}
The generator's GAN loss is a weighted sum of these discriminators' loss, \textit{i.e.},
\begin{equation*}
\begin{aligned}
&\mathcal{L}_{\text{GAN}}(G) = - \frac{\lambda_{\text{obj}}}{T} \sum_{t=1}^{T} \left( \underbrace{\log p_t^{\text{obj,un}} }_\text{obj uncond. loss} + \underbrace{ \log p_t^{\text{obj,con}} }_\text{obj cond. loss} \right)\\
& - \frac{1}{N^{\text{pat}}} \sum_{j=1}^{N^{\text{pat}}} \left( \underbrace{\log p_j^{\text{pat,un}} }_\text{uncond. loss} + \underbrace{ \lambda_{\text{txt}} \log p_j^{\text{pat,con}} }_\text{text cond. loss} + \underbrace{ \lambda_{\text{pix}} \log p_j^{\text{pix}} }_\text{shape cond. loss}\right).
\end{aligned}
\end{equation*}
Here, $T$ is the number of bounding boxes, $N^{\text{pat}}$ is the number of regular patches, $(\lambda_{\text{obj}}, \lambda_{\text{txt}}, \lambda_{\text{pix}})$ are the weights of the object-wise GAN loss, patch-wise text conditional loss and patch-wise shape conditional loss, respectively. We tried combining our discriminators with the spectral normalized projection discriminator \cite{miyato2018spectral,miyato2018cgans}, but did not see significant performance improvement. We report performance of the spectral normalized version in $\S$~\ref{sec:exp:ablation} and provide model architecture details in $\S$~\ref{sec:appendix}.
Combined with the deep multi-modal attentive similarity model (DAMSM) loss introduced in \cite{xu2017attngan}, our final image generator's loss is
\begin{equation}\label{eqn:totalloss}
\begin{aligned}
\mathcal{L}_G = \mathcal{L}_{\text{GAN}} + \lambda_{\text{DAMSM}} \mathcal{L}_{\text{DAMSM}}
\end{aligned}
\end{equation}
where $\lambda_{\text{damsm}}$ is a hyper-parameter to be tuned. Here, the DAMSM loss is a word level fine-grained image-text matching loss computed, which will be elaborated in $\S$~\ref{sec:appendix}. Based on the experiments on a held-out validation set, we set the hyperparameters in this section as: $\lambda_{\text{obj}} = 0.1, \lambda_{\text{txt}} = 0.1, \lambda_{\text{pix}} = 1$ and $\lambda_{\text{damsm}} = 100$.
\begin{remark}
Both the patch-wise and object-wise discriminators can be applied to different stages in the generation. We apply the patch-wise discriminator for every stage of the generation, following \cite{Han17stackgan2,pix2pix2017}, but only apply the object-wise discriminator at the final stage.
\end{remark}
\section{Experiments}
\label{sec:exp}
\noindent \textbf{Dataset.} We use the COCO dataset \cite{LinMBHPRDZ14} for evaluation. It contains 80 object classes, where each image is associated with object-wise annotations (\textit{i.e.}, bounding boxes and shapes) and 5 text descriptions. We use the official 2014 train (over 80K images) and validation (over 40K images) splits for training and test stages, respectively.
\begin{table}
\caption{\small The quantitative experiments. Methods marked with $0$, $1$ and $2$ respectively represent experiments using the predicted boxes and shapes, the ground-truth boxes and predicted shapes, and the ground-truth boxes and shapes. We use \textbf{bold}, $\ast$, and $\ast\ast$ to highlight the best performance under these three settings, respectively. The results of methods marked with $\dagger$ are those reported in the original papers. $\uparrow$ ($\downarrow$) means the higher (lower), the better.}
\centering
\scriptsize
\begin{tabular}[t]{|p{2.2cm}|c|c|c|}\hline
{Methods} &{Inception $\uparrow$} &{FID $\downarrow$} &{R-prcn ($\%$) $\uparrow$}\\ \hline\hline
{Obj-GAN$^{0}$} &$\mathbf{27.37 \pm 0.22}$ &$\mathbf{25.85}$ &$86.20 \pm 2.98$ \\ \hline
{Obj-GAN$^{1}$} &$27.96 \pm 0.39^{\ast}$ &$24.19^{\ast}$ &$88.36 \pm 2.82$ \\ \hline
{Obj-GAN$^{2}$} &$29.89 \pm 0.22^{\ast\ast}$ &$20.75^{\ast\ast}$ &$89.59 \pm 2.67$ \\ \hline
{P-AttnGAN w/ Lyt$^{0}$} &$18.84 \pm 0.29$ &$59.02$ &$65.71 \pm 3.74$ \\ \hline
{P-AttnGAN w/ Lyt$^{1}$} &$19.32 \pm 0.29$ &$54.96$ &$68.40 \pm 3.79$ \\ \hline
{P-AttnGAN w/ Lyt$^{2}$} &$20.81 \pm 0.16$ &$48.47$ &$70.94 \pm 3.70$ \\ \hline
{P-AttnGAN} &$26.31 \pm 0.43$ &$41.51$ &$86.71 \pm 2.97$ \\ \hline
{Obj-GAN w/ SN$^{0}$} &$26.97 \pm 0.31$ &$29.07$ &$\mathbf{86.84 \pm 2.82}$ \\ \hline
{Obj-GAN w/ SN$^{1}$} &$27.41 \pm 0.17$ &$27.26$ &$88.70 \pm 2.65^{\ast}$ \\ \hline
{Obj-GAN w/ SN$^{2}$} &$28.75 \pm 0.32$ &$23.37$ &$89.97 \pm 2.56^{\ast\ast}$ \\ \hline \hline
{Reed \textit{et al} \cite{reed2016generative}$\dagger$} &$7.88 \pm 0.07$ &n/a &n/a \\ \hline
{StackGAN \cite{Han16stackgan}$\dagger$} &$8.45 \pm 0.03$ &n/a &n/a \\ \hline
{AttnGAN \cite{xu2017attngan}} &$23.79 \pm 0.32$ &$28.76$ &$82.98 \pm 3.15$ \\ \hline
{vmGAN \cite{zhang18vmgan}$\dagger$} &$9.94 \pm 0.12$ &n/a &n/a \\ \hline
{Sg2Im \cite{johnson2018image}$\dagger$} &$6.7 \pm 0.1$ &n/a &n/a \\ \hline
{Infer \cite{hong2018inferring}$^{0}\dagger$} &$11.46 \pm 0.09$ &n/a &n/a \\ \hline
{Infer \cite{hong2018inferring}$^{1}\dagger$} &$11.94 \pm 0.09$ &n/a &n/a \\ \hline
{Infer \cite{hong2018inferring}$^{2}\dagger$} &$12.40 \pm 0.08$ &n/a &n/a \\ \hline
{Obj-GAN-SOTA$^{0}$} &$30.29 \pm 0.33$ &$25.64$ &$91.05 \pm 2.34$ \\ \hline
{Obj-GAN-SOTA$^{1}$} &$30.91 \pm 0.29$ &$24.28$ &$92.54 \pm 2.16$ \\ \hline
{Obj-GAN-SOTA$^{2}$} &$32.79 \pm 0.21$ &$21.21$ &$93.39 \pm 2.08$ \\ \hline
\end{tabular}
\label{tab:quantative}
\vspace{-13pt}
\end{table}
\noindent \textbf{Evaluation metrics.} We use the Inception score~\cite{Salimans2016} and \textit{Fr\'echet inception distance} (FID)~\cite{heusel2017gans} score as the quantitative evaluation metrics. In our experiments, we found that Inception score can be saturated, even over-fitted, while FID is a more robust measure and aligns better with human qualitative evaluation. Following~\cite{xu2017attngan}, we also use R-precision, a common evaluation metric for ranking retrieval results, to evaluate whether the generated image is well conditioned on the given text description. More specifically, given a pre-trained image-to-text retrieval model, we use generated images to query their corresponding text descriptions. First, given generated image $\hat{x}$ conditioned on sentence $s$ and 99 random sampled sentences $\{s'_i: 1\le i \le 99\}$, we rank these 100 sentences by the pre-trained image-to-text retrieval model. If the ground truth sentence $s$ is ranked highest, we count this a success retrieval. For all the images in the test dataset, we perform this retrieval task once and finally count the percentage of success retrievals as the R-precision score.
It is important to point out that none of these quantitative metrics are perfect. Better metrics are required to evaluate image generation qualities in complicated scenes. In fact, the Inception score completely fails in evaluating the semantic layout of the generated images. The R-precision score depends on the pre-trained image-to-text retrieval model it uses, and can only capture the aspects that the retrieval model is able to capture. The pre-trained model we use is still limited in capturing the relations between objects in complicated scenes, so is our R-precision score.
\noindent \textbf{Quantitative evaluation.} We compute these three metrics under two settings for the full validation dataset.
\noindent \textbf{Qualitative evaluation.} Apart from the quantitative evaluation, we also visualize the outputs of all ablative versions of Obj-GAN and the state-of-the-art methods (\textit{i.e.}, \cite{xu2017attngan}) whose pre-trained models are publicly available.
\begin{figure*}[tb]
\begin{center}
\includegraphics[width=1\linewidth]{overall_vis}
\end{center}
\vspace{-18pt}
\caption{\small The overall qualitative comparison. All images are generated without the usage of any ground-truth information.}
\label{fig:overall_qualitative_comparison}
\vspace{-17pt}
\end{figure*}
\subsection{Ablation study}
\label{sec:exp:ablation}
In this section, we first evaluate the effectiveness of the object-driven attention. Next, we compare the object-driven attention mechanism with the grid attention mechanism. Then, we evaluate the impact of the spectral normalization for Obj-GAN. We use Fig.~\ref{fig:overall_qualitative_comparison} and the higher half of Table~\ref{tab:quantative} to present the comparison among different ablative versions of Obj-GAN. Note that all ablative versions have been trained with batch size $16$ for $60$ epochs. In addition, we use the lower half of Table~\ref{tab:quantative} to show the comparison between Obj-GAN and previous methods. Finally, we validated the Obj-GAN's generalization ability on the novel text descriptions.
\noindent \textbf{Object-driven attention.} To evaluate the efficacy of the object-driven attention mechanism, we implement a baseline, named P-AttnGAN w/ Lyt, by disabling the object-driven attention mechanism in Obj-GAN. In essence, P-AttnGAN w/ Lyt can be considered as an improved version of AttnGAN with the patch-wise discriminator (abbreviated as the prefix ``P-" in name) and the modules (\textit{e.g.}, shape discriminator) for handling the conditional layout (abbreviated as ``Lyt"). Moreover, it can also be considered as a modified implementation of \cite{hong2018inferring}, which resembles their two-step (layout-image) generation. Note that there are three key differences between P-AttnGAN w/ Lyt and \cite{hong2018inferring}: (i) P-AttnGAN w/ Lyt has a multi-stage image generator that gradually increases the generated resolution and refines the generated images, while \cite{hong2018inferring} has a single-stage image generator. (ii) With the help of the grid attentive module, P-AttnGAN w/ Lyt is able to utilize the fine-grained word-level information, while \cite{hong2018inferring} conditions on the global sentence information. (iii) The third difference lies in their loss functions: P-AttnGAN w/ Lyt uses the DAMSM loss in \eqref{eqn:totalloss} to penalize the mismatch between the generated images and the input text descriptions, while \cite{hong2018inferring} uses the perceptual loss to penalize the mismatch between the generated images and the ground-truth images. As shown in Table~\ref{tab:quantative}, P-AttnGAN w/ Lyt yields higher Inception score than \cite{hong2018inferring} does.
We compare Obj-GAN with P-AttnGAN w/ Lyt under three settings, with each corresponding to a set of conditional layout input, \textit{i.e.}, the predicted boxes $\&$ shapes, the ground-truth boxes $\&$ predicted boxes, and the ground-truth boxes $\&$ shapes. As presented in Table~\ref{tab:quantative}, Obj-GAN consistently outperforms P-AttnGAN w/ Lyt on all three metrics. In Fig.~\ref{fig:comp_w_honglak}, we use the same layout as the conditional input, and compare the visual quality of their generated images. An interesting phenomenon shown in Fig.~\ref{fig:comp_w_honglak} is that both the foreground objects (\textit{e.g.}, airplane and train) and the background (\textit{e.g.}, airport and trees) textures synthesized by Obj-GAN are much richer and smoother than those using P-AttnGAN w/ Lyt. The effectiveness of the object-driven attention for the foreground objects is easy to understand. The benefits for the background textures using the object-driven attention mechanism is probably due to the fact that it implicitly provides stronger signal that distinguishes the foreground. As such, the image generator may have richer guidance and clearer emphasis when synthesizing textures for a certain region.
\noindent \textbf{Grid attention vs. object-driven attention.} We compare Obj-GAN with P-AttnGAN herein, so as to compare the effects of the object-driven and the grid attention mechanisms. In Fig.~\ref{fig:comp_w_attngan}, we show the generated image of each method as well as the corresponding attention maps aligned on the right side. In a grid attention map, the brightness of a region reflects how much this region attended to the word above the map. As for the object-driven attention map, the word above each attention map is the most attended word by the highlighted object. The highlighted region of an object-driven attention map is the object shape.
As analyzed in $\S$~\ref{subsec:imagegenerator}, the reliability of grid attention weights depends on the quality of the previous layer's image region features. This makes the grid attention unreliable sometimes, especially for complex scenes. For example, the grid attention weights in Fig.~\ref{fig:comp_w_attngan} are unreliable because they are scattered (\textit{e.g.}, the attention map for ``man") and inaccurate. However, this is not a problem for the object-driven attention mechanism, because its attention weights are directly calculated from embedding vectors of words and class labels and are independent of image features. Moreover, as shown in Fig.~\ref{fig:attention} and Equ.~\eqref{eqn:bthiden}, the impact region of the object-driven attention context vector is bounded by the object shapes, which further enhances its semantics meaningfulness. As a result, the instance-driven attention significantly improves the visual quality of the generated images, as demonstrated in Fig.~\ref{fig:comp_w_attngan}. Moreover, the performance can be further improved if the semantic layout generation is improved. In the extreme case, Obj-GAN based on ground truth layout (Obj-GAN$^{2}$) has the best visual quality (the rightmost column of Fig.~\ref{fig:comp_w_attngan}) and the best quantitative evaluation (Table~\ref{tab:quantative}).
\begin{figure}[tb]
\begin{center}
\includegraphics[width=1\linewidth]{comp_w_honglak}
\end{center}
\vspace{-16pt}
\caption{\small Qualitative comparison with P-AttnGAN w/ Lyt.}
\label{fig:comp_w_honglak}
\vspace{-13pt}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=1\linewidth]{comp_w_attngan}
\end{center}
\vspace{-16pt}
\caption{\small Qualitative comparison with P-AttnGAN. The attention maps of each method are shown beside the generated image.}
\label{fig:comp_w_attngan}
\vspace{-13pt}
\end{figure}
\noindent \textbf{Obj-GAN w/ SN vs. Obj-GAN.}
We present the comparison between the cases with or without spectral normalization in the discriminators in Table~\ref{tab:quantative} and Fig.~\ref{fig:overall_qualitative_comparison}. We observe that there is no obvious improvement on the visual quality, but slightly worse on the quantitative metrics. We show more results and discussions in $\S$~\ref{sec:appendix}.
\noindent \textbf{Comparison with previous methods.}
To compare Obj-GAN with the previous methods, initialized by the Obj-GAN models in the ablation study, we trained Obj-GAN-SOTA with batch size $64$ for 10 more epochs.
In order to evaluate AttnGAN on FID, we conducted the evaluation on the officially released pre-trained model. Note that the Sg2Im \cite{johnson2018image} focuses on generating images from scene graphs and conducted the evaluation on a different split of COCO. However, we still included Sg2Im's results to reflect the broader context of the related topic. As shown in Table~\ref{tab:quantative}, Obj-GAN-SOTA outperforms all previous methods significantly. We notice that the increment of batch size does boost the Inception score and R-precision, but does not improve FID. The possible explanation is: with a larger batch size, the DAMSM loss (a ranking loss in essence) in \eqref{eqn:totalloss} plays a more important role and improves Inception and R-precision, but it does not focus on reducing FID between the generated images and the real ones.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=1\linewidth]{generalization}
\end{center}
\vspace{-13pt}
\caption{\small Generated images for novel descriptions.}
\label{fig:generalization}
\vspace{-13pt}
\end{figure}
\noindent \textbf{Generalization ability.}
We further investigate if Obj-GAN just memorizes the scenarios in COCO or it indeed learns the relations between the objects and their surroundings. To this end, we compose several descriptions which reflect novel scenarios that are unlikely to happen in the real-world, \textit{e.g.}, \emph{a decker bus is floating on top of a lake}, or \emph{a cat is catching a frisbee}. We use Obj-GAN to synthesize images for these rare scenes. The results in Fig.~\ref{fig:generalization} further demonstrate the good generalization ability of Obj-GAN.
\section{Conclusions}
\label{sec:conclusion}
In this paper, we have presented a multi-stage Object-driven Attentive Generative Adversarial Networks (Obj-GANs) for synthesizing images with complex scenes from the text descriptions. With a novel object-driven attention layer at each stage, our generators are able to utilize the fine-grained word/object-level information to gradually refine the synthesized image. We also proposed the Fast R-CNN based object-wise discriminators, each of which is paired with a conditional input of the generator and provides object-wise discrimination signal for that condition. Our Obj-GAN significantly outperforms previous state-of-the-art GAN models on various metrics on the large-scale challenging COCO benchmark. Extensive experiments demonstrate the effectiveness and generalization ability of Obj-GAN on text-to-image generation for complex scenes.
{\small
\bibliographystyle{ieee}
|
{
"timestamp": "2019-03-01T02:01:20",
"yymm": "1902",
"arxiv_id": "1902.10740",
"language": "en",
"url": "https://arxiv.org/abs/1902.10740"
}
|
\section{\label{sec1}Derivation of the condition for continuum bands}
We find how to determine the generalized Brillouin zone $C_\beta$, which determines the continuum bands. Here we number the $2M$ solutions in the characteristic equation $f\left(\beta,E\right)=0$ so as to satisfy
\begin{equation}
\left|\beta_1\right|\leq\left|\beta_2\right|\leq\cdots\leq\left|\beta_{2M-1}\right|\leq\left|\beta_{2M}\right|.
\label{eq5S}
\end{equation}
We find that the condition to get the continuum bands can be written as
\begin{equation}
\left|\beta_M\right|=\left|\beta_{M+1}\right|.
\label{eq6S}
\end{equation}
We note that Eq.~(\ref{eq6S}) was proposed in Ref.~\onlinecite{Yao2018} in the case of $M=1$, but for general $M$ its condition has not been known so far. To get Eq.~(\ref{eq6S}), we focus on boundary conditions in a finite open chain. The wave functions satisfying the boundary conditions are given by
\begin{equation}
\psi_{n,\mu}=\sum_{j=1}^{2M}\left(\beta_j\right)^n\phi_\mu^{\left(j\right)},~(\mu=1,\cdots,q).
\label{eq7S}
\end{equation}
The equations from the boundary conditions include the $2qM$ unknown variables $\phi_\mu^{\left(j\right)}$. The real-space eigen-equation $H\ket{\psi}=E\ket{\psi}$ fixes the ratio between the values of $\phi_\mu^{\left(j\right)}$ sharing the same value of $j$. Therefore one can reduce the $2qM$ variables to the $2M$ variables $\phi_\mu^{\left(j\right)}$ with a single value of $\mu$, e.g. $\mu=1$. As a result, we can get a set of equations for the $2M$ variables $\phi_1^{\left(j\right)}~(j=1,\cdots,2M)$:
\begin{eqnarray}
A\left( \begin{array}{c}
\phi_1^{\left(1\right)} \\
\vdots \\
\phi_1^{\left(2M\right)}
\end{array}\right)=0,
\label{eq8S}
\end{eqnarray}
where $A$ is a $2M\times 2M$ matrix. Existence of its nontrivial solutions requires
\begin{equation}
\det A=0.
\label{eq9S}
\end{equation}
The form of the matrix $A$ depends on the boundary conditions and the system size. An example of these equations are give in Eqs.~(\ref{eq14})-(\ref{eq16}).
Equation~(\ref{eq9S}) is an algebraic equation for $\beta_j~(j=1,\cdots,2M)$. By solving Eqs.~(\ref{eq8S}) and (\ref{eq9S}) one can calculate eigenenergies of the system with open boundaries. One cannot analytically solve these equations for a general system size. Nonetheless, our aim is to see how the solutions for a large system size form the continuum bands. Therefore we suppose the systems size to be quite large, and consider a condition to achieve densely distributed levels. We find that Eq.~(\ref{eq9S}) is expressed in a form
\begin{equation}
\sum_{{\cal S}}g\left(\beta_{{\cal S}1},\cdots,\beta_{{\cal S}2M}\right)\left(\beta_{{\cal S}\left(M+1\right)}\cdots\beta_{{\cal S}\left(2M\right)}\right)^{L+1}=0
\label{eq:S}
\end{equation}
where $L$ is the number of unit cells in an open chain, the sum is taken over all the permutations ${\cal S}$ for $2M$ objects, and $g$ is a function of the $2M$ variables dependent on the boundary conditions but independent of $L$. We now consider behavior of its solution for large $L$. When $\left|\beta_M\right|\neq\left|\beta_{M+1}\right|$, there is only one leading term proportional to $\left(\beta_{M+1}\cdots\beta_{2M}\right)^{L+1}$ in Eq.~(\ref{eq:S}) in the limit of a large $L$. Thus it leads to a single equation for $\beta_j~(j=1,\cdots,2M)$, which does not allow continuum bands.
When $\left|\beta_M\right|=\left|\beta_{M+1}\right|$, there are two leading terms proportional to $\left(\beta_{M}\beta_{M+2}\cdots\beta_{2M}\right)^{L+1}$ and to $\left(\beta_{M+1}\beta_{M+2}\cdots\beta_{2M}\right)^{L+1}$. In such a case, we can expect that the relative phase between $\beta_M$ and $\beta_{M+1}$ can be changed almost continuously for a large $L$, producing the continuum bands.
We can see this for a specific example in Sec.~\ref{sec3} in this Supplemental Material. We note that our condition Eq.~(\ref{eq6S}) is independent of any boundary conditions.
\section{\label{sec2}Generalized Brillouin zone in Hermitian systems}
We prove that the generalized Brillouin zone becomes a unit circle in Hermitian systems. In the following, we assume that the characteristic equation $f\left(\beta,E\right)=0$ is an algebraic equation for $\beta$ of $2M$-th degree. In Hermitian systems, it can be written as
\begin{equation}
\sum_{i=-M}^Ma_i\beta^i=0,
\label{eq1}
\end{equation}
where $a_{-i}=a_i^\ast~(i=1,\cdots,M)$ and $a_0$ is real. These coefficients are functions of the eigenenergy $E$ and the hopping terms included in the tight-binding models. Here the eigenenergy $E$ is real due to Hermiticity of the Hamiltonian. Equation~(\ref{eq1}) has $2M$ solutions, and they are numbered so as to satisfy
\begin{equation}
\left|\beta_1\right|\leq\left|\beta_2\right|\leq\cdots\leq\left|\beta_{2M-1}\right|\leq\left|\beta_{2M}\right|.
\label{eq2}
\end{equation}
Since one can rewrite Eq.~(\ref{eq1}) as
\begin{equation}
\sum_{i=-M}^M\left(a_{-i}\right)^\ast\left(\beta^\ast\right)^{-i}=
\sum_{i=-M}^M a_{i}\left(\beta^\ast\right)^{-i}=0,
\label{eq3}
\end{equation}
the solutions always appear in pairs: $\left(\beta,1/\beta^\ast\right)$, and we get $\beta_{2M+1-j}=1/\beta_j^\ast$. In particular, one can get the relationship between $\beta_M$ and $\beta_{M+1}$ as $\beta_{M+1}=1/\beta_{M}^\ast$. Therefore the condition for the continuum bands $\left|\beta_M\right|=\left|\beta_{M+1}\right|$ can be rewritten as
\begin{equation}
\left|\beta_M\right|=\left|\beta_{M+1}\right|=1.
\label{eq4}
\end{equation}
Thus we can conclude that the generalized Brillouin zone becomes a unit circle in Hermitian systems.
\section{\label{sec3}Non-Hermitian SSH model}
\subsection{\label{sec3-1}Characteristic equation}
\begin{figure*}[]
\includegraphics[width=14.5cm]{00.eps}
\caption{\label{fig1}Schematic figure of the non-Hermitian SSH model. The system has the asymmetric intracell hopping amplitudes $t_1\pm\gamma_1/2$, the asymmetric nearest-neighbor intercell ones $t_2\pm\gamma_2/2$, and the symmetric next-nearest-neighbor one $t_3$. The dotted boxes indicate the unit cell.}
\end{figure*}
We give the non-Hermitian SSH model as shown in Fig.~\ref{fig1}. The Hamiltonian can be written as
\begin{eqnarray}
H&=&\sum_n\left[\left(t_1+\frac{\gamma_1}{2}\right)c_{n,{\rm A}}^\dag c_{n,{\rm B}}+\left(t_1-\frac{\gamma_1}{2}\right)c_{n,{\rm B}}^\dag c_{n,{\rm A}}\right. \nonumber\\
&&+\left(t_2+\frac{\gamma_2}{2}\right)c_{n,{\rm B}}^\dag c_{n+1,{\rm A}}+\left(t_2-\frac{\gamma_2}{2}\right)c_{n+1,{\rm A}}^\dag c_{n,{\rm B}} \nonumber\\
&&\left.+t_3\left(c_{n,{\rm A}}^\dag c_{n+1,{\rm B}}+c_{n+1,{\rm B}}^\dag c_{n,{\rm A}}\right)\right],
\label{eq5}
\end{eqnarray}
where $t_1,t_2,t_3,\gamma_1$, and $\gamma_2$ are real for simplicity. Eigenvectors are written as $\ket{\psi}=\left(\psi_{1,{\rm A}},\psi_{1,{\rm B}},\cdots,\psi_{L,{\rm A}},\psi_{L,{\rm B}}\right)^{\rm T}$ in an open chain. Then we can explicitly write the equation for $\psi_{n,\mu}$ as
\begin{eqnarray}
\left(t_2-\frac{\gamma_2}{2}\right)\psi_{n-1,{\rm B}}+\left(t_1+\frac{\gamma_1}{2}\right)\psi_{n,{\rm B}}+t_3\psi_{n+1,{\rm B}}&=&E\psi_{n,{\rm A}}, \nonumber\\
t_3\psi_{n-1,{\rm A}}+\left(t_1-\frac{\gamma_1}{2}\right)\psi_{n,{\rm A}}+\left(t_2+\frac{\gamma_2}{2}\right)\psi_{n+1,{\rm A}}&=&E\psi_{n,{\rm B}}. \nonumber\\
\label{eq6}
\end{eqnarray}
Here we take the ansatz for the wave function as a linear combination:
\begin{equation}
\psi_{n,\mu}=\sum_j\phi_{n,\mu}^{\left(j\right)},~\phi_{n,\mu}^{\left(j\right)}=\left(\beta_j\right)^n\phi_\mu^{\left(j\right)},~\left(\mu={\rm A},{\rm B}\right),
\label{eq7}
\end{equation}
where $\phi_{n,{\rm A}}^{\left(j\right)}$ and $\phi_{n,{\rm B}}^{\left(j\right)}$ takes the exponential form
\begin{equation}
\left(\phi_{n,{\rm A}}^{\left(j\right)},\phi_{n,{\rm B}}^{\left(j\right)}\right)=\beta_j^n\left(\phi_{\rm A}^{\left(j\right)},\phi_{\rm B}^{\left(j\right)}\right).
\label{eq8}
\end{equation}
By substituting the exponential form Eq.~(\ref{eq8}) into Eq.~(\ref{eq6}), one can obtain the bulk eigen-equations
\begin{eqnarray}
\left[\left(t_2-\frac{\gamma_2}{2}\right)\beta^{-1}+\left(t_1+\frac{\gamma_1}{2}\right)+t_3\beta\right]\phi_{\rm B}&=&E\phi_{\rm A}, \nonumber\\
\left[t_3\beta^{-1}+\left(t_1-\frac{\gamma_1}{2}\right)+\left(t_2+\frac{\gamma_2}{2}\right)\beta\right]\phi_{\rm A}&=&E\phi_{\rm B}. \nonumber\\
\label{eq9}
\end{eqnarray}
Therefore the characteristic equation can be written as
\begin{eqnarray}
&&\left[\left(t_2-\frac{\gamma_2}{2}\right)\beta^{-1}+\left(t_1+\frac{\gamma_1}{2}\right)+t_3\beta\right] \nonumber\\
&&\times\left[t_3\beta^{-1}+\left(t_1-\frac{\gamma_1}{2}\right)+\left(t_2+\frac{\gamma_2}{2}\right)\beta\right]=E^2.
\label{eq10}
\end{eqnarray}
We note that it is the eigenvalue equation for the generalized Bloch Hamiltonian $H\left(\beta\right)$ with $\beta\equiv{\rm e}^{ik},~k\in{\mathbb C}$. Henceforth we assume $t_2\neq\pm\gamma_2/2$ and $t_3\neq0$, meaning that Eq.~(\ref{eq10}) is a quartic equation for $\beta$, having four solutions $\beta_j~(j=1,\cdots,4)$. They are numbered so as to satisfy $\left|\beta_1\right|\leq\left|\beta_2\right|\leq\left|\beta_3\right|\leq\left|\beta_4\right|$.
\subsection{\label{sec3-2} Derivation of the boundary equation and the condition for the continuum bands}
First of all, we derive an equation coming from the boundary conditions in our model. In the following, we set the number of unit cells to be $L$. We write down the boundary condition for $\psi_{n,\mu}$:
\begin{eqnarray}
\left(t_1+\frac{\gamma_1}{2}\right)\psi_{1,{\rm B}}+t_3\psi_{2,{\rm B}}&=&E\psi_{1,{\rm A}}, \nonumber\\
\left(t_1-\frac{\gamma_1}{2}\right)\psi_{1,{\rm A}}+\left(t_2+\frac{\gamma_2}{2}\right)\psi_{2,{\rm A}}&=&E\psi_{1,{\rm B}}, \nonumber\\
\left(t_2+\frac{\gamma_2}{2}\right)\psi_{L-1,{\rm B}}+\left(t_1+\frac{\gamma_1}{2}\right)\psi_{L,{\rm B}}&=&E\psi_{L,{\rm A}}, \nonumber\\
t_3\psi_{L-1.{\rm A}}+\left(t_1-\frac{\gamma_1}{2}\right)\psi_{L,{\rm A}}&=&E\psi_{L,{\rm B}}.
\label{eq11}
\end{eqnarray}
Then, by substituting the general solution
\begin{equation}
\psi_{n,\mu}=\sum_{j=1}^4\left(\beta_j\right)^n\phi_\mu^{\left(j\right)},~\left(\mu={\rm A},{\rm B}\right)
\label{eq12}
\end{equation}
to Eq.~(\ref{eq11}), one can obtain the four equations for the eight coefficients $\phi_\mu^{\left(j\right)}~\left(j=1,\cdots,4,~\mu={\rm A},{\rm B}\right)$. By recalling that these coefficients satisfy
\begin{eqnarray}
\phi_{\rm A}^{\left(j\right)}&=&\frac{E}{t_3\beta_j^{-1}+\left(t_1-\gamma_1/2\right)+\left(t_2+\gamma_2/2\right)\beta_j}\phi_{\rm B}^{\left(j\right)}, \nonumber\\
\phi_{\rm B}^{\left(j\right)}&=&\frac{E}{\left(t_2-\gamma_2/2\right)\beta_j^{-1}+\left(t_1+\gamma_1/2\right)+t_3\beta_j}\phi_{\rm B}^{\left(j\right)}, \nonumber\\
\label{eq13}
\end{eqnarray}
from the bulk eigen-equation Eq.~(\ref{eq9}), we can reduce the problem into four linear equations for the four coefficients $\phi_{\rm A}^{\left(j\right)}~\left(j=1,\cdots,4\right)$. Therefore they can be written as
\begin{eqnarray}
\left( \begin{array}{cccc}
1 & 1 & 1 & 1 \vspace{3pt}\\
\beta_1^{L+1} & \beta_2^{L+1} & \beta_3^{L+1} & \beta_4^{L+1} \vspace{3pt}\\
X_1 & X_2 & X_3 & X_4 \vspace{3pt}\\
\beta_1^{L+1}X_1 & \beta_2^{L+1}X_2 & \beta_3^{L+1}X_3 & \beta_4^{L+1}X_4
\end{array}\right)\left( \begin{array}{c}
\phi_{\rm A}^{\left(1\right)} \vspace{3pt}\\
\phi_{\rm A}^{\left(2\right)} \vspace{3pt}\\
\phi_{\rm A}^{\left(3\right)} \vspace{3pt}\\
\phi_{\rm A}^{\left(4\right)}
\end{array}\right)=0, \nonumber\\
\label{eq14}
\end{eqnarray}
where $X_j~(j=1,\cdots,4)$ are defined as
\begin{equation}
X_j=\frac{1}{\left(t_2-\gamma_2/2\right)\beta_j^{-1}+\left(t_1+\gamma_1/2\right)+t_3\beta_j}.
\label{eq15}
\end{equation}
The condition for the determinant of the $4\times4$ matrix in Eq.~(\ref{eq14}) to vanish leads to the boundary equation as
\begin{eqnarray}
&&\left[\left(\beta_1\beta_2\right)^{L+1}+\left(\beta_3\beta_4\right)^{L+1}\right]\left(X_1-X_2\right)\left(X_3-X_4\right) \nonumber\\
&&+\left[\left(\beta_1\beta_4\right)^{L+1}+\left(\beta_2\beta_3\right)^{L+1}\right]\left(X_1-X_4\right)\left(X_2-X_3\right) \nonumber\\
&&-\left[\left(\beta_1\beta_3\right)^{L+1}+\left(\beta_2\beta_4\right)^{L+1}\right]\left(X_1-X_3\right)\left(X_2-X_4\right)=0. \nonumber\\
\label{eq16}
\end{eqnarray}
Next we demonstrate that the condition for the continuum bands is given by $\left|\beta_2\right|=\left|\beta_3\right|$. We have to consider the condition when the solutions of (\ref{eq16}) are densely distributed for a large $L$. In the previous work~\cite{Yao2018}, a simpler case of the characteristic equation being a quadratic equation for $\beta$ is studied, and the condition for the continuum bands are shown as $|\beta_1|=|\beta_2|$, i.e. the absolute values of the two solutions being equal. A natural extension of this result $\left|\beta_1\right|=\left|\beta_2\right|$ in the case of $M=2$ to a general value of $M$ is $\left|\beta_i\right|=\left|\beta_j\right|$ for some $i$, $j$ among the $2M$ solutions $\beta=\beta_j~(j=1,\cdots,2M)$. Nevertheless, we find that it is not true.
In the present case of $M=2$, if $\left|\beta_3\right|=\left|\beta_4\right|$, the only leading term in Eq.~(\ref{eq16}) is the term proportional to $\left(\beta_3\beta_4\right)^{L+1}$, leading to an equation $\left(X_1-X_2\right)\left(X_3-X_4\right)=0$ in the thermodynamic limit $L\rightarrow\infty$. By combining this equation with the characteristic equation (\ref{eq10}), the eigenenergies are restricted to discrete values, and it cannot represent continuum bands. On the other hand, if we employ a condition
\begin{equation}
\left|\beta_2\right|=\left|\beta_3\right|,
\label{eq17}
\end{equation}
Eq.~(\ref{eq16}) has two leading terms, $\left(\beta_2\beta_4\right)^{L+1}$ and $\left(\beta_3\beta_4\right)^{L+1}$ for a large $L$, and Eq.~(\ref{eq16}) can be rewritten as
\begin{equation}
\left(\frac{\beta_{2}}{\beta_3}\right)^{L+1}=\frac{\left(X_1-X_2\right)\left(X_3-X_4\right)}{\left(X_1-X_3\right)\left(X_2-X_4\right)}.
\label{eq18}
\end{equation}
For a large $L$, this equation allows a dense set of solutions when the relative phase between $\beta_2$ and $\beta_3$ is continuously changed. Therefore we conclude that Eq.~(\ref{eq17}) is an appropriate condition for the continuum bands. We here emphasize that Eq.~(\ref{eq17}) is now independent of any boundary conditions. If we change the form of the boundary condition, Eq.~(\ref{eq16}) may change; nonetheless, Eq.~(\ref{eq17}) remains the same, and together with the characteristic equation we can obtain the continuum bands.
With a special choice of parameters, the degree of the characteristic equation may become an odd number. For example, when $t_2=-\gamma_2/2$, Eq.~(\ref{eq10}) becomes a cubic equation for $\beta$. In this case, we can still regard Eq.~(\ref{eq10}) as a quartic equation with the limit $t_2\rightarrow-\gamma_2/2$ and therefore, by adding another solution $\beta=\infty$, and our result Eq.~(\ref{eq18}) holds good with four solutions $\beta_1$, $\beta_2$, $\beta_3$, $\beta_4(=\infty)$. Thus in general, when the degree of the characteristic equation is odd, one can treat it as a limiting case of an algebraic equation of an even degree, by formally adding a solution $\beta=0$ or $\beta=\infty$, depending on the system, and the condition for the continuum bands remains valid.
When $t_3=0$ in our model, the characteristic equation Eq.~(\ref{eq10}) for $\beta$ becomes a quadratic equation, and the condition for the continuum bands is $\left|\beta_1\right|=\left|\beta_2\right|$, where $\beta_1$ and $\beta_2$ are two solutions of the characteristic equation. Such cases of quadratic characteristic equations have already been studied in Ref.~\onlinecite{Yao2018}.
\subsection{\label{sec3-3}Generalized Brillouin zone}
\begin{figure}[]
\includegraphics[width=8.5cm]{BZ-SM.eps}
\caption{\label{fig:BZ-SM}Trajectories of $\beta$. (a) Trajectory of $\beta_2$ and $\beta_3$ with $\left|\beta_2\right|=\left|\beta_3\right|$ and (b) trajectories of $\beta_i$ and $\beta_j$ with $\left|\beta_i\right|=\left|\beta_j\right|$. (a) corresponds to the generalized Brillouin zone $C_\beta$. The parameters are same as Fig.~3~(c-1) in the main text: $t_1=0.3,t_2=0.5,t_3=1/5,\gamma_1=5/3$, and $\gamma_2=1/3$.}
\end{figure}
As an example we show the generalized Brillouin zone $C_\beta$ for parameters $t_1=0.3,t_2=0.5,t_3=1/5,\gamma_1=5/3$, and $\gamma_2=1/3$ in our model. We impose the condition $\left|\beta_2\right|=\left|\beta_3\right|$ and calculate the trajectories of $\beta_2$ and $\beta_3$ shown in Fig.~\ref{fig:BZ-SM}~(a). For comparison we investigate what happens if we impose a condition $\left|\beta_i\right|=\left|\beta_j\right|$ for some $i$ and $j$ among the four solutions instead. We show the trajectory of $\beta_i$ and $\beta_j$ in Fig.~\ref{fig:BZ-SM}~(b). In Ref.~\onlinecite{Yao2018}, it was suggested that $\left|\beta_i\right|=\left|\beta_j\right|$ is a necessary condition for the continuum bands. Nonetheless, it is not sufficient to get $C_\beta$, and we should restrict this condition to be $\left|\beta_2\right|=\left|\beta_3\right|$, leading to $C_\beta$ in Fig.~\ref{fig:BZ-SM}~(a).
In some cases, the generalized Brillouin zone $C_\beta$ can have cusps as seen in Fig.~\ref{fig:BZ-SM}~(a). It appears when three of the four solutions of $\beta$ share the same absolute value. Suppose $\left|\beta_1\right|<\left|\beta_2\right|=\left|\beta_3\right|<\left|\beta_4\right|$, and as we go along $C_\beta$, $\left|\beta_1\right|$ approaches $\left|\beta_2\right|=\left|\beta_3\right|$. Then, when $\left|\beta_1\right|=\left|\beta_2\right|=\left|\beta_3\right|$, the behavior of the solutions satisfying $\left|\beta_2\right|=\left|\beta_3\right|$ changes, and there appears a cusp in $C_\beta$, as one can compare Figs.~\ref{fig:BZ-SM}~(a) and (b).
\section{\label{sec4}Methods for calculating the generalized Brillouin zone}
We explain the method to get the trajectory of $\beta$ satisfying the continuum-band condition (\ref{eq17}). We first express the characteristic equation Eq.~(\ref{eq10}) as $E^2=F\left(\beta\right)$. Suppose the two solutions $\beta$ and $\beta^\prime$ have the same absolute values: $\left|\beta\right|=\left|\beta^\prime\right|$. Then we have
\begin{equation}
\beta^\prime=\beta{\rm e}^{i\theta},
\label{eq19}
\end{equation}
where $\theta$ is real. Then, by taking the difference between two equations:
\begin{equation}
E^2=F\left(\beta\right),~E^2=F\left(\beta{\rm e}^{i\theta}\right),
\label{eq20}
\end{equation}
we get
\begin{eqnarray}
&&0=\left(t_2-\frac{\gamma_2}{2}\right)t_3\beta^{-2}\left(1-{\rm e}^{-2i\theta}\right) \nonumber\\
&&+\left[\left(t_1-\frac{\gamma_1}{2}\right)\left(t_2-\frac{\gamma_2}{2}\right)+\left(t_1+\frac{\gamma_1}{2}\right)t_3\right]\beta^{-1}\left(1-{\rm e}^{-i\theta}\right) \nonumber\\
&&+\left[\left(t_1-\frac{\gamma_1}{2}\right)t_3+\left(t_1+\frac{\gamma_1}{2}\right)\left(t_2+\frac{\gamma_2}{2}\right)\right]\beta\left(1-{\rm e}^{i\theta}\right) \nonumber\\
&&+\left(t_2+\frac{\gamma_2}{2}\right)t_3\beta^2\left(1-{\rm e}^{2i\theta}\right).
\label{eq21}
\end{eqnarray}
This equation allows us to calculate $\beta$ for a given value of $\theta\in\left(0,2\pi\right)$. Then we obtain a set of values of $\beta$ that satisfies $\left|\beta\right|=\left|\beta^\prime\right|$. Here we should further constrain the values of $\beta$ by Eq.~(\ref{eq17}). Namely the absolute values of $\beta$ and $\beta^\prime$ should be the second and third largest ones among the four solutions. By selecting the values of $\beta$ and $\beta^\prime$ satisfying this condition, we can get the generalized Brillouin zone.
\section{\label{sec5}$Q$ matrix and winding number in 1D non-Hermitian systems with chiral symmetry}
\subsection{\label{sec5-1}Multi-bands model}
We focus on 1D non-Hermitian systems with chiral symmetry which have an arbitrary number of bands. In the following, the systems have a gap around $E=0$, but without exceptional points. Since the systems always have pairs of the eigenenergy $\left(E,-E\right)$ due to chiral symmetry, we can assume that the bands are composed of $N$ occupied bands with $E=-E_i~(i=1,\cdots,N)$ and $N$ unoccupied bands with $E=E_i~(i=1,\cdots,N)$. By taking an appropriate basis, the Hamiltonian in the systems can be written as the block off-diagonal form:
\begin{eqnarray}
H\left(\beta\right)=\left( \begin{array}{cc}
0 & R_+\left(\beta\right) \vspace{5pt}\\
R_-\left(\beta\right) & 0
\end{array}\right),
\label{eq23}
\end{eqnarray}
where $\beta\equiv{\rm e}^{ik},~k\in{\mathbb C}$ and $R_{\pm}\left(\beta\right)$ are $N\times N$ matrices. The chiral symmetry is expressed as
\begin{equation}
\sigma_zH(\beta)=-H(\beta)\sigma_z,\ \
\sigma_z\equiv\left( \begin{array}{cc}
\bm{1} & 0 \vspace{5pt}\\
0 & -\bm{1}
\end{array}\right),
\end{equation}
where $\bm{1}$ is an $N\times N$ identity matrix. Then the eigenvalue equations for the right and left eigenstates
\begin{eqnarray}
\ket{\psi_{R}\left(\beta\right)}&=&\left( \begin{array}{c}
\ket{a_{R}\left(\beta\right)} \vspace{5pt}\\
\ket{b_{R}\left(\beta\right)}
\end{array}\right), \nonumber \\
\bra{\psi_{L}\left(\beta\right)}&=&\left( \begin{array}{cc}
\bra{a_{L}\left(\beta\right)} & \bra{a_{R}\left(\beta\right)}
\end{array}\right),
\label{eq24}
\end{eqnarray}
are given by
\begin{eqnarray}
R_+\left(\beta\right)\ket{b_R\left(\beta\right)}&=&E\ket{a_R\left(\beta\right)}, \nonumber\\
R_-\left(\beta\right)\ket{a_R\left(\beta\right)}&=&E\ket{b_R\left(\beta\right)},
\label{eq25}
\end{eqnarray}
and
\begin{eqnarray}
\bra{a_L\left(\beta\right)}R_+\left(\beta\right)&=&E\bra{b_L\left(\beta\right)}, \nonumber\\
\bra{b_L\left(\beta\right)}R_-\left(\beta\right)&=&E\bra{a_L\left(\beta\right)},
\label{eq26}
\end{eqnarray}
respectively. For the right and left eigenstates, one can reduce Eqs.~(\ref{eq25}) and (\ref{eq26}) to
\begin{eqnarray}
R_+\left(\beta\right)R_-\left(\beta\right)\ket{a_R\left(\beta\right)}&=&E^2\ket{a_R\left(\beta\right)}, \nonumber\\
\bra{a_L\left(\beta\right)}R_+\left(\beta\right)R_-\left(\beta\right)&=&E^2\bra{a_L\left(\beta\right)},
\label{eq27}
\end{eqnarray}
respectively. Here we introduce the right and left eigenstates of the $N\times N$ matrix $R_+\left(\beta\right)R_-\left(\beta\right)$ as $\ket{a_{R/L,1}\left(\beta\right)},\cdots,\ket{a_{R/L,N}\left(\beta\right)}$, respectively, and the eigenvalues as $E_1^2\left(\beta\right),\cdots,E_N^2\left(\beta\right)$. Furthermore the right and left eigenstates satisfy
\begin{equation}
\Braket{a_{L,i}\left(\beta\right)|a_{R,j}\left(\beta\right)}=\delta_{ij}
\label{eq28}
\end{equation}
since one can take the biorthogonal basis~\cite{Brody2013,Shen2018}. Therefore we can obtain the biorthogonal eigenstates of the Hamiltonian Eq.~(\ref{eq23}) in the occupied bands for $i=1,\cdots,N$ as
\begin{eqnarray}
\ket{\psi_{R,i}\left(\beta\right)}&=&\frac{1}{\sqrt{2}}\left( \begin{array}{c}
\ket{a_{R,i}\left(\beta\right)} \vspace{5pt}\\
-\displaystyle\frac{1}{E_i\left(\beta\right)}R_-\left(\beta\right)\ket{a_{R,i}\left(\beta\right)}
\end{array}\right), \nonumber\\
\bra{\psi_{L,i}\left(\beta\right)}&=&\frac{1}{\sqrt{2}}\left( \begin{array}{cc}
\bra{a_{L,i}\left(\beta\right)} & -\displaystyle\frac{1}{E_i\left(\beta\right)}\bra{a_{L,i}R_+\left(\beta\right)}
\end{array}\right). \nonumber\\
\label{eq29}
\end{eqnarray}
Then the unoccupied eigenstates are given by $\ket{\widetilde{\psi}_{R/L,i}\left(\beta\right)}\equiv\sigma_z\ket{\psi_{R/L,i}\left(\beta\right)}$. In the systems, the $Q$ matrix $Q\left(\beta\right)$ can be defined as
\begin{equation}
Q\left(\beta\right)=\sum_{i=1}^N\left(\ket{\widetilde{\psi}_{R,i}\left(\beta\right)}\bra{\widetilde{\psi}_{L,i}\left(\beta\right)}-\ket{\psi_{R,i}\left(\beta\right)}\bra{\psi_{L,i}\left(\beta\right)}\right)
\label{eq30}
\end{equation}
and in the matrix form, it can be written as
\begin{widetext}
\begin{eqnarray}
Q\left(\beta\right)=\sum_{i=1}^N\frac{1}{E_i\left(\beta\right)}\left( \begin{array}{cc}
O & \ket{a_{R,i}\left(\beta\right)}\bra{a_{L,i}\left(\beta\right)}R_+\left(\beta\right) \vspace{5pt}\\
R_-\left(\beta\right)\ket{a_{R,i}\left(\beta\right)}\bra{a_{L,i}\left(\beta\right)} & O
\end{array}\right).
\label{eq31}
\end{eqnarray}
\end{widetext}
Here we define the matrix $q\left(\beta\right)$ and $q^{-1}\left(\beta\right)$ as
\begin{eqnarray}
q\left(\beta\right)&\equiv&\sum_{i=1}^N\frac{1}{E_i\left(\beta\right)}\ket{a_{R,i}\left(\beta\right)}\bra{a_{L,i}\left(\beta\right)}R_+\left(\beta\right), \nonumber\\
q^{-1}\left(\beta\right)&=&\sum_{i=1}^N\frac{1}{E_i\left(\beta\right)}R_-\left(\beta\right)\ket{a_{R,i}\left(\beta\right)}\bra{a_{L,i}\left(\beta\right)}, \nonumber\\
\label{eq32}
\end{eqnarray}
respectively. As a result, one can get the winding number $w$~\cite{Ryu2010,Yao2018} as
\begin{equation}
w=\frac{i}{2\pi}\int_{C_\beta}{\rm Tr}\left[{\rm d} q~q^{-1}\left(\beta\right)\right].
\label{eq33}
\end{equation}
\subsection{\label{sec5-2}Two-bands model}
In particular, we can explicitly write Eq.~(\ref{eq33}) in a two-band model. The Hamiltonian can be written as
\begin{equation}
H\left(\beta\right)=R_+\left(\beta\right)\sigma_++R_-\left(\beta\right)\sigma_-,
\label{eq34}
\end{equation}
where $\sigma_\pm=\left(\sigma_x\pm i\sigma_y\right)/2$. Then the eigenvalues are given by
\begin{equation}
E_\pm\left(\beta\right)=\pm\sqrt{R_+\left(\beta\right)R_-\left(\beta\right)}.
\label{eq35}
\end{equation}
The right and left eigenstates can be written down as
\begin{eqnarray}
\ket{u_{R,\pm}}&=&\frac{1}{\sqrt{2}\sqrt{R_+R_-}}\left( \begin{array}{c}
R_+ \vspace{5pt}\\
\sqrt{\pm R_+R_-}
\end{array}\right), \nonumber\\
\bra{u_{L,\pm}}&=&\frac{1}{\sqrt{2}\sqrt{R_+R_-}}\left( \begin{array}{cc}
R_- & \pm\sqrt{R_+R_-}
\end{array}\right), \nonumber\\
\label{eq36}
\end{eqnarray}
respectively. The subscript $+/-$ means that the eigenstates with $+/-$ have the eigenvalues $E_+$ or $E_-$, respectively. From Eq.~(\ref{eq36}), the $Q$ matrix $Q\left(\beta\right)$ can be written down by
\begin{eqnarray}
Q\left(\beta\right)&=&\ket{u_{R,+}\left(\beta\right)}\bra{u_{L,+}\left(\beta\right)}-\ket{u_{R,-}\left(\beta\right)}\bra{u_{L,-}\left(\beta\right)} \nonumber\\
&=&\frac{1}{\sqrt{R_+\left(\beta\right)R_-\left(\beta\right)}}\left( \begin{array}{cc}
0 & R_+\left(\beta\right) \\
R_-\left(\beta\right) & 0
\end{array}\right).
\label{eq37}
\end{eqnarray}
Therefore we can obtain $q=R_+/\sqrt{R_+R_-}$, and the winding number $w$ explicitly as
\begin{eqnarray}
w&=&\frac{i}{2\pi}\int_{C_\beta}{\rm d}q~q^{-1}\left(\beta\right) \nonumber\\
&=&\frac{i}{2\pi}\int_{C_\beta}{\rm d}\log q\left(\beta\right) \nonumber\\
&=&-\frac{1}{2\pi}\left[\arg q\left(\beta\right)\right]_{C_\beta} \nonumber\\
&=&-\frac{1}{2\pi}\frac{\left[\arg R_+\left(\beta\right)-\arg R_-\left(\beta\right)\right]_{C_\beta}}{2}.
\label{eq38}
\end{eqnarray}
Thus the winding number $w$ is determined by the change of the phase of $R_\pm\left(\beta\right)$ when $\beta$ goes along the generalized Brillouin zone $C_\beta$. On a complex plane, let $\ell_{\pm}$ denote the loops drawn by $R_{\pm}\left(\beta\right)$ when $\beta$ goes along $C_\beta$ in the counterclockwise way. Then $w$ is determined by the number of times that $\ell_{\pm}$ surround the origin ${\cal O}$. When neither $\ell_+$ nor $\ell_-$ surround ${\cal O}$, $w$ is zero. It takes a non-zero value when they simultaneously surround ${\cal O}$.
Here we can show the bulk-edge correspondence in this case. If parameters of the system can be continuously changed without closing the gap, the winding number $w$ does not change, and the topology of the systems remains invariant. Namely, if the systems after changing the values of the parameters have the zero-energy edge states, one can conclude that the original systems also have the zero-energy edge states. This can be proved even in non-Hermitian cases, following the proof in Hermitian systems~\cite{Ryu2002}.
Suppose we change the parameters of the system continuously, and at some values of the parameters $\ell_+$ and $\ell_-$ simultaneously pass the origin ${\cal O}$ on the ${\bm R}$ plane. At this time, the gap closing occurs because the energy eigenvalues are given by Eq.~(\ref{eq35}).
In Hermitian systems, $R_+^\ast=R_-$ holds, and two loops $\ell_+$ and $\ell_-$ are related by complex conjugation, so
\begin{equation}
\left[\arg R_-\left(\beta\right)\right]_{C_\beta}=-\left[\arg R_+\left(\beta\right)\right]_{C_\beta}.
\label{eq100}
\end{equation}
In contrast with these cases, in some non-Hermitian systems, only one of the two loops $\ell_+$ and $\ell_-$ passes the origin ${\cal O}$. In that case, only one of the two values $R_+$ or $R_-$ become $0$, leading to $E=0$, and the Hamiltonian is written as the Jordan normal form. It represents that the system has exceptional points. In Figs.~3~(a) and 4~(a) in the main text, the phase with the exceptional point extends over a finite region due to the following reason. When $R_+=0$ (or $R_-=0$) has non-real solutions, these two solutions $\beta_1^+$ and $\beta_2^+$ (or $\beta_1^-$ and $\beta_2^-$) should be complex conjugate, having the same absolute value. It is because all coefficients of $R_+=0$ (or $R_-=0$) are real. This persists as long as the following condition are satisfied: $R_+=0$ (or $R_-=0$) has non-real two solutions satisfying $\left|\beta_1^-\right|<\left|\beta_1^+\right|=\left|\beta_2^+\right|<\left|\beta_2^-\right|$ (or $\left|\beta_1^+\right|<\left|\beta_1^-\right|=\left|\beta_2^-\right|<\left|\beta_2^+\right|$). We note that the winding number is not well-defined when the system has the exceptional pints because the gap is closed.
\section{\label{sec6}Calculation on another model}
\begin{figure}[]
\includegraphics[width=8cm]{TB.eps}
\caption{\label{fig2}Schematic figure of the tight-binding model proposed in the previous work~\cite{Lee2016}. This model includes the gain on sublattice $\alpha$ and the loss on sublattice $\beta$.}
\end{figure}
We investigate the non-Hermitian tight-binding model proposed in Ref.~\onlinecite{Lee2016} as shown in Fig.~\ref{fig2}. The previous work proposed that anomalous edge states, which are localized at the either end of a finite open chain appear in this model. In this supplemental material, we reveal that this model corresponds to the non-Hermitian SSH model as shown in Fig.~\ref{fig1} with $t_3=\gamma_2=0$~\cite{Yao2018}.
First of all, we can write the Hamiltonian in this model as
\begin{eqnarray}
H&=&\sum_n\left[v\left(c_{n,\alpha}^\dag c_{n,\beta}+c_{n,\beta}^\dag c_{n,\alpha}\right)\right. \nonumber\\
&&+\frac{ir}{2}\left(c_{n+1,\alpha}^\dag c_{n,\alpha}-c_{n,\alpha}^\dag c_{n+1,\alpha}-c_{n+1,\beta}^\dag c_{n,\beta}+c_{n,\beta}^\dag c_{n+1,\beta}\right) \nonumber\\
&&+\frac{r}{2}\left(c_{n+1,\alpha}^\dag c_{n,\beta}+c_{n,\beta}^\dag c_{n+1,\alpha}+c_{n+1,\beta}^\dag c_{n,\alpha}+c_{n,\alpha}^\dag c_{n+1,\beta}\right) \nonumber\\
&&\left.+\frac{i\gamma}{2}\left(c_{n,\alpha}^\dag c_{n,\alpha}-c_{n,\beta}^\dag c_{n,\beta}\right)\right],
\label{eq39}
\end{eqnarray}
where we take $r$, $v$, and $\gamma$ to be real for simplicity. When the eigenvector of the real-space eigen-equation is written as $\ket{\psi}=\left(\psi_{1,\alpha},\psi_{1,\beta},\cdots,\psi_{L,\alpha},\psi_{L,\beta}\right)^{\rm T}$ in an open chain, the equation for $\psi_{n,\mu}$ can be written as
\begin{eqnarray}
&&\frac{ir}{2}\left(\psi_{n-1,\alpha}-\psi_{n+1,\alpha}\right)+\frac{i\gamma}{2}\psi_{n,\alpha} \nonumber\\
&&+\frac{r}{2}\left(\psi_{n-1,\beta}+\psi_{n+1,\beta}\right)+v\psi_{n,\beta}=E\psi_{n,\alpha}, \nonumber\\
&&-\frac{ir}{2}\left(\psi_{n-1,\beta}-\psi_{n+1,\beta}\right)-\frac{i\gamma}{2}\psi_{n,\beta} \nonumber\\
&&+\frac{r}{2}\left(\psi_{n-1,\alpha}+\psi_{n+1,\alpha}\right)+v\psi_{n,\alpha}=E\psi_{n,\beta}.
\label{eq40}
\end{eqnarray}
Here we take the ansatz as a linear combination:
\begin{equation}
\psi_{n,\mu}=\sum_j\phi_{n,\mu}^{\left(j\right)},~\phi_{n,\mu}^{\left(j\right)}=\left(\beta_j\right)^n\phi_\mu^{\left(j\right)},~(\mu=\alpha,\beta),
\label{eq41}
\end{equation}
where $\phi_{n,\alpha}^{\left(j\right)}$ and $\phi_{n,\beta}^{\left(j\right)}$ take the exponential form
\begin{equation}
\left(\phi^{\left(j\right)}_{n,\alpha},\phi^{\left(j\right)}_{n,\beta}\right)=\beta_j^n\left(\phi^{\left(j\right)}_\alpha,\phi^{\left(j\right)}_\beta\right).
\label{eq42}
\end{equation}
By substituting the exponential form Eq.~(\ref{eq42}) into Eq.~(\ref{eq40}), one can obtain
\begin{eqnarray}
\left[-\frac{ir}{2}\left(\beta-\beta^{-1}\right)+\frac{i\gamma}{2}\right]\phi_\alpha+\left[\frac{r}{2}\left(\beta+\beta^{-1}\right)+v\right]\phi_\beta&=&E\phi_\alpha, \nonumber\\
\left[\frac{ir}{2}\left(\beta-\beta^{-1}\right)-\frac{i\gamma}{2}\right]\phi_\beta+\left[\frac{r}{2}\left(\beta+\beta^{-1}\right)+v\right]\phi_\alpha&=&E\phi_\beta. \nonumber\\
\label{eq43}
\end{eqnarray}
Therefore the characteristic equation is given by
\begin{eqnarray}
r\left(v+\frac{\gamma}{2}\right)\beta+\left(r^2+v^2-\frac{\gamma^2}{4}-E^2\right)+r\left(v-\frac{\gamma}{2}\right)\beta^{-1}=0. \nonumber\\
\label{eq44}
\end{eqnarray}
The condition for the continuum bands can be written as $\left|\beta_1\right|=\left|\beta_2\right|$ since Eq.~(\ref{eq44}) is a quadratic equation for $\beta$. In this case, the generalized Brillouin zone becomes a circle with the radius
\begin{equation}
r\equiv\left|\beta_{1,2}\right|=\sqrt{\left|\frac{v-\gamma/2}{v+\gamma/2}\right|}
\label{eq45}
\end{equation}
which is given by Vieta's formula~\cite{Yao2018}. By substituting $\beta=r{\rm e}^{ik},~k\in{\mathbb R}$ to Eq.~(\ref{eq44}), we can get the gap-closing point as
\begin{equation}
\frac{v}{\gamma}=\pm\sqrt{\frac{1}{4}\pm\left(\frac{r}{\gamma}\right)^2}.
\label{eq46}
\end{equation}
Therefore the zero-energy edge states appear in the parameter region $v/\gamma\in\left[-\sqrt{1/4+\left(r/\gamma\right)^2},-\sqrt{1/4-\left(r/\gamma\right)^2}\right]$ and $v/\gamma\in\left[\sqrt{1/4-\left(r/\gamma\right)^2},\sqrt{1/4+\left(r/\gamma\right)^2}\right]$ as a consequence of the previous discussion. Indeed, we confirm the appearance of these zero-energy edge states in a finite open chain with $v/\gamma=0.5$. The gap closes at $v/\gamma=0,\pm\sqrt{2}/2$ in the bulk, and the zero-energy edge states appear in $v/\gamma\in\left[-1/\sqrt{2},1/\sqrt{2}\right]$. Though it seems that there are no zero-energy states around $v/\gamma=0$, this disappearance comes from a finite-size effect.
The tight-binding model in Fig.~\ref{fig2} can be regarded as the non-Hermitian SSH model~\cite{Yao2018}. Indeed, by the unitary transformation
\begin{equation}
\sigma_x\rightarrow\sigma_x,~\sigma_y\rightarrow-\sigma_z,~\sigma_z\rightarrow\sigma_y,
\label{eq47}
\end{equation}
the generalized Bloch Hamiltonian $H\left(\beta\right)$ can be rewritten as
\begin{eqnarray}
H\left(\beta\right)&=&\left[v+\frac{r}{2}\left(\beta+\beta^{-1}\right)\right]\sigma_x+\left[-\frac{ir}{2}\left(\beta-\beta^{-1}\right)+\frac{i\gamma}{2}\right]\sigma_z \nonumber\\
&\rightarrow&\left(v+\frac{\gamma}{2}+r\beta^{-1}\right)\sigma_++\left(v-\frac{\gamma}{2}+r\beta\right)\sigma_-,
\label{eq48}
\end{eqnarray}
where $\sigma_\pm=\left(\sigma_x\pm i\sigma_y\right)/2$. We note that the asymmetric hopping amplitude is caused by the gain and loss in the original system. This Hamiltonian is reduced to our model with $t_3=\gamma_2=0$.
\begin{figure}[]
\includegraphics[width=7cm]{01-c.eps}
\caption{\label{fig3}Energy bands in a finite open chain of the model given by Eq.~(\ref{eq39}). We set the parameter as $r=0.5\gamma$ and the number of unit cells as $L=50$. The gap closes at $v/\gamma=0$ and $v/\gamma=\pm c=\pm1/\sqrt{2}$. The red dashed lines represent the gap-closing points in the bulk.}
\end{figure}
\providecommand{\noopsort}[1]{}\providecommand{\singleletter}[1]{#1
|
{
"timestamp": "2019-03-01T02:12:45",
"yymm": "1902",
"arxiv_id": "1902.10958",
"language": "en",
"url": "https://arxiv.org/abs/1902.10958"
}
|
\section{Hardness}
\label{sec:hardness}
\subsection{Hardness for Shortest Paths}
\label{ssec:hardnes-shortest}
In the decision version of the redistricting problem, we are given a graph $G$, two $k$-district maps, $A$ and $B$, for some $k\in \mathbb{N}$, and an integer $L\geq 0$, and ask whether a sequence of at most $L$ switches can take $A$ into $B$. Let us denote this problem by $R(G,A,B,L)$. In this section, we show that this problem is NP-complete. We reduce NP-hardness from 3SAT. An instance of 3SAT consists of a boolean formula $\phi$ in 3CNF. Let $m$ and $n$ be the number of clauses and the number of variables, respectively, in $\phi$. We construct, for a given 3SAT instance $\phi$, a graph $G(\phi)$, two district maps $A(\phi)$ and $B(\phi)$, and a nonnegative integer $L(\phi)$. We then show that $\phi$ is satisfiable if and only if the instance $R(G(\phi),A(\phi),B(\phi),L(\phi))$ of the redistricting problem is positive.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.5\textwidth]{shortest_path_gadgets.pdf}
\caption{A variable gadget (left) and a districting pipe (right)}
\label{fig:ShortestPathGadgets}
\end{center}
\end{figure}
We construct the graph $G(\phi)$ as follows:
\begin{enumerate}\itemsep 0pt
\item For every variable $x_i$, construct a \emph{variable gadget} $G_i$, shown in Figure~\ref{fig:ShortestPathGadgets}(left).
\item For every clause $c_j$, a \emph{clause gadget} $H_j$ consists of two adjacent vertices, $c_{j,1}$ and $c_{j,2}$.
\item For every variable-clause pair $(x_i,c_j)$, if $x_i$ is a nonnegated variable in a $c_j$, insert an edge between $c_{j,2}$ and $r_{i,1}$, and if $x_i$ is a negated variable in $c_j$, insert an edge between $c_{j,2}$ and $\ell_{i,1}$.
\item Next, we add a subgraph, called a \textbf{districting pipe} $d(\phi)$, that consists of $m+n+1$ vertices. The districting pipe is a complete bipartite graph between a 2-element partite set $\{O,I\}$ and a $(m+n-1)$-element partite set.
Figure~\ref{fig:ShortestPathGadgets}(right) depicts and example where $m+n-1=5$.
\item Lastly, for each variable gadget $G_i$, insert an edge from $O$ to $\ell_{i,1}$ and $r_{i,1}$, respecively.
\end{enumerate}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.9\textwidth, angle=0]{shortest_path_graph.pdf}
\caption{Left: The graph $G(\phi)$ for the formula $\phi=(x_1\lor x_3 \lor \neg x_4) \land (\neg x_2 \lor x_3 \lor x_4)$.
The red and purple regions indicate the district in $A(\phi)$. The blue and purple regions indicated the district in $B(\phi)$. In particular, the purple regions indicate districts that are present in both $A(\phi)$ and $B(\phi)$.}
\label{fig:graphofphi}
\end{center}
\end{figure}
We now define two district maps on $G(\phi)$. Refer to Figure~\ref{fig:graphofphi}.
First, let $A(\phi)$ consist of the following districts. For each variable gadget $G_i$, we create four districts:
$\ell_i=\{\ell_{i,j}: j=1,\ldots ,5\}$, $r_i=\{r_{i,j}: j=1,\ldots , 5\}$, $\{d_{i,1},d_{i,2}\}$, and $\{u_{i,1},u_{i,2}\}$.
For each clause gadget $H_j$, we create a 2-element district $\{c_{j,1},c_{j,2}\}$. In the districting pipe, every vertex is in a singleton district, which yields $m+n+1$ singletons.
Next, we define the target district map, $B(\phi)$. For every variable gadget $G_i$, we create similar districts to $A(\phi)$, the only difference is that the district $\{d_{i,1},d_{i,2}\}$ is now split into two singletons: $\{d_{i,1}\}$ and $\{d_{i,2}\}$.
In each clause gadget $H_j$, the two vertices form singleton districts.
Lastly, the district pipe now consists of one $(m+n+1)$-vertex district.
Finally, we set $L(\phi):=4m+6n-1$. This completes the description of the instance $R(G(\phi),A(\phi),B(\phi),L(\phi))$.
\begin{theorem}\label{thm:shortest}
It is NP-complete to decide whether the length of a shortest path in $\Gamma_k(G)$ between two given district maps is below a given threshold.
\end{theorem}
The theorem is the direct consequence of Lemmas~\ref{lem:==>} and \ref{lem:<==}.
\begin{lemma}\label{lem:==>}
If there exists a satisfying truth assignment for $\phi$, then $R(G(\phi),A(\phi),B(\phi),L(\phi))$ is a positive instance.
\end{lemma}
\begin{proof}
Let $\phi$ be a boolean formula in 3CNF with $m$ clauses and $n$ variables, and let $\tau$ be a satisfying truth assignment.
We show that $A(\phi)$ can be transformed into $B(\phi)$ using $L(\phi)=4m+6n-1$ moves.
We define an open gate of $G_i$ to be $\ell_{i,1}$ if district $\ell_i$ contains the vertex $d_{i,1}$, otherwise the open gate of $G_i$ is $r_{i,1}$ by Observation~\ref{obs:cut-containment}.
\begin{enumerate}
\item For each variable $x_i$, if $\tau(x_i)=\texttt{true}$, then we expand $\ell_{i,3}$ or $\ell_{i,4}$ to $d_{i,1}$, otherwise we expand $r_{i,3}$ or $r_{i,4}$ to $d_{i,1}$ (thereby opening one of the two gates for each variable) in a total of $n$ moves.
\item For every variable $x_i$, we expand a unique district from vertex $O$ of the districting pipe $d(\phi)$ to the open gate of $G_i$. We then further expand this district to $u_{i,1}$, executing $2n$ moves. Apart from the first such district, we have to first expand a singleton district in $d(\phi)$ to $O$, and then further expand it into a variable gadget. This takes $2(n-1)$ additional moves, for a total of $2n+2(n-1)$ moves.
\item Now, for each clause $c_j$, take any variable $x_i$ that appears in a true literal in $c_j$. Such a variable exists because $\tau$ is a satisfying truth assignment. Expand a unique district from $O$ to the open gate of $x_i$, and then expand this district further to $c_{j,1}$. This takes a total of $2m$ moves. Similar to the previous argument, each of these districts need to reach vertex $O$ from their original positions, which takes $2m$ additional moves. Overall, this takes $4m$ moves.
\item Since the middle of the district pipe consisted of $m+n-1$ vertices and we moved $m+n-1$ districts out to $O$, we know that the singleton district at $I$ has expanded to all vertices in the district pipe with the exception of $O$. We now expand this $I$ district to include $O$, which takes 1 move.
\item Finally, for every variable $x_i$ we expand the district, which is next to an open gate, to now include the gate itself, thereby closing the gate; and expand the singleton district at $d_{i,2}$ to include $d_{i,1}$. Altogether this takes $2n$ moves.
\end{enumerate}
Overall, we have performed $n+2n+2(n-1)+2m+2m+1+2n=4m+6n-1=L(\phi)$ moves. These $L(\phi)$ moves transformed $A(\phi)$ to $B(\phi)$, and so the instance $R(G(\phi),A(\phi),B(\phi),L(\phi))$ is positive.
\end{proof}
\begin{lemma}\label{lem:<==}
If $R(G(\phi),A(\phi),B(\phi),L(\phi))$ is a positive instance of the redistricting problem for a boolean formula $\phi$, then there exists a satisfying truth assignment for $\phi$.
\end{lemma}
\begin{proof}
We derive lower bounds on the number of moves in any sequence of moves from $A(\phi)$ to $B(\phi)$ by making inferences from the initial and target district maps. Notice that if a district contains a leaf, then the leaf remains in the same district by Lemma~\ref{lem:leaves}. We call a district \emph{mobile} if it does not contain any leaf in $A(\Phi)$. By construction, only the $m+n+1$ districts initially in the districting pipe are mobile.
\begin{itemize}
\item[(A)] Since $u_{i,2}$ and $u_{i,1}$ are in distinct districts in $B(\phi)$, we must have a mobile district that travels to $u_{i,2}$. In order to accomplish this, we must first open one of the two gates of the variable gadget $G_i$. Opening $n$ gates, one in each variable gadget, requires at least $n$ moves.
\item[(B)] As noted above, a mobile district must travel to $u_{i,2}$ for $i=1,\ldots , n$. Moving $n$ mobile districts from $O$ to $u_{i,2}$, $i=1,\ldots , n$, requires at least $2n$ moves, and an additional $2(n-1)$ moves for $n-1$ mobile districts to reach $O$. Overall, this requires at least $2n+2(n-1)$ moves.
\item[(C)] Since each clause gadget $H_j$ consists of two districts in $B(\phi)$, a mobile district from the districting pipe must travel to $c_{j,2}$, for $j=1,\ldots, m$, which requires $4m$ moves.
\item[(D)] Because one mobile district must expand to the entire district pipe $d(\phi)$, either one mobile district expands into $I$, or the district that contains $I$ expands into $O$. In either case, this takes one additional move that has not been counted so far.
\item[(E)] Note that the gate of $G_i$ is closed and $\{d_{i,1},d_{i,2}\}$ is a 2-vertex district in $B(\phi)$, for $i=1,\ldots n$. So the district of the open gate must expand to consume its gate, and the singleton district at the leaf $d_{i,2}$ must expand into $d_{i,1}$. Together this requires a total of $2n$ moves.
\end{itemize}
Therefore, we need at least $n+2n+2(n-1)+4m+1+2n=6n+4m-1=L(\phi)$ moves to solve the redistricting problem. Since we executed exactly $L(\phi)$ moves, then we know we must have executed only the moves listed above, each of which was necessary.
Due to the fact that after opening a gate, opening the opposite gate would require additional moves, we conclude that precisely one gate opens in each variable gadget. We construct a truth assignment as follows: For every $i=1,\ldots, n$, let $\tau(x_i)=\texttt{true}$ if the left gate of the variable gadget $G_i$ opens, and $\tau(x_i)=\texttt{false}$ otherwise. Since the only way to get a district to $c_{j,2}$ was through
an open gate of one of the three literal in the clause $c_j$, then every clause is incident to an open gate of a variable gadget. Since every open gate corresponds to a true literal, at least one of the three literals is true in each clause. Henceforth, $\tau$ is a satisfying truth assignment for $\phi$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:shortest}]
Finding the shortest path in $\Gamma_k(G)$ between two given district maps is NP-hard by Lemmas~\ref{lem:<==} and \ref{lem:==>}.
The problem is also in NP: Given a sequence of $L(\phi)$ switches, we can verify in polynomial time that the switches are valid operations, and transform $A(\phi)$ to $B(\phi)$.
\end{proof}
The incontractible districts in the previous reduction are not essential for NP-hardness.
We can replace them with contractible districts, and obtain the following result.
\begin{corollary}
It is NP-complete to decide whether the length of a shortest path in $\Gamma_k'(G)$ between two given contractible district maps is below a given threshold.
\end{corollary}
\begin{proof}
In the reduction above, for a boolean formula $\phi$ in 3CNF, we constructed an instance $R(G(\phi),A(\phi),B(\phi),L(\phi))$ of the redistricting problem. We now modify the variable gadgets so that, instead of a direct edge from $\ell_{i,2}$ to $\ell_{i,3}$, we use a chain of vertices of length $L(\phi)$ between $\ell_{i,2}$ and $\ell_{i,3}$. We use a similar chain for between $r_{i,2}$ and $r_{i,3}$. We make the following modifications to the district maps $A(\phi)$ and $B(\phi)$: We extend the district of $\ell_{i,3}$ to cover the additional chain of vertices we introduced and create an entirely new district in $\ell_{i,2}$, do the same for $r_{i,3}$ and $r_{i,2}$.
By Lemma~\ref{lem:con2}, all districts are contractible. Notice that there is an additional possibility to open gates: namely by expanding the district of $\ell_{i,2}$ or $r_{i,2}$ to $\ell_{i,3}$ or $r_{i,3}$. However, since the number of moves required to do so is $L(\phi)$, this option is irrelevant for
the shortest path, since we strictly make $L(\phi)$ moves in each part of the proof.
\end{proof}
\subsection{Hardness for Connectedness}
\label{ssec:hardness-connect}
\setcounter{tocdepth}{4}
\setcounter{secnumdepth}{4}
In the \textbf{connectedness problem}, we are given a graph $G$, and two $k$-district maps, $A$ and $B$, for some $k\in \mathbb{N}$, and ask whether $A$ and $B$ are in the same component of the switch graph $\Gamma_k(G)$. Let us denote this problem by $C(G,A,B)$. In this section, we show that this problem is NP-complete. We reduce NP-hardness from 3SAT. For every boolean formula in 3CNF, we construct a graph $G$ and two $k$-district maps
that are in the same component of $\Gamma_k(G)$ if and only if the boolean formula is satisfiable.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\textwidth]{twogadgets}
\caption{A clause gadget (left), and a variable gadget (right).
The circled vertices are adjacent to both super nodes.}
\label{fig:twogadgets}
\end{center}
\end{figure}
\smallskip\noindent\textbf{Construction.}
Let $\phi$ be a boolean formula in 3CNF with $m$ clauses and $n$ variables.
\begin{enumerate}\itemsep 0pt
\item Clause Gadgets: For each clause $c_j$, construct a \textbf{clause gadget} $H_j$, which is the simplest case of a diamond (Fig.~\ref{fig:twogadgets}(left)): Create a 4-cycle
$(u_{j,1},u_{j,2},u_{j,3},u_{j,4})$,
and attach leaves, $u_{j,5}$ and $u_{j,6}$, to $u_{j,2}$ and $u_{j,4}$, respectively.
\item Variable Gadgets: For each variable $x_i$, construct a \textbf{variable gadget} $G_i$ defined as follows; refer to Fig.~\ref{fig:twogadgets}(right). Create two 5-cycles, $A=(v_{i,1},\ldots, v_{i,5})$ and $B=(v_{i,6},\ldots , v_{i,10})$, and identify vertices $v_{i,5}$ and $v_{i,8}$. Attach four leaves, $v_{i,11},\ldots, v_{i,14}$, to vertices $v_{i,1}$, $v_{i,4}$, $v_{i,7}$, and $v_{i,9}$, respectively. So far, the two pentagon forms two interconnected diamonds.
Insert two new vertices, $v_{i,15}$ and $v_{i,16}$, and connect both to $v_{i,3}$, $v_{i,5}=v_{i,8}$, and $v_{i,10}$.
Finally, attach two leaves, $v_{i,17}$ and $v_{i,18}$, to $v_{i,15}$ and $v_{i,16}$ respectively.
\item For every variable-clause pair $(x_i,c_j)$, where $x_i$ appears in $c_j$, insert an edge
from vertex $u_{j,1}$ to vertex $v_{i,2}$ or $v_{i,6}$ in $G_i$: If $x_i$ is nonnegated in $C_j$, we use vertex $v_{i,2}$, otherwise vertex $v_{i,6}$.
\item Create two special vertices, called \textbf{super nodes}, denoted $N_1$ and $N_2$.
Connect both super nodes to vertices $v_{i,1}$, $v_{i,4}$, $v_{i,7}$, $v_{i,9}$, $v_{i,15}$, $v_{i,16}$ (circled in Figure.~\ref{fig:twogadgets}) for each variable $x_i$.
\item Frame: Connect super node $N_1$ and all nodes $u_{j,3}$, $j\in\{1,\ldots,m\}$, by a cycle in an arbitrary order. Then subdivide the two edges of the cycle incident to $N_1$ by two new vertices,
and attach a leaf to each. We call the resulting cycle with two leaves the \textbf{frame}; see Figure~\ref{fig:complete}.
\item Blue Reservoir: Create a cycle with $m+n+3$ vertices, and label four consecutive vertices by $b_1,\ldots ,b_4$. Identify vertex $b_2$ with super node $N_2$; attach two leaves to $b_1$ and $b_3$, respectively; and Connect vertex $b_1$ to both $v_{i,2}$ and $v_{i,6}$ in each of the variable gadgets.
\item Garbage Collection Gadget: Create a diamond that consists of a 4-cycle, with two opposite vertices attached to leaves. Let $a_1$ and $a_2$ be the two leafless vertices of the cycle.
Identify $a_2$ with super node $N_2$; and connect $a_1$ to both $v_{i,2}$ and $v_{i,6}$ in each of the variable gadgets, and attach a path of $n+1$ vertices to it.
\end{enumerate}
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.9\textwidth]{total}
\caption{The complete construction of the graph $G(\phi)$ with the formula
$\phi=(x_1\lor x_2 \lor x_3) \land (\neg x_1 \lor \neg x_2 \lor \neg x_3)\land (\neg x_1 \lor x_2 \lor \neg x_3)$.
The circled vertices are adjacent to both super nodes.}
\label{fig:complete}
\end{center}
\end{figure}
Next, we construct two district maps, an \emph{initial} map $A(\phi)$ and a \emph{target} map $B(\phi)$.
By Lemma~\ref{lem:leaves}, the district containing the same leaves in $A(\phi)$ and $B(\phi)$ must be
correspond to each other under any sequence of switch operations. So we use the same labels for
such districts. The districts that do not contain any leaves are called \emph{mobile districts}.
Both $A(\phi)$ and $B(\phi)$ partition the vertices of each variable gadgets $G_i$ into the same three districts: $V_{i,1}=\{v_{i,1},v_{i,2},v_{i,3},v_{i,4},v_{i,11},v_{i,12}\}$,
$V_{i,2}=\{v_{i,6}, v_{i,7}, v_{i,9}, v_{i,10}, v_{i,13}, v_{i,14}\}$, and
$V_{i,3}=\{v_{i,5}, v_{i,15},v_{i,16},v_{i,17},v_{i,18}\}$.
Note that each of these districts contain two leaves.
We now specify the remaining districts of $A(\phi)$ and $B(\phi)$, respectively. The \textbf{initial district map} $A(\phi)$ is specified as follows; see Figure~\ref{fig:initial}.
\begin{itemize}\itemsep 0pt
\item Clauses: For $j=\in\{1,\ldots, m\}$, create a district $O_j=\{u_{j,1}, u_{j,2}, u_{j,4},u_{j,5},u_{j,6}\}$.
\item Frame: One district, called \texttt{frame}, contains all vertices of the frame.
\item Blue Reservoir: Create a 5-vertex district, called \texttt{blue}, that contains the super node $N_2$, its two neighbors and the two leaves in the Blue Reservoir. The remaining $m+n$ vertices in the Blue Reservoir form singleton districts, which are called \textbf{blue mobile districts}.
\item Garbage Collection Gadget: Create a 5-vertex district, called \texttt{garbage}, which contains the diamond in the Garbage Collection Gadget except for $N_2$. All $n+1$ vertices in the path attached to $a_1$ are in a single district called \texttt{purple}.
\end{itemize}
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=.75\textwidth]{total-Pi1}
\caption{The initial district map. (The circled vertices are adjacent to both super nodes.)}
\label{fig:initial}
\end{center}
\end{figure}
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=.75\textwidth]{total-Pi2}
\caption{The target district map. (The circled vertices are adjacent to both super nodes.)}
\label{fig:target}
\end{center}
\end{figure}
The \textbf{target district map} $B(\phi)$ is specified as follows; see Figure~\ref{fig:target}.
\begin{itemize}\itemsep 0pt
\item Clause Gadget: For $j\in\{1,\ldots ,m\}$, create a district $O_j=\{u_{j,2},u_{j,3},u_{j,4},u_{j,5},u_{j,6}\}$; and a singleton district $\{u_{j,1}\}$.
\item Frame: The \texttt{frame} district now contains the super node $N_1$, its two neighbors, and the two leaves of the frame. (All other vertices of the frame are in districts $O_j$, $j\in\{1,\ldots , m\}$.)
\item Blue Reservoir: The \texttt{blue} district contains all vertices of the Blue Reservoir.
\item Garbage Collection Gadget: the \texttt{garbage} district is the same as in $A(\phi)$. The \texttt{purple} district is a singleton that contains the same leaf as in $A(\phi)$; all other vertices in the path attached to $a_1$ form singleton districts.
\end{itemize}
\begin{theorem}\label{thm:connectedness}
It is NP-complete to decide whether two given $k$-district maps on a graph $G$ are in the same component of the switch graph $\Gamma_k(G)$.
\end{theorem}
The theorem is the direct consequence of Lemmas~\ref{lem:forward} and \ref{lem:backward} below.
Before proving these lemmas, we derive some basic properties of the instance $C(G(\phi),A(\phi),B(\phi))$.
In each variable gadget $G_i$, $i\in \{1,\ldots , n\}$,
we define two special states relative to the vertices $v_{i,2}$ and $v_{i,6}$ (refer to Figure~\ref{fig:twogadgets}).
We say that the \textbf{true gate is open} if $v_{i,2}$ is a cut vertex of district $V_{i,1}$; otherwise it is \textbf{closed}.
Similarly, the \textbf{false gate is open} if $v_{i,6}$ is a cut vertex of district $V_{i,2}$; otherwise it is \textbf{closed}.
Note that in the initial and target district maps, $A(\phi)$ and $B(\phi)$, both gates are closed.
\begin{lemma}\label{lem:222}
Let $\Pi$ be a district map that can be reached from $A(\phi)$ by a sequence of switches; and let $i\in \{1,\ldots, n\}$. If neither super node is contained in any of the districts $V_{i,1}$, $V_{i,2}$, and $V_{i,3}$, then precisely one of the following three cases holds:
(1) the true gate is open and the false gate is closed;
(2) the false gate is open and the true gate is closed;
(3) both gates are closed;
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:leaves}, all leaves remain in the same district as in $A(\phi)$.
By Observation~\ref{obs:cut-containment}(1), the cut vertices $v_{i,1},v_{i,4},v_{i,7},v_{i,9},v_{i,15},v_{i,16}$ are also in the same district as in $A(\phi)$. Refer to Figure~\ref{fig:twogadgets}(right).
It is enough to show that both gates cannot be opened simultaneously.
By Observation~\ref{obs:cut-containment}(2), district $V_{i,1}$ contains both $v_{i,2}$ and $v_{i,3}$, or $v_{i,5}$, or at least on one of the super nodes. Assuming that neither super node is in $V_{i,1}$ or $V_{i,2}$,
if the true gate is open then $v_{i,5}\in V_{i,1}$.
By Observation~\ref{obs:cut-containment}(2), $V_{i,2}$ must then contain $\{v_{i,6},v_{i,10}\}$.
Applying Observation~\ref{obs:cut-containment}(2) again, $V_{i,3}$ contain $v_{i,3}$.
Therefore the false gate is closed since there is a unique path connecting $v_{i,13}$ and $v_{i,14}$ in $V_{i,2}$ and it contains $v_{i,6}$.
\end{proof}
We distinguish three states of a variable gadget $G_i$: (1) \textbf{true gate is open}; (2) \textbf{false gate is open}; and
the \textbf{initial state}, where the vertices of $G_i$ are partitioned into $V_{i,1}$, $V_{i,2}$, and $V_{i,3}$ as in $A(\phi)$.
In the initial state, both gates of $G_i$ are closed, but the converse is false: there are several possible configurations in which both gates are closed. We say that a super node $N_q$, $q\in \{1,2\}$, is \textbf{available} if $N_q$ is in a district $\widehat{V}$ such that $\widehat{V}\setminus \{N_q\}$ induces a connected subgraph of $G(\phi)$ (that is, any district adjacent to $N_q$ could expand into $N_q$).
\begin{lemma}\label{lem:333}
A variable gadget can move between the three states listed above if and only if some super node is available.
\end{lemma}
\begin{proof}
(1) Assume first that neither super node is available. Let $i\in \{1,\ldots , n\}$.
We show that any transition from a district map in which one gate is open to another where the other gate is open is infeasible.
If the \textbf{true gate is open}, then $v_{i,5}\in V_{i,1}$,
$\{v_{i,6},v_{i,10}\}\subset V_{i,2}$, and $v_{i,3}\in V_{i,3}$ as shown in the proof of Lemma~\ref{lem:222}.
This configuration is uniquely defined by Observation~\ref{obs:cut-containment}(2) and, therefore, is rigid.
If the \textbf{false gate is open}, we can argue analogously.
If $G_i$ is in the initial state, then again the distribution of all vertices in $G_i$ is completely determined by Observation~\ref{obs:cut-containment}.
(2) Next assume that one of the super nodes $N_q$, $q\in \{1,2\}$, is available. We may also assume that $V_{i,1}$, $V_{i,2}$, and $V_{i,3}$ are contracted to lie entirely within $G_i$. If the \textbf{true gate is open}, we can open the false gate as follows: Expand $V_{i,2}$ into $N_q$. Then expand $V_{i,3}$ into $v_{i,10}$, $V_{i,1}$ into $v_{i,2}$ and $v_{i,3}$, and $V_{i,2}$ into $v_{i,5}$. Finally, we contract $V_{i,2}$ from $N_q$.
If the \textbf{false gate is open}, we can open the true gate analogously.
If $G_i$ is in the \textbf{initial state}, we can open the true gate as follows: Expand $V_{i,3}$ into $N_q$, then $V_{i,1}$ into $v_{i,5}$, and $V_{i,3}$ into $v_{i,3}$, and finally, contract $V_{i,3}$ from $N_q$. We can open the false gate analogously. If one of the two gates is open, the reverse sequence of these operations can reach the initial state.
\end{proof}
By Lemma~\ref{lem:con1}, a district that contains a leaf in $A(\phi)$ must contain the same leaf after any sequence of switches, and so their location in $G$ is restricted. The districts that do not contain any leaf are called \emph{mobile districts}.
\begin{lemma}\label{lem:hold}
Let $\Pi$ be a district map that can be reached from $A(\phi)$ by a sequence of switches. Then
(a) each clause gadget can hold at most one mobile district;
(b) each variable gadget $G_i$ can hold at most one mobile district if neither super node is available;
and it can hold up to three mobile districts if a super node is in $V_{i,1}$, $V_{i,2}$, or $V_{i,3}$.
\end{lemma}
\begin{proof}
Consider a clause gadget $H_j$, $j\in \{1,\ldots, m\}$. By Lemma~\ref{lem:leaves} and Observation~\ref{obs:cut-containment}(1), the leaves $u_{j,5},u_{j,6}$ and cut vertices $u_{j,2},u_{j,4}$ remain in district $O_j$. By Observation~\ref{obs:cut-containment}(1), $u_{j,1}$ or $u_{j,3}$ is in $O_j$, as well. Therefore, $H_j$ has at most one vertex that can hold a mobile district. This proves (a).
Consider a variable gadget $G_i$, $i \in \{1,\ldots , n\}$. By Lemma~\ref{lem:leaves} and Observation~\ref{obs:cut-containment}(1),
leaves and cut vertices are always in $V_{i,1}$, $V_{i,2}$, and $V_{i,3}$.
Lemma~\ref{lem:222} lists three possible cases for the distribution of the districts $V_{i,1}$, $V_{i,2}$, and $V_{i,3}$ when none of them contains a super node.
If the true (resp., false) gate is open, then only vertex $v_{i,2}$ (resp., $v_{i,6}$) can hold a mobile district.
If both gates are closed, then $V_{i,1}$, $V_{i,2}$, and $V_{i,3}$ jointly contain all vertices of $G_i$: they contain $v_{i,2}$ and $v_{i,6}$ by definition, and $\{v_{i,3}, v_{i,5}, v_{i,10}\}$ is a 3-cut that separates the leaves in $V_{i,1}$, $V_{i,2}$, and $V_{i,3}$, and all three vertices are needed
since we need three disjoint paths connecting their respective leaves.
If a super node $N_q$ is available, $q\in \{1,2\}$, then $\{v_{i,3}, v_{i,5}, v_{i,10},N_q\}$ is a 4-cut that separates the leaves in $V_{i,1}$, $V_{i,2}$, and $V_{i,3}$; so one of these vertices can hold a mobile district. In addition, $v_{i,2}$ and $v_{i,6}$ can each hold a mobile district.
\end{proof}
\begin{lemma}\label{lem:forward}
If there exists a satisfying truth assignment for $\phi$,
then $C(G(\phi),A(\phi),B(\phi))$ is a positive instance of the connectedness problem.
\end{lemma}
\begin{proof}
Let $\tau$ be a satisfying truth assignment for $\phi$. We show that there exists a sequence of switches that takes $A(\phi)$ to $B(\phi)$.
Consider the initial district map, $A(\phi)$. Note that super node $N_1$ is available. By Lemma~\ref{lem:333}, we can successively reconfigure all variable gadgets, and open the true or false gate in each according to the respective truth assignment in $\tau$.
By definition of open gates, either $v_{i,2}$ or $v_{i,6}$ is not a cut-vertex in its district.
Next, we successively modify the clause gadgets $H_j$, for $j=1,\ldots , m$. Since $\tau$ is a satisfying truth assignment, each clause $c_j$ contains a true literal. For an index $j\in \{1,\ldots, m\}$, assume that $x_i$ or $\neg x_i$ is a true literal in $c_j$. A mobile blue district at $b_4$ can expand to vertex $v_{i,2}$ or $v_{i,6}$ (depending on whether $\tau(x_i)=\texttt{true}$ or $\tau(x_i)=\texttt{false}$); and then it can further expand to vertex $u_{j,1}$.
However,
before a mobile blue district expands to $u_{j,1}$, district $O_j$ expands into vertex $u_{j,3}$ to maintain a path between the two leaves in district $O_j$ (cf.~Observation~\ref{obs:cut-containment}). Note that we can expand $O_j$ into vertex $u_{j,3}$ in any district $H_j$, since the super node $N_1$ maintains a path between the two leaves of the \texttt{frame} district (as required by Observation~\ref{obs:cut-containment}). As soon as a mobile blue district has expanded into vertex $u_{j,1}$, it contracts into this vertex such that vertex $b_4$ in the Blue Reservoir is now occupied by the next blue mobile district, and vertex $v_{i,2}$ (resp., $v_{i,6}$) in $G_i$ is occupied by district $V_{i,1}$ (resp., $V_{i,2}$) if $\tau(x_i)=\texttt{true}$ (resp., $\tau(x_i)=\texttt{false}$).
As a result, the $m$ clause gadgets each contain a singleton blue mobile district;
and the Blue Reservoir contains the remaining $n$ blue mobile districts.
Then for $i=1,\ldots, n$, we successively expand a blue mobile district from vertex $b_4$ of the Blue Reservoir to the open gate of $G_i$, and contract it into a singleton district in $G_i$ such that the next blue district expands into $b_4$. Eventually, all remaining blue mobile districts exit from the Blue Reservoir, which is then occupied by the district $\texttt{blue}$, as in $B(\phi)$.
At this point, super node $N_2$ becomes available (since every vertex on the path in the blue reservoir are in the \texttt{blue} district). Consequently, we can open the Garbage Collection Gadget by expanding the \texttt{garbage} district into super node $N_2$. Then, for $i=1,\ldots, n$, the blue mobile districts from vertex $v_{i,2}$ or $v_{i,6}$ can expand into $a_1$ in the Garbage Collection Gadget, and then further into the path of length $n+1$ attached to $a_1$. Finally, we expand district \texttt{garbage} into $a_1$; and then expand the \texttt{blue} district into $N_2$. At this point, both the Blue Reservoir and the Garbage Collection Gadget are in the target positions.
Now super node $N_2$ becomes available. We can use it to close the gates in all variable gadgets in an arbitrary order, as described in Lemma~\ref{lem:333}. Finally, expand the \texttt{blue} district to include the super node $N_2$. We have transformed $A(\phi)$ to $B(\phi)$, as required.
\end{proof}
\begin{lemma}\label{lem:backward}
If $C(G(\phi),A(\phi),B(\phi))$ is a positive instance of the connectedness problem for a boolean formula $\phi$, then there exists a satisfying truth assignment for $\phi$.
\end{lemma}
\begin{proof}
Let $\sigma$ be a sequence of switches that transforms $A(\phi)$ to $B(\phi)$. The switch operations produce a sequence of district maps $(\Pi_1,\Pi_2,\ldots , \Pi_N)$, where $\Pi_1=A(\phi)$ and $\Pi_N=B(\phi)$. For $t\in \{1,\ldots , N\}$, we say that $\Pi_t$ is the district map at \textbf{time} $t$.
The sequence $\sigma$ defines a one-to-one correspondence between the districts in $\Pi_i$ and $\Pi_{i+1}$, for $i=1,\ldots, N-1$.
By Lemma~\ref{lem:con1}, the districts that contain the same leaf in $\Pi_i$ and $\Pi_j$ correspond to each other.
We call the remaining $m+n$ districts \emph{mobile}. In $A(\phi)$, all mobile districts are singletons in the Blue Reservoir.
In $B(\phi)$, all mobile districts are singleton: there are $n$ mobile districts in the Garbage Collection Gadget, and one in each of the $m$ clause gadgets.
The sequence $\sigma$ moves $n$ mobile districts from the Blue Reservoir to the Garbage Collection Gadget. Vertex $a_1$ is the only vertex in the Garbage Collection Gadget adjacent to other parts of $G(\phi)$.
The mobile districts can pass through $a_1$, if $a_1$ is not in the \texttt{garbage} district (as in $A(\phi)$ and $B(\phi)$). Observation~\ref{obs:cut-containment}, applied for the \texttt{garbage} district, implies that if the \texttt{garbage} district does not contain $a_1$, then it contains the super node $N_2$.
Let $T_2$ be the time just before the \texttt{garbage} district expands into $N_2$ for the first time. We use the district map $\Pi_{T_2}$ at time $T_2$ to define a truth assignment for the boolean variable $\phi$. The next few paragraphs describe the district map $\Pi_{T_2}$.
Note that at time $T_2$, the Blue Reservoir cannot contain any mobile districts, by applying Observation~\ref{obs:cut-containment} for the \texttt{blue} district.
The Garbage Collection Gadget is covered by the \texttt{garbage} district, since $N_2$ has not been available before $T_2$.
There can be no mobile district in upper arc of the frame $\{u_{j,3}:j=1,\ldots, m\}$,
because every path between a mobile district in $A(\phi)$ and a $u_{j,3}$ goes through a vertex that must belong to some nonmobile district at all times by Observation~\ref{obs:cut-containment}(1).
Similarly, neither super node can be in a mobile district
Hence, all $m+n$ mobile districts are in variable and clause gadgets at $T_2$.
By Lemma~\ref{lem:hold}, each clause gadget can hold at most one mobile district,
each variable gadget can hold at most one mobile district, unless it uses a super node in which case it can hold up to 3 mobile districts. Assuming $m\geq 3$, at least one mobile district must be in a clause gadget. If a mobile district is in a clause gadget $H_j$, then vertex $u_{j,3}$ must be in district $O_i$. This forces super node $N_1$ to be locked in the \texttt{frame} district, which in turn means that neither super node is available. If neither super node is available, then every variable gadget can only hold at most one mobile district.
Since the number of mobile districts is precisely $m+n$, then every variable gadget and every clause gadget contains one mobile district.
Since every variable gadget holds one mobile district, they each have either the true or the false gate open. We use the open gates to define the truth assignment: Set $\tau(x_i)=\texttt{true}$ if and only if the true gate of $G_i$ is open.
By Lemma~\ref{lem:333}, each variable gadget used a super node to open one of its gates before time $T_2$. They each used super node $N_1$, since $N_2$ remains in the \texttt{blue} district until time $T_2$. Let $T_1$ be the last time before $T_2$ when super node $N_1$ switches districts. That is, $N_1$ remains in the same district between $T_1$ and $T_2$. Since $N_1$ is in the \texttt{frame} district at time $T_2$, is is in the \texttt{frame} district at time $T_1$, as well. Since neither super node is available between $T_1$ and $T_2$, by Lemma~\ref{lem:333}, all variable gadgets keep the same gates open that correspond to the truth assignment $\tau$.
Since at time $T_1$ super node $N_1$ becomes part of the \texttt{frame} district, the vertices $u_{j,3}$, $j\in \{1,\ldots , m\}$, must be part of the \texttt{frame} district at this time (cf. Observation~\ref{obs:cut-containment}(2)).
Then, there are no mobile districts in clause gadgets at $T_1$.
Since they each contain a mobile district at $T_2$,
such mobile district moves through a variable gadget between $T_1$ and $T_2$.
Such variable gadget must have one of its gates open and, therefore, each clause $c_j$ contains a true literal in the assignment $\tau$, as required.
\end{proof}
\section{Lower Bounds for Shortest Paths}
\label{sec:LB}
In this section, we prove lower bounds for the diameter of the switch graph $\Gamma_k(G)$, and for the length of a shortest path in $\Gamma_k(G)$ (when $\Gamma_k(G)$ need not be connected). We start with a simple construction that yields an $\Omega(kn)$ lower bound for the diameter of $\Gamma_k'(G)$ when $k\leq n/2$, which matches the upper bound of Theorem~\ref{thm:general-graphs}.
\begin{theorem}\label{thm:contractible-LB}
For every $k,n\in \mathbb{N}$, $k\leq n$, there exists a graph $G$ with $n$ vertices
such that the diameter of $\Gamma_k'(G)$ is $\Omega(k(n-k))$.
\end{theorem}
\begin{proof}
Let $G$ be a path $(v_1,\ldots, v_n)$ with $n$ vertices. Let $\Pi_1$ consist of $V_i=\{v_i\}$ for $i=1,\ldots , k-1$, and $V_k=\{v_k,\ldots, v_n\}$; and let $\Pi_2$ be the partition $W_1=\{v_1,\ldots v_{n-k}\}$ and $W_j=\{v_{n-k+j}\}$, for $j=2,\ldots, k$.
Assume that a sequence of switch operations takes $\Pi_1$ to $\Pi_2$. Since each district is nonempty at all times,
district $V_i$ is transformed into $W_i$ for all $i\in \{1,\ldots, n\}$. Each switch operation can move the rightmost vertex of at most one district, and by at most one unit. The rightmost vertex of $V_k$ remains fixed. The distance traveled by the rightmost vertices of $V_1,\ldots, V_k$ each is $n-k$. This requires at least $(k-1)(n-k)$ operations.
\end{proof}
If we connect the two endpoints of the path we get a cycle (and in particular a biconnected graph). The lower bound of Theorem~\ref{thm:contractible-LB} can be adapted to this case, which in particular implies that the diameter is not reduced even when $G$ is biconnected.
\begin{theorem}\label{thm:contractible-LB+}
For every $k,n\in \mathbb{N}$, $k\leq n$, there exists a biconnected graph $G$ with $n$ vertices
such that the diameter of $\Gamma_k(G)'$ is $\Omega(k(n-k))$.
\end{theorem}
\begin{proof}
Let $G=C_n$ be the cycle with $n$ vertices $(v_1,\ldots, v_n)$.
We construct two $k$-district maps, $\Pi_1$ and $\Pi_2$. Let $\Pi_1$ consist of $V_i=\{v_i\}$ for $i=1,\ldots , k-1$, and $V_k=\{v_k,\ldots, v_n\}$. The partition $\Pi_2$ is the copy of $\Pi_1$ rotated by $\lfloor n/2\rfloor$, that is, $W_i=\{v_{i+\lfloor n/2\rfloor}\}$ for $i=1,\ldots , k-1$, and $W_k=\{v_{k+\lfloor n/2\rfloor},\ldots, v_{n+\lfloor n/2\rfloor}\}$, where we use arithmetic modulo $n$ on the indices.
Assume that a sequence of switch operations takes $\Pi_1$ to $\Pi_2$. Note that the cyclic order of the district cannot change,
and so there is an integer $r\in \{0,\ldots , k-1\}$ such that $V_i$ is transformed to $W_{i+r\mod k}$ for all $i\in \{1,\ldots ,k\}$.
For any $r$, at least $k-2$ districts are singletons in both $\Pi_1$ and $\Pi_2$. The sum of the shortest distances
along $C_n$ between the initial and target positions is a lower bound for the number of switches.
If $r\leq \lfloor k/2\rfloor$, then the shortest distance between the initial and target positions is at least $\lfloor n/2\rfloor -r\leq \Omega(n-k)$ for the districts $V_i$, $i=1\ldots, k-1-r$; which sums to $\Omega(k(n-k))$.
If $\lfloor k/2\rfloor <r<k$, then shortest distance is at least $\lfloor n/2\rfloor -(k-r)\leq \Omega(n-k)$ for $V_i$, $i=r,\ldots, k-1$; which also sums to $\Omega(k(n-k))$.
\end{proof}
In the remainder of this section, we establish lower bounds for the diameter of a single component of $\Gamma_k(G)$, when $\Gamma_k(G)$ is disconnected (cf.~Theorem~\ref{thm:conn-test}).
\subsection{Diamonds}
\label{ssec:diamond}
A diamond is a useful construction for hardness reductions for the redistricting problem (cf.~Section~\ref{sec:hardness}). The simplest case of a diamond, with six vertices, is depicted in Figure~\ref{fig:diamond}. A diamond has two leaves: If both leaves are in the same district, then this district is incontractible by Lemma~\ref{lem:con1}, and we can encode the truth value of a variable by one of two possible paths between the leaves.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.35\textwidth]{diamond}
\caption{A diamond. If a district $V_i$ contains both leaves, $u_1$ and $u_2$, then $V_i$ contains a path between the leaves.}
\label{fig:diamond}
\end{center}
\end{figure}
Formally, a \emph{diamond} $A$ is an induced subgraph of $G$ with two cut vertices $v_1,v_2\in C(G)$ that are connected by with $m\geq 2$ interior-disjoint paths $A_1,\ldots ,A_m$, where $m-1$ of these paths contain exactly one interior node; and $v_1$ (resp., $v_2$) is the endpoint of a dangling path whose other endpoint is a leaf $u_1$ (resp., $u_2$).
Two crucial properties of diamonds are formulated in the following observation
\begin{observation}\label{obs:cut-containment}
Let $V_i$ be a district that contains both leaves $u_1$ and $u_2$.
\begin{enumerate}\itemsep -2pt
\item\label{obs1:cut-containment} Then $V_i$ contains both cut vertices, $v_1,v_2\in C(G)$, and all vertices of some path $A_1,\ldots , A_m$.
\item\label{obs2:path-availability} In a sequence of switches, if $V_i$ contains path $A_j$ and later $A_{j'}$, $j\neq j'$, then
$V_i$ contains two disjoint paths between $v_1$ and $v_2$ in some intermediate state.
\end{enumerate}
\end{observation}
We use Observation~\ref{obs:cut-containment}\eqref{obs2:path-availability} repeatedly in the hardness reductions, typically, with $m=2$ or $m=3$.
For $r\in \mathbb{N}$, a \emph{diamond chain} of length $r$ is a subgraph of $G$ with $r$ diamonds, $A_1,\ldots A_r$, where each diamond has precisely two interior-disjoint paths between its cut vertices (a \emph{left} path and a \emph{right} path), with one interior node each, and the interior vertex in the right path of $A_i$ is identified with the one in the left path of $A_{i+1}$ for $i=1,\ldots , r-1$.
Denote the interior vertex in the left path of $A_i$ by $a_i$ for $i=1,\ldots , r$; and the interior vertex of the right path of $A_r$ by $a_{r+1}$. A chain of length 6 is depicted in Figure~\ref{fig:chain}.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.5\textwidth]{diamond-chain.pdf}
\caption{A chain of 6 diamonds with two different district maps,
each containing an external singleton at a different point.}
\label{fig:chain}
\end{center}
\end{figure}
Assume that a district map $\Pi$ in which district $V_i$ contains the two leaves of diamond $A_i$, for $i=1,\ldots r$.
\begin{observation}\label{obs:unique-singleton}
In a chain of diamonds, there at most one vertex that is not in any of the districts $V_1,\ldots , V_r$,
and it is one of the vertices $a_1,\ldots a_{r+1}$.
\end{observation}
\begin{proof}
By Observation~\ref{obs:cut-containment}, each district $V_i$, $i\in \{1,\ldots ,r\}$ contains at least five vertices: both leaves and both cut vertices in the diamond $A_i$, and an interior vertex in at least one path between the cut vertices. Since the chain of diamonds has only $5r+1$ vertices, at most one vertex belong to some other district, and such a vertex is neither a leaf nor a cut vertex.
\end{proof}
We say that the diamond $A_i$ is \emph{switched to the left} (resp., \emph{right}) if $a_i\in V_i$ (resp., $a_{i+1}\in V_i$).
By Observation~\ref{obs:cut-containment}, every diamond in the chain is switched to left or right (or both).
We show that the switch position in one diamond determines all others to its left or right.
\begin{lemma}\label{lem:cascade-switch}
If a diamond $A_i$ is switched to the left ($a_i\in V_i$), then $A_j$ is also switched to the left ($a_j\in V_j$) for all $j\{1,\ldots, i-1\}$. Likewise, if $A_i$ is switched to the right ($a_{i+1}\in V_i$), then $A_j$ is also switched to the right ($a_{j+1}\in V_j$) for all $j\in\{i+1,\ldots, K\}$.
\end{lemma}
\begin{proof}
If $A_i$ is switched to the left ($a_i\in V_i$), then $A_{i-1}$ cannot be switched to the right because the districts are disjoint.
By Observation~\ref{obs:cut-containment}\eqref{obs2:path-availability}, $A_{i-1}$ must be switched to the left $(a_{i-1}\in V_{i-1}$).
Induction completes the proof. An analogous argument applies when $A_i$ is switched to the right.
\end{proof}
\begin{lemma}\label{lem:switches-across-chain}
Let $\Pi_1$ and $\Pi_2$ be two district maps such that
$a_i\notin \bigcup_{x=1}^r V_x$ in $\Pi_1$ and
$a_j\notin \bigcup_{x=1}^r V_x$ in $\Pi_2$.
Then the distance between $\Pi_1$ and $\Pi_2$ is at least $|j-i|$ in $\Gamma_k(G)$.
\end{lemma}
\begin{proof}
Assume, without loss of generality, that $i<j$. Then the diamonds between $A_i,\ldots, A_{j-1}$ are switched to the right in $\Pi_1$ and to the left in $\Pi_j$ by Lemma~\ref{lem:cascade-switch}. These $j-1$ diamonds must switch. Each diamond requires one switch to expand into its left path (the contraction from the right path not counted, since each switch contracts one district and expands another).
\end{proof}
\subsection{Incontractible}
\label{ssec:LB-incontractible}
In this section, we construct a graph $G$ and two $k$-district maps $\Pi_1$ and $\Pi_2$,
and show that they are at distance distance $\Omega (k^3 + kv)$ in $\Gamma_k(G)$.
\begin{theorem}\label{thm:LB}
For every $k,n\in \mathbb{N}$, $6k\leq n$, there exists a graph $G$ with $n$ vertices and two $k$-district maps, $\Pi_1$ and $\Pi_2$, such that the switch graph $\Gamma_k(G)$ contains a path between $\Pi_1$ and $\Pi_2$, but the length of every such path is $\Omega(k^3+kn)$.
\end{theorem}
\begin{proof}
We construct a graph $G$ in terms of three parameters, $r$, $q$, and $\ell$, and choose their values at the end of the proof.
Create two disjoint chains of diamonds, $A=(A_1,\ldots, A_r)$ and $B=(B_1,\ldots ,B_{r-1})$, of lengths $r$ and $r-1$, respectively, where $r$ is an even integer to be specified later. Denote the interior vertices of the paths in the diamonds by
$a_1,\ldots , a_{r+1}$ and $b_1,\ldots, b_r$, respectively. Insert a (spiral) path $S$ of length $2r$ on these vertices constructed as follows: Connect $a_1$ to $b_{r/2}$. For $i\in \{2,\ldots , r/2 + 1\}$, connect $a_i$ to $b_{r/2+i-1}$ and $b_{r/2-i+1}$. For $i\in \{r/2 + 2,\ldots, r + 1\}$, connect $a_i$ to $b_{i-1-r/2}$ and $b_{3r/2-i-2}$. Connect $a_{r/2+1}$ to $b_r$.
Finally, create two trees, $D_1$ and $D_2$, each consists of a path of length $q$ and
$\ell$ leaves attached to one endpoint; and identify the other endpoints of the paths
with $a_1$ and $a_{r/2+1}$, respectively.
See Fig.~\ref{fig:fans} for an example.
The total number of vertices is $n=5(2r-1)+2+2\ell+2q-2=10r+2\ell+2q-5$.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.55\textwidth]{incontract-lower.pdf}
\caption{Graph $G$ for $r=6$, with two chains of diamonds, and a spiral $S$.}
\label{fig:incont-lower}
\end{center}
\end{figure}
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.65\textwidth]{fans.pdf}
\caption{$D_1$ in $\Pi_1$, and $D_2$ in $\Pi_2$}
\label{fig:fans}
\end{center}
\end{figure}
We construct two district maps, $\Pi_1$ and $\Pi_2$ on $G$. Both $\Pi_1$ and $\Pi_2$ have $2r - 1$ diamond districts switched to the left (i.e., $a_i\in V_i$ and $b_i\in W_i$), and the last diamond ($A_r$ and $B_{r-1}$) are switched to both left and right.
These districts contain all vertices of the two chains of diamonds.
The district map $\Pi_1$ contains $\ell$ singleton districts at the leaves of $D_1$
and one district for the path in $D_1$, and one district for all vertices of $D_2$;
while $\Pi_1$ contains one district in $D_1$, $\ell$ singleton districts
at the leaves of $D_2$, and one in the path in $D_2$.
The total number of districts is $k=(2r-1)+\ell+2$.
Consider a shortest sequence $\sigma$ of switches that takes $\Pi_1$ to $\Pi_2$
(we later show that such a sequence exists). The sequence $\sigma$ defines
a one-to-one correspondence between the districts in $\Pi_1$ and $\Pi_2$.
Both $\Pi_1$ and $\Pi_2$ contain $2r-1$ identical districts in diamonds,
that each contain two leaves. By Lemma~\ref{lem:con1}, these district
are fixed in the one-to-one correspondence.
We call the remaining $\ell+2$ districts \emph{mobile}, and conclude that
$\sigma$ moves the mobile districts in $\Pi_1$ to mobile districts in $\Pi_2$.
In particular, at least $\ell$ mobile districts move from $D_1$ to $D_2$.
\begin{claim}\label{cl:shortest-path}
If a mobile district moves from $D_1$ to $D_2$, then it moves along the spiral $S$.
\end{claim}
\begin{proof}
By Observation~\ref{obs:unique-singleton}, at most one mobile district may reside in a diamond chain at a time, but cannot travel along the edges in that chain because all adjacent vertices are cut vertices. Thus the only available edges for switches are the edges of the spiral $S$.
\end{proof}
\begin{claim}\label{cl:singleton-at-a-time}
At most one mobile district is outside of $D_1$ and $D_2$ at any time.
\end{claim}
\begin{proof}
Suppose, to the contrary, that two mobile districts are outside of $D_1$ and $D_2$ at the same time.
Then they must be in different chains, by Observation~\ref{obs:unique-singleton}. Neither can move until one of them leaves to $D_1$ or $D_2$, because they prevent the diamonds in the chain from switching between left and right (cf.~Observation~\ref{obs:cut-containment}).
However, then one of the two mobile districts would have to move back to its previous position. Therefore, we could eliminate two switches from $\sigma$, contradicting the assumption that $\sigma$ is a shortest sequence that takes $\Pi_1$ to $\Pi_2$.
\end{proof}
This claim conveniently means that the shortest sequence of switches from $\Pi_1$ to $\Pi_2$
moves each mobile district through the spiral $S$ independently.
Note that there exists a sequence of switches that takes $\Pi_1$ to $\Pi_2$.
Indeed, it is enough to move $\ell$ mobile districts from $D_1$ to $D_2$.
We can move them, one at a time, along the spiral $S$.
Since the spiral $S$ alternates between chain $A$ and chain $B$,
when the mobile district is a singleton in one chain, we can reconfigure
the other chain to make the next vertex of $S$ available.
\smallskip\noindent\textbf{Analysis.}
We analyze the length of the sequence of switches $\sigma$ that takes $\Pi_1$ to $\Pi_2$.
By Claims~\ref{cl:shortest-path} and~\ref{cl:singleton-at-a-time}, we may assume that one mobile district moves from $D_1$ to $D_2$ along $S$, and then multiply by $\ell$, the number of singletons. We further break down the cost by analyzing the number of switches needed to move a mobile district from $a_i$ to $b_{r/2+i}$, for $i < r/2$.
As a mobile district travels through $S$, it visits all vertices in $\{a_1,\ldots, a_{r+1}\}$.
Between a visit to $a_i$ and $a_j$, at least $|j-i|$ diamonds in the chain $A$ must be reconfigured, using at least $|j-i|$ switches.
The summation of gaps between consecutive vertices in in $\{a_1,\ldots, a_{r+1}\}$ is an arithmetic progression that decreases from
$r$ to 1, and sums to $\sum_{i=1}^r i=\binom{r+1}{2}$. Similarly, the summation of gaps between consecutive vertices in in $\{b_1,\ldots, b_{r}\}$ is an arithmetic progression that increases from $1$ to $r-1$, and sums to $\sum_{i=1}^{r} i=\binom{r}{2}$.
The switches that reconfigure diamonds in the chain $A$ do not have any impact in the chain $B$.
Furthermore, at least $\ell-1$ mobile districts move from some leaves of $D_1$ to some leaves of $D_2$.
These mobile district have to travel through the two paths of length $q$ in $D_1$ and $D_2$, respectively.
Any switch that expands a mobile district inside $D_1$ or $D_2$ is distinct from switches that reconfigure diamonds.
Consequently, the number of switches in $\sigma$ is at least
\[
|\sigma|\geq \ell\left(\binom{r+1}{2}+\binom{r}{2}\right)+(\ell-1)2q = \Theta(\ell r^2+\ell q).
\]
Let us choose the parameters so that $q=\Theta(n)$ and $r,\ell=\Theta(k)$, and then
$|\sigma|\geq \Omega(k^3+kn)$, as claimed.
\end{proof}
\section{Introduction}
\label{sec:intro}
An \emph{electoral district} is a subdivision of territory used in the election of members to a legislative body. \emph{Gerrymandering} is the practice of drawing district boundaries with the intent to give political advantage to a particular group; it
tends to occur in electoral systems that elect one representative per district.
Detecting whether gerrymandering has been employed in designing a given district map
and producing unbiased district maps are
important problems to ensure fairness in the outcome of elections. Numerous quality measures have been proposed for the comparison of district maps~\cite{Moon,Moon18}, but none of them is known to eliminate bias. Research has focused on exploring the space of all possible district maps that meet certain basic criteria.
Since this space is computationally intractable, even for relatively small instances, randomized
algorithms play an important role in finding ``average'' district maps under suitable distributions~\cite{BGH+17}. Being an outlier may indicate that gerrymandering
has been applied in the drawing of a given map~\cite{HRM17}.
Fifield et al.~\cite{MCMC} models a district map as a vertex partition on an adjacency graph of census tracts or voting precincts. A \emph{census tract} is a small territorial subdivision used as a geographic unit in a census. Each district corresponds to a set of census tracts in the partition and must induce a connected subgraph.
Starting from a given district map, one can obtain another map by switching a subset of census tracts from one district to another. The goal is to apply such operation randomly, sampling the space of all possible district maps that meet the desired criteria.
They prove that, under some assumptions, the Markov chain produced by their experiments is ergodic, and
has a unique stationary distribution, which is approximately uniform on the space of all $k$-district maps.
Ergodicity would require the underlying sample space to be connected under the switch operation.
However, connectedness is only assumed and remains unproven in~\cite{MCMC}.
In this paper, we provide a rigorous graph-theoretic background for studying the space of district maps with a given number of districts, taking into account only the constraint that each district must be connected.
We focus on the 1-switch operation that moves precisely one vertex from one district to another.
The size of a district is an arbitrary positive integers. Note, however, that when the space of all $k$-district maps is connected, any aperiodic Markov chain is also ergodic on the subset of \emph{balanced} $k$-district maps (in which the districts have roughly the same size), which are the most relevant maps for applications. Our results are the first that establish criteria for the connectivity of this space in a graph theoretic model.
\smallskip\noindent\textbf{Our Results.}
We consider the graph-theoretic model from~\cite{MCMC}. For a graph $G$ (the adjacency graph of precincts or census tracts) and an integer $k\geq 1$, we consider the \emph{switch graph} $\Gamma_k(G)$ in which each node corresponds to the partition of $V(G)$ into $k$ nonempty subsets (districts), each of which induces a connected subgraph, and an edge corresponds to switching one vertex from one district to an adjacent district (see Section~\ref{sec:pre} for a definition).
\begin{enumerate}
\item \textbf{Connectedness.} We prove that $\Gamma_k(G)$ is connected if $G$ is biconnected (Theorem~\ref{thm:2conn-alg}), and give a combinatorial characterization of connectedness that can be tested
in $O(n+m)$ time, where $n=|V(G)|$ and $m=|E(G)|$ (Theorem~\ref{thm:conn-test}).
In general, however, it is NP-complete to decide whether two given nodes of $\Gamma_k(G)$ are in the same connected component (Theorem~\ref{thm:connectedness}).
\item \textbf{Contractible Districts.} One of our key methods to modify a district map is to contract a district into a single vertex by a sequence of switch operations. If this is feasible, we call the district \emph{contractible}; and if all districts are contractible, we call the district map \emph{contractible}. We prove that the subgraph $\Gamma_k'(G)\subset\Gamma_k(G)$ corresponding to contractible district maps is connected if $G$ is connected (Theorem~\ref{thm:general-graphs}).
\item \textbf{Diameter.} When $G$ is biconnected, the diameter of $\Gamma_k(G)$ is in $O(kn)$,
where $n=|V(G)|$ (Theorem~\ref{thm:general-graphs}), and this bound is the best possible (Theorem~\ref{thm:contractible-LB}).
When $\Gamma_k(G)$ is disconnected, the diameter of a component may be as large as $\Omega(k^3+kn)$ (Theorem~\ref{thm:LB}).
\item \textbf{Shortest Path.} Finding the shortest path between two nodes of $\Gamma_k(G)$ is NP-hard, even if $\Gamma_k(G)$ is connected, or the two nodes are known to be in the same component (Theorem~\ref{thm:shortest}).
\end{enumerate}
\ShoLong{Due to lack of space, portions of the document have been moved to the Appendix.}{}
\noindent\textbf{Related Previous Work.}
Graph partition and graph clustering algorithms~\cite{GraphPartition} are widely used in divide-and-conquer strategies.
These algorithms do not explore the space of all partitions into $k$ connected subgraphs.
Evolutionary algorithms~\cite{Sanders2012}, in this context, modify a partition by random ``mutations,'' which are successive coarsening and uncoarsening operations, rather than moving vertices from one subgraph to another.
While the adjacency graph model for district maps has been used for decades in combinatorial optimization and operations research~\cite{RiccaSS13}, the objective was finding optimal district maps under one or more criteria. Since exhaustive search is infeasible and most variants of the optimization problem are intractable~\cite{PuppeT09}, local search heuristics were suggested~\cite{RiccaS08}.
Several combinatorial results restrict $G$ to be a square grid~\cite{ApollonioBLRS09,PuppeT08}.
Heuristic and intractability results are also available for geometric variants of the optimization problem,
where districts are polygons in the plane~\cite{FleinerNT17,KMV-hardness-18,RiccaSS08}.
Elementary graph operations similar to our 1-switch operation have also been studied.
Motivated by the classical ``Fifteen'' puzzle, Wilson~\cite{WILSON1974} studied the configuration space of $t$, $t< n$, indistinguishable pebbles (a.k.a.~tokens~\cite{MonroyFHHUW12}) on the vertices of a graph $G$ with $n$ vertices, where each pebble occupies a unique vertex of $G$, and can move to any adjacent unoccupied vertex.
The occupied and unoccupied vertices partition $V(G)$ into two subsets. Crucially, the number of pebbles is fixed, and the occupied vertices need not induce a connected subgraph. Results include a combinatorial characterization of the configuration space (a.k.a.~token graph)~\cite{WILSON1974}, NP-hardness for deciding connectedness~\cite{KornhauserMS84}, finding the shortest path between two configurations~\cite{Goldreich11,RatnerW90}, and bounds on the diameter and the connectivity of the configuration space~\cite{LeanosT18,MonroyFHHUW12}. Demaine et al.~\cite{DemaineDFHIOOUY15}
considered a subgraph of the token graph, where the tokens are located at an independent set.
The diameter and shortest path in the configuration space can often be computed efficiently when
the underlying graph $G$ is a tree~\cite{AulettaMPP99,DemaineDFHIOOUY15}, or chordal~\cite{BonamyB17}.
There are a few results that require the occupied vertices to induce a connected subgraph, but they are limited to the case where $G$ is a gird~\cite{DumitrescuP06,KomuravelliSB09}, and the number of pebbles is still fixed.
Goraly et al.~\cite{GoralyH10} have later considered colored pebbles (tokens). Each color class consists of indistinguishable pebbles, unoccupied vertices are considered as one of the color classes ~\cite{FujitaNS12,YamanakaDIKKOSS15,YamanakaHKKOSUU18}: Hence all vertices in $V(G)$ are occupied and a move can swap the pebbles on two adjacent vertices. The color classes (including the ``unoccupied'' color) partition $V(G)$ into subsets. However, the cardinality of each color class remains fixed and the color classes need not induce connected subgraphs. Results, again, include combinatorial characterizations to connected configuration space~\cite{FoersterGHKSW17}, NP-completeness for the connectedness of the configuration space for $k\geq 3$ colors, and a polynomial-time algorithm for finding the shortest path for $k=2$ colors. See~\cite{BonnetMR18,MiltzowNORTU16} for recent results on the parametric complexity of these problems.
\smallskip\noindent\textbf{Organization.}
Section~\ref{sec:pre} defines the problem formally, and describes some important properties of contractible districts. Section~\ref{sec:alg} shows that $\Gamma_k(G)$ is connected if $G$ is biconnected, and $\Gamma_k'(G)$ is connected if $G$ is connected. Section~\ref{sec:LB} presents our lower bounds for the diameter of $\Gamma_k(G)$ and its components, and Section~\ref{sec:hardness} continues with our NP-hardness results.
\ShoLong{}{We conclude in Section~\ref{sec:con}.}
\section{Preliminaries}
\label{sec:pre}
Let $G=(V,E)$ be a connected graph. A \emph{$k$-district map} $\Pi$ of $G$ is a partition of $V(G)$ into disjoint nonempty subsets $\{V_1,\ldots,V_k\}$ such that the subgraph induced by $V_i$ is connected for all $i\in\{1,\ldots,k\}$.
Each subgraph induced by $V_i$ is called a \emph{district}.
We abuse the notation by writing $\Pi(v)$ for the subset in $\Pi$ that contains vertex $v$.
Given a $k$-district map $\Pi=\{V_1,\ldots,V_k\}$, and an path $(u,v,w)$ in $G$ such that $\Pi(u)=\Pi(v)\neq \Pi(w)$, a \emph{switch} (denoted \textsf{switch}$_\Pi(u,v,w)$) is an operation that returns a $k$-district map obtained from $\Pi$ by removing $v$ from the subset $\Pi(u)$ and adding it to $\Pi(w)$.
More formally, \textsf{switch}$_\Pi(u,v,w)=\Pi'=(\Pi\setminus\{\Pi(u),\Pi(w)\})\cup\{\Pi(u)\setminus\{v\},\Pi(w)\cup\{v\}\}$ if $ \Pi'$ is a $k$-district map.
Note that \textsf{switch}$_\Pi(u,v,w)$ is not defined if $\Pi(v)\setminus\{v\}$ induces a disconnected subgraph.
A switch is always reversible since if \textsf{switch}$_\Pi(u,v,w)=\Pi'$, then \textsf{switch}$_{\Pi'}(w,v,u)=\Pi$.
For every graph $G$ and integer $k$, the \emph{switch graph} $\Gamma_{k}(G)$
is the graph whose vertex set is the set of all $k$-district maps of $G$, and $\Pi_1, \Pi_2\in V(\Gamma_{k}(G))$ are connected by an edge if there exist $u,v,w\in V(G)$ such that \textsf{switch}$_{\Pi_1}(u,v,w)=\Pi_2$.
\global\def \preliminarysec {
\label{ssec:2con}
Biconnectivity plays an important role in our proofs. In particular, we rely on the concept of a \emph{block tree}, which represents the containment relation between the blocks (maximal biconnected components) and the cut vertices of a connected graph; and a \emph{SPQR tree}, which corresponds to a recursive decomposition of a biconnected graph. We review both concepts here.
\smallskip\noindent\textbf{Block Trees.} Let $G$ be a connected graph. Let $B(G)$ be the set of \emph{blocks} of $G$. (Two adjacent vertices induce a 2-connected subgraph, so a block may be a subgraph with a single edge.) Let $C(G)$ be the set of cut vertices in $G$. Then the \textbf{block tree} $T=T(G)$ is a bipartite graph, whose vertex set is $V(T)=B(G)\cup C(G)$, and $T$ contains an edge $(b,c)\in B(G)\times C(G)$ if and only if $c\in b$ (i.e., block $b$ contains vertex $c$). The definition immediately implies every leaf in $T$ corresponds to a block $b\in B(G)$ (and never a cut vertex in $C(G)$). The block tree can be computed in $O(|E(G)|)$ time and space~\cite{HopcroftT73}.
For convenience, we label every biconnected component by its vertex set (i.e., for a block $w\in B(G)$, we denote by $w$ the complete set of vertices in the block).
\smallskip\noindent\textbf{SPQR Trees.} Let $G$ be a biconnected planar graph. A deletion of a (vertex) 2-cut $\{u,v\}$ disconnects $G$ into two or more components $C_1,\ldots ,C_i$, $i\geq 2$. A \textbf{split component} of $\{u,v\}$ is either an edge $uv$ or the subgraphs of $G$ induced by $V(C_j)\cup\{u,v\}$ for $j=1,\ldots, i$.
The SPQR-tree $T_G$ of $G$ represents a recursive decomposition of $G$ defined by its 2-cuts.
A node $\mu$ of $T_G$ is associated with a multigraph called $\text{skeleton}(\mu)$ on a subset of $V(G)$, and has a \textbf{type} in $\{$S,P,R$\}$. If the type of $\mu$ is S, then $\text{skeleton}(\mu)$ is a cycle of 3 or more vertices.
If the type of $\mu$ is P, then $\text{skeleton}(\mu)$ consists of 3 or more parallel edges between a pair of vertices.
If the type of $\mu$ is R, then $\text{skeleton}(\mu)$ is a 3-connected graph on 4 or more vertices.
An edge in $\text{skeleton}(\mu)$ is \textbf{real} if it is an edge in $G$, or \textbf{virtual} otherwise.
A virtual edge connects the two vertices of a 2-cut, $u$ and $v$, and represents a subgraph of $G$
obtained in the recursive decomposition, containing a $uv$-path in $G$ that does not contain any edge in $\text{skeleton}(\mu)$.
Two nodes $\mu_1$ and $\mu_2$ of $T_G$ are adjacent if $\text{skeleton}(\mu_1)$ and $\text{skeleton}(\mu_2)$ share exactly two vertices, $u$ and $v$, that form a 2-cut in $G$.
Each virtual edge in $\text{skeleton}(\mu)$ corresponds to a pair of adjacent nodes in $T_G$.
No two S nodes (resp., no two P nodes) are adjacent. Therefore, $T_G$ is uniquely defined by $G$.
If $\mu$ is a leaf in $T_G$, then $\text{skeleton}(\mu)$ has a unique virtual edge; in particular the type of every leaf is S or R.
The SPQR tree $T_G$ has $O(|E(G)|)$ nodes and can be computed in $O(|E(G)|)$ time~\cite{BattistaT96}.
}
\ShoLong{Biconnectivity plays an important role in our proofs. In particular, we rely on the concept of a \emph{block tree}, which represents the containment relation between the blocks $B(G)$ (maximal biconnected components) and the cut vertices $C(G)$ of a connected graph.
We also rely on the \emph{SPQR tree}, which corresponds to a recursive decomposition of a biconnected graph. A more detailed review of these concepts is given in Section~\ref{ssec:2con}.}
{
\subsection{Block Trees and SPQR Trees}
\preliminarysec
}
\subsection{Contractibility}
Consider a graph $G$ and a $k$-district map $\Pi$. We say that the operation \textsf{switch}$_\Pi(u,v,w)$
\emph{contracts} $\Pi(u)$ to $\Pi(u)\setminus\{v\}$, and \emph{expands} $\Pi(w)$ to $\Pi(w)\cup\{v\}$.
A sequence of $n$ switches \emph{contracts} (resp., \emph{expands}) $V_i$ to $V_i'$ if there exists a sequence of consecutive switches that jointly contract (resp., expand) $V_i$ to $V_i'$. A subset $V_i\in\Pi$ (and its induced district) is \emph{contractible} if it can be contracted to a singleton (district of size one)
by a sequence of $|V_i|-1$ switches; otherwise it is \emph{incontractible}.
A $k$-district map is \emph{contractible} if all its districts are contractible.
\begin{lemma}\label{lem:leaves}
A switch operation cannot move a leaf of $G$ from one district to another.
\end{lemma}
\begin{proof}
Let $v\in V(G)$ be a leaf in $G$, and let $u\in V(G)$ be its unique neighbor.
Suppose, for the sake of contradiction, that a switch moves $v$ from one district to another.
Then there exists a path $(u,v,w)$ and a $k$-district map $\Pi$ for which \textsf{switch}$_\Pi(u,v,w)$ is a valid operation.
However, $v$ is a leaf, and no such path exists. The contradiction completes the proof.
\end{proof}
\begin{lemma}\label{lem:con1}
Let $T$ be the block tree of a graph $G$, and let $\Pi$ be a $k$-district map on $G$. If a district $V_\ell$ contains two leaves of $T$, say $w_i, w_j \in B(T)$, then a switch operation cannot move any vertex from $w_i\cup w_j$ to another district. Consequently, $V_\ell$ is incontractible.
\end{lemma}
\begin{proof}
Suppose, for the sake of contradiction, that $w_i\cup w_j\subseteq V_\ell$ and a switch moves some vertex $v\in w_i\cup w_j$ to another district. Then there exists a $k$-district map $\Pi$ and a path $(u,v,w)$ such that $\Pi(u)=\Pi(v)\neq \Pi(w)$, and $\Pi(u)\setminus \{v\}$ induces a connected graph. By assumption,
$w_i\cup w_j\subset \Pi(u)$. Since $w_i$ and $w_j$ are leaves in $T$, only their cut vertices can be adjacent to vertices outside of $w_i\cup w_j$. Assume w.l.o.g. that $v\in C(G)$ is the cut vertex in $w_i$. However, every path from $w_i\setminus \{v\}$ to $w_j\setminus \{v\}$ passes through $v$. Consequently $\Pi(u)\setminus \{v\}$ does not induce a connected subgraph in $G$, and \textsf{switch}$_{\Pi}(u,v,w)$ is
undefined.
Since $w_i \ne w_j$, there are at least two vertices, one from each block, that remain in $V_\ell$ after any sequence of switch operation. Consequently, $V_\ell$ cannot become a singleton.
\end{proof}
\begin{lemma}\label{lem:con2}
Let $\Pi$ be a $k$-district map on $G$, and let $V_i\in \Pi$ such that $V_i$ contains at most one leaf of the block tree $T$ of $G$. Then $V_i$ is contractible. Furthermore,
\begin{itemize}
\item if $V_i$ does not contain any leaf of the block tree, then $V_i$ can be contracted to any of its vertices;
\item if $V_i$ contains a leaf $w_j\in B(G)$ of the block tree, then $V_i$ can be contracted to a vertex $v$ if and only if $v\in w_j$ but $v$ is not a cut vertex in $C(G)$.
\end{itemize}
In both cases, a sequence of $|V_i|-1$ switches that contract $V_i$ can be computed in $O(|E(G)|)$ time.
\end{lemma}
\begin{proof}
We first prove a necessary condition on the target vertex for contraction. Assume that $V_i$ can be contracted to a vertex $t\in V_i$, and $V_i$ contains exactly one leaf $w_j\in B(G)$ of the block tree. Let $c_j$ be the unique cut vertex in $w_j$. Since since every path between $w_j\setminus \{c_j\}$ and $V_i\setminus w_j$ contains the cut vertex $c_j$, and the subgraph induced by $V_i$ must be connected at all times, no vertex in $w_j\setminus \{c_j\}$ can change districts until $c_j$ and all vertices of $V_i$ outside of $w_j$ have switched to another district. At this point, we have $V_i=w_j\setminus \{c_j\}$, consequently $t\in w_j\setminus \{c_j\}$, as required.
We next show that the above conditions are sufficient. Assume that $V_i$ and a target vertex $t\in V_i$ satisfy the above restrictions. It is enough to show that if $V_i\neq \{t\}$, there exists a vertex $v\in V_i\setminus \{t\}$, such that $v$ can be switched to another district; and $t$ and $V\setminus \{v\}$ satisfy the conditions above. Then we can successively switch all vertices in $V_i\setminus \{t\}$ to other districts until $V_i=\{t\}$, which proves that $V_i$ is contractible.
We prove that a vertex $v\in V_i\setminus \{t\}$ with the required properties exists. Let $G'$ be the subgraph induced by $V_i$. Compute the block tree of $G'$, and denote it by $T'$. Root $T'$ at the block vertex in the tree that contains $t$. We distinguish between cases.
\begin{itemize}
\item If $G'$ is not biconnected, then $G'$ contains two or more blocks. Let $w'\in B(G')$ be a leaf block in $T'$ other than the root, and let $c'\in C(G')$ be its unique cut vertex. Note that $w'$ is not a leaf block in $T$, otherwise $V_i$ would contain this leaf block, and we would assume that $t$ is a vertex of any such block. Since $w'$ is not a leaf block in $T$, it is either a subset of a non-leaf block of $T$ or a proper subset of a leaf block of $T$. In either case, there exists a vertex $v\in w'\setminus \{c'\}$ adjacent to some vertex $u\in V(G)\setminus V_i$. Since $w'$ is biconnected, $w'\setminus \{v\}$ induces a connected subgraph in $G$; consequently $V_i\setminus \{v\}$ induces a connected subgraph, as well. Therefore, $v$ can be switched to the district of $u$.
\item If $G'$ is biconnected, then $G'$ is a subgraph of some block $w\in B(G)$.
We claim that there exists a vertex $v\in V_i\setminus \{t\}$ adjacent to some vertex $u\in V(G)\setminus V_i$. To prove the claim, suppose the contrary. Then every path between $V_i\setminus \{t\}$ and $V(G)\setminus V_i$ goes through $t$. This implies that $t$ is a cut vertex, and $V_i$ is a leaf block in $G$, which contradicts our assumption for $V_i$ and $t$. This proves the claim, and $v$ can be switched to the district of $u$.
\end{itemize}
First, the invariant that $V_i$ contains at most one leaf $w\in B(T)$ is maintained. Since contraction can only remove vertices from $V_i$, the number of leaf blocks contained in $V_i$ monotonically decreases, so this invariant is maintained. Second, $t$ remains a valid choice for the contraction target. Note that if $V_i\setminus \{v\}$ contains the same leaf blocks as $V_i$, then $t$ remains a valid target. If $V_i$ contains a leaf block, say $w_j$, and $V_i\setminus \{v\}$ does not, then $v$ is the (unique) cut vertex of the block $w_j$. In this case, $t\in V_i\setminus \{v\}$, and any vertex in $V_i\setminus \{v\}$ is a valid choice for $t$. In both cases, the validity of the choice of $t$ is maintained.
This proves that $V_i$ is contractible, as required.
Our proof is constructive and leads to an efficient algorithm that successively switches every vertex in $V_i\setminus \{t\}$ to some other districts until $V_i=\{t\}$. The block trees $T$ and $T'$ can be computed in $O(|E(G)|)$ time~\cite{HopcroftT73}. While $V_i$ is contracted, we maintain the induced subgraph $G'$, and the set of edges between $V_i$ and $V(G)\setminus V_i$ in $O(|E(G)|)$ total time. While $T'$ contains two or more blocks, we can successively switch all vertices of a leaf block $w'$ that does not contain $t$ to other districts; eliminating the need for recomputing $T'$. Consequently, the overall running time of the algorithm is $O(|E(G)|)$.
\end{proof}
\begin{lemma}\label{lem:invariant}
The contractibility (resp., incontractibility) of a $k$-district map on a graph $G$ is invariant under switch operations.
\end{lemma}
\begin{proof}
Every incontractible $k$-district map contains some incontractible subset $V_i$. Lemmas~\ref{lem:con1}--\ref{lem:con2} show that a subset $V_\ell \in \Pi$ is incontractible if and only if $V_\ell$ contains at least two leaves of the block tree, say $w_i,w_j\subseteq V_\ell$.
By Lemma~\ref{lem:con2}, $w_i\cup w_j\subset V_\ell$ after any sequence of switches, so $V_i$ remains incontractible under any sequence of switches, consequently $\Pi$ remains incontractibile.
Suppose that a contractible subset $V_i\in \Pi$ becomes incontractible by some sequence of switches. Since the reverse switch operations are also valid, this implies that an incontractible district could become contractible after a sequence of switches. This contradicts the previous claim, so every contractible subset $V_i\in \Pi$ remains contractible under any sequence of switches.
\end{proof}
\section{Connectedness}
\label{sec:alg}
\ShoLong{}{In this section, we show that the switch graph $\Gamma_k(G)$ is connected when $G$ is biconnected (Sec.~\ref{ssec:2conn-alg}), and the switch graph $\Gamma_k'(G)$ over contractible instances is connected if $G$ is connected (Sec.~\ref{ssec:general-graphs}).}
\subsection{Biconnected Graphs}
\label{ssec:2conn-alg}
\ShoLong{}{This section is devoted to the proof of the following theorem.}
\begin{theorem}\label{thm:2conn-alg}
For every biconnected graph $G$ with $n$ vertices, and for every $k\in \mathbb{N}$,
the switch graph $\Gamma_k(G)$ is connected and its diameter is bounded by $O(kn)$.
\end{theorem}
\begin{proof}
Let $G$ be a biconnected graph, and $\Pi$ a $k$-district map for some $k\in \mathbb{N}$. We may assume that $k<n$, otherwise $\Gamma_k(G)$ is trivially connected. We present an algorithm (Algorithm~\ref{algo:2conn}) that performs a sequence of switch operations that transform $\Pi$ into a specific $k$-district map of $G$, that we denote by $\Pi_0$. We show that $\Pi_0$ depends only on $G$ and $k$ (but not on $\Pi$). Consequently, any two $k$-district maps can be transformed to $\Pi_0$, and $\Gamma_k(G)$ is connected.
\begin{algorithm}[htbp]
\caption{Canonical Algorithm for Biconnected Graphs}\label{algo:2conn}
\begin{algorithmic}[1]
\Procedure{Canonical}{$G,k,\Pi$}
\While {$k>1$}
\State Compute the SPQR tree $T_G$ of $G$; order the leaves by DFS; let $\mu$ be the first leaf.
\If{$\mu$ is an S node (and $\text{skeleton}(\mu)$ is a cycle with one virtual edge)}
\State Let $\text{skeleton}(\mu)=(v_1,\ldots, v_t)$, where $v_1v_t$ is the virtual edge; set $i=2$.
\While {$i<t$ and $k>1$}
\State Contract $\Pi(v_i)$ to $\{v_i\}$
\State Delete vertex $v_i$ from $G$, and put $i:=i+1$ and $k:=k-1$.
\EndWhile
\Else{ $\mu$ is an R node (and $\text{skeleton}(\mu)$ is triconnected)}
\State Let $v$ be an arbitrary vertex that is not incident to the (unique) virtual edge.
\State Contract $\Pi(v)$ to a single vertex
\State Delete vertex $v$ from $G$, and put $k:=k-1$.
\EndIf
\EndWhile
\EndProcedure
\end{algorithmic}
\end{algorithm}
\smallskip\noindent\textbf{Proof of Correctness.}
Algorithm~\ref{algo:2conn} successively contracts a district into a single vertex,
and then deletes this vertex from the graph, and the corresponding district from $\Pi$,
until the number of districts drops to 1. We need to show that each district that the algorithm
contracts into a singleton is contractible. We prove two invariants that imply this property:
\global\def \pfclaimAAA {
\begin{proof}[Proof of Claim~\ref{cl:AAA}.]
Let $\mu$ be the leaf node in line~3 of the algorithm.
If $\mu$ is an R node, then the $G$ remains biconnected after the deletion of a vertex $v$.
Assume that $\mu$ is an S node, corresponding to a cycle $(v_1,\ldots , v_t)$, $t\geq 3$, where $v_1v_t$ is the only edge that corresponds to a virtual edge. Then the deletion of all vertices in $\{v_1,\ldots, v_{t-1}\}$ produces a biconnected graph; and the deletion of a $\{v_2,\ldots ,v_i\}$, $2\leq i<t-1$, produces a biconnected graph with a dangling path $(v_{i+1},\ldots, v_t)$.
\end{proof}
}
\global\def \claimAAA {
The graph $G$ remains connected during Algorithm~\ref{algo:2conn}.
}
\begin{claim}\label{cl:AAA}
\claimAAA
\end{claim}
\ShoLong{}{\pfclaimAAA}
\begin{claim}\label{cl:BBB}
The district map $\Pi$ remains contractible during Algorithm~\ref{algo:2conn}.
\end{claim}
\global\def \pfclaimBBB {
\begin{proof}[Proof of Claim~\ref{cl:BBB}.]
In a biconnected graph, every district is contractible by Lemma~\ref{lem:con2}.
Let $\mu$ be the leaf node in line~3 of the algorithm.
If $\mu$ is a R node, then the graph $G$ remains biconnected after the deletion of a vertex, and so the $(k-1)$-district map of the remaining graph is contractible.
If $\mu$ is an S node, then $G$ obtained by deleting vertex $v_i$ is either biconnected or a biconnected graph with a dangling path $(v_{i+1},\ldots, v_t)$. In both cases, $G$ has at most one leaf. By Lemma~\ref{lem:con2}, every district that contains at most one leaf is contractible, and so the district map remains contractible.
\end{proof}
}
\ShoLong{}{\pfclaimBBB}
The following claim establishes that the switch graph $\Gamma_k(G)$ is connected since it contains a path from any district map to the district map produced by Algorithm~\ref{algo:2conn}.
\begin{claim} \label{cl:CCC}
The switch operations performed by Algorithm~\ref{algo:2conn} transform $\Pi$ to a $k$-district map that depends only on $G$ and $k$.
\end{claim}
\global\def \pfclaimCCC {
\begin{proof}[Proof of Claim~\ref{cl:CCC}.]
Denote by $\Pi_0$ the $k$-district map produced by the switch operations performed by Algorithm~\ref{algo:2conn}. When a district is contracted to a singleton, the vertex is deleted from the the graph, and so
this singleton district is in $\Pi_0$.
Since each vertex deleted from the graph $G$ was selected based on the current graph $G$, its SPQR tree, and the DFS order of its leaves, the sequence of deleted vertices depends only on $G$ and $k$. The number of deleted vertices is precisely $k-1$.
\end{proof}
}
\ShoLong{}{\pfclaimCCC}
\smallskip\noindent\textbf{Analysis.}
Algorithm~\ref{algo:2conn} successively contracts $k-1$ districts into singletons. By Lemma~\ref{lem:con2}, each of these contractions involves a sequence of $O(n)$ switches that can be computed in $O(|E(G)|)$ time.
Overall Algorithm~\ref{algo:2conn} runs in $O(k|E(G)|)$ time and performs $O(kn)$ switch operations.
For any two $k$-district maps, $\Pi_1$ and $\Pi_2$, there exists a sequence of $O(kn)$ switches that takes $\Pi_1$ to $\Pi_0$ and then to $\Pi_2$. Therefore, the diameter of $\Gamma_k(G)$ is $O(kn)$.
\end{proof}
\subsection{Algorithm for General Graphs}
\label{ssec:general-graphs}
If $G$ is a biconnected graph, then every district map is contractible by Lemma~\ref{lem:con2}, and so $\Gamma_k(G)=\Gamma_k'(G)$.
In this section, we show that if $G$ is connected, then $\Gamma_k'(G)$ is connected. That is, any contractible $k$-district map can be carried to any other by a sequence of switch operations.
\begin{theorem}\label{thm:general-graphs}
For every connected graph $G$ with $n$ vertices, and for every $k\in \mathbb{N}$,
the switch graph $\Gamma_k'(G)$ over contractible $k$-district maps
is connected and its diameter is bounded by $O(kn)$.
\end{theorem}
A crucial technical step is to move a district from one block to another, through a cut vertex. This is accomplished in the following technical lemma.
\begin{lemma}\label{lem:pushing}
Let $G$ be a connected graph with $n$ vertices that comprises two leaf blocks, $w_1,w_2\in B(G)$, and let $P$ be a path whose first (resp., last) vertex is in $w_1$ (resp., $w_2$); possibly, $P$ has a single vertex. Let $\Pi$ be a district map of $G$ in which each vertex of $P$ is a singleton district, but $w_1$ contains a district of size more than one. Then there is a sequence of $O(n)$ switches that increases number of districts in $w_1$ by one, and decreases the number districts in $w_2$ by one.
\end{lemma}
\begin{proof}
Let $c_1$ and $c_2$ be the cut vertices of $w_1$ and $w_2$, respectively, in the path $P$; possibly $c_1=c_2$. We claim that after $O(|w_1|)$ switch operations in $w_1$, we can find a path $P^*=(p_0,p_1,\ldots ,p_m)$ such that $\{p_0,p_1\}$ is a 2-vertex district in $w_1$, all other vertices in $P^*$ are singleton districts, and $p_m=c_2\in w_2$. Assuming that this is possible, we can then successively perform $\textsf{switch}(p_{i-1},p_i,p_{i+1})$ for $i=1,\ldots, m-1$, which replaces $\{p_0,p_1\}$ by two singleton districts, and produces a 2-vertex district $\{p_{m-1},p_m\}$. Finally, we contract this district to $\{p_{m-1}\}$, thereby decreasing the number of districts in $w_2$ by one. Overall, we have used $O(|w_1|)$ switches.
To prove the claim, let $G_1$ be biconnected subgraph of $G$ induced by $w_1$. Let $Q=(q_1,\ldots , q_s)$ be a shortest path in $G_1$ such that $q_1$ is in some district $V_0$ of size $|V_0|>1$ and $q_s=c_1$. Since $Q$ is a shortest path, the vertices $q_2,\ldots, q_s$ are singleton districts. If $|V_0|=2$, say $V_0=\{q_0,q_1\}$, then we can take $P^*=(q_0,q_1,\ldots ,q_s)\oplus P$, where $\oplus$ is the concatenation operation.
Assume that $|V_0|>2$. Since $G_1$ is biconnected, $V_0$ can be contracted to $\{q_1\}$ by a sequence of $|V_0|-1=O(|w_1|)$ switches (cf.~Lemma~\ref{lem:con2}). Each switch in the sequence contracts $V_0$ and expands some adjacent district. Perform the switches in this sequence until either (a) $|V_0|=2$, or (b) some singleton district $\{q_i\}$, $i=2,\ldots, s$, expands.
In both cases, we find a path $Q'=(q_i,\ldots , q_s)$, $i\in \{1,\ldots, s\}$, such that $q_i$ is in some 2-vertex district $\{q_0,q_i\}$, all other vertices in $Q'$ are singletons, and $q_s=c_1$. Consequently, we can take $P^*=(q_0,q_i,\ldots ,q_s)\oplus P$, as claimed.
\end{proof}
We can now consider the general case.
Let $G$ be a connected graph and let $k\in \mathbb{N}$. We present an algorithm (Algorithm~\ref{algo:1conn}) that transforms a given contractible $k$-district map $\Pi$ into one in pseudo-canonical form (defined below), and
then show that any two $k$-district maps in pseudo-canonical form can be transformed to each other.
Consequently, any two contractible $k$-district maps can be transformed into each other, and $\Gamma_k'(G)$ is connected.
We introduce some additional terminology. Let $T$ be a block tree of $G$. Fix an arbitrary leaf block $r\in B(G)$. We consider $T$ as a rooted tree, rooted at $r$. For a district map $\Pi$, we define a \textbf{leaf district} to be a district containing some non-root leaf block $w\in B(G)$, with the possible exception of the (unique) cut vertex of $G$ in $w$. Note that every leaf district $V_i$ corresponds to a unique leaf block (otherwise $\Pi$ would be incontractible by Lemma~\ref{lem:con1}), and we denote this block by $\text{leaf}(V_i)$. A leaf block is contractible into any vertex in $\text{leaf}(V_i)$, except for its cut vertex (cf.~Lemma~\ref{lem:con2}). Further note that a district may become a leaf district over the course of the algorithm.
\begin{figure}[htpb]
\begin{center}
\includegraphics[width=0.99\textwidth]{pseudo-canonical}
\caption{(a) A connected graph with nine blocks. (b) A pseudo-canonical 15-district maps. Five leaf districts are red and ten non-leaf districts are blue. (c) and (d): Pseudo-canonical district maps obtained from (b) by moving districts from $w_2$ to $w_1$ by successive applications of Lemma~\ref{lem:pushing}.}
\label{fig:pseudo-canonical}
\end{center}
\end{figure}
For every block $w\in B(G)$, except for the root, we define a set $\text{down}(w)$ as follows. Let $c\in C(G)$ be the parent of $w$ in $T$, let $V_i$ be the district that contains $c$, and let $\text{down}(w)$ be the set of vertices in $V_i$ that lie in $w$ or its descendants.
The set $\text{down}(w)$ is an \textbf{elbow} if $\text{down}(w)\neq \{c\}$, $V_i$ is a leaf district, and $\text{down}(w)$ does not contain the block $\text{leaf}(V_i)$.
An elbow is \textbf{maximal} if it is not contained in another elbow.
A leaf district is \textbf{elbow-free} if it does not contain any elbows.
A district map of $G$ is in \textbf{pseudo-canonical} form if every $w\in B(G)$ satisfies one of the following three mutually exclusive conditions (see Figure~\ref{fig:pseudo-canonical} for examples):
\begin{enumerate}[label=(\roman*)]\itemsep 0pt
\item \label{type:singleton} all vertices in $w$
are singleton non-leaf districts;
\item \label{type:sing+1} all vertices of $w$ are in nonleaf districts, which are contained in $w$, but not all are singletons, and all ancestor (resp., descendant) blocks of $w$ are of type (i) (resp., type~\ref{type:leaf});
\item \label{type:leaf} all vertices of $w$, with the possible exception of the parent cut-vertex of $w$, are in a leaf district and, if $w$ is not a leaf block, then this district contains the leftmost grandchild block of $w$.
\end{enumerate}
The proof of Theorem~\ref{thm:general-graphs} is the combinations of Lemmas~\ref{lem:general1} and \ref{lem:general2}.
\begin{figure}[htpb]
\begin{center}
\includegraphics[width=0.99\textwidth]{elbows}
\caption{(a) A 12-district map of a graph. The four leaf districts are red, eight nonleaf districts are blue; $c$ is the highest cut vertex in an elbow. (b), (c), and (d) show the result of Phases 1, 2, and 3 of Algorithm \ref{algo:1conn}. In Phase~3, a nonleaf district becomes a leaf district (shaded purple).}
\label{fig:elbows}
\end{center}
\end{figure}
\begin{lemma}\label{lem:general1}
Let $G$ be a connected graph with $n$ vertices and let $k\in \mathbb{N}$.
Then every contractible $k$-district map can be taken into
pseudo-canonical form by a sequence of $O(kn)$ switches.
\end{lemma}
\begin{proof}
\smallskip\noindent\textbf{Algorithm Outline.}
Let $\Pi$ be a contractible $k$-district map. Algorithm \ref{algo:1conn} (below) transforms $\Pi$ into pseudo-canonical form in three phases; refer to Figure~\ref{fig:elbows}.
Each phase processes all blocks in $B(G)$ in DFS order of the block tree $T$.
Phase~1 eliminates elbows. Phase~2 contracts leaf districts such that they are each confined to their leaf blocks.
Phase~3 contracts all nonleaf districts to singletons (or possibly turns some non-leaf districts into leaf districts).
We continue with the details.
\begin{algorithm}[htbp]
\caption{Pseudo-Canonical Algorithm for Connected Graphs}\label{algo:1conn}
\begin{algorithmic}[1]
\Procedure{Pseudo-Canonical}{$G,k,\Pi$}
\For{every non-root block $w\in B(G)$ in DFS order of $T$}
\If{$\text{down}(w)$ is an elbow}
{let $c\in C(T)$ be parent of $w$, contract $\text{down}(w)$ to $c$.}
\EndIf
\EndFor
\For{every non-leaf block $w\in B(G)$ in DFS order of $T$}
\If{$w$ intersects a leaf district,}{
\For{each leaf district $V_i$ that intersects $w$}
\State contract $w\cap V_i$ onto
cut-vertex of $w$ in the descending path of $T$ to $\text{leaf}(V_i)$;
\State apply an additional switch to contract $V_i$ out of the block $w$.
\EndFor}
\EndIf
\EndFor
\For{every block $w\in B(G)$ in DFS order of $T$}
\While{$w$ satisfies neither~\ref{type:singleton} nor~\ref{type:leaf}, and some grandchild $w'$ of $w$ is not of type~\ref{type:leaf}}
\State let $c'$ be the parent cut-vertex of $w'$, and let $V_i$ be the district containing $c'$;
\State contract $V_i$ to $\{c'\}$ preferring to expand non-leaf districts whenever possible.
\State When required, expand the leaf district in the leftmost grandchild block of $w'$.
\If{$w'$ is still not of type~\ref{type:leaf},}
\State Use Lemma~\ref{lem:pushing} with $P=(c)$ to move a district from $w'$ to $w$.
\EndIf
\EndWhile
\If{$w$ still satisfies neither~\ref{type:singleton} nor~\ref{type:leaf},}
\State{contract the district containing the parent cut-vertex $c$ of $w$ to $\{c\}$.}
\EndIf
\EndFor
\EndProcedure
\end{algorithmic}
\end{algorithm}
\smallskip\noindent\textbf{Analysis of Algorithm~2.}
Note that maximal elbows are pairwise disjoint, and every block intersects at most one maximal elbow (by the definition of $\text{down}(w)$).
Phase 1 (lines 2-3) is a for-loop over all non-root blocks. In the course of Phase~1, we maintain the invariant that if $w$ has been processed, then $\text{down}(w)$ is not an elbow. When the for-loop reaches a block $w$ where $\text{down}(w)$ is a maximal elbow, $\text{down}(w)$ is contracted to a cut vertex $c$, and produces $\text{down}(w)=\{c\}$, which is not an elbow. We also show that the contraction does not create any new elbow. Indeed, when a switch contracts $\text{down}(w)$ out of a cut vertex $c'$, then $c'$ is a descendant of $c$, and some district $V_j$ in a child $w'$ of $c'$ expands into $c'$. At this time, $c'$ becomes the highest vertex of $V_j$, and so $\text{down}(w')$ contains $\text{leaf}(V_j)$ if $V_j$ is a leaf district (hence $\text{down}(w')$ cannot be an elbow). We have shown that Phase~1 successively eliminates all elbows and does not create any new elbow. Since the maximal elbows are pairwise disjoint, the sum of their cardinalities is at most $n$, and they can be contracted with $O(n)$ switch operations. In Phases~2-3, we maintain the invariant (I1): There is no elbow in any leaf district.
Phase~2 (lines 4-8) is a for-loop over all non-leaf blocks. In the course of Phase~2, we maintain the invariant that if $w$ has been processed, then $w$ is disjoint from leaf districts. When the for-loop reaches a block $w$ that intersects a leaf district $V_i$, then
$V_i$ has no elbows by invariant (I1), and the ancestors of $w$ are disjoint from $V_i$. Consequently $V_i\cap w$ is contractible to
the child of $w$ that leads to the leaf block $\text{leaf}(V_i)$. For each leaf district $V_i$, Phase~2 uses $O(n)$ switches to contract $V_i$, and $O(kn)$ switches overall. In Phase~3, we maintain the invariant (I2): If a leaf district intersects a block, then such block is of type~\ref{type:leaf}.
Phase~3 (lines 9-17) is a for-loop over all blocks $w\in B(G)$. In the course of Phase~3, we maintain the invariant that if $w$ has been processed, it satisfies the definition of pseudo-canonical forms. Indeed, for every block $w$, the switch operations modify only $w$ or its descendants. This already implies that (I1) is maintained. Furthermore, if $w$ satisfies conditions (i) or (ii),
then the districts in $w$ remain unchanged.
Otherwise, the while-loop (lines 10-15) ensures that every district that intersects $w$ is contained in $w$.
If the conditions in the while-loop are true, then $V_i$ is a non-leaf district by (I2), and the vertices in $V_i$ lie in $w$ and its descendants.
The contractions in line 13 do not decrease the number of districts in $w$.
The preference of contractions in line 13 ensures that
(I2) is maintained for leaf districts.
Indeed, such a contraction may expand a leaf district if there is no other option:
in this case $V_i$ contains an entire block $w^*$, which is a descendant of $w$ and whose grandchildren are of type~\ref{type:leaf}; after contracting $V_i$ to the parent cut-vertex of $w^*$ expanding a leaf district, $w^*$ becomes of type~\ref{type:leaf}. Using Lemma~\ref{lem:pushing} in line 15 ensures that, eventually, $w$ is of type~\ref{type:singleton} or \ref{type:leaf}, or all its grandchildren are of type~\ref{type:leaf}. Finally, when the while loop terminates, lines 16-17 ensure that the parent cut vertex of $w$ is a singleton, and so all ancestors of $w$ comprise singletons. In Phase~3, $O(n)$ switches contract each district, which sums to $O(kn)$ switches over $k$ districts.
We have shown that Algorithm~\ref{algo:1conn} takes any input district map $\Pi$ into pseudo-canonical form. The three phases jointly use $O(kn)$ switches, as claimed.
\end{proof}
\begin{lemma}\label{lem:general2}
Let $G$ be a connected graph with $n$ vertices and let $k\in \mathbb{N}$.
For any two pseudo-canonical $k$-district maps, $\Pi_1$ and $\Pi_2$, there is
a sequence of $O(kn)$ switches that take $\Pi_1$ to $\Pi_2$.
\end{lemma}
\begin{proof}
Our proof is constructive.
For a given district map $\Pi$ in pseudo-canonical form, we assign every leaf district to the (unique) leaf block it intersects, and assign every nonleaf district to the highest block in $T$ it is contained in.
For every block $w\in B(G)$, let $d_\Pi(w)$ be the number of districts assigned to $w$ in $\Pi$. Clearly, $\sum_{w\in B(G)}d_\Pi(w)=k$.
Let $\Pi_1$ and $\Pi_2$ be $k$-district maps in pseudo-canonical form.
We first consider the case that $d_{\Pi_1}(w)=d_{\Pi_2}(w)$ for every block $w\in B(G)$.
We claim that every block is of the same type in both $\Pi_1$ and $\Pi_2$.
Traverse $T$ in post-order.
Given the types of descendants of a block $w$ notice that the type of $w$ is completely determined by the number of districts assigned to $w$.
Consequently, every block has the same type in both $\Pi_1$ and $\Pi_2$:
This implies that blocks of type~\ref{type:singleton} consists of singletons; and the union of blocks of type~\ref{type:leaf} are partitioned identically into leaf districts in both $\Pi_1$ and $\Pi_2$. Blocks of type~\ref{type:sing+1} each contain the same number of districts in both $\Pi_1$ and $\Pi_2$. These blocks are pairwise disjoint by definition. Applying Algorithm~\ref{algo:2conn} to each block of type~\ref{type:sing+1}, both $\Pi_1$ and $\Pi_2$ transform to the same district map.
Overall, this takes $O(kn)$ switches by Theorem~\ref{thm:2conn-alg}.
Next, assume that $d_{\Pi_1}(w)\neq d_{\Pi_2}(w)$ for some block $w\in B(G)$.
By the pigeonhole principle,
there exist blocks $w_1,w_2\in B(G)$ such that $d_{\Pi_1}(w_1)< d_{\Pi_2}(w_1)$ and $d_{\Pi_1}(w_2)>d_{\Pi_2}(w_2)$.
\begin{claim}\label{cl:star}
If $w_1$ (resp., $w_2$) is a highest (resp., lowest) block such that $d_{\Pi_1}(w_1)< d_{\Pi_2}(w_1)$ (resp., $d_{\Pi_1}(w_2)>d_{\Pi_2}(w_2)$), then all ancestor blocks of $w_1$ and $w_2$ are of type~\ref{type:singleton} in $\Pi_1$, and all descendant blocks of $w_1$ and $w_2$ are of type~\ref{type:leaf} in $\Pi_1$.
\end{claim}
\begin{proof}[Proof of Claim~\ref{cl:star}.]
Notice that if a block is of type~\ref{type:singleton} (resp., type~\ref{type:leaf}) then it has been assigned with the maximum (resp., minimum) number of districts that it can possibly be assigned to. Then, $w_1$ cannot be of type~\ref{type:singleton} in $\Pi_1$ and it cannot be of type~\ref{type:leaf} in $\Pi_2$.
By the definition of pseudo-canonical forms, all descendant (resp., ancestor) blocks of $w_1$ are of type~\ref{type:leaf} (resp., type~\ref{type:singleton}) in $\Pi_1$ (resp., $\Pi_2$). By the choice of $w_1$, all ancestor blocks of $w_1$ are of type~\ref{type:singleton} in $\Pi_1$.
An analogous argument proves the claim for $w_2$.
\end{proof}
We describe how to transform $\Pi_1$ into a district map $\Pi_1'$ in pseudo-canonical form such that
\begin{equation}\label{eq:potential}
\sum_{w\in B(G)}|d_{\Pi_1'}(w)-d_{\Pi_2}(w)| \leq
\sum_{w\in B(G)}|d_{\Pi_1}(w)-d_{\Pi_2}(w)| -1.
\end{equation}
Let $w_1$ and $w_2$ be blocks chosen as in Claim~\ref{cl:star}, and let $c_1$ and $c_2$ be their respective parent cut-vertices. By Claim~\ref{cl:star}, all blocks along the shortest path $P$ between $c_1$ and $c_2$ are of type~\ref{type:singleton} in $\Pi_1$, and so every vertex in the shortest path is in a singleton district.
Applying Lemma~\ref{lem:pushing} to $\Pi_1$, we can move a district from $w_2$ to $w_1$ using $O(|w_1|+|P|+|w_2|)\leq O(n)$ switches.
If $d_{\Pi_1}(w_1)=0$ (i.e., $w_1$ is of type~\ref{type:leaf} but not a leaf block) contract the leaf district out of $w_1$ by expanding the new nonleaf district that has moved into $w_1$.
If $w_2$ consists of a single (nonleaf) district, contract it onto $\{c_2\}$ while expanding the leaf district of its leftmost grandchild $w_2'$.
The number of districts assigned to block changes only in $w_1$, $w_2$, and (possibly) $w_2'$. The procedure described above increases $d(w_1)$ by one, and decreases $d(w_2)$ (and possibly $d(w_2')$) by one. The type of $w_1$ (resp., $w_2$) becomes \ref{type:singleton} or \ref{type:sing+1} (resp., \ref{type:sing+1} or \ref{type:leaf}) and, by Claim~\ref{cl:star}, $\Pi_1'$ is in pseudo-canonical form.
While $d_{\Pi_1}\neq d_{\Pi_2}$, we repeat the above procedure and set $\Pi_1=\Pi_1'$.
The while loop terminates after $O(k)$ iterations by \eqref{eq:potential}, and
it transforms $\Pi_1$ into a district map $\Pi_1''$ in pseudo-canonical form such that $\sum_{w\in B(G)}|d_{\Pi_1''}(w)-d_{\Pi_2}(w)|=0$,
which implies $d_{\Pi_1''}(w)=d_{\Pi_2}(w)$ for every $w\in B(G)$.
Then we bring $\Pi_1''$ to $\Pi_2$ with $O(kn)$ switches as noted above.
Overall, we have used $O(kn)$ switches for transforming $\Pi_1$ to $\Pi_2$, by Lemma~\ref{lem:pushing}.
This completes the proof of Lemma~\ref{lem:general2}.
\end{proof}
\subsection{Characterization of Connected Switch Graphs}
\label{ssec:char}
Using Lemmas~\ref{lem:con1}--\ref{lem:invariant} and Theorem~\ref{thm:general-graphs},
we can characterize the pairs $(G,k)$, of a connected graph $G$ and an positive integer $k$,
for which the switch graph $\Gamma_k(G)$ is connected\ShoLong{}{ (cf.~Theorem~\ref{thm:conn-test} below)}.
\begin{lemma}\label{lem:conn-char}
For a connected graph $G$ and $k\in \mathbb{N}$, the switch graph $\Gamma_k(G)$ is connected if and only if $k=1$ or every $k$-district map is contractible (i.e., $\Gamma_k(G)=\Gamma_k'(G)$).
\end{lemma}
\global\def \pflemconnchar {
\begin{proof}
The case that $k=1$ is trivial, as $\Gamma_k(G)$ is a singleton. Assume $k\geq 2$ for the remainder of the proof.
If every $k$-district map is contractible (i.e., $\Gamma_k(G)=\Gamma_k'(G)$), then $\Gamma_k'(G)$ is connected by Theorem~\ref{thm:general-graphs}, and so $\Gamma_k(G)$ is connected.
If some $k$-district maps are contractible and some are incontractible, then $\Gamma_k(G)$ is disconnected, since
there is no edge between the set of contractible and incontractible district maps by Lemma~\ref{lem:invariant}.
Finally, assume that every $k$-district map is incontractible in $G$. We show that $\Gamma_k(G)$ is disconnected.
Let $\Pi_1$ be an arbitrary $k$-district map. By Lemmas~\ref{lem:con1}--\ref{lem:con2}, some district $V_i\in \Pi_1$
contains two leaf blocks of the block graph, say $w_a,w_b\in B(G)$, with cut vertices $c_a,c_b\in C(G)$ (possible $c_a=c_b)$.
Since $G$ is connected and $k\geq 2$, there exists a district $V_j$ adjacent to $V_i$. We construct a $k$-district map $\Pi_2$
from $\Pi_1$ by replacing $V_i$ and $V_j$ with $V_i':=w_a\setminus \{c_a\}$ and $V_j':=(V_i\cup V_j)\setminus V_i'$.
Importantly, none of the districts in $\Pi_2$ contain both $w_a$ and $w_b$; and by Lemma~\ref{lem:con1},
every sequence of switch operations transforms $V_i$ to a district that contains both $w_a$ and $w_b$.
Thus $\Gamma_k(G)$ does not contain any path between $\Pi_1$ and $\Pi_2$, as required.
\end{proof}
}
\ShoLong{Proof is deferred to the Appendix.}{\pflemconnchar
}
Lemma~\ref{lem:con1} allows us to efficiently whether a connected graph $G$ admits an incontractible $k$-district map. Let $G$ be connected but not biconnected. For two leaf blocks $w_1,w_2\in B(G)$, let $P(w_1,w_2)$ denote the total number of vertices in the two blocks
along a shortest path between $w_1$ and $w_2$. Let
$M=\min\{P(w_1,w_2): w_1,w_2\in B(G) \mbox{ \rm leaf blocks}\}$.
\begin{lemma}\label{lem:conn-test}
Let $G$ be a connected graph with $n$ vertices that is connected but not biconnected, and let $k\in \mathbb{N}$.
Every $k$-district map in $G$ is contractible if and only if $n-k\leq M$.
\end{lemma}
\begin{proof}
If $n-k\leq M$, then every district in a $k$-district map contains fewer than $M$ vertices. By the definition of $M$, none of these districts can contain two leaf blocks, and, therefore, are contractible.
If $M> n-k$, then we construct a $k$-district map for $G$ in which one of the districts is incontractible.
Let $\widehat{V}\subset V(G)$ be a vertex set of minimum cardinality that contains two leaf blocks in $B(G)$ and a shortest path between then. By partitioning $V(G)\setminus \widehat{V}$ into singletons, we obtain a $\widehat{k}$-district map $\widehat{\Pi}$, where $\widehat{k}=n=M+1$, and $\widehat{V}\in \widehat{\Pi}$. Successively merge pairs of adjacent districts until the number of districts drops to $k$ (recall that $G$ is connected, so some pair of districts are always adjacent). We obtain a $k$-district map $\Pi$, where one of the districts contains $\widehat{V}$, and is incontractible by Lemma~\ref{lem:con1}, as required.
\end{proof}
\begin{lemma}\label{lem:BFS}
We can compute the value $M$ in $O(n+m)$ time, where $n=|V(G)|$ and $m=|E(G)|$.
\end{lemma}
\begin{proof}
Given a connected graph $G=(V,E)$, first compute the block tree, and modify $G$ as follows: replace each leaf block by a chain with the same number of vertices, such that one endpoint is the original cut vertex (and hence the other endpoint is a leaf), and denote by $G'$ the resulting graph. Then we run a modified multi-source BFS on $G'$, starting from the leaves. The algorithm assigns two labels to every vertex $v\in V(G')$, the \emph{level} $\ell(v)$ and a \emph{cluster} $c(v)$. Initially, each leaf $v\in V(G')$ is assigned level $\ell(v)=0$ and clusters $c(v)=v$. When the BFS visits a new vertex $v$ along an edge $uv$, it sets $\ell(v):=\ell(u)+1$ and $c(v):=c(u)$. Clearly, $\ell(v)$ is the distance from $v$ to the closest leaf in $G'$, and $c(v)$ is one such leaf. After the BFS termination, our algorithm finds an edge $uv\in E(G')$ such that $c(u)\neq c(v)$ and $\ell(u)+\ell(v)$ is minimal, and returns $\ell(u)+\ell(v)+2$.
The modified BFS runs in $O(n+m)$ time, and a desired edge $uv$ can be found in $O(m)$ additional time, so the overall running time is $O(n+m)$. It remains to prove that $M=\ell(u)+\ell(v)+2$. Note that $u$ and $v$ are at distance $\ell(u)$ and $\ell(v)$, resp., from the leaves $c(u)$ and $c(v)$. The cluster of $c(u)$ (resp., $c(v)$) contains a shortest path from $u$ to $c(u)$ (resp., from $v$ to $c(v)$), and so these shortest paths are disjoint. The concatenation of the two shortest paths is a shortest path $P'$ between the leaves $c(u)$ and $c(v)$, and it has $\ell(u)+\ell(v)+2$ vertices. The path $P'$ contains the chains incident to $u$ and $v$ in $G'$. By the definition of $G'$, these chains correspond to leaf blocks $w_1$ and $w_2$ of the same size in $G$. Consequently, $\ell(u)+\ell(v)+2 = P(w_1,w_2)$
Therefore, $M\leq \ell(u)+\ell(v)+2$.
Conversely, assume that $M=P(w_1,w_2)$ for some leaf blocks $w_1,w_2\in B(G)$. These leaf blocks correspond to chains ending in two leaves, say $w'_1$ and $w'_2$, in $G'$. By construction, the distance between $w'_1$ and $w'_2$ is $d_{G'}(w'_1,w'_2)=M-1$. Let $P'$ be a shortest path between $w'_1$ and $w'_2$. We claim that for every vertex $v'$ in $P'$, $\ell(v')$ is the minimum distance to $\{w'_1,w'_2\}$, i.e., $\ell(v')=\min\{d_{G'}(v,w_1),d_{G'}(v,w_2)\}$.
Suppose, to the contrary, that there is a vertex $v'$ in $P'$ for which $\ell(v)\neq \min\{d_{G'}(v',w'_1),d_{G'}(v',w'_2)\}$. Since $\ell(v')$ is the minimum distance to some leaf in $G'$, we have $\ell(v')=d_{G'}(v,w'_3)$ for a leaf $w'_3$, and $\ell(v')<\min\{d_{G'}(v',w'_1),d_{G'}(v',w'_2)\}$. As $v'$ is in the path $P'$,
$d_{G'}(v',w'_1)+d_{G'}(v',w'_2)=d_{G'}(w'_1,w'_2)=M-1$. By the triangle inequality, $d_{G'}(w'_1,w'_3)$ or $d_{G'}(w'_2,w'_3)$ is less than $d_{G'}(w'_1,w'_2)$,
contradicting the minimality of $P(w_1,w_2)$. Now $P'$ contains two consecutive vertices, say $u^*$ and $v^*$, such that $\ell(u^*)=d_{G'}(u^*,w'_1)$, $\ell(v^*)=d_{G'}(v^*,w'_2)$, and $c(u^*)\neq c(v^*)$. The sum of their distances to the two endpoints of $P'$ is $\ell(u^*)+\ell(v^*)=(M-1)-1=M-2$, hence $M=\ell(u^*)+\ell(v^*)+2$.
Consequently, $\ell(u)+\ell(v)+2\leq M$, as required.
\end{proof}
The combination of Theorem~\ref{thm:2conn-alg} and Lemmas~\ref{lem:conn-char}--\ref{lem:BFS}
yields the following.
\begin{theorem}\label{thm:conn-test}
For a connected graph $G$ with $n$ vertices and $k\in \mathbb{N}$, the switch graph $\Gamma_k(G)$ is connected if and only if $G$ is biconnected or $k+M\geq n$, which can be tested in
$O(n+m)$ time, where $m=|E(G)|$.
\end{theorem}
\ShoLong{}{\input{lowerBoundSec.tex}}
\ShoLong{}{\input{hard.tex}}
\section{Conclusion}
\label{sec:con}
This paper provides the theoretical foundation for using elementary switch operations to explore the configuration space $\Gamma_k(G)$ of all partitions of a given graph into $k$ nonempty subgraphs, each of which is connected. We gave a polynomial-time testable combinatorial characterization for connected configurations spaces (Theorem~\ref{thm:conn-test}).
A crucial concept in both the combinatorial characterization and the reconfiguration algorithms (Algorithms~\ref{algo:2conn} and~\ref{algo:1conn}) was \emph{contractibility}: A district is contractible if it can be reduced to a single vertex (while all $k$ districts remain connected).
In applications to electoral maps, all districts have roughly average size, say between $\frac{n}{2k}$ and $\frac{2n}{k}$, and a singleton district is impractical.
In a sense, we establish that there is a path between any two contractible district maps with average-size districts by passing through ``impractical'' district maps with singleton districts. We do not know whether singleton districts are necessary: for a constant $c\geq 1$, we can define $\Gamma_{k,c}(G)$ as the graph of $k$-district maps in which the size of every district lies in the interval $[\frac{n}{ck},\frac{cn}{k}]$. It is easy to construct examples where $\Gamma_{k,1}(G)$ has isolated vertices. Is there a constant $c>1$ such that the connectedness of $\Gamma_k(G)$ implies that $\Gamma_{k,c}(G)$ is also connected?
The problem of partitioning a graph $G$ into $k$ connected subgraphs with equal (or almost equal) number of vertices is known as the \emph{Balanced Connected $k$-Partition Problem} (BCP$_k$), which is NP-hard already for $k=2$~\cite{DyerF85}, for grids in general~\cite{BerengerNP18}, and also hard to approximate within an absolute error of $n^{1-\delta}$~\cite{Chlebikova96}.
In our model, a district maps is a partition of the vertex set into $k$ \emph{unlabeled} nonempty subsets. One could consider the \emph{labeled} variant, and define a switch graph $\Gamma^L_k(G)$ on labeled $k$-district maps. Our results do not carry over to this variant: in particular, the labeled switch graph $\Gamma_k^L(G)$ need not be connected if $G$ is biconnected. For example, if $G=C_n$ (i.e., a cycle of $n\geq 3$ vertices) and $k\geq 3$, then the cyclic order of the districts along the cycle cannot change. In the special case that $k=2$ and $G$ is biconnected, $\Gamma_2^L(G)$ is connected since we can contract a district to a singleton (cf.~Lemma~\ref{lem:con2}) and move it to any vertex while the complement remains connected.
When we move a singleton district from one vertex to another, it temporarily occupies both vertices, which should not form a 2-cut.
Contraction to a singleton is sometimes necessary in this case (one such example is $G=K_{2,m}$, $m\geq 3$, where the 2-element partite set is split between the two districts).
\bibliographystyle{plainurl}
|
{
"timestamp": "2019-03-01T02:02:37",
"yymm": "1902",
"arxiv_id": "1902.10765",
"language": "en",
"url": "https://arxiv.org/abs/1902.10765"
}
|
\section{Introduction}
Image segmentation is the process of assigning one of several
categorical labels to each pixel of an image, which is a fundamental
step in many medical image analyses. Until recently, some of the most
accurate segmentation methods were based on probabilistic mixture
models \cite{klauschen2009evaluation}. These models define a probability
distribution over an observed image ($\vec{X}$), conditioned on
unknown class labels ($\vec{Z}$) and parameters ($\vec{\theta}$).
Assuming a prior distribution over unknown variables, Bayes rule is
used to form a posterior distribution:
\begin{align}
p(\vec{Z}, \vec{\theta}\mid\vec{X})
\propto p(\vec{X}\mid\vec{Z}, \vec{\theta})~p(\vec{Z},
\vec{\theta}) ~,
\label{eq:bayes-rule}
\end{align}
which can be evaluated or approximated. Wells III \emph{et al.}
\cite{wells1996adaptive} introduced these types of models for brain
segmentation from magnetic resonance (MR) images. By assuming that the
log-transformed image intensities followed a normal distribution in the
likelihood term $p(\vec{X}\mid\vec{Z},\vec{\theta})$, they segmented
the brain into three classes: grey matter (GM), white matter (WM) and
cerebrospinal fluid (CSF).
As generative models require the data-generating process to be defined,
they can be extended to more complex joint distributions than in
\cite{wells1996adaptive}, allowing for segmentation methods robust to,
\emph{e.g.}, slice thickness, MR contrast, field strength and scanner
variability. Many of today's most widely used neuroimaging analysis
software, such as SPM \cite{ashburner2005unified}, FSL
\cite{zhang2001segmentation} and FreeSurfer \cite{fischl2004sequence},
rely on these kinds of models, and have been shown to reliably segment
a wide variety of MR data
\cite{kazemi2014quantitative,heinen2016robustness}.
However, recent advances in convolutional neural networks (CNNs) have
provided a new method for very accurate (and fast) image segmentation
\cite{long2015fully}, circumventing the need to define and invert a
potentially complex generative model. Discriminative CNNs learn a
function that maps an input (\emph{e.g.}, an MRI) to an output
(\emph{e.g.}, a segmentation) from training data, where the output is
known. They typically contain many layers, which sequentially apply
convolutions, pooling and nonlinear activation functions to the input
data. Their parameters are optimised by propagating gradients
backwards through the network (\emph{i.e.}, backpropagation). For
medical imaging, the U-net architecture \cite{ronneberger2015u} is the
most popular and now forms the basis for most top performing entries in
various medical imaging challenges aimed at segmenting, \emph{e.g.},
tumours, the whole brain or white matter
hyper-intensities\footnote{\url{braintumorsegmentation.org},
\url{wmh.isi.uu.nl}, \url{mrbrains18.isi.uu.nl}}. The more classical
segmentation frameworks based on probabilistic models seem to have met
their match.
\begin{figure}[t]
\centering
\includegraphics[trim={0cm 0.2cm 0cm
0.2cm},clip,width=0.6\textwidth]{t1-comparison}
\caption{T1-weighted MR images from two different, publicly available,
datasets: MICCAI2012 and MRBrainS18 (on which we evaluate our method).
It is evident that learning from one of these populations, and
subsequently testing on the other is very challenging. The intensities
are different by an order of magnitude, the bias is stronger in the
MRBrainS18 subject. Additionally, age related change and pathology can
be clearly seen, such as differences in ventricle size and white matter
hyper-intensities, which further complicates the learning problem.}
\label{fig:compare-t1s}
\end{figure}
Challenges on medical image segmentation can be seen as lab experiments
and -- as with new medical therapies -- there is a large gap to get
\emph{from bench to bedside}. CNNs excel in this context, factorising
the commonalities in an image population of training data, which
generalise to new data from the same population. They can struggle,
however, when faced with new data that contain unseen features
\cite{dolz20173d}, \emph{e.g.}, a different contrast (Fig.
\ref{fig:compare-t1s}). This scenario usually requires the model being
trained anew, on that unseen image contrast. In fact, even without
considering inter-individual variability (age, brain shape, pathology,
etc), a CNN-based segmentation software has yet to be presented that is
agnostic to the great variability in MR data \cite{akkus2017deep}. Lack
of such software is largely due to the limited amount of labelled data
available in medical imaging, which is a clear obstacle to their
generalisability. Some methods have been developed to address this
problem, \emph{e.g.}, intensity normalisation \cite{han2007atlas},
transfer learning \cite{van2015transfer} and batch normalisation
\cite{karani2018lifelong}. Still, none of these methods are yet general
enough to solve the task of segmenting across scanners and protocols.
Recently, approaches based on realistic data augmentation have shown promising results \cite{jog2018pulse,zhao2019data}.
In this paper, we propose an approach to bridge between the
classical, but robust, generative segmentation models and more
recent CNN based methods. The link is in the prior term of
\eqref{eq:bayes-rule}, where we encode the unknown tissue distribution
as drawn from a Markov random field (MRF). Using an MRF is in itself
nothing new; they have been used successfully for decades in order
to introduce spatial dependencies into generative segmentation models
\cite{van1999automated,agn2015brain}, relaxing the independent voxels
assumption. Here however, we instead model and learn the interactions
among neighbouring pixels by a type of CNN. This allows us to
parametrise the MRF by a more complex mathematical function than in the
regular linear case, as well as cover a larger neighbourhood than a
second-order one. The idea is that learning at the tissue level may
generalise better than learning directly from the image intensities. We
validate our approach on two publicly available datasets, acquired in
different centres, and show favourable results when applying the model
trained on one of these datasets to the other.
\subsubsection*{Related Work:}
Rather than reviewing the use of MRFs in image segmentation we will here
briefly discuss two fairly recent additions to the computer
vision field \cite{zheng2015conditional,schwing2015fully}, because they
are closely related to the method we present in the subsequent section.
The idea of both these papers is to cast the application and learning
of a conditional random field (CRF) into a CNN framework. A CRF is a
statistical modeling method that directly defines the posterior
distribution in \eqref{eq:bayes-rule}. To compute the CRF both papers
apply a mean-field approximation, which they implement in the form of a
CNN.
In contrast to the works described above, we are interested in
defining the full generative model, whilst keeping the separation
between likelihood and prior in \eqref{eq:bayes-rule}. Modelling these
two components separately allows us to include expert knowledge and
image-intensity independent prior information over the segmentation
labels. It also integrates easily with existing mixture-model-based
approaches. Furthermore, modelling the prior as an MRF, without
data-dependency in the neighbourhood model, may help in generalising
among different image populations. Finally, our model allows an
arbitrarily complex MRF distribution to be defined, including,
\emph{e.g.}, nonlinearities.
\section{Methods}
In this section we use the generative model defined by
\eqref{eq:bayes-rule} to encode an MRF over the unknown labels. We show
that computing this MRF term is analogous to the mathematical
operations performed by a CNN. We then go on to formulate learning the
MRF clique potentials as the training of a CNN. This allows us to
introduce nonlinearities and increasing complexity in the MRF
neighbourhood.
\subsubsection{Generative Model:}
The posterior in \eqref{eq:bayes-rule} allows us to estimate the
unknown tissue labels. For simplicity, we will from now on assume that
all parameters ($\vec{\theta}$) are known; we therefore only want to
infer the posterior distribution over categorical labels $\vec{Z} \in
\left\{0,1\right\}^{I \times K}$, where $I$ are the number of pixels in
the image and $K$ are the number of classes, conditioned on an observed
image $\vec{X} \in \mathbb{R}^{I\times C}$, where $C$ are the number of
channels. Modelling multi-channel images allows the use of all acquired
MR contrasts of the same subject. In practice, unknown parameters of,
\emph{e.g.}, class-wise intensity distributions would need inferring
too. Variational Bayesian (VB) inference, along with a well-chosen
mean-field approximation, allows any such model to fit within the
presented framework \cite{bishop2006pattern}.
Making use of the product rule, we may define the joint model
likelihood $p(\vec{X},\vec{Z})$ as the product of a data likelihood $p(
\vec{X}\mid\vec{Z})$ and a prior $p(\vec{Z})$. In a mixture model,
it is assumed that once labels are known, intensities are independent
across pixels and all pixels with the same label $k$ are sampled
from the same distribution $p_k(\vec{x})$. This can be written as:
\begin{align}
p(\vec{X}\mid\vec{Z}) = \prod_{i=1}^I\prod_{k=1}^K
p_k(\vec{x}_i)^{z_
{ik}} ~.
\end{align}
A common prior distribution for labels in a mixture model is the
categorical distribution, which can be stationary ($p(\vec{z}_i) =
\mathrm{Cat}\left(\vec{z}_i\mid \vec{\pi}\right)$) or non-stationary
($p(\vec{z}_i) = \mathrm{Cat}\left(\vec{z}_i\mid \vec{\pi}_i\right)$)
\cite{ashburner1997multimodal}. However, both these distributions
assume conditional independence between pixels. MRFs can be introduced
to model dependencies between pixels in a relatively tractable way by
assuming that interactions are restricted to a finite neighbourhood:
\begin{align}
p(\vec{z}_i\mid
\left\{\vec{z}_j\right\}_{j\neq i}) = p(\vec{z}_i\mid
\vec{z}_{\mathcal{N}_i}) ~,
\end{align}
where $\mathcal{N}_i$ defines pixels whose cliques contain
$\vec{z}_i$. We make the common assumption that this neighbourhood is
stationary, meaning that it is defined by relative positions with
respect to $i$: $\mathcal{N}_i = \left\{i+\delta | \delta \in
\mathcal{N}\right\}$. Here, we assume that this conditional likelihood
factorises over the neighbours and that each factor is a categorical
distribution:
\begin{align}
p(\vec{z}_i\mid\vec{z}_{\mathcal{N}_i})
= \prod_{\delta \in \mathcal{N}} \prod_{k=1}^K \prod_{l=1}^K
\left(\pi_{k,l,\delta}\right)^{z_{i,k} \cdot z_{i+\delta,l}} ~.
\end{align}
\subsubsection*{Mean-field inference:}
Despite the use of a relatively simple interaction model, the posterior
distribution over labels is intractable. Therefore, our approach is to
search for an approximate posterior distribution that factorises across
voxels:
\begin{align}
p(\vec{Z}\mid\vec{X}) \approx q(\vec{Z}) = \prod_{i=1}^I
q(\vec{z}_i) ~.
\end{align}
We use VB inference \cite{bishop2006pattern}
to iteratively find the approximate posterior $q$ that minimises its
Kullback-Leibler divergence with the true posterior distribution.
Let us assume a current approximate posterior distribution
$q(\vec{Z})~=~\prod_j\mathrm{Cat}\left(\vec{z}_j\mid\vec{r}_j\right)$;
each voxel follows a categorical distribution parameterised by
$\vec{r}_j$, which is often called a \emph{responsibility}. VB then
gives us the optimal updated distribution for factor $i$ by taking the
expected value of the joint model log-likelihood, with respect to all
other variables:
\begin{align}
\ln q^\star(\vec{z}_i) = \sum_{k=1}^K z_{ik} \left(\ln p_k(\vec{x}_i) +
\sum_{\delta\in\mathcal{N}}
\sum_{l=1}^Kr_{i+\delta,l}\ln\pi_{k,l,\delta} \right) + \mathrm{const}.
\end{align}
This distribution is again categorical with parameters:
\begin{align}
r_{ik}^\star \propto \exp\left(\ln p_k(\vec{x}_i) +
\sum_{\delta\in\mathcal{N}} \sum_{l=1}^K r_{i+\delta,l}
\ln\pi_{k,l,\delta}\right).
\end{align}
\subsubsection*{Implementation as a CNN:}
Under VB assumptions, posterior distributions should be updated one at
a time, in turn. Taking advantage of the limited support of the
neighbourhood, an efficient update scheme can be implemented by
updating at once all pixels that do not share a
neighbourhood\footnote{When $\mathcal{N}$ contains four second-order
neighbours, this corresponds to a checkerboard update scheme.}. Another
scheme can be to update all pixels at once based on the previous state
of the entire field. Drawing a parallel with linear systems, this is
comparable to Jacobi's method, while updating in turn is comparable to
the Gauss-Siedel method. In the Jacobi case, updating the labels'
expected values can be implemented as a convolution, an addition and a
softmax operation; three basic layers of CNNs:
\begin{align}
\vec{R}^\star = f(\vec{R}) = \mathrm{softmax}(\vec{C} + \vec{W} \ast
\vec{R}) ~.
\label{eq:mrf-to-cnn}
\end{align}
The matrix $\vec{C}$ contains the conditional log-likelihood terms
($\ln p_k(\vec{x}_i)$). The convolution weights $\vec{W} \in \mathbb{R}^
{\left|\mathcal{N}\right| \times K \times K}$ are equal to the log of
the MRF weights ($\ln\pi_{k,l,\delta}$) and, very importantly, their
centre is always zero. We call such filters \emph{MRF filters}, and the
combination of softmaxing and convolving an \emph{MRF layer}. These
weights are parameters of the approximate posterior distribution
$q^\star(\vec{Z})$. Note that multiple mean-field updates can be
implemented by making the MRF layer recurrent, where the output is
also the input \cite{zheng2015conditional}.
Now, let us assume that we have a set of true segmentations
$\vec{\hat{Z}}_{1\dots N}$, along with a set of approximate
distributions with parameters $\vec{R}_{1\dots N}$. One may want to
know the MRF parameters $\vec{W}$ that make the new posterior estimate
$q^\star$ with parameters $\vec{R}^\star = f(\vec{R})$ the most likely
to have generated the true segmentations. This reduces to the
optimisation problem:
\begin{align}
\vec{W}^\star = \argmax_{\vec{W}} \sum_{n=1}^N \ln q^\star(\vec{
\hat{Z}}_n \mid \vec{W}) = \argmax_{\vec{W}} \sum_{i=1}^I \sum_{k=1}^K
\hat{z}_{nik} \ln r^\star_{nik} ~,
\end{align}
which is a maximum-likelihood (ML), or risk-minimisation, problem. Note
that this objective function is the negative of what is commonly
referred to as the categorical cross-entropy loss function in
machine-learning. If the optimisation is performed by computing
gradients from a subset of random samples, this is equivalent to
optimising a CNN, with only one layer, by stochastic gradient descent.
\subsubsection*{Post-processing MRFs:}
MRFs are sometimes used to post-process segmentations, rather than as
an explicit prior in a generative model. In this case, the conditional
data term is not known, and the objective is slightly different:
approximating a factorised label distribution $q(\vec{Z}) =
\prod_{i=1}^I q(\vec{z}_i)$ that resembles the prior distribution
$p(\vec{Z})$. This can be written as finding such distribution $q$
that minimises the Kullback-Leibler divergence with the prior $p$:
\begin{align}
q^\star = \argmin_q \mathrm{KL}\left(q \middle\| p\right) ~.
\end{align}
Again, assuming all other factors fixed with $q(
\vec{z}_j)~=~
\mathrm{Cat}\left(\vec{z}_j\mid\vec{r}_j\right)$, the optimal
distribution for factor $i$ is obtained by taking the expected value of
the prior log-likelihood:
\begin{align}
\ln q^\star(\vec{z}_i) = \sum_{k=1}^K z_{ik} \left(
\sum_{\delta\in\mathcal{N}}
\sum_{l=1}^Kr_{i+\delta,l}\ln\pi_{k,l,\delta} \right) + \mathrm{const},
\end{align}
which is equivalent to dropping the conditional term in the generative
case. Equation \eqref{eq:mrf-to-cnn} is then written as $\vec{R}^\star
= \mathrm{softmax}\left(\vec{W} \ast \Vec{R}\right)$.
\subsubsection*{Nonlinear MRF:}
The conditional prior distribution
$p(\vec{z}_i\mid\vec{z}_{\mathcal{N}_i})$ that defines an MRF can, in
theory, be any strictly positive probability distribution. However, in
practice, they are usually restricted to simple log-linear functions,
which are easy to implement and efficient to compute. On the other
hand, deep neural networks allow highly nonlinear functions to be
implemented and computed efficiently. Therefore, we propose a more
complex layer based on multiple MRF filters and nonlinearities, that
implements a nonlinear MRF density. To ensure that we implement a
conditional probability, a constraint is that the input value of a
voxel may not be used to compute its posterior density. Therefore, the
first layer consists MRF filters that do not have a central weight,
and subsequent layers are of size one to avoid reintroducing the centre
value by deconvolution. We thus propose the first layer to be an MRF
filter $\vec{W} \in \mathbb{R}^{\left| \mathcal{N}\right| \times K
\times F}$, where $F$ is the number of output features. Setting $F > K$
allows the information to be decoupled into more than the initial $K$
classes and may help to capture more complex interactions. This first
convolutional layer is followed by a ReLU activation function, 1D
convolutions that keep the number of features untouched, and another
ReLU activation function. This allows features to be combined
together. A final 1D linear layer is used to recombine the information
into $K$ classes, followed by a softmax. Fig. \ref{fig:net} shows our
proposed architecture.
\begin{figure*}[t]
\centering
\includegraphics[trim={0cm 0.0cm 0cm 0.1cm},clip,width=\textwidth]{net}
\caption{An illustration of the architecture of our MRF CNN. Outlined
are the operations performed for learning to predict the centre of a
segmented pixel. The nonlinearities are introduced by the ReLU
activations. By setting the number of MRF layers to $K$ and keeping
only the final softmax layer, the linear MRF model is obtained. The
convolution kernel applied by the MRF filter is shown left of the
segmentations, with its centre constrained to be zero.}
\label{fig:net}
\end{figure*}
\subsubsection*{Implementation and training:}
In this work we set the number of MRF layers to $F=16$, we use three by
three convolutions and leaky ReLU activation functions with
$\alpha=0.1$. We optimise the CNN using the Adam optimiser. To reduce
overfitting, we augment the data in two ways: (1) by simple
left-right reflection; and (2), by sampling warps from anatomically
feasible affine transformations, followed by nearest neighbour
interpolation. Realistic affine transformations can be sampled by
parametrising them by their 12 parameter Lie group (from which the
transformation matrix can be constructed via an exponential mapping
\cite{ashburner2013symmetric}) and then learning their mean and
covariance from a large number of subjects' image headers.
\section{Validation}
This section aims to answer a series of questions: (1) does applying a
linear MRF trained by backpropagation to the output segmentations of a
generative model improve the segmentation accuracy? (2) does
complexifying the MRF distribution using numerous filters and
nonlinearities improve the segmentation accuracy compared to a linear
MRF? (3) do the learnt weights generalise to new data from an entirely
different dataset?
\subsubsection{Datasets and preprocessing:}
Our validation was performed on axial 2D slices extracted from two
publicly available
datasets\footnote{\url{my.vanderbilt.edu/masi/workshops},
\url{mrbrains18.isi.uu.nl}}:
\begin{itemize}
\item
\textbf{MICCAI2012}: T1-weighted MR scans of 30 subjects aged 18 to 96
years, (mean: 34, median: 25). The scans were manually segmented into
136 anatomical regions by Neuromorphometrics Inc. for the MICCAI 2012
challenge on multi-atlas segmentation.
\item \textbf{MRBrainS18}: Multi-sequence (T1-weighted, T1-weighted
inversion recovery and T2-FLAIR) MR scans of seven subjects, manually
segmented into ten anatomical regions. Some subjects have pathology and
they are all older than 50 years. All scans were labelled by the same
neuroanatomist.
\end{itemize}
Within each dataset, all subjects were scanned on the same scanner and
with the same sequences, whilst between datasets, the scanners and
sequences differ (Fig. \ref{fig:compare-t1s}). Both datasets have
multiple labelled brain structures, such as cortical GM, cerebellum,
ventricles, \emph{etc}. We combined these so as to obtain the same
three labels for each subject: GM, WM and OTHER ($1 - \text{GM} -
\text{WM}$). These labels were used as targets when training our model.
All T1-weighted MR scans were segmented with the algorithm implemented
in the SPM12
software\footnote{\url{www.fil.ion.ucl.ac.uk/spm/software/spm12}},
which is based on the generative model described in
\cite{ashburner2005unified}. In this model, the distribution over
categorical labels is independent across voxels, non-stationary, and
encoded by a probabilistic atlas deformed towards each subject. The
algorithm generates soft segmentations, that is, parameters of the
posterior categorical distribution over labels. We pulled the GM, WM
and OTHER classes from these segmentations. Fig. \ref{fig:results}
shows the T1-weighted image of one subject from each dataset, with its
corresponding target labels and SPM12 segmentations\footnote{Besides
disabling the final MRF clean-up, we used the default parameters of
SPM12.}.
\begin{figure*}[t]
\centering
\includegraphics[trim={0cm 0.1cm 0cm
0.1cm},clip,width=\textwidth]{results}
\caption{Example training data and results. From left to right:
T1-weighted MR image with target labels, SPM GM and WM segmentations,
results of applying the linear MRF model to the SPM segmentations,
results of applying the nonlinear MRF model to the SPM segmentations.
Below each tissue class are the corresponding Dice scores, computed
with the target labels as reference.}
\label{fig:results}
\end{figure*}
\subsubsection{Model training and evaluation:}
We trained two different models: a regular, second-order MRF
(\emph{Lin}); and a second-order nonlinear MRF (\emph{Net}). Fig.
\ref{fig:net} explains the differences in architecture between the two.
For each subject and each class, we computed the Dice score of the ML
labels obtained using SPM12 and those obtained after application of the
linear MRF and the nonlinear MRF. Statistical significance of the
observed changes was tested using two-sided Welch's \emph{t}-tests
between paired measures. Multiple comparisons were accounted for by
applying the Bonferroni correction.
We first evaluated the learning abilities of the networks. To this end,
we performed a 10-fold cross validation of the MICCAI2012 dataset, where
groups of three images were tested using a model trained on the
remaining 27 images. This yielded Dice scores for the entire MICCAI2012
dataset, which are shown in Fig. \ref{fig:dice:miccai2012}. Next, we
evaluated the generalisability of the networks, that is, what kind of
performances are obtained when the models are tested on images from an
entirely new dataset, with different imaging features. We randomly
selected models trained on one of the MICCAI2012 folds and applied them
to the images from the MRBrainS18 dataset. The results are shown in
Fig. \ref{fig:dice:mrbrains18}.
\begin{figure*}[t]
\centering
\subfloat[]{\includegraphics[trim={0cm 0.7cm 6.8cm
0.2cm},clip,scale=0.8]
{dice-all}\label{fig:dice:miccai2012}} \hspace{0.25cm}
\subfloat[]{\includegraphics[trim={8.0cm 0.7cm 0cm
0.2cm},clip,scale=0.8]
{dice-all}\label{fig:dice:mrbrains18}}
\caption{Validation of our model on the MICCAI2012 (a) and MRBrainS18
(b) datasets. Dice scores were computed for known labels (GM and WM)
and: SPM12 segmentations (SPM); and linear (Lin) and nonlinear (Net)
8-neighbour MRF applied to the SPM12 segmentations. Asterisks indicate statistical significance of paired \emph{t}-tests after Bonferroni
correction: 0.05 ($\ast$), 0.01 ($\ast\ast$), 0.001 ($\ast\ast\ast$) \& 0.0001 ($\ast\ast\ast\ast$).}
\label{fig:dice}
\end{figure*}
\subsubsection{Results:}
The 10-fold cross validation results in Fig. \ref{fig:dice:miccai2012}
show that the increase in Dice scores for both GM and WM is
statistically significant after applying either of our two MRF
CNN models. (Fig. \ref{fig:results} shows the results for a randomly
selected MICCAI2012 subject). With a mean Dice of $\{ \text{GM} =
0.867, \text{WM} = 0.921 \}$ for SPM12, and $\{ \text{GM} = 0.901,
\text{WM} = 0.929 \}$ and $\{ \text{GM} = 0.909, \text{WM} = 0.931 \}$
after applying the linear and nonlinear MRF, respectively. The
results imply that the classical generative approach of SPM12, which
currently ranks in the top 50 on the MRBrainS13 challenge
website\footnote{\url{mrbrains13.isi.uu.nl/results.php}}, could
move up quite a few positions by application of our proposed model
trained on the challenge data. As can be seen in Fig.
\ref{fig:dice:miccai2012}, for one of the subjects, all models perform
substantially worse. On closer inspection, this subject suffers from
major white matter hyper-intensities. This abnormality is currently not
handled well by the MRF CNN models, which obtain lower Dice scores than
the initial SPM12 segmentations.
Fig. \ref{fig:dice:mrbrains18} shows results when applying the models
to data from a different centre, not part of the training data (Fig.
\ref{fig:results} shows the results for a randomly selected MRBrainS18
subject). Mean Dice scores are $\{ \text{GM} = 0.722, \text{WM} = 0.816
\}$ for SPM12, and $\{ \text{GM} = 0.761, \text{WM} = 0.831 \}$ and $\{
\text{GM} = 0.755, \text{WM} = 0.829 \}$ after application of the
linear and nonlinear MRF, respectively. Application of the MRF improves
both GM and WM segmentations. The nonlinear MRF performs slightly worse
than the linear version. This result could be due to the nonlinear
model -- which possesses many more parameters than the linear model --
overfitting to the training subjects of MICCAI2012. Additionally, the
nonlinear MRF may struggle with the MRBrainS18 subjects that have
pathology (\emph{e.g.} white matter hyperintensities). Still, the fact
that Dice scores improve when applying the model to new data shows that
we can successfully improve segmenting images from different MR imaging
protocols.
\section{Discussion}
In this paper, we introduced an image segmentation method that combines
the robustness of a well-tuned generative model with some of the
outstanding learning capability of a CNN. The CNN encodes an MRF
in the prior term over the unknown labels. We evaluated the method
on annotated MR images and showed that a trained model
can be deployed on an unseen image population, with very different
characteristics from the training population. We hope
that the idea presented in this paper introduces to the medical
imaging community a principled way of bringing together probabilistic
modelling and deep learning.
In medical image analysis -- where labelled training data is sparse and
images can vary widely -- generalisability across different image
populations is one of the most important properties of learning-based
methods. However, achieving this generalisability is made difficult by
the limited amount of annotated data; the datasets we used in this
paper contained, in total, only 37 subjects. This issue may be
addressed by realistic, nonlinear data augmentation, which is able to
capture changes due to ageing and disease. Learning this variability in
shape from a large and diverse population could be a step in that
direction \cite{balbastre2018diffeomorphic}. On the other hand --
manual segmentations suffer from both intra- and inter-operator
variability, is it clinically meaningful to learn from these very
imperfect annotations? Could automatic segmentations prove more
anatomically informative than manual ones (\emph{c.f.}, Fig.
\ref{fig:results})? Semi-supervised techniques, leveraging both
labelled and unlabelled data, could be an option for making
our method less dependent on annotations (see \emph{e.g.},
\cite{roy2018quicknat}).
We chose the architecture of our proposed MRF CNN with the idea of
keeping the number of parameters low (to reduce overfitting), while
still introducing a more complex neighbourhood than in regular MRF
models. However, we did not extensively investigate different
architectures, \emph{e.g.}, activation functions and filter size. There
is therefore a possibility of improved performance by design changes to
the network. Such a change could be to hierarchically apply MRF filters
of decreasing size, which could increase neighbourhood size without
increased overfitting. Another potentially interesting idea would be
to `plug in' the MRF filters at the end of a segmentation network, such
as a U-net, emulating MRF post-processing inside the network. Finally,
we intend to integrate our model into a generative segmentation
framework and then validate its performance by comparing it to other
existing segmentation software.
\subsubsection*{Acknowledgements:} JA was funded by the EU Human Brain
Project's Grant Agreement No 785907 (SGA2). YB was funded by the MRC
and Spinal Research Charity through the ERA-NET Neuron joint call
(MR/R000050/1).
|
{
"timestamp": "2019-03-11T01:11:26",
"yymm": "1902",
"arxiv_id": "1902.10747",
"language": "en",
"url": "https://arxiv.org/abs/1902.10747"
}
|
\section{Introduction and Problem Setting}
We consider the parametric elliptic Dirichlet problem given by
\begin{align} \label{eq:PDE}
- \nabla \cdot (a({\boldsymbol{x}},{\boldsymbol{y}})\,\nabla u({\boldsymbol{x}},{\boldsymbol{y}}))
&=
f({\boldsymbol{x}}) \;\;\; \mbox{for} \,\, {\boldsymbol{x}} \in D \subset \R^d,
\;\;\; u({\boldsymbol{x}},{\boldsymbol{y}}) = 0 \;\;\; \mbox{for} \,\, {\boldsymbol{x}} \,\, \mbox{on} \,\, \partial D,
\end{align}
for $D \subset \R^d$ a bounded, convex Lipschitz polyhedron domain with boundary $\partial D$ and fixed spatial dimension
$d \in \{1,2,3\}$. The function $f$ lies in $L^2(D)$, the parametric variable ${\boldsymbol{y}} = (y_j)_{j\ge1}$ belongs to a domain $U$,
and the differential operators are understood to be with respect to the physical variable ${\boldsymbol{x}} \in D$. Here we study the ``uniform case'',
i.e., we assume that ${\boldsymbol{y}}$ is uniformly distributed on $U :=\left[-\frac12,\frac12\right]^{{\mathbb{N}}} % natural numbers {1, 2, ...}$ with uniform probability measure
$\mu({\mathrm{d}}{\boldsymbol{y}}) = \bigotimes_{j\geq 1} {\mathrm{d}} y_j = {\mathrm{d}}{\boldsymbol{y}}$.
The parametric diffusion coefficient $a({\boldsymbol{x}},{\boldsymbol{y}})$ is assumed to depend linearly on the parameters $y_j$ in the following way,
\begin{equation} \label{eq:diff_coeff}
a({\boldsymbol{x}},{\boldsymbol{y}})
=
a_0({\boldsymbol{x}}) + \sum_{j\geq 1} y_j\, \psi_j({\boldsymbol{x}})\,, \quad {\boldsymbol{x}} \in D, \quad {\boldsymbol{y}} \in U.
\end{equation}
For the variational formulation of \eqref{eq:PDE}, we consider the Sobolev space $V = H_0^1(D) $ of functions $v$
which vanish on the boundary $\partial D$ with norm
\begin{equation*}
\|v\|_V := \left( \int_D \sum_{j=1}^{d} |\partial_{x_j} v({\boldsymbol{x}})|^2 \rd{\boldsymbol{x}} \right)^{\frac12} = \|\nabla v\|_{L^2(D)}.
\end{equation*}
The corresponding dual space of bounded linear functionals on $V$ with respect to the pivot space $L^2(D)$ is
further denoted by $V^* = H^{-1}(D)$. Then, for given $f\in V^*$ and ${\boldsymbol{y}}\in U$, the weak (or variational) formulation of~\eqref{eq:PDE}
is to find $u(\cdot,{\boldsymbol{y}})\in V$ such that
\begin{equation} \label{eq:PDE_weak}
A({\boldsymbol{y}};u(\cdot,{\boldsymbol{y}}), v)
=
\langle f,v\rangle_{V^*\times V} = \int_D f({\boldsymbol{x}}) v({\boldsymbol{x}})\,\rd{\boldsymbol{x}} \quad\mbox{for all}\quad v \in V,
\end{equation}
with parametric bilinear form $A: U \times V \times V \to \R$ given by
\begin{equation} \label{eq:bilinear_form}
A({\boldsymbol{y}}; w,v)
:=
\int_D a({\boldsymbol{x}},{\boldsymbol{y}})\,\nabla w({\boldsymbol{x}})\cdot\nabla v({\boldsymbol{x}})\,\rd{\boldsymbol{x}} \quad\mbox{for all}\quad w, v\in V,
\end{equation}
and duality pairing $\langle\cdot,\cdot\rangle_{V^* \times V}$ between $V^*$ and $V$. We will often
identify elements $\varphi \in V$ with dual elements $L_\varphi \in V^*$. Indeed, for $\varphi \in V$ and $v\in V$, a bounded linear
functional is given via $L_\varphi(v) := \int_D \varphi({\boldsymbol{x}}) v({\boldsymbol{x}}) \rd{\boldsymbol{x}} = \langle \varphi,v \rangle_{L^2(D)}$ and by the Riesz representation
theorem there exists a unique representer $\widetilde{\varphi} \in V$ such that $L_\varphi(v) = \langle \widetilde\varphi,v \rangle_{L^2(D)}$
for all $v\in V$. Hence, the definition of the canonical duality pairing yields that
$\langle L_\varphi,v \rangle_{V^* \times V} = L_\varphi(v) = \langle \varphi,v \rangle_{L^2(D)}$.
Our quantity of interest is the expected value, with respect to ${\boldsymbol{y}}\in U$, of a given bounded linear
functional $G \in V^*$ applied to the solution $u(\cdot,{\boldsymbol{y}})$ of the PDE.
We therefore seek to approximate this expectation by numerically integrating $G$ applied
to a finite element approximation $u_h^s(\cdot,{\boldsymbol{y}})$ of the solution $u^s(\cdot,{\boldsymbol{y}}) \in H_0^1(D) = V$
of \eqref{eq:PDE_weak} with truncated diffusion coefficient $a({\boldsymbol{x}},({\boldsymbol{y}}_{\{1:s\}};0))$ where $\{1:s\}:=\{1,\ldots,s\}$ and we write
$({\boldsymbol{y}}_{\{1:s\}};0) = (\tilde{y}_j)_{j \ge 1}$ with $\tilde{y}_j = y_j$ for $j \in \{1:s\}$ and $\tilde{y}_j = 0$ otherwise; that is,
\begin{equation} \label{eq:QoI}
{\mathbb{E}}[G(u)]
:=
\int_{U} G(u(\cdot,{\boldsymbol{y}}))\,\mu({\mathrm{d}}{\boldsymbol{y}})
=
\int_{U} G(u(\cdot,{\boldsymbol{y}}))\,\rd{\boldsymbol{y}}
\approx
Q_N(G(u_h^s)) ,
\end{equation}
with $Q_N(\cdot)$ a linear quadrature rule using $N$ function evaluations.
The infinite-dimensional integral ${\mathbb{E}}[G(u)]$ in~\eqref{eq:QoI} is defined as
\begin{equation*}
{\mathbb{E}}[G(u)]
=
\int_{U} G(u(\cdot,{\boldsymbol{y}}))\,\rd{\boldsymbol{y}}
:=
\lim_{s\to\infty} \int_{\left[-\frac12,\frac12\right]^s} G(u(\cdot,(y_1,\ldots,y_s,0,0,\ldots)))\,\rd y_1\cdots\rd y_s
\end{equation*}
such that our integrands of interest are of the form $F({\boldsymbol{y}}) = G(u(\cdot,{\boldsymbol{y}}))$ with ${\boldsymbol{y}} \in U$. In this article,
we will employ (randomized) QMC methods of the form
\begin{equation*}
Q_N(f) = \frac 1N \sum_{k=1}^{N} F({\boldsymbol{t}}_k),
\end{equation*}
i.e., equal-weight quadrature rules with (randomly shifted) deterministic points ${\boldsymbol{t}}_1,\ldots,{\boldsymbol{t}}_{N} \in \left[-\frac12,\frac12\right]^s$. This elliptic PDE is a standard problem considered in the numerical analysis of computational methods in uncertainty quantification, see, e.g., \cite{BCM17,CDS06,DKLNS14,GHS18,HS18,K17,KN16,KSS12}.
\subsection{Existence of solutions of the variational problem}
\label{subsec:PDE}
To assure that a unique solution to the weak problem \eqref{eq:PDE_weak} exists, we need certain conditions
on the diffusion coefficient $a$. We assume
$a_0 \in L^{\infty}(D)$ and $\essinf_{{\boldsymbol{x}} \in D} a_0({\boldsymbol{x}}) > 0$,
which is equivalent to the existence of two constants $0 < a_{0,\min} \le a_{0,\max} < \infty$ such that
a.e.\ on $D$ we have
\begin{equation} \label{eq:bounds_a_1}
a_{0,\min} \le a_0({\boldsymbol{x}}) \le a_{0,\max} ,
\end{equation}
and that there exists a $\overline{\kappa} \in (0,1)$ such that
\begin{equation} \label{eq:kappa_bar}
\left\| \sum_{j \ge 1} \frac{|\psi_j|}{2 a_0} \right\|_{L^{\infty}(D)}
\le
\overline{\kappa}
<
1 .
\end{equation}
Via \eqref{eq:kappa_bar}, we obtain that $|\sum_{j \ge 1} y_j \psi_j({\boldsymbol{x}})| \le \overline{\kappa} \, a_0({\boldsymbol{x}})$ and
hence, using \eqref{eq:bounds_a_1}, almost everywhere on $D$ and for any ${\boldsymbol{y}} \in U$
\begin{align} \label{eq:bound_a_2}
0 < (1-\overline{\kappa}) \, a_{0,\min} \le a_0({\boldsymbol{x}}) + \sum_{j \ge 1} y_j \psi_j({\boldsymbol{x}}) = a({\boldsymbol{x}},{\boldsymbol{y}}) \le (1+\overline{\kappa}) \, a_{0,\max} .
\end{align}
These estimates yield the continuity and coercivity of $A({\boldsymbol{y}},\cdot,\cdot)$ defined in \eqref{eq:bilinear_form}
on $V \times V$, uniformly for all ${\boldsymbol{y}} \in U$. The Lax--Milgram theorem then ensures the existence of a unique solution
$u(\cdot,{\boldsymbol{y}})$ of the weak problem in \eqref{eq:PDE_weak}.
\subsection{Parametric regularity}
\label{subsec:PDE}
Having established the existence of unique weak parametric solutions $u(\cdot,{\boldsymbol{y}})$, we investigate their regularity in terms of the
behaviour of their mixed first-order derivatives.
Our analysis combines multiple techniques which can be found in the literature, see, e.g., \cite{CDS06,GHS18,K17,HS18,BCM17}. In particular we want to point out that our POD form bounds can take advantage of wavelet like expansions of the random field, a technique introduced in \cite{BCM17} and used to the advantage of QMC constructions by \cite{HS18} to deliver product weights to save on the construction compared to POD weights. Although we end up again with POD weights, we will save on the construction cost by making use of a special construction method, called the reduced CBC construction, which we will introduce in~Section \ref{sec:fast-reduced-CBC-POD}.
Let ${\boldsymbol{\nu}} = (\nu_j)_{j \ge 1}$ with $\nu_j \in {\mathbb{N}}} % natural numbers {1, 2, ..._0 := \{0,1,2,\ldots\}$ be a sequence of positive integers which we will refer to as a multi-index. We define the order $|{\boldsymbol{\nu}}|$ and the support ${\mathrm{supp}}({\boldsymbol{\nu}})$ as
\begin{equation*}
|{\boldsymbol{\nu}}| := \sum_{j \ge 1} \nu_j
\quad \text{and} \quad
{\mathrm{supp}}({\boldsymbol{\nu}}) := \{j \ge 1 : \nu_j > 0 \}
\end{equation*}
and introduce the sets ${\mathcal{F}}$ and ${\mathcal{F}}_1$ of finitely supported multi-indices as
\begin{equation*}
{\mathcal{F}}
:=
\{ {\boldsymbol{\nu}} \in {\mathbb{N}}} % natural numbers {1, 2, ..._0^{\mathbb{N}} : {\mathrm{supp}}({\boldsymbol{\nu}}) < \infty \}
\quad \text{and} \quad
{\mathcal{F}}_1
:=
\{ {\boldsymbol{\nu}} \in \{0,1\}^{\mathbb{N}} : {\mathrm{supp}}({\boldsymbol{\nu}}) < \infty \} ,
\end{equation*}
where ${\mathcal{F}}_1 \subseteq {\mathcal{F}}$ is the restriction containing only ${\boldsymbol{\nu}}$ with $\nu_j \in \{0,1\}$.
Then, for ${\boldsymbol{\nu}} \in {\mathcal{F}}$ denote the ${\boldsymbol{\nu}}$-th partial derivative with respect to the parametric variables ${\boldsymbol{y}} \in U$ by
\begin{equation*}
\partial^{{\boldsymbol{\nu}}}
=
\frac{\partial^{|{\boldsymbol{\nu}}|}}{\partial y_1^{\nu_1}\partial y_2^{\nu_2}\cdots},
\end{equation*}
and for a sequence ${\boldsymbol{b}} = (b_j)_{j\ge 1} \subset \R^{{\mathbb{N}}} % natural numbers {1, 2, ...}$, set ${\boldsymbol{b}}^{\boldsymbol{\nu}} := \prod_{j\ge 1} b_j^{\nu_j}$.
We further write ${\boldsymbol{\omega}} \le {\boldsymbol{\nu}}$ if $\omega_j \le \nu_j$ for all $j \ge 1$ and denote by ${\boldsymbol{e}}_i \in {\mathcal{F}}_1$ the multi-index
with components $e_j = \delta_{i,j}$.
For a fixed ${\boldsymbol{y}} \in U$, we introduce the energy norm $\|\cdot\|_{a_{\boldsymbol{y}}}^2$ in the space $V$ via
\begin{equation*}
\|v\|_{a_{\boldsymbol{y}}}^2
:=
\int_{D} a({\boldsymbol{x}},{\boldsymbol{y}}) \, |\nabla v({\boldsymbol{x}})|^2 \, \rd {\boldsymbol{x}}
\end{equation*}
for which it holds true by \eqref{eq:bound_a_2} that
\begin{equation} \label{eq:connection_norms}
(1-\overline{\kappa}) \, a_{0,\min} \|v\|_V^2 \le \|v\|_{a_{\boldsymbol{y}}}^2 \quad \text{for all} \quad v \in V .
\end{equation}
Consequently, we have that $(1-\overline{\kappa}) \, a_{0,\min} \|u(\cdot,{\boldsymbol{y}})\|_V^2 \le \|u(\cdot,{\boldsymbol{y}})\|_{a_{\boldsymbol{y}}}^2$
and hence the definition of the dual norm $\|\cdot\|_{V^*}$ yields the following initial estimate from~\eqref{eq:PDE_weak} and~\eqref{eq:bilinear_form},
\begin{align*}
\|u(\cdot,{\boldsymbol{y}})\|_{a_{\boldsymbol{y}}}^2
&=
\int_{D} a({\boldsymbol{x}},{\boldsymbol{y}}) \, |\nabla u({\boldsymbol{x}},{\boldsymbol{y}})|^2 \, \rd {\boldsymbol{x}}
=
\int_{D} f({\boldsymbol{x}}) u({\boldsymbol{x}},{\boldsymbol{y}}) \, \rd {\boldsymbol{x}} \\
&=
\langle f,u(\cdot,{\boldsymbol{y}}) \rangle_{V^* \times V}
\le
\|f\|_{V^*} \|u(\cdot,{\boldsymbol{y}})\|_V
\le
\frac{\|f\|_{V^*} \|u(\cdot,{\boldsymbol{y}})\|_{a_{\boldsymbol{y}}}}{\sqrt{(1-\overline{\kappa}) a_{0,\min}}}
\end{align*}
which gives in turn
\begin{equation} \label{est:norm_u_a}
\|u(\cdot,{\boldsymbol{y}})\|_{a_{\boldsymbol{y}}}^2
\le
\frac{\|f\|_{V^*}^2 }{(1-\overline{\kappa}) \, a_{0,\min}} .
\end{equation}
In order to exploit the decay of the norm sequence $(\|\psi_j\|_{L^\infty(D)})_{j \ge 1}$ of the basis functions, we extend
condition \eqref{eq:kappa_bar} as follows. To characterize the smoothness of the random field, we assume that there exist a sequence of reals ${\boldsymbol{b}} = (b_j)_{j\ge1}$ with $0 < b_j \le 1$ for all $j$, a constant $\kappa \in (0,1)$ and therefore also constants $\widetilde{\kappa}({\boldsymbol{\nu}}) \le \kappa$ for all ${\boldsymbol{\nu}} \in {\mathcal{F}}_1$ such that
\begin{align} \label{def:kappa}
\kappa
&:=
\left\| \sum_{j \ge 1} \frac{|\psi_j| /b_j }{2 a_0} \right\|_{L^{\infty}(D)}
<
1
,
&
\widetilde{\kappa}({\boldsymbol{\nu}}) = \left\| \sum_{j \in {\mathrm{supp}}({\boldsymbol{\nu}})} \frac{|\psi_j| / b_j}{2a_0} \right\|_{L^{\infty}(D)}
.
\end{align}
We remark that condition \eqref{eq:kappa_bar} is included in this assumption by letting $b_j = 1$ for all $j \ge 1$
and that $0 < \overline{\kappa} \le \kappa < 1$. Using the above estimations we can derive the following theorem for
the mixed first-order partial derivatives.
\begin{theorem} \label{thm:deriv_bound}
Let ${\boldsymbol{\nu}} \in {\mathcal{F}}_1$ be a multi-index of finite support and let $k \in \{0,1,\ldots,|{\boldsymbol{\nu}}|\}$.
Then, for every $f\in V^*$ and every ${\boldsymbol{y}}\in U$,
\begin{equation*}
\sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} {\boldsymbol{b}}^{-2{\boldsymbol{\omega}}} \|\partial^{{\boldsymbol{\omega}}}u(\cdot,{\boldsymbol{y}})\|_{V}^2
\le
\left( \left(\frac{2 \widetilde{\kappa}({\boldsymbol{\nu}})}{1-\overline{\kappa}}\right)^{k} \frac{\|f\|_{V^*} }{(1-\overline{\kappa}) \, a_{0,\min}} \right)^2,
\end{equation*}
with $\widetilde{\kappa}({\boldsymbol{\nu}})$ as in~\eqref{def:kappa}. Moreover, for $k = |{\boldsymbol{\nu}}|$ we obtain
\begin{equation*}
\|\partial^{{\boldsymbol{\nu}}} u(\cdot,{\boldsymbol{y}}) \|_{V}
\le
{\boldsymbol{b}}^{{\boldsymbol{\nu}}} \left(\frac{2\widetilde{\kappa}({\boldsymbol{\nu}})}{1-\overline{\kappa}}\right)^{|{\boldsymbol{\nu}}|} \frac{\|f\|_{V^*}}{(1-\overline{\kappa}) \, a_{0,\min}} .
\end{equation*}
\end{theorem}
\begin{proof}
For the special case ${\boldsymbol{\nu}} = \bszero$, the claim follows by combining \eqref{eq:connection_norms} and \eqref{est:norm_u_a}.
For ${\boldsymbol{\nu}} \in {\mathcal{F}}_1$ with $\abs{{\boldsymbol{\nu}}} > 0$, as is known from, e.g., \cite{CDS06} and \cite[Appendix]{KN16}, the linearity of $a({\boldsymbol{x}},{\boldsymbol{y}})$
gives rise to the following identity for any ${\boldsymbol{y}} \in U$:
\begin{equation} \label{eq:deriv_leibniz_form}
\|\partial^{{\boldsymbol{\nu}}} u(\cdot,{\boldsymbol{y}}) \|_{a_{\boldsymbol{y}}}^2
=
- \sum_{j \in {\mathrm{supp}}({\boldsymbol{\nu}})} \int_{D} \psi_j({\boldsymbol{x}}) \, \nabla \partial^{{\boldsymbol{\nu}} - {\boldsymbol{e}}_j} u({\boldsymbol{x}},{\boldsymbol{y}})
\cdot \nabla \partial^{{\boldsymbol{\nu}}} u({\boldsymbol{x}},{\boldsymbol{y}}) \, \rd {\boldsymbol{x}} .
\end{equation}
For sequences of $L^2(D)$-integrable functions ${\boldsymbol{f}} = (f_{{\boldsymbol{\omega}},j})_{{\boldsymbol{\omega}} \in {\mathcal{F}}, j \ge 1}$ with $f_{{\boldsymbol{\omega}},j}: D \to \R$, we define the inner product $\langle {\boldsymbol{f}}, {\boldsymbol{g}} \rangle_{{\boldsymbol{\nu}},k}$ as follows,
\begin{equation*}
\langle {\boldsymbol{f}}, {\boldsymbol{g}} \rangle_{{\boldsymbol{\nu}},k}
:=
\sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} \int_{D} \sum_{j \in {\mathrm{supp}}({\boldsymbol{\omega}})}
f_{{\boldsymbol{\omega}},j}({\boldsymbol{x}}) \, g_{{\boldsymbol{\omega}},j}({\boldsymbol{x}}) \, \rd {\boldsymbol{x}} .
\end{equation*}
We can then apply the Cauchy--Schwarz inequality to ${\boldsymbol{f}} = (f_{{\boldsymbol{\omega}},j})$ and ${\boldsymbol{g}} = (g_{{\boldsymbol{\omega}},j})$ with
$f_{{\boldsymbol{\omega}},j} = {\boldsymbol{b}}^{-{\boldsymbol{e}}_j/2} |\psi_j|^{\frac12} \, {\boldsymbol{b}}^{-({\boldsymbol{\omega}}-{\boldsymbol{e}}_j)} \nabla \partial^{{\boldsymbol{\omega}} - {\boldsymbol{e}}_j} u(\cdot,{\boldsymbol{y}})$ and $g_{{\boldsymbol{\omega}},j} = {\boldsymbol{b}}^{-{\boldsymbol{e}}_j/2} |\psi_j|^{\frac12} \, {\boldsymbol{b}}^{-{\boldsymbol{\omega}}} \nabla \partial^{{\boldsymbol{\omega}}} u(\cdot,{\boldsymbol{y}})$ to obtain,
with the help of \eqref{eq:deriv_leibniz_form},
\begin{align*}
&\sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} {\boldsymbol{b}}^{-2{\boldsymbol{\omega}}} \|\partial^{{\boldsymbol{\omega}}}u(\cdot,{\boldsymbol{y}})\|_{a_{{\boldsymbol{y}}}}^2 \\
&\quad=
-\sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} \int_{D} \sum_{j \in {\mathrm{supp}}({\boldsymbol{\omega}})} {\boldsymbol{b}}^{-{\boldsymbol{e}}_j} {\boldsymbol{b}}^{-({\boldsymbol{\omega}}-{\boldsymbol{e}}_j)} {\boldsymbol{b}}^{-{\boldsymbol{\omega}}} \psi_j({\boldsymbol{x}}) \,
\nabla \partial^{{\boldsymbol{\omega}} - {\boldsymbol{e}}_j} u({\boldsymbol{x}},{\boldsymbol{y}}) \cdot \nabla \partial^{{\boldsymbol{\omega}}} u({\boldsymbol{x}},{\boldsymbol{y}}) \, \rd {\boldsymbol{x}} \\
&\quad\le
\left( \int_{D} \sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} \sum_{j \in {\mathrm{supp}}({\boldsymbol{\omega}})} {\boldsymbol{b}}^{-{\boldsymbol{e}}_j} |\psi_j({\boldsymbol{x}})| \,
\left| {\boldsymbol{b}}^{-({\boldsymbol{\omega}}-{\boldsymbol{e}}_j)} \nabla \partial^{{\boldsymbol{\omega}} - {\boldsymbol{e}}_j} u({\boldsymbol{x}},{\boldsymbol{y}}) \right|^2 \rd {\boldsymbol{x}} \right)^{\frac12} \\
&\quad\qquad
\times\left( \int_{D} \sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} \sum_{j \in {\mathrm{supp}}({\boldsymbol{\omega}})} {\boldsymbol{b}}^{-{\boldsymbol{e}}_j} |\psi_j({\boldsymbol{x}})| \,
\left| {\boldsymbol{b}}^{-{\boldsymbol{\omega}}} \nabla \partial^{{\boldsymbol{\omega}}} u({\boldsymbol{x}},{\boldsymbol{y}}) \right|^2 \rd {\boldsymbol{x}} \right)^{\frac12} .
\end{align*}
The first of the two factors above is then bounded as follows,
\begin{align*}
&\int_{D} \sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} \sum_{j \in {\mathrm{supp}}({\boldsymbol{\omega}})} {\boldsymbol{b}}^{-{\boldsymbol{e}}_j}
|\psi_j({\boldsymbol{x}})| \,
\left| {\boldsymbol{b}}^{-({\boldsymbol{\omega}}-{\boldsymbol{e}}_j)} \nabla \partial^{{\boldsymbol{\omega}} - {\boldsymbol{e}}_j} u({\boldsymbol{x}},{\boldsymbol{y}}) \right|^2 \rd {\boldsymbol{x}} \\
&\quad=
\int_{D} \sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k-1}} \Bigg(\sum_{\substack{j \in {\mathrm{supp}}({\boldsymbol{\nu}}) \\ {\boldsymbol{\omega}} + {\boldsymbol{e}}_j \le {\boldsymbol{\nu}}}}
{\boldsymbol{b}}^{-{\boldsymbol{e}}_j} |\psi_j({\boldsymbol{x}})| \Bigg) \left| {\boldsymbol{b}}^{-{\boldsymbol{\omega}}} \nabla \partial^{{\boldsymbol{\omega}}} u({\boldsymbol{x}},{\boldsymbol{y}}) \right|^2 \rd {\boldsymbol{x}} \\
&\quad\le
\left\| \sum_{j \in {\mathrm{supp}}({\boldsymbol{\nu}})} \frac{|\psi_j| / b_j}{a(\cdot,{\boldsymbol{y}})} \right\|_{L^{\infty}(D)}
\sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k-1}} {\boldsymbol{b}}^{-2{\boldsymbol{\omega}}} \int_{D} a({\boldsymbol{x}},{\boldsymbol{y}}) \left| \nabla \partial^{{\boldsymbol{\omega}}} u({\boldsymbol{x}},{\boldsymbol{y}}) \right|^2 \rd {\boldsymbol{x}} \\
&\quad=
\left\| \sum_{j \in {\mathrm{supp}}({\boldsymbol{\nu}})} \frac{|\psi_j| / b_j}{a(\cdot,{\boldsymbol{y}})} \right\|_{L^{\infty}(D)}
\sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k-1}} {\boldsymbol{b}}^{-2{\boldsymbol{\omega}}} \|\partial^{{\boldsymbol{\omega}}}u(\cdot,{\boldsymbol{y}})\|_{a_{{\boldsymbol{y}}}}^2 ,
\end{align*}
while the other factor can be bounded trivially. Furthermore, using \eqref{eq:bound_a_2}, we have for any ${\boldsymbol{y}} \in U$
\begin{equation*}
\left\| \sum_{j \in {\mathrm{supp}}({\boldsymbol{\nu}})} \frac{|\psi_j| / b_j}{a(\cdot,{\boldsymbol{y}})} \right\|_{L^{\infty}(D)}
\le
\frac{1}{1-\overline{\kappa}} \left\| \sum_{j \in {\mathrm{supp}}({\boldsymbol{\nu}})} \frac{|\psi_j| / b_j}{a_0} \right\|_{L^{\infty}(D)} := \frac{2 \widetilde{\kappa}({\boldsymbol{\nu}})}{1-\overline{\kappa}},
\end{equation*}
so that, combining these three estimates, we obtain
\begin{align*}
&\sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} {\boldsymbol{b}}^{-2{\boldsymbol{\omega}}} \|\partial^{{\boldsymbol{\omega}}}u(\cdot,{\boldsymbol{y}})\|_{a_{{\boldsymbol{y}}}}^2 \\
&\phantom{=}\le
\frac{2 \widetilde{\kappa}({\boldsymbol{\nu}})}{1-\overline{\kappa}}
\left( \sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k-1}} {\boldsymbol{b}}^{-2{\boldsymbol{\omega}}} \|\partial^{{\boldsymbol{\omega}}}u(\cdot,{\boldsymbol{y}})\|_{a_{{\boldsymbol{y}}}}^2 \right)^{\frac12}
\left( \sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} {\boldsymbol{b}}^{-2{\boldsymbol{\omega}}} \|\partial^{{\boldsymbol{\omega}}}u(\cdot,{\boldsymbol{y}})\|_{a_{{\boldsymbol{y}}}}^2 \right)^{\frac12} .
\end{align*}
Therefore, we finally obtain that
\begin{equation*}
\sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} {\boldsymbol{b}}^{-2{\boldsymbol{\omega}}} \|\partial^{{\boldsymbol{\omega}}}u(\cdot,{\boldsymbol{y}})\|_{a_{{\boldsymbol{y}}}}^2
\le
\left(\frac{2 \widetilde{\kappa}({\boldsymbol{\nu}})}{1-\overline{\kappa}}\right)^2
\sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k-1}} {\boldsymbol{b}}^{-2{\boldsymbol{\omega}}} \|\partial^{{\boldsymbol{\omega}}}u(\cdot,{\boldsymbol{y}})\|_{a_{{\boldsymbol{y}}}}^2
\end{equation*}
which inductively gives
\begin{equation*}
\sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} {\boldsymbol{b}}^{-2{\boldsymbol{\omega}}} \|\partial^{{\boldsymbol{\omega}}}u(\cdot,{\boldsymbol{y}})\|_{a_{{\boldsymbol{y}}}}^2
\le
\left(\frac{2\widetilde{\kappa}({\boldsymbol{\nu}})}{1-\overline{\kappa}}\right)^{2k} \|u(\cdot,{\boldsymbol{y}})\|_{a_{{\boldsymbol{y}}}}^2
\le
\left(\frac{2\widetilde{\kappa}({\boldsymbol{\nu}})}{1-\overline{\kappa}}\right)^{2k} \frac{\|f\|_{V^*}^2 }{(1-\overline{\kappa}) \, a_{0,\min}} ,
\end{equation*}
where the last inequality follows from the initial estimate \eqref{est:norm_u_a}. The estimate \eqref{eq:connection_norms} then gives
\begin{align*}
\sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} {\boldsymbol{b}}^{2{\boldsymbol{\omega}}} \|\partial^{{\boldsymbol{\omega}}}u(\cdot,{\boldsymbol{y}})\|_V^2
&\le
\frac{1}{(1-\overline{\kappa}) \, a_{0,\min}} \sum_{\substack{{\boldsymbol{\omega}} \le {\boldsymbol{\nu}} \\ |{\boldsymbol{\omega}}| = k}} {\boldsymbol{b}}^{-2{\boldsymbol{\omega}}} \|\partial^{{\boldsymbol{\omega}}}u(\cdot,{\boldsymbol{y}})\|_{a_{{\boldsymbol{y}}}}^2 \\
&\le
\left(\frac{2\widetilde{\kappa}({\boldsymbol{\nu}})}{1-\overline{\kappa}}\right)^{2k} \frac{\|f\|_{V^*}^2 }{(1-\overline{\kappa})^2 \, a_{0,\min}^2},
\end{align*}
which yields the first claim. The second claim follows since the sum over the ${\boldsymbol{\omega}} \le {\boldsymbol{\nu}}$ with $|{\boldsymbol{\omega}}|=|{\boldsymbol{\nu}}|$ and ${\boldsymbol{\nu}} \in {\mathcal{F}}_1$ consists only of the term corresponding to ${\boldsymbol{\omega}}={\boldsymbol{\nu}}$.
\end{proof}
\begin{corollary} \label{cor:deriv_bound}
Under the assumptions of Theorem \ref{thm:deriv_bound}, there exists a number $\kappa(k)$ for each $k \in {\mathbb{N}}} % natural numbers {1, 2, ...$, given by
\begin{equation*}
\kappa(k) := \sup_{\substack{{\boldsymbol{\nu}} \in {\mathcal{F}}_1 \\ |{\boldsymbol{\nu}}| = k}} \widetilde{\kappa}({\boldsymbol{\nu}}) ,
\end{equation*}
such that $\widetilde{\kappa}({\boldsymbol{\nu}}) \le \kappa(k) \le \kappa < 1$ for all ${\boldsymbol{\nu}} \in {\mathcal{F}}_1$ with $|{\boldsymbol{\nu}}| = k$.
Then for ${\boldsymbol{\nu}} \in {\mathcal{F}}_1$, every $f \in V^{\ast}$, and every ${\boldsymbol{y}} \in U$, the solution $u(\cdot,{\boldsymbol{y}})$ satisfies
\begin{equation} \label{eq:bound_deriv_pod}
\|\partial^{{\boldsymbol{\nu}}} u(\cdot,{\boldsymbol{y}}) \|_{V}
\le
{\boldsymbol{b}}^{{\boldsymbol{\nu}}} \left(\frac{2\kappa(|{\boldsymbol{\nu}}|)}{1-\overline{\kappa}}\right)^{|{\boldsymbol{\nu}}|} \frac{\|f\|_{V^*}}{(1-\overline{\kappa}) \, a_{0,\min}} .
\end{equation}
\end{corollary}
Note that since $0 < \overline{\kappa} \le \kappa < 1$, the results of Theorem \ref{thm:deriv_bound} and Corollary \ref{cor:deriv_bound}
remain also valid for $\overline{\kappa}$ replaced by $\kappa$.
The obtained bounds on the mixed first-order derivatives turn out to be of product and order-dependent (so-called POD) form; that is, they are of the general form
\begin{equation} \label{eq:form_pod_bounds}
\|\partial^{{\boldsymbol{\nu}}} u(\cdot,{\boldsymbol{y}}) \|_{V}
\le
C \, {\boldsymbol{b}}^{{\boldsymbol{\nu}}} \, \Gamma(|{\boldsymbol{\nu}}|) \, \|f\|_{V^{\ast}}
\end{equation}
with a map $\Gamma: {\mathbb{N}}} % natural numbers {1, 2, ..._0 \to \R$, a sequence of reals ${\boldsymbol{b}} = (b_j)_{j \ge 1} \in \R^{{\mathbb{N}}} % natural numbers {1, 2, ...}$ and some constant $C \in \R_{+}$.
This finding motivates us to consider this special type of bounds in the following error analysis.
\section{Quasi-Monte Carlo finite element error}
We analyze the error ${\mathbb{E}}[G(u)] - Q_N(G(u_h^s))$ obtained by applying QMC
rules to the finite element approximation $u_h^s$ to approximate the expected value
\begin{equation*}
{\mathbb{E}}[G(u)] = \int_{U} G(u(\cdot,{\boldsymbol{y}}))\,\rd{\boldsymbol{y}}
.
\end{equation*}
To this end, we introduce the finite element approximation $u_h^s({\boldsymbol{x}},{\boldsymbol{y}}) := u_h({\boldsymbol{x}},({\boldsymbol{y}}_{\{1:s\}};0))$ of a solution of \eqref{eq:PDE_weak} with truncated diffusion coefficient $a({\boldsymbol{x}},({\boldsymbol{y}}_{\{1:s\}};0))$, where $u_h$ is a finite element approximation as defined in \eqref{eq:PDE_weak u_h} and $({\boldsymbol{y}}_{\{1:s\}};0)=(y_1,\ldots,y_s,0,0,\ldots)$. The overall absolute QMC finite element error is then bounded as follows
\begin{align}
&| {\mathbb{E}}[G(u)] - Q_N(G(u_h^s)) | \nonumber \\
&\quad=
| {\mathbb{E}}[G(u)] - {\mathbb{E}}[G(u^s)] + {\mathbb{E}}[G(u^s)] - {\mathbb{E}}[G(u_h^s)] + {\mathbb{E}}[G(u_h^s)] - Q_N(G(u_h^s)) | \nonumber \\
&\quad\le
|{\mathbb{E}}[G(u-u^s)]| + |{\mathbb{E}}[G(u^s-u_h^s)]| + |{\mathbb{E}}[G(u_h^s)] - Q_N(G(u_h^s))| \label{eq:error_split} .
\end{align}
The first term on the right hand side of \eqref{eq:error_split} will be referred to as (dimension) truncation error,
the second term is the finite element discretization error and the last term is the QMC quadrature error
for the integrand $u_h^s$. In the following sections we will analyze these different error terms separately.
\subsection{Finite Element Approximation}
\label{subsec:FEM}
Here, we consider the approximation of the solution $u(\cdot,{\boldsymbol{y}})$ of \eqref{eq:PDE_weak} by a finite element
approximation $u_h(\cdot,{\boldsymbol{y}})$ and assess the finite element discretization error. More specifically, denote by $\{V_h\}_{h>0}$ a family of subspaces $V_h \subset V$ of finite dimension $M_h$ such that $V_h \to V$ as $h \to 0$. We define the parametric finite element (FE) approximation as follows:
for $f\in V^*$ and given ${\boldsymbol{y}}\in U$, find $u_h(\cdot,{\boldsymbol{y}})\in V_h$ such that
\begin{equation} \label{eq:PDE_weak u_h}
A({\boldsymbol{y}};u_h(\cdot,{\boldsymbol{y}}), v_h) = \langle f,v_h\rangle_{V^*\times V}
= \int_D f({\boldsymbol{x}}) v_h({\boldsymbol{x}})\,\rd{\boldsymbol{x}} \quad\mbox{for all}\quad v_h\in V_h .
\end{equation}
To establish convergence of the finite element approximations, we need some further conditions on $a({\boldsymbol{x}},{\boldsymbol{y}})$.
To this end, we define the space $W^{1,\infty}(D) \subseteq L^\infty(D)$ endowed with the norm
$\|v\|_{W^{1,\infty}(D)} = \max\{ \|v\|_{L^{\infty}(D)}, \|\nabla v\|_{L^{\infty}(D)} \}$ and require that
\begin{equation} \label{eq:cond_fem}
a_0 \in W^{1,\infty}(D) \quad \text{and} \quad \sum_{j \ge 1} \|\psi_j\|_{W^{1,\infty}(D)} < \infty .
\end{equation}
Under these conditions and using that $f \in L^2(D)$, it was proven in \cite[Theorems 7.1 and 7.2]{KSS12} that for any ${\boldsymbol{y}} \in U$
the approximations $u_h(\cdot,{\boldsymbol{y}})$ satisfy
\begin{equation*} \label{eq:uh_bound}
\|u(\cdot,{\boldsymbol{y}}) - u_h(\cdot,{\boldsymbol{y}})\|_V
\le
C_1 \,h\, \|f\|_{L^2} .
\end{equation*}
In addition, if (the representer of) the bounded linear functional $G \in V^*$ lies in $L^2(D)$ we have for any ${\boldsymbol{y}} \in U$,
as $h \to 0$,
\begin{align} \label{eq:IG_uh_bound}
|G(u(\cdot,{\boldsymbol{y}})) - G(u_h(\cdot,{\boldsymbol{y}}))| \nonumber
&\le
C_2 \,h^2\, \|f\|_{L^2}\, \|G\|_{L^2}, \\
|{\mathbb{E}}[G(u(\cdot,{\boldsymbol{y}}) - u_h(\cdot,{\boldsymbol{y}}))]| \,
&\le
C_3 \,h^2\, \|f\|_{L^2}\, \|G\|_{L^2},
\end{align}
where the constants $C_1,C_2,C_3 > 0$ are independent of $h$ and ${\boldsymbol{y}}$. Since the above statements hold true for any ${\boldsymbol{y}} \in U$, they remain also valid for $u^s({\boldsymbol{x}},{\boldsymbol{y}}) := u({\boldsymbol{x}},({\boldsymbol{y}}_{\{1:s\}};0))$ and $u_h^s({\boldsymbol{x}},{\boldsymbol{y}}) := u_h({\boldsymbol{x}},({\boldsymbol{y}}_{\{1:s\}};0))$.
\subsection{Dimension Truncation}
\label{subsec:Truncation}
For every $s \in {\mathbb{N}}} % natural numbers {1, 2, ...$ and ${\boldsymbol{y}} \in U$, we formally define the solution of the parametric weak problem \eqref{eq:PDE_weak} corresponding to the diffusion coefficient $a({\boldsymbol{x}},({\boldsymbol{y}}_{\{1:s\}};0))$ with sum truncated to $s$ terms as
\begin{equation} \label{eq:truncated_sol}
u^s(\cdot,{\boldsymbol{y}}) := u(\cdot, ({\boldsymbol{y}}_{\{1:s\}};0)) .
\end{equation}
In \cite[Proposition 5.1]{GHS18} it was shown that for the solution $u^s$ the following error estimates are satisfied.
\begin{theorem}
Let $\overline{\kappa} \in (0,1)$ be such that \eqref{eq:kappa_bar} is satisfied and assume furthermore that there exists a
sequence of reals ${\boldsymbol{b}} = (b_j)_{j\ge1}$ with $0 < b_j \le 1$ for all $j$ and a constant $\kappa \in [\overline{\kappa},1)$ as defined
in \eqref{def:kappa}. Then, for every ${\boldsymbol{y}} \in U$ and each $s \in {\mathbb{N}}} % natural numbers {1, 2, ...$
\begin{equation*}
\|u(\cdot,{\boldsymbol{y}}) - u^s(\cdot,{\boldsymbol{y}}) \|_V
\le
\frac{a_{0,\max} \, \|f\|_{V^*}}{( a_{0,\min} (1-\overline{\kappa}))^2 } \sup_{j \ge s+1} b_j .
\end{equation*}
Moreover, if it holds for $\kappa$ that $\frac{\kappa \, a_{0,\max}}{(1-\overline{\kappa}) \, a_{0,\min}} \sup_{j \ge s+1} b_j < 1$,
then for every $G \in V^*$ we have
\begin{multline} \label{eq:truncation_IG_u}
\left| {\mathbb{E}}[G(u)] - \int_{\left[-\frac12,\frac12\right]^s} G(u^s(\cdot,({\boldsymbol{y}}_{\{1:s\}};0))) \,\rd{\boldsymbol{y}}_{\{1:s\}} \right|
\\\le
\frac{\|G\|_{V^*} \, \|f\|_{V^*}}{(1-\overline{\kappa}) \, a_{0,\min} - a_{0,\max} \, \kappa \sup_{j \ge s+1} b_j}
\left(\frac{a_{0,\max}}{(1-\overline{\kappa}) \, a_{0,\min}} \kappa \sup_{j \ge s+1} b_j \right)^2 .
\end{multline}
\end{theorem}
In the following subsection, we will discuss how to approximate the finite-dimensional integral of solutions
of the form \eqref{eq:truncated_sol} by means of QMC methods.
\subsection{Quasi-Monte Carlo Integration}
\label{subsec:QMC}
For a real-valued function $F:[-\tfrac12,\tfrac12]^s \to \R$ defined over the $s$-dimensional
unit cube centered at the origin, we consider the approximation of the integral $I_s(F)$ by $N$-point QMC
rules $Q_N(F)$, i.e.,
\begin{equation*}
I_s(F)
:= \int_{[-\frac12,\frac12]^s} F({\boldsymbol{y}})\, \rd{\boldsymbol{y}}
\,\approx\,
\frac1N \sum_{k=1}^N F({\boldsymbol{t}}_{k})
=:
Q_N(F) ,
\end{equation*}
with quadrature points ${\boldsymbol{t}}_1,\ldots,{\boldsymbol{t}}_N \in [-\tfrac12,\tfrac12]^s$.
As a quality criterion of such a rule, we define the worst-case error for QMC integration in some Banach space ${\mathcal{H}}$ as
\begin{equation*}
e^{\text{wor}}({\boldsymbol{t}}_1,\ldots,{\boldsymbol{t}}_N)
:=
\sup_{\substack{F\in{\mathcal{H}}\\ \|F\|_{{\mathcal{H}}}\le 1}} |I_s(F) - Q_N(F)| .
\end{equation*}
In this article, we consider randomly shifted rank-1 lattice rules as randomized QMC rules, with underlying points of the form
\begin{equation*}
\widetilde{{\boldsymbol{t}}}_k({\boldsymbol{\Delta}}) = \left\{(k {\boldsymbol{z}})/N + {\boldsymbol{\Delta}} \right\} - \left(1/2,\ldots,1/2\right), \quad k=1,\ldots,N ,
\end{equation*}
with generating vector ${\boldsymbol{z}} \in \Z^s$, uniform random shift ${\boldsymbol{\Delta}} \in [0,1]^s$ and component-wise applied fractional part,
denoted by $\{ {\boldsymbol{x}} \}$. For simplicity, we denote the worst-case error using a shifted lattice rule with generating
vector ${\boldsymbol{z}}$ and shift ${\boldsymbol{\Delta}}$ by $e_{N,s}({\boldsymbol{z}},{\boldsymbol{\Delta}})$.
For randomly shifted QMC rules, the probabilistic error bound
\begin{equation*}
\sqrt{{\mathbb{E}}_{{\boldsymbol{\Delta}}}\left[ |I_s(F) - Q_N(F)|^2 \right]}
\le
\widehat{e}_{N,s}({\boldsymbol{z}}) \, \|F\|_{{\mathcal{H}}},
\end{equation*}
holds for all $F \in {\mathcal{H}}$, with shift-averaged worst-case error
\begin{equation*}
\widehat{e}_{N,s}({\boldsymbol{z}})
:=
\left(\int_{[0,1]^s} e^2_{N,s}({\boldsymbol{z}},{\boldsymbol{\Delta}}) \,\rd{\boldsymbol{\Delta}} \right)^{1/2} .
\end{equation*}
As function space ${\mathcal{H}}$ for our integrands $F$, we consider the weighted, unanchored Sobolev space
${\mathcal{W}}_{s,{\boldsymbol{\gamma}}}$, which is a Hilbert space of functions defined over
$[-\frac{1}{2},\frac{1}{2}]^s$ with square integrable mixed first derivatives and general non-negative weights
${\boldsymbol{\gamma}} = (\gamma_{{\mathfrak{u}}})_{{\mathfrak{u}} \subseteq \{1:s\}}$. More precisely, the norm for $F \in {\mathcal{W}}_{s,{\boldsymbol{\gamma}}}$
is given by
\begin{equation} \label{eq:sob_norm}
\|F\|_{{\mathcal{W}}_{s,{\boldsymbol{\gamma}}}}
:=
\left(
\sum_{{\mathfrak{u}}\subseteq\{1:s\}}
\gamma_{\mathfrak{u}}^{-1}
\int_{[-\frac{1}{2},\frac{1}{2}]^{|{\mathfrak{u}}|}}
\left(
\int_{[-\frac{1}{2},\frac{1}{2}]^{s-|{\mathfrak{u}}|}}
\frac{\partial^{|{\mathfrak{u}}|}F}{\partial {\boldsymbol{y}}_{\mathfrak{u}}}({\boldsymbol{y}}_{\mathfrak{u}};{\boldsymbol{y}}_{-{\mathfrak{u}}})
\,\rd{\boldsymbol{y}}_{-{\mathfrak{u}}}
\right)^2 \,
\rd{\boldsymbol{y}}_{\mathfrak{u}}
\right)^{1/2},
\end{equation}
where $\{1:s\} := \{1,\ldots,s\}$, $\frac{\partial^{|{\mathfrak{u}}|}F}{\partial {\boldsymbol{y}}_{\mathfrak{u}}}$ denotes the mixed first
derivative with respect to the variables ${\boldsymbol{y}}_{\mathfrak{u}} = (y_j)_{j\in{\mathfrak{u}}}$ and we set
${\boldsymbol{y}}_{-{\mathfrak{u}}} = (y_j)_{j\in\{1:s\}\setminus{\mathfrak{u}}}$. \\
For the efficient construction of good lattice rule generating vectors, we consider the so-called reduced
component-by-component (CBC) construction introduced in \cite{DKLP15}.
For $b \in {\mathbb{N}}} % natural numbers {1, 2, ...$ and $m \in {\mathbb{N}}} % natural numbers {1, 2, ..._0$, we define the group of units of integers modulo $b^m$ via
\begin{equation*}
\ZZ_{b^m}^{\times} := \left\{z\in \Z_{b^m}: \gcd(z,b^m)=1\right\},
\end{equation*}
and note that $\ZZ_{b^0}^{\times}=\ZZ_{1}^{\times}=\{0\}$ since $\gcd(0,1)=1$.
Henceforth, let $b$ be prime and recall that then, for $m \ge 1$, $|\ZZ_{b^m}^{\times}|=\varphi(b^m)=b^{m-1} \varphi(b)$
and $|\ZZ_{b}^\times| = \varphi(b) = (b-1)$, where $\varphi$ is Euler's totient function.
Let ${\boldsymbol{w}}:=(w_j)_{j\ge 1}$ be a non-decreasing sequence of integers in ${\mathbb{N}}} % natural numbers {1, 2, ..._0$, the elements of which we will refer to as reduction indices.
In the reduced CBC algorithm the components $\widetilde{z}_j$ of the generating vector $\widetilde{{\boldsymbol{z}}}$ of the lattice rule will be taken as multiples of $b^{w_j}$.
In \cite{DKLP15}, the reduced CBC construction was introduced to construct rank-1 lattice rules for $1$-periodic
functions in a weighted Korobov space ${\mathcal{H}}(K_{s,\alpha,{\boldsymbol{\gamma}}})$ of smoothness $\alpha$
(see, e.g., \cite{SW01}). We denote the worst-case error in ${\mathcal{H}}(K_{s,\alpha,{\boldsymbol{\gamma}}})$ using a
rank-1 lattice rule with generating vector ${\boldsymbol{z}}$ by $e_{N,s}({\boldsymbol{z}})$. Following \cite{DKLP15}, the reduced CBC construction
is then given in Algorithm~\ref{alg:RedCBCAlg}.
{\centering
\begin{minipage}{\linewidth}
\begin{algorithm}[H]
\small
\caption{\small Reduced component-by-component construction}
\label{alg:RedCBCAlg}
\textbf{Input:} Prime power $N=b^m$ with $m \in {\mathbb{N}}} % natural numbers {1, 2, ..._0$ and integer reduction indices $0 \le w_1 \le \cdots \le w_s$. \\[1.75mm]
For $j$ from $1$ to $s$ and as long as $w_j<m$ do:
\begin{itemize}
\item[]
\begin{itemize}
\item[$\bullet$] Select $z_j \in \ZZ_{b^{m-w_j}}^{\times}$ such that
\vspace{-7pt}
\begin{equation*}
z_j := \argmin_{z \in \ZZ_{b^{m-w_j}}^{\times}} e^2_{N,j}(b^{w_1} z_1,\ldots,b^{w_{j-1}} z_{j-1}, b^{w_j} z) .
\end{equation*}
\end{itemize}
\end{itemize}
Set all remaining $z_j := 0$ (for $j$ with $w_j \ge m$). \\[1.75mm]
\textbf{Return:} Generating vector $\widetilde{{\boldsymbol{z}}}:=(b^{w_1} z_1,\ldots, b^{w_s} z_s)$ for $N=b^m$.
\end{algorithm}
\end{minipage}
}
\par
\vspace{5pt}
The following theorem, proven in \cite{DKLP15}, states that the algorithm yields generating
vectors with a small integration error for general weights $\gamma_{{\mathfrak{u}}}$ in the Korobov space.
\begin{theorem} \label{thm:redcbc_korobov}
For a prime power $N=b^m$ let $\widetilde{{\boldsymbol{z}}}=(b^{w_1} z_1,\ldots, b^{w_s} z_s)$ be constructed according to Algorithm~\ref{alg:RedCBCAlg} with integer reduction indices $0 \le w_1 \le \cdots \le w_s$. Then for every $d\in \{1:s\}$ and every $\lambda \in(1/\alpha,1]$ it holds for the worst-case error in the Korobov space ${\mathcal{H}}(K_{s,\alpha,{\boldsymbol{\gamma}}})$ with $\alpha > 1$ that
\begin{equation*}
e_{N,d}^2(b^{w_1} z_1,\ldots, b^{w_d} z_d)
\le
\left(\sum_{\emptyset\neq{\mathfrak{u}}\subseteq \{1:d\}} \gamma_{\mathfrak{u}}^\lambda
\, (2\zeta (\alpha\lambda))^{|{\mathfrak{u}}|} \, b^{\min\{m,\max_{j\in{\mathfrak{u}}}w_j\}}\right)^{\frac{1}{\lambda}}
\left(\frac{2}{N}\right)^{\frac1\lambda}.
\end{equation*}
\end{theorem}
This theorem can be extended to the weighted unanchored Sobolev space
${\mathcal{W}}_{s,{\boldsymbol{\gamma}}}$ using randomly shifted lattice rules as follows.
\begin{theorem} \label{thm:QMC_sh_lat_red_CBC}
For a prime power $N = b^m$, $m \in {\mathbb{N}}} % natural numbers {1, 2, ..._0$, and for $F \in {\mathcal{W}}_{s,{\boldsymbol{\gamma}}}$ belonging to the weighted unanchored Sobolev space defined over $[-\frac12,\frac12]^s$ with weights ${\boldsymbol{\gamma}} = (\gamma_{{\mathfrak{u}}})_{{\mathfrak{u}} \subseteq \{1:s\}}$, a randomly shifted lattice rule can be constructed by the reduced CBC algorithm, see Algorithm~\ref{alg:RedCBCAlg}, such that for all $\lambda\in (1/2,1]$,
\begin{multline*}
\sqrt{{\mathbb{E}}_{{\boldsymbol{\Delta}}}\left[ |I_s(F) - Q_N(F)|^2 \right]} \\
\le
\left(
\sum_{\emptyset\ne{\mathfrak{u}}\subseteq\{1:s\}} \gamma_{\mathfrak{u}}^\lambda\,
\varrho^{|{\mathfrak{u}}|}(\lambda) \, b^{\min\{m,\max_{j \in {\mathfrak{u}}} w_j\}}
\right)^{1/(2\lambda)}
\left( \frac2N \right)^{1/(2\lambda)} \, \|F\|_{{\mathcal{W}}_{s,{\boldsymbol{\gamma}}}},
\end{multline*}
with integer reduction indices $0 \le w_1 \le \cdots \le w_s$ and
$\varrho(\lambda) = 2\zeta(2\lambda) (2\pi^2)^{-\lambda}$.
\end{theorem}
\begin{proof}
Using Theorem~\ref{thm:redcbc_korobov} and the connection that the shift-averaged kernel of the Sobolev space equals the kernel of the Korobov space
${\mathcal{H}}(K_{s,\alpha,\widetilde{{\boldsymbol{\gamma}}}})$ with $\alpha = 2$ and weights $\widetilde{\gamma}_{\mathfrak{u}} = \gamma_{\mathfrak{u}} / (2\pi^2)^{|{\mathfrak{u}}|}$, see,
e.g., \cite{DKS13,N2014}, the result follows from
\begin{equation*}
\sqrt{{\mathbb{E}}_{{\boldsymbol{\Delta}}}\left[ |I(F) - Q_N(F)|^2 \right]}
\le
\sqrt{{\mathbb{E}}_{{\boldsymbol{\Delta}}}\left[ e^2_{N,s}({\boldsymbol{z}},{\boldsymbol{\Delta}}) \, \|F\|_{{\mathcal{W}}_{s,{\boldsymbol{\gamma}}}}^2 \right]}
=
\widehat{e}_{N,s}({\boldsymbol{z}}) \, \|F\|_{{\mathcal{W}}_{s,{\boldsymbol{\gamma}}}} . \qedhere
\end{equation*}
\end{proof}
It follows that we can construct the lattice rule in the weighted Korobov space using the connection mentioned in the proof of the previous theorem.
\subsection{Implementation of the reduced CBC algorithm}
\label{sec:fast-reduced-CBC-POD}
Similar to other variants of the CBC construction, we present a fast version of the reduced CBC method for POD weights in Algorithm~\ref{alg:RedCBCAlgPOD} for which Theorems~\ref{thm:redcbc_korobov} and~\ref{thm:QMC_sh_lat_red_CBC} still hold. The full derivation of Algorithm~\ref{alg:RedCBCAlgPOD} is given in Section~\ref{sec:appendix},
here we only introduce the necessary notation.
The squared worst-case error for POD weights ${\boldsymbol{\gamma}} = (\gamma_{{\mathfrak{u}}})_{{\mathfrak{u}} \subseteq \{1:s\}}$ with
$\gamma_{{\mathfrak{u}}} = \Gamma(|{\mathfrak{u}}|) \prod_{j \in {\mathfrak{u}}} \gamma_j$ and $\gamma_\emptyset = 1$ in the weighted Korobov space
${\mathcal{H}}(K_{s,\alpha,{\boldsymbol{\gamma}}})$ with $\alpha > 1$ can be written as
\begin{equation*}
e^2_{N,s}({\boldsymbol{z}})
=
\frac1N \sum_{k=0}^{N-1} \sum_{\ell=1}^{s} \sum_{\substack{{\mathfrak{u}} \subseteq \{1:s\} \\ \abs{{\mathfrak{u}}}=\ell}}
\Gamma(\ell) \prod_{j \in {\mathfrak{u}}} \gamma_j \, \omega\!\left(\left\{\frac{k z_j}{N}\right\}\right) ,
\end{equation*}
where $\omega(x) = \sum_{0 \ne h \in \Z} {\mathrm{e}}^{2\pi {\mathrm{i}} \, h x} / \abs{h}^{\alpha}$, see, e.g., \cite{DKS13,N2014}, and for $n \in {\mathbb{N}}} % natural numbers {1, 2, ...$ we define $\Omega_n$ as
\begin{equation*}
\Omega_{n}
:=
\left[ \omega\!\left( \frac{k z \bmod n}{n} \right) \right]_{\substack{z \in \ZZ_{n}^{\times} \\ k \in \ZZ_{n}}}
\in \R^{\varphi(n) \times n}
.
\end{equation*}
We assume that the values of the function $\omega$ can be computed at unit cost.
For integers $0 \le w' \le w'' \le m$ and given base~$b$ we define the ``fold and sum'' operator, which divides a length $b^{m-w'}$
vector into blocks of equal length $b^{m-w''}$ and sums them up, i.e.,
\begin{align}\label{eq:fold-and-sum}
P_{w'',w'}^m
:
\R^{b^{m-w'}} \to \R^{b^{m-w''}}
:
P_{w'',w'}^m \, {\boldsymbol{v}}
=
\bigl[ \; \underbrace{I_{b^{m-w''}} | \cdots | I_{b^{m-w''}}}_{b^{w''-w'} \text{ times}} \; \bigr] \, {\boldsymbol{v}}
,
\end{align}
where $\cramped{I_{b^{m-w''}}}$ is the identity matrix of size $\cramped{b^{m-w''} \times b^{m-w''}}$.
The computational cost of applying $P_{w'',w'}^m$ is the length of the input vector $\cramped{{\mathcal{O}}(b^{m-w'})}$.
It should be clear that $\cramped{P_{w''',w''}^m \, P_{w'',w'}^m \, {\boldsymbol{v}} = P_{w''',w'}^m \, {\boldsymbol{v}}}$ for $0 \le w' \le w'' \le w''' \le m$.
In step~\ref{step-update} of Algorithm \ref{alg:RedCBCAlgPOD} the notation $.*$ denotes the element-wise product of
two vectors and $\Omega_{b^{m-w_j}}(z_j,:)$ means to take the row corresponding to $z=z_j$ from the matrix. Furthermore,
Algorithm \ref{alg:RedCBCAlgPOD} includes an optional step in which the reduction indices are adjusted in case $w_1 > 0$, the
auxiliary variable $w_0=0$ is introduced to satisfy the recurrence relation.
The standard fast CBC algorithm for POD weights has a complexity of ${\mathcal{O}}(s \, N \log N + s^2 N)$, see, e.g., \cite{DKS13,N2014}.
The cost of our new algorithm can be substantially lower as is stated in the following theorem. We stress that the presented algorithm
is the first realization of the reduced CBC construction for POD weights. Our new algorithm improves upon the one stated in \cite{DKLP15}
which only considers product weights, but the same technique can be used there since POD weights are more general and include product weights.
\begin{theorem} \label{thm:complexity_redcbc}
Given a sequence of integer reduction indices $0 \le w_1 \le w_2 \le \cdots$, the reduced CBC algorithm for a prime power $N = b^m$ points in $s$ dimensions
as specified in Algorithm~\ref{alg:RedCBCAlgPOD} can construct a lattice rule with near optimal worst-case error as in Theorem~\ref{thm:QMC_sh_lat_red_CBC} with an arithmetic cost of
\begin{equation*}
{\mathcal{O}}\!\left(\sum_{j=1}^{\min\{s,s^{\ast}\}} (m-w_j+j) \, b^{m-w_j} \right),
\end{equation*}
where $s^*$ is defined to be the largest integer such that $w_{s^*} < m$. The memory cost is ${\mathcal{O}}(\sum_{j=1}^{\min\{s,s^{\ast}\}} b^{m-w_j})$.
In case of product weights ${\mathcal{O}}(\sum_{j=1}^{\min\{s,s^{\ast}\}} (m-w_j) \, b^{m-w_j})$ operations are required for the construction with
memory ${\mathcal{O}}(b^{m-w_1})$.
\end{theorem}
\begin{proof}
We refer to Algorithm~\ref{alg:RedCBCAlgPOD}.
Step~\ref{step-qbar} can be calculated in ${\mathcal{O}}(j\,b^{m-w_{j-1}})$ operations (and we may assume $w_0 = w_1$ since the case $w_1 > 0$
can be reduced to the case $w_1 = 0$). The matrix-vector multiplication in step~\ref{step-mv} can be done by exploiting the block-circulant structure to obtain a fast matrix-vector product by FFTs at a cost of ${\mathcal{O}}((m-w_j) \, b^{m-w_j})$, see, e.g., \cite{CKN06,CN06b}.
We ignore the possible saving by pre-computation of FFTs on the first columns of the blocks in the matrices $\Omega_{b^{m-w_j}}$ as this has cost ${\mathcal{O}}((m-w_1) \, b^{m-w_1})$ and therefore is already included in the cost of step~\ref{step-mv}. Finally, the vectors ${\boldsymbol{q}}_{j,\ell}$ for $\ell=1,\ldots,j$ in step~\ref{step-update} can be calculated in ${\mathcal{O}}(j \,b^{m-w_{j-1}})$. To obtain the total complexity we remark that the applications of the ``fold and sum'' operator, marked by the square brackets could be performed in iteration $j-1$ such that the cost of steps~\ref{step-qbar} and~\ref{step-update} in iteration $j$ are only ${\mathcal{O}}(j \, b^{m-w_j})$ instead of ${\mathcal{O}}(j \, b^{m-w_{j-1}})$. The cost of the
additional fold and sum to prepare for iteration $j$ in iteration $j-1$, which can be performed after step~\ref{step-update}, is then equal to the cost of step~\ref{step-update} in
that iteration. Since we can assume $w_0 = w_1$ we obtain the claimed construction cost. Note that the algorithm is written in such a way that the vectors ${\boldsymbol{q}}_{j,\ell-1}$ can be reused for storing the vectors ${\boldsymbol{q}}_{j,\ell}$ (which might be smaller). Similarly for the vectors $\overline{{\boldsymbol{q}}}_j$.
Therefore the memory cost is ${\mathcal{O}}(\sum_{j=1}^{\min\{s,s^{\ast}\}} b^{m-w_j})$. The result for product weights can be obtained similarly, see, e.g., \cite{N2014}.
\end{proof}
{\centering
\begin{minipage}{\linewidth}
\begin{algorithm}[H]
\small
\caption{\small Fast reduced CBC construction for POD weights}
\label{alg:RedCBCAlgPOD}
\textbf{Input:} Prime power $N=b^m$ with $m \in {\mathbb{N}}} % natural numbers {1, 2, ..._0$, integer reduction indices $0 \le w_1 \le \cdots \le w_s$, \\
and weights $\Gamma(\ell)$, $\ell \in {\mathbb{N}}} % natural numbers {1, 2, ..._0$ with $\Gamma(0) = 1$, and $\gamma_j$, $j \in {\mathbb{N}}} % natural numbers {1, 2, ...$ such that $\gamma_{\mathfrak{u}} = \Gamma(|{\mathfrak{u}}|) \prod_{j\in{\mathfrak{u}}} \gamma_j$.
\\[2mm]
Optional: Adjust $m := \max\{0, m - w_1\}$ and for $j$ from $s$ down to $1$ adjust $w_j := w_j - w_1$.
\\[1mm]
Set ${\boldsymbol{q}}_{0,0}:=\mathbf{1}_{b^m}$ and ${\boldsymbol{q}}_{0,1}:=\mathbf{0}_{b^m}$, set $w_0:=0$.
\\[1mm]
For $j$ from $1$ to $s$ and as long as $w_j < m$ do:
\begin{enumerate}
\item\label{step-qbar} Set $\overline{{\boldsymbol{q}}}_j := \sum_{\ell=1}^{j} \frac{\Gamma(\ell)}{\Gamma(\ell-1)} \left[ P_{w_j,w_{j-1}}^{m} {\boldsymbol{q}}_{j-1,\ell-1} \right] \in \R^{b^{m-w_j}}$
(with ${\boldsymbol{q}}_{j-1,\ell-1} \in \R^{b^{m-w_{j-1}}}$).
\item\label{step-mv} Calculate ${\boldsymbol{T}}_j := \Omega_{b^{m-w_j}} \, \overline{{\boldsymbol{q}}}_j \in \R^{\varphi(b^{m-w_j})}$ by exploiting the block-circulant structure
of the matrix $\Omega_{b^{m-w_j}}$ using FFTs.
\item\label{step-argmin} Set $z_j := \argmin_{z \in \ZZ_{b^{m-w_j}}^{\times}} {\boldsymbol{T}}_j(z)$, with ${\boldsymbol{T}}_j(z)$ the component corresponding to $z$.
\item\label{step-update} Set ${\boldsymbol{q}}_{j,0} := \mathbf{1}_{b^{m-w_j}}$ and ${\boldsymbol{q}}_{j,j+1} := \mathbf{0}_{b^{m-w_j}}$ and for $\ell$ from $j$ down to $1$ set
\begin{equation*}
{\boldsymbol{q}}_{j,\ell} := \left[ P_{w_j,w_{j-1}}^{m} {\boldsymbol{q}}_{j-1,\ell} \right] + \frac{\Gamma(\ell)}{\Gamma(\ell-1)} \gamma_j \, \Omega_{b^{m-w_j}}(z_j,:) \, {.*} \, \left[ P_{w_j,w_{j-1}}^{m} {\boldsymbol{q}}_{j-1,\ell-1} \right] \in \R^{b^{m-w_j}}.
\end{equation*}
\item\label{step-wce} Optional: Calculate squared worst-case error by $e^2_j := \frac1{b^m} \sum_{k \in \Z_{b^{m-w_j}}} \sum_{\ell=1}^j {\boldsymbol{q}}_{j,\ell}(k)$.
\end{enumerate}
Set all remaining $z_j := 0$ (for $j$ with $w_j \ge m$). \\[1.75mm]
\textbf{Return:} Generating vector $\widetilde{{\boldsymbol{z}}}:=(b^{w_1} z_1,\ldots, b^{w_s} z_s)$ for $N=b^m$. \\
(Note: the $w_j$'s and $m$ might have been adjusted to make $w_1=0$.)
\end{algorithm}
\end{minipage}
}
\section{QMC finite element error analysis}
We now combine the results of the previous subsections to analyze the overall QMC finite element error. We consider
the root mean square error (RMSE) given by
\begin{equation*}
e_{N,s,h}^{\text{RMSE}}(G(u))
:=
\sqrt{{\mathbb{E}}_{{\boldsymbol{\Delta}}}\left[|{\mathbb{E}}[G(u)] - Q_N(G(u_h^s))|^2\right]}
.
\end{equation*}
The error ${\mathbb{E}}[G(u)] - Q_N(G(u_h^s))$ can be written as
\begin{align*}
{\mathbb{E}}[G(u)] - Q_N(G(u_h^s))
&=
{\mathbb{E}}[G(u)] - I_s(G(u_h^s)) + I_s(G(u_h^s)) - Q_N(G(u_h^s))
\end{align*}
such that due to the fact that ${\mathbb{E}}_{{\boldsymbol{\Delta}}}(Q_N(f)) = I_s(f)$ for any integrand $f$ we obtain
\begin{align*}
{\mathbb{E}}_{{\boldsymbol{\Delta}}}\left[({\mathbb{E}}[G(u)] - Q_N(G(u_h^s)))^2\right]
&=
({\mathbb{E}}[G(u)] - I_s(G(u_h^s)))^2 + {\mathbb{E}}_{{\boldsymbol{\Delta}}}\left[(I_s - Q_N)^2(G(u_h^s))\right] \\
&\phantom{= } +
2 ({\mathbb{E}}[G(u)] - I_s(G(u_h^s))) \, {\mathbb{E}}_{{\boldsymbol{\Delta}}}\left[(I_s - Q_N)(G(u_h^s))\right] \\
&=
({\mathbb{E}}[G(u)] - I_s(G(u_h^s)))^2 + {\mathbb{E}}_{{\boldsymbol{\Delta}}}\left[(I_s - Q_N)^2(G(u_h^s))\right] .
\end{align*}
Then, noting that ${\mathbb{E}}[G(u)] - I_s(G(u_h^s)) = {\mathbb{E}}[G(u)] - I_s(G(u^s)) + I_s(G(u^s)) - I_s(G(u_h^s))$,
\begin{align*}
({\mathbb{E}}[G(u)] - I_s(G(u_h^s)))^2
&=
({\mathbb{E}}[G(u)] - I_s(G(u^s)))^2 + (I_s(G(u^s)) - I_s(G(u_h^s)))^2 \\
&\phantom{= } +
2 ({\mathbb{E}}[G(u)] - I_s(G(u^s))) (I_s(G(u^s)) - I_s(G(u_h^s)))
\end{align*}
and since for general $x,y \in \RR$ it holds that $2 x y \le x^2 + y^2$, we obtain
furthermore
\begin{align*}
({\mathbb{E}}[G(u)] - I_s(G(u_h^s)))^2
&\le
2 ({\mathbb{E}}[G(u)] - I_s(G(u^s)))^2 + 2 (I_s(G(u^s)) - I_s(G(u_h^s)))^2 .
\end{align*}
From the previous subsections we can then use \eqref{eq:truncation_IG_u} for the truncation part,
\eqref{eq:IG_uh_bound}, which holds for general ${\boldsymbol{y}} \in U$ and thus also for $y_{\{1:s\}}$, for the finite element error,
and Theorem \ref{thm:QMC_sh_lat_red_CBC} for the QMC integration error to obtain the following error bound
for the mean square error ${\mathbb{E}}_{{\boldsymbol{\Delta}}}[ |{\mathbb{E}}[G(u)] - Q_N(G(u_h^s))|^2 ] =: e^{\text{MSE}}_{N,s,h}(G(u))$,
\begin{align} \label{eq:combined_error}
e^{\text{MSE}}_{N,s,h}(G(u))
&\le
K_1 \|f\|^2_{V^*}\|G\|^2_{V^*} \left(\frac{1}{(1-\overline{\kappa}) \, a_{0,\min} - a_{0,\max} \, \kappa \sup_{j \ge s+1} b_j} \right)^2 \nonumber \\
&\phantom{\le}\times \left(\frac{a_{0,\max}}{(1-\overline{\kappa}) \, a_{0,\min}} \kappa \sup_{j \ge s+1} b_j \right)^4
+ K_2 \|f\|^2_{L^2}\|G\|^2_{L^2} \, h^4 \\
&\phantom{\le}+ \left(
\sum_{\emptyset\ne{\mathfrak{u}}\subseteq\{1:s\}} \gamma_{\mathfrak{u}}^\lambda\,
\varrho^{|{\mathfrak{u}}|}(\lambda) \, b^{\min\{m,\max_{j \in {\mathfrak{u}}} w_j\}}
\right)^{1/\lambda}
\left( \frac2N \right)^{1/\lambda} \, \|G (u_h^s)\|_{{\mathcal{W}}_{s,{\boldsymbol{\gamma}}}}^2 \nonumber
\end{align}
for some constants $K_1, K_2 \in \R_{+}$ and provided that
$\frac{a_{0,\max}}{(1-\overline{\kappa}) \, a_{0,\min}} \kappa \, \sup_{j \ge s+1} b_j < 1$.
\subsection{Derivative bounds of POD form}
In the following we assume that we have general bounds on the mixed partial derivatives
$\partial^{{\boldsymbol{\nu}}} u(\cdot,{\boldsymbol{y}})$ which are of POD form; that is,
\begin{equation} \label{eq:form_pod_bounds}
\|\partial^{{\boldsymbol{\nu}}} u(\cdot,{\boldsymbol{y}}) \|_{V}
\le
C \, \widetilde{{\boldsymbol{b}}}^{{\boldsymbol{\nu}}} \, \Gamma(|{\boldsymbol{\nu}}|) \, \|f\|_{V^{\ast}}
\end{equation}
with a map $\Gamma: {\mathbb{N}}} % natural numbers {1, 2, ..._0 \to \R$, a sequence of reals $\widetilde{{\boldsymbol{b}}} = (\widetilde{b}_j)_{j \ge 1} \in \R^{{\mathbb{N}}} % natural numbers {1, 2, ...}$ and some constant $C \in \R_{+}$.
Such bounds can be found in the literature and we provided a new derivation in Theorem~\ref{thm:deriv_bound} also leading to POD weights.
For bounding the norm $\|G (u_h^s)\|_{{\mathcal{W}}_{s,{\boldsymbol{\gamma}}}}$, we can then use \eqref{eq:form_pod_bounds} and the definition in \eqref{eq:sob_norm} to proceed as outlined in \cite{KN16}, to obtain the estimate
\begin{equation} \label{eq:bound_functional}
\|G(u_h^s)\|_{{\mathcal{W}}_{s,{\boldsymbol{\gamma}}}}
\le
C \, \|f\|_{V^*} \|G\|_{V^*}
\Bigg(
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \frac{\Gamma(|{\mathfrak{u}}|)^2 \prod_{j\in{\mathfrak{u}}} \widetilde{b}_j^2}{\gamma_{\mathfrak{u}}}
\Bigg)^{1/2}\,.
\end{equation}
Denoting ${\boldsymbol{w}}:=(w_j)_{j\ge 1}$ and using \eqref{eq:bound_functional}, the contribution of the quadrature error
to the mean square error $e_{N,h,s}^{\text{MSE}}(G(u))$ can be upper bounded by
\begin{equation} \label{eq:intermquaderror}
\begin{aligned}
&\left(
\sum_{\emptyset\ne{\mathfrak{u}}\subseteq\{1:s\}} \gamma_{\mathfrak{u}}^\lambda\,
\varrho^{|{\mathfrak{u}}|}(\lambda) \, b^{\min\{m,\max_{j \in {\mathfrak{u}}} w_j\}}
\right)^{1/\lambda}
\left( \frac2N \right)^{1/\lambda} \, \|G (u_h^s)\|^2_{{\mathcal{W}}_{s,{\boldsymbol{\gamma}}}} \\
&\qquad\le
C \, \|f\|_{V^*} \|G\|_{V^*} \, C_{{\boldsymbol{\gamma}},{\boldsymbol{w}},\lambda} \left( \frac2N \right)^{1/\lambda} ,
\end{aligned}
\end{equation}
where we define
\begin{equation*}
C_{{\boldsymbol{\gamma}},{\boldsymbol{w}},\lambda}
:=
\left(
\sum_{\emptyset\ne{\mathfrak{u}}\subseteq\{1:s\}} \gamma_{\mathfrak{u}}^\lambda\,
\varrho^{|{\mathfrak{u}}|}(\lambda) \, b^{\min\{m,\max_{j \in {\mathfrak{u}}} w_j\}}
\right)^{1/\lambda}
\left(
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \frac{\Gamma(|{\mathfrak{u}}|)^2 \prod_{j\in{\mathfrak{u}}} \widetilde{b}_j^2}{\gamma_{\mathfrak{u}}}
\right) .
\end{equation*}
The term $C_{{\boldsymbol{\gamma}},{\boldsymbol{w}},\lambda}$ can be bounded as
\begin{equation*}
C_{{\boldsymbol{\gamma}},{\boldsymbol{w}},\lambda}\le
\left(
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \gamma_{\mathfrak{u}}^\lambda\,
\varrho^{|{\mathfrak{u}}|}(\lambda) \, b^{\sum_{j \in {\mathfrak{u}}} w_j - \sum_{\ell=1}^{|{\mathfrak{u}}|-1} w_{\ell}}
\right)^{1/\lambda}
\left(
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \frac{\Gamma(|{\mathfrak{u}}|)^2 \prod_{j\in{\mathfrak{u}}} \widetilde{b}_j^2}{\gamma_{\mathfrak{u}}}
\right) .
\end{equation*}
Due to \cite[Lemma~6.2]{KSS12} the latter term is minimized by choosing the weights
$\gamma_{\mathfrak{u}}$ as
\begin{equation} \label{eq:form_weights}
\gamma_{\mathfrak{u}}
:=
\left(\frac{\Gamma(|{\mathfrak{u}}|)^2\, \prod_{j\in{\mathfrak{u}}} \widetilde{b}_j^2 \, \prod_{\ell=1}^{|{\mathfrak{u}}|-1} b^{w_{\ell}}}{\prod_{j\in{\mathfrak{u}}} \rho (\lambda)\, b^{w_j}}\right)^{1/(1+\lambda)}.
\end{equation}
Then we set
\begin{equation*}
A_{\lambda}
:=
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \gamma_{\mathfrak{u}}^\lambda\,
\varrho^{|{\mathfrak{u}}|}(\lambda) \, b^{\sum_{j \in {\mathfrak{u}}} w_j - \sum_{\ell=1}^{|{\mathfrak{u}}|-1} w_{\ell}}
=
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \left[ \left(\frac{\Gamma(|{\mathfrak{u}}|)^{2\lambda}}{\prod_{\ell=1}^{|{\mathfrak{u}}|-1} b^{w_{\ell}}}\right)
\left(\prod_{j \in {\mathfrak{u}}} \rho(\lambda) \, \widetilde{b}_j^{2\lambda} \, b^{w_j} \right)
\right]^\frac{1}{1+\lambda}
\end{equation*}
and easily see that also
\begin{equation*}
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \gamma_{\mathfrak{u}}^{-1} \left(\Gamma(|{\mathfrak{u}}|)^2 \prod_{j\in{\mathfrak{u}}} \widetilde{b}_j^2\right)
= A_\lambda,
\end{equation*}
which implies that $C_{{\boldsymbol{\gamma}},{\boldsymbol{w}},\lambda} \le A_\lambda^{1 + 1/\lambda}$.
We demonstrate how the term $A_{\lambda}$ can be estimated for the derivative bounds
derived in Section \ref{subsec:PDE}.
In view of Theorem \ref{thm:deriv_bound}, assume in the following that
\begin{equation}\label{eq:assumpThm1}
\Gamma (\abs{{\mathfrak{u}}})=\kappa^{\abs{{\mathfrak{u}}}},
\quad
\widetilde{b}_j=\frac{2\,b_j}{1-\kappa},
\quad
\sum_{j=1}^\infty \left(b_j b^{w_j}\right)^{p}<\infty \quad \text{for} \quad p \in (0,1) .
\end{equation}
Note that we could also choose $\Gamma (\abs{{\mathfrak{u}}})=\kappa (\abs{{\mathfrak{u}}})^{\abs{{\mathfrak{u}}}}$ above,
in which case the subsequent estimate of $A_\lambda$ can be done analogously, but
to make the argument less technical, we consider the slightly coarser variant
$\Gamma (\abs{{\mathfrak{u}}})=\kappa^{\abs{{\mathfrak{u}}}}$ here.
In this case,
\[
A_\lambda
=
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \left[\kappa^{\abs{{\mathfrak{u}}}}\right]^{\frac{2\lambda}{1+\lambda}}
\left(\prod_{\ell=1}^{|{\mathfrak{u}}|-1} b^{\frac{-w_{\ell}}{2\lambda}}\right)^\frac{2\lambda}{1+\lambda}
\prod_{j\in{\mathfrak{u}}} \left(\left(\frac{2\,b_j}{1-\kappa}\right)^{2\lambda}\, b^{w_j}\, \rho (\lambda)\right)^{\frac{1}{1+\lambda}}.
\]
Note that, as $\lambda \le 1$, it holds that $b^{\frac{-w_{\ell}}{2\lambda}} \le b^{\frac{-w_{\ell}}{2}}$ and hence
\[
A_\lambda
\le
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \left(\kappa^{\abs{{\mathfrak{u}}}}
\prod_{\ell=1}^{|{\mathfrak{u}}|-1} b^{\frac{-w_{\ell}}{2}}\right)^\frac{2\lambda}{1+\lambda}
\prod_{j\in{\mathfrak{u}}} \left(\left(\frac{2\,b_j}{1-\kappa}\right)^{2\lambda}\, b^{w_j}\, \rho (\lambda)\right)^{\frac{1}{1+\lambda}}.
\]
We now proceed similarly to the proof of Theorem 6.4 in \cite{KSS12}. Let $(\alpha_j)_{j\ge 1}$ be a sequence of positive reals,
to be specified below, which satisfies $\Sigma:=\sum_{j=1}^\infty \alpha_j<\infty$.
Dividing and multiplying by $\prod_{j\in{\mathfrak{u}}} \alpha_j^{(2\lambda)/(1+\lambda)}$,
and applying H\"{o}lder's inequality with conjugate components $p=(1+\lambda)/(2\lambda)$ and $p^*=(1+\lambda)/(1-\lambda)$,
\begin{eqnarray*}
A_\lambda
&\le&
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \left(\kappa^{\abs{{\mathfrak{u}}}}
\prod_{\ell=1}^{|{\mathfrak{u}}|-1} b^{\frac{-w_{\ell}}{2}}\right)^\frac{2\lambda}{1+\lambda}
\left(\prod_{j\in{\mathfrak{u}}} \alpha_j^{\frac{2\lambda}{1+\lambda}}\right)
\prod_{j\in{\mathfrak{u}}} \left(\left(\frac{2\,b_j}{1-\kappa}\right)^{2\lambda}\, b^{w_j}\, \rho (\lambda)/\alpha_j^{2\lambda}\right)^{\frac{1}{1+\lambda}}\\
&\le&
\left(\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \kappa^{\abs{{\mathfrak{u}}}}
\left(\prod_{\ell=1}^{|{\mathfrak{u}}|-1} b^{\frac{-w_{\ell}}{2}}\right) \prod_{j\in{\mathfrak{u}}} \alpha_j\right)^\frac{2\lambda}{1+\lambda} \\
&&\times
\left(\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \prod_{j\in{\mathfrak{u}}}
\left(\left(\frac{2\,b_j}{1-\kappa}\right)^{2\lambda}\, b^{w_j}\, \rho (\lambda)/\alpha_j^{2\lambda}\right)^{\frac{1}{1-\lambda}}
\right)^{\frac{1-\lambda}{1+\lambda}} = B^{\frac{2\lambda}{1+\lambda}} \cdot \widetilde{B}^{\frac{1-\lambda}{1+\lambda}} ,
\end{eqnarray*}
where we define
\[
B
:=
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \kappa^{\abs{{\mathfrak{u}}}}
\left(\prod_{\ell=1}^{|{\mathfrak{u}}|-1} b^{\frac{-w_{\ell}}{2}}\right) \prod_{j\in{\mathfrak{u}}} \alpha_j,
\quad
\widetilde{B}
:=
\sum_{{\mathfrak{u}}\subseteq\{1:s\}} \prod_{j\in{\mathfrak{u}}}
\left(\left(\frac{2\,b_j}{1-\kappa}\right)^{2\lambda}\, \frac{b^{w_j}\, \rho (\lambda)}{\alpha_j^{2\lambda}} \right)^{\frac{1}{1-\lambda}} .
\]
For the first factor we estimate
\begin{eqnarray*}
B
&\le&
\sum_{{\mathfrak{u}}:\, \abs{{\mathfrak{u}}}<\infty} \kappa^{\abs{{\mathfrak{u}}}} \prod_{\ell=1}^{|{\mathfrak{u}}|-1} b^{\frac{-w_{\ell}}{2}} \prod_{j\in{\mathfrak{u}}} \alpha_j
=
\sum_{k=1}^\infty \left( \kappa^{k} \prod_{\ell=1}^{k-1} b^{\frac{-w_{\ell}}{2}}\right)
\sum_{\substack{{\mathfrak{u}}:\, \abs{{\mathfrak{u}}}<\infty\\ \abs{{\mathfrak{u}}}=k}} \prod_{j\in{\mathfrak{u}}} \alpha_j \\
&\le&
\sum_{k=1}^\infty \left( \kappa^{k} \prod_{\ell=1}^{k-1} b^{\frac{-w_{\ell}}{2}}\right)\frac{1}{k!}
\sum_{{\boldsymbol{u}} \in{\mathbb{N}}} % natural numbers {1, 2, ...^k} \prod_{i=1}^k \alpha_{u_i}
=
\sum_{k=1}^\infty \left( \kappa^{k} \prod_{\ell=1}^{k-1} b^{\frac{-w_{\ell}}{2}}\right)\frac{1}{k!} \Sigma^k.
\end{eqnarray*}
By the ratio test, the latter expression is finite if we choose $(\alpha_j)_{j\ge 1}$ such that
$L:=\sup_{k\in{\mathbb{N}}} % natural numbers {1, 2, ...} \kappa\,b^{\frac{-w_k}{2}}(k+1)^{-1}=\kappa b^{\frac{-w_1}{2}}/2<1/\Sigma$.
Hence we assume that $(\alpha_j)_{j\ge 1}$ is chosen such that indeed $L<1/\Sigma$.
Note that $L$ is small if $\kappa$ is small, which means that $\Sigma$ can be allowed to
be large in this case. Consider now the term
\begin{eqnarray*}
\widetilde{B} &\le& \sum_{{\mathfrak{u}}:\, \abs{{\mathfrak{u}}}<\infty}
\prod_{j\in{\mathfrak{u}}}
\left(\left(\frac{2\,b_j}{1-\kappa}\right)^{2\lambda}\, b^{w_j}\, \rho (\lambda)/\alpha_j^{2\lambda}\right)^{\frac{1}{1-\lambda}}\\
&\le & \exp\left(\sum_{j=1}^\infty
\left(\left(\frac{2\,b_j}{1-\kappa}\right)^{2\lambda}\, b^{w_j}\, \rho (\lambda)/\alpha_j^{2\lambda}\right)^{\frac{1}{1-\lambda}}\right)\\
&\le& \exp \left(\sum_{j=1}^\infty \left(\frac{1}{1-\kappa}\right)^{\frac{2\lambda}{1-\lambda}}(\rho(\lambda))^{\frac{1}{1-\lambda}}4^\lambda
\left(b_j b^{w_j}\frac{1}{\alpha_j}\right)^{\frac{2\lambda}{1-\lambda}}\right)\\
&=& \exp\left(\left(1-\kappa\right)^{\frac{-2\lambda}{1-\lambda}}(\rho(\lambda))^{\frac{1}{1-\lambda}}4^\lambda
\sum_{j=1}^\infty \left(b_j b^{w_j}\alpha_j^{-1}\right)^{\frac{2\lambda}{1-\lambda}}\right).
\end{eqnarray*}
We require
\begin{equation}\label{eq:condAlambda}
L<1/\Sigma=1/\sum_{j=1}^\infty \alpha_j\quad\mbox{and}
\quad \sum_{j=1}^\infty \left(b_j b^{w_j}\alpha_j^{-1}\right)^{\frac{2\lambda}{1-\lambda}}<\infty.
\end{equation}
To this end, we choose
$\alpha_j:=\frac{\left(b_j b^{w_j}\right)^{p}}{\theta}$, where
$\frac{\theta}{L}>\sum_{j=1}^\infty \left(b_j b^{w_j}\right)^{p}$.
Then,
\begin{eqnarray}\label{eq:Alambdabound}
A_\lambda &\le & \left(\sum_{k=1}^\infty \left( \kappa^{k}
\prod_{\ell=1}^{k-1} b^{\frac{-w_{\ell}}{2}}\right)\frac{1}{k!}
\Sigma^k\right)^\frac{2\lambda}{1+\lambda}\nonumber\\
&&\times \exp\left(\frac{1-\lambda}{1+\lambda}
\left(\frac{1}{1-\kappa}\right)^{\frac{2\lambda}{1-\lambda}}(\rho(\lambda))^{\frac{1}{1-\lambda}}4^\lambda
\sum_{j=1}^\infty \left(b_j b^{w_j}\frac{1}{\alpha_j}\right)^{\frac{2\lambda}{1-\lambda}}\right)
\end{eqnarray}
as long as we choose $\lambda$ such that
\begin{equation}\label{eq:chooselambda}
\sum_{j=1}^\infty \left(b_j b^{w_j}\alpha_j^{-1}\right)^{2\lambda/(1-\lambda)}<\infty.
\end{equation}
We denote the upper bound in \eqref{eq:Alambdabound} by $\overline{A} (\lambda)$. Similarly to what
is done in \cite[Proof of Theorem 6.4]{KSS12}, we see that Condition \eqref{eq:chooselambda} is satisfied if
$\lambda\ge \frac{p}{2-p}$.
Again, similarly to \cite[Proof of Theorem 6.4]{KSS12} we see that the latter can be achieved by choosing
\begin{equation}\label{eq:lambdap}
\lambda_p=\begin{cases}
1/(2-2\delta)\quad\mbox{for some $\delta\in (0,1/2)$} & \mbox{if $p\in (0,2/3]$},\\
p/(2-p) & \mbox{if $p\in (2/3,1)$}.
\end{cases}
\end{equation}
Hence by choosing $\lambda$ equal to $\lambda_p$, we get an efficient bound
on $C_{{\boldsymbol{\gamma}},{\boldsymbol{w}},\lambda_p} = A_{\lambda_p}^{1 + 1/\lambda_p}$,
as long as the $w_j$ are chosen to guarantee convergence of $\sum_{j=1}^\infty \left(b_j b^{w_j}\right)^{p}$.
\section{Combined error bound}
The derivation in the previous section leads to the following result.
\begin{theorem}
Given the PDE in \eqref{eq:PDE} for which we characterized the regularity of the random field by a sequence of
$b_j$ with sparsity $p \in (0,1)$ and determined a sequence of $w_j$ such that $\sum_{j=1}^\infty (b_j \, b^{w_j})^p<\infty$, we can construct the generating vector for an $N$-point randomized lattice rule using the reduced CBC algorithm (Algorithm \ref{alg:RedCBCAlgPOD}), at the cost of $\mathcal{O}(\sum_{j=1}^{\min\{s,s^{\ast}\}} (m-w_j+j) \, b^{m-w_j})$ operations, such that, assuming that \eqref{def:kappa}, \eqref{eq:cond_fem} and $\frac{\kappa \, a_{0,\max}}{(1-\overline{\kappa}) \, a_{0,\min}} \sup_{j \ge s+1} b_j < 1$ hold, we obtain an upper bound
\begin{equation}\label{eq:combined_error2}
e^{\text{MSE}}_{N,s,h}(G(u)) \lesssim \left(\sup_{j \ge s+1} b_j \right)^2 + h^4 + \left( \frac2N \right)^{1/\lambda_p},
\end{equation}
where the implied constant is independent of $s$, $h$ and $N$.
\end{theorem}
Observe that if the $w_j$ increase sufficiently fast, the construction cost of Algorithm \ref{alg:RedCBCAlgPOD} does not depend anymore on the increasing dimensionality. Further note that the first term on the right-hand side of \eqref{eq:combined_error2} is small if $\sup_{j \ge s+1} b_j$ is small, and, since we assumed that $b_j$ must tend to zero by assumption \eqref{eq:assumpThm1}, we can shrink the first summand by choosing $s$ sufficiently large. By choosing $h$ sufficiently small, and $N$ sufficiently large, we can also make the other two summands in the overall error bound small.
Note that $\sup_{j\ge s+1} b_j\le \sum_{j\ge s+1} b_j$, and that \eqref{eq:assumpThm1}
yields $\sum_{j=1}^\infty b_j^p <\infty$, which implies that one can use the machinery developed in \cite{KSS12}
to obtain a cost analysis similar to \cite[Theorem 8.1]{KSS12}.
Note, in particular, that it is sufficient to choose $N$ of order
$\mathcal{O}(\varepsilon^{-\lambda_p /2})$, independently of $s$, to meet an error threshold of $\varepsilon$.
\section{Derivation of the fast reduced CBC algorithm}
\label{sec:appendix}
Finally in this last section the derivation of the fast reduced CBC algorithm for POD weights in Algorithm~\ref{alg:RedCBCAlgPOD} is given.
For prime $b$ and $m \in {\mathbb{N}}} % natural numbers {1, 2, ...$ let $N = b^m$.
Consider a generating vector $\widetilde{{\boldsymbol{z}}} = (b^{w_1} z_1,\ldots, b^{w_d} z_d)$ with $z_j \in \Z_{b^{m-w_j}}^\times$ and integer $0 \le w_j \le m$ for each $j = 1,\ldots,d$.
Furthermore, for an integer $0 \le w' \le m$, the squared worst-case error can be written as
\begin{align*}
e_{b^m,d}^2(\widetilde{{\boldsymbol{z}}})
&=
\frac{1}{b^m} \sum_{k \in \Z_{b^m}} \sum_{\ell=1}^{d} \sum_{\substack{{\mathfrak{u}} \subseteq \{1:d\} \\ |{\mathfrak{u}}| = \ell}}
\Gamma(\ell)
\prod_{j \in {\mathfrak{u}}} \gamma_j \, \omega\!\left(\frac{k \, b^{w_j} z_j \bmod b^m}{b^m}\right)
\\
&=
\frac{1}{b^m} \sum_{k \in \Z_{b^m}} \sum_{\ell=1}^{d} \sum_{\substack{{\mathfrak{u}} \subseteq \{1:d\} \\ |{\mathfrak{u}}| = \ell}}
\Gamma(\ell)
\prod_{j \in {\mathfrak{u}}} \gamma_j \, \omega\!\left(\frac{k \, z_j \bmod b^{m-w_j}}{b^{m-w_j}}\right)
\\
&=
\frac{1}{b^m} \sum_{k' \in \Z_{b^{m-w'}}} \sum_{\ell=1}^{d}
\underbrace{\sum_{t \in \Z_{b^{w'}}} \sum_{\substack{{\mathfrak{u}} \subseteq \{1:d\} \\ |{\mathfrak{u}}| = \ell}}
\Gamma(\ell)
\prod_{j \in {\mathfrak{u}}} \gamma_j \, \omega\!\left(\frac{(k' + t \, b^{m-w'}) \, z_j \bmod b^{m-w_j}}{b^{m-w_j}}\right)}_{=: q_{d,\ell,w'}(k') \text{ for } k' \in \Z_{b^{m-w'}}}
\\
&=
\frac{1}{b^m} \sum_{k' \in \Z_{b^{m-w'}}} \sum_{\ell=1}^{d} q_{d,\ell,w'}(k')
.
\end{align*}
We note that this holds for any integer $0 \le w' \le m$ and, in particular, for $w=0$ this is the vector being used in the normal fast CBC algorithm.
We now write the error in terms of the previous error, as is standard for CBC algorithms, by splitting the expression into subsets ${\mathfrak{u}} \subseteq \{1:d\}$
for which $d \not\in {\mathfrak{u}}$ and $d \in {\mathfrak{u}}$, to obtain
\begin{align*}
&
e_{b^m,d}^2(\widetilde{{\boldsymbol{z}}})
=
e^2_{b^m,d-1}(\widetilde{z}_1,\ldots,\widetilde{z}_{d-1}) +
\\
&\quad
\frac{1}{b^m} \sum_{k \in \Z_{b^m}}
\sum_{\ell=0}^{d-1} \frac{\Gamma(\ell+1)}{\Gamma(\ell)}
\!\!\sum_{\substack{{\mathfrak{u}} \subseteq \{1:d-1\} \\ |{\mathfrak{u}}| = \ell}} \Gamma(\ell)
\prod_{j \in {\mathfrak{u}}} \gamma_j \, \omega\!\left(\frac{k \, z_j \bmod b^{m-w_j}}{b^{m-w_j}}\right)
\,
\gamma_d \, \omega\!\left(\frac{k \, z_d \bmod b^{m-w_d}}{b^{m-w_d}}\right)
.
\end{align*}
Since the choice of $z_d \in \Z_{b^{m-w_d}}^\times$ is modulo~$b^{m-w_d}$, we can make a judicious choice for splitting up $k = k' + t\,b^{m-w_d}$
for which the effect of dimension~$d$ (for a choice of $z_d$) is then constant for all $t \in \Z_{b^{w_d}}$. We obtain
\begin{align}\label{eq:e2-split}
e_{b^m,d}^2(\widetilde{{\boldsymbol{z}}})
&=
e^2_{b^m,d-1}
+
\frac{1}{b^m} \sum_{k' \in \Z_{b^{m-w_d}}}
\sum_{\ell=0}^{d-1} \frac{\Gamma(\ell+1)}{\Gamma(\ell)}
q_{d-1,\ell,w_d}(k')
\,
\gamma_d \, \omega\!\left(\frac{k' \, z_d \bmod b^{m-w_d}}{b^{m-w_d}}\right)
.
\end{align}
Then we observe that for all $0 \le w_{d-1} \le w_d \le m$, with $k' \in \Z_{b^{m-w_d}}$, writing $t = t' + t'' \, b^{w_d-w_{d-1}} \in \Z_{b^{w_d}}$
with $t' \in \Z_{b^{w_d-w_{d-1}}}$ and $t'' \in \Z_{b^{w_{d-1}}}$, leads to
\begin{align*}
q_{d-1,\ell,w_d}(k')
&=
\sum_{t' \in \Z_{b^{w_d-w_{d-1}}}} \sum_{t'' \in \Z_{b^{w_{d-1}}}}
\\
&\qquad\quad
\sum_{\substack{{\mathfrak{u}} \subseteq \{1:d-1\} \\ |{\mathfrak{u}}| = \ell}}
\Gamma(\ell)
\prod_{j \in {\mathfrak{u}}} \gamma_j \, \omega\!\left(\frac{(k' + (t' + t'' \, b^{w_d-w_{d-1}}) \, b^{m-w_d}) \, z_j \bmod b^{m-w_j}}{b^{m-w_j}}\right)
\\
&=
\sum_{t' \in \Z_{b^{w_d-w_{d-1}}}} q_{d-1,\ell,w_{d-1}}(k' + t' \, b^{m-w_d})
,
\end{align*}
where $k'' = k' + t' \, b^{m-w_d} \in \Z_{b^{m-w_{d-1}}}$ as required for $q_{d-1,\ell,w_{d-1}}(k'')$.
Note that this is the property of the ``fold and sum'' operator as introduced in~\eqref{eq:fold-and-sum} and mentioned there.
Using matrix-vector notation, we rewrite the expression in \eqref{eq:e2-split} for all $z_d \in \Z_{b^{m-w_d}}^\times$ as
\begin{align*}
{\boldsymbol{e}}_{b^m,d}^2
&=
e_{b^m,d-1}^2
+
\frac{\gamma_d}{b^m} \, \Omega_{b^{m-w_d}}
\left( \sum_{\ell=0}^{d-1} \frac{\Gamma(\ell+1)}{\Gamma(\ell)}
\left[ P_{w_d,w_{d-1}}^m \, {\boldsymbol{q}}_{d-1,\ell,w_{d-1}} \right] \right) ,
\end{align*}
where ${\boldsymbol{e}}_{b^m,d}^2 \in \R^{\varphi(b^{m-w_d})}$ is the vector with components $e_{b^m,d}^2(b^{w_1} z_1, \ldots, b^{w_d} z_d)$ for all $z_d \in \Z_{b^{m-w_d}}^\times$.
After $z_d$ has been selected we can calculate (for $\ell=1,\ldots,d$)
\begin{align*}
{\boldsymbol{q}}_{d,\ell,w_d}
=
\left[ P_{w_d,w_{d-1}}^m \, {\boldsymbol{q}}_{d-1,\ell,w_{d-1}} \right]
+
\frac{\Gamma(\ell)}{\Gamma(\ell-1)} \gamma_d \,
\Omega_{b^{m-w_d}}(z_d,:) \, {.*} \,
\left[ P_{w_d,w_{d-1}}^m \, {\boldsymbol{q}}_{d-1,\ell-1,w_{d-1}} \right] .
\end{align*}
In Algorithm~\ref{alg:RedCBCAlgPOD} the vectors ${\boldsymbol{q}}_{j,\ell,w_j}$ are denoted by just ${\boldsymbol{q}}_{j,\ell}$.
|
{
"timestamp": "2019-03-01T02:18:07",
"yymm": "1902",
"arxiv_id": "1902.11068",
"language": "en",
"url": "https://arxiv.org/abs/1902.11068"
}
|
\section{Introduction}
\emph{Deep neural networks} (DNNs) have reached an unprecedented level of predictive accuracy in several real-world application domains (e.g., text processing \cite{conneau2016very,zhang2015character}, image recognition \cite{he2016deep,Huang2017DenselyCC}, and video analysis \cite{liu2019lstm,xu2018dual}) due to their capability of approximating any universal function \cite{goodfellow2016deep}. However, DNNs are often difficult to train due in a large part to the vanishing gradient phenomenon~\cite{glorot2010understanding,ioffe2015batch}.
The \emph{residual network} (ResNet)~\cite{he2016deep} is proposed to alleviate this issue using its key component known as the skip connection, which creates ``bypass'' path for information propagation \cite{he2016identityv2}.
Nevertheless, it remains challenging to achieve robustness in training a very deep DNN.
As the DNN becomes deeper, it requires more careful tuning of the model hyperparameters (e.g., learning rate) and initialization of the model weights to perform well. This issue can be mitigated using normalization techniques such as \emph{batch normalization} (BN)~\cite{ioffe2015batch}, layer normalization~\cite{lei2016layer}, and group normalization~\cite{wu2018group}, among which BN is most widely used.
BN normalizes the inputs of each layer to enable a robust training of DNN. It has been shown that BN provides a smoothing effect on the optimization landscape \cite{2018arXiv180511604S}, thus ameliorating the issue of a vanishing gradient \cite{ioffe2015batch}.
Unfortunately, we have empirically observed that even when a very deep DNN is coupled with BN, it can potentially (and surprisingly) experience an exploding gradient in its deeper layers, thus impairing
its robustness in training.
As a result, our experiments in Section~\ref{exp:training-robust} reveal that BN does not help in training ResNets with depths $49$ and $110$ robustly on the \textit{AG-NEWS} text and the \textit{CIFAR-$10$} vision datasets, respectively.
Furthermore, noisy data (e.g., images with mosaics and texts with spelling errors) can adversely impact the training procedure of a DNN with BN, thus degrading its robustness in generalization.
This paper presents a simple yet principled approach to boosting the robustness of ResNet in both training and generalization, which is motivated by a dynamical systems perspective~\cite{ChenNeuralODE,lu2017beyond,ruthotto2018deep}. Namely, we can view a DNN from the perspective of a partial differential equation (PDE). This naturally inspires us to characterize ResNet by an explicit Euler method. Theoretically, we show that the step factor $h$ in the Euler method can serve to control the robustness of ResNet in both training and generalization. Specifically, we prove that a small $h$ can benefit the training robustness from the view of back-propagation; from the view of forward-propagation, the small $h$ can help the generalization robustness. We conduct experiments on both vision and text datasets, i.e., \mbox{\textit{CIFAR-10}}~\cite{krizhevsky2009learning} and \textit{AG-NEWS}~\cite{zhang2015character}. The comprehensive experiments demonstrate that small $h$ can benefit the training and generalization robustness. We confirm that the use of a small $h$ can benefit the training robustness without using BN as well. To sum up, the empirical results validate the efficacy of small $h$ parameter in the goal towards robust ResNet.
\section{Related literature}
Before delving into robust ResNet (Section \ref{section:Towards Robust ResNet}), we provide an overview of several prerequisites, including ResNet, batch normalization and partial differential equations.
\subsection{Brief summary of ResNet}
Residual network (ResNet) is a variant of the deep neural network, which exhibits compelling accuracy and convergence properties~\cite{he2016deep}. Its key component is the skip connection. Thus, ResNet consists of many stacked residual blocks, and its general form is shown below:
\begin{align*}
& \mathbf{y}_n = \mathbf{x}_n + \mathcal{F} (\mathbf{x}_n);\\
& \mathbf{x}_{n+1} = I(\mathbf{y}_n),
\end{align*}
where $\mathbf{x}_n$ and $\mathbf{x}_{n+1}$ are the input and output of the $n$-th block, $\mathcal{F}$ is a residual block containing feature transformation, e.g., convolutional operation or affine transformation, and $I$ is an element-wise operation, e.g., ReLU function~\cite{Nair2010} or identity mapping~\cite{he2016identityv2}.
Based on the core idea of the skip connection, variants of ResNet have been proposed for specific tasks, e.g., DenseNet for image recognition~\cite{Huang2017DenselyCC}, and VDCNN with shortcut connections for text classification~\cite{conneau2016very}.
\subsection{Batch normalization}
Batch normalization (BN) has been widely adopted for training deep neural networks~\cite{ioffe2015batch}, which normalizes layer input over each training mini-batch. Specifically, for the input of a layer with dimension $d$, $\mathbf{x} = (x^{(1)},\ldots,x^{(k)},\ldots,x^{(d)})$, BN first normalizes each scalar feature $\hat{x}^{(k)} = \frac{x^{(k)} - \mu^{(k)}}{\sigma^{(k)}}$ independently, where $\mu^{(k)} = \mathrm{E}[x^{(k)}]$ and $\sigma^{(k)} = \sqrt{\mathrm{Var}[x^{(k}]}$ are computed over a mini-batch of size $m$. Then, it performs an affine transformation of each normalized value by $\text{BN}(x^{(k)}) = \gamma^{(k)}\hat{x}^{(k)} + \beta ^{(k)}$, where $\gamma = (\gamma^{(1)},\ldots,\gamma^{(k)},\ldots,\gamma^{(d)})$ and $\beta = (\beta^{(1)},\ldots,\beta^{(k)},\ldots,\beta^{(d)}) $ are learned during training.
BN has been shown to smooth the optimization landscape of deep neural network \cite{2018arXiv180511604S}. This offers more stable gradients for robust training of deep neural network, thus effectively ameliorating the vanishing gradient problem~\cite{ioffe2015batch}.
\subsection{Characterizing deep neural network by a partial differential equation}
Deep neural networks (DNN) represent the underlying nonlinear feature transformations, so that the transformed features can be matched with their target values, such as categorical labels for classification and continuous quantities for regression. Namely, DNN transforms the input data through multiple layers, and the feature in the last layer should be linearly separable \cite{haber2017stable}.
Turning to the dynamical systems view, a partial differential equation (PDE) can characterize the motion of particles \cite{atkinson2008introduction,ascher2008numerical}. Motivated by this viewpoint, we can characterize the feature transformations in DNN as a system of the first-order PDE, which can be formulated as:
\begin{definition} Feature transformations in DNN can be characterized as the first-order partial differential equation (PDE):
\begin{equation}
\dot{{\mathbf{x}}} = f(t, \mathbf{x}),
\end{equation}
where $\dot{{\mathbf{x}}} = \frac{\partial \mathbf{x}}{\partial t} $, $f: T \times X \rightarrow R^d$, $T \subseteq R^+$, and $X \subseteq R^d$.
$t \in T$ is the time along the feature transformations, in which $t \geq 0 $.
$\mathbf{x} \in X $ is the feature vector of dimension $d$.
\end{definition}
Given an initial data $\mathbf{x}_{0}$ as the input, a PDE gives rise to the initial value problem (\textbf{IVP}):
\begin{equation}
\dot{\mathbf{x}} = f(t, \mathbf{x} ) ;\ \mathbf{x}(0) = \mathbf{x}_{0},
\end{equation}
where $\mathbf{x}_{0} $ is a specified initial input of feature vector. $\mathbf{x}(t)$ is the transformed feature at the time $t$.
\begin{figure}[tp!]
\includegraphics[scale = 0.12]{vis_Euler_method/PDE.pdf}
\caption{Given an IVP, we provide its analytic solution and approximation by an explicit Euler method with different step factors $h$. }
\label{fig:vis_euler_method}
\end{figure}
A solution to an IVP is a function of time $t$. For example, we have IVP $\dot{x} = -2.3 x, x(0) = 1.0$. Thus, we can easily derive its analytic solution $x(t) = e ^{-2.3 t}$ as shown in Figure~\ref{fig:vis_euler_method}. However, for more complicated PDEs, it is not always feasible to obtain an analytic solution. Rather, their solutions are approximated by numerical methods, of which the explicit Euler method (shown in Definition \ref{Euler-Method} in next section) is the most classic example.
\section{Towards Robust ResNet} \label{section:Towards Robust ResNet}
Residual network (ResNet) is one exemplar of DNN. ResNet can be described by the explicit Euler method (Definition~\ref{Euler-Method}), which naturally brings us a Euler-viewed ResNet (Eq.~\eqref{prol-Euler-viewed_resnet}). Inspired by the stability of the Euler method, we emphasize the efficacy of the step factor $h$ for ResNet, which enables ResNet to have smoother feature transformations. Such smoothness means that feature transitions take a small change at every adjacent block during the forward propagation. This smoothness has two-fold benefits: preventing information explosion over the depth (Section~\ref{sec:training-robustness}), and filtering noise in the input features (Section~\ref{sec:generalization-robustness}). All these together drive the significant robustness gains in the ResNet.
\subsection{Connections of ResNet with the Euler method}
Consider the following definition of the explicit Euler method.
\begin{definition}\label{Euler-Method} The explicit Euler method \cite{atkinson2008introduction,ascher2008numerical} can be represented as follows.
\begin{equation}
\mathbf{x}_{n+1} = \mathbf{x}_{n} + h {f}(t_n, \mathbf{x}_n),
\label{eq:h}
\end{equation}
where $\mathbf{x}_{n}$ is the approximation of $\mathbf{x}(t_n)$.
\end{definition}
Given the solution $\mathbf{x}(t_n)$ at some time $t_n$, the PDE $\dot{\mathbf{x}} = f(t, \mathbf{x} )$ tells ``in which direction to continue". At time $t_n$, the explicit Euler method computes this direction $f(t_n, \mathbf{x}_n)$, and follows it when a small time step changes ($t_n \rightarrow t_n + h$).
In order to obtain the reasonable approximation, the step factor $h$ in Euler's method has to be chosen to be ``small enough''. Its magnitude depends on the PDEs and its specified initial inputs. For instance, in Figure \ref{fig:vis_euler_method}, we leverage Euler method with various step factors $h$ to approximate the IVP $\dot{x} = -2.3x$, $x(0) = 1.0$.
It is clearly shown that the approximation of $x$ fluctuates more with large $h$, hence the smaller $h$ approximates the true function better.
\paragraph{Euler-viewed ResNet:} Taking $h = 1.0$ and ${f}(t_n, \mathbf{x}_n) =\mathcal{F} (\mathbf{x}_n )$, we can use the Euler method to characterize the forward propagation of the original ResNet structure \cite{haber2017stable}. Specifically,
to generalize ResNet by the Euler method (Eq.~\eqref{eq:h}), we characterize the forward propagation of an Euler-viewed ResNet as follows.
Following the conventional analysis \cite{he2016identityv2}, ResNet is a stacked structure \cite{he2016deep}, where its residual block with step factor $h \in R^+$ performs the following computation:
\begin{align} \label{prol-Euler-viewed_resnet}
& \mathbf{y}_n = \mathbf{x}_n + h \mathcal{F} (\mathbf{x}_n, W_n, \text{BN}_n); \nonumber \\
& \mathbf{x}_{n+1} = I(\mathbf{y}_n).
\end{align}
At layer $n$, $\mathbf{x}_n$ is the input features to the $n$-th Residual block $\mathcal{F}$. $\text{BN}_n$ are the parameters of batch normalization, which applies directly after (or before) the transformation function $W_n$, i.e., convolutional operations or affine transformation matrix. The function $I$ is an element-wise operation, e.g., identity mapping or ReLu.
\iffalse
\begin{theorem} (Stability of Euler Method) \cite{ascher2008numerical} The explicit Euler method is stable if $h$ is chosen sufficiently small, and satisfies
\begin{equation}
\max |1 + h \{ \lambda ({J}_i) \} | \leq 1,
\label{eq:eig}
\end{equation}
where $\{\lambda ({J}_i)\}$ is the set of eigenvalue of the Jacobian matrix of ${f}(t_i, \mathbf{x}_i)$ with respect to the features $\mathbf{x}_i$, at the $i$th approximation step, and the maximum is taken over the real part of the eigenvalues in the set.
\end{theorem}
\fi
As analysed in the chapter ``analysis of the Euler method''~\cite{butcher2016numerical}, for any fixed time interval $t$, the Euler method approximation is better refined with smaller $h$. However, it is not obvious whether small $h$ can helps on robustness of deep ResNet in terms of training and generalization.
To answer this question, we prove that the reduced step factor $h$ in Eq.~\eqref{prol-Euler-viewed_resnet} can boost its robustness. Specifically, the benefits of this new parameter $h$ are as follows. (a) Training robustness (Section ~\ref{sec:training-robustness}): Small $h$ helps the information back-propagation during the training, and thus prevents its explosion over depth. (b) Generalization robustness (Section~\ref{sec:generalization-robustness}): With greater smoothness in feature forward-propagation, the noise in the features is reduced rather than amplified, thus mitigating the negative effects of noise and enhancing generalization. We will now theoretically explore the efficacy of small $h$ for training and generalization robustness.
\subsection{Training robustness: Importance of small $h$ on information back-propagation}\label{sec:training-robustness}
To simplify the analysis, let $I$ be the identity mapping operation from Eq.~\eqref{prol-Euler-viewed_resnet}, i.e., $\mathbf{x}_{n+1} = \mathbf{y}_n$. Recursively, we have
\begin{equation}
\mathbf{x}_N = \mathbf{x}_n + h \sum_{i=n}^{N-1} \mathcal{F} (\mathbf{x}_i, W_i, BN_i)\label{recursive_resblock},
\end{equation}
for any deeper block $N$ and any shallower block $n$.
Eq.~\eqref{recursive_resblock} is used to analyse the information back-propagation. Denoting the loss function as $L$, by the chain rule, we have:
\begin{equation}
\frac{\partial L}{\partial \mathbf{x}_n} =
\frac{\partial L}{\partial \mathbf{x}_N} \frac{\partial \mathbf{x}_N}{\partial \mathbf{x}_n} =
\frac{\partial L}{\partial \mathbf{x}_N} \big (1+ h \frac{\partial}{\partial \mathbf{x}_n} \sum_{i=n}^{i= N-1} \mathcal{F} (\mathbf{x}_i, W_i, BN_i) \big). \label{back_prop}
\end{equation}
Eq.~\eqref{back_prop} shows that the back-propagated information $\frac{\partial L}{\partial \mathbf{x}_n}$ can be decomposed into two additive terms: a term $\frac{\partial L}{\partial \mathbf{x}_N} $ (the first term) that propagates information directly without going through any weight layers, and a term
$ h \frac{\partial L}{\partial \mathbf{x}_N} \big ( \frac{\partial}{\partial \mathbf{x}_n} \sum_{i=n}^{i= N-1} \mathcal{F} (\mathbf{x}_i, W_i, BN_i) \big )$ (the second term), which propagates through weight layers between $n$ and $N$.
The first term as stated by \cite{he2016identityv2} ensures that information $\frac{\partial L}{\partial \mathbf{x}_n}$ does not vanish. However, the second term can blow-up the information $\frac{\partial L}{\partial \mathbf{x}_n}$, especially when the weights of layers are large. The standard approach for tackling this is to apply batch normalization (BN) after transforming functions \cite{ioffe2015batch}.
Let us first examine the effects of BN. For analysis purposes, we examine one residual block, taking $N = n+1$, $\mathcal{F} (\mathbf{x}_n) = \text{BN}(W_n \mathbf{x}_n)$, $\mathbf{x}'_n = W_n \mathbf{x}_n $, and $\text{BN}(\mathbf{x}'_n) = \gamma \frac{\mathbf{x}'_n - \Bar{\mu}}{\Bar{\sigma}} + \beta$, where BN is an element-wise operation. For a batch size of $m$, there are $m$ transformed $\mathbf{x}'_n$ with mean vector $\Bar{\mu} = (\mu_1, \mu_2, \dots, \mu_m)^T$ and standard deviation vector $\Bar{\sigma} = (\sigma_1, \sigma_2, \dots, \sigma_m)^T$. Thus, we get
\begin{equation}
\frac{\partial \mathcal{F} }{\partial \mathbf{x}_n} = W_n \frac{\gamma}{\Bar{\sigma}}. \label{one_block_get_gradient}
\end{equation}
Taking $\sigma_n$ as the smallest one among elements in $\Bar{\sigma }$, we get
\begin{equation}
|| \frac{\partial \mathcal{F} }{\partial \mathbf{x}_n} || \leq \frac{\gamma}{\sigma_n} ||W_n||.
\end{equation}
To sum up, we have
\begin{equation}
\frac{\partial L}{\partial \mathbf{x}_n} \leq
\frac{\partial L}{\partial \mathbf{x}_{n+1}} \big (1+ h (\frac{\gamma}{\sigma_n} ||W_n||) \big ). \label{one_block_judge_depth}
\end{equation}
As observed by \cite{2018arXiv180511604S}, $\Bar{\sigma}$ tends to be large, thus BN gives the positive effect of constraining the explosive back-propagated information (Eq.~\eqref{one_block_judge_depth}). However, when networks go deeper, $\Bar{\sigma}$ tends to be highly uncontrolled in practice, and the back-propagated information is still accumulated over the depth and can again blow up. For example, Figure~\ref{judge_depth} (Section~\ref{exp:training-robust}) shows that, when we train depth-110 ResNet on \textit{CIFAR-10} and depth-49 ResNet on \textit{AG-NEWS}, the performance is significantly worse compared to their shallower counterparts. Thus, a reduced $h$ can serve to re-constrain the explosive back-propagated information.
Hence, even without BN, reducing $h$ can also serve to stabilize the training procedure. In other words, our reduced $h$ offers enhanced robustness even without BN. We carry out experiments supporting this observation in Section \ref{section:judge_small_h_without_BN}.
Let us take a special case to illustrate the importance of small $h$ on information back-propagation.
\begin{proposition}
let $n=0$ and $N = D$, where $D$ is the number of last residual block.
Supposing at any transformation block $i$, we have $|| \frac{\partial \mathcal{F}}{\partial \mathbf{x}_i} || \leq \mathcal{W}$, where $\mathcal{W}$ upper bounds the effects of BN, i.e., $\frac{\gamma}{\sigma}||W|| \leq \mathcal{W}$.
Then, we have $ \frac{\partial L}{\partial \mathbf{x}_0} \leq \frac{\partial L}{\partial \mathbf{x}_D} (1 - h + h (1+\mathcal{W})^D)$.
\end{proposition}
\begin{proof}
\begin{align}
\frac{\partial L}{\partial \mathbf{x}_0} &= \frac{\partial L}{\partial \mathbf{x}_D} \big (1+ h \frac{\partial}{\partial \mathbf{x}_n} \sum_{i=0}^{i= D-1} \mathcal{F} (\mathbf{x}_i) \big ); \\ \nonumber
\frac{\partial L}{\partial \mathbf{x}_0} &\leq \frac{\partial L}{\partial \mathbf{x}_D} (1+ h \sum_{i=0}^{D-1}(\mathcal{W} (1+\mathcal{W})^i) ) \\ \nonumber
&= \frac{\partial L}{\partial \mathbf{x}_D} (1-h + h (1+\mathcal{W})^D ).
\end{align}
\end{proof}
Note that $1+\mathcal{W}$ is bigger than 1, and the back-propagated information explodes exponentially w.r.t the depth $D$. This hurts the training robustness of ResNet. However, reducing $h$ can give extra control of the back-propagated information, since the increasing term $(1+\mathcal{W})^D$ will be constrained by $h$ and the back-propagated information $\frac{\partial L}{\partial \mathbf{x}_0}$ will not explode. This guides us that, when ResNet becomes deeper, we should reduce $h$. This also informs us that, even without BN, training a deeper network is still possible.
\subsection{Generalization robustness: Importance of small $h$ on information forward-propagation}\label{sec:generalization-robustness}
It is obvious that the reduced $h$ gives extra control of the feature transformations in ResNet. Namely, it makes feature transformations smoother, i.e., $||\mathbf{x}_{n+1} -\mathbf{x}_{n} || $ is smaller, as features go through ResNet block by block. In other words, it forces $\mathbf{x}_{n+1}$ to take a reduced change compared with $\mathbf{x}_{n}$.
More importantly, as features forward propagate through the deep ResNet, negative effects of the input noise are reduced and constrained over the depth. We theoretically show that a reduced $h$ can help stabilize the output $f (\mathbf{x})\in R^m$ of ResNet against the input noise, where $\mathbf{x}$ can be a vector of pixels or vectorization of texts. Suppose we have a perturbed copy $\mathbf{x}_{\epsilon}$ of $\mathbf{x}$, we expect $f(\mathbf{x}_{\epsilon})$ is close to $f(\mathbf{x})$, that is
\begin{equation}
\forall \mathbf{x}: \mathbf{x} \approx \mathbf{x}_{\epsilon} \implies
f(\mathbf{x}) \approx f(\mathbf{x}_{\epsilon}). \nonumber
\end{equation}
Let $\mathbf{x}_N$ be the transformed feature of ResNet starting at input feature $\mathbf{x}_0$, and $\mathbf{x}^{\epsilon}_{N}$ starting at $\mathbf{x}^{\epsilon}_0$, where $\mathbf{x}^{\epsilon}_0$ is a perturbed copy of $\mathbf{x}_0$, i.e., $||\mathbf{x}^{\epsilon}_0 - \mathbf{x}_0|| \leq \epsilon$.
\begin{align} \label{theoretical_judge_noise}
||\mathbf{x}^{\epsilon}_{N} - \mathbf{x}_N || &=
|| \mathbf{x}^{\epsilon}_0 + h \sum_{i=0}^{N-1}\mathcal{F}(\mathbf{x}^{\epsilon}_i) - \mathbf{x}_0 - h \sum_{i=0}^{N-1} \mathcal{F}(\mathbf{x}_i) || \\ \nonumber &=
||\epsilon + h (\sum_{i=0}^{N-1} \mathcal{F}(\mathbf{x}^{\epsilon}_i) - \sum_{i=0}^{N-1}\mathcal{F}(\mathbf{x}_i) ) || \\ \nonumber &\leq
\epsilon + h \sum_{i=0}^{N-1} || \mathcal{F}(\mathbf{x}^{\epsilon}_i) - \mathcal{F}(\mathbf{x}_i)||.
\end{align}
Eq.~\eqref{theoretical_judge_noise} shows that the noise with input features is amplified along the depth of ResNet. However, with the introduction of a reduced $h$ (e.g., from 1 to 0.1), we limit the noise amplification, and provide the extra capacity for filtering out the input noise.
Let us take a special case to illustrate the importance of small $h$ on information forward-propagation.
\begin{proposition}
Consider a deeper block $N = \mathrm{D}$ as the last residual block in ResNet, and suppose at any intermediate transformation $i$, $|| \mathcal{F}(\mathbf{x}^{\epsilon}_i) - \mathcal{F}(\mathbf{x}_i) || \leq \mathcal{W} $. The noise at layer $D$ is denoted by $||\mathbf{x}^{\epsilon}_{D} - \mathbf{x}_D|| = \epsilon_{D}$, thus we have
\begin{equation} \label{eliminate_noise}
\epsilon_{D} \leq \epsilon + hD \mathcal{W}.
\end{equation}
\end{proposition}
Its proof follows directly from the Eq.~\eqref{theoretical_judge_noise}.
Eq.~\eqref{eliminate_noise} informs us that the small $h$ can compensate the negative effects of noise accumulated over the depth $D$. In other words, when ResNet is deeper (larger $D$), we expect $h$ to be smaller; while when ResNet is shallower, we allow $h$ to be larger.
Here, we should be informed that, for a given depth of ResNet, we cannot reduce step factor $h$ infinitely small. In the limiting case, if we take $h = 0$, there would be no transformations from initial feature $\mathbf{x}_0$ to the final state. In this case, the noise during the transformation is perfectly bounded, but it also smooths out all transformations.
In addition, if $h$ is too small, we probably need more layers (increase depth) of ResNet to get enough approximations, so that the transformed features can be matched with their target values. In this case, we increase the computational cost by introducing more layers.
\section{Experiments}
In this section, we conduct experiments on the vision dataset \textit{CIFAR-10} \cite{krizhevsky2009learning} and the text dataset \textit{AG-NEWS} \cite{zhang2015character}.
We also employ a synthetic binary dataset \textit{TWO-MOON} in Section~\ref{exp:generalization-robustness} to illustrate input noise filtering along the forward propagation. We fix our step factor $h=0.1$ and compare it with the original ResNet, which corresponds to $h=1.0$, in Sections \ref{exp:training-robust} and \ref{exp:generalization-robustness}. We discuss how to select the step factor $h$ in Section \ref{exp:selection-of-h}.
For the vision dataset \textit{CIFAR-10}, the residual block $\mathcal{F}$ contains two 2-D convolutional operations~\cite{he2016deep}. For the text dataset \textit{AG-NEWS}, the residual block $\mathcal{F}$ contains two 1-D convolutional operations~\cite{conneau2016very}. For the synthetic dataset \textit{TWO-MOON}, the residual block $\mathcal{F}$ contains two affine transformation matrices.
To tackle the dimensional mismatch in the shortcut connections of ResNet, we adopt the same practice as in ~\cite{he2016deep} for \textit{CIFAR-10} and ~\cite{conneau2016very} for \textit{AG-NEWS}, which use convolutional layers of the kernel of size one to match the dimensions.
\subsection{Small $h$ helps training robustness}\label{exp:training-robust}
\paragraph{Small $h$ helps on deeper networks:}
\begin{figure} [tp!]
\includegraphics[scale = 0.12]{{cifar-10/no_noise/SGD_Momentum_judge_depth/cifar-10}.pdf} \includegraphics[scale=0.12]{{nlp_plots/nlp_torch_stability/nlp_no_noise_depth/ag_news}.pdf}
\caption{To verify the effectiveness of small $h$ on increasing the robustness of the training procedure of deep ResNet, we compare ResNet with reduced step $h$ ($h=0.1$) and the original ResNet (corresponding to $h=1.0$) over various depths (other hyperparameters remain the same). Each configuration has 5 trials with different random seeds. We provide median test accuracy with the number of epochs on the datasets. The standard deviation is plotted as the shaded color.}
\label{judge_depth}
\end{figure}
Figure~\ref{judge_depth} shows that, in both vision and text datasets, our method ($h = 0.1$) outperforms the existing method (corresponding to $h = 1.0$), especially when the networks go deeper. In particular, in ResNet with depth-110 for \textit{CIFAR-10} and ResNet with depth-49 for \textit{AG-NEWS}, the existing method ($h = 1.0$) fails to train the networks properly, resulting in $45\%$ and $25\%$ of test accuracy, respectively.
It is noted that, the existing method degrades over the increasing depth of ResNet. For example, in \textit{CIFAR-10} experiments, ResNet with depth-56 and depth-110 perform worse than their counterparts ResNet with depth-32 and depth-44. In \textit{AG-NEWS} experiments, ResNet with depth-29 and depth-49 perform worse than ResNet with depth-9 and depth-17. However, our method ($h = 0.1$) reduces the variance of test accuracy over the increasing depth. For example, in \textit{CIFAR-10} experiments, our method shows a lower variance over the depth, with test accuracy improving from $70\% $ to $79\%$ from depth 32 to 110.
The blue shaded area ($h = 0.1$) is much thinner compared with the red one ($h = 1.0$). This shows that our method ($h = 0.1$) has a smaller variance of the test accuracy over different random seeds.
This demonstrates that small $h$ offers training robustness.
To sum up, as theoretically shown in Section \ref{sec:training-robustness}, the reduced $h$ can prevent the explosion of back-propagated information over the depth, thus making the training procedure of ResNet more robust.
\paragraph{Small $h$ helps networks without BN:}
\label{section:judge_small_h_without_BN}
Figure~\ref{judge_BN} shows that, without using BN, training plain ResNet ($h=1.0$) is unstable and exhibits large variance for both vision \textit{CIFAR-10} and text \textit{AG-NEWS} datasets. Particularly, without using BN, even training a shallow ResNet (depth-32) on \textit{CIFAR-10} fails at almost every training trail. However, with reduced $h = 0.1$, training performance improves significantly and exhibits low variance. As theoretically shown in Section \ref{sec:training-robustness}, reduced $h$ has beneficial effects on top of BN. This can help the back-propagated information by preventing its explosion, and thus enhance the training robustness of deep networks.
\begin{figure}[tp!]
\includegraphics[scale = 0.12]{{cifar-10/no_noise/SGD_Momentum_judge_BN/cifar-10}.pdf}
\includegraphics[scale=0.12]{{nlp_plots/nlp_torch_stability/nlp_no_noise_BN/ag_news}.pdf}
\caption{
To verify the effectiveness of small $h$ on improving robustness of the training procedure without batch normalization (BN), we compare ResNet with reduced step $h$ ($h = 0.1$) and the original ResNet (corresponding to $h = 1.0$) without BN (other hyperparameters remain the same). Each configuration has 5 trials with different random seeds. We provide median test accuracy with the number of epochs on the datasets. The standard deviation is plotted as the shaded color.}
\label{judge_BN}
\end{figure}
\subsection{Small $h$ helps generalization robustness}
\label{exp:generalization-robustness}
\paragraph{Synthetic data for noise filtering:} To give insights on why small $h$ can help the generalization robustness of ResNet, let us first consider a synthetic data example of using ResNet for binary classification task, namely separating noisy ``red and blue'' points in a 2-D plane. We train a vanilla ResNet with $h = 1.0$ and ResNet with $h = 0.1$ on the middle of Figure~\ref{fig:moon}, and perform the feature transformations on the test set (the right of Figure~\ref{fig:moon}).
\begin{figure}[h!]
\centering
\includegraphics[scale=0.23]{vis_moon_data/moon_clean.pdf}
\includegraphics[scale = 0.23]{vis_moon_data/train_moon.pdf}
\includegraphics[scale = 0.23]{vis_moon_data/test_moon.pdf}
\caption{Left: Synthetic binary data \textit{TWO-MOON} (clean underlying data). Middle: Noisy training data with perturbed input features. Right: Noisy test data. Both noisy training and test data come from the same underlying distribution with the same noise-adding process.}
\label{fig:moon}
\end{figure}
\begin{figure}[h!]
\includegraphics[scale = 0.15]{vis_moon_data/res_h_1.0/0.pdf}
\includegraphics[scale = 0.15]{vis_moon_data/res_h_1.0/5.pdf}
\includegraphics[scale = 0.15]{vis_moon_data/res_h_1.0/10.pdf}
\includegraphics[scale = 0.15]{vis_moon_data/res_h_1.0/90.pdf}
\includegraphics[scale = 0.15]{vis_moon_data/res_h_1.0/100.pdf}
\caption{To illustrate the feature transformations through the vanilla ResNet (no BN, $h=1.0$) on the synthetic 2-dim binary data of Figure \ref{fig:moon} (right) , we plot transformed features from selected layers 0, 5, 10, 90, 100 of a deep (100-layer) ResNet. Axes ranges increase and shift to visualize diverging features with the increasing depth.
}
\label{fig:moonResNet}
\end{figure}
In Figure~\ref{fig:moonResNet}, the features are transformed through forward-propagation in vanilla ResNet (no BN, $h=1.0$). The noise in the input features leads to the mixing of red and blue points, sabotaging the generalization of ResNet. The reason for this phenomenon is that, with large $h$, features undergo violent transformations between two adjacent blocks, and negative effects of noise are amplified along the depth.
\begin{figure}[h!]
\includegraphics[scale = 0.15]{vis_moon_data/res_h_0.1_deep/0.pdf}
\includegraphics[scale = 0.15]{vis_moon_data/res_h_0.1_deep/50.pdf}
\includegraphics[scale = 0.15]{vis_moon_data/res_h_0.1_deep/100.pdf}
\includegraphics[scale = 0.15]{vis_moon_data/res_h_0.1_deep/450.pdf}
\includegraphics[scale = 0.15]{vis_moon_data/res_h_0.1_deep/500.pdf}
\caption{Feature transformations through the ResNet without BN but with step factor of $h = 0.1$ on the synthetic 2-dim data of Figure~\ref{fig:moon} (right).
We plot transformed features from selected layers 0, 50, 100, 450, 500 of a deeper (500-layer) ResNet.
Axes ranges shift to better visualize features with the increasing depth.
}
\label{fig:moon_data_res_h=0.1}
\end{figure}
On the other hand, Figure~\ref{fig:moon_data_res_h=0.1} shows that, with small $h$ features undergo smooth transformations at every adjacent residual block, and thus the input noise is gradually filtered out, which leads to the correct classification. As stated in Section~\ref{sec:generalization-robustness}, small $h$ can help bounding the negative effects of noise in input features.
With small $h$, noise amplification along the depth is effectively constrained.
We also made animations visualizing the transformations and conducted experiments comparing the effects of BN on smoothing out the noise. Interested readers may go to the \href{https://www.dropbox.com/sh/majyu9d69r1y1ya/AAD-fJ-KmbK9-eHQRn_7d8PEa?dl=0}{Dropbox link} for reference.
\begin{figure}[h!]
\centering
\includegraphics[scale = 0.10]{{cifar-10/exp_v2/ResNet110_SGD_momentum/cifar-10_110}.pdf}
\includegraphics[scale=0.10]{{nlp_plots/nlp_torch_noisy_v2/nlp_torch_noisy_lr_0.01/ag_news_49}.pdf}
\caption{To verify the effectiveness of small $h$ on generalization robustness of ResNet, we train on noisy input data with various noise level, and compare ResNet with reduced step factor ($h= 0.1$) and original ResNet ($h= 1$) (other hyperparameters remain the same.). We provide the best test accuracy with number of epochs on clean input data of both visual and text dataset \textit{CIFAR-10} (depth-110 ResNet) and \textit{AG-NEWS} (depth-49 ResNet). Each configuration takes 5 trails each with different random seeds. The standard deviation is plotted as the shaded color (For the better visualization, the shaded color in \textit{CIFAR-10} is $0.4 \times$ std.).
}
\label{fig:real-world}
\end{figure}
\paragraph{Real-world data for noise filtering:}
We train ResNet with $h = 0.1$ and $h = 1.0$ on noisy data (i.e., input perturbations), and test it on clean data. For the training dataset \textit{CIFAR-10}, we inject Gaussian noise at every normalized pixel with zero mean and a different standard deviation to represent different noise levels. For the training dataset \textit{AG-NEWS}, we choose different proportions (different noise levels) of characters randomly in the texts and alter them.
Figure~\ref{fig:real-world} shows that, at different noise levels, our method ($h = 0.1$) continuously outperforms the original method (corresponding to $h = 1.0$). In particular, for \textit{CIFAR-10} at the noise levels 0.01 and 0.1, our method outperforms the original method by a large margin around $9\%$ and $17\%$ respectively. We observe that ResNet with reduced $h$ has the smaller variance compared to its counterpart under different noise levels. In other words, our method is robust to training on noisy input by bounding the negative effects of noise. By taking the smooth transformations, it gradually filters out the noise in the input features along the forward propagation of ResNet. Thus, our method offers better generalization robustness.
\subsection{How to select step size $h$}\label{exp:selection-of-h}
Last but not least, we perform a grid search of $h$ from $0.001$ to $1.0$ to explore optimal training and generalization robustness.
\begin{figure}[h!]
\includegraphics[scale = 0.12]{{judge_different_h/cifar-10/SGD_Momentum_no_noise_judge_different_h/cifar-10}.pdf}
\includegraphics[scale = 0.12]{{judge_different_h/ag_news/nlp_no_noise_fixed_depth_eval_different_h/ag_news}.pdf}
\caption{To choose the appropriate $h$ for a fixed depth of ResNet, we perform a grid search of $h$ for \textit{CIFAR-10} (depth-110 ResNet) and for \textit{AG-NEWS} (depth-49 ResNet) (other hyperparameters remain the same).
We provide the median test accuracy with the number of epochs.
Each configuration takes 5 trails with different random seeds.
The standard deviation is plotted as the shaded color (For the better visualization, the shaded color in \textit{CIFAR-10} is $0.3 \times$ std.).
}
\label{fig:training-robust-h}
\end{figure}
\paragraph{Selection of $h$ for training robustness:}
To search $h$ for training robustness, we train fixed depth ResNet with various $h$.
Figure~\ref{fig:training-robust-h} shows that the proper $h$ for \textit{CIFAR-10} (depth-110 ResNet) is near $0.1$. The proper $h$ for \textit{AG-NEWS} (depth-49 ResNet) is between $0.1$ and $0.5$.
For a given depth, we should choose smaller $h$ but not too small. If $h$ is very small, e.g., $h=0.001$ and $0.01$, it smooths out useful transformations, thus leading to undesirable performance.
\begin{figure}[h!]
\includegraphics[scale = 0.12]{{judge_different_h/cifar-10/SGD_Momentum_noisy_judge_different_h/cifar-10}.pdf}
\includegraphics[scale = 0.12]{{judge_different_h/ag_news/nlp_noisy_fixed_depth_eval_different_h/ag_news}.pdf}
\caption{To choose the appropriate $h$ for a fixed depth of ResNet on noisy input, we perform a grid search of $h$ for \textit{CIFAR-10} (depth-110 ResNet, noise level = 0.1) and for \textit{AG-NEWS} (depth-49 ResNet, noise level = 0.5) (Other hyperparameters remain the same.). We provide the best test accuracy with number of epochs on clean input data.
Each configuration takes 5 trials with different random seeds.
The standard deviation is plotted as the shaded color (For the better visualization, the shaded color in \textit{CIFAR-10} is $0.2 \times$ std.).
}
\label{fig:generalization-robust-h}
\end{figure}
\paragraph{Selection of $h$ for generalization robustness:}
To search $h$ for generalization robustness, we train ResNet with various $h$ on the noisy input data.
Figure~\ref{fig:generalization-robust-h} shows that for noisy \textit{CIFAR-10} (depth-110 ResNet, noise level $= 0.1$), the proper $h$ should be between $0.1$ and $0.5$, and for noisy \textit{AG-NEWS} (depth-49 ResNet, noise level $= 0.5$), the proper $h$ should be between $0.1$ and $0.8$.
Too small h, e.g., h = 0.001, 0.01 leads to undesirable performance.
As discussed in Section 3.3, h cannot be too small in order to get enough approximations to match the target value at the final layer of ResNet.
\section{Conclusion}
This paper proposed a simple but principled approach to enhance the robustness of ResNet. Motivated by the dynamical system view, we characterize ResNet by an explicit Euler method. We theoretically find that the step factor $h$ in the Euler method can control the robustness of ResNet in both training and generalization. From the view of back-propagation, we prove that a small $h$ can benefit the training robustness, while from the view of forward-propagation, we prove that a small $h$ can help the generalization robustness. We conduct comprehensive experiments on vision \mbox{\textit{CIFAR-10}} and text \textit{AG-NEWS} datasets. Experiments confirm that small $h$ can benefit the training and generalization robustness. Future work can explore several promising directions:
(a) How to transfer the experience of small $h$ to other network structures, e.g., RNN for natural language processing \cite{cho2014learning},
(b) How to handle the noisy labels $y$ \cite{han2018co}, and
(c) Other means for choosing the step size $h$, e.g., using Bayesian optimization \cite{hennig2012entropy,srinivas2009gaussian,daxberger2017distributed,hoang2018decentralized}.
\clearpage
\bibliographystyle{unsrt}
|
{
"timestamp": "2019-03-01T02:09:12",
"yymm": "1902",
"arxiv_id": "1902.10887",
"language": "en",
"url": "https://arxiv.org/abs/1902.10887"
}
|
\section{Introduction} \label{sec-intro}
Gauss discovered the arithmetico-geometric mean ($ \AGM $) at the age of $ 15 $. Starting with two positive real numbers $ x $ and $ y $, Gauss considered the sequences $ \{x_n\} $ and $ \{y_n\} $ of arithmetic and geometric means $$ x_0 =x, y_0 = y , x_n = \dfrac{x_{n-1}+y_{n-1}}{2}, y_n = \sqrt{x_{n-1} y_{n-1}}, {\text{ for }} n \geq 1 .$$
Then Gauss defined $ \AGM (x,y) $ to be the common limit of the sequences $ \{x_n\} $ and $ \{y_n\} $, i.e.,
\begin{align}
\AGM (x,y) = \lim\limits_{n \to \infty} x_n = \lim\limits_{n \to \infty} y_n.
\end{align}
For an engaging historical account on $ \AGM $ and its applications in mathematics readers are referred to \cite{almkvist1988gauss},\cite{cox1997arithmetic}.
In this paper, we ask if there exist a group law $ * $, which is compatible with $ \AGM $. Before proceeding further we give some definitions relevant to this work.
\begin{definition}[\textbf{Mean}] \label{def:mean}
Let $ S $ be a set equipped with a binary operation $ m $. It is said that $ m $ is a \textit{mean}, if it satisfies the following
\begin{enumerate} [label=(\subscript{M}{{\arabic*}})]
\item \hspace{1.5cm} $ m(x,x) = x $,
\item \hspace{1.5cm} $ m(x,y) = m(y,x) $,
\item \hspace{1.5cm} $ m(x,y) = m(z,y) \implies x = z. $
\end{enumerate}
\end{definition}
\begin{definition}[\textbf{Compatibility of binary operations}] \label{def:compatible}
Let $ S $ be a set equipped with a binary mean operation $ m $ and another binary operation $ * $.
The binary mean operation $ m $, and the binary operation $ * $, are said to be \textit{compatible} with each other, if $ m(x,y) * m(x,y) = x * y $ for all $ x, y \in S $.
\end{definition}
Here we find conditions on the mean $ m $ which force any compatible operation $ * $ to be a group operation.
Let $ \displaystyle \AM(x,y) = \frac{x+y}{2} $ be the arithmetic mean of $ x,y \in \mathbb{R} $ with $ + $ being the usual addition in $\mathbb{R} $. Then clearly $ \AM(x,y)+ \AM(x,y)=x+y $, therefore, the classical arithmetic mean $ \AM(x,y)$ is compatible with the group law of $ + $ in $\mathbb{R} $, in the sense of Def.~$\ref{def:compatible} $. Similarly, the geometric mean $ \GM $ is also compatible with the group law of multiplication in positive reals. Similarly, it can be verified that the harmonic mean $ \displaystyle h(x,y) = \frac{2xy}{x+y} $ is compatible with the group law $\displaystyle x*y = \frac{xy}{x+y} $. It is then natural to consider if there exists any such group operation over $ \mathbb{R}^{+} $, which is compatible with the arithmetic-geometric mean ($ \AGM $) of Gauss. In other words, we want to address the question, if there exists a group operation $ * $, such that $ \AGM(x,y)* \AGM(x,y)=x*y $. Using one of Jacobi's theta functions, Shinji Tanimoto has successfully constructed a non-associative loop operation $ \star $ (c.f. \cite{tanimoto2007noveljp}, \cite{tanimoto2007novel}, Sec.~$ \ref{subsec:Tanimoto} $ below) that is compatible with $ \AGM $. However, no group law $ * $ compatible with $ \AGM $ is known to exist. Indeed, we prove that no such group law $ * $ can exist, which is compatible with $\AGM $.
\subsection{A non-associative loop operation compatible with $ \AGM $}\label{subsec:Tanimoto}
Now we recall the binary operation $ \star $ introduced by Shinji Tanimoto in \cite{tanimoto2007noveljp}, \cite{tanimoto2007novel}.
\begin{definition}[Tanimoto, \cite{tanimoto2007noveljp}, \cite{tanimoto2007novel}] \label{def:starTanimoto}
{\it
For any two positive numbers $x$ and $y,$ choose a unique $q~(-1 <q<1)$ such that
$1 / {\rm agm}(x,y) = \theta^2(q)$. Here,
$\theta$ is one of the Jacobi's theta functions:
\[
\theta(q) = \sum_{n = - \infty}^{+ \infty} q^{n^2} = 1 + 2\sum_{n = 1}^{\infty} q^{n^2}.
\]
Then define
\begin{eqnarray}
x \star y = \theta^2(-q)/\theta^2(q).
\end{eqnarray}
}
\end{definition}
We also recall the following theorems from \cite{tanimoto2007novel}, which describe the properties of the $ \star $ operation. We note that here variables $x, y$ are positive real numbers.
\noindent
\begin{theorem}[Tanimoto, \cite{tanimoto2007novel}]\label{thm:Tanimoto1}
{\it The operation $\star$ defined above satisfies the following properties.}\\
~{\bf (A)} $1 \star x = x$ {\rm for all} $x$. {\it Hence {\rm 1} is the unit element of the operation}. \\
~{\bf (B)} $x \star x = y \star y$ {\it implies} $x = y$. \\
~{\bf (C)} $x \star y = {\rm agm} (x, y) \star {\rm agm} (x, y)$. {\it Thus the mean with respect to
the operation is the $ \AGM $.}
\end{theorem}
\begin{theorem}[Tanimoto, \cite{tanimoto2007novel}] \label{thm:Tanimoto2}
{\it The operation $\star$ satisfies the following algebraic properties.}\\
~{\bf (D)} $a \star x = a \star y$ {\it implies} $x = y$ ({\it a cancellation law}). \\
~{\bf (E)} $(ax) \star (ay) = a \star (a(x \star y))$ {\it for any} $a, x, y$ ({\it a distributive law}).\\
~{\bf (F)} {\it If $z = x \star y$, then} $y = x(x^{-1} \star (x^{-1}z))$. {\it In particular, the inverse of $x$
with respect to the operation is} $x(x^{-1} \star x^{-1})$.
\end{theorem}
Finally, we note that Tanimoto claims that the $\star $ operation is not associative (although, he does not give any example).\footnote{It can easily be verified that $ \AGM(\AGM(1,2), \AGM(3,4)) \neq \AGM(\AGM(1,3), \AGM(2,4)) $. Theorem \ref{thm_main}, then implies that $ \star $ is not associative.}
\section{Main results}
Now we are ready to prove our claim that there does not exist any group law $ * $, that is compatible with $ \AGM $ in the sense of the Def.~$ \ref{def:compatible} $.
In this direction, first we prove the following theorem.
\begin{theorem} \label{thm_main}
Let $ m(x,y) $ be a binary operation defined over positive reals satisfying the following:
\begin{enumerate} [label=(\subscript{M}{{\arabic*}})]
\item \hspace{1.5cm} $ m(x,x) = x $, \label{cond_one}
\item \hspace{1.5cm} $ m(x,y) = m(y,x) $, \label{cond_two}
\item \hspace{1.5cm} $ m(x,y) = m(z,y) \implies x = z $, \label{cond_three}
\item \hspace{1.5cm} $ m(x,y) * m(x,y) = x * y $ \qquad (Gauss' Functional Equation),\label{cond_four}
\item \hspace{1.5cm} $ e*x = x $, \label{lem:idempotent}
\item \hspace{1.5cm} $ x * x = y * y \implies x = y. $ \label{cond_six}
\end{enumerate}
Then $ m $ is medial, i.e., $ m(m(x,y),m(z,u)) = m(m(x,z),m(y,u)) $ if and only if the $ * $ operation is associative.
\end{theorem}
Before proving Theorem \ref{thm_main}, we state and prove the following lemmas.
\begin{lemma}
Under the hypothesis $ (M_1) $-$ (M_6) $ of Theorem~\ref{thm_main}, we have the following results.
\begin{enumerate}[]
\item $ m(x,y) = m(e,x*y)$, \label{lem:tostar}
\item $ x*y = y*x $. \label{lem:commute}
\end{enumerate}
\end{lemma}
\begin{proof}
The lemma follows from the following calculations.
\begin{enumerate}[]
\item We have
\begin{align*}
m(x,y)*m(x,y) &= x*y \qquad (\text{from \ref{cond_four}}) \\
&= e*(x*y) \qquad (\text{from \ref{lem:idempotent}}) \\
&= m(e,x*y)*m(e,x*y) \qquad (\text{from \ref{cond_four}}) .
\end{align*}
Now the result follows from \ref{cond_six}.
\item $ x*y = m(x,y)*m(x,y) = m(y,x)*m(y,x) = y*x .$
\end{enumerate}
\end{proof}
\begin{lemma}\label{lem:cond_five}
Assume the hypothesis $ (M_1) $-$ (M_6) $ of Theorem~\ref{thm_main}. Also assume either $ * $ is associative, or $ m $ is medial. Then
\begin{align}
m(x,e) * m(e,y) = m(x,y).
\end{align}
\end{lemma}
\begin{proof}
First we assume that $ * $ is associative. Then the desired conclusion follows from the following calculation and \ref{cond_six}.
\begin{align*}
&(m(x,e) * m(e,y))*(m(x,e) * m(e,y)) \\
&= m(x,e) * m(e,y)*m(x,e) * m(e,y) \qquad (\text{from the associativity of $ * $}) \\
&= m(x,e) * m(x,e)*m(e,y) * m(e,y) \qquad (\text{from Lemma 1, part \ref{lem:commute}}) \\
&= (m(x,e) * m(x,e))*(m(e,y) * m(e,y)) \qquad (\text{from the associativity of $ * $}) \\
&= (x*e)*(e*y) \qquad (\text{from \ref{cond_four}}) \\
&= x*y \qquad (\text{from Lemma 1, part \ref{lem:commute} and \ref{cond_four}}) \\
& = m(x,y) *m(x,y) \qquad (\text{from \ref{cond_four}})
\end{align*}
Next we assume that $ m $ is medial, i.e., $ m(m(x,y),m(z,u)) = m(m(x,z),m(y,u)) $.
Then we have
\begin{align}
& \, \,m(m(x,y),m(z,u))* m(m(x,y),m(z,u)) \nonumber\\
&= m(m(x,z),m(y,u)) * m(m(x,z),m(y,u)) \nonumber \\
\implies & m(x,y) * m(z,u) = m(x,z)*m(y,u) \qquad (\text{from \ref{cond_four}}) \label{eq_one} \\
\implies & m(x,y) * m(e,e) = m(x,e)*m(y,e) \qquad (\text{put $ z=u=e $}) \nonumber \\
\implies & m(x,y) * e = m(x,e)*m(e,y) \qquad (\text{from \ref{cond_one} and \ref{cond_two}}) \nonumber \\
\implies & m(x,y) = m(x,e)*m(e,y) \qquad (\text{from \ref{cond_two} and Lemma 1, \ref{lem:commute}}) \nonumber
\end{align}
\end{proof}
\paragraph{\textbf{Proof of Theorem~\ref{thm_main}}}
\begin{proof}
Assume that $ * $ is associative. Then
\begin{align}
& \, \,m(m(x,y),m(z,u))* m(m(x,y),m(z,u)) \nonumber \\
&= m(x,y) * m(z,u) \qquad (\text{from \ref{cond_four}}) \nonumber \\
& = (m(x,e)*m(e,y))*(m(z,e)*m(e,u)) \qquad (\text{from Lemma \ref{lem:cond_five}}) \nonumber \\
&= m(x,e)*m(e,y)*m(z,e)*m(e,u) \qquad (\text{from the associativity of $ * $}) \nonumber \\
&= m(x,e)*m(z,e)*m(e,y)*m(e,u) \qquad (\text{from Lemma 1, part \ref{lem:commute}}) \nonumber\\
&= m(x,e)*m(e,z)*m(y,e)*m(e,u) \qquad (\text{from \ref{cond_two}}) \nonumber \\
&= (m(x,e)*m(e,z))*(m(y,e)*m(e,u)) \qquad (\text{from the associativity of $ * $}) \nonumber\\
&= m(x,z)*m(y,u) \qquad (\text{from Lemma \ref{lem:cond_five}}) \nonumber \\
&= m(m(x,z),m(y,u))* m(m(x,z),m(y,u)) \qquad (\text{from \ref{cond_four}}) \nonumber.
\end{align}
This proves one direction of the theorem, as \ref{cond_six} now implies that $ m $ is medial, i.e., $ m(m(x,y),m(z,u)) = m(m(x,z),m(y,u)) $.\\
Next to prove the other direction assume that $$ m(m(x,y),m(z,u)) = m(m(x,z),m(y,u)) .$$
Then from Eq.~$ \eqref{eq_one} $ we have
\begin{align*}
m(x,y) * m(z,u) = m(x,u)*m(z,y).
\end{align*}
For $ x=e $, the above relation becomes
\begin{align} \label{eq:basic1}
m(e,y) * m(z,u) = m(e,u)*m(z,y).
\end{align}
Now,
\begin{align}\label{eq:basic2}
m(e,y) * m(z,u) =& m(e,y) *m(e,z*u) \qquad (\text{from Lemma 1, part \ref{lem:tostar} }) \nonumber\\
=& m(y,e) *m(e,z*u) \qquad (\text{from \ref{cond_two}}) \nonumber\\
=& m(y,z*u) \qquad (\text{from Lemma \ref{lem:cond_five}}) \nonumber \\
=& m(e,y*(z*u)). \qquad (\text{from Lemma 1, part \ref{lem:tostar} })
\end{align}
Similarly,
\begin{align} \label{eq:basic3}
m(e,u) * m(z,y) = m(e,u*(z*y)).
\end{align}
From Eq.~$ \eqref{eq:basic1}$, Eq.~$ \eqref{eq:basic2}$, and Eq.~$ \eqref{eq:basic3} $, we get $$m(e,y*(z*u)) = m(e,u*(z*y)) .$$
\begin{align*}
m(e,y*(z*u)) &= m(e,u*(z*y)) \\
\implies y*(z*u) &= u*(z*y) \qquad (\text{from \ref{cond_three}}) \\
\implies y*(z*u) &= u*(y*z) \qquad (\text{from Lemma 1, part \ref{lem:commute} }) \\
\implies y*(z*u) &= (y*z)*u \qquad (\text{from Lemma 1, part \ref{lem:commute} })
\end{align*}
This completes the proof.
\end{proof}
\begin{corollary} [of Theorem \ref{thm_main}]
There does not exist any group law $ * $, that is compatible with $ \AGM $.
\end{corollary}
\begin{proof}
From the definition of $ \AGM $, it is obvious that $ \AGM(x,x) =x $ and $ \AGM(x,y) = \AGM(y,x) $. Further, if $\AGM(x,y) = \AGM(x,z) $, then $$ \AGM(x,y) \star \AGM(x,y) = \AGM(x,z) \star \AGM(x,z) \implies x \star y = x \star z \implies y = z,$$ from Theorem~$ \ref{thm:Tanimoto1} $\,(C) and Theorem~$ \ref{thm:Tanimoto2} $\,(D).
Therefore, $ \AGM $ is a mean operation in the sense of Def.~$ \ref{def:mean} $.
Further, the $ \star $ operation defined by Tanimoto (see Def.~$ \ref{def:starTanimoto} $) is not associative, and moreover, $ \AGM $ and $ \star $ satisfy the hypothesis of Theorem~$ \ref{thm_main} $, from the definition of $ \star $, and by virtues of Theorem~$ \ref{thm:Tanimoto1} $ and Theorem~$ \ref{thm:Tanimoto2} $ (we note that in \cite{tanimoto2007novel}, our identity element $ e $ is represented by $ 1 $). Therefore, from Theorem~$ \ref{thm_main} $, it follows that $ \AGM $ is not medial (alternatively, from a direct numerical computation it can be verified that $ \AGM $ is not medial). But then, Theorem~$ \ref{thm_main} $ also implies that $ \AGM $ can not be compatible with any $ * $ operation which is associative and satisfies \ref{cond_four}-\ref{cond_six}. Therefore, there can not exist any group law $ * $, that is compatible with $ \AGM $.
\end{proof}
Suppose for a mean $ m $, if $ m(m(x,y),m(x,z)) = m(x,m(x,z)) $, then the mean $ m $ is said to be self-distributive. If $ (x*x)*(y*z) = x*y*(x*z) $ then $ * $ is called Moufang.
It is easy to see that in the above proofs, the full force of associativity (or, for that matter the medial law) is not used. Indeed, `associativity' and `medial' in Theorem~\ref{thm_main}, can be replaced by `Moufang' and `self-distributive', respectively and the proof of the theorem still remains valid.
\begin{theorem} \label{thm:Moufang}
For a mean $ m $ and satisfying $ (M_1) $-$ (M_6) $ of Theorem~\ref{thm_main},
$ m $ is self-distributive, i.e., $ m(m(x,y),m(x,z)) = m(x,m(x,z)) $ if and only if the $ * $ operation is Moufang.
\end{theorem}
One can easily verify (for example by using Mathematica) that
$$ \AGM(\AGM(1,2),\AGM(1,3)) \neq \AGM(1,\AGM(2,3)) .$$
Hence, Gauss' Functional Equation for $ \AGM $ can not be solved even among Moufang loops.
Although, we have remarked earlier that the proof of Theorem~\ref{thm:Moufang} follows on the same line as Theorem~\ref{thm_main}, we are enclosing an automated proof of this theorem by using Prover9 \cite{prover9}, in the Appendix, for readers interested in automated reasoning.\\
\section{Appendix}
\noindent\textbf{Moufang identity implies self-distrtibutivity.}
\begin{verbatim}
1 m(x,m(y,z)) = m(m(x,y),m(x,z)) # label(goal). [].
3 m(x,y) = m(y,x). [].
5 m(x,y) * m(x,y) = x * y. [].
6 x * x != y * y | x = y. [].
7 x * e = x. [].
8 (x * y) * (x * z) = (x * x) * (y * z). [].
9 m(m(c1,c2),m(c1,c3)) != m(c1,m(c2,c3)). [1].
10 m(c1,m(c2,c3)) != m(m(c1,c2),m(c1,c3)). [9].
15 m(x,y) * m(y,x) = y * x. [3,5].
16 x * y = y * x. [3,5,15].
17 x * y != z * z | m(x,y) = z. [5,6].
23 c1 * m(c2,c3) != m(c1,c2) * m(c1,c3). [6,10,5,5].
32 c1 * m(c3,c2) != m(c1,c2) * m(c1,c3). [3,23].
48 e * x = x. [16,7].
50 (x * y) * (z * x) = (x * x) * (y * z). [16,8].
79 c1 * m(c3,c2) != m(c2,c1) * m(c1,c3). [3,32].
130 m(e,x * x) = x. [17,48].
132 m(x * x,y * y) = x * y. [17,8].
160 m(e,x * y) = m(x,y). [5,130].
221 c1 * m(c3,c2) != m(c1,c3) * m(c2,c1). [16,79].
293 m(x * y,z * z) = m(x,y) * z. [5,132].
294 m(x * x,y * z) = x * m(y,z). [5,132].
662 m(x * y,z * x) = x * m(y,z). [50,160,160,294].
1706 m(x * y,z * u) = m(x,y) * m(z,u). [5,293].
1748 m(x,y) * m(z,x) = x * m(y,z). [662,1706].
1749 $F. [1748,221].
\end{verbatim}
\noindent\textbf{Self-distributivity implies Moufang identity.}
\begin{verbatim}
1 (x * y) * (x * z) = (x * x) * (y * z) # label(non_clause) # label(goal). [].
2 m(x,x) = x. [].
3 m(x,y) = m(y,x). [].
4 m(x,y) != m(z,y) | x = z. [].
5 m(x,y) * m(x,y) = x * y. [].
6 x * x != y * y | x = y. [].
7 x * e = x. [].
8 m(x,m(y,z)) = m(m(x,y),m(x,z)). [].
9 m(m(x,y),m(x,z)) = m(x,m(y,z)). [8].
10 (c1 * c2) * (c1 * c3) != (c1 * c1) * (c2 * c3). [1].
13 m(x,y) != m(z,x) | y = z. [3,4].
15 m(x,y) * m(y,x) = y * x. [3,5].
16 x * y = y * x. [3,5,15].
17 x * y != z * z | m(x,y) = z. [5,6].
22 m(x,y) * m(x,z) = x * m(y,z). [9,5,9,5].
24 e * x = x. [16,7].
26 m(x,y) != m(x,z) | y = z. [3,13].
29 m(e,x * x) = x. [17,24].
33 x * x != y | m(e,y) = x. [24,17].
41 m(e,x) != y | y * y = x. [29,26].
55 x != y | y * y = x * x. [29,41].
56 m(e,x * y) = m(x,y). [33,22,2].
58 m(x * x,y) = x * m(e,y). [29,22,24].
68 m(x,e) != m(y,z) | y * z = x. [56,13].
74 m(x,y) * m(e,z) = m(x * y,z). [56,22,24].
79 m(x,y * y) = y * m(e,x). [58,3].
80 m(x * x,y) = x * m(y,e). [3,58].
99 m(x * x,y * z) = x * m(y,z). [56,58].
134 m(x,y * y) = y * m(x,e). [3,79].
153 m(x,x * y) = x * m(y,e). [80,22,22,2,3].
220 m(x * x,y) = m(x,x * y). [153,80].
225 m(x,y * y) = m(y,y * x). [153,134].
240 m(x,x * (y * z)) = x * m(y,z). [99,220].
338 x * (y * y) = y * (y * x). [55,225,22,2,22,2].
427 (x * x) * y = x * (x * y). [338,16].
448 (c1 * c2) * (c1 * c3) != c1 * (c1 * (c2 * c3)). [10,427].
504 m(c1 * c2,c1 * c3) != c1 * m(c2,c3). [68,448,3,56,240].
550 m(x,y) * m(z,u) = m(x * y,z * u). [56,74].
568 m(x * y,x * z) = x * m(y,z). [22,550].
569 $F. [568,504].
\end{verbatim}
\bibliographystyle{plain}
|
{
"timestamp": "2019-03-01T02:04:23",
"yymm": "1902",
"arxiv_id": "1902.10809",
"language": "en",
"url": "https://arxiv.org/abs/1902.10809"
}
|
\section{\label{sec:Intro}Introduction}
Semiconductor-superconductor hybrid junctions have generated significant interest over the last decade. In particular, III-V semiconductor (InAs/InSb) nanowires in proximity to an s-wave superconductor have been extensively studied as a platform for topological superconductivity\cite{Zhang,Mourik,Deng1557,Chene1701476,ADas_2012,Rokhinson_2012}. Majorana bound states (MBSs) emerge as zero energy edge excitations in a gapped bulk spectrum of the topological superconducting nanowire \cite{Majorana,Majorana2006,Kitaev_2001,BeenakkerReview2,Sau_2010,Sau_2010_2,Alicea_2010,Alicea_2012,LutchynPRL}. Signatures of MBS have been reported as a zero-bias conductance peak in tunnelling experiments \cite{Zhang,Mourik,Deng1557,Chene1701476,ADas_2012,Wimmer_2011}. The $4\pi$ Majorana-Josephson effect has been predicted and observed in nanowire Josephson junctions tuned to the topologically non-trivial regime \cite{Kitaev_2001, Rokhinson_2012}.
With the massive progress being made with nanowire setups, it is anticipated that the focus will shift from the detection to the demonstration of non-Abelian statistics and finally to topological quantum information processing\cite{Alicea_2011,Sarma_2015,Aasen_2016,Hyart,Karzig_2017,Plugge_2017}. The 4-$\pi$ Josephson effect forms the basis of braiding and readout schemes of a recent topological qubit proposal\cite{Stenger_2019}.
Josephson junctions based on semiconductor-superconductor hybrids form the basis for microwave quantum circuity\cite{Lasrsen}, and superconducting qubits\cite{deLange,Hassler_2011}. They afford an attractive alternative for a scalable computing architecture with the possibility of an all-electric qubit control\cite{Kringhoj,Lasrsen,Casparis}.
Several studies have focused on the structure of transverse subbands and magnetoconductance due to radial confinement in semiconductor nanowires \cite{Hernandez,Blomers,Aritra}. Recent experiments study the critical current as a function of the magnetic field and gate voltage in nanowire Josephson junctions tuned to the few-subband regime\cite{IQC,Frolov,Szombati}. For a magnetic field oriented along the nanowire axis, \citeauthor{Frolov} measured a strong suppression of the critical current at fields on the order of 100 mT in InSb weaklinks with NbTiN contacts. At higher fields, the critical current exhibited local minima (nodes). Similar results were obtained by \citeauthor{IQC} for InAs-Nb Josephson junctions. Unlike the Fraunhofer diffraction in wide planar junctions, the critical current nodes were aperiodic in the magnetic field, and highly sensitive to local fluctuations in the gate voltage. Motivated by these experiments, the object of this paper is to theoretically analyze few-mode nanowire Josephson junctions in a magnetic field oriented along the nanowire axis.
We thus employ the Keldysh {Non-Equilibrium Green's Function formalism} (NEGF)\cite{Keldysh, datta_1995, datta_2005, DuBois, Rammer, NEGFChina} to model quasiparticle transport in the junction, and compute the evolution of the critical current as a function of the axial field and chemical potential. Based on the simulations, we attribute the observed oscillations to the interference of the transverse subbands in the nanowire. These results are crucial in the design of Majorana setups\cite{Aasen_2016,Hyart,Karzig_2017,Plugge_2017} and in interpreting experiments, particularly for those based on critical current measurements\cite{SanJose}.
Quantum transport traditionally involves excited states and the use of a variant of the Landauer-B{\"u}ttiker's scattering theory\cite{datta_1995,BEENAKKER19911,Aniket} for performing transport calculations. This essentially involves solving the Schr{\"o}dinger equation and an appropriate treatment of the boundary conditions. In a superconductor, however, zero-bias transport is essentially a ground state phenomenon supported by Cooper pairs condensed at the fermi level\cite{DeGennes,Tinkham}. \citeauthor{BTK} generalised the scattering theory approach to hybrid semiconductor-superconductor junctions by solving the Bogoliubov-de Gennes equation across the N-S interface\cite{BTK}. Beenakker applied this formalism for mesoscopic N-S junctions, thus providing a multichannel generalization of Blonder's results\cite{Beenakker}. This technique has been prevalent in the literature\cite{IQC1,Bagwell} ever since and it forms the basis for numerous simulation packages such as Kwant \cite{Kwant}. Despite its benefits, the scattering theory approach is not very convenient in dealing with disordered junctions. While phase-coherent scattering processes can be included via random on-site potentials, it is difficult to model phase-relaxing interactions. Moreover, this formalism becomes intractable whenever a self-consistent determination of the order parameter becomes necessary. This self-consistent computation can be performed using the correlation Green's function\cite{LevyYeyatiPRB, LevyYeyatiPRL}, and various scattering mechanisms such as electron-electron, electron-phonon interactions can be included through suitable self-energy operators in the NEGF formalism.
This paper is organised as follows. We start with the Bogolubov-de Gennes mean-field description of a one-dimensional nanowire Josephson junction (SNS).
In Sec.~\ref{sec:Model}, we describe the junction in a tight-binding model and outline the key aspects of the NEGF formalism. The details of this formalism have been relegated to Appendix~\ref{sec:app}. Employing the NEGF formalism, we compute the Andreev Bound State (ABS) spectrum and Current-Phase-Relationship (CPR) for this junction.
Previous work almost exclusively focused on the Andreev Approximation regime, which assumes the chemical potential of the nanowire ($\mu$) to be much larger than the superconducting order parameter ($\Delta_0$), i.e, $\mu \gg \Delta_0$. We go beyond this Andreev approximation limit and investigate the bound states which anti-cross at a superconducting phase difference of $\pi$ between the leads.
In Sec.~\ref{sec:radialconf} we model three-dimensional Josephson junctions with transverse angular momentum subbands.
The radial confinement gives rise to transverse angular momentum subbands which pick up characteristic phases in a magnetic field. Section~\ref{sec:labelling} details the procedure we follow to label these angular momentum subbands. In Sec.~\ref{sec:oscillations}, we reproduce the critical supercurrent oscillations in the presence of an axial magnetic field. Our results confirm these observed oscillations to be arising from the interference between orbital channels of the junction.
With the aim of gaining a thorough understanding of the experiments, we consider scattering processes in the nanowire and study the effect of an onsite disorder potential, gate voltage fluctuations and phase-breaking processes on the critical current oscillations.
\section{\label{sec:Model} Formalism}
Superconducting correlations are induced in a proximitised semiconductor by electron-hole conversions at the interface, a process known as Andreev reflection\cite{Andreev,Andreev2}.
Low bias transport in normal (N)-superconductor (S) junctions involves Andreev reflections at the interface.
\begin{figure}[!htbp]
\includegraphics[width=0.4\textwidth,keepaspectratio]{SNS_3d}
\caption{Schematic of the nanowire Josephson junction, with a semi-infinite superconducting (S) leads (blue), and a normal (N) device region (green). The red spheres form the discretized lattice model. The length of the nanowire can be increased by adding more layers of the N-region. A large potential $U$ on the inner lattice points (yellow sites) confines the particles to surface of the nanowire. The transverse square cross-section is 60 nm wide.
}
\label{fig:SNS_3d}
\end{figure}
We first consider a one-dimensional SNS junction consisting of a semiconductor nanowire with supercondcuting contacts. We model this system using the Bogolubov-de Gennes (BdG) mean-filed Hamiltonian within the tight-binding approximation, $H = \mathcal{H}_0 + \mathcal{H}_{\text{p}}$, where
\begin{align}
\mathcal{H}_0 &= \int dz \sum_{\sigma}\psi_{\sigma}^{\dagger}(z)\left(-\frac{\hbar^2}{2m^*}\partial_z^2 + V(z) - \mu \right)\psi_{\sigma}(z)\\
\mathcal{H}_p &= \int dz \psi_{\uparrow}^{\dagger}(z)\Delta(z)\psi_{\downarrow}^{\dagger}(z) + h.c.
\label{eq:model}
\end{align}
$\mathcal{H}_0$ is the single-particle effective Hamiltonian, $\psi_{\sigma}$ is the field operator with spin index $\sigma
\in \{\uparrow,\downarrow \}$, $m^*$ is the electron effective mass, and $V$ models a potential energy induced in the junction. The chemical potential is defined as the energy difference between the lowest occupied subband and the Fermi energy, and is denoted by $\mu$. We assume an identical effective mass in the N and S regions thus neglecting the Fermi wave-vector mismatch at the interface. $\Delta(z)$ is the superconducting order parameter along the junction, which we assume to be constant with jump-discontinuities at the N/S interfaces
\begin{equation}
\Delta(z) = \mathcal{\theta}(z)\Delta_0e^{i\chi_L} + \mathcal{\theta}(L-z)\Delta_0 e^{i\chi_R}
\end{equation}
where $\theta(x)$ is the unit step function at $x=0$, $\chi_{L,R}$ is the superconducting phase of the left and right leads respectively, and $\phi=\chi_L-\chi_R$ is the phase difference.
In the $\left[\psi_{\uparrow}^{\dagger}(r), \psi_{\downarrow}(r)\right]$ Nambu basis, we have the BdG equation
\begin{equation}
\begin{bmatrix}
\mathcal{H}_0 & \Delta(z) \\
\Delta^*(z) & -\mathcal{H}_0^*
\end{bmatrix}
\begin{bmatrix}
u(z) \\
v(z)
\end{bmatrix}
=
E
\begin{bmatrix}
u(z) \\
v(z)
\end{bmatrix}
\label{eq:BdG}
\end{equation}
The device is divided into three parts -- a normal semiconductor section with a length $L$ extended over $z \in [0,L]$, and semi-infinite superconducting contacts extending to $ z = \pm \infty$ on either side of the semiconductor (Fig.~\ref{fig:SNS_1d}).
We discretise the continuum model of Eq.~\ref{eq:model} into a lattice model with a spacing of $a$. This is shown in Fig.~\ref{fig:SNS_1d}. The superconductors are modelled as semi-infinite leads, while the number of lattice points in the normal region controls the length of the nanowire.
The on-site tight-binding parameters in the normal and superconducting regions, in the Nambu representation are \\
\begin{equation}
\alpha_N=
\begin{bmatrix}
2t-\mu & 0\\
0 & -2t+\mu
\end{bmatrix}
\end{equation}
\begin{equation}
\alpha_S=
\begin{bmatrix}
2t-\mu & \Delta\\
\Delta^{*} & -2t+\mu
\end{bmatrix}
\label{eq:alpha}
\end{equation}
\\
where, $t = \frac{\hbar^2}{2m^*a^2}$ is the nearest neighbour tight binding hopping parameter.
The hopping matrix is given by
\begin{equation}
\beta =
\begin{bmatrix}
-t & 0\\
0 & t
\end{bmatrix}
\label{eq:beta}
\end{equation}
The device Hamiltonian is subsequently written as
\begin{equation}
H = \sum_i^n c_i^{\dagger}\alpha_{N/S}c_i + \sum_{|i-j|=1}^n c_i^{\dagger}\beta c_j
\end{equation}
where $c^{\dagger}_i$ is the creation operator of the Nambu spinor $\left[\psi_{\uparrow}^{\dagger}(r), \psi_{\downarrow}(r)\right]$ at site $i$, and $n=L/a$ is the number of sites in the device.
The Hamiltonian of the normal region can be written in the general form
\begin{equation}
{H} = \mqty(\alpha_N & \beta & 0 & \dots & 0 \\ \beta^{\dagger} & \alpha_N & \beta & 0 & 0 \\ 0 & \beta^{\dagger}& \alpha_N & \beta & \vdots \\ \vdots & 0 & \ddots & \ddots & \beta \\ 0 & \dots & \dots & \beta^{\dagger} & \alpha_N)\label{eq:Hamiltonian}\end{equation}
\subsection{\label{sec:SNS}Andreev Bound States in SNS Junctions}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.4\textwidth,keepaspectratio]{SNS}
\caption{Schematic of the SNS junction, with a semi-infinite superconducting leads, and an N device region. The length of the nanowire is given by the number of lattice points ($L=Na$) in the N-region. $\alpha$ and $\beta$ are the tight-binding onsite and nearest neighbour coupling terms respectively. $\Delta_0$ and $\chi$ is the magnitude and phase of the superconducting order parameter.}
\label{fig:SNS_1d}
\end{figure}
Andreev reflections at the N/S interfaces give rise to Andreev bound states in the semiconductor. We use the NEGF formalism to compute these bound state energies as a function of the superconducting phase difference ($\phi$) of the leads.
The retarded Green's function in the energy domain is given by
\begin{equation}
G^r(E) = \left(E\mathbb{I} + i \eta-\mathcal{H}-\Sigma_1-\Sigma_2\right)^{-1}
\end{equation}
where $E$ denotes the energy, $\mathbb{I}$ is the identity matrix and $\eta$ is an infinitesimal real constant. The Hamiltonian $\mathcal{H}$ is given by Eq.~\ref{eq:Hamiltonian}. The self-energy terms $\Sigma_{1,2}$ model the coupling of the device to the semi-infinite leads. The self-energy is not hermitian, and its anti-hermitian part is responsible for the finite lifetime of the electron in the device. This subsequently contributes to broadening the energy levels in the device. The self-energies are computed using the surface-Green's function, which requires an iterative procedure as outlined in Appendix~\ref{sec:app}.
We compute the density of states ($d$) in the nanowire as the trace of the spectral Green's function
\begin{equation}
d(E) = \Tr \left(A(E)\right) = \Tr \left [i \left(G^r(E) - G^a(E)\right) \right ]
\end{equation}
The real-valued singularities of the density of states are the Andreev bound state (ABS) energies. This is computed as a function of the phase difference ($\phi$) of the order parameter of the contacts and is shown in Fig.~\ref{fig:ABS_AA}. The parameters for this computation are consistent with the Andreev approximation\cite{Andreev,Andreev2,BeenakkerReview,Ashida} ($\mu \gg \Delta_0$). As discussed in Appendix~\ref{sec:BeyondAA}, the breakdown of this approximation is manifested as an avoided level crossing in the ABS spectrum.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.4\textwidth,keepaspectratio]{AndreevApprox_mu30Delta}
\caption{Andreev Bound state spectrum as a function of the superconductor phase difference for a clean 1-dimensional SNS junction. The junction is tuned into the Andreev Approximation regime with $\mu = 30\Delta_0$.}
\label{fig:ABS_AA}
\end{figure}
\subsection{Current Phase Relationship}
The current-phase relationship (CPR) links the macroscopic current flow in the junction to the phase gradient of the superconducting order parameter\cite{Josephson}. The traditional approach to computing the CPR involves a demarcation of the bound state and continuum currents. The bound state current involves transport in the sub-gap energy range ($\abs{E} < \Delta_0$) while the continuum current is supported by the continuous energy spectrum outside the gap. Once the ABS spectrum is computed from scattering theory, a thermodynamic relation is used to calculate the bound state current, and the transmission formalism is used for the continuum current. The total current is the sum of the bound state and continuum currents\cite{Kulik}.
By contrast, when using the NEGF formalism the current-energy density can be computed at contact $i$, as a function of the phase difference $\phi$ using the current operator\cite{datta_1995,datta_2005}
\begin{equation}
J_i(E) = \frac{2e}{h}f(E)\text{Tr}\left[\Re\left( G^a(E)\Sigma_i^a(E)-G^r(E)\Sigma_i^r(E) \right)\tau_z\right]
\end{equation}
where $f(E) = 1/\left(\exp\left(E/k_BT\right)+1\right)$ is the Fermi-Dirac occupation probability for a given energy level and $k_B$ is the Boltzmann constant. $G^{r(a)}$ and $\Sigma_i^{r(a)}$ are the retarded (advanced) Green's function and contact $i$ self-energy respectively. To incorporate the opposite charge of electrons and holes we use the Pauli-z operator ($\tau_z$) in the particle-hole Nambu space. This current operator is reviewed in the Appendix~\ref{sec:app}. The total current at a phase difference $\phi$ is then given by
\begin{equation}
I(\phi) = \int_{-\infty}^{\infty}J_i(E)dE
\end{equation}
Figure~\ref{fig:subfigures} compares the current phase-relations for a long junction ($L>\xi_0$) as calculated from ideal scattering theory and NEGF. Ideal scattering theory neglects normal reflections at the N/S interfaces and computes bound states that cross at $\phi=\pi$. Hence, there's a discontinuity at $\phi=\pi$ in the CPR calculated using this method. The NEGF result is expected to match the scattering theory exactly in the $\mu \gg \Delta_0$ limit.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.4\textwidth,keepaspectratio]{Figure5}
\caption{The total current-phase-relation in an SNS junction computed from the NEGF current operator (red) is compared with the ideal-scattering theory (blue) calculations. The nanowire length L=400 nm, chemical potential $\mu=30\Delta_0$ and healing length $\xi=222$ nm. Ideal scattering theory neglects normal reflections at the N/S interfaces. We see that we recover the expected saw-tooth profile for the total current using the NEGF formalism.
}%
\label{fig:subfigures}
\end{figure}
\section{\label{sec:radialconf}Transverse subbands in Josephson Junctions}
We now consider a more realistic three-dimensional model of the junction with a magnetic field along the nanowire axis, parallel to the direction of current flow. Figure~\ref{fig:SNS_3d} illustrates a discrete lattice model of the three-dimensional nanowire. The junction is along the $z$-direction and the transverse subbands are on the $x$-$y$ plane. In III-V semiconductors (InAs, InN) the charge carriers are typically confined close to the surface due to a positive surface potential, forming a surface accumulation layer. In accordance with this, we use a shell conduction model by including a large surface confining potential $U$
at the core of the nanowire (yellow sites in Fig.~\ref{fig:SNS_3d}). The radial confinement and azimuthal periodicity of the nanowire gives rise to transverse subbands.
The single-electron Hamiltonian of the nanowire is
\begin{equation}
\mathcal{H}_0 = -\mu + \frac{-\hbar^2}{2m^*}\frac{\partial^2}{\partial z^2} + \mathcal{H}_T + U
\end{equation}
where $z$ is the longitudinal direction, $U$ is the surface confinement potential, and $\mathcal{H}_T$ is the Hamiltonian of the transverse modes.
The rotational symmetry of the nanowire about the longitudinal axis results in angular momentum $\left(\ell \right)$ subbands. This is because
\begin{equation}
[\mathcal{H}_0,L_z] = 0
\end{equation}
and hence $ \ell$ is a good quantum number. The $\ell$ subbands are eigenstates of the $L_z$ operator, labelled by their eigenvalue
\begin{equation}
\hat{L}_z\ket{\ell} = \hbar\ell\ket{\ell}
\label{eq:l}
\end{equation}
The azimuthal motion of the Andreev quasiparticles couples with the applied magnetic field, resulting in a quasiparticle phase pickup. Oscillations in the maximal supercurrent (critical current) with field have been measured by \citeauthor{IQC}, and \citeauthor{Frolov}. Unlike the Fraunhofer interference in wide planar junctions, the field is aligned with the current and the oscillations do not show any periodicity.
Using Peierls substitution, we include the orbital effect of the vector potential in the phase of the transverse hopping. For a constant magnetic field along the $z-$direction, the vector potential can be written as
\begin{equation}
\mathbf{A} = B \cdot x\mathbf{\hat{y}}
\label{eq:A}
\end{equation}
Within the tight-binding approximation, the on-site and hopping matrices in the particle-hole Nambu space are given by
\begin{equation}
\alpha_{N/S}=
\begin{bmatrix}
{h} & \Delta_{N/S}\\
\Delta_{N/S}^* & -{h}^
\end{bmatrix}
\label{eq:alpha3d}
\end{equation}
where ${h} = 2t_x+2t_z+\abs{t_y}\left[2+(2\pi n_x\Phi_a)^2\right]-\mu$
\begin{equation}
\beta_{x,z} =
\begin{bmatrix}
-t_{x,z} & 0\\
0 & t_{x,z}
\end{bmatrix}
\end{equation}
\begin{equation}
\beta_y =
\begin{bmatrix}
-t_ye^{i2\pi n_x\Phi_a} & 0\\
0 & t_ye^{-i2\pi n_x\Phi_a}
\end{bmatrix}
\label{eq:betaxz}
\end{equation}
where $\Phi_a$ is the flux quanta per unit cell, and $n_x$ is the lattice site index in the $x-$direction. This factor alters the on-site energy ($\alpha_{N/S}$) and contributes a phase to the hopping term corresponding to the gauge chosen for the vector potential (Eq.~\ref{eq:A}).
\subsection{Andreev Bound States in a magnetic field}
Figures~\ref{fig:DOS_nofield}, and~\ref{fig:DOS_field} plot the subgap density of states as obtained from the spectral Green's function for a nanowire with an InAs effective mass $m^*$ = 0.023$m_e$\cite{NAKWASKI19951} ($m_e$ is the bare electron mass), radius R = 30 nm and chemical potential $\mu=10 \Delta_0$. As described in Appendix~\ref{sec:BeyondAA}, the bound states anti-cross at $\phi=\pi$ due to normal reflections at the N/S interfaces. A normalised flux of $\Phi = 0.1$ is applied in Fig.~\ref{fig:DOS_field}, which lifts the degeneracy of the $\pm \ell$ subbands.
\begin{figure}[!htb]
\begin{center}
\subfigure[]{%
\label{fig:DOS_nofield}
\includegraphics[width=0.4\textwidth]{Figure6A}
}\\%
\subfigure[]{%
\label{fig:DOS_field} \includegraphics[width=0.4\textwidth]{Figure6B.png}
\end{center}
\caption{%
The Andreev bound state spectrum as a function of the superconductor phase difference of the in the (a) absence and (b) presence of a magnetic field ($\Phi=0.5$) in a three-dimensional model of an SNS Josephson junction. The nanowire length $L=8$ nm, chemical potential $\mu=10\Delta_0$ and healing length $\xi = 128$ nm.
}%
\end{figure}
The chemical potential is adjusted to populate a few subbands. In Fig.~\ref{fig:DOS_field} we see that the $\ell=0$ subband remains unaffected, while the $\ell \neq 0 $ subbands split in the presence of an axial flux. The process used to label the subbands is described in Sec.~\ref{sec:labelling}.
The characteristic length-scale associated with an occupied subband is called the healing length $(\xi_{\ell})$\cite{Kulik,Bagwell}, and is given by
\begin{equation}
\xi_{\ell} = \frac{\hbar v_{F,\ell}}{2\Delta_0}
\end{equation}
$v_{F,\ell}$ is the fermi-velocity and is given by
\begin{equation}
v_{F,\ell} = \sqrt{2\left(\mu - \frac{\hbar^2}{2m^*R^2}\left(\ell^2+\Phi^2 \right)\right)/m^*}
\label{eq:fermi_vel}
\end{equation}
$\Phi$ is the normalised flux through the nanowire cross section.
We have a ``short junction'' when the nanowire is shorter than the healing length ($L<\xi_{\ell}$). Note that the healing length depends on the angular momentum quantum number, and the classification of the junction (as long/short) is subband dependent.
We now describe a procedure to label the angular-momentum subbands using the correlation Green's function ($G^n$).
\subsection{\label{sec:labelling}Labelling the Angular Momentum Subbands}
The angular momentum subbands are characterised by Eq.~\ref{eq:l}, however, we do not have access to the wave-functions in a numerical simulation. We do have the correlation Green's function $-iG^< = G^n$ which gives the particle-hole density per-unit energy. Using this, we find the expectation of the $L_z$ operator as a function of energy
\begin{equation}
\langle \hat{L}_z \rangle = \frac{\Tr\left(G^n\cdot L_z\right)}{\Tr\left(G^n\right)}
\label{eq:ExLz}
\end{equation}
We first need to construct the $\hat{L}_z$ operator in a discrete lattice model. In the Cartesian coordinate system, the $\hat{L}_z$ operator is written as
\begin{equation}
\hat{L}_z = \hat{x}\cdot \hat{p}_y - \hat{y}\cdot \hat{p}_x
\label{eq:Lz}
\end{equation}
The position operators $\hat{x}, \hat{y}$ are diagonal in the tight-binding basis, with each entry a multiple of the lattice constant ($a$). For example, if we consider two points along the $x$ and $y$ axis, and 1 in the $z$ direction, we have the following position operators
\begin{equation}
\mathbb{I}_x \otimes \hat{y} =
a\left(\begin{matrix}
1 & 0 & 0 & 0\\
0 & 2 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 2\\
\end{matrix}\right)
\hspace{1cm}
\hat{x}\otimes\mathbb{I}_y =
a\left(\begin{matrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{matrix}\right)
\label{eq:x}
\end{equation}
The momentum operators can be written as
\begin{equation}
\hat{p}_i = -i\frac{ m}{\hbar}[\hat{x}_i,\mathcal{H}_0]
\label{eq:p}
\end{equation}
where the subscript $i$ is used to denote $x,y$ basis of the position and momentum operators. Using Eq.~\ref{eq:Lz},~\ref{eq:x},~\ref{eq:p}, we write the $\hat{L}_z$ operator and using Eq.~\ref{eq:ExLz} we find the expectation of $\hat{L}_z$ as a function of energy.
We employ the above procedure to label the angular momentum of the Andreev bound states in an SNS junction. We fix the chemical potential to give us three occupied $\ell$ subbands ($\ell = 0, \pm 1$). The bound state spectrum as a function of the phase difference of the order parameter ($\phi$) is plotted in Fig.~\ref{fig:ABS0Flux}. In Fig.~\ref{fig:ABS_label_0Flux} we show the labelling of the states at $\phi = \pi$. Next, we turn on an axial magnetic flux through the nanowire. The $\ell = \pm 1$ subband states split (Fig.~\ref{fig:ABS01Flux}), and states at $\phi=\pi$ is labelled in Fig.~\ref{fig:ABS_label_01Flux}.
In the next section we explain the importance of including only the $\ell = 0$ subbands in the superconducting contacts.
\begin{figure*}[!htb]
\begin{center}
\subfigure[Subgap density of states vs $\phi$]{%
\label{fig:ABS0Flux}
\includegraphics[width=0.4\textwidth]{Figure7A}
}%
\subfigure[Subgap density of states vs $\phi$]{%
\label{fig:ABS_label_0Flux}
\includegraphics[width=0.4\textwidth]{Figure7C}
}\\%
\subfigure[Subgap density of states at $\phi=\pi$]{%
\label{fig:ABS01Flux}
\includegraphics[width=0.4\textwidth]{Figure7D}
}%
\subfigure[Subgap density of states at $\phi=\pi$]{%
\label{fig:ABS_label_01Flux}
\includegraphics[width=0.4\textwidth]{Figure7B}
}%
\end{center}
\caption{%
The Andreev bound state spectrum for an SNS junction with three occupied subbands is plotted in (a) no magnetic field and (b) an axial magnetic field with $\Phi=0.01$. (c) The density of states (Red)and angular momentum quantum number (Blue) is plotted for at $\phi=\pi$ for the ABS spectrum in (a). The Andreev bound states are broadened, and hence the blue $\langle L_z \rangle$ peaks don’t represent the angular momentum quantum number in a quantitatively accurate manner. (d) same as (c), but for the ABS spectrum in (b) with an axial magnetic flux of $\Phi=0.01$. The nanowire length $L=8$ nm, chemical potential $\mu=5\Delta_0$ and healing length $\xi = 90$ nm.
}%
\end{figure*}
\subsection{Zero angular momentum subband in the Superconductor}
The BdG Hamiltonian in a superconductor is given by
\begin{equation}
\begin{bmatrix}
H_0 &\Delta \\
\Delta^* & -H_0^*
\end{bmatrix}
\begin{bmatrix}
u\\v
\end{bmatrix}
=
E
\begin{bmatrix}
u\\v
\end{bmatrix}
\label{eq:BdGSC}
\end{equation}
For a cylindrical geometry with an azimuthal vector potential, $H_0$ is given by
\begin{equation}
H_0 = -\frac{\hbar^2}{2m^*}\frac{\partial^2}{\partial z^2} + \frac{1}{2m^*}\left(-i\hbar \frac{1}{R}\frac{\partial}{\partial \theta} -eA_{\theta}\right)^2 - \mu
\end{equation}
As discussed, the radial confinement due to the nanowire's cylindrical geometry gives rise to angular momentum subbands labelled by $\ell$.
We will analyze the eigenenergies of this superconductor in the presence and absence of a magnetic field.\\
\textbf{Case 1: Zero Magnetic Field} $\mathbf{B= 0, A = 0}$\\
Using the ansatz wavefunction $\exp(ik_zz)\exp(i\ell\theta)$, the diagonal elements of the BdG Hamiltonian can be written as
$$h_{\ell} = \frac{\hbar^2k_z^2}{2m^*} + \frac{\hbar^2\ell^2}{2m^*R^2} - \mu = \frac{\hbar^2k_z^2}{2m^*} - \mu_{\ell}$$
where $\mu_{\ell}$ is the effective chemical potential
\begin{equation}\mu_{\ell} = \mu-\frac{\hbar^2\ell^2}{2m^*R^2} \label{eq:mu_eff} \end{equation}
The BdG Hamiltonian simplifies to
\begin{equation}
\mathcal{H}_{BdG} =
\begin{bmatrix}
h_{\ell} &\Delta \\
\Delta^* & -h_{\ell}
\end{bmatrix}
\end{equation}
and its eigenvalues $E$ are given by
\begin{equation}
E=\pm \sqrt{h_{\ell}^2 + \Delta_0^2}
\end{equation}
This is just the superconductor dispersion relation, with a gap of $\Delta_0$ on either side of the fermi level. \\
\textbf{Case 2: Constant Magnetic Field} $\mathbf{B}=B_z\hat{z}$\\
In the Coulomb gauge we can write the vector potential for this magnetic field as
\begin{equation}
\mathbf{A} = A_{\theta}\hat{\bm{\theta}}
\end{equation}
From Stoke's law
\begin{equation}
\oint{\mathbf{A}\cdot Rd\bm{\theta}} = \int{\mathbf{B}\cdot d\mathbf{A}}
\end{equation}
Exploiting the symmetry of the cylindrical geometry, the above equation can be simplified to
\begin{equation}
A_{\theta} = \frac{\Phi\hbar}{e R}
\end{equation}
Using the same ansatz $\exp(ik_zz)\exp(i\ell\theta)$, the diagonal elements can be written as
\begin{equation}
\zeta^e_{\ell} = \frac{\hbar^2k_z^2}{2m^*} + \frac{\hbar^2\left(\ell-\Phi\right)^2}{2m^*R^2} - \mu = \frac{\hbar^2k_z^2}{2m^*}-\mu_{\ell} - \mathcal{E}_{\ell}
\label{eq:zeta_e}
\end{equation}
\begin{equation}
\zeta^h_{\ell} = \frac{\hbar^2k_z^2}{2m^*} + \frac{\hbar^2\left(\ell+\Phi\right)^2}{2m^*R^2} - \mu = \frac{\hbar^2k_z^2}{2m^*}-\mu_{\ell} + \mathcal{E}_{\ell}
\label{eq:zeta_h}
\end{equation}
for the electron and hole parts respectively. The effective chemical potential $\mu_{\ell}$ is defined in Eq.~\ref{eq:mu_eff}, and the field-coupling term $\mathcal{E}_{\ell} = \frac{\hbar^2\left(2\ell \Phi\right)}{2m^*R^2}$.
We note that
\begin{equation}
\zeta^{e (h)}_{\ell} = h_{\ell} \mp \mathcal{E}_{\ell}
\end{equation}
The BdG Hamiltonian can then be written as
\begin{equation}
\mathcal{H}_{BdG} =
\begin{bmatrix}
h_{\ell} &\Delta \\
\Delta^* & -h_{\ell}
\end{bmatrix}
- \mathcal{E}_{\ell} \mathbb{I}
\end{equation}
and the eigenvalues $E$ are given by
\begin{equation}
E=\pm \sqrt{h_{\ell}^2 + \Delta_0^2} - \mathcal{E}_{\ell}
\end{equation}
Thus, we see that a magnetic field ``shifts'' the superconducting gap. It is no longer centred at the fermi level.
\begin{figure}[!htb]
\begin{center}
\subfigure[$\ell \neq 0$ in the contacts]{%
\label{fig:lneq0}
\includegraphics[width=0.4\textwidth]{Figure8A}
}\\%
\subfigure[$\ell = 0$ in the contacts]{%
\label{fig:leq0}
\includegraphics[width=0.4\textwidth]{Figure8B}
\end{center}
\caption{%
The Andreev bound state spectrum in an SNS unction for a nanowire length $L= 160$ nm, chemical potential $\mu=30\Delta_0$ and healing length $\xi_0=222$ nm. We observe a shift of the superconducting gap when we have (a) $\ell \neq 0$ in the contacts. In (b), we constrain the contacts to have $\ell = 0$ and confirm that the gap does not shift.
}%
\end{figure}
While this may be a good model for a superconducting ``nanowire'', experimental setups usually involve flat contacts which naturally support only the $\ell = 0$ subband. For $\ell=0$, we have $\mathcal{E}_{\ell}=0$, and thus the superconducting gap stays fixed.
From Eqs.~\ref{eq:zeta_e},~\ref{eq:zeta_h}, the effective chemical potential for electron-like (hole-like) particles in the $\ell$ subband in the N-section is given by
\begin{equation}
\mu_{\ell}^{e(h)} = \mu - \frac{\hbar^2}{2m^*R^2}\left(\ell \mp \Phi\right)^2
\label{eq:mu_eff2}
\end{equation}
The electron and hole wavenumbers can then be written as a function of energy $(E)$
\begin{equation}
k^{e(h)}_{\ell}(E) = \frac{\sqrt{2m^*}}{\hbar}\sqrt{\mu^{e(h)}_{\ell} \pm E}
\end{equation}
\subsection{\label{sec:1d_eff} The 1-dimensional effective subband model}
As outlined above, it is important to ensure that we only have the $\ell = 0$ subbands in the contacts. We also note from Eqs.~\ref{eq:mu_eff},~\ref{eq:mu_eff2} that we can incorporate the effect of the angular momentum subbands via an effective potential $\mu_{\ell}$, and a field-coupling term $\mathcal{E}_{\ell}$.
The tight-binding Hamiltonian of the nanowire can be written as
\begin{equation}
\mathcal{H}^0 = \begin{bmatrix}
A & B & 0 & \dots & 0 \\
B^{\dagger} & A & B & 0 & 0\\
0 & \ddots & \ddots & \ddots & 0\\
\vdots & 0 & \ddots & \ddots & B\\
0 & 0 & 0 & B^{\dagger} & A
\end{bmatrix}
\label{eq:H0}
\end{equation}
\begin{equation}
A = \left(\alpha_{-l} \oplus \dots \oplus \alpha_{+l} \right) \text{ and } B= \mathbb{I}_{N_l} \otimes \beta
\label{eq:AB}
\end{equation}
\begin{equation}
\alpha_{l} =
\begin{bmatrix}
2t_z - \mu + t_y\left(l-\Phi\right)^2 & 0 \\
0 & -2t_z + \mu-t_y\left(l+\Phi\right)^2
\end{bmatrix}
\label{eq:alpha_l}
\end{equation}
\begin{equation}
\beta =
\begin{bmatrix}
-t_z & 0 \\
0 & t_z
\end{bmatrix}
\label{eq:beta_l}
\end{equation}
where $N_l$ is the number of subbands. (For Example, $N_{\ell}=3 \implies $ ($\ell={-1,0,1}$)) and $\mathbb{I}_n$ is the $n \times n$ identity matrix.
Meanwhile, the Hamiltonian of the contacts takes a similar form with
\begin{equation}
A = \left(\mathbb{I}_{N_{\ell}} \otimes \alpha_{0} \right) \text{ and } B= \mathbb{I}_{N_{\ell}} \otimes \beta
\end{equation}
\section{\label{sec:oscillations} Supercurrent Oscillations}
We compute the CPR of an SNS junction at finite magnetic fields, within the shell conduction model and a radius of 30 nm. Temperature is set to $T=100$ mK in all the simulations. In Fig.~\ref{fig:CPR1_B}, we show the current phase relations as a function of the magnetic flux for a single occupied subband. Since only the $\ell=0$ subband is populated, there is no phase shift in the ABS, and the CPR retains its saw-tooth shape with a maximum near $\phi=\pi$. We plot the critical current as a function of the flux in Fig.~\ref{fig:CC_1subband}. The gradual fall in the critical current can be attributed to the decrease in average quasiparticle momentum with increasing flux, as shown in Eq.~\ref{eq:fermi_vel}. Eventually, at $\Phi = 4.04$ the band depopulates ($\min\left(\mu_{\ell}^e,\mu_{\ell}^h\right) = 0$) and the current falls to zero. We observe in Fig.~\ref{fig:CC_1subband} that the critical current does not monotonically decrease to zero, particularly for $\Phi \in [3,4]$. The appearance of these small oscillations is due to the interference with the quasiparticles normally reflected from the N/S interfaces.
As discussed in Appendix~\ref{sec:BeyondAA}, the discontinuity in the density of states gives rise to normal reflections. These reflected quasiparticles interfere and result in the non-monotonic decrease of the single subband critical current.
Next, we consider the case when three subbands are occupied ($|\ell| \leq 1$). The magnetic field evolution of the current phase relation is plotted in Fig.~\ref{fig:CPR3_B}. Since the ABS for the $|\ell| = 1$ subbands are phase shifted in presence of a flux (Fig.~\ref{fig:leq0}), the total current is no longer maximum near $\phi = \pi$. The current in the junction is the sum over the individual subband currents, and hence the flux-dependent phase shift results in an interference pattern. The phase shift in a subband CPR is proportional to the difference in the electron-hole wavenumbers $(k_{\ell}^e-k_{\ell}^h)$, and the length $L$ of the junction. Hence, the fluxes at which the subband currents constructively interfere need not occur at integer multiples of the flux quantum ($\Phi_0=h/e$).
In Fig.~\ref{fig:CC_3subband} we plot the critical current for three occupied subbands as a function of the axial flux. We see several oscillations of the critical current before the $|\ell| = 1$ subbands depopulate at $\Phi = 3.04$. At zero flux, the CPR of each subband is maximum near $\phi = \pi$ and hence they all add up constructively. Each subband contributes equally to the critical current.
As the flux is increased, the electron-hole pairs in the $|\ell| = 1$ subbands pickup a phase and the subband CPRs no longer interfere constructively. Consequently, the critical current decreases with flux. At $\Phi=0.72$, the critical current switches phase from $\phi<\pi$ to $\phi>\pi$ and the current increases again. This increase persists till $\Phi =1.08$ at which point the current is maximum near $\phi=2\pi$. At this flux, the $|\ell| = 1$ subband current peaks near $\phi=2\pi$ while it is negligible near $\phi=\pi$. Hence, this secondary peak, which only involves contribution from $|\ell|=1$ subbands is approximately a two-third of the primary peak and corresponds to a phase pickup of $\pi$ in the aforementioned subbands. Finally, as noted earlier, the magnitudes of the primary and secondary peaks progressively diminish due to the decrease in average quasiparticle velocity.
\begin{figure*}[!htb]
\begin{center}
\subfigure[ $\ell = 0$
{\label{fig:CPR1_B}
\includegraphics[width=0.4\textwidth]{SingleSubband_CPR}
}%
\subfigure[ $|\ell| = 0,1$
{\label{fig:CPR3_B}
\includegraphics[width=0.4\textwidth]{ThreeSubband_CPR}
}\\%
\subfigure[ $|\ell| = 0$
{\label{fig:CC_1subband}
\includegraphics[width=0.33\textwidth]{SingleSubband}
}%
\subfigure[ $|\ell| = 0,1,2$
{\label{fig:CC_3subband}
\includegraphics[width=0.33\textwidth]{ThreeSubband}
\subfigure[ $|\ell| = 0,1$
{ \label{fig:CC_5subband}
\includegraphics[width=0.33\textwidth]{FiveSubband}
\end{center}
\caption{%
The current-phase-relation (CPR) in presence of an applied magnetic flux is plotted in a (a) one subband model and (b) three subband model. The critical current is plotted as a function of the applied magnetic flux in a (c) one subband, (d) three subband and (e) five subband model. The black vertical dotted lines in (d) and (e) denote the depopulation of the subbands, in a descending order of the angular momentum quantum number. The CPRs in (b) are the sum over the subband currents. The $|\ell|=1,2$ subbands pick up a phase proportional to the difference in the quasiparticle momenta and hence the critical current oscillates with the applied flux. The simulations were performed for $L=160$ nm, $\mu=30\Delta_0$, $\xi_0=222$ nm.}%
\end{figure*}
Finally, we consider the situation when five subbands are occupied ($|\ell| \leq 2$). The critical current is plotted as a function of the magnetic flux in Fig.~\ref{fig:CC_5subband}. Once again, at $\Phi=0$ the subband currents are all in-phase and constructively interfere to give a maximum. In presence of a magnetic field, the quasiparticles in the $|\ell|=1$ and $|\ell|=2$ subbands pick up different phases and hence, they do not appear to constructively interfere again in presence of a magnetic field to recover the zero field critical current.
The absence of such oscillations with a single occupied subband (Fig.~\ref{fig:CC_1subband}) confirms the subband supercurrent interference as the causal agent.
\subsection{Effect of Disorder}
In order to simulate experimentally relevant conditions, we include a random uncorrelated onsite disorder potential $u \in [-W,W]$ in the semiconductor. This models phase-coherent scattering events in the junction. We parameterise the disorder by the mean free path ($\lambda_{mf}$), which is estimated from the disorder-averaged normal state conductance ($g$) using the following relation
\begin{equation}
g = \frac{2e^2}{h}N_{\ell}\frac{1}{\left(1+L/\lambda_{mf}\right)}
\label{eq:mfp
\end{equation}
$N_{\ell}$ is the number of subbands and $L$ is the length of the junction.
In Fig.~\ref{fig:SNSDisorder} we plot the critical current oscillations in a nanowire for particular realisations of the disorder. While the initial decay and the oscillations are still present, the secondary maxima are suppressed. In a clean nanowire with a saw-tooth CPR (which peaks near $\phi=\pi$ at zero field), at a magnetic flux $\Phi^*$ the $|\ell|=1$ subbands pickup a phase of $\pi$ and their CPRs peak near $\phi=2\pi$. The $\ell=0$ subband retains its sawtooth CPR with a negligible current near $\phi=2\pi$. This results in the secondary maximum.
Upon adding disorder to the nanowire we depart from this saw-tooth CPR, tending towards a sinusoidal CPR which peaks further away from $\phi=\pi$ at zero field. Thus, there exists no $\Phi^*$ at which the $|\ell| = 1$ subband current peaks while the $\ell=0$ subband current is negligible. As a consequence of the sinusoidal CPR, on picking up a phase of $\pi$ the $|\ell| = 1$ subbands destructively interfere with the $\ell = 0$ subband, and this causes the suppression of the secondary maxima.
\begin{figure*}[!htb]
\begin{center}
\subfigure[$\lambda_{mf} = 30$ nm, $L = 160$ nm
{%
\label{fig:CC_L160_l30}
\includegraphics[width=0.4\textwidth,keepaspectratio]{ThreeSubband_lmf-30_it2}
\subfigure[$\lambda_{mf} = 30$ nm, $L = 160$ nm]
{%
\includegraphics[width=0.4\textwidth,keepaspectratio]{ThreeSubband_lmf-30_it7}
}\\%
\subfigure[$\lambda_{mf} = 80$ nm, $L = 160$ nm]
{%
\includegraphics[width=0.4\textwidth,keepaspectratio]{ThreeSubband_lmf-77_it7}
}%
\subfigure[$\lambda_{mf} = 30$ nm, $L = 240$ nm]
{%
\includegraphics[width=0.4\textwidth,keepaspectratio]{ThreeSubband_L240-avg}
}%
\end{center}
\caption{%
The critical current oscillations for a disordered SNS Josephson junction. The disorder is parameterised by the mean-free path ($\lambda_{mf}$) which is calculated from the normal state disorder-averaged conductance using Eq.~\ref{eq:mfp}. Each sub-figure shows the critical current evolution for a given realisation of the disorder potential, and is labelled by the mean-free path $\lambda_{mf}$} and nanowire length $L$. The chemical potential $\mu=30\Delta_0$ for all the plots. (a) and (b) plot the oscillations for two different realisations of a random disorder potential resulting in $\lambda_{mf}= 30$ nm.
\label{fig:SNSDisorder}
\end{figure*}
As shown in Fig.~\ref{fig:CleanDis}, the first crticial current node in a disordered junction occurs at a lower magnetic field as compared to the clean nanowire. In the presence of scatterers the effective path traversed by the quasiparticles increases and hence, the subbands destructively interfere at a lower flux.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.4\textwidth]{nodes}
\end{center}
\caption{%
The critical current as a function of the applied magnetic flux for a clean and disordered ($\lambda_{mf} = 30, 77$ nm) junction. We observe that the first node occurs at a lower field as compared to a clean junction.}%
\label{fig:CleanDis}
\end{figure}
Furthermore, as shown by \citeauthor{Frolov}, the essential effect of disorder can be observed by the dependence of the critical current oscillations on the gate voltage. As shown in Fig.~\ref{fig:clean} for the clean nanowire, small variations in the gate voltage hardly cause any fluctuations in the oscillations. This is because small changes in the chemical potential do not change the number of occupied subbands and only weakly affects the quasiparticle transmission through the junction. However, in a disordered nanowire with a small mean free path, the quasiparticles traverse a longer path in the nanowire and hence, the critical current oscillations are significantly affected by the gate voltage. This is shown in Figs.~\ref{fig:dis1},\ref{fig:dis2} for two disorder realisations.
From Figs.~\ref{fig:SNSDisorder},\ref{fig:EffDis1} we infer that the critical current oscillations are highly sensitive to the gate voltage and the particular realisation of the disorder. Thus, a macroscopic current measurement indirectly gives us information about the microscopic specifics of the junction. However, while our model provides a qualitative understanding of the oscillations, the high sensitivity w.r.t. the microscopic parameters renders a quantitative description of the experiment very difficult.
\begin{figure}[!htb]
\begin{center}
\subfigure[Clean Junction]{%
\label{fig:clean}
\includegraphics[width=0.4\textwidth]{ThreeSubband_mu3033_Clean}
}\\%
%
\subfigure[$\lambda_{mf} = 30$ nm, $L = 160$ nm ]{%
\label{fig:dis1}
\includegraphics[width=0.4\textwidth]{ThreeSubband_mu3033_1}
}\\%
\subfigure[$\lambda_{mf} = 30$ nm, $L = 160$ nm]{%
\label{fig:dis2}
\includegraphics[width=0.4\textwidth]{ThreeSubband_mu3033_2}
}
\end{center}
\caption{%
Small fluctuations in the gate voltage change the chemical potential ($\mu$) between the blue ($\mu = 30\Delta_0$) and red ($\mu=33\Delta_0$) curves. $\Delta_0$ is the superconducting order parameter. We observe small variations in the oscillations on slightly varying the chemical potential in (a) a clean junction. For a disordered junction in (b),(c) we observe larger fluctuations in the oscillations in response to small variations in the chemical potential. Two instances of the disorder potential are shown.
}%
\label{fig:EffDis1}
\end{figure}
\subsection{Dephasing in the nanowire}
The analysis presented in the previous sections described the phase-coherent flow of quasiparticles in the junction. We now include phase-breaking processes \cite{Golizadeh,datta_2005}, which may arise from the interaction of an electron with the surrounding bath of phonons, or other electrons. Molecular beam epitaxy grown InAs nanowires typically have a phase relaxation length of the order of a few-hundred nanometers $l_{\phi} \sim 300$ nm\cite{Estevez_2010,Blomers2}. Hence, for a junction with $L=160$ nm, we only include elastic electron-phonon scattering processes. Adopting a homogeneous model, we assume an identical electron-phonon coupling strength $D_0 = 0.001$ eV$^2$ throughout the nanowire.
We subsume these processes within the NEGF formalism by including a self-energy term for the phonon bath $\Sigma^r_s$, proportional to the Green's function and the electron-phonon coupling strength. This calls for a self-consistent computation of the Green's function and the bath self-energy \cite{Golizadeh}
\begin{align}
G^r(E) &= \left(E\mathbb{I} + i \eta-\mathcal{H}-\Sigma_1-\Sigma_2 - \Sigma_s \right)^{-1}\\
\Sigma_s &= D \times G^r(E)
\end{align}
where $\times$ denotes element by element multiplication, and $D=D_0\mathbb{I}$. This model corresponds to spatially uncorrelated electron-phonon interactions, and relaxes both the phase and momentum of the quasiparticles in the nanowire.
The critical current oscillations in the presence of elastic electron-phonon interactions are shown in Fig.~\ref{fig:PBProcesses}. Figure~\ref{fig:C&NC} includes the disorder potential profile from Fig.~\ref{fig:CC_L160_l30} in addition to the phase-breaking processes. The oscillations in Fig.~\ref{fig:PBProcesses} are in excellent qualitative agreement with the experiments by \citeauthor{IQC} and \citeauthor{Frolov}. Thus, we infer that phase-breaking processes play a non-negligible role in III-V semiconductor nanowire Josephson junctions.
\begin{figure}[!htb]
\begin{center}
\subfigure[]{%
\label{fig:onlyNC}
\includegraphics[width=0.4\textwidth]{OnlyNonCoherent.png}
}\\%
%
\subfigure[]{%
\label{fig:C&NC}
\includegraphics[width=0.4\textwidth]{NonCoherent_Coherent.png}
}
\end{center}
\caption{%
Critical current oscillations with phase-breaking scattering processes in the nanowire. The electron-phonon interaction is parameterised by the coupling strength $D_0 = 0.001$ eV$^2$. (a) The nanowire is free of disorder and only phase-breaking processes are involved. (b) The red curve corresponds to a nanowire with an onsite potential distribution from Fig.~\ref{fig:CC_L160_l30} in addition to the phase-breaking processes. The blue curve is identical to (a). The simulations were performed for $L=160$ nm, $\mu=30\Delta_0$, $D_0=0.001$ eV$^2$.
}%
\label{fig:PBProcesses}
\end{figure}
\section{Conclusion}
In this paper we employed the Keldysh Non-Equilibrium Green's Function formalism to model quantum transport in semiconductor nanowire Josephson junctions. We analyzed semiconductor nanowire Josephson junctions using a three-dimensional discrete lattice model described by the Bogolubov-de Gennes Hamiltonian in the tight-binding approximation, and computed the Andreev bound state spectrum and current-phase relations. We went beyond the Andreev approximation limit and investigated the avoided level crossing in the ABS spectrum. Our results confirm the measured critical current oscillations to arise from the subband supercurrent interference in presence of an axial magnetic field. The phase picked up by the quasiparticles depend on the difference of their wavenumbers, the length of the junction and the angular momentum quantum number. Thus, the oscillations do not show any periodicity in the flux quantum. We included phase-coherent scattering to model a disordered junction and investigated its effect on the critical current oscillations. We observe that the oscillations in the disordered junction are highly sensitive to the realisation of the random disorder potential, and on small fluctuations of the gate voltage. This high sensitivity makes a quantitative description of the experiment a challenging task. Nevertheless, a macroscopic current measurement conveys valuable information about the microscopic profile of the junction. Finally, we include elastic dephasing in the nanowire by modelling weak electron-phonon interactions. We observe an excellent qualtitive match of our results with the experiment, and this underscores the role played by phase-breaking processes in III-V nanowire Josephson junctions.
\indent {\it{Acknowledgements: }} The authors BM and PS would like to thank Prof. Supriyo Datta, Prof. Kantimay Das Gupta and Abhishek Sharma for useful discussions throughout this work. This work is an outcome of the Research and Development work undertaken in the project under the Visvesvaraya PhD Scheme of Ministry of Electronics and Information Technology, Government of India, being implemented by Digital India Corporation (formerly Media Lab Asia). This work was also supported by the Science and Engineering Research Board (SERB) of the Government of India under Grant number EMR/2017/002853.
KG and JB acknowledge funding from the Natural Sciences and Engineering Research Council of Canada (NSERC).
|
{
"timestamp": "2019-03-01T02:12:04",
"yymm": "1902",
"arxiv_id": "1902.10947",
"language": "en",
"url": "https://arxiv.org/abs/1902.10947"
}
|
\section{Introduction}
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The style file \verb'mnras.cls' can be used to approximate the final appearance of the journal, and provides numerous features to simplify the preparation of papers.
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\begin{verbatim}
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\section{Preparing and submitting a paper}
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Rename the file, update the information on the title page, and then work on the text of your paper.
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\label{sec:options}
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\begin{verbatim}
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\end{equation}
By default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \verb'\numberwithin{equation}{section}' to the preamble.
It is also possible to produce un-numbered equations by using the \LaTeX\ built-in \verb'\['\textellipsis\verb'\]' and \verb'$$'\textellipsis\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.
\subsection{Special symbols}
\begin{table}
\caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}
\label{tab:anysymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\sun' & \sun & Sun, solar\\[2pt]
\verb'\earth' & \earth & Earth, terrestrial\\[2pt]
\verb'\micron' & \micron & microns\\[2pt]
\verb'\degr' & \degr & degrees\\[2pt]
\verb'\arcmin' & \arcmin & arcminutes\\[2pt]
\verb'\arcsec' & \arcsec & arcseconds\\[2pt]
\verb'\fdg' & \fdg & fraction of a degree\\[2pt]
\verb'\farcm' & \farcm & fraction of an arcminute\\[2pt]
\verb'\farcs' & \farcs & fraction of an arcsecond\\[2pt]
\verb'\fd' & \fd & fraction of a day\\[2pt]
\verb'\fh' & \fh & fraction of an hour\\[2pt]
\verb'\fm' & \fm & fraction of a minute\\[2pt]
\verb'\fs' & \fs & fraction of a second\\[2pt]
\verb'\fp' & \fp & fraction of a period\\[2pt]
\verb'\diameter' & \diameter & diameter\\[2pt]
\verb'\sq' & \sq & square, Q.E.D.\\[2pt]
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Additional commands for mathematical symbols. These can only be used in maths mode.}
\label{tab:mathssymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\upi' & $\upi$ & upright pi\\[2pt]
\verb'\umu' & $\umu$ & upright mu\\[2pt]
\verb'\upartial' & $\upartial$ & upright partial derivative\\[2pt]
\verb'\lid' & $\lid$ & less than or equal to\\[2pt]
\verb'\gid' & $\gid$ & greater than or equal to\\[2pt]
\verb'\la' & $\la$ & less than of order\\[2pt]
\verb'\ga' & $\ga$ & greater than of order\\[2pt]
\verb'\loa' & $\loa$ & less than approximately\\[2pt]
\verb'\goa' & $\goa$ & greater than approximately\\[2pt]
\verb'\cor' & $\cor$ & corresponds to\\[2pt]
\verb'\sol' & $\sol$ & similar to or less than\\[2pt]
\verb'\sog' & $\sog$ & similar to or greater than\\[2pt]
\verb'\lse' & $\lse$ & less than or homotopic to \\[2pt]
\verb'\gse' & $\gse$ & greater than or homotopic to\\[2pt]
\verb'\getsto' & $\getsto$ & from over to\\[2pt]
\verb'\grole' & $\grole$ & greater over less\\[2pt]
\verb'\leogr' & $\leogr$ & less over greater\\
\hline
\end{tabular}
\end{table}
Some additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\ref{tab:anysymbols}--\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.
Many other mathematical symbols are also available, either built into \LaTeX\ or via additional packages. If you want to insert a specific symbol but don't know the \LaTeX\ command, we recommend using the Detexify website\footnote{\url{http://detexify.kirelabs.org}}.
Sometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.
To produce bold symbols in mathematics, use \verb'\bmath' for simple variables, and the \verb'bm' package for more complex symbols (see section~\ref{sec:packages}). Vectors are set in bold italic, using \verb'\mathbfit{}'.
For matrices, use \verb'\mathbfss{}' to produce a bold sans-serif font e.g. \mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\nabla$ (del, used in gradients, divergence etc.) use \verb'$\nabla$'.
\subsection{Ions}
A new \verb'\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.
For example, to typeset singly ionised calcium use \verb'\ion{Ca}{ii}', which produces \ion{Ca}{ii}.
\section{Figures and tables}
\label{sec:fig_table}
Figures and tables (collectively called `floats') are mostly the same as built into \LaTeX.
\subsection{Basic examples}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
Figures are inserted in the usual way using a \verb'figure' environment and \verb'\includegraphics'. The example Figure~\ref{fig:example} was generated using the code:
\begin{verbatim}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
\end{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
The example Table~\ref{tab:example} was generated using the code:
\begin{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
\subsection{Captions and placement}
Captions go \emph{above} tables but \emph{below} figures, as in the examples above.
The \LaTeX\ float placement commands \verb'[htbp]' are intentionally disabled.
Layout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.
Simply place the \LaTeX\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.
By default a figure or table will occupy one column of the page.
To produce a wider version which covers both columns, use the \verb'figure*' or \verb'table*' environment.
If a figure or table is too long to fit on a single page it can be split it into several parts.
Create an additional figure or table which uses \verb'\contcaption{}' instead of \verb'\caption{}'.
This will automatically correct the numbering and add `\emph{continued}' at the start of the caption.
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
Table~\ref{tab:continued} was generated using the code:
\begin{verbatim}
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
To produce a landscape figure or table, use the \verb'pdflscape' package and the \verb'landscape' environment.
The landscape Table~\ref{tab:landscape} was produced using the code:
\begin{verbatim}
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & ...\\
Unit & Unit & ...\\
\hline
Data & Data & ...\\
Data & Data & ...\\
...\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\end{verbatim}
Unfortunately this method will force a page break before the table appears.
More complicated solutions are possible, but authors shouldn't worry about this.
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\
Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\
\hline
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\section{References and citations}
\subsection{Cross-referencing}
The usual \LaTeX\ commands \verb'\label{}' and \verb'\ref{}' can be used for cross-referencing within the same paper.
We recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.
This ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).
It is best to give each section, figure and table a logical label.
For example, Table~\ref{tab:mathssymbols} has the label \verb'tab:mathssymbols', whilst section~\ref{sec:packages} has the label \verb'sec:packages'.
Add the label \emph{after} the section or caption command, as in the examples in sections~\ref{sec:sections} and \ref{sec:fig_table}.
Enter the cross-reference with a non-breaking space between the type of object and the number, like this: \verb'see Figure~\ref{fig:example}'.
The \verb'\autoref{}' command can be used to automatically fill out the type of object, saving on typing.
It also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.
For example, \verb'\autoref{tab:journal_abbr}' produces \autoref{tab:journal_abbr}.
\subsection{Citations}
\label{sec:cite}
MNRAS uses the Harvard -- author (year) -- citation style, e.g. \citet{author2013}.
This is implemented in \LaTeX\ via the \verb'natbib' package, which in turn is included via the \verb'usenatbib' package option (see section~\ref{sec:options}), which should be used in all papers.
Each entry in the reference list has a `key' (see section~\ref{sec:ref_list}) which is used to generate citations.
There are two basic \verb'natbib' commands:
\begin{description}
\item \verb'\citet{key}' produces an in-text citation: \citet{author2013}
\item \verb'\citep{key}' produces a bracketed (parenthetical) citation: \citep{author2013}
\end{description}
Citations will include clickable links to the relevant entry in the reference list, if supported by your \LaTeX\ compiler.
\defcitealias{smith2014}{Paper~I}
\begin{table*}
\caption{Common citation commands, provided by the \texttt{natbib} package.}
\label{tab:natbib}
\begin{tabular}{lll}
\hline
Command & Ouput & Note\\
\hline
\verb'\citet{key}' & \citet{smith2014} & \\
\verb'\citep{key}' & \citep{smith2014} & \\
\verb'\citep{key,key2}' & \citep{smith2014,jones2015} & Multiple papers\\
\verb'\citet[table 4]{key}' & \citet[table 4]{smith2014} & \\
\verb'\citep[see][figure 7]{key}' & \citep[see][figure 7]{smith2014} & \\
\verb'\citealt{key}' & \citealt{smith2014} & For use with manual brackets\\
\verb'\citeauthor{key}' & \citeauthor{smith2014} & If already cited in close proximity\\
\verb'\defcitealias{key}{Paper~I}' & & Define an alias (doesn't work in floats)\\
\verb'\citetalias{key}' & \citetalias{smith2014} & \\
\verb'\citepalias{key}' & \citepalias{smith2014} & \\
\hline
\end{tabular}
\end{table*}
There are a number of other \verb'natbib' commands which can be used for more complicated citations.
The most commonly used ones are listed in Table~\ref{tab:natbib}.
For full guidance on their use, consult the \verb'natbib' documentation\footnote{\url{http://www.ctan.org/pkg/natbib}}.
If a reference has several authors, \verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \bibtex\ (see section~\ref{sec:ref_list}) then this is handled automatically. If not, the \verb'\citet*{}' and \verb'\citep*{}' commands can be used at the first citation to include all of the authors.
\subsection{The list of references}
\label{sec:ref_list}
It is possible to enter references manually using the usual \LaTeX\ commands, but we strongly encourage authors to use \bibtex\ instead.
\bibtex\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.
An MNRAS \bibtex\ style file, \verb'mnras.bst', is distributed as part of this package.
The rest of this section will assume you are using \bibtex.
References are entered into a separate \verb'.bib' file in standard \bibtex\ formatting.
This can be done manually, or there are several software packages which make editing the \verb'.bib' file much easier.
We particularly recommend \textsc{JabRef}\footnote{\url{http://jabref.sourceforge.net/}}, which works on all major operating systems.
\bibtex\ entries can be obtained from the NASA Astrophysics Data System\footnote{\label{foot:ads}\url{http://adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.
Simply copy this into your \verb'.bib' file or into the `BibTeX source' tab in \textsc{JabRef}.
Each entry in the \verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.
Simply cite it in the usual way, as described in section~\ref{sec:cite}, using the specified key.
Compile the paper as usual, but add an extra step to run the \texttt{bibtex} command.
Consult the documentation for your compiler or latex distribution.
Correct formatting of the reference list will be handled by \bibtex\ in almost all cases, provided that the correct information was entered into the \verb'.bib' file.
Note that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.
If in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\ref{foot:itas}}$ for the current guidelines on how to format the list of references.
\section{Appendices and online material}
To start an appendix, simply place the \verb'
\section{Introduction}
\label{introduction}
Protoplanetary discs (`Proplyds') are thin Keplerian discs around pre-main-sequence stars \citep{1994ApJ...429..781S} and are the birth places of planets. Proplyds form as a result of angular momentum conservation during the gravitational collapse of clouds when stars are forming. Within the discs, dust can coagulate to form a range of objects from pebbles to planets.
The evolution and dispersal of protoplanetary discs controls the planet formation process, and observations suggest that disc lifetimes are between $\approx 3 - 5$ Myr \citep[e.g.][]{1995Natur.373..494Z,2001ApJ...553L.153H,2006ApJ...651.1177P,2018MNRAS.477.5191R}. Internal processes remove mass from the protoplanetary disc and, after several Myr, disc accretion slows significantly to the point that these processes begin removing more mass than can be replaced, leading to very rapid disc dispersal \citep{2001MNRAS.328..485C, 2011MNRAS.412...13O}.
Proplyd host stars do not form in isolation, but rather in clusters and associations with stellar densities that exceed that of the Galactic field by a few orders of magnitude \citep{2003ARA&A..41...57L,2010MNRAS.409L..54B}. Tens to thousands of stars can form in these regions that are a fraction of a parsec in size \citep{2000prpl.conf..151C}. Observations of young star-forming regions have revealed that stars form in filamentary structures \citep{2010A&A...518L.102A, 2014MNRAS.439.3275W}, resulting in hierarchical spatial distributions. The net motion of stars within these regions indicate that the structures are often collapsing (i.e. sub-virial) \citep{2006A&A...445..979P,2015ApJ...799..136F,2018arXiv180702115K}.
The initial densities of star-forming regions are difficult to determine, and span a wide range \citep{2010MNRAS.409L..54B,2012MNRAS.421.2025K}, but many are thought to be at least $\sim$ 100 M$_{\odot}$ pc$^{-3}$ at the epoch of star formation \citep{2014MNRAS.445.4037P}. \cite{2014MNRAS.445.4037P} shows that two regions with similar present-day densities at present times may have originally had very different initial densities because initially dense regions expand much faster than lower-density regions due to two-body relaxation. They compare the present-day stellar densities and amount of spatial substructure in seven star-forming regions, including the Orion Nebula Cluster and Upper Scorpius -- which both contain massive stars that could act as external photoionising sources -- to infer the likely range of initial stellar densities in each of these star-forming regions and all are consistent with having an initial density in the range 10 -- 1000 M$_{\odot}$ pc$^3$.
External processes, such as close stellar interactions, can also cause proplyds to be truncated or destroyed, as well as disrupting the orbits of fledgling planets \citep{2000A&A...362..968A, 2001MNRAS.323..785B, 2001MNRAS.325..449S, 2006ApJ...641..504A, 2008A&A...488..191O, 2012MNRAS.419.2448P, 2014MNRAS.441.2094R, 2015A&A...577A.115V, 2016MNRAS.457..313P, 2018arXiv180400013W}. The density of the star forming region will affect the rate of stellar interactions, with stars in low-density environments experiencing fewer dynamical interactions than higher density environments \citep{2014MNRAS.438..639W, 2010MNRAS.409L..54B}. Furthermore, star-forming regions can contain massive stars (> 15\,M$_{\odot}$), whose intense far ultra-violet (FUV) and extreme ultra-violet (EUV) radiation fields are significantly higher than those in the interstellar medium \citep{2000A&A...362..968A,2004ApJ...611..360A, 2008ApJ...675.1361F}. This high-energy radiation heats the gaseous material of the upper layers of the disc until the thermal energy of the heated layer exceeds the gravitational potential of the disc, causing it to escape as a photoevaporative wind \citep{1994ApJ...428..654H,1998ApJ...499..758J}. This mass loss will affect the evolution of protoplanetary discs, and reduce the reservoir of material available to form gas giant planets \citep{2018MNRAS.475.5460H}.
The effects of external photoevaporation appear to be observed in nearby star-forming regions, such as the Orion Nebula Cluster (ONC) \citep{1996AJ....111.1977M, 2016ApJ...826...16E, 2018ApJ...860...77E} and $\sigma$ Orionis \citep{2017AJ....153..240A}. The ONC has been preferentially observed due to its proximity to Earth ($\sim$415 pc) and because the discs can be viewed in silhouette due to the bright nebulous background. Studies have also shown that 80 -- 85\% of stars within the ONC host protoplanetary discs \citep{2000AJ....119.2919B, 2000AJ....120.3162L}, making the ONC a favorable target for studying disc evolution and dispersal \citep[though see][for an alternative interpretation which posits that the ONC proplyds are merely ionisation fronts of material left over from discs that are almost destroyed]{2007MNRAS.376.1350C}.
Due to two-body and violent relaxation, initially highly substructured star-forming regions can evolve to smooth and centrally concentrated clusters after only a few Myr \citep{2010MNRAS.407.1098A, 2014MNRAS.438..620P}. Furthermore, two clusters that presently have similar densities may have had very different initial densities because initially very dense clusters expand faster than lower density counterparts. As mentioned before, the initial density will affect the rate at which protoplanetary discs are disrupted and destroyed due to stellar interactions. However, how much the initial density and substructure of a star-forming region affects the rate of protoplanetary disc dispersal due to external photoevaporation has yet to be studied.
Previous studies into the effects of external photoevaporation on protoplanetary discs in star-forming regions have tended to calculate the background UV radiation without directly calculating the disc mass-loss \citep{2000A&A...362..968A, 2004ApJ...611..360A}. \cite{2001MNRAS.325..449S} did calculate mass-loss rates in simulations specifically tailored to match the ONC, but assumed rather low stellar densities ($\sim$40\,M$_\odot$\,pc$^{-3}$), whereas \citet{2014MNRAS.445.4037P} suggests that the initial density of the ONC may have been much higher (>100\,M$_\odot$\,pc$^{-3}$).
These previous studies of external photoevaporation used spherically smooth spatial distributions with primordial mass segregation to model the environment of the ONC as observed today. However, observations of star forming regions show that stars form in highly substructured filamentary environments, where the stars are moving with subvirial velocities. The initial net motion and spatial structure of a star-forming region will govern its future evolution, and by extension the degree to which planet formation is hindered. These initial conditions lead to dynamical mass segregation on timescales of the age of the ONC \citep{2010MNRAS.407.1098A,2014MNRAS.438..620P}, negating the requirement for primordial mass segregation \citep{1998MNRAS.295..691B}.
Here, we focus on initial conditions that more closely reflect observations of young star forming regions \citep{2004MNRAS.348..589C}, and determine how much external photoevaporation affects the evolution of protoplanetary discs.. Therefore, we do not centrally concentrate our massive stars, but randomly distribute them in our simulated star-forming regions. We run suites of simulations that cover a range of initial conditions, with varying initial density, spatial distribution and net bulk motion (virial ratio). We then calculate and compare the mass-loss rates due to external photoevaporation for each set of initial conditions.
The paper is organised as follows. In Section~\ref{method} we describe our $N$-body simulations, protoplanetary disc assumptions and our external photoevaporation prescription; in Section~\ref{results} we present our results; we provide a discussion in Section~\ref{discussion} and we conclude in Section~\ref{conclusion}.
\section{Method}
\label{method}
In this section we describe our method to select low-mass star-forming regions containing massive stars, before describing the subsequent $N$-body and stellar evolution of these regions.
\subsection{Creating low-mass star-forming regions}
We adopt two different masses ($100$ or $1000$M$_{\odot}$) for our star-forming regions and populate these regions with stars drawn randomly from the initial mass function (IMF) parameterised in \citet{2013MNRAS.429.1725M}, which has a probability density function of the form:
\begin{equation}
p(m) \propto \left(\frac{m}{\mu}\right)^{-\alpha}\left(1 + \left(\frac{m}{\mu}\right)^{1 - \alpha}\right)^{-\beta}
\label{imf}.
\end{equation}
Here, $\mu = 0.2$\,M$_\odot$ is the average stellar mass, $\alpha = 2.3$ is the \citet{1955ApJ...121..161S} power-law exponent for higher mass stars, and $\beta = 1.4$ is used to describe the slope of the IMF for low-mass objects \citep*[which also deviates from the log-normal form;][]{2010ARA&A..48..339B}. Finally, we sample from this IMF within the mass range $m_{\rm low} = 0.1$\,M$_\odot$ to $m_{\rm up} = 50$\,M$_\odot$.
We use a ``soft-sampling'' technique for sampling the IMF \citep{2006ApJ...648..572E}, which implies that the only formal limit on the most massive star that can form is that of the upper limit of the IMF \citep{2007MNRAS.380.1271P}. From an ensemble of Monte Carlo simulations, we find that typically, a $1000$M$_{\odot}$ star-forming region will contain five massive stars (M$_{\star} > 15$M$_{\odot}$).
Low-mass star-forming regions rarely contain massive stars; however, if the only limit on the mass of the star that can form is the total mass of the star-forming region itself then occasionally we would expect a low-mass star-forming region to contain one or more massive stars and such regions are observed \citep[e.g. $\gamma^2$ Vel, a low-mass region containing at least two massive stars,][]{2014A&A...563A..94J}. Note that we are not explicitly attempting to model the $\gamma^2$ Vel star-forming region, which harbours a dense (sub)cluster within a more diffuse region. Instead, we are pointing out that observational examples of low-mass star-forming regions such as this and $\sigma$~Orionis occasionally contain massive stars and their photoevaporative effects on discs in these low-mass star-forming regions could be important.
In order to demonstrate the importance of low-mass star-forming regions, let us consider the demographics of star-forming regions from randomly sampling the underlying probability distributions. The observed mass function of star-forming regions follows a power-law of the form
\begin{equation}
N \propto M_{\rm cl}^{-2},
\label{cmf}
\end{equation}
where $N$ is the number of star-forming regions with mass $M_{\rm cl}$ \citep{2003ARA&A..41...57L}, which implies that there many more low-mass star-forming regions compared to high-mass regions.
We follow the procedure in \citet{2007MNRAS.380.1271P} and \citet{2017MNRAS.464.4318N} and sample $1 \times 10^6$ star-forming regions in the mass range 50 -- $10^5$\,M$_\odot$. Of these, $\sim$1200 lie in the mass range 1000$\pm$10\,M$_\odot$, and $\sim$ 15\,000 lie in the mass range 100$\pm$10\,M$_\odot$.
We then randomly populate our star-forming regions with stars drawn from Eqn.~\ref{imf}, until the total mass of stars equals or exceeds the chosen star-forming region mass from Eqn.~\ref{cmf}.
\cite{2017MNRAS.464.4318N} found that $\sim$ 10 per cent of low-mass star-forming regions contain at least one massive star ($>$15\,M$_\odot$) when using the ``soft-sampling'' technique described above, and 1\,per cent of low-mass regions contain two massive stars. Furthermore, when taking into account the decreasing probability of forming a high-mass star-forming region (Eqn.~\ref{cmf}), the number of low-mass ($M_{\rm cl} = 100 \pm 1$\,M$_\odot$) regions containing at least one massive star is $\sim$3100, which is actually greater than the total number of high-mass ($M_{\rm cl} = 1000 \pm 10$\,M$_\odot$) star-forming regions (1200). Of these 1200 high-mass regions, $\sim$1000 contain at least one massive ($>$15\,M$_\odot$) star.
If we translate these numbers into the total number of stars that may be affected by photoevaporation, the average number of stars in our $M_{\rm cl} = 100 \pm 1$\,M$_\odot$ star-forming regions containing at least one massive star is $\sim$110, so in the 3100 low-mass regions that contain at least one massive star there are $\sim$341\,000 stars in total. The average number of stars in our high mass star-forming regions ($M_{\rm cl} = 1000 \pm 10$\,M$_\odot$) is $\sim$1710, and so the 1000 regions that contain at least one massive star host a total of $\sim$1\,700\,000 stars that could be affected by photoevaporation. In short, the fraction of \emph{stars} originating in low-mass star-forming regions containing at least one massive star is $\sim$20\,per cent of the total number of stars originating from high-mass regions containing at least one massive star. If we stipulate that the high-mass regions must contain three or more massive stars, only $\sim$580 regions out of 1200 fulfil this criteria and host a total of 986\,000 stars. The fraction of stars originating in low-mass star-forming regions containing at least one massive star is $\sim$35\,per cent of the total number of stars originating from high-mass regions containing at least three massive stars.
Crucially, this makes no assumption about the disruption and dissolution of these star-forming regions, and how many stars from each type of region eventually enter the Galactic field. The Galactic potential will influence the destruction of low-mass star-forming regions much more than high-mass regions \citep{2008gady.book.....B}, which take longer to dissolve into the Galactic field (and some remain as long-lived open clusters). Therefore, the majority of planet-hosting Field stars may come from lower-mass regions.
Given their significant contribution to the integrated stellar mass function, we therefore also investigate low-mass star-forming regions ($100$\,M$_{\odot}$) that contain either one or two massive stars -- these represent an unusual sampling of the IMF but allow us to investigate the effects of photoevaporation in less populous star-forming regions.
Hence we have three different star-forming region set-ups; a 100\,M$_{\odot}$ region with one massive star (38\,M$_\odot$), a 100\,M$_{\odot}$ region with two massive stars (42 and 23\,M$_\odot$) and one 1000 M$_{\odot}$ region with 5 massive stars (43, 33, 26, 17 and 17\,M$_{\odot}$). These regions were selected as the median outcomes of Monte Carlo sampling of $1 \times 10^6$ star-forming regions \citep{2017MNRAS.464.4318N}, and then filled with stellar masses drawn from the IMF \citep{2013MNRAS.429.1725M}. We then selected the median regions in terms of the total number of stars from within the mass ranges of 100\,$\pm1$M$_\odot$ and 1000\,$\pm10$M$_\odot$, with the stipulation that they had to contain massive stars. For the 100\,M$_\odot$ mass regions, we specifically selected the median region containing one and two massive stars. For the 1000\,M$_\odot$ we selected the average cluster that contained three or more massive stars.
The external photoevaporation prescriptions we will adopt in this work are those from \citet{2001MNRAS.325..449S} which only weakly depend on the adopted stellar IMF, but in a future paper we will assign a FUV flux and an EUV flux to each intermediate/high-mass star based on its mass and then determine the respective fluxes incident on every low-mass star and use the recent FRIED grid of models \citep{2018MNRAS.481..452H} to determine mass-loss for individual discs.
\subsection{$N$-body simulations}
Our simulations are created with initial substructure by following the box-fractal method in \cite{2004A&A...413..929G}. We use a range of fractal dimensions for varying amounts of substructure: $D = 1.6$ (highly sub-structured), $D = 2.0$ (moderately sub-structured), and $D = 3.0$ (smooth). The method also correlates stellar velocities on local scales so that nearby stars have similar velocities, but more distant stars can have a wide range of different velocities.
The velocities are then scaled to the virial ratio $\alpha_{\rm vir}$, where $\alpha_{\rm vir} = T/|\Omega|$; $T$ is the total kinetic energy and $\Omega$ is the total potential energy of the stars. A range of virial ratios are used: $\alpha_{\rm vir} = 0.3$ (sub-virial, or collapsing), $\alpha_{\rm vir} = 0.5$ (virial equilibrium), and $\alpha_{\rm vir} = 0.7$ (supervirial, or expanding). Note that this virial ratio determines the net bulk motion, i.e.\,\,whether the star-forming region will collapse or expand. The correlated velocities on local scales mean that the local velocity dispersion can be subvirial, facilitating a violent relaxation process as the star-forming region evolves. Such local subvirial velocity dispersions are observed in the earliest stages of star formation \citep{1981MNRAS.194..809L,2006A&A...445..979P, 2015ApJ...799..136F}.
We create star-forming regions with stellar densities of 100\,M$_{\odot}$\,pc$^{-3}$ or 10\,M$_{\odot}$\,pc$^{-3}$ for the 1000\,M$_\odot$ regions; for the 100\,M$_\odot$ regions we set an initial density of 100\,M$_{\odot}$\,pc$^{-3}$. Such densities bracket the range observed in present-day star-forming regions \citep{2010MNRAS.409L..54B} as well as allowing for potentially higher \emph{primordial} densities \citep{2014MNRAS.445.4037P}.
Finally, we created a set of simulations with a Plummer sphere distribution \citep{1911MNRAS..71..460P} to facilitate comparisons with previous studies. We use the same IMF from our 1000 M$_{\odot}$ simulations to create two clusters with Plummer sphere distributions that have initial densities of 10 and 100 M$_{\odot}$ pc$^{-3}$.
We evolve each of our star-forming regions for 10 Myr using the \texttt{kira} integrator within the \texttt{Starlab} environment \citep{2001MNRAS.321..199P}. Stellar evolution is implemented using the \texttt{SeBa} look-up tables \citep{1996A&A...309..179P}. No binary or multiple stellar systems are included in these simulations. To gauge the amount of stochasticity in the disc photoevaporation, we run 20 realisations of the same initial conditions, identical apart from the random number seed used to assign the positions and velocities.
\subsection{Protoplanetary discs and external photoevaporation}
The mass loss rate of discs at a certain distance from a neighboring massive star is determined by the strength of the star's FUV ($h\nu$ < 13.6 eV) and EUV ($h\nu$ > 13.6 eV) fluxes at that distance. Mass loss due to FUV photons is caused by heating the circumstellar disc, which creates an unbound neutral layer that can drift towards the ionization front, where it meets the EUV field. FUV is independent of the distance from the massive star because the only requirement is that the FUV flux is strong enough to heat the disc above its escape velocity. With EUV, the mass loss rate depends on the EUV flux and so is directly dependent on the distance from the massive star(s).
We use the same prescriptions for FUV and EUV photoevaporation as \cite{2001MNRAS.325..449S}:
\begin{equation}
\dot{M}_{\rm FUV} \approx 2 \times 10^{-9}r_{d} \,\,{\rm M_{\odot} yr^{-1}},
\end{equation}
\begin{equation}
\dot{M}_{\rm EUV} \approx 8 \times 10^{-12}r^{3/2}_{d}\frac{\Phi_{i}}{d^2} \,\,{\rm M_{\odot} yr^{-1}},
\end{equation}
where $r_{d}$ is the radius of the disc in astronomical units, $\Phi_{i}$ is the ionizing EUV photon luminosity from each massive star in units of 10$^{49}$ s$^{-1}$ and $d$ is the distance from the massive star in parsecs. The UV photon rate ($\Phi_{i}$) for the massive stars (> 15M$_{\odot}$) is dependent on stellar mass and we use the values from \citet{1996ApJ...460..914V} and \citet{2003ApJ...599.1333S}.
These photoevaporation rates were derived assuming a disc density profile $\Sigma \propto r_d^{-2}$ \citep{2000prpl.conf..401H,2009apsf.book.....H}; however, our analysis does not take into account the evolution of the surface density profile if the disc radius were to change significantly.
Observations of star forming regions show that disc radii can extend to several 100s AU \citep[e.g.][]{2018arXiv180305923A}. However, the typical initial radius of proplyds is still debated in the literature and therefore we sample a wide range of initial radii: 10, 50, 100, 200 and 1000 AU. We adopt a single value for each disc radius, focusing primarily on 100 AU discs, and then repeat the analysis for the five values. In reality, the radius of the disc will change due to internal processes such as viscous evolution, and due to internal and external photoevaporation, but we are unable to account for this (and the changing disc density profile) in our $N$-body simulations.
The initial disc masses are also debated, with theoretical constraints from the Minimum Mass Solar Nebula \citep[MMSN,][]{1977Ap&SS..51..153W, 1981PThPS..70...35H} and observations \citep{2013ApJ...771..129A,2016ApJ...828...46A} suggesting an upper limit of $M_{\rm disc} = 0.01$\,M$_{\star}$. Following the example in \cite{2001MNRAS.325..449S}, the initial disc masses in our simulation are 10 per cent of the host star mass ($M_{\rm disc}$ = 0.1 M$_{\star}$). Current observations suggest that disc masses are $\sim$1 per cent of the host star mass. We select 10 per cent so that we are sampling the upper range of the disc masses. While we do not account for accretion onto the protoplanetary discs, our discs are large enough in mass that we can neglect the accretion onto the disc as it will be minimal in comparison. For completeness, we ran a set of simulations where the disc masses were 1 per cent of the stellar host mass, which is more consistent with the MMSN estimates.
We subtract mass from our discs according to Eqns.~2~and~3. The rate of photoevaporation due to EUV radiation is dependent on distance from the ionising source, $d$, whereas the photoevaporation rate due to FUV is largely independent of distance from the source \citep{1999ApJ...515..669S}. Following \citet{1999ApJ...515..669S} and \citet{2001MNRAS.325..449S} we apply Eqn.~2 if the disc-hosting stars are within 0.3\,pc of the ionising source, noting that this distance is calibrated to models where $\theta^1$C~Ori is the most massive star (40\,M$_\odot$), which is commensurate with the most massive star in our simulations. However, we note that this is likely an underestimate of the amount of photoevaporation due to FUV fields in star-forming regions \citep{2016MNRAS.457.3593F,2018MNRAS.475.5460H}.
\section{Results}
\label{results}
We focus on 1000 M$_{\odot}$ star-forming regions, which typically contain $\sim$5 massive stars (M$_{\star} > 15$M$_{\odot}$) that act as photoionising sources \citep[c.f.][]{2001MNRAS.325..449S}. Our 1000 M$_{\odot}$ cluster contains 5 massive stars; 43.2, 32.7, 25.7 and two 17 M$_{\odot}$ with $\Phi_{i}$ values of 1.1, 0.47, 0.19 and $\sim$ 0.013 respectively. We focus on the results for two different initial stellar densities, $\sim$ 10 M$_{\odot}$ pc$^{-3}$ and $\sim$ 100 M$_{\odot}$ pc$^{-3}$ and, apart from the final section, the assumed initial mass for every disc is $M_{\rm disc} = 0.1$\,M$_{\star}$.
We present the results from varying different initial properties within the star forming regions, focusing on the spatial distribution (fractal dimension, $D$) and net bulk motion (virial ratio, $\alpha_{\rm vir}$). We focus on protoplanetary discs that have a radius of 100 AU, however the effects of external photoevaporation on discs with different radii and mass are discussed later in Sections~\ref{dispersal rates} and \ref{disc masses}.
We compare our results to the observed disc fractions in both \cite{2001ApJ...553L.153H} and \cite{2018MNRAS.477.5191R} using ages from the models in \cite{2000A&A...358..593S}. We discuss the caveats associated with these models in Section~4.
We later present two low mass clusters (100 M$_{\odot}$) with an initial density of $\sim$ 100 M$_{\odot}$ pc$^{-3}$ that are subvirial ($\alpha_{\rm vir} = 0.3$) and highly substructured ($D =$ 1.6). Our clusters contain either one (38\,M$_\odot$) or two (42\,M$_\odot$ and 23\,M$_\odot$) massive stars. The corresponding $\Phi$ values are $\Phi = 0.76$ and $\Phi = 1.01, 0.11$ respectively.
\subsection{Substructure in star-forming regions}
We first present the results from four simulations of star forming regions, where in each simulation the star forming region has a different initial spatial distribution; $D =$ 1.6 (highly substructured), $D = 2.0$ (moderately substructured), $D = 3.0$ (smooth) and a Plummer sphere spatial distribution. In all simulations the star-forming region is subvirial ($\alpha_{\rm vir} = 0.3$).
In Fig.~1 we show the average fraction of stars that have retained their (100 AU) discs from 20 runs of each simulation, where the initial substructure of the star-forming region is varied. We present the results for regions with two different initial stellar densities; 10 and 100 M$_{\odot}$ pc$^{-3}$ respectively.
\begin{figure*}
\begin{center}
\setlength{\subfigcapskip}{10pt}
\subfigure[Density = 10 M$_{\odot}$ pc$^{-3}$]{\label{high_1000_f}{\includegraphics[scale=0.433]{discs_with_time_comparison_substructure_10_M_sun_pc_-3.png}}}
\subfigure[Density = 100 M$_{\odot}$ pc$^{-3}$]{\label{low_1000_f}{\includegraphics[scale=0.433]{discs_with_time_comparison_substructure_100_M_sun_pc_-3.png}}}
\caption{The average percentage of stars retaining their 100 AU disc over time within a sub-virial ($\alpha_{\rm vir} = 0.3$) cluster. The amount of substructure in the star-forming region is varied from highly substructured ($D = 1.6$) to smooth and centrally concentrated (Plummer Sphere). Two different initial densities (10 and 100 M$_{\odot}$ pc$^{-3}$) are considered. Each coloured line represents a different initial spatial distribution. The red data points are observational values from \protect\cite{2001ApJ...553L.153H}. The grey data points are from \protect\cite{2018MNRAS.477.5191R} using stellar ages from the models in \protect\cite{2000A&A...358..593S}. The colored shaded regions show the complete range of values from the 20 runs for each set of initial conditions.}
\end{center}
\end{figure*}
Fig.~1(a) shows the results from a star forming region with an initial density of 10 M$_{\odot}$ pc$^{-3}$. Within highly sub-structured regions ($D = 1.6$), the time taken for half of the stars to lose their discs due to external photoevaporation is 2.12 Myr. In moderately sub-structured regions ($D = 2.0$), this time increases to 2.60 Myr. However, the average percentage of remaining discs with time in both remain relatively similar throughout the 10 Myr. For regions with an initially smooth and spherical distribution ($D = 3.0$), the time taken for half of the discs to disperse is 3.62 Myr. Discs within Plummer spheres have the longest lifetimes (3.85\,Myr), with an average of $\sim$ 29.7 per cent of discs surviving for longer than 10 Myr.
\citet{1911MNRAS..71..460P} models (and other models that describe smooth star clusters such as a \citet{1966AJ.....71...64K} profile or an \citet{1987ApJ...323...54E} profile) are intended the model dynamically relaxed systems, whereas young star-forming regions have yet to relax. Therefore, even a smooth box fractal ($D = 3.0$) contains kinematic substructure, which causes the dynamical evolution of such a region to be more violent than a smooth Plummer sphere. It is therefore unsurprising that fewer discs survive in kinematically substructured fractal regions than in Plummer spheres with a similar density.
Fig.~1(b) shows the results for star forming regions with an initial density of 100 M$_{\odot}$ pc$^{-3}$. For discs in the highly sub-structured regions ($D = 1.6$), the time taken for half of the stars within the cluster to lose their protoplanetary discs is 0.87 Myr. The majority of discs within the highly substructured region ($D = 1.6$) are dispersed after 10 Myr, with $\sim$ 6 per cent surviving for the length of the simulation. The majority of discs within smooth, spherical regions are also dispersed within a short time frame, with only $\sim$ 3 per cent remaining after 10 Myr.
In Table 1 we summarise the average time taken for half of the stars in a star forming region to lose their discs for each spatial distribution and Table 2 summarises the percentage of discs remaining at the end of the 10 Myr simulation.
In the low density simulations, regions with more spatial substructure photoevaporate discs faster than smoother regions of a comparable density (Fig.~1(a)) . The reason for this is that the more substructured regions are initially further from dynamical equilibrium than the smooth regions, and low-mass stars undergo more close interactions with the high mass stars as the regions relax.
Interestingly, in the high-density simulations (Fig.~1(b)), whilst the fraction of discs remaining after 10\,Myr is lower than in the low density simulations, the initially more substructured star-forming regions contain more discs than the smooth regions after 10\,Myr (and their disc-destruction half-life is longer, see Table 1). We attribute this to the higher ejection rate of massive stars in dense, substructured star-forming regions \citep{2014MNRAS.438..620P}, which means that some of the ionising sources are no longer near the majority of the proplyds as early as 1\,Myr after the start of dynamical evolution.
\begin{table*}
\centering
\caption{The time taken for half of stars within the star-forming region to lose the gas from their 100 AU protoplanetary discs in a 1000 M$_{\odot}$ sub-virial ($\alpha_{\rm vir} = 0.3$) region with two different initial densities; 10 and 100 M$_{\odot}$ pc$^{-3}$. Four different spatial distributions are analysed; $D = 1.6$ (highly sub-structured), $D = 2.0$ (moderately sub-structured), $D = 3.0$ (smooth), and a Plummer sphere distribution. The highest and lowest values from the 20 different runs are included.}
\begin{tabular}{ccccc}
\hline
& \multicolumn{2}{c|}{Half life of protoplanetary discs} \\
Fractal dimension ($D$) & $\rho = $10 M$_{\odot}$ pc$^{-3}$ & $\rho = $100 M$_{\odot}$ pc$^{-3}$\\
\hline
1.6 & 2.12 $\pm$ $^{0.51}_{1.11}$ Myr & 0.87 $\pm$ $^{0.50}_{0.49}$ Myr \\ \hline
2.0 & 2.60 $\pm$ $^{1.36}_{0.62}$ Myr & 0.67 $\pm$ $^{0.21}_{0.22}$ Myr \\ \hline
3.0 & 3.62 $\pm$ $^{1.68}_{0.89}$ Myr & 0.65 $\pm$ $^{0.10}_{0.16}$ Myr\\ \hline
Plummer Sphere & 3.85 $\pm$ $^{3.70}_{1.34}$ Myr & 0.84 $\pm$ $^{0.90}_{0.29}$ Myr \\
\hline
\end{tabular}
\end{table*}
\begin{table*}
\centering
\caption{The average percentage of 100 AU discs remaining after 10 Myr within a sub-virial ($\alpha_{\rm vir} = 0.3$) star forming region from 20 realisations of each simulation. The amount of substructure is varied from highly substructured ($D = 1.6$) to a smooth and centrally concentrated Plummer sphere. The highest and lowest values from the 20 different runs are included. Two different initial densities (10 and 100 M$_{\odot}$ pc$^{-3}$) are considered.}
\begin{tabular}{ccccc}
\hline
& \multicolumn{2}{c|}{Percentage of discs remaining after 10 Myr} \\
Fractal dimension ($D$) & $\rho = $10 M$_{\odot}$ pc$^{-3}$ & $\rho$ = 100 M$_{\odot}$ pc$^{-3}$\\
\hline
1.6 & 16.40 $\pm$ $^{4.58}_{9.8}$ \% & 5.99 $\pm$ $^{2.88}_{2.67}$ \% \\ \hline
2.0 & 17.75 $\pm$ $^{17.03}_{4.04}$ \% & 2.27 $\pm$ $^{4.04}_{0.92}$ \% \\ \hline
3.0 & 21.60 $\pm$ $^{16.63}_{10.88}$ \% & 1.35 $\pm$ $^{0.63}_{0.42}$ \% \\ \hline
Plummer Sphere & 29.70 $\pm$ $^{18.95}_{14.98}$ \% & 2.81 $\pm$ $^{2.36}_{1.63}$ \% \\
\hline
\end{tabular}
\end{table*}
\subsection{Virial ratio}
We explore how changing the net bulk motion of the star-forming region affects the rate of disc dispersal due to external photoevaporation. We run simulations of our star-forming region with three different virial ratios; 0.3 (subvirial, or collapsing), 0.5 (virial equilibrium), and 0.7 (supervirial, or expanding). We keep the fractal dimension constant, adopting $D$ = 2.0 and as before we analyse simulations with two different initial densities; 10 M$_{\odot}$ pc$^{-3}$ and 100 M$_{\odot}$ pc$^{-3}$.
In Table 3 we summarise the average time taken for half of the stars in each region to lose their (100 AU) discs for a given bulk virial ratio. Table 4 shows the percentage of discs remaining at the end of the 10 Myr simulation. Fig.~2 shows the the average fraction of stars that have retained their discs from the 20 runs of each simulation for a star forming region where we vary the initial bulk motion (virial ratio)
\begin{figure*}
\begin{center}
\setlength{\subfigcapskip}{10pt}
\subfigure[Density = 10 M$_{\odot}$ pc$^{-3}$]{\label{low_1000_f}{\includegraphics[scale=0.2775]{discs_with_time_comparison_virial_ratio_10_M_sun_pc_-3.png}}}
\subfigure[Density = 100 M$_{\odot}$ pc$^{-3}$]{\label{high_1000_f}{\includegraphics[scale=0.433]{discs_with_time_comparison_virial_ratio_100_M_sun_pc_-3.png}}}
\caption{The average percentage of stars retaining their 100 AU disc with time in a 1000 M$_{\odot}$, moderately substructured ($D = 2.0$) star forming region with an initial density of 10 and 100 M$_{\odot}$ pc$^{-3}$. Each coloured line represents a different virial ratio. The red data points are observational values from \protect\cite{2001ApJ...553L.153H}. The grey data points are from \protect\cite{2018MNRAS.477.5191R} using ages from the stellar model in \protect\cite{2000A&A...358..593S}. The shaded regions show all values between the maximum and minimum values from all 20 runs of the simulations.}
\end{center}
\end{figure*}
Fig.~2 (a) shows the average mass loss rate in a star forming region with an initial density of 10 M$_{\odot}$ pc$^{-3}$. The time taken for half of the stars within a collapsing (sub-virial) star-forming region to lose their discs is 2.60 Myr. In regions that are expanding (supervirial), this time increases to 3.53 Myr. The percentage of discs within the subvirial region after 10 Myr is 17.75\%, in comparison to discs within an expanding region where the percentage rises to 32.67\%.
The initial net bulk motion of low density star-forming regions affects the amount of discs that are photoevaporated due to external radiation, with subvirial regions evaporating more discs at a faster rate than either virialised or supervirial regions.
Fig.~2(b) shows the results for a star-forming region with an initial density of 100 M$_{\odot}$ pc$^{-3}$. The time taken for half the stars within a collapsing region to lose their discs is 0.67 Myr. This time is similar for regions in virial equilibrium and expanding regions (0.68 and 0.63 Myr respectively). The lower disc half-life for the supervirial regions could again be due to massive stars being ejected in the (sub)virial regions. The percentage of discs remaining after 10 Myr in sub-virial star forming regions is 2.27\% whereas in regions where the net motion is expansive, this is increased to 5.00\%.
\begin{table*}
\centering
\caption{The time taken for half of stars within the cluster to lose the gas within their 100 AU protoplanetary discs in a 1000 M$_{\odot}$, moderately sub-structured ($D = 2.0$) star forming region for two different initial densities; 10 and 100 M$_{\odot}$ pc$^{-3}$. Three different virial ratios are analysed: $\alpha_{\rm vir} = 0.3$ (sub-virial, or collapsing), $\alpha_{\rm vir} = 0.5$ (virial equilibrium), and $\alpha_{\rm vir} = 0.7$ (super virial, or expanding).}
\begin{tabular}{ccccc}
\hline
& \multicolumn{2}{c|}{Half life of protoplanetary discs} \\
Virial Ratio ($\alpha_{\rm vir}$) & $\rho = $10 M$_{\odot}$ pc$^{-3}$ & $\rho = $100 M$_{\odot}$ pc$^{-3}$\\
\hline
0.3 & 2.60 $\pm$ $^{1.36}_{0.62}$ Myr & 0.67 $\pm$ $^{0.21}_{0.22}$ Myr \\ \hline
0.5 & 3.10 $\pm$ $^{2.73}_{1.15}$ Myr & 0.68 $\pm$ $^{0.66}_{0.35}$ Myr \\ \hline
0.7 & 3.53 $\pm$ $^{1.72}_{1.40}$ Myr & 0.63 $\pm$ $^{0.57}_{0.23}$ Myr\\
\hline
\end{tabular}
\end{table*}
\begin{table*}
\centering
\caption{The average percentage from 20 runs of simulations of 100 AU discs remaining after 10 Myr within a moderately substrutured ($D = 2.0$) cluster. The bulk motion (virial ratio) of the star-forming region is varied, from collapsing (sub-virial, $\alpha_{\rm vir} = 0.3$) to expanding (super virial, $\alpha_{\rm vir} = 0.7$). The highest and lowest values from the 20 different runs are included.}
\begin{tabular}{ccccc}
\hline
& \multicolumn{2}{c|}{Percentage of discs remaining after 10 Myr} \\
Virial Ratio ($\alpha_{\rm vir}$) & $\rho = $10 M$_{\odot}$ pc$^{-3}$ & $\rho = $100 M$_{\odot}$ pc$^{-3}$\\
\hline
0.3 & 17.75 $\pm$ $^{17.03}_{4.04}$ \% & 2.27 $\pm$ $^{4.04}_{0.92}$ \% \\ \hline
0.5 & 29.77 $\pm$ $^{15.73}_{17.95}$ \% & 4.16 $\pm$ $^{15.14}_{2.06}$ \% \\ \hline
0.7 & 32.67 $\pm$ $^{10.9}_{16.14}$ \% & 5.00 $\pm$ $^{8.04}_{3.23}$ \% \\
\hline
\end{tabular}
\end{table*}
\subsection{Disc radii}
\label{dispersal rates}
Here we present the rates of disc dispersal for different initial disc radii in a star-forming region with two different initial densities (10 and 100 $M_{\odot}$ pc$^{-3}$). The region has a fractal dimension of $D = 2.0$ (moderately substructured) and a viral ratio of $\alpha_{\rm vir} = 0.3$ (sub-virial).
Fig.~3 shows the percentage of protoplanetary discs with initial radii ranging between 10 -- 1000\,AU that have some remaining mass over 10 Myr in a 1000 M$_{\odot}$ star-forming region with different initial stellar densities; 10 M$_{\odot}$ pc$^{-3}$ in Fig~3(a) and 100 M$_{\odot}$ pc$^{-3}$ in Fig.~3(b).
\begin{figure*}
\begin{center}
\setlength{\subfigcapskip}{10pt}
\subfigure[Density = 10 M$_{\odot}$ pc$^{-3}$]{\label{low_1000_f}{\includegraphics[scale=0.275]{percentage_of_discs_a0_3d2_0r8_1000_f.png}}}
\subfigure[Density = 100 M$_{\odot}$ pc$^{-3}$]{\label{high_1000_f}{\includegraphics[scale=0.275]{percentage_of_discs_a0_3d2_0r3_5_1000_f.png}}}
\caption{The percentage of total remaining discs over time for a 1000 M$_{\odot}$ star-forming region with an initial density of $\sim$10 and $100$ M$_{\odot}$ pc$^{-3}$ (panels a and b respectively). The cluster has is moderately sub-structured ($D = 2.0$) and is sub-virial ($\alpha_{vir} = 0.3$) and Each colour represents a different initial disc radius. The disc masses are 10 per cent of the host star mass. The multiple coloured lines are each a single run of the 20 simulation runs. The black data points are observational values from \protect\cite{2001ApJ...553L.153H}.}
\end{center}
\end{figure*}
In the lower-density star-forming regions (Fig.~3(a)), the time taken for half of the 100 AU discs to be completely photoevaporated is 2.60 Myr. Discs with radii of 10 AU have much greater lifetimes, with an average of $\sim$77 per cent of discs surviving the full length of the simulation. The majority of discs with very large radii (1000 AU) are still depleted within very short timescales. Disc depletion rates begin to switch off after $\sim$4 Myr due to a combination of a large decrease in density of the star forming region, which peaks at $\sim$2 Myr, and the death of the most massive star at 4.33 Myr.
Fig.~3(b) shows that the majority (90\,per cent or more) of discs with radii $>$ 10\,AU are completely photoevaporated before the end of the 10 Myr simulation in moderately dense star-forming regions. The time taken for half of the stars in the region to lose their 100 AU discs is $0.67$ Myr. The vast majority of the largest discs (1000 AU) are photoevaporated completely within 2 Myr, with half of the stars in the region losing their discs within < 0.1 Myr.
We also ran simulations for low mass star-forming regions (100 M$_{\odot}$) with an initial density of $\sim$100 M$_{\odot}$ pc$^{-3}$. These low-mass regions contain two massive stars (42\,M$_\odot$ and 23\,M$_\odot$), which represents an unusual sampling of the IMF \citep{2007MNRAS.380.1271P}, but is observed in nature \citep[e.g. $\gamma^2$ Vel,][]{2014A&A...563A..94J}. In these low mass regions, half of discs with radii of 100 AU dissipated in 0.51 Myr. This is comparable to the 1000 M$_{\odot}$ regions with a similar density. We will further explore the effects of different stellar IMFs on disc dispersal in a future paper.
\subsection{Disc Masses}
\label{disc masses}
We have assumed that the disc masses are 10\,per cent of the host star's mass, which is likely to be an overestimate and various studies suggest that the disc mass is as low as 1\,per cent of the host star's mass \citep{1977Ap&SS..51..153W, 1981PThPS..70...35H, 2013ApJ...771..129A}.
In Fig.~4 we show the results for a star-forming region with our two different initial densities (10 and 100 M$_{\odot}$ pc$^{-3}$ respectively), where the initial disc masses are set to $M_{\rm disc} = 0.01$\,M$_{\star}$.
\begin{figure*}
\begin{center}
\setlength{\subfigcapskip}{10pt}
\subfigure[Density = 10 M$_{\odot}$ pc$^{-3}$]{\label{low_1000_f}{\includegraphics[scale=0.275]{percentage_of_discs_1_perc_a0_3d2_0r8_1000_f.png}}}
\subfigure[Density = 100 M$_{\odot}$ pc$^{-3}$]{\label{high_1000_f}{\includegraphics[scale=0.275]{percentage_of_discs_1_perc_a0_3d2_0r3_5_1000_f.png}}}
\caption{The percentage of total remaining discs over time for a star forming region of 1000 M$_{\odot}$ with an initial density of $\sim$10 and 100 M$_{\odot}$ pc$^{-3}$ respectively, a fractal dimension of $D = 2.0$ and a virial ratio of $\alpha_{\rm vir} = 0.3$. The initial disc masses are 1 per cent of the host star mass. Each colour represents a different disc radius. The multiple coloured lines are each a single run of the 20 simulation runs. The black data points are from observational values from \protect\cite{2001ApJ...553L.153H}.}
\end{center}
\end{figure*}
Fig. ~4(a) shows that on average the time taken for half of the stars within the low density star forming region to lose their 100 AU disc is $0.71$ Myr, less than half of the time taken for discs with 10 per cent of the mass of their stellar host. For discs with a radii of 10 AU, the half life is 3.31 Myr.
The timescale for half of the 100 AU discs to dissipate in the moderately dense (100 M$_{\odot}$\,pc$^{-3}$) star forming region (see Fig.~ 4b) is $\sim$ 0.14 Myr. For discs with a radius of 10 AU, the half life is $\sim$ 0.84 Myr. Less than 5 per cent of 10 AU discs survive for more than 3 Myr.
\begin{table*}
\centering
\caption{The time taken for half of stars in a star forming region to lose the gas within their 100 AU protoplanetary discs in a 1000 M$_{\odot}$, moderately sub-structured ($D = 2.0$) region for two different initial densities; 10 and 100 M$_{\odot}$ pc$^{-3}$ and two different masses of disc, 10 per cent and 1 per cent. Three different virial ratios are analysed: $\alpha_{\rm vir} = 0.3$ (sub-virial, or collapsing), $\alpha_{\rm vir} = 0.5$ (virial equilibrium), and $\alpha_{\rm vir} = 0.7$ (super virial, or expanding). The highest and lowest values from the 20 different runs are included.}
\begin{tabular}{ccccc}
\hline
& \multicolumn{4}{c|}{Half life of cluster protoplanetary discs} \\
& \multicolumn{2}{c|}{Disc mass = 0.1\,M$_{\star}$} & \multicolumn{2}{c|}{Disc mass = 0.01\,M$_{\star}$} \\
Disc Radius (AU) & $\rho = $10 M$_{\odot}$ pc$^{-3}$ & $\rho = $100 M$_{\odot}$ pc$^{-3}$ & $\rho = $10 M$_{\odot}$ pc$^{-3}$ & $\rho = $100 M$_{\odot}$ pc$^{-3}$ \\
\hline
10 & > 50\% remaining & 3.92 $\pm$ $^{5.91}_{1.87}$ Myr & 3.31 $\pm$ $^{5.16}_{0.9}$ Myr & 0.84 $\pm$ $^{0.22}_{0.27}$ Myr \\ \hline
50 & 3.94 $\pm$ $^{5.96}_{0.82}$ Myr & 1.04 $\pm$ $^{0.40}_{0.24}$ Myr & 1.22 $\pm$ $^{1.18}_{0.35}$ Myr & 0.28 $\pm$ $^{0.07}_{0.09}$ Myr \\ \hline
100 & 2.60 $\pm$ $^{1.36}_{0.62}$ Myr & 0.67 $\pm$ $^{0.21}_{0.22}$ Myr & 0.71 $\pm$ $^{0.96}_{0.21}$ Myr & 0.14 $\pm$ $^{0.05}_{0.05}$ Myr \\ \hline
200 & 1.55 $\pm$ $^{1.34}_{0.44}$ Myr & 0.36 $\pm$ $^{0.09}_{0.11}$ Myr & 0.39 $\pm$ $^{0.58}_{0.12}$ Myr & 0.06 $\pm$ $^{0.03}_{0.02}$ Myr \\ \hline
1000 & 0.37 $\pm$ $^{0.55}_{0.11}$ Myr & 0.06 $\pm$ $^{0.02}_{0.02}$ Myr & 0.15 $\pm$ $^{0.20}_{0.05}$ Myr & 0.02 $\pm$ $^{0.02}_{0.01}$ Myr \\
\hline
\end{tabular}
\end{table*}
\begin{table*}
\centering
\caption{The average percentage of 100 AU discs remaining after 10 Myr within a moderately substructured ($D = 2.0$) star forming region for two different initial densities, ($10$ and $100$ M$_{\odot}$ pc$^{-3}$), with two different initial disc masses, 10 per cent and 1 per cent the mass of the host star. The bulk motion (virial ratio) of the star forming region is varied, from collapsing (sub-virial, $\alpha_{\rm vir} = 0.3$) to expanding (super-virial, $\alpha_{\rm vir} = 0.7$). The highest and lowest values from the 20 different runs are included.}
\begin{tabular}{ccccc}
\hline
& \multicolumn{4}{c|}{Percentage of discs remaining after 10 Myr} \\
& \multicolumn{2}{c|}{Disc mass = 0.1\,M$_{\star}$} & \multicolumn{2}{c|}{Disc mass = 0.01\,M$_{\star}$} \\
Disc Radius (AU) & $\rho = $10 M$_{\odot}$ pc$^{-3}$ & $\rho = $100 M$_{\odot}$ pc$^{-3}$ & $\rho = $10 M$_{\odot}$ pc$^{-3}$ & $\rho = $100 M$_{\odot}$ pc$^{-3}$\\
\hline
10 & 77.29 $\pm$ $^{12.79}_{4.71}$ \% & 39.49 $\pm$ $^{14.34}_{14.17}$ \% & 28.81 $\pm$ $^{19.13}_{9.09}$ \% & 3.99 $\pm$ $^{6.65}_{2.18}$ \% \\ \hline
50 & 34.44 $\pm$ $^{18.08}_{7.57}$ \% & 6.73 $\pm$ $^{5.97}_{3.62}$ \% & 4.79 $\pm$ $^{9.34}_{1.43}$ \% & 0.40 $\pm$ $^{1.53}_{0.23}$ \% \\ \hline
100 & 17.75 $\pm$ $^{17.03}_{4.04}$ \% & 2.27 $\pm$ $^{4.04}_{0.92}$ \% & 2.27 $\pm$ $^{3.91}_{1.13}$ \% & 0.08 $\pm$ $^{0.59}_{0.08}$ \% \\ \hline
200 & 7.12 $\pm$ $^{10.79}_{1.82}$ \% & 0.74 $\pm$ $^{1.95}_{0.36}$ \% & 1.09 $\pm$ $^{1.81}_{14.17}$ \% & 0.00 $\pm$ $^{0.38}_{0.00}$ \% \\ \hline
1000 & 1.09 $\pm$ $^{1.73}_{0.9}$ \% & 0.0 $\pm$ $^{0.38}_{0.00}$ \% & 0.29 $\pm$ $^{0.51}_{0.29}$ \% & 0.00 $\pm$ $^{0.04}_{0.00}$ \% \\
\hline
\end{tabular}
\end{table*}
\subsection{Mass of star-forming regions}
We also ran simulations for two different low mass star-forming regions (100 M$_{\odot}$) with an initial density of $\sim$ 100 M$_{\odot}$ pc$^{-3}$, which were sub-virial ($\alpha_{vir}$ = 0.3) and substructured ($D$ = 1.6). These low-mass regions contain one (38\,M$_\odot$) or two (42\,M$_\odot$ and 23\,M$_\odot$) massive stars, which represents an unusual sampling of the IMF \citep{2007MNRAS.380.1271P}, but is observed in nature \citep[e.g. $\gamma^2$ Vel,][]{2014A&A...563A..94J}. Our expectation from randomly sampling the IMF is that 10\,per cent of all star-forming regions can host a massive star, and 1\,per cent of regions will host two massive stars. We note that the lack of massive star(s) in \emph{any} star-forming region would preclude disc destruction from photoevaporation, though as discussed in Section 2 it is unclear which type of star-forming region (in terms of total mass, $M_{\rm cl}$) contributes the most (planet hosting) stars to the Galactic field.
In both of these low mass regions, half of discs with radii of 100 AU dissipated before $\sim$ 1 Myr (Fig.~\ref{fig:clus_hist}). The time taken for half of the discs to be destroyed in a region with one massive star is 0.95 Myr. This time is reduced to 0.37 Myr for the cluster with 2 massive stars.
At the end of the 10 Myr simulation, 15.5 per cent of discs within the region with one massive star are surviving. Within the region containing 2 massive stars, less than 5 per cent of discs are remaining, double the number of discs remaining in higher mass regions with the same initial conditions.
\begin{figure}
\includegraphics[width=\columnwidth, scale = 5]{discs_with_time_comparison_substructure_low_mass_100_M_sun_pc_-3.png}
\caption{The median percentage of protoplanetary discs (100 AU) remaining with time for two 100 M$_{\odot}$ clusters with initial densities of 100 M$_{\odot}$ pc$^{-3}$ but different numbers of massive stars. The green line shows values for a cluster with 1 massive star (> 15 M$_{\odot}$) and the orange a cluster with 2 massive stars. The red data points are observational values from \protect\cite{2001ApJ...553L.153H}. The grey data points are from \protect\cite{2018MNRAS.477.5191R} using ages from the stellar model in \protect\cite{2000A&A...358..593S}. The colored shaded regions show the complete range of values from the 20 runs for each set of the different clusters.}
\label{fig:clus_hist}
\end{figure}
\section{Discussion}
\label{discussion}
The initial conditions of a star-forming region will affect the rate at which protoplanetary discs are photoevaporated due to the radiation from nearby massive stars. The initial substructure and net bulk motion of a star-forming region impacts the rate of disc dispersal.
\subsection{Changing the initial conditions of star-forming regions}
In our low-density simulations, highly substructured ($D = 1.6$) regions disperse half of the protoplanetary discs within 1.51 Myr, more than twice as fast as smooth ($D = 3.0$) regions. In simulations with a Plummer sphere distribution, more than 30 per cent of discs remain at the end of the 10 Myr simulation, almost double that of discs within highly substructured clusters.
At these low densities, the degree of substructure matters because a more substructured star-forming region is further from dynamical equilibrium than a smooth region. When this occurs, a low-mass star is likely to have more close encounters with a massive ionising star than in a smooth region.
In moderately dense initial conditions (100 M$_{\odot}$ pc$^{-3}$), the difference in the fraction of discs that are photoevaporated between different initial spatial distributions decreases greatly, although regions with a Plummer sphere distribution retain more of their discs than regions with initial substructure. However, the average of all runs indicates that the amount of initial substructure has little effect on the survival rates of discs at these densities and fewer than 50 per cent of discs remain after 1 Myr.
The effect of changing the net bulk motion of the star-forming region has a similar impact on the rate of disc dispersal as the initial substructure has. For low density regions (10 M$_{\odot}$ pc$^{-3}$), the difference between the amount of discs surviving within a collapsing and an expanding star forming region is $\sim$ 15 per cent, with the collapsing regions enabling more photoevaporation than in expanding regions. Again, approximately double the number of discs remain in expanding clusters than in collapsing clusters. In moderately dense clusters it is similar, with the difference being $\sim$ 3 per cent.
For low mass star-forming regions (100 M$_{\odot}$), disc dispersal rates are similar to those of in higher mass regions. Whilst the UV field strength can vary due to different realisations of the IMF \citep{2008ApJ...675.1361F}, these low mass regions show that the mere presence of a high mass star (> 15\,M$_{\odot}$) will cause disc lifetimes to be shortened dramatically.
Our simulations are set up to mimic the observations of star formation in filaments, where the pre-stellar cores have subvirial motion \citep{1981MNRAS.194..809L, 2015ApJ...799..136F}. The local velocity dispersion is therefore always subvirial to some degree, and because mass-loss due to photoevaporation is so fast (eqns.~2~and~3), most of the photoevaporation occurs during the substructured phase of a star-forming region.
Previous studies investigating the effects of external photoevaporation on disc dispersal rates assumed smooth and centrally concentrated spatial distributions \citep{2001MNRAS.325..449S,2004ApJ...611..360A,2006ApJ...641..504A,2018MNRAS.478.2700W}, replicating environments like the present-day conditions of the ONC. However, using the present-day spatial and kinematic distributions to model star clusters may not accurately replicate the dynamical history of the star-forming region from which the cluster formed \citep{2014MNRAS.438..620P}.
We cannot provide a direct comparison with the work by \citet{2008ApJ...675.1361F} and \citet{2018MNRAS.478.2700W} because these authors assume initially smooth initial conditions, and we have shown that the severity of photoevaporation depends on the degree of initial substructure, as well as the initial positions of the most massive stars. We have distributed the massive stars randomly within the substructure, and after dynamical evolution these massive stars migrate towards the centre of the star cluster as it forms \citep{2010MNRAS.407.1098A,2014MNRAS.438..620P,2016MNRAS.459L.119P}.
Furthermore, we have used the photoevaporation prescription from \citet{2001MNRAS.325..449S}, rather than determine the photoevaporation rate from the EUV/FUV fluxes as a function of the flux in the interstellar medium \citep[the so-called $G_0$ value, $1.6 \times 10^{-3}$\,erg\,s$^{-1}$\,cm$^{-2}$][]{1968BAN....19..421H}. In comparison with \citet{2001MNRAS.325..449S}, we find that discs are destroyed earlier in ONC-type regions because the initial densities are higher \citep[in line with current observations,][]{2014MNRAS.445.4037P}, and the star-forming regions are substructured \citep{2004MNRAS.348..589C}.
An initially highly substructured star-forming region can become smooth and centrally concentrated within a few Myr due to a combination of violent and two body relaxation. Protoplanetary discs in these highly substructured environments will be photoevaporated at faster rates than discs within initially smooth regions. Even though they will both appear smooth within a few Myr, the percentage of discs remaining, and possibly the population of planets within the regions, will vary greatly.
The initial density of the cluster has the largest effect on the disc dispersal rate due to external photoevaporation. The `moderately-dense' clusters reflect the likely initial densities of many star-forming regions \citep{2014MNRAS.445.4037P}. However, we find that these `moderately-dense' environments are very destructive for protoplanetary discs and evaporate discs at rates faster than suggested by observations (compare the black points in Figs.~1~and~2 with our simulated data). Our results suggest that protoplanetary discs (or at least their gas content) would always be significantly depleted in moderately dense (100 M$_{\odot}$ pc$^{-3}$) star-forming regions, if those regions contain massive stars.
\cite{2001ApJ...553L.153H} finds that the fraction of disc-hosting stars in young star-forming regions falls to 50\,per cent after $\sim$ 3 Myr whereas \cite{2018MNRAS.477.5191R} find that after only $\sim$ 2 Myr half of the discs remain in their observed regions. In comparison, more than half of the discs in our simulations that are within dense environments are destroyed within $\sim$1 Myr. One interesting data point in the \cite{2001ApJ...553L.153H} sample is the ONC. With an age of $\sim$1 Myr, the centre of the ONC contains 4 massive stars and a density of $\sim$400 M$_{\odot}$ pc$^{-3}$ \citep{1998ApJ...492..540H}. Observations of this part of the ONC suggest that $\sim$80 -- 85 per cent of stars within the cluster are surrounded by bright ionization fronts, interpreted to be discs, with radii of $\sim$1000 AU \citep{2000AJ....119.2919B,2000AJ....120.3162L}.
The age of our simulated regions, where 80 per cent of stars still possess a 100 AU disc with some mass, is 0.48 Myr -- likely to be less than half the age of the ONC. This suggests that the massive stars within the ONC should have destroyed the majority of 100 AU discs. From similar arguments, \citet{2007MNRAS.376.1350C} concluded that the possible discs in the ONC with radii >10\,AU are likely to be merely ionisation fronts, containing little mass. Our simulations with different initial disc radii show that the radius of the disc will greatly affect the rate at which it is photoevaporated \citep[see also][]{2007MNRAS.376.1350C} due to the dependence on disc radii within the FUV and EUV photoevaporation prescriptions.
Recent surveys suggest that most stars in the Galactic field host planets, and many of these are gas or ice giants \citep{2011arXiv1109.2497M}. This implies that the majority of planet forming discs were able to survive a significant amount of time in their birth environment. Our simulations suggest that this is only possible in low-density regions that contain no photoionising sources (i.e.\,massive stars). Therefore, (giant) planet formation must occur on very rapid timescales (<1\,Myr), or stars that host giant planets must have formed in very benign environments.
Many observed protoplanetary discs are located in low-mass, low-density star-forming regions \citep{2013ApJ...771..129A,2018arXiv180305923A} and would be unaffected by external photoevaporation. However, many star-forming regions are typically moderate-density ($\sim$100 M$_{\odot}$ pc$^{-3}$) environments \citep{2014MNRAS.445.4037P} and our results suggest that the majority of protoplanetary discs in star-forming regions with these densities do not survive for long enough periods of time to form giant planets.
\subsection{Caveats}
There are several caveats to our results, which we discuss below.
The effects of external EUV radiation on protoplanetary discs can be reduced when thick winds are present, caused by FUV heating of the disc \citep{2014prpl.conf..475A}. However, the majority of the disc mass loss occurs due to FUV radiation. We repeated our analysis without EUV photoevaporation and find the disc dispersal rates to be similar.
It is possible that we are overestimating the amount of photoevaporation from massive stars. However, recent research suggests that the prescriptions used here are actually underestimating the amount of FUV radiation that discs receive \citep{2016MNRAS.457.3593F,2018MNRAS.475.5460H}. As FUV is the dominant source of external photoevaporation, the protoplanetary discs in our simulations could dissipate on even shorter timescales.
Star formation is an inherently inefficient process, with typically only $\sim$30\,per cent of the mass of a giant molecular cloud converted into stars. Young star-forming regions are observed to contain a large amount of dust and gas, which could shield the proplyds from significant photoevaporation. At these early stages the stellar density within the substructure is highest, and is therefore when the largest percentage of stars are in closest proximity to the massive stars. However, hydrodynamical simulations of star-forming regions show that massive stars blow large ($\sim$pc-scale) cavities within the gas on short time scales \citep{2013MNRAS.430..234D}, and so low-mass disc-hosting stars that would be affected by EUV/FUV radiation will likely reside in the cavities blown out by the massive stars. If the gas and dust could shield the disc, this would protect them for a very short period of time \citep{2015ApJ...804...29G}. Whether this grace period would be long enough to allow gas-rich giant planets to form is uncertain.
Given that most star-forming regions have stellar densities above a few M$_\odot$\,pc$^{-3}$ \citep{2010MNRAS.409L..54B}, external photoevaporation will detrimentally affect protoplanetary discs in any star-forming region that contains massive stars. This implies that star-forming regions that do not contain massive stars are more likely to form giant planets, but we note that massive stars appear necessary in order to deliver short-lived radioisotopes to the young Solar System \citep{2018PrPNP.102....1L}. The number of massive stars in a star-forming region appears to only be limited by the mass of the star-forming cloud \citep{2007MNRAS.380.1271P}, but this also means that low-mass star forming regions ($<10^4$M$_\odot$) stochastically sample the IMF, meaning that our simulations cannot be described as `typical' star-forming regions.
Quantifying disc dispersal is further complicated by how difficult it is to determine the ages of young stars, especially before 1 Myr \citep{2000A&A...358..593S}. We use the stellar ages from the \cite{2000A&A...358..593S} model. However, models of pre-main sequence stellar evolution calculate different ages depending on the physics that is implemented. Of the three models presented in \cite{2018MNRAS.477.5191R} we use the ages from \cite{2000A&A...358..593S} so that we are comparing the lower end of cluster ages to our simulations. The average stellar age calculated for the clusters in \cite{2018MNRAS.477.5191R} is significantly shorter than in more recent models from \citet{2016A&A...593A..99F}. By using these lower age limits, we more than halve the possible average life times of the discs within the observed clusters.
Furthermore, recent work by \citet{2013MNRAS.434..806B} suggests that the ages of pre-main sequence stars may be underestimated by a factor of two, meaning that the observed discs \citep[e.g.][]{2001ApJ...553L.153H} could be a factor of two older. This would make it even more surprising that discs would remain around low-mass stars, if those stars form in regions containing massive stars.
There is also the question of how quickly the photoionising massive stars form. In the competitive accretion models \citep{2001MNRAS.323..785B}, massive stars gradually gain in mass over $\sim$ 1 Myr \citep{2010ApJ...709...27W}, suggesting high-mass stars form later than low-mass stars \citep{2014prpl.conf..149T}, which would in turn decrease the amount of time low-mass stars spend near the photoionising sources \citep{2012MNRAS.424..377D, 2014MNRAS.442..694D}. In our simulations all stars form simultaneously, and therefore the disc-hosting low-mass stars do not have this grace period, which would increase disc lifetimes.
The growth of planetesimals into planets can be greatly accelerated by the accretion of cm-scale pebbles. \cite{2017AREPS..45..359J} show that once a 10$^{-2}\,M_{\earth}$ planetesimal has formed it can grow to Jupiter mass in 1 Myr when starting as far out as about 15 AU. An initial phase of accreting pebbles forms a 10$\,M_{\earth}$ core in about 0.8\,Myr, which then undergoes runaway gas accretion to reach Jupiter mass. Such processes potentially allow close-in giant planets to be formed even in the relatively hostile conditions that we consider here.
However, photoevaporation by the central star can cause large amounts of mass loss in the inner disc, potentially affecting giant planet formation \citep{2014prpl.conf..475A}. Grain size also has a significant effect on disc dispersal rates. Mass loss occurs much more quickly when grain growth has occurred because the FUV radiation can penetrate deeper into the disc \citep{2016MNRAS.457.3593F}.
Discs that can survive in moderately dense environments have small radii (10 -- 50 AU). This is because of the disc radius dependency in the external photoevaporation prescriptions. Internal UV radiation can cause significant mass loss and erosion of the disc within short time scales \citep[1 Myr,][]{2008ApJ...683..287G}. The timescale for internal disc dispersal is very short ($10^{5}$ yr), with a UV switch being triggered due to the slowing of accretion onto the inner 10 AU of the disc \citep{2001MNRAS.328..485C}, also calling into question the survivability of small discs.
Our disc radii are fixed, but in reality disc radii change with time, often in an inside-out fashion where the initial radius is small (and not as susceptible to photoevaporation) compared to later in the disc's life. We include several different disc radii to help visualise what happens for different disc initial conditions, but we cannot model the full viscous evolution in our post-processing analysis.
In our simulations, we have elected to keep the stellar IMF constant across different realisations of the spatial and kinematic initial conditions of our star-forming regions. The reasons for this are two-fold. First, we wish to isolate the possible effects of stochastic dynamical evolution \citep{2010MNRAS.407.1098A,2012MNRAS.424..272P,2014MNRAS.438..620P}, which could lead to different photoevaporation rates even if the ionising flux from massive stars were kept constant. The uncertainties shown by the shaded regions in Figs.~1,~2,~and~5 show this stochasticity for the same initial conditions. Second, the photoevaporation prescriptions we adopt \citep[following][]{2001MNRAS.325..449S} are actually quite insensitive to the mass of the most massive stars (but rather depend on whether the massive stars are present or not).
However, \citet{2008ApJ...675.1361F} show that the FUV and EUV fluxes can vary if the stellar IMF is extremely top-heavy and contains more massive stars than expected on average. In a forthcoming paper we will calculate the EUV and FUV fluxes in our substructured star-forming regions and use recent the FRIED models of disc photoevaporation from \citet{2018MNRAS.481..452H} to determine mass-loss based on these models, and whether it depends strongly on stochastic sampling of the stellar IMF.
Similarly, in some of our simulations the massive stars are ejected early on, which is a common occurrence in simulations of dense star-forming regions \citep[][Schoettler et al., in prep.]{2010MNRAS.407.1098A,2016A&A...590A.107O}. We will also quantify the effects of these ejections on the fraction of surviving protoplanetary discs in an upcoming paper.
The majority of discs observed with ALMA have been located in low-mass, low-density star-forming regions. Current observations suggest that the majority of stars form in moderately dense ($\sim$100 M$_{\odot}$ pc$^{-3}$) environments \citep{2014MNRAS.445.4037P}. However, the majority of protoplanetary discs in clusters with these densities do not survive for long enough periods of time to form planets, as planet formation is thought to take place over a few million years \citep{1996Icar..124...62P}. The fact that the majority of stars have planetary systems around them poses important questions as a result of the discrepancies that seemingly arise. This may indicate that the majority of stars form in low mass clusters where there are few to no high mass stars.
We adopt initial disc masses that are 10 per cent the mass of the host star, which is likely to be a large overestimate. When looking at more realistic values (1 per cent), discs are destroyed on even shorter timescales. However, it should be noted that accretion and internal photoevaporation will have much larger effects on disc mass evolution for these lower mass discs.
\section{Conclusions}
\label{conclusion}
We have calculated the mass loss due to external photoevaporation of protoplanetary discs in $N$-body simulations of the evolution of star-forming regions. We ran a suite of simulations where we vary the initial spatial structure, bulk motion and initial density of the regions. We compared our simulations that more closely represent observed star-forming regions (subvirial, substructured) with those of primordially mass segregated, spherical clusters, similar to those used in previous studies of external photoevaporation.
The parameter that most affects rates of disc dispersal is the initial density of the star-forming region. The majority of protoplanetary discs within simulated regions that mimic the conditions in nearby star-forming regions are dispersed due to external photoevaporation within very short time scales. In moderately dense ($\sim$100 M$_{\odot}$ pc$^{-3}$) star-forming regions which have moderate levels of substructure ($D = 2.0$) and are collapsing ($\alpha_{\rm vir} = 0.3$), we find the time taken for half of 100 AU discs to dissipate is significantly shorter (3 times less) than suggested in observational studies \citep{2001ApJ...553L.153H}. Lower density clusters ($\sim$10 M$_{\odot}$ pc$^{-3}$) allow discs to survive long enough to match observations of disc lifetimes, although the half-life of 100 AU discs is still less than that found by \cite{2001ApJ...553L.153H}.
The initial spatial distribution of the star-forming region also affects the rate of protoplanetary disc dispersal due to external photoevaporation. The degree to which initial substructure affects disc dispersal rates depends on the initial density. In moderately dense ($\sim$100 M$_{\odot}$ pc$^{-3}$) regions the effects are washed out, but in lower-density regions ($\sim$10 M$_{\odot}$ pc$^{-3}$) we find that the more fractal and clumpy a star-forming region is, the higher the rate of disc dispersal. This is due to violent relaxation and the rapid increase in density (sometimes up to an order of a magnitude) of the star forming region within a short amount of time. As most star forming regions appear to have a high degree of substructure, it is important for future studies of disc dispersal to take the initial conditions into consideration due to external photoevaporation in dense environments.
The virial ratio of the star forming region affects the rate of disc dispersal in a similar way to substructure. Regions that have a low initial density and are collapsing photoevaporate more discs on average than clusters which are expanding. The effects of varying the initial net bulk motion in moderately density clusters is negligible.
The majority of observed stars in the Galactic field host planetary systems, implying their protoplanetary discs survived long enough for formation to take place. There are three possible scenarios to resolve this apparent tension between observations and our simulations:
i) The majority of planets may not form in moderately dense star-forming regions ($\sim$100 M$_{\odot}$ pc$^{-3}$); rather, they would form in low density regions with no photoionising massive stars present. Many of the protoplanetary discs have been in observed in these low-density ambient environments \citep{2018arXiv180305923A}, but significant numbers of protoplanetary discs (or at least their remnants) have been observed in dense, hostile regions like the ONC \citep{1996AJ....111.1977M}.
ii) If some planets do form in dense, clustered environments containing massive stars (such as the ONC), then this suggests that giant planet formation must happen on very short time scales (less than 1 -- 2 Myr), or be confined to discs with radii significantly smaller than the orbit of Neptune in our Solar System. \cite{2017AREPS..45..359J} show that giant planet formation can occur on these timescales once large enough planetesimals have formed. However, internal photoevaporation processes can deplete the inner disc and set limits on the formation time of giant planets \citep{2014prpl.conf..475A}.
iii) The current calculations of mass-loss in discs due to external photoevaporation are severely overestimating the detrimental effects of EUV and FUV radiation. However, recent research \citep{2016MNRAS.457.3593F,2018MNRAS.475.5460H} suggests that photoevaporative mass-loss rates caused by FUV radiation may be underestimated, and our calculations also underestimated the effects as we adopt conservatively high initial disc masses.
\section*{Acknowledgements}
We thank A. R. Williams for useful discussions and S. Habergham-Mawson for comments. RBN is partially supported by an STFC studentship. RJP acknowledges support from the Royal Society in the form of a Dorothy Hodgkin Fellowship. RC and MBD are supported by grant 2014.0017, ``IMPACT", from the Knut and Alice Wallenberg Foundation.
\bibliographystyle{mnras}
|
{
"timestamp": "2019-03-01T02:21:05",
"yymm": "1902",
"arxiv_id": "1902.11094",
"language": "en",
"url": "https://arxiv.org/abs/1902.11094"
}
|
\section{INTRODUCTION}
Diversity of physical properties of magnets provides a fascinating playground in condensed matter physics.
When we explore this exciting arena, what is interesting to note is that there are many restrictions imposed by the symmetry of the magnetic structures.
For example, it has been well known that the structure of linear response tensors are determined by the magnetic point group\cite{Birss1962,Birss1964,Kleiner1966,Rivera2009,Szaller2013,Seemann2015,Suzuki2017}.
Furthermore, identifying the order parameter for the magnetic phase is most useful for deeper understanding of physical phenomena.
There has been many studies to investigate the relation between the
order parameters in a particular magnetic structure and macroscopic
phenomena such as anomalous Hall (AH)
effect~\cite{Shindou2001,Suzuki2017,Smejkal2019}, electromagnetic (EM)
effect~\cite{Ederer2007,Spaldin2008,Spaldin2013,Hayami2014,Hayami2018,Watanabe2018}, and optical responses~\cite{Higo2018}.
Thus we have a significant chance to specify or even design a magnet exhibiting desired physical properties by investigating the order parameters which characterize the magnetic structures.
For this purpose, the multipole expansion of the magnetic structure is an efficient and powerful approach.
Indeed, the multipole moments inherent in the magnetic structure, which we call cluster multipole moments~\cite{Suzuki2017}, have played a crucial role as a key order parameter in a variety of studies: EM effect has been discussed in terms of the magnetic (M) rank-0 monopole and rank-2 quadrupole as well as the magnetic toroidal (MT) rank-1 dipole as the order parameter to characterize the specific magnetic structures~\cite{Ederer2007,Spaldin2008,Spaldin2013,Thoele2016}.
The relation between parity odd multipoles and electromagnetic effect
has recently been investigated based on generalized forms of the multipole expansions for magnetic distributions~\cite{Hayami2018,Watanabe2018}.
It has also been shown that the rank-3 M multipole (octupole) plays a key role for a large AH effect~\cite{Nakatsuji2015,Kiyohara2016,Nayak2016}, anomalous Nernst effect~\cite{Ikhlas2017}, and magneto-optical Kerr effect~\cite{Higo2018} in the coplanar antiferromagnets Mn$_3$$Z$ ($Z$=Sn, Ge)~\cite{Suzuki2017}.
In these studies, the interplay between the physical properties and the magnetic structure through cluster multipoles has been investigated extensively.
However, there has been no concrete scheme to make a complete basis set of cluster multipoles for a given crystal structure.
In this paper, we propose a scheme to generate cluster multipoles which form a complete orthonormal basis set for arbitrary magnetic structures.
Here, we introduce a virtual atomic cluster, which depends only on the crystallographic point group of the system.
We define unambiguously the magnetic configurations corresponding to the symmetry-adapted multipoles in the atomic cluster.
The obtained magnetic configurations are mapped to the original crystal structure with the magnetic point group symmetry preserved.
The generated complete basis set for the magnetic structure in crystal is useful to measure the symmetry breaking as an order parameter according to the magnetic point group~\cite{Suzuki2017}.
Although we restrict ourselves to the case with ``uniform'' magnetic structures characterized by the ordering vector ${\bm q}=0$ in this paper, an extension to cases with nonzero $\bm{q}$ ordering vectors is straightforward.
To demonstrate the efficiency of the present scheme, we apply the cluster multipole expansion of magnetic structures to pyrochlore and hexagonal $AB$O$_3$ crystal structures.
For the pyrochlore structure, it is shown that the all-in all-out magnetic structure corresponds to a
M octupole, and two-in two-out and one-in three-out magnetic structures are expressed by the linear combinations of the M dipole and octupole that
belong to the same irreducible representation (IREP) of the crystallographic point group.
The two-in two-out and one-in three-out magnetic structures can be transformed continuously to the pure antiferromagnetic structures, indicating that the antiferromagnetic structures without net magnetization yield the AH effect.
For the hexagonal $AB$O$_3$ structure, higher rank multipoles such as MT quadrupole and M octupole are necessary to describe the magnetic structures exhibiting the AH and EM effects within the uniform magnetic ordering.
\begin{center}
\begin{figure*}[t]
\includegraphics[width=0.9\linewidth]{GenerationFlow.pdf}
\caption{Outline of the generation procedure for 3$N_{{\rm atom}}$ complete basis set of multipole magnetic structure classified by the crystallographic point group.}
\label{Fig:GenerationFlow}
\end{figure*}
\end{center}
\section{Multipole expansion of magnetic structures in crystal}
\subsection{Outline of multipole expansion of magnetic structures in crystal}
In this paper, we present a method to generate an orthonormal complete basis set of magnetic structures corresponding to the cluster multipoles classified according to the IREPs of the crystallographic point group.
To make free from confusion and ambiguity, we define some words used in this paper.
``Atomic cluster'' is used for the atoms transformed to each other only by the rotation operations of a point group and distinguished from ``crystal'' which assumes a periodicity for an atomic configuration.
``Magnetic configuration'' is used for the alignment of magnetic dipole moments on atoms of the atomic cluster and distinguished from ``magnetic structure'' which is for the alignment of magnetic dipole moments on atoms in periodic crystal.
A magnetic configuration (a magnetic structure) characterized by the M and/or MT multipoles is called multipole magnetic configuration (multipole magnetic structure).
The outline of the generation procedure for the orthonormal multipole
magnetic structures in crystal is following (see also Fig.~\ref{Fig:GenerationFlow}):
\begin{enumerate}
\item Set a virtual atomic cluster corresponding to the crystallographic point group of a target crystal.
\item Generate magnetic configurations corresponding to the symmetrized M and MT multipoles in the virtual atomic cluster.
\item Map the magnetic dipole moments on the atoms in the virtual atomic cluster to those on the crystallographically equivalent atoms in crystal, which represent the multipole magnetic structures.
\item Orthonormalize the bases of multipole magnetic structures by using
the Gram-Schmidt orthogonalization procedure.
\end{enumerate}
Details for the procedure will be explained in Sec.~\ref{Sec:VirtualCluster}, \ref{Sec:Mapping}, and \ref{Sec:Orthonormalization}.
With this method, all the generated orthonormal magnetic structures are characterized by the symmetry-adapted multipoles.
Thus, the multipoles can be a useful measure of symmetry breaking of the crystallographic point group in the presence of the uniform magnetic orders.
It is a natural extension of the conventional dipole magnetization which measures the symmetry breaking of the ferromagnetic order.
The multipole expansion for magnetic structures in crystal is very efficient to investigate the relation between the magnetic structures and physical properties beyond the symmetry analysis.
\subsection{Multipole magnetic configurations in atomic clusters}
The multipole expansion of the vector gauge potential is given under the Coulomb gauge $\bm{\nabla}\cdot{\bm A}({\bm r})=0$ as follows:
\begin{align}
{\bm A}({\bm r})=\sum_{\ell m}\biggl(b_{\ell}M_{\ell m} \frac{{\bm Y}_{\ell m}^{\ell}(\hat{{\bm r}})}{r^{\ell+1}} + c_{\ell} T_{\ell m}\frac{{\bm Y}_{\ell m}^{\ell+1}(\hat{{\bm r}})}{r^{\ell+2}} \biggr),
\label{Eq:VecPot}
\end{align}
where ${\bm Y}_{\ell m}^{\ell'}(\hat{{\bm r}})$ ($\ell\ge 1$, $-\ell\le m \le \ell$, $\ell'=\ell-1, \ell,\ell+1$) is the vector spherical harmonics that transforms as conventional scalar spherical harmonics $Y_{\ell m}(\hat{{\bm r}})$ for rotation operation with its orbital angular momentum $\ell'$~\cite{Blatt1991,Kusunose2008,Varshalovich1988}.
The expansion coefficients in Eq.~(\ref{Eq:VecPot})
are the so-called M multipole, $M_{\ell m}$, and MT multipole, $T_{\ell m}$, respectively ($b_{\ell}$ and $c_{\ell}$ are introduced for convenience).
The M and MT multipoles around a single magnetic ion in the unit of Bohr magneton are defined as
\begin{align}
M_{\ell m}&=\sqrt{\frac{4\pi}{2\ell+1}}\sum_{j}\biggl(\frac{2{\bm\ell}_j}{\ell +1}+{\bm \sigma_j}\biggr)\cdot \bm{O}_{lm}(\bm{r}_{j}),
\label{Eq:FullMP}
\\
T_{\ell m}&=\sqrt{\frac{4\pi}{2\ell+1}}\sum_{j}\biggl\{\frac{{\bm r}_j}{\ell +1}\times\biggl(\frac{2{\bm \ell}_j}{\ell +2}+{\bm \sigma}_j\biggr)\biggr\} \cdot
\bm{O}_{lm}(\bm{r}_{j}),
\label{Eq:FullTD}
\end{align}
with
\begin{align}
\bm{O}_{lm}(\bm{r})\equiv \bm{\nabla}\left[ r^{\ell}Y^{*}_{\ell m}(\hat{\bm{r}}) \right],
\label{eq:opry}
\end{align}
where ${\bm \ell}_j$ and ${\bm\sigma}_j$ are the orbital and spin angular momentum of an electron at ${\bm r}_j$.
Here, $r=|\bm{r}|$ and $\hat{\bm{r}}=\bm{r}/r$.
Since the vector spherical harmonics ${\bm Y}_{\ell m}^{\ell'}$ are an orthonormal complete basis set of a vector function on a sphere, the M and MT multipoles $M_{\ell m}$ and $T_{\ell m}$ represent the arbitrary angular dependence of the magnetization distribution on a single magnetic ion.
Focusing on the spin part of Eqs.~(\ref{Eq:FullMP}) and (\ref{Eq:FullTD}), the M and MT multipoles can be extended straightforwardly to characterize the classical magnetic configurations $\{{\bm m}_i \}$ in an atomic cluster whose atoms are transformed to each other through the point-group symmetry operations with respect to the symmetry center of the atomic cluster.
The explicit definitions are given as
\begin{align}
M_{\ell m}&\equiv \sqrt{\frac{4\pi}{2\ell+1}}\sum_{i=1}^{N_{\rm atom}}{\bm m}_i\cdot
\bm{O}_{lm}(\bm{R}_{i}), \\
T_{\ell m}&\equiv \frac{1}{\ell+1}\sqrt{\frac{4\pi}{2\ell+1}}\sum_{i=1}^{N_{\rm atom}}({\bm R}_i\times {\bm m}_{i})\cdot
\bm{O}_{lm}(\bm{R}_{i}),
\end{align}
where ${\bm R}_{i}$ is the position vector of $i$-th atom and $N_{\rm atom}$ is the number of atoms in the atomic cluster.
These multipoles have the same transformation property with that of the spherical harmonics $Y_{\ell m}$ for rotation operation of the point group, and hence they are classified according to the IREPs of the point group~\cite{Hayami2018}.
The spherical harmonics are usually symmetrized according to the IREPs as
\begin{align}
\mathcal{Y}_{\ell \gamma}(\hat{{\bm r}}) =\sum_{m} c^{\gamma}_{\ell m} Y_{\ell m}(\hat{{\bm r}}),
\label{Eq:Spherical}
\end{align}
where $\gamma$ runs from 1 to $2\ell+1$ in order to distinguish the IREP and its component including multiplicity which is necessary when the same IREPs are multiply appeared in the same rank $\ell$.
The coefficients $c^{\gamma}_{\ell m}$ of the symmetrized spherical harmonics $\mathcal{Y}_{\ell \gamma}(\hat{{\bm r}})$ are tabulated in Ref.~\onlinecite{Kusunose2008} for instance, where $c^{\gamma}_{\ell m}$ are chosen so that $\mathcal{Y}_{\ell \gamma}(\hat{{\bm r}})$ is real.
This is always possible in the presence of the time-reversal symmetry.
The symmetry-adapted multipoles are thus reexpressed as follows:
\begin{align}
M_{\ell \gamma}&=\sum_{i=1}^{N_{\rm atom}} {\bm u}_{\ell\gamma i}^{M}\cdot {\bm m}_i,
\label{Eq:CMP} \\
T_{\ell \gamma}&=\sum_{i=1}^{N_{\rm atom}} {\bm u}_{\ell\gamma i}^{T}\cdot {\bm m}_i,
\label{Eq:CTD}
\end{align}
where
\begin{align}
{\bm u}^{M}_{\ell\gamma i}&=\sqrt{\frac{4\pi}{2\ell+1}}\bm{\mathcal{O}}_{l\gamma}(\bm{R}_i),
\label{Eq:base_mp} \\
{\bm u}^{T}_{\ell\gamma i}&=\frac{1}{\ell+1}\sqrt{\frac{4\pi}{2\ell+1}}
\left(\bm{\mathcal{O}}_{l\gamma}(\bm{R}_i)\times {\bm R}_i\right),
\label{Eq:base_td}
\end{align}
and $\bm{\mathcal{O}}_{l\gamma}$ is given by replacing $Y_{lm}^{*}$ in Eq.~(\ref{eq:opry}) with $\mathcal{Y}_{l\gamma}$.
In generating an orthogonal basis set of the magnetic structures based on the symmetry, the conventional projection operator method requires nontrivial trials to find out the suitable trial functions, and become complicated when the crystal contains many magnetic atoms.
Moreover, that procedure has large ambiguity for low-symmetry crystals, and for the complicated magnetic structures, it would be difficult to find out the correspondence between the magnetic structures and symmetry-adapted multipoles.
On the contrary, our new method is highly advantageous since it is possible to automatically generate a complete orthonormal basis set of magnetic structures, and it is, by definition, based on the symmetry-adapted multipoles.
Here, we comment on the so-called M monopole or magnetic flux configuration that corresponds to a magnetic configuration in which all the magnetic dipole moments point to the center of the atomic cluster.
The monopole magnetic configuration is parity odd and invariant
under all the rotational symmetry operations of the point group.
Such a monopole magnetic configuration can be defined as~\cite{Spaldin2013,Thole2016}:
\begin{align}
{\bm u}^{M}_{01i}\equiv\frac{{\bm R}_i}{R_i^{2}}.
\label{Eq:base_mono}
\end{align}
In this paper, however, we do not include this type of M monopole in the cluster multipole expansion since the ordinary multipole expansion in Eq.~(\ref{Eq:VecPot}) does not contain the monopole term and the corresponding magnetic configuration always appears as a higher-rank multipole of the parity odd fully symmetric
IREP, as will be shown in the case of hexagonal $AB$O$_3$ in Sec.~\ref{Sec:HexABO3}.
\subsection{Virtual atomic clusters for crystallographic point groups}
\label{Sec:VirtualCluster}
\begin{figure}[tb]
\includegraphics[width=0.8\linewidth]{VirtualClusters.pdf}
\caption{Examples of virtual atomic clusters for (a) $O_{h}$, (b) $C_{4h}$, (c) $D_{6h}$, and (d) $C_{6v}$ point groups.}
\label{Fig:VirtualClusters}
\end{figure}
In crystal, crystallographically equivalent atoms are transformed to
each other by combinations of point group symmetry
operation, $\mathcal{R}_i$, non-primitive translation, ${\bm \tau}_j$,
and primitive translation, ${\bm T}_j$, which constitute
symmetry operations of space group.
According to Neumann's principle, the appearance of macroscopic phenomena such as AH and EM effects are determined by crystallographic point group, whose symmetry operations consist only of the rotation part $\mathcal{R}_i$ of the space group.
Therefore, in order to discuss the macroscopic phenomena induced by the antiferromagnetic order, it is necessary to define appropriate order parameters for antiferromagnetic order reflecting the symmetry breaking of the crystallographic point group.
The crystallographic point group corresponding to the space group of crystal is more definitely defined as follows.
The space group $\mathcal{G}$ is decomposed as
\begin{align}
{\mathcal G} = \sum_{i=1}^{N_{\rm coset}}\{\mathcal{R}_{i}|\bm{\tau}_{i}\}{\mathcal H}T,
\label{Eq:cosetSG}
\end{align}
where the subgroup $\mathcal{H}$ consists only of the pure rotation symmetry operations $\{h_1, h_2, \ldots,h_{N_{0}}\}$, $T$ is the group consisting of lattice translations ${\bm T}_{j}$, and $\{\mathcal{R}_{i}|\bm{\tau}_{i}\}$ are representative elements of ${\mathcal G}$ with rotation operation $\mathcal{R}_i$ ($\mathcal{R}_{1}=E$ (identity operation)) and non-primitive translation operation $\bm{\tau}_{i}$ ($\bm{\tau}_{1}=\bm{0}$).
$N_{\rm coset}$ is the number of cosets.
Note that the terms with $i\ge 2$ in Eq.~(\ref{Eq:cosetSG}) exist only in nonsymmorphic space groups.
The crystallographic point group ${\mathcal P}$ corresponding to the space group ${\mathcal G}$ is then defined as
\begin{align}
{\mathcal P} = \sum_{i=1}^{N_{\rm coset}}\{\mathcal{R}_{i}|{\bm 0} \}{\mathcal H}.
\label{Eq:cosetCPG}
\end{align}
Magnetic configurations in
an atomic cluster is unambiguously defined by multipoles through Eqs.~(\ref{Eq:CMP}) and (\ref{Eq:CTD}).
It is highly contrast to a direct generation of magnetic structures in crystal, which is not straightforward since the atoms in crystal are related not only by rotation operations but also by translation operations as we discussed in Ref.~\onlinecite{Suzuki2017}.
To avoid difficulties in the direct generation of magnetic structures in crystal, our strategy to generate the symmetry-adapted magnetic structures in crystal is to generate the multipole magnetic configurations at first in a virtual atomic cluster defined under the point group ${\mathcal P}$, and then map the magnetic configurations onto the magnetic structure in crystal so as to preserve the magnetic point group symmetry of the multipole configurations.
The virtual atomic cluster is defined as an atomic cluster consisting of the same number of atoms as the symmetry operations in the crystallographic point group ${\mathcal P}$,
in which their atomic positions are given as the general Wyckoff positions of the corresponding symmorphic space group,
as listed in Table~\ref{Tab:VirtualCluster} for representative point groups.
In the virtual atomic cluster, the magnetic configurations corresponding to the M and MT multipoles, Eqs.~(\ref{Eq:CMP}) and (\ref{Eq:CTD}), are obtained by Eqs.~(\ref{Eq:base_mp}) and (\ref{Eq:base_td}), respectively, with respect to the origin of the virtual atomic cluster.
The examples of the virtual atomic clusters are depicted in Fig.~\ref{Fig:VirtualClusters} for $O_h$, $C_{4h}$, $D_{6h}$, $C_{6v}$ point groups.
Note that the virtual atomic cluster has ambiguity due to the choice of the parameter $x$, $y$, $z$, as seen in the Table~\ref{Tab:VirtualCluster}.
This leads to an arbitrariness of the magnetic configurations in the virtual atomic cluster.
This arbitrariness is largely reduced when the magnetic configuration is mapped onto the atoms at high symmetry sites in crystal, as explained in Sec.~\ref{Sec:Mapping}.
\begin{center}
\begin{table*}[tb]
\caption{List of atomic positions of virtual atomic clusters for representative crystallographic point groups.
Those for other point groups are found in Ref.~\onlinecite{internationaltables2002} as general Wyckoff positions of symmorphic space groups as well as the listed point groups.
}
\begin{tabular}{cl} \hline \hline
& \multicolumn{1}{c}{atomic positions of virtual atomic clusters} \\
\hline
$O_h$ & (1) $x,y,z$ (2) $\bar{x}, \bar{y}, z$ (3) $\bar{x},y,\bar{z}$ (4) $x, \bar{y}, \bar{z}$ (5) $z,x,y$ (6)
$z, \bar{x}, \bar{y}$ (7) $\bar{z}, \bar{x},y$ (8) $\bar{z},x,
\bar{y}$ \\
& (9) $y, z,x$ (10) $\bar{y}, z, \bar{x}$ (11) $y, \bar{z},
\bar{x}$ (12) $\bar{y}, \bar{z},x$ (13) $y,x,\bar{z}$ (14)
$\bar{y}, \bar{x}, \bar{z}$ (15) $y, \bar{x}, z$ (16) $\bar{y},x,z$ \\
& (17) $x, z,\bar{y}$ (18) $\bar{x}, z,y$ (19) $\bar{x}, \bar{z}, \bar{y}$ (20) $x,\bar{z},y$ (21) $z,y,\bar{x}$ (22) $z,\bar{y},x$ (23) $\bar{z},y,x$ (24) $\bar{z},\bar{y},\bar{x}$ \\
& (25) $\bar{x},\bar{y},\bar{z}$ (26) $x, y,\bar{z}$ (27) $x,\bar{y},
z$ (28)$\bar{x},y,z$ (29) $\bar{z},\bar{x},\bar{y}$ (30) $\bar{z},
x, y$ (31) $z, x,\bar{y}$ (32) $z,\bar{x},y$ \\
& (33) $\bar{y},\bar{z},\bar{x}$ (34) $y,\bar{z}, x$ (35) $\bar{y}, z, x$ (36) $y, z,\bar{x}$ (37) $\bar{y},\bar{x}, z$ (38) $y, x, z$ (39) $\bar{y}, x,\bar{z}$ (40) $y,\bar{x},\bar{z}$ \\
& (41) $\bar{x},\bar{z}, y$ (42) $x,\bar{z},\bar{y}$ (43) $x, z, y$ (44) $\bar{x}, z,\bar{y}$
(45) $\bar{z},\bar{y}, x$ (46) $\bar{z}, y,\bar{x}$ (47)
$z,\bar{y},\bar{x}$ (48) $z,y,x$ \\
\hline
$O$ & (1) $x,y,z$ (2) $\bar{x}, \bar{y}, z$ (3) $\bar{x},y,\bar{z}$ (4) $x, \bar{y}, \bar{z}$ (5) $z,x,y$ (6)
$z, \bar{x}, \bar{y}$ (7) $\bar{z}, \bar{x},y$ (8) $\bar{z},x,
\bar{y}$ \\
& (9) $y, z,x$ (10) $\bar{y}, z, \bar{x}$ (11) $y, \bar{z},
\bar{x}$ (12) $\bar{y}, \bar{z},x$ (13) $y,x,\bar{z}$ (14)
$\bar{y}, \bar{x}, \bar{z}$ (15) $y, \bar{x}, z$ (16) $\bar{y},x,z$ \\
& (17) $x, z,\bar{y}$ (18) $\bar{x}, z,y$ (19) $\bar{x}, \bar{z}, \bar{y}$ (20) $x,\bar{z},y$ (21) $z,y,\bar{x}$ (22) $z,\bar{y},x$ (23) $\bar{z},y,x$ (24) $\bar{z},\bar{y},\bar{x}$ \\
\hline
$T_{d}$ & (1) $x,y, z $ (2) $ \bar{x}, \bar{y}, z $ (3) $ \bar{x},y, \bar{z} $ (4) $ x, \bar{y}, \bar{z}
$ (5) $ z,x,y $ (6) $ z, \bar{x}, \bar{y} $ (7) $ \bar{z}, \bar{x},y $ (8) $ \bar{z},x, \bar{y}$ \\
& (9) $ y, z,x $ (10) $ \bar{y}, z, \bar{x} $ (11) $ y, \bar{z}, \bar{x} $ (12) $ \bar{y}, \bar{z},x
$ (13) $ y,x, z $ (14) $ \bar{y}, \bar{x}, z $ (15) $ y, \bar{x}, \bar{z} $ (16) $ \bar{y},x, \bar{z}$ \\
& (17) $ x, z,y $ (18) $ \bar{x}, z, \bar{y} $ (19) $ \bar{x}, \bar{z},y $ (20) $ x, \bar{z}, \bar{y}
$ (21) $ z,y,x $ (22) $ z, \bar{y}, \bar{x} $ (23) $ \bar{z},y, \bar{x}
$ (24) $ \bar{z}, \bar{y},x $ \\
\hline
$T_{h}$ & (1) $ x,y, z $ (2) $ \bar{x}, \bar{y}, z $ (3) $ \bar{x},y,\bar{z} $ (4) $ x,\bar{y},\bar{z}
$ (5) $ z,x,y $ (6) $ z,\bar{x},\bar{y} $ (7) $ \bar{z},\bar{x},y $ (8)
$ \bar{z},x,\bar{y}$ \\
& (9) $ y,z,x $ (10) $ \bar{y}, z,\bar{x} $ (11) $ y,\bar{z},\bar{x} $ (12) $ \bar{y},\bar{z},x
$ (13) $ \bar{x}, \bar{y}, \bar{z} $ (14) $ x,y,\bar{z} $ (15) $ x,\bar{y}, z $ (16) $ \bar{x},y, z
$ \\
& (17) $ \bar{z}, \bar{x}, \bar{y} $ (18) $ \bar{z},x,y $ (19) $ z,x, \bar{y} $ (20) $ z, \bar{x},y
$ (21) $ \bar{y}, \bar{z}, \bar{x} $ (22) $ y, \bar{z},x $ (23) $ \bar{y}, z,x $ (24) $ y, z,\bar{x}$ \\
\hline
$C_{4h}$ & (1) $x,y,z$ (2) $\bar{x}, \bar{y}, z$ (3) $\bar{y}, x, z$ (4) $y, \bar{x}, z$
(5) $\bar{x},\bar{y},\bar{z}$ (6) $x, y, \bar{z}$ (7) $y,
\bar{x}, \bar{z}$ (8) $\bar{y}, x, \bar{z}$ \\
\hline
$D_{6h}$ & (1) $x,y, z$ (2) $\bar{y},x-y, z$ (3)$\bar{x}+y, \bar{x}, z$
(4) $\bar{x}, \bar{y}, z$ (5) $y, \bar{x}+y, z$ (6) $x-y,x, z$
(7) $y,x,\bar{z}$ (8) $x-y, \bar{y},\bar{z}$ \\
& (9) $\bar{x}, \bar{x}+y, \bar{z}$ (10) $\bar{y}, \bar{x}, \bar{z}$ (11) $\bar{x}+y,y, \bar{z}$ (12) $x,x-y, \bar{z}$ (13) $\bar{x}, \bar{y}, \bar{z}$ (14) $y, \bar{x}+y, \bar{z}$ (15) $x-y, x, \bar{z}$ (16) $x,y,\bar{z}$ \\
& (17) $\bar{y},z-y,\bar{z}$ (18) $\bar{x}+y, \bar{x}, \bar{z}$ (19)
$\bar{y},\bar{x}, z$ (20) $\bar{x}+y,y,z $(21) $x,x-y,z$ (22)
$y,x, z$ (23) $x-y,\bar{y},z$ (24) $\bar{x}, \bar{x}+y,z$ \\
\hline
$C_{6v}$ & (1) $x,y,z$ (2) $\bar{y}, x-y, z$ (3) $\bar{x}+y,\bar{x}, z$
(4) $\bar{x}, \bar{y}, z$ (5) $y, \bar{x}+y, z$ (6) $x-y, x, z$
(7) $\bar{y}, \bar{x}, z$ (8) $\bar{x}+y, y, z$, \\
& (9) $x, x-y, z$ (10) $y, x, z$ (11) $x-y, \bar{y}, z$ (12) $\bar{x}, \bar{x}+y, z$ \\
\hline
$D_{3d}$ & (1) $x,y,z$ (2) $\bar{y},x-y, z$ (3) $\bar{x}+y, \bar{x}, z$
(4) $\bar{y}, \bar{x}, \bar{z}$ (5) $\bar{x}+y,y,\bar{z}$ (6) $x, x-y,\bar{z}$ (7) $\bar{x}, \bar{y}, \bar{z}$ (8) $ y, \bar{x}+y, \bar{z}$ \\
& (9) $ x-y,x, \bar{z} $ (10) $ y,x,z$ (11) $ x-y, \bar{y}, z $ (12) $
\bar{x}, \bar{x}+y, z$\\
\hline
\end{tabular}
\label{Tab:VirtualCluster}
\end{table*}
\end{center}
\subsection{Mapping of multipoles from virtual atomic cluster to crystal}
\label{Sec:Mapping}
The purpose of this section is to obtain the magnetic structures whose
transformation property for the magnetic point group operations is
the same with that of multipole configurations in the virtual atomic cluster.
For this purpose, we first choose one atom in the virtual atomic cluster and one in crystal and set the same M dipole moments on these atoms by setting a mapping from the atoms in virtual atomic cluster to that in crystal.
The whole mapping between the atoms in the virtual atomic cluster and
the symmetrically equivalent atoms in crystal is obtained by identifying
the atoms transformed by the point group symmetry operations, $R_i$, in
the virtual atomic cluster with the atoms transformed by the symmetry
operations of space group, $\{E|\bm{T}_{j}\}\{R_i|\bm{\tau}_i\}h_{k}$, in the crystal for the initially chosen atoms in both systems.
The mapping from the atoms in the virtual atomic cluster to the symmetrically equivalent atoms in crystal obviously does not have one-to-one correspondence.
The M dipole moment on an atom in crystal is obtained by summing up the M dipole moments on the all of the corresponding atoms in the virtual atomic cluster, as will be discussed in Sec.~\ref{Sec:SimpleEx}.
The generated magnetic structures in crystal are fully characterized by the symmetry-adapted multipoles in the virtual atomic cluster through the mapping, which we call the multipole magnetic structures.
We note that the correspondence between generated magnetic structures and multipole configurations depends on the choice of the first mapping between the atoms in virtual atomic cluster and that in crystal.
It occurs especially for low-symmetry crystallographic point groups having the multiple symmetrized bases within the same IREPs and rank.
In this case, we have to specify the representative atoms in the virtual atomic cluster and in crystal to identify the correspondence between the magnetic structures and multipole configurations.
The other arbitrariness for the multipole structures arises from a choice of the parameter ($x$, $y$, $z$) of virtual atomic cluster as listed in Table~\ref{Tab:VirtualCluster}, as already mentioned in Sec.~\ref{Sec:VirtualCluster}.
The advantage to introduce the virtual atomic cluster is that once we generate the multipole magnetic configurations in the virtual atomic cluster, we can systematically generate complete orthonormal basis sets for arbitrary crystal structures by appropriate mappings, in which they share the common
point group between the atomic cluster and crystal.
This is in a high contrast with our previous scheme proposed in Ref.~\onlinecite{Suzuki2017} to identify the multipoles in crystal.
In the previous scheme, the atomic cluster is defined not in a virtual one but directly for the atoms in the crystal.
We identified $N_{\rm coset}$ clusters in which one consist of the atoms transformed to each other by the point group operations of $\mathcal{H}$ in Eq.~(\ref{Eq:cosetSG}) and the others are determined from transformation of $\{\mathcal{R}_i|\bm{\tau}_i\}\mathcal{H}$ for the cluster in the crystal unit cell.
The macroscopic multipole moment was obtained by summing up the multipole moments in each cluster
in accordance with a nonsymmorphic space group of Mn$_3$$Z$ ($Z$=Sn, Ge).
However, there are some cases that the higher rank multipoles are ill defined, especially when the magnetic atoms are located at high symmetry sites.
The present scheme does not cause this problem and is advantageous in its efficiency since the arbitrariness of the generated orthonormal magnetic structures is largely reduced as compared with the conventional method using projection operator~\cite{Bertaut1968, Bertaut1981}, whose generated magnetic structures highly depend on the choice of the trial functions to be operated.
Based on the present procedure, we can systematically and automatically generate magnetic structures corresponding to the multipoles according to the IREPs of the crystallographic point group of the focusing crystal.
\subsection{Orthonormalization of basis set for magnetic structure in crystal}
\label{Sec:Orthonormalization}
The multipole expansion in Eq.~(\ref{Eq:VecPot}) requires infinite number of components to express magnetization distribution in continuous space.
In contrast, the multipole expansion in magnetic structures that does not break the crystal periodicity is represented by a linear combination of 3$N_{\rm atom}$ orthogonal basis set, where $N_{\rm atom}$ is the number of atoms in a crystal unit cell.
In this subsection, we explain how to obtain the 3$N_{\rm atom}$ orthonormal basis set which is sufficient to express a uniform magnetic order.
For notational convenience, we introduce the vector notation for uniform magnetic structures, $\{{\bm a}\}=({\bm a}_1,{\bm a}_2,\cdots,{\bm a}_{N_{\rm atom}})$, where ${\bm a}_{i}$ represents the three component vector on the $i$-th atom in the crystal unit cell.
Usually, the magnetic structure $\{{\bm m}_i\}$ is decomposed into the ferromagnetic part $\{{\bm m}_i^{\rm FM}\}$, where ${\bm m}_i^{\rm FM}=\sum_i{\bm m}_i/N_{\rm atom}$, and the rest antiferromagnetic part $\{{\bm m}_{i}^{\rm AFM}\}$, where ${\bm m}_{i}^{\rm AFM}={\bm m}_{i}-{\bm m}_i^{\rm FM}$\cite{Ederer2007}.
Such a decomposition for the magnetic structure is generalized to obtain the 3$N_{\rm atom}$ orthonormal complete basis set by using Gram-Schmidt orthonormalization procedure as follows:
\begin{align}
\{{\bm e}_{1\gamma}^{1}\}&\equiv \frac{\{{\bm u}^{1}_{1\gamma}\}}{\sqrt{(\{{\bm u}^{1}_{1\gamma}\}\cdot\{{\bm u}^{1}_{1\gamma}\})}},
\label{Eq:GS_initial}\\
\{{\bm v}_{\ell\gamma}^{\mu}\} &=\{{\bm u}_{\ell\gamma}^{\mu}\}-\sum_{\ell'=1}^{\ell}\sum_{\mu'=1}^{\mu}\sum_{\gamma'=1}^{\gamma-1}(\{{\bm u}_{\ell\gamma}^{\mu'}\}\cdot\{{\bm e}_{\ell'\gamma'}^{\mu'}\})\{{\bm e}_{\ell'\gamma'}^{\mu'}\},
\label{Eq:GS_Orthogonal}\\
\{{\bm e}_{\ell\gamma}^{\mu}\}&=\frac{\{{\bm v}_{\ell\gamma}^{\mu}\}}{\sqrt{(\{{\bm v}_{\ell\gamma}^{\mu}\}\cdot\{{\bm v}_{\ell\gamma}^{\mu}\})}},
\label{Eq:GS_Orthonormal}
\end{align}
where $\mu=1$ and $2$ represent the M and MT multipoles, respectively.
The initial ${\bm u}^{1}_{1\gamma}$ in Eq.~(\ref{Eq:GS_initial}) is set as the M dipole moments, $M_{x}$, $M_{y}$, and $M_{z}$ for $\gamma=1,2,3$, respectively.
The iterated calculation of the Gram-Schmidt orthonormalization procedure, Eqs.~(\ref{Eq:GS_Orthogonal}) and (\ref{Eq:GS_Orthonormal}), starting from the lower-rank multipoles to higher ones as shown in Fig.~\ref{Fig:GenerationFlow}, automatically generates orthonormal complete basis set of the uniform magnetic structures classified according to the IREPs of crystallographic point group.
After orthonormalization procedure, the magnetic structures corresponding to the higher-rank multipole may not be the pure multipole with definite rank as in Eq.~(\ref{Eq:GS_Orthogonal}).
However, since the subtraction in Eq.~(\ref{Eq:GS_Orthogonal}) is to eliminate the overlap between the highest-rank multipole and the lower ones, and $\{{\bm e}_{\ell\gamma}^{\mu}\}$ in Eq.~(\ref{Eq:GS_Orthonormal}) always contains $\{{\bm u}_{\ell\gamma}^{\mu}\}$, it can be regarded as the rank-$\ell$ multipole.
The generated finite norm
of vectors $\{{\bm e}_{\ell\gamma}^{\mu}\}$ are stored sequentially as $\{{\bm e}^{i}\}$ with the sequential indices $i=1,...,3N_{\rm atom}$.
With this procedure, the ferromagnetic structures $\{{\bm e}^{i}\}$ with $i\le 3$ are orthogonal to the antiferromagnetic structures with $i>3$.
The obtained magnetic structure basis set are orthonormal, i.e., $(\{{\bm e}^{i}\}\cdot \{{\bm e}^{j}\})=\delta_{ij}$.
Since the obtained basis set is complete for the uniform magnetic structures, arbitrary uniform magnetic structure can be expressed as $\{{\bm m}\}=\sum_{i=1}^{3N_{\rm atom}}c_{i}\{{\bm e}^{i}\}$ with $c_{i}=(\{{\bm m}\}\cdot \{{\bm e}^{i}\})$.
Note that the relation $\sum_{i}^{3N_{\rm atom}}|c_{i}|^2=\sum_{i}^{N_{\rm atom}}|{\bm m}_{i}|^2$ holds.
\section{Examples of multipole expansion for crystals}
\subsection{Simple examples}
\label{Sec:SimpleEx}
We here illustrate the correspondence between the symmetry-adapted multipole magnetic configurations in the virtual atomic cluster and magnetic structures in crystal.
We take two simple examples, i.e., the atoms placed at 8$l$ and 2$e$
Wyckoff sites in symmorphic space group $P4/m$ (No.~83), and 8$k$ and
2$a$ Wyckoff sites in nonsymmorphic space group $P4_{2}/m$ (No.~84).
The crystallographic point group of both space groups is $C_{4h}$, which corresponds to a virtual atomic cluster consisting of eight atoms transformed to each other through the symmetry operations
as shown in Fig.~\ref{Fig:VirtualClusters} (b).
In Fig.~\ref{Fig:R2Toroid_VC}, two fold rotation $C_{2}$ transforms the atom 1 to atom 2, four fold rotation $C_{4}$ to atom 3, $C_{4}^{-1}$ to atom 4, space inversion $I$ to atom 5, $IC_{2}$ to atom 6, $IC_{4}$ to atom 7, and $IC_{4}^{-1}$ to atom 8.
These transformation relations provide mapping from the atoms in the virtual atomic cluster to symmetrically equivalent atoms in the crystal as discussed in Sec.~\ref{Sec:Mapping}.
Figures~\ref{Fig:R2Toroid_VC} and \ref{Fig:R2Toroid_Cryst} show the relation between the atoms in the virtual atomic cluster and the atoms at the Wyckoff sites in crystal.
\begin{figure}[tb]
\includegraphics[width=0.8\linewidth]{rank2_toroidal_C4h.pdf}
\caption{
Magnetic configuration of the MT quadrupole in $B_{g}$ IREP of $C_{4h}$ point group in the virtual atomic cluster, placing an atom labeled 1 at (0,0.3,0.3), which is obtained by Eq.~(\ref{Eq:base_td}) with $\mathcal{Y}_{25}$ in Eq.~(\ref{Eq:symY25}).}
\label{Fig:R2Toroid_VC}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=0.9\linewidth]{Mapped_toroidal.pdf}
\caption{MT quadrupoles in crystals belonging to $P4/m$ and $P4_{2}/m$ space groups, which are obtained by mapping from the $C_{4h}$ virtual atomic cluster as shown in Fig.~\ref{Fig:R2Toroid_VC}.}
\label{Fig:R2Toroid_Cryst}
\end{figure}
For the $C_{4h}$ point group, the rank-2 spherical harmonics are symmetrized as
\begin{align}
\mathcal{Y}_{21} &= Y_{20}\quad (A_g), \cr
\mathcal{Y}_{22} &= \frac{1}{\sqrt{2}}(Y_{2-1}-Y_{21})\quad (E_g), \cr
\mathcal{Y}_{23} &= \frac{1}{\sqrt{2}i}(Y_{2-1}+Y_{21})\quad (E_g),
\label{Eq:symY25} \\
\mathcal{Y}_{24} &= \frac{1}{\sqrt{2}}(Y_{2-2}+Y_{22})\quad (B_g), \cr
\mathcal{Y}_{25} &= \frac{1}{\sqrt{2}i}(Y_{2-2}-Y_{22})\quad (B_g). \nonumber
\end{align}
The multipole configurations are calculated from Eq.~(\ref{Eq:base_mp}) and (\ref{Eq:base_td}) on the virtual atomic cluster as shown in Fig.~\ref{Fig:VirtualClusters}(b), where the atom 1 is placed at (0,0.3,0,3).
Figure~\ref{Fig:R2Toroid_VC} shows the magnetic configuration generated from Eq.~(\ref{Eq:base_td}) with the symmetrized spherical harmonics $\mathcal{Y}_{25}$ in Eq.~(\ref{Eq:symY25}), corresponding to the MT quadrupole, $T_{xy}$, classified to $B_{g}$ IREP of $C_{4h}$ point group~\cite{Hayami2018}.
This magnetic configuration in the virtual atomic cluster is mapped to symmetrically equivalent atoms, denoted by the signatures of Wyckoff positions, to obtain the corresponding magnetic structures in crystals as shown in Fig.~\ref{Fig:R2Toroid_Cryst}.
For the general Wyckoff positions in each crystal ($8l$ site of
$P4/m$ and $8k$ site of $P4_{2}/m$), the magnetic structures are straightforwardly obtained through the mapping (Fig.~\ref{Fig:R2Toroid_Cryst}).
Meanwhile, for $2e$ site of $P4/m$ and $2a$ site of $P4_{2}/m$, multiple atoms in the virtual cluster are mapped to the same atoms in crystal, leading to the cancellation of the M dipole moment that is consistent with the symmetry in these Wyckoff sites.
\subsection{Pyrochlore structure}
\begin{table}[tb]
\caption{
Relation between the multipoles corresponding to the orthonormal magnetic structures for pyrochlore crystal structure and IREP for the crystallographic point group $O_{h}$, as well as the magnetic point group with its principal axis.
The active component of the AH conductivity (AHC) is also shown.
}
\begin{tabular}{cccccc} \hline\hline
No. & IREP & multipole & MPG & P. axis & AHC \\
\hline
1 & $T_{1g}$ & $M_{x}$ & $4/mm'm'$ & [100] & $\sigma_{yz}$ \\
2 & & $M_{y}$ & $4/mm'm'$ & [010] & $\sigma_{zx}$ \\
3 & & $M_{z}$ & $4/mm'm'$ & [001] & $\sigma_{xy}$ \\ \hline
4 & $E_{g}$ & $T_{v}$ & $4/mmm$ & [001] & --- \\
5 & & $T_{u}$ & $4'/mmm'$ & [001] & --- \\ \hline
6 & $T_{2g}$ & $T_{yz}$ & $4'/mm'm$ & [100] & ---\footnotemark[1] \\
7 & & $T_{zx}$ & $4'/mm'm$ & [010] & ---\footnotemark[1] \\
8 & & $T_{xy}$ & $4'/mm'm$ & [001] & ---\footnotemark[1] \\ \hline
9 & $A_{2g}$ & $M_{xyz}$ & $m\bar{3}m'$ & [001] & --- \\ \hline
10 & $T_{1g}$ & $M_{x}^{\alpha}$ & $4/mm'm'$ & [100] & $\sigma_{yz}$ \\
11 & & $M_{y}^{\alpha}$ & $4/mm'm'$ & [010] & $\sigma_{zx}$ \\
12 & & $M_{z}^{\alpha}$ & $4/mm'm'$ & [001] & $\sigma_{xy}$ \\
\hline
\end{tabular}
\footnotetext[1]{The multipole magnetic structures characterized by one of $T_{yz}$, $T_{zx}$, and $T_{xy}$ does not induce the AH effect, but those obtained by linear combinations of these MT quadrupoles can induce the AH conductivity in general due to the absence of rotation symmetries.}
\label{Tab:Pyrochlore_CMP_sym}
\end{table}
\begin{figure}[tb]
\includegraphics[width=0.9\linewidth]{Pyrochlore.pdf}
\caption{Multipole magnetic structures in pyrochlore structure characterized by the IREP of $O_h$ point group and its rank.
Notations of multipoles are adapted from Tables in Ref.~\onlinecite{Hayami2018}.}
\label{Fig:Pyrochlore}
\end{figure}
Pyrochlore crystal structure belongs to the space group $Fd\bar{3}m$ (No.~227), whose crystallographic point group $O_h$ leads to the virtual atomic cluster as shown in Fig.~\ref{Fig:VirtualClusters}(a).
The generation procedure of orthonormal magnetic structures gives twelve orthonormal bases characterized by the M dipole and octupole, and MT quadrupole as shown in Table~\ref{Tab:Pyrochlore_CMP_sym}.
In the pyrochlore structure, the inversion symmetry breaking is accompanied by the breaking of the translation symmetries since the two atoms related by the space inversion are also transformed by the commensurate translation of the crystal.
Therefore, a uniform magnetic structure
must preserve the space inversion symmetry, restricting the parity-even multipole structures without EM effects.
The generated magnetic structure coincides with the basis set obtained in earlier work.~\cite{Wills2006}.
\begin{figure}[tb]
\includegraphics[width=0.9\linewidth]{Decomposition.pdf}
\caption{The multipole expansion of all-in all-out, two-in two-out and one-in three-out magnetic structures in pyrochlore crystal structure.
}
\label{Fig:MPexpansion_Pyrochlore}
\end{figure}
The pyrochlore compounds are known to show a variety of magnetic order depending on the atomic constitutions, such as all-in all-out, two-in two-out, and one-in three-out magnetic orders.
The relation between the observed magnetic structures and the M multipoles are shown in Table~\ref{Tab:RealMaterials}.
The corresponding uniform magnetic orders are shown in Fig.~\ref{Fig:Pyrochlore}.
All-in all-out magnetic structures, reported in Cd$_2$Os$_2$O$_7$ and Er$_2$Ti$_2$O$_7$, correspond to $M_{xyz}$-octupole in Fig.~\ref{Fig:Pyrochlore}.
The magnetic point group of the all-in all-out structure prohibits to induce AH effect.
Meanwhile, two-in two-out magnetic structure is expressed by a linear combination of the M dipole and octupole belonging to $T_{1g}$ IREP as shown in Fig.~\ref{Fig:MPexpansion_Pyrochlore}.
The two-in two-out magnetic structure is allowed to induce the AH effect by the magnetic point group symmetry.
Since arbitrary linear combination does not change the symmetry, pure $T_{1g}$ octupole structure $M_{z}^{\alpha}$ without net magnetization can also induce the AH effect.
This fact may be relevant to the mechanism of the AH effect of Pr$_2$Ir$_2$O$_7$ without net magnetization~\cite{Machida2010,Nakatsuji2011}.
The order parameter for the magnetic structures of Yb$_2$Ti$_2$O$_7$, Yb$_2$Sn$_2$O$_7$, and Tm$_2$Mn$_2$O$_7$ are also characterized by the $T_{1g}$ multipoles, but in these cases, the M dipole moment is dominant~\cite{Gaudet2016,Yaouanc2013,Pomjakushina2015}.
\begin{center}
\begin{table*}[tb]
\caption{Relation between the experimentally observed magnetic structures and the M multipoles in pyrochlore and hexagonal $AB$O$_3$ crystal structures.
$M_{[111]}=(M_{x}+M_{y}+M_{z})/\sqrt{3}$ and $M_{[111]}^{\alpha}$ is similarly defined.
}
\begin{tabular}{cccc} \hline\hline
Multipole & IREP & Name & Materials \\ \hline\hline
Pyrochlore \\ \hline
$M_{xyz}$ & $A_{2g}$ & all-in all-out & Cd$_2$Os$_2$O$_7$~\cite{Yamaura2012}, Er$_2$Ti$_2$O$_7$~\cite{Poole2007} \\
$T_{xy}$ & $T_{2g}$ & --- & Gd$_2$Sn$_2$O$_7$~\cite{Wills2006}, Er$_2$Ru$_2$O$_7$~\cite{Taira2003} \\
($M_{z}$, $M_{z}^{\alpha}$) & $T_{1g}$ & 2-in 2-out & Ho$_2$Ru$_2$O$_7$\cite{Wiebe2004}, Tb$_2$Sn$_2$O$_7$\cite{Mirebeau2005} \\
$M_{z}^{\alpha}=2\sqrt{2}M_{z}$ & & & \\
($M_{z}$, $M_{z}^{\alpha}$) & $T_{1g}$ & --- & Yb$_{2}$Ti$_{2}$O$_{7}$\cite{Gaudet2016}, Yb$_{2}$Sn$_{2}$O$_{7}$~\cite{Yaouanc2013} \\
$M_{z}$ dominant & & & Tm$_{2}$Mn$_{2}$O$_{7}$\cite{Pomjakushina2015} \\
($M_{[111]}$, $M_{[111]}^{\alpha}$, $M_{xyz}$) & $T_{1g}$ & 1-in 3-out & Tb$_2$Ti$_2$O$_7$\footnotemark[2]~\cite{Sazonov2013} \\ \hline\hline
Hexagonal $AB$O$_3$ \\ \hline
$M_{u}$ & $A_{2}$ & --- & LuFeO$_3$~\cite{Disseler2015}, ScMnO$_3$ (75-129K)~\cite{Munoz2000} \\
$M_{3b}$ & $B_{1}$ & --- & HoMnO$_3$ (below 40K)~\cite{Brown2006} \\
$M_{3a}$ & $B_{2}$ & --- & HoMnO$_3$ (40-75K)~\cite{Brown2006}, YbMnO$_3$~\cite{Fabreges2008} \\
($M_{u}$, $T_{z}$) & $A_{2}\oplus A_{1}$ & --- & ScMnO$_3$ (below 75K)~\cite{Munoz2000} \\
($M_{3b}$, $M_{3a}$) & $B_{1}\oplus B_{2}$ & --- & YMnO$_3$~\cite{Brown2006} \\
\hline
\end{tabular}
\footnotetext[2]{Stabilized under magnetic fields above 5T.}
\label{Tab:RealMaterials}
\end{table*}
\end{center}
\subsection{Hexagonal $AB$O$_3$}
\label{Sec:HexABO3}
\begin{table}[tb]
\caption{
Relation between the multipoles corresponding to the orthonormal magnetic structures for hexagonal $AB$O$_3$ and IREP of $C_{6v}$, as well as the magnetic point group (MPG) with its principal axis. The active components of the AH conductivity (AHC) and EM tensor are also shown.
}
\begin{tabular}{ccccccc} \hline\hline
No. & IREP& multipole & MPG & P. axis & AHC & EM \\
\hline
1 & $E_{1}$ & $M_{x}$ & $mm'2'$ & [100] & $\sigma_{yz}$ & $\alpha_{xz}$, $\alpha_{zx}$ \\
2 & & $M_{y}$ & $m'm2'$ & [100] & $\sigma_{zx}$ & $\alpha_{yz}$, $\alpha_{zy}$ \\ \hline
3 & $A_{2}$ & $M_{z}$ & $6m'm'$ & [001] & $\sigma_{xy}$ & $\alpha_{xx}=\alpha_{yy}$,$\alpha_{zz}$ \\ \hline
4 & $E_{1}$ & $T_{x}$ & $m'm2'$ & [100] & $\sigma_{zx}$ & $\alpha_{yz}$, $\alpha_{zy}$ \\
5 & & $T_{y}$ & $mm'2'$ & [100] & $\sigma_{yz}$ & $\alpha_{xz}$, $\alpha_{zx}$ \\ \hline
6 & $A_{1}$ & $T_{z}$ & $6mm$ & [001] & --- & $\alpha_{xy}=-\alpha_{yx}$ \\ \hline
7 & $A_{2}$ & $M_{u}$ & $6m'm'$ & [001] & $\sigma_{xy}$ & $\alpha_{xx}=\alpha_{yy},\alpha_{zz}$ \\ \hline
8 & $E_{2}$ & $M_{v}$ & $m'm'2$ & [100] & $\sigma_{xy}$ & $\alpha_{xx}$,$\alpha_{yy}$,$\alpha_{zz}$ \\
9 & & $M_{xy}$ & $mm2$ & [100] & --- & $\alpha_{xy}$, $\alpha_{yx}$ \\ \hline
10 & $E_{1}$ & $T_{yz}$ & $mm'2'$ & [100] & $\sigma_{yz}$ & $\alpha_{xz}$, $\alpha_{zx}$ \\
11 & & $T_{zx}$ & $m'm2'$ & [100] & $\sigma_{zx}$ & $\alpha_{yz}$, $\alpha_{zy}$ \\ \hline
12 & $E_{2}$ & $T_{v}$ & $mm2$ & [100] & --- & $\alpha_{xy}$, $\alpha_{yx}$ \\
13 & & $T_{xy}$ & $m'm'2$ & [100] & $\sigma_{xy}$ & $\alpha_{xx}$,$\alpha_{yy}$,$\alpha_{zz}$\\ \hline
14 & $B_{1}$ & $M_{3b}$ & $6'mm'$ & [001] & --- & --- \\ \hline
15 & $B_{2}$ & $M_{3a}$ & $6'm'm$ & [001] & --- & --- \\ \hline
16 & $B_{1}$ & $T_{3a}$ & $6'mm'$ & [001] & --- & --- \\ \hline
17 & $E_{2}$ & $T_{xyz}$ & $m'm'2$ & [100] & $\sigma_{xy}$ & $\alpha_{xx}$,$\alpha_{yy}$,$\alpha_{zz}$ \\
18 & & $T_{z}^{\beta}$ & $mm2$ & [100] & --- & $\alpha_{xy}$, $\alpha_{yx}$ \\
\hline
\end{tabular}
\label{Tab:HexagonalABO3_CMP_sym}
\end{table}
Hexagonal $AB$O$_3$ compounds belong to $P6_{3}cm$ (No.~185) space group, whose crystallographic point group is $C_{6v}$, and the corresponding virtual cluster is
shown in Fig.~\ref{Fig:VirtualClusters}(d).
The magnetic structure generation gives eighteen magnetic structures as shown in Fig.~\ref{Fig:ABO3_multipole}.
The present scheme generates the real basis set, while it contains the complex expressions for two dimensional IREPs, $E_{1}$ and $E_{2}$, in the previous work~\cite{Munoz2000}.
In Table~\ref{Tab:HexagonalABO3_CMP_sym}, we summarize the relation between the multipole magnetic structures and possible AH and EM effects induced under uniform magnetic order.
Table~\ref{Tab:HexagonalABO3_CMP_sym} shows that the higher-rank multipoles are necessary to fully describe the EM effects.
The magnetic order characterized by $M_{u}$ quadrupole is recognized in LuFeO$_3$, and in a temperature range of 75K-129K in ScMnO$_3$~\cite{Munoz2000}.
The $M_{u}$ quadrupole induces the diagonal components in the EM tensor, $\alpha_{xx}=\alpha_{yy}$ and $\alpha_{zz}$.
The M quadrupole belongs to $A_2$ IREP, which is the same as that of $M_{z}$ dipole in the $C_{6v}$ crystallographic point group, and hence it can also induce the AH conductivity $\sigma_{xy}$ as shown in Table~\ref{Tab:RealMaterials}.
We note that the {$M_{u}$} quadrupole structure is also regarded as the monopole magnetic structure characterized by $M_{0}$ since the same magnetic structure is obtained by using Eq.~(\ref{Eq:base_mono}) with the appropriate mapping from the virtual cluster to the crystal unit cell.
The magnetic structures observed in HoMnO$_3$~\cite{Brown2006} and YbMnO$_3$~\cite{Fabreges2008} are characterized by $M_{3a}$ and $M_{3b}$ octupoles, which do not induce the EM effect (Table~\ref{Tab:HexagonalABO3_CMP_sym}).
The magnetic structures in YMnO$_3$ and the low-temperature phase in ScMnO$_3$ in Table~\ref{Tab:RealMaterials} are characterized by linear combinations of the multipole structures with different IREPs, which lowers the magnetic point group symmetry, and may lead to additional finite components in the AH conductivity and EM {\rm tensors.}
For instance, the ($M_u$, $T_z$) multipole structure of low-temperature phase of ScMnO$_3$ only preserves 6 fold rotation symmetry along $z$-axis and can have finite $\sigma_{xy}$ for the AH conductivity~\cite{Kleiner1966,Seemann2015,Suzuki2017}, and finite $\alpha_{xx}=\alpha_{yy}$, $\alpha_{zz}$, and $\alpha_{xy}=-\alpha_{yx}$ for the EM coefficients~\cite{Birss1964,Rivera2009}.
Meanwhile, the ($M_{3b}$, $M_{3a}$) multipole structure of YMnO$_3$ preserves the magnetic point group symmetry operation of 6 fold rotation along $z$-axis combined with the time reversal operation, and no AH and EM effects are expected.
\begin{center}
\begin{figure*}[tb]
\includegraphics[width=0.9\linewidth]{ABO3_hexagonal_all.pdf}
\caption{Multipole magnetic structures in $AB$O$_3$ characterized by the IREP of $C_{6v}$ point group and its rank.
Notations of multipoles are adapted from Tables in Ref.~\onlinecite{Hayami2018}.}
\label{Fig:ABO3_multipole}
\end{figure*}
\end{center}
\section{Summary}
We have proposed a scheme to efficiently generate the symmetry-adapted orthonormal magnetic structures in crystallographic point group by introducing a virtual atomic cluster to perform the multipole expansion.
With this method, we can obtain magnetic structure that is fully characterized by the M and MT multipole as the suitable order parameters.
We have introduced a virtual atomic cluster to obtain the magnetic structures preserving the magnetic point group symmetry of the multipoles.
We have applied the proposed method to pyrochlore and hexagonal $AB$O$_3$ crystal structures.
For the pyrochlore crystal structure, we have investigated all-in all-out, two-in two-out, and one-in three-out magnetic structures and found that the two-in two-out and one-in three-out magnetic structures are able to be transformed continuously to the pure antiferromagnetic (octupole) structures without net magnetization, leading to the AH effect.
For the hexagonal $AB$O$_3$ crystal structure, the expression of the EM effect is fully identified by the multipole magnetic structures.
The proposed scheme paves a way to generate the multipole magnetic structures in compatible with the crystallographic point group, which is essential for macroscopic phenomena following Neumann's principle, and is useful to search for desired functional magnetic materials.
\section*{Acknowledgments}
This work was supported by JSPS KAKENHI Grant Numbers JP15K17713 (MTS),
JP15H05883 (J-Physics) (MTS), JP18H04230 (MTS), JP16H04021 (MTS), JP15K05176(HK),
JP15H05885 (J-Physics) (HK), JP16H06590(SH), JP18H04296
(J-Physics) (SH), JP18K13488 (SH), JP16H06345 (RA), JST
PREST (MTS), CREST Grant Number JPMJCR15Q5 (RA\& MTS).
T. N. is supported by RIKEN Special Postdoctoral Researchers Program.
\bibliographystyle{apsrev}
|
{
"timestamp": "2019-03-01T02:04:46",
"yymm": "1902",
"arxiv_id": "1902.10819",
"language": "en",
"url": "https://arxiv.org/abs/1902.10819"
}
|
\chapter{Composite Photographs}
\label{app:appendix1}
\begin{figure*}[!htbp]
\centering
\centerline{\includegraphics[width=6.5in]{composite_female}}
\caption{Composite gay and straight female faces.}
\label{fig:composite_female}%
\end{figure*}
\begin{figure*}[!htbp]
\centering
\centerline{\includegraphics[width=6.5in]{composite_male}}
\caption{Composite gay and straight male faces.}
\label{fig:composite_male}%
\end{figure*}
Figures \ref{fig:composite_female} and \ref{fig:composite_male} show composite facial images of gay and straight subjects (Composites are shown separately for asian and white subjects).
To create the composites, images were first classified into broad racial groups. Composite images were only created for well represented groups (white and asian subjects). For each group the photographs were ranked by the ML Model~1 (deep learning classifier) from Study 1. The ranking orders photographs from those ``least likely to be gay'' (straight) to ``most likely to be gay''. Then photos were filtered to have a yaw angle of less than 11 degrees. Finally the top 150 ranked and bottom 150 ranked photographs were composed together to make up composite images \footnote{The compositor failed to recognize faces in some images but at least 100 images were used for every composite photograph}.
Photographs were composed by first locating the position of the main facial features in each photograph and then averaging them to create a target face. Then Delaunay triangulation was used to map each source image to the destination image. Each triangle in each source image was copied to the destination triangle using affine transformation and bilinear interpolation. Finally the pixels were averaged to obtain the composite facial image.
\chapter{Hue, Saturation and Brightness
\label{app:appendix2}
\begin{figure*}[!htbp]
\centering
\centerline{\includegraphics{hue_density}}
\caption{Hue}%
\label{fig:hue_density}%
\end{figure*}
Figures~\ref{fig:hue_density}, \ref{fig:saturation_density} and \ref{fig:brightness_density} plot the hue, saturation and brightness distributions for the highly blurred facial images used by Model~3 (Section \ref{model:blurred}).
To generate these distributions each 5$\plh$5 pixel blurred image was converted into the HSV (Hue, Saturation and Value) colour space \cite{foley}. Then the number of occurrences across all the images for a particular value were summed to create a density distribution. For saturation and brightness, values have an intensity between 0 and 255. The hues are mapped to a scale from 0 to 180 which wraps around at each end.
Each figure has two plots, one for females and one for males. Each plot shows the distribution of the colour metric in each category (Straight and Gay).
\begin{figure*}[!htbp]
\centering
\centerline{\includegraphics[width=6.2in]{saturation_density}}
\caption{Saturation}%
\label{fig:saturation_density}%
\end{figure*}
\begin{figure*}[!htbp]
\centering
\centerline{\includegraphics[width=6.2in]{brightness_density}}%
\caption{Brightness}%
\label{fig:brightness_density}%
\end{figure*}
\chapter{The Second Chapter}
\label{chap:second}
The same structure as before, including section, subsections and sub-subsections. Make sure that you follow the same conventions throughout, to avoid confusing the reader. Always remember to include a summary.
\section{The First Section}
\label{sec:second:first_sec}
\section{The Second Section}
\label{sec:second:second_sec}
\subsection{A Subsection}
\label{sec:second:second_sec:one}
\subsection{Another Subsection}
\label{sec:second:second_sec:two}
\section{Summary}
\label{sec:second:summary}
\chapter{Introduction}
\label{chap:introduction}
\pagestyle{headings}
\pagenumbering{arabic}
\setcounter{page}{1}
\graphicspath{{chapters/introduction/figures/}}
\graphicspath{{./chapters/introduction/figures/}}
The ability to predict sexual orientation from facial images using machine learning (ML) techniques has serious consequences for gay men and women. To verify previous results by Wang and Kosinski (W\&K) \cite{wang_kosinski}, a replication study is undertaken with an independent dataset.
\section{Motivation}
This type of research falls within the discipline of Social Psychology, which studies how people act, think and feel in the context of society. Part of the field is concerned with how humans perceive other humans. Some of these studies investigate how well humans are able to perceive \textit{unambiguous} features of another person (such as their gender or age) and others focus on \textit{ambiguous} features that are not readily perceived in daily life, such as a person's sexual orientation or their political affiliation \cite{political_aff}.
While there have been previous studies demonstrating that humans have some skill at guessing someone's sexual orientation from various types of information \cite{ambady_hallahan_conner}, W\&K were the first to utilize the availability of new deep learning algorithms and online data sources to show that ML models are able to predict someone's sexual orientation from a photograph of their face \cite{wang_kosinski}.
Two of their models used ML techniques to predict sexual orientation from a photograph of a face. One used deep neural networks (DNN) \cite[Study 1a]{wang_kosinski} to extract features from the cropped facial image (see Figure \ref{fig:figure_fpp_crop}). The second study used only facial morphology \cite[Study 3]{wang_kosinski}. Facial morphology refers to the shape and position of the main facial features (such as eyes and nose) and the outline of the face (see Figure \ref{fig:figure_fpp_example} for an example of the information available in the facial morphology studies). The images were gathered from online dating profiles and from Facebook. The data was limited to white individuals from the United States.
For comparison, W\&K \cite{wang_kosinski} also ran an experiment in which they tested humans' ability to detect sexual orientation from the same photographs. They found that humans are able to predict sexual orientation with modest success, achieving an accuracy measured by the Area Under the Curve (AUC) of AUC=.61 for male images and AUC=.54 for female images. Both their ML models outperformed the humans. The deep learning classifier had an accuracy of AUC=.81 for males and AUC=.71 for females \cite[Study 1a]{wang_kosinski}. The facial morphology classifier scored AUC=.85 for males and AUC=.70 for females \cite[Study~4]{wang_kosinski}.
\newline
\begin{figure*}[!b]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics{john_fpp_rectangle}%
\caption{The rectangle used to crop out the face.}
\label{fig:intro_sfig1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics{john_cropped}%
\caption{Cropped facial image}
\label{fig:intro_sfig2}
\end{subfigure}
\caption{The facial rectangle used to identify the face and the resulting cropped face used as an input by the deep learning ML models.}
\label{fig:figure_fpp_crop}
\end{figure*}
\indent W\&K note that their second model can predict sexual orientation from the individual components of the face, such as the contour of the face or the mouth. In particular, the facial contour predicts sexual orientation with an accuracy of AUC=.75 for men and AUC=.63 for women. Based on this outcome, W\&K make the claim that this validates a theory of sexual orientation called prenatal hormone theory (PHT) \cite{allen_gorski, jannini, udry}. PHT predicts that gay people will have ``gender-atypical'' facial morphology due to their exposure to differing hormone environments in the womb. They argue that this finding is unlikely to be due to different styles of presentation or grooming because it is quite difficult to alter the contour of one's face \cite{wang_kosinski}.
The theory that one can detect sexual orientation from facial differences due to a biological cause has been challenged by Ag{\"u}era y Arcas {\it et al} \cite{stereotypes}, who claim that there are several plausible explanations for these facial differences which do not rely on a biological origin. They note that there are many visual signals expressed through the face which are indicative of sexual orientation, such as the presence of eyewear, facial hair and the use of makeup. To get further insight into possible differences in presentation between gay and straight people, they built a questionnaire that asked the participants about their lifestyle and grooming habits. They found that only eight yes/no questions were sufficient to determine a person's sexual orientation with an accuracy of AUC=.70 for females \cite{stereotypes}.
\begin{figure*}[!b]
\centering
\centerline{
\includegraphics[width=2.8in]{john_fpp_landmarks}}%
\caption{The facial morphology ``landmarks'' used as input by the facial morphology ML models.}%
\label{fig:figure_fpp_example}%
\end{figure*}%
\newpage
\indent Ag{\"u}era y Arcas {\it et al} proceed to question whether pre-natal hormonal exposure is responsible for facial differences in photographs. They argue that the angle that a photograph is taken at, especially when the person is taking a picture of themselves, has the effect of ``enlarging the chin, shortening the nose, shrinking the forehead, and attenuating the smile''. These differences could possibly result in either a deep learning classifier or a facial morphology classifier learning sexual orientation from differences due to the angle of the photograph and not any biological differences \cite{stereotypes}.
They go on to point out that there are obvious superficial differences between the gay and straight faces generated from W\&K's dataset (see \cite[Study 2]{wang_kosinski} for their composite images). Straight women in the composite image appear to wear more makeup than gay women and are less likely to wear glasses. Straight males are less likely to wear glasses and are more likely to have clearly visible facial hair \cite{stereotypes}.
To demonstrate how an alteration of appearance can make one appear ``more stereotypically gay'' or ``more stereotypically straight'', the researchers took photographs of themselves in which they altered their presentations \cite{stereotypes}. Figure \ref{fig:figure_altered_presentation} demonstrates these changes in appearance. The male photographs on the left are taken from a lower angle, have no eyewear and feature more facial hair. The female photograph on the left is shown with makeup and is taken from a higher angle. The photograph on the right has eyewear and is taken from a lower angle.
\thispagestyle{empty}
\noindent
\hspace*{\fill}
\begin{minipage}[t]{0.8\textwidth
\centering
\centerline{
\includegraphics[width=2.3in]{male1_left}%
\includegraphics[width=2.3in]{male1_right}}%
\centerline{
\includegraphics[width=2.3in]{male2_left}%
\includegraphics[width=2.3in]{male2_right}}%
\centerline{
\includegraphics[width=2.3in]{female_left}%
\includegraphics[width=2.3in]{female_right}}%
\captionof{figure}{Portraits of the same individual with altered presentation.}
\label{fig:figure_altered_presentation}
\end{minipage}
\hspace*{\fill}
\vspace{\fill}
\pagebreak
\pagestyle{headings}
\section{Objectives}
The main objective of this study is to replicate the results of previous studies that used ML techniques to predict sexual orientation from facial images. Previous results show that ML techniques are able to do so better than humans can. These studies are repeated with a new dataset that is not controlled for race or country.
A new ML model based on highly blurred images is also introduced to investigate whether the information present in the predominant colours in the face and immediate background are predictive of sexual orientation.
The following factors are investigated to see whether the ML models are relying on them to make their predictions:
\begin{itemize}
\item the \textit{head pose} (or angle at which the photograph is taken at),%
\item the presence of facial hair or eyewear.%
\end{itemize}
\section{Contributions}
This work makes the following contributions in the area of social psychology:
\begin{itemize}
\item The study replicates previous studies using ML techniques to predict sexual orientation from facial images. It is shown that both deep learning classifiers and facial morphology classifiers trained on photographs from dating profiles are able to predict sexual orientation, and that they do so better than humans can. The models use a new dataset not limited by race or country.
\item The study introduces a new ML model that tests whether sexual orientation can be predicted from a highly blurred facial image. It is shown that the predominant colour information present in the face and background of a highly blurred facial image is predictive of sexual orientation.
\item The study demonstrates that intentional alterations to one's appearance to fit gay and straight stereotypes do not alter the sexual orientation label generated by the ML models.
\item The study shows that the \textit{head pose} is not correlated with sexual orientation.
\item The study shows that the models are still able to predict sexual orientation even while controlling for the presence or absence of facial hair and eyewear.
\end{itemize}
\section{Thesis outline}
The remainder of this thesis is organized as follows:
\begin{itemize}
\item \textbf{Chapter \ref{chap:ethical}} discusses ethical issues related to this research
\item \textbf{Chapter \ref{chap:literature}} reviews previous work
\item \textbf{Chapter \ref{chap:dataset_ml_models}} describes the methods used to collect the dataset, and three ML models to predict sexual orientation:
\begin{itemize}
\item Dataset of facial images
\item \textbf{Model 1}: Deep Neural Network (DNN) classifier
\item \textbf{Model 2}: Facial morphology classifier
\item \textbf{Model 3}: Highly blurred image classifier
\end{itemize}
\item \textbf{Chapter \ref{chap:methods}} Describes the four experiments performed:
\begin{itemize}
\item \textbf{Study \hyperref[study:study1_ml]{1}}: Three ML models (Model~1, 2 and 3) are used to predict sexual orientation from facial images.
\item \textbf{Study \hyperref[study:study2_altered_presentation]{2}}: Using the portraits in Figure \ref{fig:figure_altered_presentation}, ML models 1, 2 and 3 are evaluated to see whether alterations in presentation would result in changes to the predicted sexual orientation\footnote{Permission was granted by the authors to reproduce their photographs and to test the models on them.}.
\item \textbf{Study \hyperref[study:study3_headpose]{3}}: Tests whether \textit{head pose} (or the angle that a photograph is taken at) is correlated with sexual orientation.
\item \textbf{Study \hyperref[study:study4_facial_hair_eyewear]{4}}: Tests whether the classifiers are still able to predict sexual orientation while controlling for two superficial features: facial hair or eyewear.
\end{itemize}
\item \textbf{Chapter \ref{chap:results}} presents the results
\item \textbf{Chapter \ref{chap:discussion}} discusses the results and compares the findings to W\&K.
\end{itemize}
\chapter{Ethical Considerations}
\label{chap:ethical}
This chapter discusses some of the ethical issues that arise from using ML to predict people's sexual orientation. Section~\ref{ethics:rationale} discusses risks to privacy. Section~\ref{ethics:abuses} gives brief examples of historical abuses by scientists studying bodily features. Section~\ref{ethics:confidentiality} contains a note about the confidentiality of the data used in this study.
\section{Rationale for study}
\label{ethics:rationale}
After the publication of W\&K's work \cite{wang_kosinski} there was vociferous critique of their motives and their findings \cite{stereotypes, read_persons_character, glaad}. Todorov said that the implications of using AI\footnote{AI refers to Artificial Intelligence, which is equivalent to machine learning in this context.} to do ``face reading'' are ``morally abhorrent'' \cite{read_persons_character}. He argues that even if these algorithms are not capable of ``reading'' attributes like intelligence, political orientation and criminal inclination from photographs, merely talking about the possibility might encourage repressive governments to try employ these techniques \cite{read_persons_character}. We would add that the very idea that a government is capable of invading one's privacy in this way is in itself enough to create fear in targeted groups.
\subsection{Risks to privacy}
Kosinski argues that the world should be made aware of these techniques and the danger they pose. It has been demonstrated that besides the information betrayed by our photographs, digital trails such as who we link to on social media are also predictive of sexual orientation \cite{jernigan}. Kosinski says that ``there is an urgent need for making policymakers, the general public, and gay communities aware of the risks that they might be facing already'' \cite{wang_kosinski}.
We agree that the new risks society faces as a result of public digital profiles is too great to ignore. It's important to know whether our digital profiles do or do not reveal information like our sexual orientation or political affiliation. It's also important for people to know how to protect themselves. Todorov argues that the new ML techniques are only exposing what humans can tell anyway \cite{read_persons_character}, but even if the machines are only as good as humans, there is a fundamental difference when these processes can be automated by machines. Automated systems have the ability to operate at a completely different scale and with high fidelity over extended periods of time.
\subsubsection*{Automated surveillance}
The main new risk today is of automated surveillance by organisations most interested in uncovering people's intimate traits. Although easily within reach of mass surveillance states like the United States and China \cite{surveillance, surveillance_china}, automated tools for surveillance can easily be built with off-the-shelf technology by small organisations today.
\subsubsection*{Risks to gay people}
In many countries the threat of a loss of privacy regarding intimate traits is very serious. In South Africa, for example, practices such as ``corrective rape'' pose a grave threat~\cite{corrective_rape}. Worldwide there are at least eight countries in which homosexuality or homosexual acts are punishable by death \cite{death_penalty}.
\section{Historical abuses by scientists}
\label{ethics:abuses}
As described by W\&K \cite{wang_kosinski} there is a long history of the use of bodily features in a pseudo-scientific way to stigmatize, disenfranchise and marginalize people. In South Africa in particular there is a history of `race science' exemplified by the bodily measurements of 133 ``coloured'' males at the Zoology Department of Stellenbosch University in 1937 \cite{handri_walters}. These experiments ultimately led to the ``recognition of this population as a separate racial group that could be subjected to the laws of the apartheid state'' \cite{handri_walters}. In South Africa there is also a history of human rights abuses against homosexual people by health practitioners in academia during the apartheid era \cite{aversion}.
\subsubsection{Physiognomy}
Physiognomy is the reading of a person's character from their face \cite{read_persons_character}. This idea has a long history reaching back to the time of the ancient world. Today the scientific community rejects physiognomy as pseudo-science inspired by superstition and racism~\cite{jenkinson1997}. A recent example of physiognomy as pseudo-science is a Chinese study claiming to be able to detect criminality from identity photographs \cite{chinese_criminality}.
\section{Confidentiality and privacy in this study}
\label{ethics:confidentiality}
All of the data collected for this study is used to present results in aggregate only, and all data linked to individuals has been kept private. We have also not disclosed the sources of the data.
\iffalse
This study has been approved by the Faculty Committee for Research Ethics and Integrity (Faculty of Engineering, Built Environment \& Information Technology).
\fi
\chapter{Literature Review}
\label{chap:literature}
The study of sexual orientation and whether it can be perceived by others is not new. Section~\ref{lit:previous} briefly summarises previously published results. Section~\ref{lit:ml} describes the first use of ML to predict sexual orientation from a large dataset of photographs. It is not known how people or machines are able to predict sexual orientation from facial differences. Section~\ref{lit:biological} presents a claim for biological origins while Section~\ref{lit:rebuttal} contains a rebuttal of this idea.
\section{Previous work studying the prediction of sexual orientation}
\label{lit:previous}
Ambady {\it et al} \cite{ambady_hallahan_conner} in 1999 described how it is possible for humans to determine the sexual orientation of an individual through brief exposure to visual or audio clips with probability greater than chance. They summarise previous research which indicates that dynamic information (such as body movement, voice or video) provides more information than static features such as those present in a photograph. Their study finds that dynamic information is indeed superior to static information with the best accuracy of 70\% coming from silent video clips that are ten seconds long \cite{ambady_hallahan_conner}.
A later study by Rule and Ambady in 2007 showed that it is possible to perceive the sexual orientation of someone from a very brief exposure to a photograph of that person. By testing exposures differing in length of time they determined that an exposure as short as 50 ms is sufficient for a better than chance determination of sexual orientation. Results for longer exposures were not significantly different from the 50 ms exposure~\cite{brief_exposures}.
Olivola and Todorov \cite{olivola_2010b} reported in 2010 on the ability of website users to guess personal characteristics from a photograph. For most variables they found that people ignored the \textit{base rate} of the non-uniform distribution of that variable. For sexual orientation, they established via a survey that people assume that 90\% of people are heterosexual. However, when asked to predict sexual orientation from a photograph they ignored this \textit{base rate} and predicted a heterosexual sexual orientation 60\% of the time. Participants only achieved better than chance scores when the base rate approached equiprobability (50\%), for variables such as ``Is this person in a long term relationship?'' or ``Does this person have a college degree?''.
In 2013 Lyons also tested the ability of humans to predict sexual orientation from greyscale dating profile photographs and found that humans could do this with better than chance results \cite{lyons}.
In 2015 Cox {\it et al} \cite{coxdevine} found that human judges do not have a better than chance ability to predict sexual orientation from dating profile photographs.
\section{Use of machine learning to predict sexual orientation}
\label{lit:ml}
In 2017 a study by W\&K \cite{wang_kosinski} used static photographs of individuals collected from dating sites and Facebook to test whether humans could accurately judge sexual orientation from the photographs. They found that humans were able to achieve a 61\% success rate for classifying men \cite{wang_kosinski}. They then used a pre-trained deep learning neural network~\cite{deep_learning} on the same dataset and found that the resulting classifier was able to determine sexual orientation 81\% of the time for men and 71\% for women from a single image of the subject. When five images of the subject were used, the success rate increased to 91\% and 83\% respectively \cite[Study 1a]{wang_kosinski}.
The DNN they used is a pre-trained ML model that was originally built to identify someone from a photograph of their face. It was trained on a large number of photographs of celebrities. By modifying the model they were able to use the features it had already learned for a new task: predicting sexual orientation.
W\&K also used another pre-trained ML model \cite{faceplusplus} to derive the positions and shape of the main facial features (eyebrows, eyes, nose, mouth and facial contour). From these features they created a second model that also successfully predicted sexual orientation (male AUC=.85 and female AUC=.70 for five images per subject) \cite[Study 3]{wang_kosinski}.
\section{Claims of a biological origin for facial differences}
\label{lit:biological}
Placing emphasis on the ability of their second classifier to differentiate between gay and straight individuals based on the facial contour alone, W\&K claim that this result supports a theory called prenatal hormone theory (PHT) \cite{allen_gorski, jannini, udry}. PHT claims that same-gender sexual orientation in adults is a result of exposure to particular hormone environments before birth. W\&K claim that the theory predicts ``gender atypical'' faces for gay men and women (more feminine faces for gay men and more masculine faces for gay women) \cite{wang_kosinski}.
\section{Rebuttal of Wang and Kosinski}
\label{lit:rebuttal}
Gelman {\it et al} point out that the type of classifier created by W\&K (and replicated in this study) is not useful in predicting sexual orientation for the general population because gay people are a minority and thus the classifier would have to be very accurate to be effective \cite{gelman}. They also criticize the use of binary categories (straight and gay) as an oversimplification of sexual orientation. To better communicate the uncertainty in the classifier results, they suggest that a third category should be reported when the classifier is unsure of the result \cite{gelman}.
Ag{\"u}era y Arcas {\it et al} \cite{stereotypes} responded to W\&K and provided alternative explanations for their findings. Instead of linking the ability to detect sexual orientation from facial images to a biological origin, they argue that ML models are learning from superficial features that have been hiding in plain sight \cite{stereotypes}. By inspecting the composite images of gay and straight males and females published by W\&K \cite[Study 1c]{wang_kosinski} they note the following apparent differences:
\begin{itemize}
\item The composite straight female face has eyeshadow, the gay female face does not.
\item Glasses (eyewear) are visible on both gay portraits but not the straight ones.
\item Straight males appear to have more and darker facial hair.
\item The gay male composite has a brighter face than the straight male and the straight female has a brighter face than the gay female composite.
\end{itemize}
To investigate whether there are any ``group preferences'' in grooming, presentation and lifestyle, they created a questionnaire with 77 questions asking respondents questions such as ``Do you wear eyeshadow?'',``Do you wear glasses?'', ``Do you have a beard?'' and other questions about gender and sexual orientation. Their survey polled 8000 Americans (the same demographic that W\&K used for their experiments).
They found that there is indeed a greater preference for wearing makeup and eyeshadow among the straight females polled relative to the gay females. Similarly, they found that gay males and females were more likely to report wearing glasses than straight men. In addition, they were also more likely to report that they like the way that they look in glasses.
Gay males reported being less likely to have ``serious facial hair'' than straight males (with the exception of gay males around 40 years old). Straight males also reported that they were more likely to ``work outdoors'' than gay males (which would explain less brightness in their faces).
All of these questionnaire responses match with the observations of the differences seen in W\&K's composite images. These apparent facial differences might then be the result of lifestyle, presentation and grooming preferences that differ between gay and straight groups.
The differences mentioned above are likely to be detected by a DNN such as the one used by W\&K for their first study. The second study, however, relies on facial morphology which should be insensitive to the kinds of differences described above. Ag{\"u}era y Arcas {\it et al} \cite{stereotypes} argue that the differences detected by the facial morphology classifier might simply be due to a relative difference in the pitch that a photograph is taken at.
Returning to the composite images, Ag{\"u}era y Arcas {\it et al} \cite{stereotypes} point out that the nostrils (which appear as dark ovals) are more prominent for photographs of straight males and gay females. They postulate that this might be due to a preference for taking pictures from a higher angle in the other two groups. Taking pictures from a higher angle also changes the shape of features such as the eyebrows and eyes, and this could easily be detected by a facial morphology classifier \cite{stereotypes}.
\section{Summary}
This chapter reviewed prior work studying whether it is possible for humans to predict sexual orientation from photographs or other types of recording (such as videos). W\&K claim that sexual orientation is detectable from faces due to biological differences consistent with PHT. Ag{\"u}era y Arcas {\it et al} rebut this claim and argue that the facial differences are due to lifestyle choices and differences in presentation and grooming.
\chapter{Dataset and Machine Learning Models}
\label{chap:dataset_ml_models}
This chapter describes the dataset and machine learning models used in this study. Section~\ref{section_dataset} explains how the dating profile images were collected, cleaned and labelled. The remaining sections describe the three ML models. Section~\ref{model:vgg} describes Model~1, based on a deep neural network. Section~\ref{model:fpp} describes Model~2, based on facial morphology. Section~\ref{model:blurred} describes Model~3, trained on highly blurred images.
\section{Dataset of facial images}
\label{section_dataset}
\subsection{Dataset acquisition}
To replicate the experiments by W\&K \cite{wang_kosinski}, an independent dataset of facial images was collected from online dating sites using similar criteria.
Dating sites were used as a source for facial images because they are currently\footnote{in 2018} the most practical source of images for this kind of study. By stating their own gender and expressing an interest in another gender, dating profiles provide a convenient set of photographs labelled with a proxy for each person's sexual orientation.
Dating sites provide the opportunity to gather larger sample sizes than have previously been available to conventional studies which have to gather portraits individually, or manually extract them from a published source such as a newspaper.
For each dating profile retrieved, the following information was captured:
\begin{itemize}
\item One or more photographs of that person
\item The gender of the person
\item The gender that they are seeking
\item The age of the person
\end{itemize}
\subsubsection*{Terminology}
In this chapter and in the rest of this work we use the word `gay' to refer to someone who has expressed an interest in someone with the same gender (same sex attracted). We use the word `straight' to refer to someone who has expressed an interest in someone of a different gender (opposite sex attracted)\footnote{We use terminology suggested by the American Psychological Association \cite{avoiding_heterosexual_bias}}. The sources that we used to gather the data generally only allow people to identify themselves as belonging to one binary gender (male or female). We excluded users that expressed an interest in more than one gender.
Each photograph (facial image) is labelled with the gender of the subject and the gender of the person that they are seeking.
Every photograph then falls into one of four basic categories:
\begin{itemize}
\item Gay Female
\item Straight Female
\item Gay Male
\item Straight Male
\end{itemize}
The first dating site used to gather images, \textbf{Site A}, did not have a sufficient number of gay males and gay females. To increase the number of samples, images were collected from two additional sites. \textbf{Site B} caters to gay males only. \textbf{Site C} caters to gay females only. To protect the privacy of the subjects, the origin of the data sources are not disclosed.
In total about 500,000 photographs were retrieved. Table \ref{data_source_table1} lists the number of photographs downloaded in each category for each data source.
\begin{table}[hb]
\centering
\begin{tabular}{llll}
\hline
\rowcolor[HTML]{000000}
\multicolumn{1}{|l|}{\cellcolor[HTML]{000000}{\color[HTML]{FFFFFF} }} & \multicolumn{1}{l|}{\cellcolor[HTML]{000000}{\color[HTML]{FFFFFF} \textbf{Site A}}} & \multicolumn{1}{l|}{\cellcolor[HTML]{000000}{\color[HTML]{FFFFFF} \textbf{Site B}}} & \multicolumn{1}{l|}{\cellcolor[HTML]{000000}{\color[HTML]{FFFFFF} \textbf{Site C}}} \\ \hline%
Female - gay & 5951 & & 75768 \\ \hline%
Female - straight & 292588 & & \\ \hline%
Male - gay & 11030 & 104198 & \\ \hline%
Male - straight & 72818 & & \\ \hline%
\end{tabular}
\caption{Number of photographs retrieved in each category from each data source}%
\label{data_source_table1}%
\end{table}%
\subsection{Dataset cleaning and facial extraction}
For each photograph and associated user profile, a number of filtering and cleaning steps were applied. Only photographs for users aged 18 to 35 were included \footnote{This is the same age range used by W\&K\cite{wang_kosinski}}. Users that had missing or invalid data, such as missing age or gender labels, were removed.
Faces were automatically extracted from each photograph by using an external face recognition model called \textit{Face++}\footnote{Face++ can be accessed at \url{https://www.faceplusplus.com}}. \textit{Face++} generates a rectangle identifying the face (see Figure \ref{fig:figure_fpp_crop}) and facial metrics that describe facial ``landmarks" (see Figure \ref{fig:figure_fpp_example}). It also returns a \textit{head pose} for the face, comprised of the pitch, roll and yaw angles.
To avoid ambiguity, all photographs that did not contain exactly one face were discarded. Based on the facial metrics, photographs were excluded that:
\begin{itemize}
\item Have a pitch of greater than 10 degrees or a yaw of greater than 15 degrees
\item Have a distance between the eyes of less than 40 pixels
\item Do not contain all of the facial landmarks normally detected by \textit{Face++}
\end{itemize}
These steps exclude images where the subject is not facing the camera, is not close to the camera or whose face is partially occluded.
\subsection{Manual filtering and labelling}
After the automatic filtering processes each image was inspected manually and rejected if it did not meet the following criteria:
\begin{itemize}
\item Each image must be of a single adult person (often users upload pictures of animals or children).
\item The image should be clear and not blurred or overexposed.
\item The person in the image should match the category assigned to them (for example, verify that the picture is of a male).
\item The photograph must not be digitally transformed or stylized in an obvious way, such as by the addition of digitally generated sunglasses or noses.
\item The photograph must not be of a celebrity.
\item The subject's face must not be obscured or occluded in the photograph\footnote{In some cases a mobile phone is visible in the lower half of the face when subjects are photographing themselves in a mirror}.
\end{itemize}
In addition, photographs were rejected if it was obvious that the user had mislabelled their own gender. This happens quite frequently, and we assume that it is because some users may not be computer literate (or literate in English). It is possible that they use the default gender setting when creating their account (although we observed this phenomenon for both males and females). This is different from people who purposely present themselves as belonging to a gender different to their sex assigned at birth.
\subsubsection*{Facial hair}
Each image was also labelled with an indicator marking the presence (or absence) of significant facial hair. For the purposes of this study, ``facial hair'' or ``significant facial hair'' is defined as either:
\begin{itemize}
\item Clearly visible hair (several millimetres in length) in the moustache, beard or cheek areas
\item Short stubble in the moustache, beard or cheek areas
\end{itemize}
The purpose of this definition is to test whether a grooming preference for facial hair is what is being recognized by the ML models and used to predict sexual orientation. None of the female subjects exhibited facial hair.
\subsubsection*{Eyewear}
Similarly, it was recorded whether the subject in each image was or was not wearing eyewear. In this study spectacles, sunglasses and mock or novelty spectacles are considered to qualify as eyewear. Images with digitally generated spectacles were discarded.
\subsubsection*{Filtering and labelling methodology}
To filter and label the photographs, software was developed to review the photographs. Photographs were randomly selected from each category for the reviewing process in batches of a hundred. Reviewers could not see the stated sexual orientation for each photograph. This process was repeated for the different categories until there were roughly 5000 reviewed images available for each category.
Each cropped face was presented to the user along with the stated gender of the subject. Users had the option to view the full photograph. For each photograph, an option to approve or reject the photograph was presented (with neither option selected by default). If approved, options to label the presence or absence of facial hair and eyewear were presented (none selected by default). A photograph was only approved if the facial hair and eyewear sections were completed.
\subsection{Final photograph count after automatic and manual filtering}
In Table \ref{data_source_table2}, the final number of photographs are shown in each category after the automatic filtering process and manual approval or rejection by a human.
\begin{table}[!ht]%
\centering
\begin{tabular}{ll}
\hline
\rowcolor[HTML]{000000}%
\multicolumn{1}{|l|}{\cellcolor[HTML]{000000}{\color[HTML]{FFFFFF} }} & \multicolumn{1}{l|}{\cellcolor[HTML]{000000}{\color[HTML]{FFFFFF} \textbf{Total}}} \\ \hline
Female - gay & 5132 \\ \hline
Female - straight & 5406 \\ \hline
Male - gay & 5706 \\ \hline
Male - straight & 4666 \\ \hline
\end{tabular}%
\caption{Total number of photographs reviewed and approved by a human in each category}%
\label{data_source_table2}%
\end{table}
\section{Model 1: Deep neural network classifier}
\label{model:vgg}
We recreate the DNN model used by W\&K to predict sexual orientation from facial images \cite[Study 1a]{wang_kosinski}.
\subsection{VGGFace}
This model uses VGGFace \cite{vgg16}, a pre-trained deep learning neural network, to extract features from facial images, and then trains a logistic regression model on these features to predict the sexual orientation of the subject of the image.
VGGFace is a convolutional neural network that was developed to recognise individuals from pictures of their faces. It was trained on one million photographs of 2,622 different celebrities. Although the neural network was originally developed to identify a specific person from a facial image, by removing the last layer of the network we are able to obtain the facial features that the model uses for its final classification layer.
These features produced by a DNN are generally not interpretable by humans, but can be thought of as a numerical representation of a face.
\subsection{Model pipeline}
The input to the model is a cropped facial image extracted by \textit{Face++} \cite{faceplusplus}. See Figure~\ref{fig:figure_fpp_crop} for an example of a cropped face. The image is then scaled down to a 224 by 224 pixel image.
Next we used VGG16 \cite{vgg16}, a variant of VGGFace, to generate 4,096 features from each facial image.
We perform dimensionality reduction on the features using principal components analysis (PCA) \cite{pca}. Each image is represented by 500 principal components.
Each image is associated with a sexual orientation label (gay or straight) derived from an individual's reported gender and dating interest (see Section~\ref{section_dataset}).
To predict the sexual orientation of the individual in each image, we trained a logistic regression model using the principal components as independent variables and the sexual orientation labels as dependent variables. Males and females were modelled separately. This method is the same as that used by W\&K except that they used singular value decomposition (SVD) instead of PCA for dimensionality reduction \cite{wang_kosinski}.
\section{Model 2: Facial morphology classifier}
\label{model:fpp}
We recreate the facial morphology model used by W\&K to predict sexual orientation from facial images \cite[Study 3]{wang_kosinski}.
\subsection{Face++}
This model uses \textit{Face++}\footnote{Face++ can be accessed at \url{https://www.faceplusplus.com}} \cite{faceplusplus}, an external model accessible as a service, to extract facial ``landmarks" for each face. It then uses distances derived from these landmarks to train a logistic regression model to predict the sexual orientation of the subject of the image.
The landmarks are facial metrics that describe the position of facial features on the face. \textit{Face++} returns a fixed number of landmark points for a face. The landmarks are grouped into several components:
\begin{itemize}
\item[--] Contour (19 points)
\item[--] Mouth (18 points)
\item[--] Eyebrows (16 points)
\item[--] Eyes (20 points)
\item[--] Nose (10 points)
\end{itemize}
Figure \ref{fig:figure_fpp_example} demonstrates the landmarks for each facial component. Each component consists of ten or more points, and there are a total of 83 points for the whole face. Logistic regression classifiers were created for each individual facial component, and a combined classifier from all the components for the whole face.
\subsection{Model pipeline}
Each facial image is passed to \textit{Face++} to generate landmarks for the face. To generate features suitable for ML the same method as used by W\&K \cite{wang_kosinski} is followed. The distance from each point to every other point within each component is calculated. For example, since the nose has 10 points, $ 10 \times 9 = 90 $ euclidean distances are calculated for this component. All the distances are scaled by dividing them by the distance between the centre of the eyes.
We used PCA on the features. For the combined classifier which incorporates all the facial components, we reduced the number of components to 500.
Each image is associated with a sexual orientation label (gay or straight) derived from an individual's reported gender and dating interest (see Section~\ref{section_dataset}).
To predict the sexual orientation of the individual in each image, we trained a logistic regression model using the principal components as independent variables and the sexual orientation labels as dependent variables. Males and females were modelled separately. This method is the same as that used by W\&K except that they used singular value decomposition (SVD) instead of PCA for dimensionality reduction \cite{wang_kosinski}.
\section{Model 3: Highly blurred image classifier}
\label{model:blurred}
We create a new ML model to predict sexual orientation from facial images using a highly blurred image of the face.
The purpose of this model is to test whether the colour information alone is predictive of sexual orientation. The cropped and blurred image that is presented to the classifier contains no information about the shape or size of the facial features. This contrasts with the facial morphology classifier which relies solely on the shape of the facial features.
\newcommand{\plh}{%
{\ooalign{$\phantom{0}$\cr\hidewidth$\scriptstyle\times$\cr}}%
}
\newcommand{\PLH}{{\mkern-2mu\times\mkern-2mu}}%
Two types of blurred image are created, the first is a 5$\plh$5 pixel image containing 25~colours, and the second is a 1 pixel image containing a single colour value.
\subsection{Blurring}
\FloatBarrier
Cropped facial images (Figure \ref{fig:figure_fpp_crop}) are passed through a blurring algorithm\footnote{Images were blurred using ImageMagick's \textit{scale} operator \cite{imagemagick_scale_operator}.} to generate a highly blurred image. Figure \ref{fig:figure_study_3_blurred_image} shows an example of the original cropped facial image and the resulting blurred 5$\plh$5 pixel and 1~pixel images. The blurred images are shown enlarged because the actual 5$\plh$5 pixel and 1~pixel images are very small.
Since each pixel has three RGB colour channels, the resulting images have ($25~\times~3~=~75$) and ($1 \times 3 = 3$) features per image respectively.
\FloatBarrier
\begin{figure}[!h]
\vspace{0.2in}
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics{john_cropped_orig}%
\caption{Cropped facial image}%
\label{fig:sfig1}%
\end{subfigure}%
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics{john_cropped_blurred_enlarged}%
\caption{$5\plh5$ pixel blurred image}%
\label{fig:sfig2}%
\end{subfigure}%
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics{john_cropped_swatch_enlarged}%
\caption{1 pixel blurred image}%
\label{fig:sfig3}%
\end{subfigure}%
\caption{The cropped facial image and derived blurred photos (blurred images are shown enlarged).}%
\label{fig:figure_study_3_blurred_image}%
\end{figure}
\subsection{Model pipeline}
Each cropped facial image is blurred to produce 75 or 3 features, for 5$\plh$5 pixel and 1~pixel images, respectively.
Each image is associated with a sexual orientation label (gay or straight) derived from an individual's reported gender and dating interest (see Section~\ref{section_dataset}).
To predict the sexual orientation of the individual in each image, we trained a logistic regression model using the features as independent variables and the sexual orientation labels as dependent variables. Males and females were modelled separately.
\section{Summary}
This chapter described the collection, cleaning and labelling process for the dataset of dating profile images. It also described the model pipelines for the three different ML models used in this study. The next chapter describes the methodology for the experiments performed using this dataset and these ML models.
\FloatBarrier
\chapter{Methodology}
\label{chap:methods}
Using the dataset and ML models from Chapter \ref{chap:dataset_ml_models}, four experiments are performed to investigate whether it is possible to predict sexual orientation from a facial image, and which properties of the facial images might make that possible.
Study \hyperref[study:study1_ml]{1} in Section~\ref{study:study1_ml} uses the three ML models to predict sexual orientation from facial images. Study \hyperref[study:study2_altered_presentation]{2} in Section~\ref{study:study2_altered_presentation} evaluates whether alterations to presentation result in changes to the predicted sexual orientation. Study \hyperref[study:study3_headpose]{3} in Section~\ref{study:study3_headpose} tests whether the \textit{head pose} (or the angle that a photograph is taken at) is correlated with sexual orientation. Study \hyperref[study:study4_facial_hair_eyewear]{4} in Section~\ref{study:study4_facial_hair_eyewear} controls for facial hair and eyewear as potential confounders while predicting sexual orientation.
\section{Study 1: Prediction of sexual orientation from facial images using machine learning models}
\label{study:study1_ml}
To replicate W\&K's ML studies \cite{wang_kosinski} we repeated their experiments using a deep learning classifier (Model~\hyperref[model:vgg]{1}) and a facial morphology classifier (Model~\hyperref[model:fpp]{2}) on an independent dataset (see Section~\ref{section_dataset}). We also created a new ML model that tests whether sexual orientation can be predicted from a highly blurred facial image (Model~\hyperref[model:blurred]{3}).
For all three models we use the same methods for training and scoring, discussed below.
\label{section:statistical_methods}
\subsubsection*{Model training}
Each base dataset contains samples which are labelled as being in either a positive or negative class. For example, in this study, the positive class might be ``Gay Male'' and the negative class ``Straight Male''. It doesn't matter which label is used as the positive class. To balance out the number of samples in each class, samples are randomly selected without replacement from each class in an alternate fashion until there are no samples left in one of the classes. The final dataset used for training thus has 50\% of its samples coming from the positive class and 50\% from the negative class.
To avoid overfitting, a stratified 20-fold cross validation technique is used. The data is randomly split into 20 parts, whilst preserving the 50/50 split between classes in each part. For each part a logistic regression model is trained on the remaining 19 parts and then used to generate a prediction score for every sample in that part.
The logistic regression model is configured to use L1 regularization \cite{lasso} with a default regression strength parameter C=1.0. A grid search was used to check that the default parameter performed best.
\subsubsection*{Model scoring}
\label{section:model_scoring}
The prediction score for each sample, which lies between 0 and 1, is used to predict whether the sample falls in the positive or the negative class. A value below the threshold of 0.5 places the sample in the positive class, all other values fall in the negative class.
Using the predicted labels and the actual labels for each sample, a false positive rate and a true positive rate is calculated. These two values then allow us to calculate the area under the curve (AUC) of the receiver operating characteristic (ROC) \cite{roc_auc}.
The ROC AUC score represents the probability that when given one randomly chosen positive instance and one randomly chosen negative instance, the classifier will correctly identify the positive instance.
Note that the ROC AUC score used here is invariant to the threshold used for the classifer \cite{roc_auc}.
\subsubsection*{Accuracy for increasing numbers of images per subject}
Some of the subjects in the dataset have more than one facial image associated with them. To assess how the accuracy of the model changes with more than one image available per subject, we constructed accuracy scores for 1 to 5 images. To create these scores, we used the following procedure:
For $n$ from 1 to 5:
\begin{itemize}
\item[--] Select all subjects with at least $n$ photos.
\item[--] For each subject randomly select $n$ photos.
\item[--] For each subject average the prediction scores for the photos.
\item[--] Score the model by comparing the averaged predicted class per subject against their actual class label.
\end{itemize}
Prediction scores were logit transformed before averaging, and the reverse transformation was applied after averaging \cite{logit}.
Finally, we have 5 scores representing the accuracy of the models when presented with $n$ images per subject. The scores for higher numbers of facial images vary due to the random sampling of photos in the scoring procedure, so we repeat it ten times and average the results.
\section{Study 2: Evaluate machine learning models with altered presentation}
\label{study:study2_altered_presentation}
To test whether an individual's change in presentation changes the predicted sexual orientation labels, we evaluated each pair of portraits in Figure \ref{fig:figure_altered_presentation} using the ML models tested in this study\footnote{Permission was granted by the authors to reproduce their photographs and to test the models on them.}.
\subsubsection*{Differences in presentation}
Each pair of photographs has a significant difference in pitch of more than 11 degrees. All the images on the left are taken without eyewear, while those on the right do have eyewear. The male portraits are shown with more facial hair on the left than on the right. The male portraits are taken from a low angle on the left and a high angle on the right. The female portrait on the left is shown with makeup and is taken from a high angle. The female portrait on the right has no makeup and is taken from a level angle.
These changes in presentation are in line with Ag{\"u}era y Arcas {\it et al}'s survey of 8000 Americans showing that straight males up to the age of 35 are more likely to report having serious facial hair than gay males. A preference for how they look in glasses and a higher prevalence of wearing glasses is reported by both gay males and gay females. Gay women report wearing makeup less often than straight women \cite{stereotypes}.
To test whether the models predict the same sexual orientation label for each pair of photographs we evaluated all six photographs using the three models from Chapter \ref{chap:dataset_ml_models}:
\begin{itemize}
\item \textbf{Model 1}: Deep learning classifier
\item \textbf{Model 2}: Facial morphology classifier
\item \textbf{Model 3}: Blurred image classifier
\end{itemize}
This produces 18 sexual orientation predictions, or 9 pairs of predictions.
We test the outcome to see if the predicted labels are pairwise consistent (does the model predict the same label for two images of the same individual?), and whether the labels are consistent between models (do the models agree on the predicted sexual orientation of an individual?).
\subsubsection*{Probability of pairwise consistency}
The probability of the models being pairwise consistent by chance is calculated as follows. The probability that a model will predict a pairwise consistent label for a pair of photographs is ${ {4\choose 2} = \frac{1}{2}}$. This is because there are four possible pairs of labels: (gay, gay), (gay, straight), (straight, straight) and (straight, gay). Two out of these four, (gay, gay) and (straight, straight), indicate that the model considers the subjects of the photographs to have the same sexual orientation.
Since we have three pairs of photographs, the probability of the model being pairwise consistent for all three pairs is: $$ \left(\frac{1}{2}\right)^3 = \frac{1}{8} $$
Then, if we consider the models to be independent, the probability of all three models being pairwise consistent is: $$ \left(\frac{1}{8}\right)^3 = \frac{1}{512} $$
\subsubsection*{Probability of consistent predictions across models}
Again assuming that the models are independent, the probability of all three models agreeing on the same sexual orientation label for a pair of photographs is $ \frac{2}{64} $. This is because the models have to predict either ((straight, straight), (straight, straight), (straight, straight)) or ((gay, gay), (gay, gay), (gay, gay)) out of $ 64 $ combinations. The probability of two out of three models agreeing on the label is $ \frac{6}{64} $. The remaining probability is $ \frac{56}{64} $. These probabilities are used to calculate how likely it is that two, three or no models agree on the sexual orientation label for the three pairs of photographs.
\section{Study 3: Test correlation of sexual orientation with head pose}
\label{study:study3_headpose}
To test whether the deep learning and facial morphology classifiers might be predicting sexual orientation from the \textit{head pose}, we test whether the three \textit{head pose} angles in the dataset are correlated with sexual orientation.
The \textit{Face++} service identifies three \textit{head pose} angles for each photograph: pitch, roll and yaw.
For each of these angles we measured the correlation between the angle and the binary category label (straight or gay) using a Point-Biserial Correlation \cite{pointbiserial}. We checked that the distribution for each angle is normally distributed using a Shapiro-Wilk test (the test was performed for each continuous variable in each binary category) \cite{shapiro}.
In addition, we investigated whether there might be a non-parametric relationship between the \textit{head pose} angles and sexual orientation. To do this we calculated the maximal information coefficient \cite{mic_reshef}.
All correlations were measured separately for males and females.
\section{Study 4: Test prediction of sexual orientation while controlling for facial hair and eyewear}
\label{study:study4_facial_hair_eyewear}
To test whether the ML models in Study 1 are learning to predict sexual orientation from acquired features such as the presence of facial hair or eyewear, the dataset was filtered so that these features occur with equal probability. Each photograph was manually labelled to indicate whether the subject had facial hair (\textbf{A}) or was wearing eyewear (\textbf{B}), as described in Section~\ref{section_dataset}.
\subsection*{A. Facial hair}
The ML experiments in Study 1 were repeated, but the dataset was filtered so that each photograph in each category has a 50\% chance of having facial hair or not. We only tested facial hair for male subjects. None of the female photographs exhibited significant facial hair.
We created a logistic regression model to predict sexual orientation from a cropped facial image in the same manner as in Study 1, the only difference is that the models were trained on this reduced dataset.
Table \ref{table:facial_hair_counts} shows the number of photographs in each category evenly balanced between those with and without facial hair.
\begin{table}[hb]
\centering
\begin{tabular}{lll}
\hline
\rowcolor[HTML]{000000}
{\color[HTML]{FFFFFF} } & {\color[HTML]{FFFFFF} Facial Hair} & {\color[HTML]{FFFFFF} No Facial Hair} \\ \hline
Male - gay & 1969 & 1969 \\
Male - straight & 1969 & 1969 \\ \hline
\end{tabular}
\caption{Number of photographs in each category with and without facial hair}%
\label{table:facial_hair_counts}%
\end{table}
\subsection*{B. Eyewear}
A separate experiment was performed where the photographs in each category had a 50\% chance of having eyewear. Since such a small proportion of straight females have eyewear in their photographs (4\% versus 20\% for straight men), the sample size was too small to include females.
Table \ref{table:eyewear_counts} shows the number of photographs in each category evenly balanced between those with and without eyewear.
As with \textbf{A}), we used the same methodology to predict sexual orientation as in Study~1 but with the reduced dataset.
\begin{table}[hb]
\centering
\begin{tabular}{lll}
\hline
\rowcolor[HTML]{000000}
{\color[HTML]{FFFFFF} } & {\color[HTML]{FFFFFF} Eyewear} & {\color[HTML]{FFFFFF} No Eyewear} \\ \hline
Male - gay & 965 & 965 \\
Male - straight & 965 & 965 \\%\\
\hline
\end{tabular}
\caption{Number of photographs in each category with and without eyewear}%
\label{table:eyewear_counts}%
\end{table}
\section{Summary}
This chapter described the methodology for four experiments investigating whether it is possible to predict sexual orientation from facial images and which properties of these images might make that possible. The next section presents the obtained results.
\chapter{Results}
\label{chap:results}
\graphicspath{{figures/}}
\graphicspath{{chapters/introduction/figures/}}
This chapter presents some of the statistical properties of the final dataset (Section~\ref{results:dataset}), and the detailed results for each study from Chapter~\ref{chap:methods}. Section~\ref{results:study1} contains the results for each of the three ML models evaluated on the dataset of dating profile images. Section~\ref{results:study2} presents the results for the three ML models evaluated on photographs with intentionally altered presentation. Section~\ref{results:study3} contains the evaluation of whether the \textit{head pose} is correlated with sexual orientation. Section~\ref{results:study4} describes the results when predicting sexual orientation while controlling for facial hair and eyewear. Section~\ref{results:summary} briefly summarises the results of each experiment.
\section{Dataset}
\label{results:dataset}
Statistical properties of the final dataset are presented in this section.
\subsubsection*{Photograph count}
The final dataset used for the studies has 20,910 facial images of 10,372 gay and straight (55\%/46\%) men and women (49\%/51\%), see Table \ref{table:results_final_count} for details. Figure \ref{fig:figure_histogram_images_per_subject} shows the number of photographs per subject for males and females.
\subsubsection*{Skew by age}
The dataset has different age distributions for straight and gay subjects. Figure \ref{fig:age_density} shows that in the dataset gay females are younger than straight females. A similar skew exists for males.
The female age densities show steep peaks at around 20 years for gay subjects and~25 for straight subjects. The male distributions are more evenly spread out.
\begin{figure*}[!tp]
\centering
\centerline{\includegraphics[width=6in]{number_photos_per_user}}%
\caption{Number of photographs per subject}%
\label{fig:figure_histogram_images_per_subject}%
\vspace{0.2in}
\centerline{\includegraphics[width=6in]{age_density_final}}%
\caption{Age distribution of females and males by sexual orientation.}%
\label{fig:age_density}%
\end{figure*}
\subsubsection*{Composite facial images}
Appendix \ref{app:appendix1} shows composite photographs comparing averaged gay and straight faces from this dataset.
\begin{table}[!t]
\centering
\begin{tabular}{lcccc}
\hline
& \multicolumn{2}{c}{Females} & \multicolumn{2}{c}{Males} \\ \cline{2-5}
& Gay & Straight & Gay & Straight \\ \hline
Unique users & 3760 & 3515 & 4738 & 3418 \\
Median age (IQR) & 24 (23--27) & 23 (20--27) & 24 (21--28) & 26 (23--30) \\
Total images & 5132 & 5406 & 4666 & 5706 \\
Users with at least: & & & & \\
\,\,\,\,\,\,\,\,1 image & 3760 & 3515 & 4738 & 3418 \\
\,\,\,\,\,\,\,\,2 images & 853 & 1103 & 652 & 604 \\
\,\,\,\,\,\,\,\,3 images & 270 & 458 & 147 & 246 \\
\,\,\,\,\,\,\,\,4 images & 92 & 173 & 72 & 128 \\
\,\,\,\,\,\,\,\,5 images & 45 & 76 & 36 & 70 \\ \hline
\end{tabular}
\caption{Frequencies of users and facial images and the median and interquartile range (IQR) for ages.}
\label{table:results_final_count}
\end{table}
\clearpage
\section{Study 1: Prediction of sexual orientation from facial images using machine learning models}
\label{results:study1}
The results for the three ML models are presented.
\subsection{Model 1: Deep neural network classifier}
The classifier based on DNN features (Model~1) has a ROC AUC score of AUC=.68 for males and AUC=.77 for females when using one facial image to predict sexual orientation.
Figure \ref{fig:figure_1_vgg_auc_scores_by_num_images} shows the AUC scores for Model~1 with increasing numbers (from 1 to 5) of images per subject. The accuracy with which it predicts sexual orientation increases with more images per subject. At three images per subject the classifier scores AUC=.78 for males and AUC=.88 for females.
\begin{figure*}[t]
\centering
\centerline{\includegraphics[width=5.1in]{figure_1_vgg_auc_scores_by_num_images}}%
\caption{Accuracy of the DNN classifiers (Model~1) when provided with increasing numbers of images per subject.}%
\label{fig:figure_1_vgg_auc_scores_by_num_images}%
\end{figure*}
The accuracy decreases with more than three images. This is due to the very small sample size available for subjects with four and five images available in the dataset (see~Table~\ref{table:results_final_count}). The performance values for four and five subjects also vary significantly every time that the classifier is scored, due to the random sample taken during the scoring procedure (see Section~\ref{section:model_scoring}). For example, when evaluating four images per subject the minimum and maximum scores recorded for female subjects were AUC=.84 and AUC=.87, respectively (s.d. 0.01). For males with four images the minimum and maximum were AUC=.78 and AUC=.83, respectively (s.d. 0.01). For the remaining studies only scores up to three images per subject are reported.
Figure \ref{fig:study_1_confusion_matrix} shows the confusion matrices\footnote{A confusion matrix shows the proportion of true positives, false positives, true negatives and false negatives in four quadrants.} for Model~1's female and male classifiers predicting sexual orientation from three images per subject.
\begin{figure*}[!t]
\centering
\centerline{\includegraphics[width=6in]{vgg_confusion_matrix}
\caption{Confusion matrices for the DNN classifier (Model~1) when provided with three facial images per subject.}%
\label{fig:study_1_confusion_matrix}%
\end{figure*}
\subsection{Model 2: Facial morphology classifier}
\label{results:study2}
\begin{figure*}[!htbp]
\centering
\centerline{\includegraphics{figure_2_vgg_and_fpp_scores_by_component}}
\caption{Accuracy of the facial morphology classifiers (Model~2) when provided with three images per subject. For comparison the accuracy of the deep learning classifier, Model~1, is shown on top (for three images per subject).}
\label{fig:figure_2_vgg_and_fpp_scores_by_component}%
\end{figure*}
The classifier based on facial morphology (Model~2) has a ROC AUC score of AUC=.62 for males and AUC=.72 for females when predicting sexual orientation from the whole face using one facial image per subject. When provided with three images per subject, Model~2 scored AUC=.68 for males and AUC=.81 for females.
Figure \ref{fig:figure_2_vgg_and_fpp_scores_by_component} shows the AUC scores for each facial component when provided with three images per subject. The results from Model~1 are shown on top for reference.
For males the eyes and eyebrows are most predictive of sexual orientation and the nose has no predictive value. For females the eyes are most predictive and the facial contour least predictive.
As with Model~1 for this dataset, the results for females are significantly higher than for males.
\subsubsection*{Limitations}
The facial morphology features used by this model may be sensitive to \textit{head pose} \cite{stereotypes}. They may also be sensitive to smiling or the overall facial expression.
\subsection{Model 3: Highly blurred image classifier}
\label{results:study3}
The classifier based on highly blurred images (Model~3) has a ROC AUC score of AUC=.63 for males and AUC=.72 for females when predicting sexual orientation from one 5$\plh$5 pixel image. When provided with three images per subject, this increases to male AUC=.63 and female AUC=.82.
Figure \ref{fig:figure_blurred_1_scores_by_num_images} shows the AUC scores for Model~3 with increasing numbers (from 1 to 3) of images per subject. It shows results for both 5$\plh$5 pixel images and 1 pixel images.
The classifier derived from 5$\plh$5 pixel images shows accuracy comparable with the facial morphology classifier (Model~2).
\begin{figure*}[!htbp]
\centering
\centerline{\includegraphics{figure_blurred_1_scores_by_num_images}}%
\caption{Accuracy of the classifier derived from Highly Blurred Images (Model~3). Scores are shown with increasing numbers of images per subject.}
\label{fig:figure_blurred_1_scores_by_num_images}%
\end{figure*}%
\section{Study 2: Evaluate machine learning models with altered presentation}
\label{results:study2}
All three ML models were evaluated on pairs of photographs with intentionally altered presentation. The predicted sexual orientation labels are show in Figure \ref{fig:figure_three_subjects_three_models} using symbols to depict matching labels. All three models predicted pairwise consistent labels for the three pairs of photographs in Figure \ref{fig:figure_altered_presentation}.
\begin{figure*}[!htbp]
\centering
\centerline{\includegraphics[width=4.5in]{three_subjects_three_models1}}%
\caption{Classification results for each facial image. Original photographs are shown in Figures \ref{fig:figure_altered_presentation}. Each model was used to predict sexual orientation for the pair of photographs for each subject. Matched symbols for a subject indicate a pairwise consistent prediction by that model. Matching symbols per subject across models indicate agreement on sexual orientation by the models for that subject.}%
\label{fig:figure_three_subjects_three_models}%
\end{figure*}%
\subsubsection*{Probability of pairwise consistency}
Assuming that the models are independent and unbiased there is a
\begin{equation}
\left(\frac{1}{8}\right)^3 = \frac{1}{512} = 0.2\%
\end{equation}
chance of all three models predicting a pairwise consistent label for all pairs of photographs.
\subsubsection*{Probability of consistent predictions across models}
The sexual orientation labels for each subject were also predicted consistently across all the models (with the exception of Model~3 for Subject~2).
With the same assumptions, there is a
\begin{equation}
\frac{2}{64} \times \frac{2}{64} \times \frac{6}{64} = 0.01\%
\end{equation}
chance of all three models agreeing on the same pairs of sexual orientation labels for two subjects and two models agreeing for the other subject.
\subsection{Limitations}
This experiment had a very small sample size of only six facial images. The Highly Blurred Image classifier (Model~3) could possibly benefit from the fact that these photographs are taken under similar conditions with similar lighting, similar backgrounds and the same colour of clothing.
\section{Study 3: Test correlation of sexual orientation with head pose}
\label{results:study3}
No strong correlations were found between any of the \textit{head pose} angles and the sexual orientation of the subjects.
\begin{table}[!htbp]
\centering
\begin{tabular}{ccccc}
\cline{1-3}
\rowcolor[HTML]{000000}
\multicolumn{1}{|c|}{\cellcolor[HTML]{000000}{\color[HTML]{FFFFFF} \textbf{Angle type}}} & \multicolumn{2}{c|}{\cellcolor[HTML]{000000}{\color[HTML]{FFFFFF} \textbf{Correlation}}} & \multicolumn{2}{c}{\cellcolor[HTML]{000000}{\color[HTML]{FFFFFF} \textbf{MIC}}} \\ \cline{1-3}
\rowcolor[HTML]{000000}
{\color[HTML]{FFFFFF} } & {\color[HTML]{FFFFFF} Female} & {\color[HTML]{FFFFFF} Male} & {\color[HTML]{FFFFFF} Female} & {\color[HTML]{FFFFFF} Male} \\
\multicolumn{1}{c|}{Pitch} & \multicolumn{1}{c|}{0.10} & \multicolumn{1}{c|}{0.03} & \multicolumn{1}{c|}{0.07} & 0.06 \\
\multicolumn{1}{c|}{Roll} & \multicolumn{1}{c|}{0.05} & \multicolumn{1}{c|}{0.16} & \multicolumn{1}{c|}{0.07} & 0.08 \\
\multicolumn{1}{c|}{Yaw} & \multicolumn{1}{c|}{0.03} & \multicolumn{1}{c|}{-0.01} & \multicolumn{1}{c|}{0.06} & 0.06
\end{tabular}
\caption{Correlation and maximal information coefficient (MIC) relative to sexual orientation for each \textit{head pose} angle.}
\label{table:angle_correlations}
\end{table}
Table \ref{table:angle_correlations} lists the Point-Biserial correlation \cite{pointbiserial} and maximal information coefficient~\cite{mic_reshef} for each type of \textit{head pose} angle (pitch, roll and yaw) with the sexual orientation of the subject.
All correlations had a p-value of less than 0.05, except for the uncorrelated yaw angle for males.
Although the correlations are weak, it is possible to see differences related to sexual orientation in the distribution plots. Figure \ref{fig:figure_angle_categories} plots the distributions of the pitch, roll and yaw angles of the \textit{head pose} for gay and straight subjects\footnote{Only photographs with pitch angles of less than 10 degrees and yaw angles less than 15 degrees were included in the dataset. The roll angles were clipped at -40 degrees and 40 degrees for these plots.}.
Although both categories of females favour photographs taken from a higher angle (greater pitch), it is more pronounced in straight females. Straight females are also more likely to tilt (roll) their heads. There also appears to be a small difference in preference for turning the chin to one side (yaw).
Gay males have a slight preference for a greater pitch and are more likely to tilt their heads (roll).
\subsection*{Limitations}
The correlation test only evaluated each angle individually against the sexual orientation category. It could be that all three angles together show a greater correlation with sexual orientation.
The \textit{head pose} information is provided by the \textit{Face++} model and it is assumed that this information is accurate.
\begin{figure*}[!htbp]
\centering
\centerline{\includegraphics{angle_density_final}}
\caption{Distribution of \textbf{pitch}, \textbf{yaw} and \textbf{roll} angles by sexual orientation.}
\label{fig:figure_angle_categories}%
\end{figure*}
\section{Study 4: Test prediction of sexual orientation while controlling for facial hair and eyewear}
\label{results:study4}
Figure \ref{fig:figure_all_model_scores_facial_hair} shows that when gay male subjects and straight male subjects are evenly split between those with and those without \textbf{facial hair}, the classifier is still able to predict sexual orientation accurately (male AUC=.67; for three images per subject AUC=.77).
The same applies when the subjects are evenly split between those with and without \textbf{eyewear} in each category (male AUC=.67; for three images per subject AUC=.78).
\begin{figure*}[!h]
\centering
\centerline{\includegraphics{figure_all_model_scores_facial_hair}}
\caption{Accuracy in predicting sexual orientation for males evenly split between those with and without \textbf{facial hair} and \textbf{eyewear}. Scores were generated using Model~1 and are shown for an increasing number of images available per subject.}%
\label{fig:figure_all_model_scores_facial_hair}%
\end{figure*}
\clearpage
\section{Summary of results}
\label{results:summary}
A summary of the results for each study is presented.
\subsubsection*{Study 1}
The results for Study 1 show that ML models are capable of predicting sexual orientation from facial images, specifically, photographs from online dating profiles. The two ML models (Model~\hyperref[model:vgg]{1} and Model~\hyperref[model:fpp]{2}) replicating experiments by W\&K \cite{wang_kosinski} using a new dataset show that it is possible to predict sexual orientation from both photographs (see~Figure~\ref{fig:figure_fpp_crop}) and facial morphology (see Figure \ref{fig:figure_fpp_example}).
In addition, the new blurred image classifier (Model~\hyperref[model:blurred]{3}) demonstrates that the colour information contained in a blurred photograph of the face (see Figure \ref{fig:figure_study_3_blurred_image}) is also predictive of sexual orientation.
Figure \ref{fig:figure_all_model_scores} compares the performance for all three models, for both male and female datasets. ROC AUC scores are shown for increasing numbers of photographs per subject
\begin{figure*}[!ht]
\centering
\centerline{\includegraphics{figure_all_model_scores}}%
\caption{Accuracy predicting sexual orientation for Models 1, 2 and 3. Scores are shown for increasing number of images per subject.}%
\label{fig:figure_all_model_scores}%
\end{figure*}
\subsubsection*{Study 2}
The models above were tested on pairs of photographs in which individuals modified their appearance (see Figure \ref{fig:figure_altered_presentation}) to appear more stereotypically gay or straight. Changes to presentation included large changes in pitch and changes to facial hair, eyewear and makeup. All three classifiers were pairwise consistent in predicting the same sexual orientation label for both photographs in each pair.
\subsubsection*{Study 3}
The angles of the head in the photograph making up the \textit{head pose} (pitch, roll and yaw) are not correlated with the subject's sexual orientation.
\subsubsection*{Study 4}
When male subjects are evenly split in each category between those with facial hair and those without facial hair, the classifier is still able to predict sexual orientation accurately (male AUC=.68). Similarly for eyewear, when male subjects are evenly split between those with and those without, the classifier is still able to predict sexual orientation accurately (male AUC=.66).
\subsubsection*{Summary}
\label{sec:intro1}
\addcontentsline{toc}{subsection}{\nameref{sec:intro1}}
This chapter presented the results for the four experiments described in Chapter \ref{chap:methods}. Chapter \ref{chap:discussion} discusses how the results of the replication study compare with that of the original study and what the results of the other experiments reveal about the ability to predict sexual orientation.
\chapter{Discussion}
\label{chap:discussion}
This chapter discusses the results obtained for studies 1-4.
\subsubsection*{Replication and comparison with previous work}
The previous experiments by W\&K replicated here show that ML techniques can predict sexual orientation from facial images. They can also do so better than the human judges tested in their control experiment (human judges scored male AUC=.61 and female AUC=.54 in their experiment \cite{wang_kosinski}).
Despite the smaller dataset (this study has about 20,000 images, where W\&K used 35,000), the models in this study have broadly similar accuracy. However, the DNN model in this study is more accurate for females than for males (Model~1 male AUC=.78 and female AUC=.88 for three images per subject). W\&K's model performs better for males than females (W\&K male AUC=.88 and female AUC=.76 for three images per subject).
Similarly for facial morphology, the classifier in this study performs better for females than for males (Model~2 male AUC=.68 and female AUC=.81 for three images per subject). W\&K's model performs better for males than females (W\&K male AUC=.85 and female AUC=.70 for five images per subject).
Each facial component in this study (except for noses for males) was also predictive of sexual orientation, but the relative ranking compared to W\&K's results was completely different. In W\&K's study, the facial contour component was as good or better than all the other components. In this study, the eyes were the best predictor, and the contour was as poor as or was the least predictive component.
\subsubsection*{Differences in datasets}
While W\&K limited their study to white subjects from the United States, this study included facial images from individuals from many different countries, representing several broad racial groups. Most of the images in this study are of whites and asians. The subjects are younger than those in W\&K (see Table \ref{table:results_final_count}).
A significant difference between this dataset and W\&K's is that the data used in this study has some number of straight transgender females. It is possible that differences in either their facial morphology or presentation (or both) make it easier for the female classifiers to distinguish gay facial images from straight images. This could explain why this study's ML models all perform better for the female dataset than for males.
When comparing the composites in Appendix \ref{app:appendix1} against the composites published by W\&K \cite{wang_kosinski}, the ``baseball cap'' effect of a darkened forehead is noticeably absent in this study's composites.
\subsubsection*{Differences in presentation versus biological differences}
It is difficult to determine whether the ML models employed in this study are predicting sexual orientation from differences that are of a biological nature (such as the shape of the face and facial features) or differences in presentation.
The results from Study 2 indicate that a deliberate change in presentation is not sufficient to change the sexual orientation prediction label for the three models used in this study. Furthermore, all the models (with one exception) agreed on the sexual orientation label for the three individuals tested.
Study 3 also found no correlation between \textit{head pose} angles and sexual orientation, implying that the classifiers are relying on additional information present in each facial image.
The fact that a model is able to predict sexual orientation from facial morphology (Model~2) implies either a biological difference, or an apparent difference due to \textit{head pose}, or some consistent difference in facial expression (such as smiling). Straight females in Appendix \ref{app:appendix1} appear to have a slightly more pronounced smile than their gay counterparts. Gay males in the composite images have clearly more pronounced smiles than their straight counterparts.
In the composite images it can be seen that the gay males have slightly thinner faces (and smaller jaws) than the straight males. Gay females have slightly wider faces and jaws than the straight females. These differences may have a biological origin, but they are also consistent with the pitch angle differences plotted in Figure \ref{fig:figure_angle_categories}.
\subsubsection*{Blurred photographs}
Model 3 showed that a model trained on a highly blurred facial image (Figure \ref{fig:figure_study_3_blurred_image}) is able to predict sexual orientation. The 1 pixel based classifier performed relative poorly, achieving scores barely better than chance for one subject per image. However, the 5$\plh$5~pixel model achieved a score of AUC=.63 for males and AUC=.72 for females using a single image per subject.
This indicates that there is a significant amount of information in the brightness, hue or other colour related information in the blurred images from this dataset that is predictive of sexual orientation. Inspection of the composites in Appendix \ref{app:appendix1} show noticeably brighter faces for straight females and gay males. Straight females also appear to wear brighter, pinker makeup on their lips.
Appendix \ref{app:appendix2} contains plots for the hue, saturation and brightness \cite{foley} distributions of the blurred images, and compares gay and straight sexual orientations. There is a clear difference in the brightness distributions for both females and males. The saturation values are also different for females.
Although the classifier is able to learn to predict sexual orientation from blurred photographs, the underlying cause for such differences is unknown. They could be due to a biological difference such as a difference in facial brightness. It could also be due to a group preference for makeup, or perhaps related to how the photograph is taken. Some people might use a mobile phone to take photographs for dating profiles and others might have them taken in a professionally lit photographic studio. The types of post-processing applied to photographs might vary between groups \cite{gelman}. It is also possible that photographs from different types of mobile phones or those that are uploaded to different dating websites are processed with different image compression algorithms and that there are artifacts resulting from these methods that are easily detectable by ML models.
\subsubsection*{Controls for facial hair and eyewear}
The dataset used in this study confirms some of the trends regarding facial hair and eyewear investigated by Ag{\"u}era y Arcas {\it et al} \cite{stereotypes}. In the composite images more signs of facial hair are visible for straight white males as compared to gay white males. In the dataset for this study straight males are more likely to have facial hair (57\% versus 44\% for gay males). It is also possible to observe traces of spectacles on the gay males and females. In this dataset gay and straight males are equally likely to wear eyewear (20\%). Straight females are highly unlikely to wear eyewear (4\%) versus gay females (20\%).
Despite this preference for facial hair in straight males, Study 4 demonstrates that the models are still able to predict sexual orientation even when the proportion of subjects with facial hair in each category has been balanced out. The same applies for eyewear. This implies that neither differences in facial hair or eyewear alone are responsible for the ability to detect sexual orientation.
\chapter{Conclusion}
\label{chap:conclusion}
By means of a replication study, this work set out to investigate whether it is possible to predict sexual orientation from facial images. The replication results for two of W\&K's ML models verified the ability to predict sexual orientation from dating profile photographs. While the dataset used in this study produces better results for females than males, the accuracies of the models are broadly similar to that reported by W\&K~\cite{wang_kosinski}.
This study also demonstrated that a new ML model using highly blurred facial images is capable of predicting sexual orientation. This model relies on consistent differences in the colour information (such as hue, saturation and brightness) in the facial images to be able to distinguish between gay and straight subjects.
By testing the models on portraits intentionally altered with changes to makeup, facial hair, eyewear and the \textit{head pose}, it has been shown that the models are invariant to these changes. Furthermore, they are still capable of predicting sexual orientation while controlling for facial hair or eyewear.
These results leave open the question of how much the prediction of sexual orientation is influenced by biological features such as facial morphology, and how much by differences in presentation, grooming and lifestyle. Future work on this topic might investigate more precisely what role facial morphology has in predicting sexual orientation. It would also be useful to test whether ML models learn to predict sexual orientation from makeup. The nature of the relationship between colour information (such as facial brightness) with sexual orientation is not clear, future work might explore this.
\chapter{The First Chapter}
\label{chap:first}
Explain what the chapter focusses on. Be brief, and only focus on the main theme of the chapter. Also reference any previous chapters that link to the theme of this chapter.
Then, outline the remaining sections and what each covers in a broad sense. Use labeled references like these: Section~\ref{sec:first:first_sec}, Section~\ref{sec:first:second_sec} and Section~\ref{sec:first:summary}. Note that all labels in this document follow a convention, but you are free to choose whatever labels you want to.
\section{A First Section}
\label{sec:first:first_sec}
The text may include sections, subsections and sub-subsections. Refer to the ``Not So Short Introduction to \LaTeXe'' for more details on what types of organisation are available to you. Include labels for each section, subsection and sub-sub-section, so that you can easily reference them in your text, as was shown in the previous section (this also ensures that your numbering will still be correct, even if you add additional sections and subsections at a later date). Any undefined labels that you reference will not cause the document build process to fail. Instead, they are replaced by question marks, and will generate warnings while compiling. Watch out for these before you submit work.
Equations are included as follows:
\begin{equation}
\label{eq:eqn}
\eta(t)=t+c
\end{equation}
Make sure that you define all the symbols you use in your equations. Equations are referenced in the text as follows: Equation~(\ref{eq:eqn}).
\begin{figure}
\centering{\includegraphics{chapters/chapter1/figures/fig1}}
\caption[A short caption, for the figure list]{A long caption, for under the figure.}
\label{fig:fig1}
\end{figure}
You may include figures, such as Figure~\ref{fig:fig1}. Note the short caption has no period at the end of it, while the long caption does (in other words, always provide the short caption without the period, even if the text is the same as the long caption). We recommend using Xfig (which is an open source tool) if you can for your diagrams. Make sure that the files you create are in encapsulated postscript (eps) format. Note that you must not specify the file extension when including the image (this is because PDF generation automatically converts eps image files to PDF files).
\begin{graph}
\centering{\includegraphics{chapters/chapter1/graphs/graph1}}
\vspace{10pt}
\caption[A short graph caption]{A long graph caption.}
\label{grf:graph1}
\end{graph}
You may also provide graphs, such as Graph~\ref{grf:graph1}. The captions follow the same principal as for figures. We recommend using the very flexible gnuplot (which is open source software) to generate your graphs.
\begin{table}
\caption[A short table caption]{A long table caption.}
\centering{
\begin{tabular}[t]{|l|c|c|}
\hline
\textbf{Parameter} & \textbf{Symbol} & \textbf{Domain} \\
\hline
First & $A$ & $[2,\infty)$ \\
Second & $B$ & $[0.0,\infty)$ \\
Third & $C$ & $(0.0,\infty)$ \\
\end{tabular}
}
\label{tab:table1}
\end{table}
You may include tables, such as Table~\ref{tab:table1}. The captions follow the same principal as for figures and graphs. You may use a package called \texttt{hhline} to add fancy borders to your tables. Please refer to the package documentation for details.
\begin{algorithm}
\centering{\fbox{\parbox[]{155mm}{
\setlength{\parindent}{0.3cm}
\vspace{0.3cm}
\small{
\indent Initialise all variables
\par\textbf{repeat:}
\par\hspace{1cm}Select a pattern, and load it
\par\hspace{1cm}\textbf{for all} {\em values $v$ in the pattern\/} \textbf{do}
\par\hspace{1cm}\hspace{1cm}Analyse the pattern
\par\hspace{1cm}\textbf{end for}
\par\textbf{until} {\em stopping criteria are met}
}
\vspace{0.3cm}
}}}
\caption[A short algorithm caption]{A long algorithm caption.}
\label{alg:algorithm1}
\end{algorithm}
Finally, you may add algorithms, such as Algorithm~\ref{alg:algorithm1}. You are free to typeset these as you see fit, or use one of the many algorithm-related packages that are available. In the future, we will try to develop a nice way of generating pseudocode in a standard format like what you see in Algorithm~\ref{alg:algorithm1}. Again, the captioning conventions are the same as for figures, graphs and tables.
All floating bodies (i.e.\ figures, graphs, tables and algorithms) are automatically positioned by \LaTeX. You can change this behaviour by modifying a number of fractions that govern when and where \LaTeX\ can position floating bodies. These are located in \texttt{dissertation.tex}, and there are a several you can modify (e.g.\ \texttt{topfraction}, \texttt{bottomfraction}, \texttt{textfraction} and \texttt{floatpagefraction}). Note that any floating bodies that cannot be placed throughout the text will group together on ``float pages''.
\section{A Second Section}
\label{sec:first:second_sec}
This is how a second section would appear in your document. You cannot nest sections in one another (to get that effect, use sub-section). It may improve the flow of your writing if you introduce the subsections before jumping into them, so that the reader has an idea of what is coming. Again, use the subsection's labels to ensure that the correct numbering is used throughout.
\subsection{A Subsection}
\label{sec:first:second_sec:sub}
You may place subsections as well. Again, these cannot be nested in one another. Consider the overall structure of your dissertation very carefully before writing. Try to avoid too many short subsections, and split your discussion logically between sections and subsections.
\subsubsection{A Sub-Subsection}
\label{sec:first:second_sec:sub:subsub}
You may even add a third level of headings, called sub-subsections (although these are not numbered, or listed on the contents page). It pays to carefully consider whether you will be including sub-subsections. If the text for such a section is short, and there aren't many topics to cover, bulleted lists often look better. However, if the text needs to be broken up into paragraphs, rather use sub-subsections. Also, try to avoid too many sub-subsections, since they cannot be referenced properly, and may become confusing to the reader if they stretch over too many pages. Also, note that while you can label a sub-subsection (just like a section or subsection), referencing the label will simply provide the number of the subsection within which the sub-subsection is located.
You can include acronyms in your dissertation as follows: \useacronym{AI}, \useacronym{ANN} and \useacronym{CI}. The acronyms should all be defined in \texttt{dissertation.tex}, using \texttt{newacronym}. Note that on the first use of the acronym, the full term is used, with the acronym following it in parentheses. On subsequent uses (such as \useacronym{AI}), only the acronym is given.
While it is not at all necessary, you might like to include index terms in your dissertation. You can do this by providing index terms in the text when you mention the topic (e.g.\ widgets\index{Widgets}). You can make an index term bold in the index if it's a primary page reference (e.g.\ for artificial intelligence\index{Artificial Intelligence|idxbf}). If you index the same term again, it will show up as a non-bold secondary reference. You can provide references to other index terms in the \texttt{dissertation.tex} file. There are many more options, but these are outlined properly in the \texttt{makeindx} documentation.
\section{Summary}
\label{sec:first:summary}
This section should follow all the previous ones forming the main body of the chapter. Provide an outline of the previous sections of this chapter, explaining what each dealt with. Provide section references using labels, as you did for the outline at the start of the chapter.
Give a brief synopsis of what the following chapter will cover, in a general sense. This will improve the logical flow of your work from one chapter to the next.
\chapter{Conclusions}
\label{chap:conclusions}
Provide an introduction, stating that this chapter summarises the conclusions of your work, and consider future directions that related research could take. Again, reference Sections~\ref{sec:conclusions:conclusion_summary} and \ref{sec:conclusions:future_work}.
\section{Summary of Conclusions}
\label{sec:conclusions:conclusion_summary}
Summarise your conclusions here. You should consider each of the objectives you listed in the introduction, and explain how each was met, providing a discussion on what your specific findings were for each. Also mention what novel contributions (these might include a new taxonomy, model, algorithm, or empirical results not previously published) your work has introduced to the field while the various objectives were being addressed.
\section{Future Work}
\label{sec:conclusions:future_work}
Enumerate the future work that you could foresee developing from the work you have done here. Mention areas you could not focus on, or possible extensions to your work. It is a good idea to be thorough, since you increase your chances of being referenced by other researchers who follow up on your work, even if you do not do so yourself. You may consider writing this as a bulleted list, if you mention many aspects.
|
{
"timestamp": "2019-03-01T02:01:19",
"yymm": "1902",
"arxiv_id": "1902.10739",
"language": "en",
"url": "https://arxiv.org/abs/1902.10739"
}
|
\section{Introduction}
The non-perturbative behavior of the most common supersymmetric gauge theories was analyzed and elucidated by Seiberg and his collaborators in a series of groundbreaking works \cite{ADS}, \cite{Seiberg1}, \cite{Seiberg2}. Among the discussed cases was also $SU(N)$ supersymmetric gauge theory with $N_f$ matter supermultiplets in the fundamental and antifundamental representations. This model is of the utmost importance because of the similarities with the more mundane QCD.
By using the tools of holomorphicity and dualities between different regimes the behavior of the theory as a function of the number of flavors $N_f$ was elucidated by pointing out when various phases occur and when chiral symmetry breaking takes place. It is known that in general for suppersymmetric QCD supersymmetry is not dynamically broken except for the case $N_f< N$ which is problematic. If a supersymmetric gauge theory displays or not dynamical supersymmetry breaking is crucial for the construction of supersymmetric extensions of the standard model of elementary particles. In most cases an additional supersymmetry breaking sector must be introduced through soft terms.
The case $N_f<N$ is critical. The theory has a global symmetry $SU(N_f)_L\times SU(N_f)_R\times U(1)_B\times U(1)_R$ with a squark assignment under $U(1)_B \times U(1)_R$ of ($1, \frac{N_f-N}{N_f}$) (and of anti squarks ($-1,\frac{N_f-N}{N_f}$)). The quarks assignment is ($1,-\frac{N_f}{N}$) (with the antiquarks assignment ($-1,-\frac{N}{N_f}$)). There are in general $D$ flat directions in the moduli space. The vacuum expectations values for the squarks can be brought into the diagonal form:
\begin{eqnarray}
\langle \tilde{\Phi}^*\rangle =\langle \Phi\rangle =
\left(
\begin{array}{ccccc}
v_1&0&0&...&0\\
0&v_2&0&...&0\\
0&0&0&...&v_f\\
...&...&...&...&...\\
0&0&0&...&0\\
0&0&0&...&0
\end{array}
\right).
\label{assign45534}
\end{eqnarray}
where the matrix has the dimension $N\times N_f$.
It can be shown \cite{ADS} \cite{Seiberg1}, \cite{Seiberg2} that in the low energy regime the theory is described by the $ADS$ superpotential:
\begin{eqnarray}
W_{ADS}=(N-N_f)\Bigg(\frac{\Lambda^{3N-N_f}}{\det M}\Bigg)^{\frac{1}{N-N_f}},
\label{ADsuperpot66645}
\end{eqnarray}
where $\Lambda$ is the holomorphic scale of the theory and $M$ is a $N_f\times N_f$ matrix field that describes the uneaten chiral supermultiplets $M^j_i=\tilde{\Phi}^{jn}\Phi_{ni}$.
Is then evident from Eq. (\ref{ADsuperpot66645}) that the scalar potential is zero for $M=\infty$. Then the theory may be assumed with dynamical supersymmetry breaking because the zero potential is obtained or without dynamical supersymmetry breaking because the zero potential is obtained for $M\rightarrow \infty$. One says that there is no ground state of the theory. It is known that if supersymmetry is dynamically broken for some values of the coupling constant it is in general broken. Moreover in this case the Witten index $(-1)^F=n_{b0}-n_{F0}$ \cite{Witten} which is given by the difference between the bosonic zero modes and fermionic modes should be zero.
At this point we need to consider what through decades of study one can learn from regular QCD. Below some scale the theory confines and bound states of quarks, the mesons and the baryons form. Moreover it is perfectly possible that the mesons and baryons may contain more than the common number of quarks which are two for mesons and three for baryons. In particular bound states of tetraquark mesons may exist \cite{Jaffe}, \cite{Jora}. This is in general a good assumption that helps explain the unusual inverted mass spectrum of the scalar mesons.
Then it is natural to wonder: what happens if in the low energy description of supersymmetric QCD one introduces additional degrees of freedom, mesons states composed of $2N_f-2$ squarks (and the corresponding quark structure). Could this help elucidate some of the aspects still unclear regarding the behavior of the theory for $N_f<N$?
In this work we will show that although this idea might seem artificial and redundant it can properly and consistently be implemented in the supersymmetric QCD with $N_f<N$. Moreover in this context one can show without any doubt that the theory has the supersymmetry unbroken for $N_f>\frac{N}{2}$ whereas for $N_f\leq\frac{N}{2}$ the supersymmetry is broken. Section II contains the description of the states and the alternative superpotential that can be constructed out of these. In section III we show in which cases supersymmetry is dynamically broken and in which it is not. The conclusions are drawn in section IV.
\section{The superpotential}
We introduce additional meson fields made of $2N_f-2$ squarks:
\begin{eqnarray}
M^{\prime\prime i_1}_{j_1}=\epsilon^{i_1i_2...i_{N_f}}\epsilon_{j_1j_2...j_{N_f}}M^{\dagger j_2}_{i_2}...M^{\dagger j_{N_f}}_{i_{N_f}}.
\label{res534428}
\end{eqnarray}
One can construct many holomorphic invariant out of $M$ and $M'$ but it is sufficient to consider only two of them:
\begin{eqnarray}
&&I_2={\rm Tr}MM^{\prime\prime \dagger}
\nonumber\\
&&I_3=\det[M^{\prime \prime \dagger}].
\label{finalres554663}
\end{eqnarray}
Note that by construction these invariants are still holomorphic. For the simplicity of the notation we redefine $M'=M^{\prime\prime\dagger}$.
We need to match the correct degrees of freedom. It is known that for $N_f<N$ there are generically $N_f^2$ matter degrees of freedom left is the gauge group is broken down to $SU(N-N_f)$. Then if we introduce two matrices $M$ and $M'$ there are $2N_f^2$ degrees of freedom and one needs $N_f^2$ constraints. We will consider these as obtained from the equation of motion for the fields $M$. Then one needs to introduce in the superpotentail the combination $tI_1-I_2$ where $t=(N_f-1)!$. One can check,
\begin{eqnarray}
\frac{\partial (tI_1-I_2)}{\partial M^{i_1}_{j_1}}=\epsilon^{i_1i_2...i_{N_f}}\epsilon_{j_1j_2...j_{N_f}}M^{ j_2}_{i_2}...M^{ j_{N_f}}_{i_{N_f}}-M^{\prime i_1}_{j_1}=0.
\label{constr66453}
\end{eqnarray}
This leads to the $N_f^2$ wanted constraints.
We will present here the final version of the superpotential we propose obtained through matching the correct behavior under anomalies and the correct quantum numbers by analogy with the construction of the $ADS$ superpotential:
\begin{eqnarray}
W=(N-N_f)(I_1t-I_2)^{-\frac{N_f}{N-N_f}}I_3^{\frac{1}{N-N_f}}(\Lambda^b)^{\frac{1}{N-N_f}}(-1)^{\frac{N_f}{N-N_f}}N_f^{\frac{N_f}{N-N_f}}=H(I_1t-I_2)^{-\frac{N_f}{N-N_f}}I_3^{\frac{1}{N-N_f}}.
\label{superpot64553}
\end{eqnarray}
Here $I_1=\det M$.
It is obvious that the potential in Eq. (\ref{superpot64553}) is similar in many ways with the $ADS$ superpotential and that satisfies all the requirements.
Here we will check only the consistency of the superpotential in the case one flavor is decoupled. We give mass to the flavor $N_f$ by adding a mass term:
\begin{eqnarray}
W'=W+yM^{N_f}_{N_f}.
\label{dec885774}
\end{eqnarray}
Then one can write directly the form of the matrices (without repeating known calculations known from the standard case):
\begin{eqnarray}
M=
\left(
\begin{array}{cc}
\tilde{M}&0\\
0&y.
\end{array}
\right).
\label{fomrm7}
\end{eqnarray}
Moreover:
\begin{eqnarray}
M'=
\left(
\begin{array}{cc}
(N_f-1)\tilde{M}^{\prime}y&0\\
0&y_2
\end{array}
\right).
\label{fomrmofmprime}
\end{eqnarray}
Here we needed to take into account a decoupling limit that preserves the general expression for the field $M'$. Then one can write the relations between the old and the new invariants for $N_f-1$ flavors:
\begin{eqnarray}
&&I_1=\tilde{I_1}y
\nonumber\\
&&I_2=(N_f-1)\tilde{I_2}y+yy_2
\nonumber\\
&&I_3=(N_f-1)^{N_f-1}y^{N_f-1}\tilde{I_3}y_2.
\label{expr657888}
\end{eqnarray}
Then,
\begin{eqnarray}
&&W=(N-N_f)(I_1t-I_2)^{-\frac{N_f}{N-N_f}}I_3^{\frac{1}{N-N_f}}(\Lambda^b)^{\frac{1}{N-N_f}}\times
\nonumber\\
&&(-1)^{\frac{N_f}{N-N_f}}N_f^{\frac{N_f}{N-N_f}}=
\nonumber\\
&&[\tilde{I}_1yt-(N_f-1)\tilde{I}_2y-yy_2]^{-\frac{N_f}{N-N_f}}\tilde{I}_3^{\frac{1}{N-N_f}}(N_f-1)^{\frac{N_f-1}{N-N_f}}y_2^{\frac{1}{N-N_f}}y^{\frac{N_f-1}{N-N_f}}\times
\nonumber\\
&&(-1)^{\frac{N_f}{N-N_f}}N_f^{\frac{N_f}{N-N_f}}=
\nonumber\\
&&X_2y^{\frac{1}{N_f-N}}.
\label{exprfinal64554}
\end{eqnarray}
The decoupling conditions are:
\begin{eqnarray}
&&\frac{\partial W}{\partial y_2}=\frac{\partial W'} {\partial y_2}=0
\nonumber\\
&&\frac{\partial W'}{\partial y}=\frac{\partial W}{\partial y}+m=0.
\label{twoexpr774665}
\end{eqnarray}
Then the first equation in Eq. (\ref{twoexpr774665}) leads to:
\begin{eqnarray}
y_2=-\tilde{I_1}(N_f-2)!+\tilde{I_2}.
\label{firstrez664553}
\end{eqnarray}
The second equation in Eq. (\ref{twoexpr774665}) yields:
\begin{eqnarray}
&&W'=\frac{N+1-N_f}{N-N_f}W
\nonumber\\
&&y^{\frac{1}{N_f-N}}=X_2^{\frac{1}{N_f-1-N}}m^{-\frac{1}{N_f-1-N}}(N-N_f)^{-\frac{1}{N_f-N-1}}.
\label{rez663552}
\end{eqnarray}
By introducing $y$ and $y_2$ from Eqs. (\ref{firstrez664553}) and (\ref{rez663552}) into Eq. (\ref{exprfinal64554}) and after some calculations one obtains:
\begin{eqnarray}
W'=(N_f+1-N)(N_f-1)^{\frac{N_f-1}{N-N_f+1}}(-1)^{\frac{N_f-1}{N-N_f+1}}[\tilde{I}_1(N_f-2)!-\tilde{I_2}]^{-\frac{N_f-1}{N+1-N_f}}\tilde{I}_3^{\frac{1}{N+1-N_f}}.
\label{finalrez775664}
\end{eqnarray}
\section{Dynamical supersymmetry breaking and the new superpotential}
First we note that requiring that the massless degree of freedom satisfy some constraint is equivalent with the introduction of a large mass term for the same degrees of freedom. In our approach the main degrees of freedom were the field $M'$ such that it makes sense to introduce a large mass term for ($M$) (Alternatively one may consider $M$ the main degrees of freedom with the same main results). The new superpotential will be:
\begin{eqnarray}
W_1=W +m^j_iM^i_j.
\label{newsup76885}
\end{eqnarray}
We are interested in studying the vacuum. For that one needs to solve:
\begin{eqnarray}
&&\frac{\partial W_1}{\partial M^{i}_j}=
H(-\frac{N_f}{N-N_f})[\frac{I_1t}{M^j_i}-M^{\prime j_i}][I_1t-I_2]^{-\frac{N_f}{N-N_f}-1}I_3^{\frac{1}{N-N_f}}+m^j_i=0
\nonumber\\
&&\frac{\partial W_1}{\partial M^{\prime i}_j}=H\frac{1}{N-N_f}[N_f\frac{M^j_i}{I_1t-I_2}+\frac{1}{M^{\prime j}_i}][I_1t-I_2]^{-\frac{N_f}{N-N_f}}I_3^{\frac{1}{N-N_f}}=0.
\label{twosolve6664553}
\end{eqnarray}
Now the set-up of the problem is as such the fields $M$ get eliminated. Note that $M$ and $M'$ are two arbitrary matrices that in general do not commute. This means that one can consider one of the matrices diagonal but cannot assume that the other is also diagonal. We will consider $M$ diagonal. Since all elements of $M$ have masses which in the end will be taken to infinity one can verify through direct calculations or by simple analogy with the behavior of the $W_{ADS}$ superpotential that the matrix $M$ must have all components equal to zero. Then the first equation in (\ref{twosolve6664553}) is clearly verified. One can write the second equation in (\ref{twosolve6664553}) as:
\begin{eqnarray}
[N_fM^j_i+(I_1t-I_2)\frac{1}{M^{\prime j}_i}][I_1t-I_2]^{-\frac{N_f}{N-N_f}-1}I_3^{\frac{1}{N-N_f}}=0.
\label{res64553}
\end{eqnarray}
The first element in the first square bracket is zero. Then the second must also be zero:
\begin{eqnarray}
\frac{1}{M^{\prime j}_i}[I_1t-I_2]^{-\frac{N_f}{N-N_f}}I_3^{\frac{1}{N-N_f}}=0.
\label{seceq4243}
\end{eqnarray}
The equation above must be true for both diagonal and off diagonal elements of $M'$. Then it is necessary that:
\begin{eqnarray}
\frac{1}{M^{\prime j}_i}I_3^{\frac{1}{N-N_f}}=0.
\label{result56647}
\end{eqnarray}
Consider $I_3=\infty$. We assume without loss of generality that in order for the determinant to be zero (and by the power of symmetry) either the diagonal elements are infinite and the off-diagonal elements are zero or the reciprocal. Then one must have for the first case $\frac{1}{N-N_f}-\frac{1}{F}<0$ and also $\frac{1}{N-N_f}+\frac{1}{N_f}<0$. The reversed case leads to the same conclusion. It is clear that these conditions cannot be satisfied so $I_3$ cannot be infinite. Then we assume $I_3=0$. The following conditions must be satisfied for this case: $\frac{1}{N-N_f}-\frac{1}{F}>0$ and $\frac{1}{N-N_f}+\frac{1}{N_f}>0$. These constraints are satisfied for
$N_f>\frac{N}{2}$. There is no other solution. Then one can tune the other eigenvalues accordingly to satisfy both equations in (\ref{twosolve6664553}).
We conclude that for $N_f>\frac{N}{2}$ there is a zero of the potential for finite values of the fields $M'$ and thus supersymmetry is not dynamically broken. For $N_f\leq\frac{N}{2}$ there is no zero of the potential even for infinite vacuum expectation values so supersymmetry is clearly broken.
\section{Conclusions}
In \cite{ADS}, \cite{Seiberg1}, \cite{Seiberg2} it was shown that for supersymmetric QCD with $N_f<N$ flavors the low energy regime is described by the $ADS$ superpotential. Considering all the anomalies and charges this potential is unique. However there is a slew of superpotentials that one can construct if one introduces additional degrees of freedom. Of course the additional degrees of freedom come with constraint equations or with large mass terms.
Here we introduced one single additional field $M^{\prime i}_j$ described in Eq. (\ref{res534428}) and two possible invariants that contain this field. Then we were able to build a consistent superpotential out of these degrees of freedom.
One might wonder: why reconsider a matter already settled? The reason is that properties of the theory that are not manifest in terms of some set of variables may become evident in terms of a different one. In our case the clear determination of the situations when supersymmetry is dynamically broken and when it is not was at the core of all our endeavours.
In the present work we showed that for $N_f>\frac{N}{2}$ the theory displays a fully supersymmetric vacuum for finite values of the fields therefore there is no dynamical breaking of the supersymmetry. For $N_f\leq\frac{N}{2}$ there is no value of the fields, finite or infinite for which the scalar potential is zero. Consequently supersymmetry is dynamically broken. Thus in terms of the new variables the unwanted behavior of the $W_{ADS}$ superpotential where the vacuum is obtained for fields at infinity is not encountered.
One can introduce new version of the superpotential in terms of additional degrees of freedom also for the cases with $N_f\geq N$. But this is a topic of discussion for another work.
|
{
"timestamp": "2019-03-01T02:18:38",
"yymm": "1902",
"arxiv_id": "1902.11076",
"language": "en",
"url": "https://arxiv.org/abs/1902.11076"
}
|
\section{Introduction}
Recently, research of analyzing graphs with machine learning has received more and more attention, mainly focusing on node classification~\citep{kipf2016semi}, link prediction~\citep{zhu2016max} and clustering tasks~\citep{fortunato2010community}. Graph convolutions, as the transformation of traditional convolutions from Euclidean domain to non-Euclidean domain, have been leveraged to design Graph Neural Networks to deal with a wide range of graph-based machine learning tasks.
Graph Convolutional Networks~(GCNs)~\citep{kipf2016semi} generalize convolutional neural networks~(CNNs) to graph structured data from the perspective of spectral theory based on prior works~\citep{bruna2013spectral,defferrard2016convolutional}. It has been demonstrated that GCN and its variants~\citep{hamilton2017inductive,velickovic2017graph,dai2018learning,chen2017stochastic} significantly outperform traditional multi-layer perceptron~(MLP) models and prior graph embedding approaches~\citep{tang2015line,perozzi2014deepwalk,grover2016node2vec}.
However, there are still many deficits on GCNs, thus in this paper we propose to apply VAT on GCNs to tackle these drawbacks of GCNs. Particularly, we firstly highlight the importance of VAT on GCNs from the following aspects, which construct the motivation of our approaches.
\noindent \textbf{Lacking the Leverage of Unlabeled Data for GCNs.} The optimization of GCNs is solely based on the labeled nodes. Concretely speaking, GCNs directly distribute gradient information over the entire labeled set of nodes from the supervised loss. Due to the lack of loss on unlabeled data, the parameters that are not involved in the receptive field may not be updated~\citep{chen2017stochastic}, resulting in the inefficiency of information propagation of GCNs.
\noindent \textbf{The Smoothness of GCNs.} \cite{bruna2013spectral} firstly define the spectral convolutional operation on graphs and point out that adding the smoothness constraint on the spectrum of the filters improves classification results, since the filters are enforced to have better spatial localization. \cite{defferrard2016convolutional} utilize Chebyshev polynomials to approximate the spectral convolutions and also state that spectral convolutions rely on the smoothness in Fourier domain. Since GCNs are established on spectral theory mentioned above and are equivalent to Symmetrical Laplacian Smoothing~\citep{li2018deeper}, the performance of GCN actually heavily depends on the effect of its smoothness.
\noindent \textbf{Effect of Regularization in Semi-Supervised Learning.} Regularization plays a crucial role in semi-supervised learning including graph-based learning tasks. On the one hand, by introducing regularization, a model can make full use of unlabeled data, thus enhancing the performance in semi-supervised learning. On the other hand, regularization can also be regarded as prior knowledge that can smooth the posterior output. For GCN model, a good regularization can not only leverage the unlabeled data to refine its optimization, but only benefit the smoothness of GCNs, resulting in a improved generalization performance.
\noindent \textbf{Virtual Adversarial Regularization on GCNs.} Virtual Adversarial Training~(VAT)~\citep{miyato2018virtual} smartly performs adversarial training without label information to impose a local smoothness on the classifier, which is especially beneficial to semi-supervised learning. In particular, VAT endeavors to smooth the model anisotropically in the direction in which the model is the most sensitive, i.e., the adversarial direction, to improve the generalization performance of a model. In addition, the existence of robustness issue in GCNs has been explored in recent works~\citep{zugner2018adversarial,zugner2018adversarial2}, allowing VAT on graph-based learning task.
Due to the fact that VAT has been successfully applied on semi-supervised image classification~\citep{miyato2018virtual,yu2018tangent} and text classification~\citep{miyato2016adversarial}, a natural question could be asked: \textit{Can we utilize the efficacy of VAT to improve the performance of GCNs in semi-supervised node classification?}
Following this motivation, in our paper, we formally introduce VAT regularization on the original supervised loss of GCNs in semi-supervised node classification task. Concretely speaking, firstly, a detailed theoretical analysis of GCNs focusing on the first-order approximation of local spectral convolutions and the obtained Symmetric Laplacian Smoothing~\citep{li2018deeper} is provided to demystify how GCNs work in semi-supervised learning. Moreover, based on the motivation described above, we elaborate the process of applying VAT on GCNs in a theoretical way by additionally imposing virtual adversarial loss on the basic loss of GCNs, resulting in GCNVAT algorithm framework. Next, due to the sparse property of node features, in the realization of our method, we actually add virtual adversarial perturbations on sparse and dense features, respectively, and attain the GCNSVAT and GCNDVAT algorithms. Finally, in the experimental part, we demonstrate the effectiveness of the two approaches under different training sizes and refine a theoretical analysis on the sensitivity to the hyper-parameters on VAT, facilitating us to apply our approaches in real applications involving graph-based machine learning tasks. In summary, the contributions of the paper are listed below:
\begin{itemize}
\item To the best of our knowledge, we are the first to focus on applying better regularization on original GCN to refine its generalization performance.
\item We are the first to successfully transfer the efficacy of Virtual Adversarial Training~(VAT) to the semi-supervised node classification on graphs and point out the difference compared with image and text classification setting.
\item We refine the sensitivity analysis of hyper-parameters in GCNSVAT and GCNDVAT algorithms, facilitating the deployment of our methods in real scenarios.
\end{itemize}
\section{GCNs with Virtual Adversarial Training}
In this section, we will elaborate how the GCNs work in semi-supervised learning and how to utilize the virtual adversarial training to improve the local smoothness of GCNs.
\subsection{Semi-Supervised Classification with GCNs}
Graph Convolutional Networks~(GCNs) are derived from first-order approximation of localized spectral filters~\citep{kipf2016semi} and are finally equivalent to Symmetric Laplacian Smoothing~\citep{li2018deeper}. Firstly, we denote a graph by $G=(V,E)$, where $V$ is the vertex set and $E$ is the edge set. $X$ and $A$ are the features and adjacent matrix of the graph, respectively and $D=\text{diag}(d_1,d_2,\cdots,d_n)$ denotes the degree matrix of $A$, where $d_i=\sum_{j}a_{ij}$ is the degree of vertex $i$.
\paragraph{First-Order Approximation} GCNs are based on the graph spectral theory. For efficient computation, \cite{defferrard2016convolutional} approximate the spectral filter $g_{\theta}$ with Chebyshev polynomials up to $K^{th}$ order:
\begin{equation} \begin{aligned}
g_{\theta^\prime}(\Lambda)= \sum_{k=0}^{K-1} {\theta_k^\prime T_k(\Lambda) },
\end{aligned} \end{equation}
where $\Lambda$ is the eigenvalues matrix of normalized graph Laplacian $L=I_N-D^{-\frac{1}{2}}AD^{-\frac{1}{2}}$. $T_k$ is the Chebyshev polynomials and $\theta_k^\prime$ is a vector of Chebyshev coefficients. Further, \cite{kipf2016semi} simplified this model by limiting $K=1$ and approximated $\lambda_{max}$ by 2. Then the first-order approximation of spectral graph convolution is defined as:
\begin{equation} \begin{aligned}
g_\theta \star x = \theta(I_N+D^{-\frac{1}{2}}AD^{-\frac{1}{2}}) x,
\end{aligned} \end{equation}
where $\theta$ is the only Chebyshev coefficients left. Through the normalization trick, the final form of graph convolutional networks with two layers in GCNs~\citep{kipf2016semi} is:
\begin{equation} \begin{aligned}
Z=f(X,A)=\text{softmax}(\hat{A} \ \text{ReLU}(\hat{A}XW^{(0)}) W^{(1)}),
\end{aligned} \end{equation}
where $\hat{A}=\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}, \tilde{A}=A+I$. $\tilde{D}$ is the degree matrix of $\tilde{A}$. $Z$ is the obtained embedding matrix from nodes, $W^{(0)}$ is the input-to-hidden weight matrix and $W^{(1)}$ is the hidden-to-output weight matrix.
\paragraph{Symmetric Laplacian Smoothing} \cite{li2018deeper} point out the reason why the GCNs work lies in the Symmetric Laplacian Smoothing of this spectral convolutional type. We simplify it as follows:
\begin{equation} \begin{aligned}
{\bf z}_i=\sum_{j}^{}\frac{\tilde{a}_{ij}}{\sqrt{\tilde{d}_i}\sqrt{\tilde{d}_j}}{\bf x}_j \ \ \ (\text{for} \ 1\le i \le n),
\end{aligned} \end{equation}
where $\bf z_i$ is the first-layer embedding of node $i$ from features $\bf x$ and corresponding matrix formulation is as follows:
\begin{equation} \begin{aligned}
Z^{(1)} = \tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}X,
\end{aligned} \end{equation}
where $Z^{(1)}$ is the one-layer embedding matrix of feature matrix $X$.
\paragraph{Optimization} Finally, the loss function is defined as the cross entropy error over all labeled nodes:
\begin{equation} \begin{aligned}
\mathcal{L}_0 = -\sum_{l \in \mathcal{Y}_L} \sum_{f=1}^{F} Y_{lf} Z_{lf},
\end{aligned} \end{equation}
where $\mathcal{Y}_L$ is the set of node indices that have labels. In fact, the performance of GCNs heavily depends on the efficiency of this Laplacian Smoothing Convolutions, which has been demonstrated in \citep{li2018deeper,kipf2016semi}. Therefore, how to design a good regularization to refine the smoothness of GCNs plays a crucial role for the improvement of performance for GCNs.
\subsection{Virtual Adversarial Training in GCNs}
Virtual Adversarial Training~(VAT)~\citep{miyato2018virtual} is a regularization method that trains the output distribution to be isotropically smooth around each input data point by selectively smoothing the model in its most anisotropic direction, namely adversarial direction. In this section, we apply VAT on GCNs to improve the local smoothness of GCNs.
\paragraph{Assumptions} Firstly, both VAT and GCNs mainly focus on semi-supervised setting, in which two assumptions should be implicitly met~\citep{yu2018tangent}:
\begin{itemize}
\item \textbf{Manifold Assumption.} The observed data $x$ presented in high dimensional space is with high probability concentrated in the vicinity of some underlying manifold with much lower dimensional space.
\item \textbf{Smoothness Assumption.} If two points $x_1, x_2 \in \mathcal{M}$ are close in manifold distance, then the conditional probability $p(y|x_1)$ and $p(y|x_2)$ should be similar. In other words, the true classifier, or the true condition distribution $p(y|x)$ varies smoothly along the underlying manifold $\mathcal{M}$.
\end{itemize}
In the node classification task, GCNs, which involve the graph embedding process, also implicitly conform to these assumptions. There is underlying manifold in the process of graph embedding and the conditional distribution of embedding vectors are expected to vary smoothly along the underlying manifold. In this way, we are capable of utilizing VAT to smooth the embedding of nodes in the adversarial direction to improve the generalization of GCNs.
\paragraph{Difference of VAT on Graph and Image, Text.} Traditional VAT~\citep{miyato2018virtual} is proposed on image classification while VAT on text classification~\citep{miyato2016adversarial} is applied on word embedding vectors of each word. For VAT on graphs, we simply apply VAT on the features of nodes for easy implementation. Additionally, another obvious difference lies in that the relation between each node is not independent for the node classification task compared with image and text classification. The classification result of each node not only depends on the feature itself but also the features of its neighbors, resulting in the \textit{Propagation Effect} of perturbations on feature of each node. We use $\mathcal{D}_l$ and $\mathcal{D}_{ul}$ to denote dataset with labeled nodes and unlabeled nodes respectively. $\overline{x}$ represents features excluding feature $x$ of current node.
\paragraph{Adversarial Training in GCNs} Here we formally define the adversarial training in GCNs, where adversarial perturbations are solely added on features of labeled nodes:
\begin{equation} \begin{aligned}
\min_{\theta} \max_{r,\|r\|\leq\epsilon} D\left[q(y|x_l, \overline{x}, A), p(y|x_l+r, \overline{x}, A;\theta)\right],
\end{aligned} \end{equation}
where $D[q,p]$ measures the divergence between two distributions $q$ and $p$. $q(y|x_l, \overline{X}, A)$ is the true distribution of output labels, usually one hot vector $h(y;y_l)$ and $p(y|x_l+r, \overline{x}, A)=f(X,A)$ denotes the predicted distribution by GCNs. $x_l$ represents the feature of current labeled node and $r$ represents the adversarial perturbation on the feature $x_l$. When the true distribution is denoted by one hot vector $h(y;y_l)$, the perturbation $r_{\rm adv}$ in $L_{2}$ norm can be linearly approximated:
\begin{equation} \begin{aligned}
r_{\rm adv} \approx \epsilon \frac{g}{\|g\|_2},\ {\rm where}\ g=\nabla_{x_l} D\left[h(y;y_l), p(y|x_l, \overline{x}, A;\theta)\right].
\end{aligned} \end{equation}
\paragraph{Virtual Adversarial Loss}
In order to utilize the unlabeled data, we are expected to evaluate the true conditional probability $q(y|x_l, \overline{x}, A)$. Therefore, we use the current estimate $p(y|x,\overline{x}, A;\hat{\theta})$ in place of $q(y|x, \overline{x}, A)$.
\begin{equation} \begin{aligned}
\min_{\theta} \max_{r,\|r\|\leq\epsilon} D\left[p(y|x,\overline{x}, A;\hat{\theta}), p(y|x+r,\overline{x}, A;\theta)\right]
\end{aligned} \end{equation}
Then virtual adversarial regularization is constructed from inner max loss:
\begin{equation}\begin{aligned}
\mathcal{R}_{\rm vadv}(x, \mathcal{D}_l, \mathcal{D}_{ul}, \theta) = & \\ \max_{r,\|r\|\leq\epsilon} &D\left[p(y|x,\overline{x}, A;\hat{\theta}), p(y|x+r,\overline{x}, A;\theta)\right]
\end{aligned} \end{equation}
The final regularization term we propose in this
study is the average of $\mathcal{R}_{\rm vadv}(x, \mathcal{D}_l, \mathcal{D}_{ul}, \theta)$ over all input nodes:
\begin{equation}
\mathbb{E}_{x\sim \mathcal{D}}\mathcal{R}_{\rm vadv}=\frac{1}{N_l + N_{ul}}\sum_{x \in \mathcal{D}_l, \mathcal{D}_{ul}} \mathcal{R}_{\rm vadv}(x, \mathcal{D}_l, \mathcal{D}_{ul}, \theta)
\end{equation}
\paragraph{Virtual Adversarial Training}
The full objective function is thus given by:
\begin{equation}
\min_{\theta} \mathcal{L}_0 + \alpha\mathbb{E}_{x\sim \mathcal{D}}\mathcal{R}_{\rm vadv},
\end{equation}
where $\mathcal{L}_0$ is constructed from labeled nodes in GCNs, $\alpha$ denotes the regularization coefficient and VAT regularization is crafted from both labeled and unlabeled nodes.
\subsection{Fast Approximation of VAT in GCNs}
The key of VAT in GCNs is the approximation of $r_{vadv}$ where
\begin{equation} \begin{aligned}
r_{\rm vadv} = \mathop{\arg\max}_{r,\|r\|\leq\epsilon} D\left[p(y|x,\overline{x}, A;\hat{\theta}), p(y|x+r,\overline{x}, A;\hat{\theta})\right].
\end{aligned} \end{equation}
\paragraph{Second-Order Approximation} Just like the situation in traditional VAT, the evaluation of GCNs with VAT cannot be performed with the linear approximation since:
\begin{equation} \begin{aligned}
&D\left[p(y|x,\overline{x}, A;\hat{\theta}), p(y|x+r,\overline{x}, A; \hat{\theta})\right] \\&\approx D\left[p(y|x,\overline{x}, A;\hat{\theta}), p(y|x+r,\overline{x}, A; \hat{\theta})\right]|_{r=0}\\
&+ r^\top \nabla_r D\left[p(y|x,\overline{x}, A; \hat{\theta}), p(y|x+r,\overline{x}, A; \hat{\theta})\right]|_{r=0} \\
&= 0 + 0 = 0
\end{aligned} \end{equation}
Therefore, a second-order approximation is needed:
\begin{equation} \begin{aligned}
D(r,x,\overline{x}, A;\hat{\theta}) &\approx& \frac{1}{2}r^{T}H(x,\overline{x}, A; \hat{\theta}) r,
\end{aligned} \end{equation}
where $ H(x,\overline{x}, A; \hat{\theta}) := \nabla_{r}^2 D(r,x,\overline{x}, A;\hat{\theta})|_{r=0}$. Then the evaluation of $r_{\rm vadv}$ can be approximated by:
\begin{equation} \begin{aligned}
r_{\rm vadv} &\approx \arg \mathop {\rm max}\limits_r \{r^{T} H(x, \hat{\theta}) r; ~\|r\|_2 \leq \epsilon\}\\
&= \epsilon \overline{u(x, \overline{x}, A;\hat{\theta})},
\end{aligned} \end{equation}
where $u(x,\overline{x}, A;\hat{\theta})$ is the first dominant eigenvector of $H(x,\overline{x}, A;\hat{\theta})$ with magnitude 1.
\paragraph{Power Iteration Approximation} Following VAT, we also apply power iteration approximation for first dominant eigenvector $u(x,\overline{x}, A;\hat{\theta})$:
\begin{equation} \begin{aligned}
d \leftarrow \overline{H d},
\end{aligned} \end{equation}
where $d$ is initialized as a randomly sampled unit vector and can finally converge to $u$.
\paragraph{Finite Difference Approximation} We also employ finite difference approximation for $H$:
\begin{equation} \begin{aligned}
H d &\approx \frac{\nabla_{r} D(r,x, \overline{x}, A;\hat{\theta}) |_{r=\xi d} - \nabla_{r} D(r,x,\overline{x}, A;\hat{\theta})|_{r=0}}{\xi}\\
&= \frac{\nabla_{r}D(r,x,\overline{x}, A;\hat{\theta})|_{r=\xi d}}{\xi},
\end{aligned} \end{equation}
After the two approximations, $r_{\rm vadv}$ is evaluated by:
\begin{equation} \begin{aligned}
d \leftarrow \overline{\nabla_{r} D(r,x,\overline{x}, A;\hat{\theta})|_{r=\xi d}}.
\end{aligned} \end{equation}
As the demonstration in traditional VAT~\citep{miyato2018virtual}, $K=1$ is sufficient to achieve good performance of VAT in GCNs. Thus, the final approximation of $r_{\rm vadv}$ is:
\begin{equation} \begin{aligned}
&r_{\rm vadv} \approx \epsilon \frac{g}{\|g\|_2} \\
&{\rm where}\ g=\nabla_r D\left[p(y|x,\overline{x}, A;\hat{\theta}), p(y|x+r,\overline{x}, A; \hat{\theta})\right]\Big|_{r=\xi d}.
\end{aligned} \end{equation}
\section{Algorithm}
In this section, we will elaborate our Graph Convolutional Networks with Virtual Adversarial Training (GCNVAT) Algorithm. Algorithm~\ref{alg:GCNVAT} summarizes the procedures of the computation of mini-batch SGD for GCNs with VAT algorithm.
\begin{algorithm}[ht]
\caption{Mini-batch SGD for GCNVAT Framework}
\label{alg:GCNVAT}
\textbf{Input}: Features Matrix $X$, Adjacent Matrix $A$. Graph Convolution Network $f_{\theta}$\\
\textbf{Output}: Graph Embedding $Z=f_{\theta}(X,A)$
\begin{algorithmic}[1]
\STATE Choose $M$ samples of $x^{(i)} (i=1,\dots,M)$ from dataset $\mathcal{D}$ at random.
\STATE Compute the predicted distribution of current GCNs:
$$p(y|x_l, \overline{x}, A;\hat{\theta}) \leftarrow f_{\hat{\theta}}(X,A)$$
\STATE \textbf{\% Step 1: Fast Approximation of $r_{\rm vadv}$}
\STATE Generate a random unit vector $d^{(i)} \in R^{I}$ using an iid Gaussian distribution.
\STATE Calculate $r_{\rm vadv}$ via taking the gradient of $D$ with respect to $r$ on $r=\xi d^{(i)}$ on each input data point $x^{(i)}$:
\begin{equation} \begin{aligned}
&g^{(i)} \leftarrow \\
&\nabla_{r} D\left[p(y|x^{(i)},\overline{x}, A; \hat{\theta}), p(y|x^{(i)}+r,\overline{x}, A;\hat{\theta})\right]\Big|_{r=\xi d^{(i)}} \nonumber
\end{aligned} \end{equation}
\STATE Evaluation of $r_{\rm vadv}$:
$$r_{\rm vadv}^{(i)} \leftarrow \epsilon g^{(i)}/\|g^{(i)}\|_2$$
\STATE \textbf{\% Step 2: Evaluation of Virtual Adversarial Loss}
\begin{equation} \begin{aligned}
&\mathbb{E}_{x\sim \mathcal{D}}\mathcal{R}_{\rm vadv} = \\
& \nabla_{\theta} \left(\frac{1}{M}\sum_{i=1}^M D\left[p(y|x^{(i)}; \hat{\theta}), p(y|x^{(i)}+r_{{\rm vadv}}^{(i)}; \theta)\right]\right)\Bigg|_{\theta=\hat{\theta}} \nonumber
\end{aligned} \end{equation}
\STATE \textbf{\% Step 3: Virtual Adversarial Training}
\STATE Compute the supervised loss $\mathcal{L}_0$ of GCNs:
\begin{equation} \begin{aligned}
\mathcal{L}_0 = -\sum_{l \in \mathcal{Y}_L} \sum_{f=1}^{F} Y_{lf} Z_{lf} \nonumber
\end{aligned} \end{equation}
\STATE Update $\theta$ by optimizing the full objective function $\mathcal{L}$:
\begin{equation}
\mathcal{L}=\mathcal{L}_0 + \alpha\mathbb{E}_{x\sim \mathcal{D}}\mathcal{R}_{\rm vadv} \nonumber
\end{equation}
\end{algorithmic}
\end{algorithm}
Our GCNVAT Algorithm Framework is economical in computation since the derivative of the full objective function can be computed with at most three sets of propagation in total. Specifically speaking, firstly, by initializing the random unit vector $d^{(i)}$ in mini-batch and computing the gradient of divergence between predicted distribution of GCNs and that with the initial perturbation, we can evaluate the fast approximated $r_{\rm vadv}$, which is involved in the first set of back propagation. Secondly, after the computation of $r_{\rm vadv}$, we are able to compute the average virtual adversarial loss in the mini-batch and optimize this loss under fixed $r_{\rm vadv}$, which incorporates the second set of back propagation. Finally, the third back propagation is related to the original supervised loss based on labeled nodes in GCNs. All in all, by this GCNs with VAT algorithm including three sets of back propagation, we are capable of imposing the local adversarial regularization on the original supervised loss of GCNs through smoothing the posterior distribution of the model in the most adversarial direction, thereby improving the generalization of original GCNs.
\paragraph{GCNSVAT and GCNDVAT} In the real scenarios, there are usually sparse features for each node especially for a large graph, which are involved in the computation of sparse tensor. In this case, in the implementation of our GCNVAT algorithm framework, we customize two similar GCNVAT methods for different properties of node features. For GCN Sparse VAT~(GCNSVAT), we only apply virtual adversarial perturbations on the specific sparse elements in feature of each node, which may save much computation time especially for high-dimensional feature vectors. For GCN Dense VAT~(GCNDVAT), we actually perturb each element in feature by transforming the the sparse feature matrix to a dense one.
\section{Experiments}
In the experimental part, we conduct extensive experiments to demonstrate the effectiveness of our GCNSVAT and GCNDVAT algorithms. Firstly, we test the performance of both algorithms under different label rates compared with the original GCN. Then we make another comparison under the standard semi-supervised setting with other state-of-the-art approaches. Finally, a sensitivity analysis of hyper-parameters is provided for broad deployment of our method in real applications.
\paragraph{Experimental Setup} For the graph dataset, we select the three commonly used citation networks: CiteSeer, Cora and PubMed~\citep{sen2008collective}. Dateset statistics are summarized in Table~\ref{table_dataset}. For all methods involved in GCNs, we use the same hyper-parameters as in \citep{kipf2016semi}: learning
rate of 0.01, 0.5 dropout rate, 2 convolutional layers, and 16 hidden units without
validation set for fair comparison. As for the hyper-parameters, we fix regularization coefficient $\alpha=1.0$ and only change the perturbation magnitude $\epsilon$ to control the regularization effect under different training sizes, which is further discussed later in the sensitivity analysis part. All the results are the mean accuracy of 10 runs to avoid stochastic effect.
\begin{table}[htbp]
\centering
\begin{tabular}{lrrrrr}
\hline
\textbf{Dateset} & \textbf{Nodes} & \textbf{Edges} & \textbf{Classes} & \textbf{Features} & \makecell{\bf Label \\ \bf Rate} \\
\hline
CiteSeer & 3327& 4732& 6& 3703& 3.6\%\\
Cora & 2708& 5429& 7& 1433& 5.2\%\\
PubMed & 19717& 44338& 3& 500& 0.3\%\\
\hline
\end{tabular}
\caption{Dateset statistics}
\label{table_dataset}
\end{table}
\subsection{Effect under Different Training Sizes}
To verify the consistent effectiveness of our two methods on the improvement of generalization performance, we compare GCNSVAT and GCNDVAT algorithms with original GCN method~\citep{kipf2016semi} under different training sizes across the three datasets and the results can be observed in Figure~\ref{labelrate}.
\begin{figure}[htbp]
\centering
\centering\includegraphics[width=.5\textwidth,trim=100 0 80 30,clip]{labelrate.pdf}
\caption{Classification Accuracies of GCNSVAT and GCNDVAT algorithms compared with GCN across three datasets.}
\label{labelrate}
\end{figure}
As illustrated in Figure~\ref{labelrate}, GCNSVAT~(the red line) and GCNDVAT~(the blue line) outperform original GCN~(the black line) consistently under all tested label rates. Actually, it is important to note that with the increasing of label rates, the regularization effect imposed by VAT on GCNs diminishes in both approaches since the improvement from regularization based on unlabeled data is decreasing. In other words, the superior performance of GCN with Virtual Adversarial Training are especially significant when there are few training sizes. Fortunately, in real scenarios, it is common to observe graphs with a small number of labeled nodes, thereby our algorithms are especially practical in these applications.
\noindent \textbf{Choice of GCNSVAT and GCNDVAT.} GCNDVAT performs consistently better in comparison with GCNSVAT even though GCNDVAT requires extra computation cost related to perturbations in the entire feature space. As for the reason, we argue that continuous perturbations in features facilitate the effect of VAT than discrete perturbations in sparse features. However, in the scenarios where the graph are large-scaled and their features are sparse, it is more appropriate to utilize GCNSVAT from the perspective of economical computation.
\begin{table}[ht]
\centering\textbf{Cora}
\begin{tabular}{lrrrrrr}
\hline
\textbf{Rates} & 0.5\% & 1\% & 2\% & 3\% & 4\% & 5\%\\
\hline
\makecell{\textbf{GCN}}&\makecell{36.2\\(0.11)}&\makecell{40.6\\(0.08)}&\makecell{69.0\\(0.05)}&\makecell{75.2\\(0.03)}&\makecell{78.2\\(0.1)}&\makecell{78.4\\(0.01)}\\
\hline
\makecell{\textbf{GCN}\\\textbf{SVAT}}&\makecell{43.6\\(0.10)}&\makecell{53.9\\(0.08)}&\makecell{71.4\\(0.05)}&\makecell{75.6\\(0.02)}&\makecell{78.3\\(0.01)}&\makecell{78.5\\(0.01)}\\
\hline
\makecell{\textbf{GCN}\\\textbf{DVAT}}&\makecell{\bf49.0\\(0.10)}&\makecell{\bf61.8\\(0.06)}&\makecell{\bf71.9\\(0.03)}&\makecell{\bf75.9\\(0.02)}&\makecell{\bf78.4\\(0.01)}&\makecell{\bf78.6\\(0.01)}\\
\hline
\end{tabular}
\caption{Classification Accuracies on Cora with different label rates. Numbers in bracket are the standard deviation of accuracies.}
\label{table_weakly_cora}
\centering{\textbf{CiteSeer}}
\begin{tabular}{lrrrrrr}
\hline
\textbf{Rates} & 0.5\% & 1\% & 2\% & 3\% & 4\% & 5\%\\
\hline
\makecell{\textbf{GCN}}&\makecell{36.1\\(0.09)}&\makecell{45.7\\(0.04)}&\makecell{64.3\\(0.04)}&\makecell{68.1\\(0.02)}&\makecell{69.1\\(0.01)}&\makecell{70.3\\(0.01)} \\
\hline
\makecell{\textbf{GCN}\\\textbf{SVAT}}&\makecell{47.0\\(0.08)}&\makecell{52.4\\(0.02)}&\makecell{65.8\\(0.02)}&\makecell{68.6\\(0.01)}&\makecell{69.5\\(0.01)}&\makecell{70.7\\(0.01)} \\
\hline
\makecell{\textbf{GCN}\\\textbf{DVAT}}&\makecell{\bf51.5\\(0.07)}&\makecell{\bf58.5\\(0.03)}&\makecell{\bf67.4\\(0.01)}&\makecell{\bf69.2\\(0.01)}&\makecell{\bf70.8\\(0.01)}&\makecell{\bf71.3\\(0.01)}\\
\hline
\end{tabular}
\caption{Classification Accuracies on CiteSeer with different label rates. Numbers in bracket are the standard deviation of accuracies.}
\label{table_weakly_citeseer}
\centering\textbf{PubMed}
\begin{tabular}{lrrrrr}
\hline
\textbf{Rates} & 0.03\% & 0.05\%& 0.1\%& 0.2\% & 0.3\%\\
\hline
\makecell{\textbf{GCN}}&\makecell{46.3\\(0.08)}&\makecell{56.1\\(0.10)}&\makecell{63.3\\(0.06)}&\makecell{70.4\\(0.04)}&\makecell{77.1\\(0.02)}\\
\hline
\makecell{\textbf{GCN}\\\textbf{SVAT}}&\makecell{52.1\\(0.06)}&\makecell{56.9\\(0.08)}&\makecell{63.5\\(0.07)}&\makecell{71.2\\(0.04)}&\makecell{77.2\\(0.02)}\\
\hline
\makecell{\textbf{GCN}\\\textbf{DVAT}}&\makecell{\bf53.3\\(0.06)}&\makecell{\bf58.6\\(0.06)}&\makecell{\bf66.3\\(0.05)}&\makecell{\bf72.2\\(0.03)}&\makecell{\bf77.3\\(0.02)}\\
\hline
\end{tabular}
\caption{Classification Accuracies on PubMed with different label rates. Numbers in bracket are the standard deviation of accuracies.}
\label{table_weakly_pubmed}
\end{table}
More specifically, we list the detailed performances of GCNSVAT and GCNDVAT compared with original GCN under different label rates, which are exhibited in Tables~\ref{table_weakly_cora},~\ref{table_weakly_citeseer} and~\ref{table_weakly_pubmed}, respectively. We report the mean accuracy of 10 runs. The results in tables provide a more sufficient evidence for the effectiveness of our two methods.
\subsection{Effect on Standard Semi-Supervised Learning}
Apart from the experiments under different training sizes, we also test the performance of GCNSVAT and GCNDVAT algorithms in standard semi-supervised setting with standard label rates listed in Table~\ref{table_dataset}. Particularly, we compare our methods with other state-of-the-art methods on the node classification task under standard label rate and the results of baselines are referred from \citep{kipf2016semi}.
\begin{table}[htbp]
\centering
\begin{tabular}{lrrrrrr}
\hline
\textbf{Method} & \textbf{CiteSeer} & \textbf{Cora} & \textbf{PubMed} \\
\hline
\textbf{ManiReg} & 60.1& 59.5& 70.7\\
\textbf{SemiEmb} & 59.6& 59.0& 71.7\\
\textbf{LP} & 45.3& 68.0& 63.0\\
\textbf{DeepWalk} & 43.2& 67.2& 65.3\\
\textbf{Planetoid} & 64.7& 75.7& 77.2\\
\textbf{GCN} & 68.4& 78.4& 77.3\\
\hline
\textbf{GCNSVAT} & 68.7& 78.5 & 77.5 \\
\textbf{GCNDVAT} &\bf{69.3}&\bf{78.6 }&\bf{77.6}\\
\hline
\end{tabular}
\caption{Accuracy under 20 Labels per Class across three datasets.}
\label{table_standard}
\end{table}
From Table~\ref{table_standard}, it turns out that our GCNDVAT algorithm exhibits the state-of-the-art performance though the improvement are not apparent compared with that in few training sizes, while our GCNSVAT algorithm also shares a similar performance. Through the extensive experiments in semi-supervised learning, we demonstrate thoroughly that VAT suffices to improve the generalization performance of GCNs by additionally providing an adversarial regularization both in semi-supervised setting with few labeled nodes and standard semi-supervised setting.
\subsection{Sensitivity Analysis of Hyper-parameters}
One of the notable advantage of VAT in GCNs is that there are just two scalar-valued hyper-parameters: (1) the perturbation magnitude $\epsilon$ that constraints the norm of adversarial perturbation and (2) the regularization coefficient $\alpha$ that controls the balance between supervised loss $\mathcal{L}_0$ and virtual adversarial loss $\mathbb{E}_{x\sim \mathcal{D}}\mathcal{R}_{\rm vadv}$. We refine the analysis in original VAT~\citep{miyato2018virtual} and theoretically demonstrate the total loss is more sensitive to $\epsilon$ rather than $\alpha$ in the regularization control of GCNs with VAT setting.
Consider the second approximation of virtual adversarial regularization:
\begin{equation} \begin{aligned}
\mathcal{R}_{\rm vadv}(x, \mathcal{D}_l, \mathcal{D}_{ul}, \theta)=&\max_r\{D(r, x, \overline{x}, A;\theta); \|r\|_2 \leq \epsilon\} \\ \approx & \max_r\{\frac{1}{2}r^{T}H(x, \overline{x}, A;\theta)r; \|r\|_2 \leq \epsilon\} \\
=&\frac{1}{2} \epsilon^2 \lambda_1(x, \overline{x}, A;\theta),
\end{aligned} \end{equation}
where $\lambda_1(x,\overline{x}, A; \theta)$ is the dominant eigenvalue of Hessian matrix $H(x, \overline{x}, A;\theta)$ of $D$. Substituting this into the objective function, we obtain
\begin{equation} \begin{aligned}
&\mathcal{L}_0 + \alpha \mathbb{E}_{x\sim \mathcal{D}}\mathcal{R}_{\rm vadv} \\
=&\mathcal{L}_0 + \alpha\frac{1}{N_l+N_{ul}}\sum_{x_* \in \mathcal{D}_l, \mathcal{D}_{ul}}\mathcal{R}_{\rm vadv}(x, \mathcal{D}_l, \mathcal{D}_{ul}, \theta) \\
\approx &\mathcal{L}_0 + \frac{1}{2}\alpha\epsilon^2 \frac{1}{N_l+N_{ul}}\sum_{x_* \in \mathcal{D}_l, \mathcal{D}_{ul}}\lambda_1(x_*, \overline{x}, A;\theta).
\end{aligned} \end{equation}
Thus, the strength of regularization is approximately proportional to $\alpha$ and $\epsilon^2$. In consideration of the regularization term is more sensitive to the change of $\epsilon$, in our experiments we just tune the perturbation $\epsilon$ to control the regularization by fixing $\alpha=1$ for both methods.
Further, we present the tendency between the selected optimal $\epsilon$ and label rates. As for the different label rates, it is natural to expect that GCNs with VAT under lower label rate requires larger VAT regularization, yielding the urge for larger optimal $\epsilon$. We empirically verify this conclusion in Figure~\ref{figure_epsilon}.
\begin{figure}[htbp]
\centering
\centering\includegraphics[width=.4\textwidth,trim=120 20 100 30,clip]{epsilon.pdf}
\caption{Sensitivity analysis of epsilon $\epsilon$ on two methods.}
\label{figure_epsilon}
\end{figure}
From Figure~\ref{figure_epsilon}, it is easy to observe that with the increasing of label rates, there is a descending trend of optimal $\epsilon$ for both GCNSVAT and GCNDVAT across three datasets. It meets our expectation since large VAT regularization are more expected for GCNs under lower label rates to obtain the optimal generalization of GCNs. In addition, the optimal $\epsilon$ parameter in GCNSVAT under the same label rate tends to be higher than that in GCNDVAT, especially when the label rate is lower. The reason is obvious because GCNSVAT only applies perturbations on specific elements of sparse feature for each node, thus requiring larger perturbations on those features to get similar regularization effect compared with GCNDVAT.
\section{Discussions and Conclusion}
GCNs with Virtual Adversarial Training is established on the adversarial training on GCNs, which in our paper is simply constrained in the adversarial perturbations on the features of nodes. However, there may exists a better form of adversarial training in GCNs by additionally considering the change of sensitive edges with respects to the output performance. Therefore, incorporating a better form of Virtual Adversarial Training into graphs allows better improvement of generalization of GCNs. Besides, how to combine VAT with other form Graph Neural Networks especially in inductive setting, is also worthwhile to explore in the future.
In our paper, we impose VAT regularization on the original supervised loss of GCN to enhance its generalization in semi-supervised learning, resulting in GCNSVAT and GCNDVAT, whose perturbations are added in sparse and dense features, respectively. Particularly, we apply VAT on GCNs in a theoretical way by additionally imposing virtual adversarial loss on the basic supervised loss of GCNs. Then we empirically demonstrate the improvement caused by the VAT regularization under different training sizes across three datasets. Our endeavour validates that smoothing anisotropic direction on the posterior distribution of GCNs suffices to improve the Symmetric Laplacian Smoothing of original GCN model.
|
{
"timestamp": "2019-03-01T02:17:10",
"yymm": "1902",
"arxiv_id": "1902.11045",
"language": "en",
"url": "https://arxiv.org/abs/1902.11045"
}
|
\section{Introduction}
The paper study parabolic equation
\begin{equation}\label{eqa}
\begin{split}
(\Delta -q(x,t)-\partial_t)u(x,t)=A(u(x,t))
\end{split}
\end{equation}
on Riemannian manifold $M$ evolving by the geometric flow
\begin{equation}\label{eqb}
\begin{split}
\frac{\partial }{\partial t}g(t)=2h(t),
\end{split}
\end{equation}
where $(x,t)\in M\times [0,T]$, $q(x,t)$ is a function on $M\times [0,T]$ of $C^2$ in $x$-variables and $C^1$ in $t$-variable, $A(u)$ is a function of $C^2$ in $u$, and $h(t)$ is a symmetric $(0,2)$-tensor field on $(M,g(t))$. A important example would be the case where $h(t)=-\Ric (t)$ and $g(t)$ is a solution of the Ricci flow introduced by R.S. Hamilton \cite{hamilton1982Three}. We will give some gradient estimates and Harnack inequalities for positive solutions of equation \eqref{eqa}.
The study of gradient estimates for parabolic equations originated with the work of P. Li and S.-T. Yau \cite{li1986parabolic}. They prove a space-time gradient estimate for positive solutions of the heat equation on a complete manifold. By integrating the gradient estimate along a space-time path, a Harnack inequality was derived. Therefore, Li--Yau inequality is often called differential Harnack inequality. It is easy to see that the above space-time estimate will become an elliptic type gradient estimate for a time-independent solution (see \cite{cheng1975differential}). But the elliptic type estimate cannot hold for a time-dependent solution in general, this can be seen from the form of the fundamental solution of the heat equation in $\mathbb R^n$. However, in 1993, R.S. Hamilton \cite{hamilton1993matrix} established an elliptic type gradient estimate for positive solutions of the heat equation on compact manifolds. It is worth noting that the noncompact version of Hamilton's estimate is not true even for $\mathbb R^n$ (see \cite[Remark~1.1]{souplet2006sharp}). Nevertheless, for complete noncompact manifolds, P. Souplet and Q.S. Zhang \cite{souplet2006sharp} obtained an elliptic type gradient estimate for a bounded positive solution of the heat equation after inserting a necessary logarithmic correction term. Li--Yau type and Hamilton--Souplet--Zhang type gradient estimates have been obtained for other nonlinear equations on manifolds, see for example \cite{chen2009gradient,chen2016gradient,dung2015gradient,li1991Gradient,li2015li,ma2006gradient,negrin1995Gradient,ruan2007elliptic,wu2015elliptic,wu2017Elliptic,yang2008gradient} and the references therein.
On the other hand, gradient estimates are very powerful tools in geometric analysis. For instance, R.S. Hamilton \cite{hamilton1993The,hamilton1995Harnack} established differential Harnack inequalities for the Ricci flow and the mean curvature flow. These results have important applications in the singularity analysis. Over the past two decades, many authors used similar techniques to prove gradient estimates and Harnack inequalities for geometric flows. The list of relevant references includes but is not limited to \cite{bailesteanu2010Gradient,cao2009Differential,guo2014Harnack,ishida2014Geometric,li2018li,li2016Harnack,liu2009Gradient,ni2004Ricci,sun2011Gradient,zhang2006Some,zhao2016Gradient}. In this paper, we follow the work of J. Sun \cite{sun2011Gradient} and M. Bailesteanu et al. \cite{bailesteanu2010Gradient}, and focus on the system \eqref{eqa}--\eqref{eqb}.
Now we give some remarks on equation \eqref{eqa}. When $A(u)=au\log u$, the nonlinear elliptic equation corresponding to \eqref{eqa} is related to the gradient Ricci soliton. When $A(u)=au^\beta $, the nonlinear elliptic equation corresponding to \eqref{eqa} is related to the Yamabe-type equation. In general, the parabolic equation \eqref{eqa} is the so-called reaction-diffusion equation, which can be found in many mathematical models in physics, chemistry and biology (see \cite{rothe1984Global,Smoller1983Shock}), where $qu+A(u)$ and $\Delta u$ are the reaction term and the diffusion term, respectively. The reaction-diffusion equations are very important objects in pure and applied mathematics.
In \cite{chen2018Li}, Q. Chen and the author studied the equation \eqref{eqa} with a convection term on a complete manifold with a fixed metric. Here, we establish some gradient estimates for positive solutions of \eqref{eqa} under geometric flow \eqref{eqb}, which are richer and sharper than \cite{chen2018Li}.
The rest of this paper is organized as follows.
In Section~\ref{sec2}, we establish space-time gradient estimates for positive solution of \eqref{eqa}. We firstly consider that $M$ is a complete noncompact manifold without boundary. A local and a global estimate were established, see Theorem~\ref{tha} and Corollary~\ref{coa}. Next, the case that $M$ is closed is also deal with. In this case, inspired by \cite{bailesteanu2010Gradient}, we obtain a sharper estimate than \cite[Theorem~6]{sun2011Gradient}, see Theorem~\ref{thb}. We also give the corresponding Harnack inequalities in the above two cases, see Corollary~\ref{coc}.
In Section~\ref{sec3}, we consider the case that the solution is bounded, and establish elliptic type gradient estimates of local and global versions, see Theorem~\ref{thc} and Corollary~\ref{cob}. The elliptic type Harnack inequality is also obtained, see Corollary~\ref{cod}.
Finally, in Section~\ref{sec4}, we give some applications and explanations of these gradient estimates in some specific cases.
Throughout the paper, we denote by $n$ the dimension of the manifold $M$, and by $d(x,y,t)$ the geodesic distance between $x,y\in M$ under $g(t)$. When we say that $u(x,t)$ is a solution to the equation \eqref{eqa}, we mean $u$ is a solution which is smooth in $x$-variables and $t$-variable. In addition, we have to give some notations for the convenience of writing. Let $f=\log u$ and $\hat A(f)=\frac{A(u)}{u}$. Then
\[
\hat A_f=A'(u)-A(u)/u,\quad \hat A_{ff}=uA''(u)-A'(u)+A(u)/u.
\]
For $u>0$ we define several nonnegative real numbers (some of $\lambda , \Lambda ,\Sigma ,\kappa $ are allowed to be infinite) as follows:
\begin{align*}
&\lambda_{2R}:=-\min_{Q_{2R,T}}\hat A_f^-=-\min \left\{0,\min_{Q_{2R,T}}(A'(u)-A(u)/u)\right\},\\
&\Lambda_{2R}:=\max_{Q_{2R,T}}\hat A_f^+=\max \left\{0,\max_{Q_{2R,T}}(A'(u)-A(u)/u)\right\},\\
&\Sigma_{2R}:=\max_{Q_{2R,T}}\hat A_{ff}^+=\max \left\{0,\max_{Q_{2R,T}}(uA''(u)-A'(u)+A(u)/u)\right\}\\
&\kappa_{2R}:=-\min \left\{0,\min_{Q_{2R,T}}\left(A'(u)-A(u)/u\right),\min_{Q_{2R,T}}A'(u)\right\}
\end{align*}
and
\begin{align*}
&\lambda :=-\inf_{M\times [0,T]}\hat A_f^-=-\min \left\{0,\inf_{M\times [0,T]}(A'(u)-A(u)/u)\right\},\\
&\Lambda :=\sup_{M\times [0,T]}\hat A_f^+=\max \left\{0,\sup_{M\times [0,T]}(A'(u)-A(u)/u)\right\},\\
&\Sigma :=\sup_{M\times [0,T]}\hat A_{ff}^+=\max \left\{0,\sup_{M\times [0,T]}(uA''(u)-A'(u)+A(u)/u)\right\}\\
&\kappa :=-\min \left\{0,\inf_{Q_{2R,T}}\left(A'(u)-A(u)/u\right),\inf_{Q_{2R,T}}A'(u)\right\}.
\end{align*}
Here, we denote by $v^+=\max \{0,v\}$ and $v^-=\min \{0,v\}$ the positive part and the negative part of a function $v$. Notice that if $M$ is compact, then $\lambda ,\Lambda ,\Sigma $ and $\kappa $ must be finite.
\section{space-time gradient estimates for positive solutions}\label{sec2}
Firstly, we have the following local space-time gradient estimate for \eqref{eqa}--\eqref{eqb}.
\begin{thm}\label{tha}
Let $(M,g(0))$ be a complete Riemannian manifold, and let $g(t)$ evolves by \eqref{eqb} for $t\in [0,T]$. Given $x_0$ and $R>0$, let $u$ be a positive solution to \eqref{eqa} in the cube $Q_{2R,T}:=\{(x,t):d(x,x_0,t)\le 2R, 0\le t\le T\}$. Suppose that there exist constants $K_1, K_2, K_3, K_4,\gamma ,\theta \ge 0$ such that
\[
\Ric \ge -K_1g,\quad -K_2g\le h\le K_3g,\quad |\nabla h|\le K_4
\]
and
\[
|\nabla q|\le \gamma_{2R},\quad \Delta q\le \theta_{2R}
\]
on $Q_{2R,T}$. Then for any $\alpha >1$ and $0<\varepsilon <1$, we have
\begin{equation}\label{eql}
\begin{split}
&\frac{|\nabla u(x,t)|^2}{u^2(x,t)}-\alpha \frac{u_t(x,t)}{u(x,t)}-\alpha q(x,t)-\alpha \frac{A(u(x,t))}{u(x,t)}\\
\le &\frac{n\alpha^2}{t}+\frac{C\alpha^2}{R^2}\left(\frac{\alpha^2}{\alpha -1}+\sqrt{K_1}R\right)+C\alpha^2K_2+n\alpha^2\lambda_{2R}\\
&+\Bigg\{n\alpha^2\bigg[\alpha \theta_{2R}+n\alpha^2\max \{K_2^2,K_3^2\}\\
&\ \qquad +\frac{n\alpha^2}{4(1-\varepsilon )(\alpha -1)^2}\Big((\alpha -1)\Lambda_{2R}+\alpha \Sigma_{2R}+2(K_1+(\alpha -1)K_3+K_4)\Big)^2\\
&\ \qquad +\frac{9}{8}n\alpha^2K_4+\frac{3}{4}\left(\frac{4n\alpha^2}{\varepsilon }\right)^{\frac{1}{3}}(\alpha -1)^{\frac{2}{3}}\gamma_{2R}^{\frac{4}{3}}\bigg]\Bigg\}^{\frac{1}{2}}
\end{split}
\end{equation}
on $Q_{R,T}$, where $C$ is a constant that depends only on $n$.
\end{thm}
\begin{rem}
We see that Theorem~\ref{tha} covers \cite[Theorem~1]{sun2011Gradient}. In fact, when $q(x,t)=A(u)=0$, from Theorem~\ref{tha} we can get
\begin{equation*}
\begin{split}
\frac{|\nabla u(x,t)|^2}{u^2(x,t)}-\alpha \frac{u_t(x,t)}{u(x,t)}\le &\frac{n\alpha^2}{t}+\frac{C\alpha^2}{R^2}\left(\frac{\alpha^2}{\alpha -1}+\sqrt{K_1}R\right)+C\alpha^2K_2\\
&+\Bigg\{n\alpha^2\bigg[n\alpha^2\max \{K_2^2,K_3^2\}+\frac{9}{8}n\alpha^2K_4\\
&+\frac{n\alpha^2}{(1-\varepsilon )(\alpha -1)^2}(K_1+(\alpha -1)K_3+K_4)^2\bigg]\Bigg\}^{\frac{1}{2}}.
\end{split}
\end{equation*}
Let $\varepsilon \to 0+$, we thus get
\begin{equation*}
\begin{split}
\frac{|\nabla u(x,t)|^2}{u^2(x,t)}-\alpha \frac{u_t(x,t)}{u(x,t)}\le &\frac{n\alpha^2}{t}+\frac{C\alpha^2}{R^2}\left(R\sqrt{K_1}+\frac{\alpha^2}{\alpha -1}\right)+C\alpha^2K_2\\
&+\frac{n\alpha^2}{\alpha -1}(K_1+(\alpha -1)K_3+K_4)\\
&+n\alpha^2\left(\max \{K_2,K_3\}+\sqrt{9K_4/8}\right)^2.
\end{split}
\end{equation*}
\end{rem}
To prove Theorem~\ref{tha}, we need the following two lemmas. Let $f=\log u$, by \eqref{eqa} we know that $f$ satisfies
\begin{equation}\label{eqc}
\begin{split}
\Delta f=f_t+q+\frac{A(u)}{u}-|\nabla f|^2=f_t+q+\hat A(f)-|\nabla f|^2
\end{split}
\end{equation}
Set $F=t(|\nabla f|^2-\alpha f_t-\alpha q-\alpha \hat A)$. We have
\begin{lem}[Lemma~3 in \cite{sun2011Gradient}]\label{lea}
Suppose the metric evolves by \eqref{eqb}. Then for any smooth function $f$, we have
\[
\frac{\partial }{\partial t}|\nabla f|^2=-2h(\nabla f,\nabla f)+2\langle \nabla f,\nabla (f_t)\rangle
\]
and
\begin{equation*}
\begin{split}
(\Delta f)_t=\Delta(f_t)-2\langle h,\Hess f\rangle -2\langle \Div h-\tfrac{1}{2}\nabla ({\tr}_gh),\nabla f\rangle ,
\end{split}
\end{equation*}
where $\Div h$ is the divergence of $h$.
\end{lem}
\begin{lem}\label{leb}
Let $(M,g(t))$ satisfies the hypotheses of Theorem~\ref{tha}. Then for any $\delta \in(0,\frac{1}{\alpha })$, we have
\begin{equation}\label{eqd}
\begin{split}
(\Delta -\partial_t)F
\ge &\frac{2(1-\delta \alpha)t}{n}(|\nabla f|^2-f_t-q-\hat A)^2-\frac{F}{t}-2\langle \nabla f, \nabla F\rangle \\
&+\alpha t\hat A_f(|\nabla f|^2-f_t-q-\hat A)-2(\alpha -1)t\langle \nabla f,\nabla q\rangle \\
&-t(2(\alpha -1)\hat A_f+\alpha \hat A_{ff}+2K_1+2(\alpha -1)K_3)|\nabla f|^2\\
&-3\alpha t\sqrt{n}K_4|\nabla f|-\frac{\alpha tn}{2\delta }\max \{K_2^2,K_3^2\}-\alpha t\Delta q.
\end{split}
\end{equation}
\end{lem}
\begin{proof}
By the Bochner formula, \eqref{eqc} and Lemma~\ref{lea}, we calculate
\begin{equation*}
\begin{split}
\Delta F=&2t|\Hess f|^2+2t\Ric (\nabla f,\nabla f)+2t\langle \nabla f,\nabla \Delta f\rangle \\
&-\alpha t\Delta (f_t)-\alpha t\Delta q-\alpha t\hat A_f\Delta f-\alpha t\hat A_{ff}|\nabla f|^2\\
=&2t|\Hess f|^2+2t\Ric (\nabla f,\nabla f)+2t\langle \nabla f,\nabla \Delta f\rangle \\
&-\alpha t(\Delta f)_t-2\alpha t\langle h,\Hess f\rangle -2\alpha t\langle \Div h-\tfrac{1}{2}\nabla ({\tr}_gh),\nabla f\rangle \\
&-\alpha t\Delta q-\alpha ta\hat A_f\Delta f-\alpha ta\hat A_{ff}|\nabla f|^2.
\end{split}
\end{equation*}
By \eqref{eqc} and the definition of $F$ we have
\[
\nabla \Delta f=-\frac{\nabla F}{t}-(\alpha -1)(\nabla (f_t)+\nabla q+\hat A_f\nabla f)
\]
and
\[
(\Delta f)_t=\frac{F}{t^2}-\frac{F_t}{t}-(\alpha -1)(f_{tt}+q_t+\hat A_ff_t).
\]
By Lemma~\ref{lea} we also have
\begin{equation*}
\begin{split}
F_t=&|\nabla f|^2-\alpha f_t-\alpha q-\alpha \hat A\\
&+2t\langle \nabla f,(\nabla f)_t\rangle -\alpha t(f_{tt}+q_t+\hat A_ff_t)\\
=&|\nabla f|^2-\alpha f_t-\alpha q-\alpha \hat A\\
&+2t\langle \nabla f,\nabla (f_t)\rangle -2th(\nabla f,\nabla f)-\alpha t(f_{tt}+q_t+\hat A_ff_t).
\end{split}
\end{equation*}
It follows the above equalities that
\begin{equation}\label{eqe}
\begin{split}
(\Delta -\partial_t)F=&2t|\Hess f|^2+2t\Ric (\nabla f,\nabla f)-\frac{F}{t}-2\langle \nabla f,\nabla F\rangle \\
&-2\alpha t\langle h,\Hess f\rangle -2\alpha t\langle \Div h-\tfrac{1}{2}\nabla ({\tr}_gh),\nabla f\rangle \\
&+\alpha t\hat A_f(|\nabla f|^2-f_t-q-\hat A)-\alpha t\Delta q-2(\alpha -1)t\langle \nabla f,\nabla q\rangle \\
&-2(\alpha -1)th(\nabla f,\nabla f)-2(\alpha -1)t\hat A_f|\nabla f|^2-\alpha t\hat A_{ff}|\nabla f|^2
\end{split}
\end{equation}
The assumption $-K_2g\le h\le K_3g$ implies
\[
|h|^2\le n\max \{K_2^2,K_3^2\}.
\]
By Young's inequality,
\begin{equation}\label{eqf}
\begin{split}
\langle h,\Hess f\rangle \le &\delta |\Hess f|^2+\frac{1}{4\delta }|h|^2\\
\le &\delta |\Hess f|^2+\frac{n}{4\delta }\max \{K_2^2,K_3^2\}
\end{split}
\end{equation}
for any $\delta \in (0,\frac{1}{\alpha})$. We also have
\begin{equation}\label{eqg}
\begin{split}
|\Div h-\tfrac{1}{2}\nabla ({\tr}_gh)|=&|g^{ij}\nabla_ih_{jl}-\tfrac{1}{2}g^{ij}\nabla_lh_{ij}|\\
\le &\frac{3}{2}|g||\nabla h|\le \frac{3}{2}\sqrt{n}K_4.
\end{split}
\end{equation}
On the other hand,
\begin{equation}\label{eqh}
\begin{split}
|\Hess f|^2\ge \frac{1}{n}(\Delta f)^2=\frac{1}{n}(|\nabla f|^2-f_t-q-\hat A)^2.
\end{split}
\end{equation}
Substituting \eqref{eqf}, \eqref{eqg} and \eqref{eqh} into \eqref{eqe} and using the assumptions on bounds of $\Ric $ and $h$, we obtain the final inequality \eqref{eqd}.
\end{proof}
\begin{proof}[The proof of Theorem~\ref{tha}]
By the assumption of bounds of Ricci tensor and the evolution of the metric, we know that $g(t)$ is uniformly equivalent to the initial metric $g(0)$ (see \cite[Corollary~6.11]{chow2006Hamilton}), that is,
\[
e^{-2K_2T}g(0)\le g(t)\le e^{2K_3T}g(0).
\]
Then we know that $(M,g(t))$ is also complete for $t\in [0,T]$.
Let $\phi \in C^\infty ((0,+\infty ])$,
\begin{equation*}
\phi (s)=
\begin{cases}
1, & s\in [0,1],\\
0, & s\in [2,+\infty )
\end{cases}
\end{equation*}
satisfies $\phi(s)\in [0,1], \phi'(s)\le 0, \phi''(s)\ge -C_1$ and $\frac{|\phi'(s)|^2}{\phi(s)}\le C_1$, where $C_1$ is an absolute constant. Define
\[
\eta(x,t)=\phi\left(\frac{r(x,t)}{R}\right),
\]
where $r(x,t)=d(x,x_0,t)$. Using the argument of \cite{calabi1958An}, we can assume that the function $\eta(x,t)$ is $C^2$ with support in $Q_{2R,T}$.
Define $G=\eta F$. For any $T_1\in (0,T]$, let $(x_1,t_1)\in Q_{2R,T_1}$ at which $G$ attains its maximum, and without loss of generality, we can assume $G(x_1,t_1)>0$, and then $\eta(x_1,t_1)>0$ and $F(x_1,t_1)>0$. Hence, at $(x_1,t_1)$, we have
\[
\nabla G=0,\quad \Delta G\le 0,\quad \partial_tG\ge 0.
\]
Hence, we obtain
\begin{equation}\label{eqi}
\begin{split}
\nabla F=-\frac{F}{\eta }\nabla \eta
\end{split}
\end{equation}
and
\begin{equation}\label{eqj}
\begin{split}
0\ge &(\Delta -\partial_t)G\\
=&F(\Delta -\partial_t)\eta +\eta (\Delta -\partial_t)F+2\langle \nabla \eta ,\nabla F\rangle .
\end{split}
\end{equation}
By the properties of $\phi $ and the Laplacian comparison theorem, we have
\[
\frac{|\nabla \eta |^2}{\eta }\le \frac{C_1}{R^2}
\]
and
\begin{equation*}
\begin{split}
\Delta \eta =&\phi'\frac{\Delta r}{R}+\phi''\frac{|\nabla r|^2}{R^2}\\
\ge &-\frac{\sqrt{C_1}}{R}\sqrt{(n-1)K_1}\coth \left(\sqrt{\tfrac{K_1}{n-1}}R\right)-\frac{C_1}{R^2}\\
\ge &-\frac{(n-1)\sqrt{C_1}}{R^2}-\frac{\sqrt{(n-1)C_1K_1}}{R}-\frac{C_1}{R^2}.
\end{split}
\end{equation*}
By \cite[p. 494]{sun2011Gradient}, there exist a constant $C_2$ such that
\[
-F\eta_t\ge -C_2K_2F.
\]
Substituting the above three inequalities into \eqref{eqj} and using \eqref{eqi}, we obtain
\begin{equation}\label{eqk}
\begin{split}
0\ge &-\left(\frac{(n-1)\sqrt{C_1}}{R^2}+\frac{\sqrt{(n-1)C_1K_1}}{R}+\frac{3C_1}{R^2}+C_2K_2\right)F+\eta (\Delta -\partial_t)F\\
=:&\eta (\Delta -\partial_t)F-\left(\frac{C_3(n)}{R^2}+\frac{C_4(n)}{R}\sqrt{K_1}+C_2K_2\right)F.
\end{split}
\end{equation}
Let $\mu =\frac{|\nabla f(x_1,t_1)|^2}{F(x_1,t_1)}$, then, at $(x_1,t_1)$ we have
\[
\eta \langle \nabla f,\nabla F\rangle =-F\langle \nabla f,\nabla \eta \rangle \le \frac{\sqrt{C_1}}{R}\eta^{\frac{1}{2}}F|\nabla f|
\]
and
\[
|\nabla f|^2-f_t-q-a\hat A=\left(\mu -\frac{t_1\mu -1}{t_1\alpha }\right)F.
\]
Therefore, at $(x_1,t_1)$, by Lemma~\ref{leb} and \eqref{eqk}, and using the inequality
\begin{equation}\label{eqo}
\begin{split}
3\alpha \sqrt{n}K_4|\nabla f|\le 2K_4|\nabla f|^2+\frac{9}{8}n\alpha^2K_4,
\end{split}
\end{equation}
we obtain
\begin{equation*}
\begin{split}
0\ge &\frac{2(1-\delta \alpha )t_1}{n}\eta \left(\mu -\frac{t_1\mu -1}{t_1\alpha }\right)^2F^2-\frac{\eta F}{t_1}-\frac{2\sqrt{C_1}}{R}\eta^{\frac{1}{2}}\mu^{\frac{1}{2}}F^{\frac{3}{2}}\\
&+\alpha t_1\eta \hat A_f\left(\mu -\frac{t_1\mu -1}{t_1\alpha }\right)F-\alpha t_1\eta \Delta q-2(\alpha -1)t_1\eta \langle \nabla f,\nabla q\rangle \\
&-t_1\eta (2(\alpha -1)\hat A_f+\alpha \hat A_{ff}+2(K_1+(\alpha -1)K_3+K_4))\mu F\\
&-\frac{9}{8}t_1\eta n\alpha^2K_4-\frac{t_1\eta n\alpha }{2\delta }\max \{K_2^2,K_3^2\}\\
&-\left(\tfrac{C_3(n)}{R^2}+\tfrac{C_4(n)}{R}\sqrt{K_1}+C_2K_2\right)F.
\end{split}
\end{equation*}
Multiplying through by $t_1\eta $, we conclude that
\begin{equation*}
\begin{split}
0\ge &\frac{2(1-\delta \alpha )}{n\alpha^2}(1+(\alpha -1)t_1\mu )^2G^2-\eta G-\frac{2\sqrt{C_1}}{R}t_1\mu^{\frac{1}{2}}G^{\frac{3}{2}}\\
&+t_1\eta \hat A_f(1+(\alpha -1)t_1\mu )G-\alpha t_1^2\eta^2\Delta q-2(\alpha -1)t_1^2\eta^2\langle \nabla f,\nabla q\rangle \\
&-t_1^2\eta[2(\alpha -1)\hat A_f+\alpha \hat A_{ff}+2(K_1+(\alpha -1)K_3+K_4)]\mu G\\
&-\frac{9}{8}t_1^2\eta^2n\alpha^2K_4-\frac{t_1^2\eta^2n\alpha }{2\delta }\max \{K_2^2,K_3^2\}\\
&-t_1\left(\tfrac{C_3(n)}{R^2}+\tfrac{C_4(n)}{R}\sqrt{K_1}+C_2K_2\right)G\\
=&\frac{2(1-\delta \alpha )}{n\alpha^2}(1+(\alpha -1)t_1\mu )^2G^2-\eta G-\frac{2\sqrt{C_1}}{R}t_1\mu^{\frac{1}{2}}G^{\frac{3}{2}}\\
&+t_1\eta \hat A_fG-\alpha t_1^2\eta^2\Delta q-2(\alpha -1)t_1^2\eta^2\langle \nabla f,\nabla q\rangle \\
&-t_1^2\eta[(\alpha -1)\hat A_f+\alpha \hat A_{ff}+2(K_1+(\alpha -1)K_3+K_4)]\mu G\\
&-\frac{9}{8}t_1^2\eta^2n\alpha^2K_4-\frac{t_1^2\eta^2n\alpha }{2\delta }\max \{K_2^2,K_3^2\}\\
&-t_1\left(\tfrac{C_3(n)}{R^2}+\tfrac{C_4(n)}{R}\sqrt{K_1}+C_2K_2\right)G
\end{split}
\end{equation*}
Noticing that $0<\eta(x_1,t_1) \le 1$, from the above inequalities we obtain
\begin{equation*}
\begin{split}
0\ge &\frac{2(1-\delta \alpha )}{n\alpha^2}G^2+\frac{4(1-\delta \alpha )}{n\alpha^2}(\alpha -1)t_1\mu G^2\\
&+\frac{2(1-\delta \alpha )}{n\alpha^2}(\alpha -1)^2t_1^2\mu^2G^2-G-t_1\lambda_{2R}G-\frac{2\sqrt{C_1}}{R}t_1\mu^{\frac{1}{2}}G^{\frac{3}{2}}\\
&-t_1^2\Big[(\alpha -1)\Lambda_{2R}+\alpha \Sigma_{2R}+2(K_1+(\alpha -1)K_3+K_4)\Big]\mu G\\
&-2t_1^2(\alpha -1)\gamma_{2R}\mu^{\frac{1}{2}}G^{\frac{1}{2}}-\frac{9}{8}t_1^2n\alpha^2K_4-\frac{t_1^2n\alpha }{2\delta }\max \{K_2^2,K_3^2\}\\
&-\alpha t_1^2\theta_{2R}-t_1\left(\tfrac{C_3(n)}{R^2}+\tfrac{C_4(n)}{R}\sqrt{K_1}+C_2K_2\right)G.
\end{split}
\end{equation*}
By Young's inequality, we have
\[
\frac{2\sqrt{C_1}}{R}\mu^{\frac{1}{2}}G^{\frac{3}{2}}\le \frac{4(1-\delta \alpha )}{n\alpha^2}(\alpha -1)\mu G^2+\frac{n\alpha^2C_1G}{8(1-\delta \alpha )(\alpha -1)R^2},
\]
\begin{equation*}
\begin{split}
2(\alpha -1)\gamma_{2R}\mu^{\frac{1}{2}}G^{\frac{1}{2}}\le \frac{2(1-\delta \alpha )\varepsilon }{n\alpha^2}(\alpha -1)^2\mu^2G^2+\frac{3}{4}\left(\frac{2n\alpha^2}{(1-\delta \alpha )\varepsilon }\right)^{\frac{1}{3}}(\alpha -1)^{\frac{2}{3}}\gamma_{2R}^{\frac{4}{3}}
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
&\Big[(\alpha -1)\Lambda_{2R}+\alpha \Sigma_{2R}+2(K_1+(\alpha -1)K_3+K_4)\Big]\mu G\\
\le &\frac{2(1-\delta \alpha )(1-\varepsilon )}{n\alpha^2}(\alpha -1)^2\mu^2G^2\\
&+\frac{n\alpha^2}{8(1-\delta \alpha )(1-\varepsilon )(\alpha -1)^2}\Big[(\alpha -1)\Lambda_{2R}+\alpha \Sigma_{2R}+2(K_1+(\alpha -1)K_3+K_4)\Big]^2,
\end{split}
\end{equation*}
where $\varepsilon \in (0,1)$ is an arbitrary constant. Combining the above four inequalities, there exists a constant $C_5(n)$ that depends only on $n$, such that
\begin{equation*}
\begin{split}
0\ge &\frac{2(1-\delta \alpha )}{n\alpha^2}G^2\\
&-\left[1+t_1\left(\lambda_{2R}+\frac{C_5(n)}{R^2}\left(\frac{\alpha^2}{(1-\delta \alpha )(\alpha -1)}+\sqrt{K_1}R\right)+C_2K_2\right)\right]G\\
&-t_1^2\alpha \theta_{2R}-t_1^2n\alpha^2\max \{K_2^2,K_3^2\}\\
&-t_1^2\frac{n\alpha^2}{8(1-\delta \alpha )(1-\varepsilon )(\alpha -1)^2}\\
&\cdot \Big[(\alpha -1)\Lambda_{2R}+\alpha \Sigma_{2R}+2(K_1+(\alpha -1)K_3+K_4)\Big]^2\\
&-\frac{9}{8}t_1^2n\alpha^2K_4-\frac{3}{4}t_1^2\left(\frac{2n\alpha^2}{(1-\delta \alpha )\varepsilon }\right)^{\frac{1}{3}}(\alpha -1)^{\frac{2}{3}}\gamma_{2R}^{\frac{4}{3}}.
\end{split}
\end{equation*}
For a positive number $a$ and two nonnegative numbers $b,c$, from the inequality $ax^2-bx-c\le 0$ we have $x\le \frac{b}{a}+\sqrt{\frac{c}{a}}$. Hence, we obtain
\begin{equation*}
\begin{split}
G\le &\frac{n\alpha^2}{2(1-\delta \alpha )}\left[1+t_1\left(\lambda_{2R}+\frac{C_5(n)}{R^2}\left(\frac{\alpha^2}{(1-\delta \alpha )(\alpha -1)}+\sqrt{K_1}R\right)+C_2K_2\right)\right]\\
&+t_1\Bigg\{n\alpha^2\bigg[\alpha \theta_{2R}+n\alpha^2\max \{K_2^2,K_3^2\}\\
&\ \qquad +\frac{n\alpha^2}{8(1-\delta \alpha )(1-\varepsilon )(\alpha -1)^2}\\
&\quad \quad \ \cdot \Big[(\alpha -1)\Lambda_{2R}+\alpha \Sigma_{2R}+2(K_1+(\alpha -1)K_3+K_4)\Big]^2\\
&\ \qquad +\frac{9}{8}n\alpha^2K_4+\frac{3}{4}\left(\frac{2n\alpha^2}{(1-\delta \alpha )\varepsilon }\right)^{\frac{1}{3}}(\alpha -1)^{\frac{2}{3}}\gamma_{2R}^{\frac{4}{3}}\bigg]\Bigg\}^{\frac{1}{2}}.
\end{split}
\end{equation*}
Now, by taking $\delta =\frac{1}{2\alpha }$, and noticing that $d(x,x_0,T_1)\le R$ implies $\eta(x,T_1)=1$, we can get
\begin{equation*}
\begin{split}
&(|\nabla f|^2-\alpha f_1-\alpha q_t-\alpha \hat A)(x,T_1)=\frac{F(x,T_1)}{T_1}\le \frac{G(x_1,t_1)}{T_1}\\
\le &\frac{n\alpha^2}{T_1}+\frac{C(n)\alpha^2}{R^2}\left(\frac{\alpha^2}{\alpha -1}+\sqrt{K_1}R\right)+C(n)\alpha^2K_2+n\alpha^2\lambda_{2R}\\
&+\Bigg\{n\alpha^2\bigg[\alpha \theta_{2R}+n\alpha^2\max \{K_2^2,K_3^2\}\\
&\ \qquad +\frac{n\alpha^2}{4(1-\varepsilon )(\alpha -1)^2}\Big((\alpha -1)\Lambda_{2R}+\alpha \Sigma_{2R}+2(K_1+(\alpha -1)K_3+K_4)\Big)^2\\
&\ \qquad +\frac{9}{8}n\alpha^2K_4+\frac{3}{4}\left(\frac{4n\alpha^2}{\varepsilon }\right)^{\frac{1}{3}}(\alpha -1)^{\frac{2}{3}}\gamma_{2R}^{\frac{4}{3}}\bigg]\Bigg\}^{\frac{1}{2}},
\end{split}
\end{equation*}
where $C(n)$ is an appropriate constant that depends only $n$. Since $T_1$ is arbitrary, we complete the proof.
\end{proof}
\begin{rem}
In the above proof, if we use $x\le \frac{1}{2a}\left(b+\sqrt{b^2+4ac}\right)$ instead of $x\le \frac{b}{a}+\sqrt{\frac{c}{a}}$, then a more appropriate $\delta $ may give a sharper estimate.
\end{rem}
From the above local estimate, we get a global one:
\begin{cor}\label{coa}
Let $(M,g(0))$ be a complete noncompact Riemannian manifold without boundary, and let $g(t)$ evolves by \eqref{eqb} for $t\in [0,T]$. Suppose that there exist constant $K_1, K_2, K_3, K_4, \gamma , \theta \ge 0$ such that
\[
\Ric \ge -K_1g,\quad -K_2g\le h\le K_3g,\quad |\nabla h|\le K_4
\]
and
\[
|\nabla q|\le \gamma ,\quad \Delta q\le \theta .
\]
If $u$ is a positive solution to \eqref{eqa}, then for any $\alpha >1$, we have
\begin{equation}\label{eqm}
\begin{split}
&\frac{|\nabla u(x,t)|^2}{u^2(x,t)}-\alpha \frac{u_t(x,t)}{u(x,t)}-\alpha q(x,t)-\alpha \frac{A(u(x,t))}{u(x,t)}\\
\le &\frac{n\alpha^2}{t}+C'\left(K_1+K_2+K_3+K_4+\sqrt{K_4}+\sqrt{\theta }+\gamma^{\frac{2}{3}}+\lambda +\Lambda +\Sigma \right)
\end{split}
\end{equation}
on $M\times [0,T]$, where $C'$ is a constant that depends only on $n,\alpha $.
\end{cor}
\begin{proof}
By the uniform equivalence of $g(t)$, we know that $(M,g(t))$ is complete noncompact for $t\in [0,T]$. Now we choose $\varepsilon =\varepsilon_0$ in \eqref{eql}, where $\varepsilon_0$ is an arbitrary fixed number in $(0,1)$. Let $R\to +\infty $ in \eqref{eql}, and using the inequality $\sqrt{x+y}\le \sqrt{x}+\sqrt{y}$ holds for any $x,y\ge 0$, we complete the proof.
\end{proof}
We now consider the case that the manifold $M$ is closed. By Lemma~\ref{leb}, we have a global gradient estimate on a closed Riemannian manifold.
\begin{thm}\label{thb}
Let $(M,g(t))$ be a closed Riemannian manifold, where $g(t)$ evolves by \eqref{eqb} for $t\in [0,T]$ and satisfies
\[
\Ric \ge -K_1g,\quad -K_2g\le h\le K_3g,\quad |\nabla h|\le K_4.
\]
If $u$ is a positive solution to \eqref{eqa}, and $q(x,t)$ satisfies
\[
|\nabla q|\le \gamma ,\quad \Delta q\le \theta .
\]
Then for any $\alpha >1$, we have
\begin{equation}\label{eqn}
\begin{split}
&\frac{|\nabla u(x,t)|^2}{u^2(x,t)}-\alpha \frac{u_t(x,t)}{u(x,t)}-\alpha q(x,t)-\alpha \frac{A(u(x,t))}{u(x,t)}\\
\le &\frac{n\alpha^2}{2t}+\frac{n\alpha^2}{\alpha -1}(K_1+(\alpha -1)K_3+K_4)+n\alpha^2\left(\max \{K_2,K_3\}+\frac{3}{4}\sqrt{2K_4}\right)\\
&+\alpha^{\frac{3}{2}}\sqrt{n\theta }+\left(\frac{n\alpha^2}{2}(\alpha -1)^{\frac{1}{2}}+\sqrt{n}\alpha (\alpha -1)^{\frac{1}{4}}\right)\gamma^{\frac{2}{3}}+\frac{n\alpha^2}{2}(\lambda +\Lambda )+\frac{n\alpha^3}{2(\alpha -1)}\Sigma
\end{split}
\end{equation}
on $M\times (0,T]$.
\end{thm}
\begin{proof}
We use the same symbols $f, F$ as above. Set
\begin{equation*}
\begin{split}
\bar F(x,t)=&F(x,t)-\frac{n\alpha^2}{\alpha -1}(K_1+(\alpha -1)K_3+K_4)t\\
&-n\alpha^2\left(\max \{K_2,K_3\}+\frac{3}{4}\sqrt{2K_4}\right)t-\alpha^{\frac{3}{2}}\sqrt{n\theta }t\\
&-\left(\frac{n\alpha^2}{2}(\alpha -1)^{\frac{1}{2}}+\sqrt{n}\alpha (\alpha -1)^{\frac{1}{4}}\right)\gamma^{\frac{2}{3}}t-\frac{n\alpha^2}{2}(\lambda +\Lambda )t-\frac{n\alpha^3}{2(\alpha -1)}\Sigma t.
\end{split}
\end{equation*}
If $\bar F(x,t)\le \frac{n\alpha^2}{2}$ for any $(x,t)\in M\times (0,T]$, the proof is complete.
If \eqref{eqn} doesn't hold, then at the maximal point $(x_0,t_0)$ of $\bar F(x,t)$, we have
\[
\bar F(x_0,t_0)>\frac{n\alpha^2}{2}.
\]
As $\bar F(x,0)=0$, we know that $t_0>0$ here. Then applying the maximum principle, we have
\[
\nabla \bar F(x_0,t_0)=0,\quad \Delta \bar F(x_0,t_0)=0,\quad \partial_t\bar F(x_0,t_0)=0.
\]
Therefore, we obtain
\[
0\ge (\Delta -\partial_t)\bar F\ge (\Delta -\partial_t)F.
\]
Using Lemma~\ref{leb}, inequality \eqref{eqo} and the fact that
\begin{equation*}
\begin{split}
|\nabla f|^2-f_t-q-\hat A=&\frac{1}{\alpha }(|\nabla f|^2-\alpha f_t-\alpha q-\alpha \hat A)+\frac{\alpha -1}{\alpha }|\nabla f|^2\\
=&\frac{1}{\alpha }\frac{F}{t_0}+\frac{\alpha -1}{\alpha }|\nabla f|^2,
\end{split}
\end{equation*}
we obtain
\begin{equation*}
\begin{split}
0\ge &\frac{2(1-\delta \alpha )t_0}{n\alpha^2}\left(\frac{F}{t_0}\right)^2+\frac{4(1-\delta \alpha )(\alpha -1)t_0}{n\alpha^2}|\nabla f|^2\frac{F}{t_0}\\
&+\frac{2(1-\delta \alpha )(\alpha -1)^2t_0}{n\alpha^2}|\nabla f|^4-\frac{F}{t_0}-t_0\lambda \frac{F}{t_0}\\
&-2t_0(\alpha -1)\gamma |\nabla f|-\frac{9}{8}t_0n\alpha^2K_4-t_0\alpha \theta -\frac{t_0n\alpha }{2\delta }\max \{K_2^2,K_3^2\}\\
&-t_0\left((\alpha -1)\Lambda +\alpha \Sigma +2(K_1+(\alpha -1)K_3+K_4)\right)|\nabla f|^2.
\end{split}
\end{equation*}
By
\begin{equation*}
\begin{split}
\frac{F}{t_0}=&\frac{\bar F}{t_0}+\frac{n\alpha^2}{\alpha -1}(K_1+(\alpha -1)K_3+K_4)+\frac{3}{2}n\alpha^2\sqrt{K_4}\\
&+\left(\frac{n\alpha^2}{2}(\alpha -1)^{\frac{1}{2}}+\sqrt{2n}\alpha (\alpha -1)^{\frac{1}{4}}\right)\gamma^{\frac{2}{3}}\\
&+n\alpha^2A_1+\frac{n\alpha^3}{2(\alpha -1)}A_2>0,
\end{split}
\end{equation*}
\[
2t_0(\alpha -1)\gamma |\nabla f|^2\le t_0(\alpha -1)^{\frac{3}{2}}\gamma^{\frac{2}{3}}|\nabla f|^2+t_0(\alpha -1)^{\frac{1}{2}}\gamma^{\frac{4}{3}},
\]
and using the inequality $ax^2-bx\ge -\frac{b^2}{4a}$ holds for $a>0, b\ge 0$, we obtain
\begin{equation*}
\begin{split}
0\ge &\frac{2(1-\delta \alpha )t_0}{n\alpha^2}\left(\frac{F}{t_0}\right)^2-\frac{F}{t_0}-t_0\lambda \frac{F}{t_0}-\frac{9}{8}t_0n\alpha^2K_4-t_0\alpha \theta \\
&-\frac{t_0n\alpha }{2\delta }\max \{K_2^2,K_3^2\}-t_0(\alpha -1)^{\frac{1}{2}}\gamma^{\frac{4}{3}}-\frac{t_0n\alpha^2}{8(1-\delta \alpha )(\alpha -1)^2}E^2,
\end{split}
\end{equation*}
where
\[
E=(\alpha -1)^{\frac{3}{2}}\gamma^{\frac{2}{3}}+(\alpha -1)\Lambda +\alpha \Sigma +2(K_1+(\alpha -1)K_3+K_4).
\]
For a positive number $a$ and two nonnegative numbers $b,c$, from the inequality $ax^2-bx-c\le 0$ we have
\[
x\le \frac{1}{2a}\left(b+\sqrt{b^2+4ac}\right).
\]
Hence, we obtain
\begin{equation*}
\begin{split}
\frac{F}{t_0}\le &\frac{n\alpha^2}{4(1-\delta \alpha )t_0}\Bigg\{1+t_0\lambda +\bigg[(1+t_0\lambda )^2\\
&+\frac{8(1-\delta \alpha )t_0}{n\alpha^2}\bigg(\frac{n\alpha^2t_0}{8(1-\delta \alpha )(\alpha -1)^2}E^2\\
&+t_0(\alpha -1)^{\frac{1}{2}}\gamma^{\frac{4}{3}}+\frac{9}{8}t_0n\alpha^2K_4+t_0\alpha \theta +\frac{t_0n\alpha }{2\delta }\max \{K_2^2,K_3^2\}\bigg)\bigg]^{\frac{1}{2}}\Bigg\}.
\end{split}
\end{equation*}
Using the inequality $\sqrt{x+y}\le \sqrt{x}+\sqrt{y}$ holds for any $x,y\ge 0$, we obtain
\begin{equation*}
\begin{split}
\frac{F}{t_0}\le &\frac{n\alpha^2}{4(1-\delta \alpha )t_0}\Bigg\{1+t_0\lambda +\bigg[(1+t_0\lambda )^2+\frac{4(1-\delta \alpha )t_0^2}{\delta \alpha }\max \{K_2^2,K_3^2\}\bigg]^{\frac{1}{2}}\Bigg\}\\
&+\frac{n\alpha^2}{4(1-\delta \alpha )t_0}\Bigg\{\frac{8(1-\delta \alpha )t_0^2}{n\alpha^2}\bigg[\frac{n\alpha^2}{8(1-\delta \alpha )(\alpha -1)^2}E^2\\
&+(\alpha -1)^{\frac{1}{2}}\gamma^{\frac{4}{3}}+\frac{9}{8}n\alpha^2K_4+\alpha \theta \bigg]\Bigg\}^{\frac{1}{2}}.
\end{split}
\end{equation*}
Now, by taking $\delta =\frac{t_0\max \{K_2,K_3\}}{1+t_0\lambda +2t_0\max \{K_2,K_3\}}\cdot \frac{1}{\alpha }\in (0,\frac{1}{\alpha })$ for $\max \{K_2,K_3\}\ne 0$. However, from Lemma~\ref{leb} we know that we can choose $\delta =0$ if $K_2=K_3=0$. Therefore, in any case, we can get
\begin{equation*}
\begin{split}
&\frac{n\alpha^2}{4(1-\delta \alpha )t_0}\Bigg\{1+t_0\lambda +\bigg[(1+t_0\lambda )^2+\frac{4(1-\delta \alpha )t_0^2}{\delta \alpha }\max \{K_2^2,K_3^2\}\bigg]^{\frac{1}{2}}\Bigg\}\\
=&\frac{n\alpha^2}{2t_0}+\frac{n\alpha^2}{2}\lambda +n\alpha^2\max \{K_2,K_3\}
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
&\frac{n\alpha^2}{4(1-\delta \alpha )t_0}\left(\frac{8(1-\delta \alpha )t_0^2}{n\alpha^2}\right)^{\frac{1}{2}}=\left(\frac{1+t_0\lambda +2t_0\max \{K_2,K_3\}}{2(1+t_0\lambda +t_0\max \{K_2,K_3\})}\right)^{\frac{1}{2}}\sqrt{n}\alpha \le \sqrt{n}\alpha .
\end{split}
\end{equation*}
Therefore, again according to $\sqrt{x+y}\le \sqrt{x}+\sqrt{y}$, we obtain
\begin{equation*}
\begin{split}
\frac{F}{t_0}\le &\frac{n\alpha^2}{2t_0}+\frac{n\alpha^2}{2}\lambda +n\alpha^2\max \{K_2,K_3\}\\
&+\sqrt{n}\alpha \Bigg\{\frac{\sqrt{n}\alpha }{2\sqrt{2}(\alpha -1)}\sqrt{\frac{1+t_0\lambda +2t_0\max \{K_2,K_3\}}{1+t_0\lambda +t_0\max \{K_2,K_3\}}}E\\
&+(\alpha -1)^{\frac{1}{4}}\gamma^{\frac{2}{3}}+\frac{3}{2\sqrt{2}}\sqrt{n}\alpha \sqrt{K_4}+\sqrt{\alpha \theta }\Bigg\}\\
\le &\frac{n\alpha^2}{2t_0}+\frac{n\alpha^2}{2}\lambda +n\alpha^2\max \{K_2,K_3\}\\
&+\frac{n\alpha^2}{2(\alpha -1)}E+\sqrt{n}\alpha (\alpha -1)^{\frac{1}{4}}\gamma^{\frac{2}{3}}+\frac{3}{4}n\alpha^2\sqrt{2K_4}+\alpha^{\frac{3}{2}}\sqrt{n\theta }.
\end{split}
\end{equation*}
Substituting $E$ into the above inequality yields
\begin{equation*}
\begin{split}
\frac{F}{t_0}\le &\frac{n\alpha^2}{2t_0}+\frac{n\alpha^2}{\alpha -1}(K_1+(\alpha -1)K_3+K_4)\\
&+n\alpha^2\left(\max \{K_2,K_3\}+\frac{3}{4}\sqrt{2K_4}\right)+\alpha^{\frac{3}{2}}\sqrt{n\theta }\\
&+\left(\frac{n\alpha^2}{2}(\alpha -1)^{\frac{1}{2}}+\sqrt{n}\alpha (\alpha -1)^{\frac{1}{4}}\right)\gamma^{\frac{2}{3}}+\frac{n\alpha^2}{2}(\lambda +\Lambda )+\frac{n\alpha^3}{2(\alpha -1)}\Sigma .
\end{split}
\end{equation*}
This implies that $\bar F(x_0,t_0)\le \frac{n\alpha^2}{2}$, in contradiction with our assumption. So \eqref{eqn} holds.
\end{proof}
\begin{rem}
In \cite[Theorem~6]{sun2011Gradient}, The coefficient of $\frac{1}{t}$ in the right hand side of the gradient inequality is $n\alpha^2$. We see Theorem~\ref{thb} extends and improves Sun's estimate.
\end{rem}
\begin{rem}\label{rea}
In Theorem~\ref{tha} if $K_1=K_4=\Sigma_{2R}=0$, we can let $\alpha \to 1$. Similarly, in Corollary~\ref{coa} and Theorem~\ref{thb}, if $K_1=K_4=\Sigma =0$, we can also let $\alpha \to 1$.
\end{rem}
Similar to \cite[Corollary~8]{sun2011Gradient}, integrating the gradient estimate in space-time as in \cite{li1986parabolic} or \cite{guenther2002The}, we can derive the following parabolic Harnack type inequality.
\begin{cor}\label{coc}
Let $(M,g(0))$ be a complete noncompact Riemannian manifold without boundary or a closed Riemannian manifold. Assume that $g(t)$ evolves by \eqref{eqb} for $t\in [0,T]$ and satisfies
\[
\Ric \ge -K_1g,\quad -K_2g\le h\le K_3g,\quad |\nabla h|\le K_4.
\]
If $u$ is a positive solution to \eqref{eqa}, and $q(x,t)$ satisfies
\[
|\nabla q|\le \gamma ,\quad \Delta q\le \theta .
\]
Then for any $(x_1,t_1), (x_2,t_2)$ in $M\times (0,T]$ such that $t_1<t_2$, we have
\begin{equation}\label{eqt}
\begin{split}
u(x_1,t_1)\le u(x_2,t_2)\left(\frac{t_2}{t_1}\right)^{\tau n\alpha }\exp \left(\frac{\alpha Z }{4(t_2-t_1)}+C\frac{t_2-t_1}{\alpha }K\right),
\end{split}
\end{equation}
for any $\alpha >1$, where
\begin{equation*}
\tau =
\begin{cases}
1, & \mbox{if}\ (M,g(0))\ \mbox{is complete noncompact without boundary},\\
\frac{1}{2}, & \mbox{if}\ (M,g(0))\ \mbox{is closed,}
\end{cases}
\end{equation*}
\[K=K_1+K_2+K_3+K_4+\sqrt{K_4}+\gamma +\sqrt{\theta }+\gamma^{\frac{2}{3}}+\lambda +\Lambda +\Sigma ,
\]
$C$ is a constant that depends only on $n,\alpha $, and
\[
Z=\inf_{\zeta }\int_0^1|\zeta'(s)|_{\sigma(s)}^2ds
\]
is the infimum over smooth curves $\zeta $ jointing $x_2$ and $x_1$ ($\zeta(0)=x_2$, $\zeta(1)=x_1$) of the averaged square velocity of $\zeta $ measured at time $\sigma(s)=(1-s)t_2+st_1$.
\end{cor}
\begin{proof}
The gradient estimate in Corollary~\ref{coa} and Theorem~\ref{thb} can both be written as
\[
\frac{|\nabla u(x,t)|^2}{u^2(x,t)}-\alpha \frac{u_t(x,t)}{u(x,t)}\le \frac{\tau n\alpha^2}{t}+C(n,\alpha )K,
\]
for any $\alpha >1$. Take any curve $\zeta $ satisfying the assumption and define
\[
l(s)=\log u(\zeta(s),\sigma(s)).
\]
Then $l(0)=\log u(x_2,t_2)$ and $l(1)=\log u(x_1,t_1)$. A direct computation yields
\begin{equation*}
\begin{split}
\frac{dl(s)}{ds}=&(t_2-t_1)\left(\frac{\nabla u}{u}\frac{\zeta'(s)}{t_2-t_1}-\frac{u_t}{u}\right)\\
\le &\frac{\alpha |\zeta'(s)|_\sigma^2}{4(t_2-t_1)}+\frac{t_2-t_1}{\alpha }\left(\frac{\tau n\alpha^2}{\sigma(s)}+CK\right).
\end{split}
\end{equation*}
Integrating this inequality over $\zeta(s)$, we have
\begin{equation*}
\begin{split}
\log \frac{u(x_1,t_1)}{u(x_2,t_2)}=&\int_0^1\frac{dl(s)}{ds}ds\\
\le &\int_0^1\frac{\alpha |\zeta'(s)|_\sigma^2}{4(t_2-t_1)}+C\frac{t_2-t_1}{\alpha }K+\tau n\alpha \log \frac{t_2}{t_1},
\end{split}
\end{equation*}
which implies the corollary.
\end{proof}
\section{Elliptic type gradient estimates for bounded positive solutions}\label{sec3}
Now we establish elliptic type gradient estimates for \eqref{eqa}--\eqref{eqb}. Firstly we give the local version.
\begin{thm}\label{thc}
Let $(M,g(t))$ be a complete solution to \eqref{eqb} for $t\in [0,T]$ and let $u$ be a positive solution to \eqref{eqa}. Suppose that there exist constants $L>0$ and $K\ge 0$, such that $u\le L$ and
\[
\Ric \ge -K_1g,\quad h\ge -K_2g
\]
on $Q_{2R,T}$. Then we have
\begin{equation}\label{eqp}
\begin{split}
\frac{|\nabla u(x,t)|}{u(x,t)}\le \tilde C\left(\frac{1}{\sqrt{t}}+\frac{1}{R}+\sqrt{H}\right)\left(1+\log \frac{L}{u(x,t)}\right)
\end{split}
\end{equation}
on $Q_{R,T}$, where $\tilde C$ is a constant that depends only on $n$ and
\[
H=K_1+K_2+\max_{Q_{2R,T}}(|q|-q)+\max_{Q_{2R,T}}\frac{|\nabla q|}{\sqrt{|q|}}+\kappa_{2R}.
\]
\end{thm}
\begin{rem}
In \cite[Theorem~1.9]{chen2018Li}, the authors gave a elliptic type gradient estimate for bounded positive solutions of \eqref{eqa} with a convection term on a complete manifold, where the metric does not depend on time. In the estimate of \cite[Theorem~1.9]{chen2018Li}, the upper bound induced by the term $A(u)$ is
\[
-\min \left\{0,\min_{Q_{2R,T}}(A'(u)-A(u)/u)\right\}-\min \left\{0,\min_{Q_{2R,T}}(A(u)/u)\right\}
\]
instead of $\kappa_{2R}$ here. Compare with \cite[Theorem~1.9]{chen2018Li}, we see that our estimate \eqref{eqp} is sharper. In fact, in general, for real numbers $x,y$, a direct calculation yields
\[
\min \{0,x\}+\min \{0,y\}\le \min \{0,x+y\}.
\]
Similarly, for two functions $f,g$ on the same domain $D$, the following obvious fact holds:
\[
\min_Df+\min_Dg\le \min_D(f+g).
\]
By the above two inequalities we obtain
\begin{equation*}
\begin{split}
&\min \left\{0,\min_{Q_{2R,T}}(A'(u)-A(u)/u)\right\}+\min \left\{0,\min_{Q_{2R,T}}(A(u)/u)\right\}\\
\le &\min \left\{0,\min_{Q_{2R,T}}\left(A'(u)-A(u)/u\right)+\min_{Q_{2R,T}}\left(A(u)/u\right)\right\}\\
\le &\min \left\{0,\min_{Q_{2R,T}}A'(u)\right\}.
\end{split}
\end{equation*}
On the other hand, it is obvious that
\begin{equation*}
\begin{split}
&\min \left\{0,\min_{Q_{2R,T}}(A'(u)-A(u)/u)\right\}+\min \left\{0,\min_{Q_{2R,T}}(A(u)/u)\right\}\\
\le &\min \left\{0,\min_{Q_{2R,T}}(A'(u)-A(u)/u)\right\}.
\end{split}
\end{equation*}
In conclusion,
\begin{equation*}
\begin{split}
&\min \left\{0,\min_{Q_{2R,T}}(A'(u)-A(u)/u)\right\}+\min \left\{0,\min_{Q_{2R,T}}(A(u)/u)\right\}\\
\le &\min \left\{0,\min_{Q_{2R,T}}(A'(u)-A(u)/u),\min_{Q_{2R,T}}A'(u)\right\}=-\kappa_{2R}.
\end{split}
\end{equation*}
That is,
\[
\kappa_{2R}\le -\min \left\{0,\min_{Q_{2R,T}}(A'(u)-A(u)/u)\right\}-\min \left\{0,\min_{Q_{2R,T}}(A(u)/u)\right\}.
\]
And we will see in the proof of Theorem~\ref{thc} that this sharper estimate comes from a more careful treatment of the term $\hat A_f+\frac{\hat A(f)}{1-f}$. However, the treatment we give here is not necessarily optimal. It is possible that a sharper estimate will be applied to more equations.
\end{rem}
Now we are ready to prove Theorem~\ref{thc}. Noticing that if $0<u\le L$ is a solution to \eqref{eqa}, then $\tilde u=\frac{u}{L}$ is a solution to the equation
\[
(\Delta -q(x,t)-\partial_t)u(x,t)=\frac{1}{L}A(L\tilde u(x,t))
\]
and $0<\tilde u\le 1$. Hence, we can assume that $0<u\le 1$ in the proof of Theorem~\ref{thc}. Similar to the proof of Theorem~\ref{tha}, we need a auxiliary lemma. We still set $f=\log u\le 0$ and $\hat A(f)=\frac{A(u)}{u}$. In this case, we define $w=|\nabla \log(1-f)|^2$ and $F(x,t)=tw(x,t)$.
\begin{lem}\label{lec}
Let $(M,g(t))$ be a complete solution to \eqref{eqb} for $t\in [0,T]$ and let $u\in (0,1]$ be a solution to \eqref{eqa}. Suppose that there exists a constant $K\ge 0$, such that
\[
\Ric +h\ge -Kg
\]
on $Q_{2R,T}$. Then we have
\begin{equation}\label{eqq}
\begin{split}
(\Delta -\partial_t)F\ge &\frac{2(1-f)}{t}F^2-2\left(K-\hat A_f-\frac{q}{1-f}-\frac{\hat A(f)}{1-f}\right)F\\
&-\frac{F}{t}-2f\langle \nabla \log(1-f),\nabla F\rangle -\frac{2t}{1-f}\langle \nabla \log(1-f),\nabla q\rangle .
\end{split}
\end{equation}
on $Q_{2R,T}$.
\end{lem}
\begin{proof}
By the Bochner formula we have
\begin{equation*}
\begin{split}
\Delta F=&2t|\Hess \log(1-f)|^2+2t\Ric (\nabla \log(1-f),\nabla \log(1-f))\\
&+2t\langle \nabla \log(1-f),\nabla \Delta \log(1-f)\rangle .
\end{split}
\end{equation*}
However, by \eqref{eqc},
\begin{equation*}
\begin{split}
\Delta \log(1-f)=&-\frac{\Delta f}{1-f}-\frac{|\nabla f|^2}{(1-f)^2}=-fw-\frac{f_t+q+\hat A}{1-f}.
\end{split}
\end{equation*}
Therefore, we obtain
\begin{equation*}
\begin{split}
\Delta F=&2t|\Hess \log(1-f)|^2+2t\Ric (\nabla \log(1-f),\nabla \log(1-f))\\
&+\frac{2(1-f)}{t}F^2-2f\langle \nabla \log(1-f),\nabla F\rangle \\
&+\frac{2}{1-f}(f_t+q+\hat A)F-\frac{2t}{1-f}\langle \nabla \log(1-f),\nabla (f_t)\rangle \\
&-\frac{2t}{1-f}\langle \nabla \log(1-f),\nabla q\rangle -\frac{2t\hat A_f}{1-f}\langle \nabla \log(1-f),\nabla f\rangle .
\end{split}
\end{equation*}
On the other hand, by the first equality of Lemma~\ref{lea},
\begin{equation*}
\begin{split}
F_t=&-2th(\nabla \log(1-f),\nabla \log(1-f))\\
&+2t\langle \nabla \log(1-f),\nabla ((\log(1-f))_t)\rangle +\frac{F}{t}\\
=&-2th(\nabla \log(1-f),\nabla \log(1-f))\\
&-\frac{2t}{1-f}\langle \nabla \log(1-f),\nabla (f_t)\rangle +\frac{2f_tF}{1-f}+\frac{F}{t}.
\end{split}
\end{equation*}
Combining the above two equalities, we get
\begin{equation*}
\begin{split}
(\Delta -\partial_t)F\ge &\frac{2(1-f)}{t}F^2+2t(\Ric +h)(\nabla \log(1-f),\nabla \log(1-f))\\
&+2\hat A_fF+\frac{2}{1-f}(q+\hat A)F-2f\langle \nabla \log(1-f),\nabla F\rangle \\
&-\frac{2t}{1-f}\langle \nabla \log(1-f),\nabla q\rangle -\frac{F}{t}.
\end{split}
\end{equation*}
The lemma follows from the assumption on bound of $\Ric +h$.
\end{proof}
\begin{rem}
It is easy to see that we don't need any assumption on the Ricci tensor if geometric flow \eqref{eqb} is the Ricci flow, i.e., $h=-\Ric$.
\end{rem}
\begin{proof}[The proof of Theorem~\ref{thc}]
Choosing $\phi $ and $\eta $ as in the proof of Theorem~\ref{tha}. For any $T_1\in (0,T]$, let $(x_1,t_1)\in Q_{2R,T_1}$, at which $G(x,t)=\eta(x,t)F(x,t)$ attains its maximum, and without loss of generality, we can assume $G(x_1,t_1)>0$, and then $\eta(x_1,t_1)>0$ and $F(x_1,t_1)>0$. By Lemma~\ref{lec} and a similar argument as in the proof of Theorem~\ref{tha}, we have at $(x_1,t_1)$,
\begin{equation*}
\begin{split}
0\ge &\frac{2\eta (1-f)}{t_1}F^2-2\eta \left(K_1+K_2-\hat A_f-\frac{q}{1-f}-\frac{\hat A(f)}{1-f}\right)F\\
&-2\eta f\langle \nabla \log(1-f),\nabla F\rangle -\frac{2t_1\eta }{1-f}\langle \nabla \log(1-f),\nabla q\rangle \\
&-\frac{\eta F}{t_1}-\left(\frac{C_3(n)}{R^2}+\frac{C_4(n)}{R}\sqrt{K_1}+C_2K_2\right)F\\
=&\frac{2\eta (1-f)}{t_1}F^2-2\eta \left(K_1+K_2-\hat A_f-\frac{q}{1-f}-\frac{\hat A(f)}{1-f}\right)F\\
&+2f\langle \nabla \log(1-f),\nabla \eta \rangle F-\frac{2t_1\eta }{1-f}\langle \nabla \log(1-f),\nabla q\rangle \\
&-\frac{\eta F}{t_1}-\left(\frac{C_3(n)}{R^2}+\frac{C_4(n)}{R}\sqrt{K_1}+C_2K_2\right)F.
\end{split}
\end{equation*}
Multiplying both sides of the above inequality by $t_1\eta $, we have
\begin{equation*}
\begin{split}
0\ge &2(1-f)G^2-2t_1\eta \left(K_1+K_2-\hat A_f-\frac{q}{1-f}-\frac{\hat A(f)}{1-f}\right)G\\
&+2t_1f\langle \nabla \log(1-f),\nabla \eta \rangle G-\frac{2t_1^2\eta^2}{1-f}\langle \nabla \log(1-f),\nabla q\rangle \\
&-\eta G-t_1\left(\frac{C_3(n)}{R^2}+\frac{C_4(n)}{R}\sqrt{K_1}+C_2K_2\right)G.
\end{split}
\end{equation*}
Noticing that $f\le 0$, by Young's inequality,
\begin{equation*}
\begin{split}
-2t_1f\langle \nabla \log(1-f),\nabla \eta \rangle G\le &-2ft_1^{\frac{1}{2}}\frac{|\nabla \eta |}{\eta^{\frac{1}{2}}}G^{\frac{3}{2}}\le -2ft_1\frac{\sqrt{C_1}}{R}G^{\frac{3}{2}}\\
\le &(1-f)G^2+\frac{27t_1^2}{16}\frac{C_1^2}{R^4}\frac{f^4}{(1-f)^3}
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\langle \nabla \log(1-f),\nabla q\rangle \le |\nabla q|w^{\frac{1}{2}}\le |q|w+\frac{|\nabla q|^2}{4|q|}.
\end{split}
\end{equation*}
Combining the above three inequalities we have
\begin{equation*}
\begin{split}
0\ge &(1-f)G^2-2t_1\eta \left(K_1+K_2-\hat A_f-\frac{q}{1-f}-\frac{\hat A(f)}{1-f}\right)G\\
&-\frac{27t_1^2}{16}\frac{C_1^2}{R^4}\frac{f^4}{(1-f)^3}-\frac{2t_1\eta}{1-f}|q|G-\frac{t_1^2\eta2}{2(1-f)}\frac{|\nabla q|^2}{|q|}\\
&-\eta G-t_1\left(\frac{C_3(n)}{R^2}+\frac{C_4(n)}{R}\sqrt{K_1}+C_2K_2\right)G.
\end{split}
\end{equation*}
From $0\le \frac{1}{1-f}<1$ and $0<\frac{-f}{1-f}\le 1$, we see that
\begin{equation*}
\begin{split}
\hat A_f+\frac{\hat A(f)}{1-f}=&\frac{-f}{1-f}\hat A_f+\frac{1}{1-f}\left(\hat A_f+\hat A(f)\right)\\
\ge &\frac{-f}{1-f}\min \left\{0,\min_{Q_{2R,T}}\hat A_f\right\}+\frac{1}{1-f}\min \left\{0,\min_{Q_{2R,T}}A'(u)\right\}\\
\ge &\min \left\{0,\min_{Q_{2R,T}}\hat A_f,\min_{Q_{2R,T}}A'(u)\right\}\\
=&-\kappa_{2R}.
\end{split}
\end{equation*}
By $0<\eta(x_1,t_1)\le 1$, we get
\begin{equation*}
\begin{split}
0\ge &(1-f)G^2-G-2t_1\left(K_1+K_2+\max_{Q_{2R,T}}(|q|-q)+\kappa_{2R}\right)G\\
&-t_1\left(\frac{C_3(n)}{R^2}+\frac{C_4(n)}{R}\sqrt{K_1}+C_2K_2\right)G-\frac{27t_1^2}{16}\frac{C_1^2}{R^4}\frac{f^4}{(1-f)^3}-\frac{t_1^2}{2}\max_{Q_{2R,T}}\frac{|\nabla q|^2}{|q|}.
\end{split}
\end{equation*}
Applying the quadratic formula and the inequality of arithmetic and geometric means
\[
\frac{\sqrt{K_1}}{R}\le \frac{1}{2R^2}+\frac{K_1}{2},
\]
and noticing the fact $0\le \frac{-f}{1-f}<1$ again, we obtain
\begin{equation*}
\begin{split}
G\le &C_6(n)\left(1+t_1\left(\frac{1}{R^2}+H\right)\right),
\end{split}
\end{equation*}
where $C_6(n)$ is a constant that depends only on $n$ and
\[
H=K_1+K_2+\max_{Q_{2R,T}}(|q|-q)+\max_{Q_{2R,T}}\frac{|\nabla q|}{\sqrt{|q|}}+\kappa_{2R}.
\]
Noticing that $d(x,x_0,T_1)\le R$ implies $\eta(x,T_1)=1$, we can get
\begin{equation*}
\begin{split}
w(x,T_1)=\frac{F(x,T_1)}{T_1}\le \frac{G(x_1,t_1)}{T_1}\le C_6(n)\left(\frac{1}{T_1}+\frac{1}{R^2}+H\right).
\end{split}
\end{equation*}
Since $T_1$ is arbitrary, and using $\sqrt{x+y}\le \sqrt{x}+\sqrt{y}$, we complete the proof.
\end{proof}
Similar to Corollary~\ref{coa}, when $(M,g(0))$ is a complete noncompact Riemannian manifold without boundary and $g(t)$ evolves by \eqref{eqb}, we can obtain a global estimate from Theorem~\ref{thc} by taking $R\to 0$.
\begin{cor}\label{cob}
Let $(M,g(t))$ be a complete solution to \eqref{eqb} for $t\in [0,T]$ and $(M,g(0))$ be a complete noncompact Riemannian manifold without boundary. Let $u$ be a positive solution to \eqref{eqa}. Suppose that there exist constants $L>0$ and $K\ge 0$, such that $u\le L$ and
\[
\Ric \ge -K_1g,\quad h\ge -K_2g.
\]
Then we have
\begin{equation}\label{eqp}
\begin{split}
\frac{|\nabla u(x,t)|}{u(x,t)}\le \tilde C\left(\frac{1}{\sqrt{t}}+\sqrt{H}\right)\left(1+\log \frac{L}{u(x,t)}\right)
\end{split}
\end{equation}
on $M\times [0,T]$, where $\tilde C$ as in Theorem~\ref{thc} and
\[
H=K_1+K_2+\sup_{M\times [0,T]}(|q|-q)+\sup_{M\times [0,T]}\frac{|\nabla q|}{\sqrt{|q|}}+\kappa .
\]
\end{cor}
The following corollary gives a elliptic Harnack inequality by integrating the elliptic type gradient estimate \eqref{eqp} in space only. Unlike Corollary~\ref{coc}, this inequality can compare the function values at two spatial points at the same time, but inequality \eqref{eqt} cannot.
\begin{cor}\label{cod}
Let $(M,g(t))$ be a complete solution to \eqref{eqb} for $t\in [0,T]$ and $(M,g(0))$ be a complete noncompact Riemannian manifold without boundary. Let $u$ be a positive solution to \eqref{eqa}. Suppose that there exist constants $L>0$ and $K\ge 0$, such that $u\le L$ and
\[
\Ric \ge -K_1g,\quad h\ge -K_2g.
\]
Then for any $x_1,x_2\in M$, we have
\begin{equation}\label{equ}
\begin{split}
u(x_1,t)\ge u(x_2,t)^{\mathcal H}\exp ((1-\mathcal H)(1+\log L)).
\end{split}
\end{equation}
in each $t\in [0,T]$. Here, $\mathcal H=\exp \left(\tilde C\left(\frac{1}{\sqrt{t}}+\sqrt{H}\right)d(x_1,x_2,t)\right)$,
where $\tilde C$ as in Theorem~\ref{thc} and
\[
H=K_1+K_2+\sup_{M\times [0,T]}(|q|-q)+\sup_{M\times [0,T]}\frac{|\nabla q|}{\sqrt{|q|}}+\kappa .
\]
\end{cor}
\begin{proof}
For any fixed $t$ and any $x_1,x_2\in M$, let $\zeta :[0,1]\to M$ is the geodesic of minimal length, which connecting $x_2$ and $x_1$, $\zeta(0)=x_2$ and $\zeta(1)=x_1$. Let $f=\log u$ and
\[
l(s)=\log(1+\log L-f(\zeta(s),t)).
\]
By Corollary~\ref{cob} we have
\begin{equation*}
\begin{split}
\frac{dl(s)}{ds}=&\frac{-\langle (\nabla f)(\zeta(s),t),\zeta'(s)\rangle }{1+\log L-f(\zeta(s),t)}\\
\le &|\zeta'(s)|\cdot \frac{|\nabla f|(\zeta(s),t)}{1+\log L-f(\zeta(s),t)}\\
\le &\tilde C|\zeta'(s)|\left(\frac{1}{\sqrt{t}}+\sqrt{H}\right).
\end{split}
\end{equation*}
Integrating this inequality over $\zeta(s)$, we have
\begin{equation*}
\begin{split}
\log \frac{1+\log L-f(x_1,t)}{1+\log L-f(x_2,t)}\le \tilde C\left(\frac{1}{\sqrt{t}}+\sqrt{H}\right)d(x_1,x_2,t).
\end{split}
\end{equation*}
From this inequality, inequality \eqref{equ} can be obtained through a simple calculation.
\end{proof}
\section{Applications}\label{sec4}
We will give some applications of gradient estimates in section~2 and section~3 to some special equations. In some cases, we also take the geometric flow as the Ricci flow, i.e., $h=-Ric $ in \eqref{eqb}.
\subsection{Applications of space-time gradient estimates}
In this subsection, we focus on applications of space-time gradient estimate for positive solutions. In this case, we see that $\lambda ,\Lambda ,\Sigma $ are not necessarily finite in \eqref{eqm}. For example, if we choose $A(u)=u^p$ for $p>0$, then $\lambda ,\Lambda ,\Sigma $ are all multiples of $\sup_{M\times [0,T]}u^{p-1}$ and they are not all zero if $p\ne 1$. But $\sup_{M\times [0,T]}u^{p-1}$ is not necessarily finite, unless $u$ is bounded. For the case that $q=0$ and $A(u)=u^p$, the reader can also refer to \cite{li2016Harnack,li2018li,zhao2016Gradient}.
Naturally, for general unbounded positive function $u$, we want to know when $\lambda ,\Lambda $ and $\Sigma $ are finite. Since $u$ is a positive solution to \eqref{eqa}, $A(u)$ can be written as $u\cdot \frac{A(u)}{u}$, which is $A(u)=uH(u)$ for some $C^2$ function $H(u)$. As pointed by Q. Chen and the author in \cite[Remark~1.8]{chen2018Li}, $\lambda ,\Lambda ,\Sigma <+\infty $ implies
\[
|H(u)|\le C_0\log u,\quad \mbox{as}\ u\to +\infty.
\]
When $A(u)=au\log u$, A direct computation yields $\Sigma =0$ and
\begin{equation*}
\begin{cases}
\lambda_{2R}=\lambda \equiv 0,\quad \Lambda_{2R}=\Lambda \equiv a,\quad &\mbox{if}\ a\ge 0,\\
\lambda_{2R}=\lambda \equiv -a ,\quad \Lambda_{2R}=\Lambda \equiv 0,\quad &\mbox{if}\ a\le 0.
\end{cases}
\end{equation*}
Therefore, we can obtain that local and global gradient estimates for positive solutions of the equation
\begin{equation}\label{eqr}
\begin{split}
(\Delta -q(x,t)-\partial_t)u(x,t)=au(x,t)\log (u(x,t)),\quad a\in \mathbb R
\end{split}
\end{equation}
from Theorem~\ref{tha} and Corollary~\ref{coa}.
\begin{rem}
By the asymptotic behavior of $H$, we can find many examples that satisfy $\lambda ,\Lambda ,\Sigma <+\infty $. Such as $H(u)=\frac{P_k(u)}{P_l(u)}$, where $P_k, P_l$ are polynomials of degree $k,l$, respectively and $k<l$. Hence we can obtain gradient estimates for positive solution of the following series of equations
\[
(\Delta -q(x,t)-\partial_t)u(x,t)=au(x,t)\frac{P_k(u(x,t))}{P_l(u(x,t))},\quad a\in \mathbb R,\ k<l.
\]
For instance, if we choose $k=0, l=1$, so $A(u)=\frac{u}{u+1}$, then $\lambda =\frac{1}{4}, \Lambda =0$ and $\Sigma =\frac{8}{27}$.
\end{rem}
On the other hand, as mentioned in Remark~\ref{rea}, if $A(u)=au\log u$, then $\Sigma =0$. In addition, we take $h=-\Ric $. In this case, we don't need the assumption on the bound $|\nabla h|$ since the contracted second Bianchi identity. At this time, when $\alpha \to 1$, we can also let $\varepsilon \to 0$ in local estimate \eqref{eql}, and then we are arriving at
\begin{cor}
Let $(M,g(0))$ be a complete Riemannian manifold, and let $g(t)$ evolves by the Ricci flow for $t\in [0,T]$. Suppose that there exist constants $K_2, \theta \ge 0$ such that
\[
0\le \Ric \le K_2g
\]
and
\[
\Delta q\le \theta_{2R}
\]
on $Q_{2R,T}$. If $u$ is a positive solution to \eqref{eqr}. Then on $Q_{R,T}$, we have
\begin{enumerate}
\item for $a\ge 0$,
\begin{equation*}
\begin{split}
\frac{|\nabla u(x,t)|^2}{u^2(x,t)}-\frac{u_t(x,t)}{u(x,t)}-q(x,t)-\frac{A(u(x,t))}{u(x,t)}
\le \frac{n}{t}+(C+n)K_2+\sqrt{n\theta }+\frac{n}{2}a;
\end{split}
\end{equation*}
\item for $a<0$,
\begin{equation*}
\begin{split}
\frac{|\nabla u(x,t)|^2}{u^2(x,t)}-\frac{u_t(x,t)}{u(x,t)}-q(x,t)-\frac{A(u(x,t))}{u(x,t)}
\le \frac{n}{t}+(C+n)K_2+\sqrt{n\theta }-na,
\end{split}
\end{equation*}
\end{enumerate}
where $C$ as in Theorem~\ref{tha}.
\end{cor}
From the above local estimate, we have immediately
\begin{cor}
Let $(M,g(0))$ be a complete noncompact Riemannian manifold without boundary, and let $g(t)$ evolves by the Ricci flow for $t\in [0,T]$. Suppose that there exist constants $K_2, \theta \ge 0$ such that
\[
0\le \Ric \le K_2g
\]
and
\[
\Delta q\le \theta .
\]
If $u$ is a positive solution to \eqref{eqr}. Then we have
\begin{equation*}
\begin{split}
\frac{|\nabla u(x,t)|^2}{u^2(x,t)}-\frac{u_t(x,t)}{u(x,t)}-q(x,t)-\frac{A(u(x,t))}{u(x,t)}
\le \frac{n}{t}+C''\left(K_2+\sqrt{\theta }+|a|\right)
\end{split}
\end{equation*}
on $M\times [0,T]$, where $C''$ is a constant that depends only on $n$.
\end{cor}
When the manifold is closed, we also have
\begin{cor}
Let $(M,g(t))$ be a closed Riemannian manifold, where $g(t)$ evolves by \eqref{eqb} for $t\in [0,T]$ and satisfies
\[
0\le \Ric \le K_2g.
\]
If $u$ is a positive solution to the equation
\[
(\Delta -q(x,t)-\partial_t)u(x,t)=au(x,t)\log (u(x,t)),
\]
and $q(x,t)$ satisfies
\[
\Delta q\le \theta .
\]
Then we have
\begin{equation*}
\begin{split}
\frac{|\nabla u(x,t)|^2}{u^2(x,t)}-\frac{u_t(x,t)}{u(x,t)}-q(x,t)-\frac{A(u(x,t))}{u(x,t)}\le \frac{n}{2t}+nK_2+\sqrt{n\theta }+\frac{n}{2}|a|
\end{split}
\end{equation*}
on $M\times (0,T]$.
\end{cor}
\subsection{Applications of elliptic type gradient estimates}
Now we give some applications of elliptic type gradient estimates for bounded positive solutions. Since we are dealing with bounded positive solutions, $A(u)$ that satisfies the conditions $\kappa <+\infty $ is easy to find.
We will consider that elliptic type gradient estimates for bounded positive solutions of the equation
\begin{equation}\label{eqs}
\begin{split}
(\Delta -q(x,t)-\partial_t)u(x,t)=au(x,t)^\beta ,\quad a\in \mathbb R,\ \beta \in (-\infty ,0]\cup [1,+\infty).
\end{split}
\end{equation}
In order not to be redundant, we only give the global estimate here, and the local one is omitted.
\begin{cor}
Let $(M,g(t))$ be a complete solution to \eqref{eqb} for $t\in [0,T]$ and $(M,g(0))$ be a complete noncompact Riemannian manifold without boundary. Let $u$ be a positive solution to \eqref{eqs}. Suppose that there exist constants $L>0$ and $K\ge 0$, such that $u\le L$ and
\[
\Ric \ge -K_1g,\quad h\ge -K_2g.
\]
Then on $M\times [0,T]$, we have
\begin{equation*}
\begin{split}
\frac{|\nabla u(x,t)|}{u(x,t)}\le \tilde C\left(\frac{1}{\sqrt{t}}+\sqrt{H'}\right)\left(1+\log \frac{L}{u(x,t)}\right),
\end{split}
\end{equation*}
where $\tilde C$ as in Theorem~\ref{thc} and
\[
H'=K_1+K_2+\sup_{M\times [0,T]}(|q|-q)+\sup_{M\times [0,T]}\frac{|\nabla q|}{\sqrt{|q|}}+\kappa_0
\]
with
\begin{equation*}
\kappa_0=
\begin{cases}
\frac{\sign a-1}{2}a\beta L^{\beta -1},\ &\mbox{if}\ a\in \mathbb R,\ \beta \ge 1,\\
0,\ &\mbox{if}\ a\le 0,\ \beta \le 0,\\
a(1-\beta )\left(\inf_{M\times [0,T]}u(x,t)\right)^{\beta -1} ,\ &\mbox{if}\ a\ge 0,\ \beta \le 0.
\end{cases}
\end{equation*}
Here, $\sign a$ is the sign function, which is $1, 0, -1$ if $a>0, =0, <0$, respectively.
\end{cor}
\begin{proof}
From Corollary~\ref{cob}, we just have to compute $\kappa $. By the definition, we have
\begin{equation*}
\begin{split}
\kappa =&-\min \left\{0,\inf_{M\times [0,T]}(a\beta u^{\beta -1}),\inf_{M\times [0,T]}(a(\beta -1)u^{\beta -1})\right\}\\
=&
\begin{cases}
0,\ &\mbox{if}\ a\ge 0,\ \beta \ge 1,\\
-a\beta L^{\beta -1},\ &\mbox{if}\ a\le 0,\ \beta \ge 1,\\
0,\ &\mbox{if}\ a\le 0,\ \beta \le 0,\\
a(1-\beta )\left(\inf_{M\times [0,T]}u(x,t)\right)^{\beta -1} ,\ &\mbox{if}\ a\ge 0,\ \beta \le 0.
\end{cases}
\end{split}
\end{equation*}
Therefore, we obtain the corollary.
\end{proof}
In particular, when $q(x,t)=const.$, the term $qu(x,t)$ can be combined by $au(x,t)^\beta $, so we get
\begin{cor}
Let $(M,g(t))$ be a complete solution to \eqref{eqb} for $t\in [0,T]$ and $(M,g(0))$ be a complete noncompact Riemannian manifold without boundary. Let $u$ be a positive solution to
\[
(\Delta -\partial_t)u(x,t)=au(x,t)^\beta ,\quad a\in \mathbb R,\ \beta \in (-\infty ,0]\cup [1,+\infty).
\]
Suppose that there exist constants $L>0$ and $K\ge 0$, such that $u\le L$ and
\[
\Ric \ge -K_1g,\quad h\ge -K_2g.
\]
Then on $M\times [0,T]$, we have
\begin{equation*}
\begin{split}
\frac{|\nabla u(x,t)|}{u(x,t)}\le \tilde C\left(\frac{1}{\sqrt{t}}+\sqrt{K_1+K_2+\kappa_0}\right)\left(1+\log \frac{L}{u(x,t)}\right),
\end{split}
\end{equation*}
where $\tilde C$ as in Theorem~\ref{thc} and
\begin{equation*}
\kappa_0=
\begin{cases}
\frac{\sign a-1}{2}a\beta L^{\beta -1},\ &\mbox{if}\ a\in \mathbb R,\ \beta \ge 1,\\
0,\ &\mbox{if}\ a\le 0,\ \beta \le 0,\\
a(1-\beta )\left(\inf_{M\times [0,T]}u(x,t)\right)^{\beta -1} ,\ &\mbox{if}\ a\ge 0,\ \beta \le 0.
\end{cases}
\end{equation*}
Here, $\sign a$ is the sign function, which is $1, 0, -1$ if $a>0, =0, <0$, respectively.
\end{cor}
\begin{rem}
For each of these specific equations that appear in this section, we also have the corresponding Harnack inequality, which we will not write them all down here.
\end{rem}
\bibliographystyle{abbrv}
|
{
"timestamp": "2019-04-22T02:13:42",
"yymm": "1902",
"arxiv_id": "1902.11013",
"language": "en",
"url": "https://arxiv.org/abs/1902.11013"
}
|
\section{Introduction}
\begin{figure*}[t]
\centering
\includegraphics[width = 0.92\linewidth]{ppt/overview.pdf}
\caption{\small
Framework using triplet loss to train face recognition model: total $P*K$ images are sampled for $P$ persons with $K$ images each person. The sampled images are mapped into feature vectors through deep convolutional network. The indexs of triplet pairs are computed by a hard example mining process based on the feature vectors and the responding triplet feature pairs can be gathered. Finally, the features of triplet pairs are inputed into triplet loss to train the CNN.
}
\label{fig:method_show}
\end{figure*}
Face recognition has achieved significant improvement due to the power of deep representation through convolutional neural network. Convolutional neural network(CNN) based method first encodes the image which contains face into deep presentation and then apply the loss function to train the CNN such that the distance of feature vectors of the same persons is smaller than that of the different persons. Amost all of these loss functions can be divided into two categeries: 1) softmax classification based loss and its variants such as Sphere face~\cite{liu2017sphereface}, Arcface~\cite{arcface} and Cosface~\cite{wang2018cosface}; 2) metric learning based loss such as contrastive loss~\cite{sun2014deep} and Triplet loss~\cite{schroff2015facenet}. The previous one recently draw more attentions and has achieved great progress with the development of angular softmax loss and larger margin softmax loss to enhance the discriminative power of softmax loss. However, the last one is hard to train and heavily depends on people's experiences of hard example mining due to high computional complexity such as $O(N^3)$ of triplet loss for a dataset with $N$ samples.
In this work, we present an efficient implentation of triplet loss on face recognition task and conduct serveral experiments to analyze the factors that influence the training of triplet loss. The overview of our implementation can be seen in figure~\ref{fig:method_show}. Unlike the softmax loss, data sample of triplet loss need ensure that the valid triplet pair can be constructed as much as possible. In order to achieve this, we follow ~\cite{hermans2017defense} and the total $P*K$ images are sampled for $P$ persons with $K$ images each. The sampled images are mapped into feature vector $F^p_k$ through deep convolutional network such as Resnet~\cite{he2016deep} or Mobilenet~\cite{howard2017mobilenets}. Then the triplet pairs are selected by a hard mining process based on the embedded feature vectors. For convenience of implementation, we first get the indexs of triplet pairs and then gather their responding features. Finally, the gathered features of triplet pairs are fed into triplet loss to train the CNN.
The contributions of our paper can be summarized as: 1) we present an efficient implementation of our proposed framework which provides variant choices for the component of our proposed framewrok such as different CNN feature extractor, different triplet pair selection method as in figure~\ref{fig:method_show} and also support multi-gpus to accelerate training process. 2) we practically analyze a series of factors that influence training of triplet loss by experiments.
\section{Related Work}
\textbf{FaceNet: A Unified Embedding for Face Recognition and Clustering}~\cite{schroff2015facenet}. FaceNet uses traiplet loss to train the CNN model and mines semi-hard examples to train the triplet loss. It utilized on a dataset with 8M identities and trained on a cluster of cpu for thousands of hours. FaceNet achieves remarkable result on LFW~\cite{LFWTech} while it's not practical because it relys on such large dataset and need a large mount of time to train
\textbf{In Defense of the Triplet Loss for Person Re-Identification}~\cite{hermans2017defense}. This work applys triplet loss on person re-identification problem. It constructs image batch efficiently by sampling $P$ persons with $K$ images each. It also proposed batch hard and batch all hard example mining strategies. This work achieves the start-of-the-art in person re-identification.
\section{Framework}
The overview of our proposed framework can be viewed as figrue~\ref{fig:method_show}. The framework include the following module: data sample, feature embedding, triplet selection and triplet loss. We will describe them in details as follow.
\textbf{Data Sample}. Following ~\cite{hermans2017defense}, a batch of input images consists of $P$ persons and each person includes $K$ images. So in each iteration, we sample total $P*K$ images. Using such data sampling method, it's convenient to select valid triplet pairs and mine hard examples.
\textbf{Feature Extraction}. We use MobileFacenets~\cite{chen2018mobilefacenets} to extract the feature $F$ of the input image $x$ as a deep representation. We also fix the feature $||F_i||=1$ by L2-normalization. It only has about 6.0MB parameters and can be infered very fast as well.
\textbf{Triplet Loss}
Through the feature extraction powered by CNN, the input image $x$ can be mapped into a feature vector with $d$ dimension, and the map function is denoted as $f(x)$. The goal of the triplet loss is to make sure that the feature vector $f^a_i(x)$ of image $x^a_i$(called $anchor$) is close to $f^p_i(x)$ of image $x^p_i$(called $positive$) which has the same identity as image $x^a_i$ while $f^a_i(x)$ is far away from $f^n_i(x)$ of the image $x^n_i(x)$(called $negative$) that has the different identity as $x^a_i$. We can formulate this loss as:
\begin{equation}
\|f^a_i(x) -f^p_i(x)\|^2_2 + \alpha < \|f^a_i(x) -f^n_i(x)\|^2_2
\end{equation}
where $\alpha$ is the margin to avoid the collapse of $f_i(x)$. And the loss that will be optimized can become $L$
\begin{equation}
L=max(0,\|f^a_i(x) -f^p_i(x)\|^2_2 + \alpha - \|f^a_i(x) -f^n_i(x)\|^2_2)
\label{opt_loss}
\end{equation}
\textbf{Triplet Selection}
Triplet selection aims to choice the valid triplet $(i,j,k)$ which is used as input of triplet loss. The valid triplet means that $i$, $j$ have the identity and $i$, $k$ have different identity. As see in \textbf{Data sample}, the input is composed of $P$ persons with $K$ images each and total $B=P*K$ images. In order to obtain the all possible valid triplets, we iterate each image $x^k_a$ as anchor in person $k$ and any other image $x^k_p$ in person $k$ can be positive. Thus the negetive can be from all images of other persons except k. We summary this as algothrim ~\ref{all_triplets}. So there are about $O(P*K*P*K)$ valid triplet pairs but not every triplet pair can contribute to the triplet loss. As we see in formula \ref{opt_loss}, only the triplet pair that satisfies $\|f^a_i(x) -f^p_i(x)\|^2_2 + \alpha - \|f^a_i(x) -f^n_i(x)\|^2_2 > 0$ has loss value. We develop algorithm to mine such triplet pairs. We also develop different kinds of strategies to mine the 'hard' examples based on algorithm ~\ref{all_triplets}, and our experiment shows that these hard mining strategies can achieve better performanc for triplet loss. We summary all these mining strategies as follow.
\begin{itemize}
\item \textbf{Batch All}. As we can see in formula ~\ref{opt_loss}, only the triplet pair $(i,j,k)$ that satisify $\|f_i(x) -f_j(x)\|^2_2 + \alpha > \|f_i(x) -f_k(x)\|^2_2$ has the loss value. So we choice all of these triplet pairs as "hard" examples. We just modify algorithm~\ref{all_triplets} a little to obtain the Batch All algorithm \ref{batch_all}.
\item \textbf{Batch Random} If there are many negatives for some $anchor$ and $positive$, we randomly select a negative. That can be destribed as algorithm \ref{batch_random}.
\item \textbf{Batch Min Min} There may be many negatives for some $(anchor, positive)$ and we select the $negative$ that has the least distance with $anchor$. There may be also many $positives$ for some $anchor$, and we continue to select the $positive$ in which the responding $negative$ has least distance with $anchor$. That can be shown as algorithm \ref{batch_min_min}.
\item \textbf{Batch Min Max} There may be many $negatives$ for some $(anchor, positive)$ and we select the $negative$ that has the least distance with $anchor$. There may be also many positives for some $anchor$, and we just select the $positive $in which the responding $negative$ has the biggest distance with $anchor$. That can be shown as algorithm \ref{batch_min_max}.
\item \textbf{Batch Hardest} There may be many valid triplet pairs for some person, and we select only one pair in which $negative$ has the least distance with $anchor$. This can be seen as algorithm \ref{batch_hardest}.
\end{itemize}
\begin{algorithm}
\caption{Select all possible valid triplets}
\label{all_triplets}
\begin{algorithmic}[1]
\State {Random choice $B$ input images for $P$ persons with $K$ images each}
\State {Initilize list $T$ to hold all selected triplet pairs}
\For {Each person $p \in [1,P]$}
\For {Each image $i$ in person $p$ as anchor}
\For {Each image $j$ in person $p$ and $j!=i$ as positive }
\For {Each image $k \in [1,B]$ and $k$ is not in person $p$}
\State {T.append((i,j,k))}
\EndFor
\EndFor
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{Batch All}
\label{batch_all}
\begin{algorithmic}[1]
\State {Random choice $B$ input images for $P$ persons with $K$ images each}
\State {Forwarding $B$ input images to obtain the feature pool $F$}
\State {Compute distance matrix $M_{B\times B}$ between $F$}
\State {Initilize list $T$ to hold all selected triplet pairs}
\For {Each person $p \in [1,P]$}
\For {Each image $i$ in person $p$ as anchor}
\For {Each image $j$ in person $p$ and $j!=i$ as positive }
\For {Each image $k \in [1,B]$ and $k$ is not in person $p$ as negative}
\If {$M(i,j)+ \alpha > M(i,k)$}
\State {T.append((i,j,k))}
\EndIf
\EndFor
\EndFor
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{Batch Random}
\label{batch_random}
\begin{algorithmic}[1]
\State {Random choice $B$ input images for $P$ persons with $K$ images each}
\State {Forwarding $B$ input images to obtain the feature pool $F$}
\State {Compute distance matrix $M_{B\times B}$ between $F$}
\State {Initilize list $T$ to hold all selected triplet pairs}
\For {Each person $p \in [1,P]$}
\For {Each image $i$ in person $j$ as anchor}
\For {Each image $j$ in person $p$ and $j!=i$ as positive }
\State {Initilize list t}
\For {Each image $k \in [1,B]$ and $k$ is not in person $p$ as negative}
\If {$M(i,j)+ \alpha > M(i,k)$}
\State {t.append((i,j,k)}
\EndIf
\EndFor
\State {Random choice $k$ in t and T.append($k$)}
\EndFor
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{Batch Min Min}
\label{batch_min_min}
\begin{algorithmic}[1]
\State {Random choice $B$ input images for $P$ persons with $K$ images each}
\State {Forwarding $B$ input images to obtain the feature pool $F$}
\State {Compute distance matrix $M_{B\times B}$ between $F$}
\State {Initilize list $T$ to hold all selected triplet pairs}
\For {Each person $p \in [1,P]$}
\For {Each image $i$ in person $p$ as anchor}
\State {Initilize list t}
\For {Each image $j$ in person $p$ and $j!=i$ as positive }
\For {Each image $k \in [1,B]$ and $k$ is not in person $p$ as negative}
\If {$M(i,j)+ \alpha > M(i,k)$}
\State {t.append((i,j,k))}
\EndIf
\EndFor
\EndFor
\State {$k\_min\_min = \arg\min\limits_{(i,j,k)}{\{M(i,k)|(i,j,k) \in t\}}$}
\State {T.append(k\_min\_min)}
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{Batch Min Max}
\label{batch_min_max}
\begin{algorithmic}[1]
\State {Random choice $B$ input images for $P$ persons with $K$ images each}
\State {Forwarding $B$ input images to obtain the feature pool $F$}
\State {Compute distance matrix $M_{B\times B}$ between $F$}
\State {Initilize list $T$ to hold all selected triplet pairs}
\For {Each person $p \in [1,P]$}
\For {Each image $i$ in person $p$ as anchor}
\State {Initilize list t1}
\For {Each image $j$ in person $p$ and $j!=i$ as positive }
\State {Initilize list t2}
\For {Each image $k \in [1,B]$ and $k$ is not person $p$ as negative}
\If {$M(i,j)+ \alpha > M(i,k)$}
\State {t2.append((i,j,k))}
\EndIf
\EndFor
\State {$k\_min = \arg\min\limits_{(i,j,k)}{\{M(i,k)|(i,j,k) \in t2\}}$}
\State {t1.append($k\_min$)}
\EndFor
\State {$k\_min\_max = \arg\max\limits_{(i,j,k)}{\{M(i,k)|(i,j,k) \in t1\}}$}
\State {T.append($k\_min\_max$)}
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{Batch Hardest}
\label{batch_hardest}
\begin{algorithmic}[1]
\State {Random choice $B$ input images for $P$ persons with $K$ images each}
\State {Forwarding $B$ input images to obtain the feature pool $F$}
\State {Compute distance matrix $M_{B\times B}$ between $F$}
\State {Initilize list $T$ to hold all selected triplet pairs}
\For {Each person $p \in [1,P]$}
\State {Initilize list t}
\For {Each image $i$ in person $p$ as anchor}
\
\For {Each image $j$ in person $p$ and $j!=i$ as positive }
\For {Each image $k \in [1,B]$ and $k$ is not in person $p$ as negative}
\If {$M(i,j)+ \alpha > M(i,k)$}
\State {t.append((i,j,k))}
\EndIf
\EndFor
\EndFor
\EndFor
\State {$k\_min\_min = \arg\min\limits_{(i,j,k)}{\{M(i,k)|(i,j,k) \in t\}}$}
\State {T.append(\_min\_min)}
\EndFor
\end{algorithmic}
\end{algorithm}
\textbf{Mining methods}. The triplet selection is based on a pool of the feature vectors $F$. There are serveral methods to obtain the pool of feature vectors with size B.
\begin{itemize}
\item \textbf{Online mining.} We obtain the pool of features by forwarding a batch of input images with size of B once a time.
\item \textbf{Offline mining.} We forword all images in dataset to get the pool of features and select triplet pairs. Then we train these triplet pairs by a sequence of iterations.
\item \textbf{Semi-online mining.} We generate the pool of features by forwording CNN model in serval iterations like $10$ times and then select triplet pairs. That can choice more triplet pairs while it doesn't consume too much time.
\end{itemize}
\section{Experiment}
We train all models on CASIA-Webface~\cite{yi2014learning} and test them on LFW. We first train the model with Softmax and CosFace respectively, as our pretrained model. Then all models based on triplet loss are optimized by ADAGRAD optimizer with learning rate $0.001$ and $\alpha$ is set as $0.2$.
\textbf{ Results with different mining strategies}. We first pretrain CNN model with softmax classifer. Then we finetune the deep CNN model using triplet loss with different mining strategies. Every model is trained with $60k$ iterations and each iteration is optimized with batch size $210$. The results of triplet loss with different mining strategy can be seen as table~\ref{tab_strategy_exp}. All mining strategies can boost the performance of softmax classifer. The $BH\_min\_min$ strategy and $BH\_min\_max$ strategy improve performace more than BR and BA, while $B\_hardest$ is close to $BH\_min\_min$ and $BH\_min\_max$. The $BH\_min\_min$ strategy and $BH\_min\_max$ strategy is 'hard' strategy but not the 'hardest' strategy, which may make it easy to train for triplet loss.
\begin{table}[h]{}
\centering
\begin{tabular}{|c|c|}
\hline
strategy & acc(\%) \\\hline
softmax pretrain & 97.1 \\\hline
BH\_Min\_Min & 98.0 \\\hline
BH\_Min\_Max & 98.0 \\\hline
BH\_Hardest & 97.9 \\\hline
BH\_Random & 97.8 \\\hline
BH\_All & 97.5\\
\hline
\end{tabular}
\caption{Accuracy on LFW with different mining strategies.}\label{tab_strategy_exp}
\end{table}
\textbf{ Results with different initial models}. We also compare the models with different initial methods. We pretained two model with softmax classifer and CosFace respectively. Then we used these pretrained model as initial models and finetined it using triplet loss with $BH\_min\_max$ strategy which shows best performance in all strategies. The result can be viewed as table~\ref{tab_pretrain_exp}. The model with pretrain is more better than that without pretrain. The pretrained model gives a good start for triplet loss, which is essential for training of triplet loss. We also can see that pretrained model with CosFace is better than softmax because it has a better initialization.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|}
\hline
pretrain& iters & acc(\%) \\\hline
softmax pretrain & 6w &97.1 \\\hline
cosface pretrain & 6w &98.3 \\\hline
with softmax & 6w &98.0 \\\hline
with cosface & 6w &98.6 \\\hline
without pretrain & 6w & 92.2 \\
\hline
\end{tabular}
\caption{Accuracy on LFW with different initial models.}\label{tab_pretrain_exp}
\end{table}
\textbf{Results with different ($P$,$K$) combinations}. In this experiment, we use the pretrained model trained by softmax and finetue it by triplet loss with the $BH\_min\_max$ strategy. The result of different combination can be seen as tabel~\ref{tab_pk_exp}. When we keep $B$($B=P*K$) as constant ($210$). From the table \ref{tab_pk_exp}, we can know that the larger the $P$ is, the better the performance is. For larger $P$, the $anchor$ can see more $negative$ from other persons, which may avoid the model trapped in local optimization.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|}
\hline
P & K & acc(\%) \\\hline
42 & 5 & 98.0 \\\hline
30 & 7 & 98.0 \\\hline
14 & 15 & 97.7 \\\hline
10 & 21 & 97.5 \\
\hline
\end{tabular}
\caption{Accuracy on LFW with different $P$ and $K$ settings.}\label{tab_pk_exp}
\end{table}
\textbf{Results with different mining methods}. In this experiment, we compare the models trained in online and semi-online. All models are pretrained by Softmax and finetued with ${Batch\_min\_max}$ strategy. In simi-online, we forwarded the model by $10$ iterations with $210$ images each and then selected the triple pairs based on features of $10$ iterations. The results can be seen as table~\ref{tab_train_exp}. The sime-online training is better than online. This may be understood by that semi-online training increase the $P$ to improve the preformance as we see in tabel \ref{tab_pk_exp}. We demostrate this by add a experiment with multi-gpus. We increase batch size to $210*4$ by rising $P$ from $30$ to $30*4$ with 4 gpus. The accuracy of mutli-gpus is $98.3\%$, a similar accuracy with semi-online. We can use semi-online method to acheive the approaching performance of mutli-gpus when our computing resource is limited.
\begin{table}[h]
\centering
{}
\begin{tabular}{|c|c|}
\hline
train & acc(\%) \\\hline
softmax pretrain & 97.1 \\\hline
online & 98.0 \\\hline
semi-online & 98.2 \\\hline
online with multi-gpus & 98.3 \\
\hline
\end{tabular}
\caption{Accuracy on LFW with different mining methods.}\label{tab_train_exp}
\end{table}
\section{Conclusion}
We present an efficient implementation based on triplet loss for face recognition. We analyze the important factors that influence the performance of triplet loss by experiment. The results of experiment shows: 1) the pretrained model is very important for training CNN model with triplet loss; 2) hard example mining is essecial and we proposal two new mining methods: $BH\_min\_min$ and $BH\_min\_max$; 3) we can improve the performance by inreasing $P$ and a better way to do this is to train model with multi-gpus or in semi-online if the computing resource is limited.
\textbf{Acknowledgements.} We thank the supports from Key Lab of Intelligent Information Processing of Chinese Academy of Sciences and TAL-AILab.
{\small
\bibliographystyle{ieee}
|
{
"timestamp": "2019-03-01T02:15:16",
"yymm": "1902",
"arxiv_id": "1902.11007",
"language": "en",
"url": "https://arxiv.org/abs/1902.11007"
}
|
\section{Introduction}
In this work we attempt to tackle the issue of dark energy \citep[see, e.g.,][]{Peebles:2003:RMP:} by considering just usual baryonic matter in an ever-expanding Universe. We try to keep the investigation's assumptions as general as possible. Thus, we do not specify the equation of state (EOS) and we avoid to limit the study to a specific spacetime. In this framework the baryonic matter is described by an irrotational relativistic perfect fluid. For our analysis we follow a perfect fluid formalism introduced by \citet{Lichnerowicz:1967:RHM:} and \citet{Carter:1979:AGN:}, which in recent works was employed mainly for neutron stars \citep[see, e.g.,][]{Gourgoulhon:2006:EAS:,Markakis:2016:PRD:}.
In particular, we consider a perfect fluid in an equilibrium configuration with proper energy
density $\epsilon$. The state of the fluid depends on two parameters, which can
be taken to be the rest-mass density $\rho$ and specific entropy (entropy per unit rest-mass) $s$.
Then the EOS of the fluid is given by a function
\begin{align} \label{eq:EOSi}
\epsilon = \epsilon (\rho,s )\, .
\end{align}
From Eq.~\eqref{eq:EOSi} one can derive the first law of thermodynamics:
\begin{align}\label{eq:1stlaw}
d\epsilon= \mu \frac{d \rho}{m_b} + T d(s \rho)\, ,
\end{align}
where $m_{\rm{b}}$ denotes the rest mass of a baryon and $\mu$ is the baryon chemical potential.
The pressure $p$ and specific enthalpy
$h$ are functions of $\rho$ and $s$ entirely determined by Eq.~\eqref{eq:EOSi}:
\begin{align}
p &= - \epsilon + \rho\, T\, s + \frac{\mu}{m_{\rm{b}}}\rho \label{eq:pEOS}\, , \\
h &:= \frac{{\epsilon + p}}{\rho } = \frac{\mu }{{{m_{\rm{b}}}}} + Ts\, \label{eq:enth}\, .
\end{align}
Note that Eq.~\eqref{eq:pEOS} can be obtained by the extensivity property of the energy density, while the second equality of Eq.~\eqref{eq:enth} comes from Eq.~\eqref{eq:pEOS}. Now Eqs.~\eqref{eq:1stlaw} and \eqref{eq:enth} yield the thermodynamic relations
\begin{align}
d\epsilon &= h\, d\rho + \rho\, T\, ds \label{eq:de}\, , \\
dp &= \rho\, dh - \rho\, T\, ds \label{eq:dp}\, .
\end{align}
Moreover, writing $h = h(\rho ,s)$ and differentiating yields
\begin{equation} \label{eq:dh}
dh = \frac{{hc_{\rm{s}}^2}}{\rho }d\rho + {\left. {\frac{{\partial h}}{{\partial s}}} \right|_{\rho}}ds\, ,
\end{equation}
where
\begin{equation} \label{eq:cs2}
c_{\rm{s}}^2 = {\left. {\frac{{\partial p}}{{\partial \epsilon }}} \right|_s} = \frac{\rho }{h}{\left. {\frac{{\partial h}}{{\partial \rho }}} \right|_s}
\end{equation}
is the sound speed. In order to ensure causal evolution, given the upper bound
for signal propagation set by the speed of light, physically admissible fluids
should have
\begin{align} \label{eq:SpSoL}
0\lesssim s_m^2\leq c_s^2\leq 1,
\end{align}
where $s_m^2$ is an arbitrarily close to zero cut-off value for the speed of sound.
A simple perfect fluid is characterized by the energy-momentum tensor
\begin{align}
{T_{\alpha}}^{\beta} = h\, \rho\, u_\alpha u^\beta + p\, {g_{\alpha}}^{\beta}
= \left( \epsilon + p \right)\, u_\alpha u^\beta + p\, {g_{\alpha}}^{\beta}\ ,
\end{align}
where $ {g_{\alpha \beta }}$ is the spacetime metric and $u^{\mu}$ is the
timelike vector tangent to the fluid's flow, satisfying the normalization
condition ${u^\alpha }{u_\alpha } = - 1$. Such energy-momentum tensor is the
source in Einstein's field equations (EFE) ${G_{\alpha}}^{\beta} = {T_{\alpha}}^{\beta}\,$, which are assumed to hold throughout this work.
By taking the covariant divergence of EFE, the doubly contracted Bianchi
identities $\nabla_{\beta}{G_{\alpha}}^{\beta}\equiv0$ assure the covariant
conservation of energy-momentum
\begin{equation}
\nabla_{\beta} {T_{\alpha}}^{\beta}=0\, ,\label{emcons}
\end{equation}
which is the relativistic version of Euler equation. Using Eq.~\eqref{eq:dp} with variation evaluated along the flow lines
($d \rightarrow u^{\alpha}\nabla_{\alpha}$) and thanks to the normalization of
the timelike vector $u^{\alpha}$, eq.\eqref{emcons} takes the form
\begin{equation} \label{ValenciaT}
\nabla_\alpha T^\alpha_{\,\,\,\,\beta} =p_\beta\nabla_\alpha (\rho u^\alpha)+
\rho[ u^{\alpha} \Omega_{\alpha \beta}- T \nabla_\beta s ]= 0\, ,
\end{equation}
where $ {p_\alpha } = h{u_\alpha }$ is the {\it canonical momentum} of a fluid element, and its exterior derivative $ \Omega_{\alpha \beta}:=\nabla_\alpha {\rm{}} p_\beta-\nabla_\beta {\rm{}} p_\alpha\,$
is the {\it canonical vorticity 2-form}. If we assume the rest-mass (or baryon) conservation
\begin{equation} \label{eq:continuityeqn}
\nabla_\alpha (\rho u^\alpha)=0\, ,
\end{equation}
eq.~\eqref{ValenciaT} yields the relativistic Euler
equation in the canonical form:
\begin{equation} \label{eq:EulerCanonical}
u^{\alpha} \Omega_{\alpha \beta}= T \nabla_\beta s\, .
\end{equation}
Contraction of eq.~\eqref{eq:EulerCanonical} with the four-velocity
${u^\beta }$ makes the left-hand side vanish identically.\footnote{This is
because the left-hand side, after contraction with $u^{\beta}$, ends up being a
product of the symmetric term $u^{\alpha}u^{\beta}$ with the antisymmetric 2-form
$\Omega_{\alpha \beta}$.} Hence the specific entropy is constant along the flow lines:
\begin{equation} \label{eq:adiabatic}
{u^\alpha }{\nabla _\alpha }s = 0\, .
\end{equation}
This reflects the fact that the Euler equation describes {\it adiabatic flows},
{\it i.e.} there are no heat fluxes in the fluid
nor particle production. The adiabatic character of the fluid as expressed by
Eq.~\eqref{eq:adiabatic} is a consequence of assuming rest-mass conservation
Eq.~\eqref{eq:continuityeqn}.
\section{Thermodynamical relations for an irrotational fluid} \label{sec:ThermoRel}
The condition for irrotational fluid flow is $\Omega_{\alpha \beta}=0$,
and implies through Eq.~\eqref{eq:EulerCanonical} that the specific entropy is
constant, {\it i.e.} $\text{d} s=0$. The fundamental relations Eqs.~\eqref{eq:de}-\eqref{eq:dh} reduce to
\begin{align}
d h &=\frac{h\, c_{\rm{s}}^2 }{\rho}\, d\rho\, \label{eq:drho} ,\\
d \epsilon &=h\, d\rho\label{eq:depsilon}\, ,\\
d p &=\rho\, dh\, . \label{eq:dpdh}
\end{align}
Using the limits set by Eq.~\eqref{eq:SpSoL} and making the reasonable assumption
that the rest-mass density is a positive quantity, since we consider fluid
composed only of baryonic matter, we arrive through Eq.~\eqref{eq:drho} to
\begin{align} \label{eq:drhoInt}
\int_{\rho_1}^{~\rho} \frac{s_m^2 d\rho'}{\rho'} \le
\int_{\rho_1}^{~\rho} \frac{c_s^2 d\rho'}{\rho'} \le \int_{\rho_1}^{~\rho} \frac{d\rho'}{\rho'} \quad
\Rightarrow \quad \left(\frac{\rho}{\rho_1}\right)^{s_m^2} \le\frac{h}{h_1}\le \frac{\rho}{\rho_1}\, ,
\end{align}
where index ``1'' refers to the integration constants of the specific fluid with
equation of state described by the speed of sound $c_s^2$, not by the lower
bound and upper bounds of Eq.~\eqref{eq:SpSoL}. Note that we have assumed that
$d\rho>0$. Eq.~\eqref{eq:drhoInt} implies
$ \displaystyle \left(\frac{\rho}{\rho_1}\right)^{s_m^2-1}\le 1$ ,
which gives that $\rho_1\le\rho$, since $s_m^2<1$, i.e. the integration constant
$\rho_1$ corresponds to the minimum of the allowed values for the rest-mass
density of the fluid. Moreover, inequality~\eqref{eq:drhoInt} implies that
$h/h_1>0$. At this point we do not make any assumption about the sign of the
specific enthalpy.
Because of Eq.~\eqref{eq:depsilon}, Eq.~\eqref{eq:drhoInt} results in
\begin{align} \label{eq:depsilonInt}
&\frac{1}{\rho_1^{s_m^2}}\int_{\rho_1}^{~\rho} \rho'^{s_m^2} d\rho'
\le \frac{1}{h_1} \int_{\rho_1}^{~\rho} h d\rho' \le \frac{1}{\rho_1}\int_{\rho_1}^{~\rho} \rho' d\rho' \nonumber\\
\Rightarrow &\frac{\rho_1}{1+{s_m^2}}\left[\left(\frac{\rho}{\rho_1}\right)^{s_m^2+1}-1\right] \le
\frac{\epsilon-\epsilon_1}{h_1} \le \frac{\rho_1}{2}\left[\left(\frac{\rho}{\rho_1}\right)^{2}-1\right]\, ,
\end{align}
where $\displaystyle \int_{\rho_1}^{\rho} h d\rho'=\int_{\epsilon_1}^{\epsilon} d\epsilon'$ was employed.
From Eqs.~\eqref{eq:drho} and \eqref{eq:dpdh} we get
\begin{align} \label{eq:dpdrho}
dp=c_s^2 h d\rho\, .
\end{align}
Taking into account Eq.~\eqref{eq:dpdrho}, from Eq.~\eqref{eq:drhoInt} and
Eq.~\eqref{eq:SpSoL} we have
\begin{align} \label{eq:dpInt}
&\frac{s_m^2 }{\rho_1^{s_m^2}}\int_{\rho_1}^{~\rho} \rho'^{s_m^2} d\rho'
\le \frac{1}{h_1} \int_{\rho_1}^{~\rho} c_s^2 h d\rho' \le \frac{1}{\rho_1}\int_{\rho_1}^{~\rho} \rho' d\rho' \nonumber\\
\Rightarrow &\frac{s_m^2 \rho_1}{1+{s_m^2}}\left[\left(\frac{\rho}{\rho_1}\right)^{s_m^2+1}-1\right] \le
\frac{p-p_1}{h_1} \le \frac{\rho_1}{2}\left[\left(\frac{\rho}{\rho_1}\right)^{2}-1\right]\, ,
\end{align}
where $\displaystyle \int_{\rho_1}^{\rho} c_s^2 h d\rho'=\int_{p_1}^{p} dp'$ was employed.
Since $\rho \ge \rho_1$, inequality~\eqref{eq:dpInt} gives that
$(p-p_1)/h_1\ge 0$, while inequality~\eqref{eq:depsilonInt} gives that
$(\epsilon-\epsilon_1)/h_1\ge 0$. For $\rho=\rho_1$,
Eqs.~\eqref{eq:drhoInt},~\eqref{eq:depsilonInt}, \eqref{eq:dpInt} reduce to
$h=h_1$, $\epsilon=\epsilon_1$, $p=p_1$ respectively, which is trivial but
self-consistent.
\subsubsection{Assuming constant speed of sound}
Assuming $c_{s}^2 $ is independent of specific enthalpy, i.e. constant, then
by following similar steps as for arriving to the inequalities~\eqref{eq:drhoInt},~\eqref{eq:depsilonInt},~\eqref{eq:dpInt},
we get
\begin{align}
\epsilon - \epsilon_1 &= \frac{1}{1+c_s^2}\, \rho_1\, h_1\, \left[\left( \frac{\rho}{\rho_1} \right)^{1+c_s^2} - 1 \right]\, ,\label{eq:epsilon} \\
p - p_1 &= \frac{c_s^2}{1+c_s^2}\, \rho_1\, h_1\, \left[\left( \frac{\rho}{\rho_1} \right)^{1+c_s^2} - 1 \right]\label{eq:pressure}\, ,
\end{align}
which leads to
\begin{equation}\label{eq:eos}
p = c_s^2 \left(\epsilon - \epsilon_1 \right)\, +p_1\, .
\end{equation}
Note that if one changes the equation of the state of the fluid, i.e. $c_{s}^2 $,
the integration constants denoted with ``1'' change as well.
\section{Asymptotic behaviors}
\subsection{Rest-mass density}
The rest mass conservation~\eqref{eq:continuityeqn} can be rewritten as:
\begin{align} \label{eq:continuityeqn2}
\dot{\rho}+\rho~\theta=0\, ,
\end{align}
where $\theta=\nabla_{\alpha}u^{\alpha}$ is the expansion scalar of the congruence
$u^{\alpha}$, $\dot{\ }={u^\alpha }{\nabla _\alpha }$ denotes the derivative with
respect to a relevant time parameter $t$ along the congruence $u^{\alpha}$.
Integrating Eq.~\eqref{eq:continuityeqn2} along the time parameter $t$ leads to
\begin{equation}\label{eq:restmass_exp}
\rho = \rho_0\ e^{-\int_{t_{0}}^t \theta(t')\, dt'}\, ,
\end{equation}
with initial condition $\rho(t_0)=\rho_0$.
\begin{prop} \label{prop:restmass}
For a perfect fluid moving along an expanding congruence
with conserved positive rest-mass, the rest-mass density vanishes asymptotically, $\rho\rightarrow 0^{+}$, in the limit $t\rightarrow\infty$.
\end{prop}
\begin{proof}
Since we have an expanding congruence, there exists a $k>0$, such
that $\theta\ge k$. Eq.~\eqref{eq:restmass_exp} then leads to
\begin{align}
\rho = \rho_0\ e^{-\int_{t_{0}}^t \theta(t')\, dt'} \le \rho_0\ e^{- \int_{t_{0}}^t k\, dt'}
= \rho_0\ e^{-k(t-t_0)}\rightarrow 0\quad \text{for}\quad t\rightarrow\infty\, .
\end{align}
Since $\rho > 0$, one has $\rho\rightarrow 0^{+} $ for $t\rightarrow\infty$,
i.e. the rest mass density asymptotically vanishes.
\end{proof}
Proposition~\ref{prop:restmass} and the fact that $\rho_1\le\rho$ suggests that
$\rho_1$ must be an infinitesimally small positive quantity, i.e. $\rho_1\equiv 0^{+}$.
Moreover, Proposition~\ref{prop:restmass} implies that for $t\rightarrow\infty$
Eqs.~\eqref{eq:epsilon},~\eqref{eq:pressure} derived for a fluid with constant
non-zero speed of sound lead to
\begin{align}
\epsilon - \epsilon_1 &\simeq -\frac{1}{1+c_s^2}\, \rho_1\, h_1\, , \label{eq:epsilonInf} \\
p - p_1 &\simeq -\frac{c_s^2}{1+c_s^2}\, \rho_1\, h_1\, \label{eq:pressureInf}\, .
\end{align}
To show an interesting implication of these relations, let us fix the constants of
integration by considering the vanishing pressure limit, $p_1 =0$. In this limit,
one typically imposes that the specific enthalpy is equal to unity. Then, the relation $\epsilon+p=\rho h$, for $p=p_1=0$ and $h=h_1=1$,
implies
\begin{equation}
\epsilon_1 =\rho_1 \label{eq:enthalpyB}\, .
\end{equation}
With these constraints on the constants, we obtain the following expressions for
Eqs.~\eqref{eq:epsilonInf},~\eqref{eq:pressureInf}:
\begin{align}
p &\simeq -\frac{\epsilon_1\, c_s^2}{1+c_s^2}\, ,\label{eq:pressBCS}\\
\epsilon &\simeq \frac{\epsilon_1\, c_s^2}{1+c_s^2}\, .\label{eq:energyBCS}
\end{align}
It is immediately evident that Eq.~\eqref{eq:energyBCS} represents
a constant positive contribution to the energy density for any $c_s^2>0$,
if $\epsilon_1=\rho_1>0$. In a cosmological context such term behaves like a
{\it cosmological constant}, since $p=-\epsilon$. This has been already noticed for the case of the stiff fluid $(c_{\textrm{s}}=1)$ by \citet{Christodoulou:1995:ARRMA:}.
Applying proposition~\ref{prop:restmass} on the inequalities~\eqref{eq:depsilonInt},~\eqref{eq:dpInt}
and using the \eqref{eq:enthalpyB} choice for fixing the constants, we arrive at:
\begin{align}
- \frac{\epsilon_1 s_m^2}{1+s_m^2} &\lesssim p \lesssim -\frac{\epsilon_1}{2}\, ,\label{eq:pressBIS}\\
\frac{\epsilon_1 s_m^2}{1+s_m^2} &\lesssim \epsilon \lesssim \frac{\epsilon_1}{2}\, .\label{eq:energyBIS}
\end{align}
Eq.~\eqref{eq:energyBIS} still implies a constant positive contribution to the
energy density for $t\rightarrow\infty$, but Eq.~\eqref{eq:pressBIS} is only
possible if $\epsilon_1=0$, since $s_m^2\ll 1$. Thus, we are led to
$\epsilon_1=0$, which means that Eqs.~\eqref{eq:pressBIS},~\eqref{eq:energyBIS}
respectively lead to $p \simeq \epsilon \simeq 0$. Moreover, since the above
inequalities include the constant speed case as a subcase, then $\epsilon_1=0$
for Eqs.~\eqref{eq:pressBCS},~\eqref{eq:energyBCS}, so they do not imply the
existence of a cosmological constant. On the other hand, this result might be
suggesting that the choice~\eqref{eq:enthalpyB} we have made to fix the constants
is not the proper one.
In fact if we do not fix the constants, according to Proposition~\ref{prop:restmass}
the inequalities~\eqref{eq:drhoInt},~\eqref{eq:depsilonInt},~\eqref{eq:dpInt}
reduce to
\begin{align}
\frac{h}{h_1} &\simeq 0 ,\label{eq:enthalpyBIA}\\
-\frac{\rho_1}{1+{s_m^2}} &\lesssim \frac{\epsilon-\epsilon_1}{h_1} \lesssim -\frac{\rho_1}{2}\, , \label{eq:energyBIA}\\
- \frac{s_m^2 \rho_1}{1+{s_m^2}} &\lesssim \frac{p-p_1}{h_1} \lesssim -\frac{\rho_1}{2}\, . \label{eq:pressBIA}
\end{align}
Again because of $s_m^2\ll 1$, Eq.~\eqref{eq:pressBIA} can hold only if
$\rho_1$ is exactly zero. Note that even if $s_m^2$ was equal to zero
$\rho_1$ had to be zero as well. By not allowing the rest mass energy density to
acquire the zero value, we have arrived to a contradiction. If one would allow it,
then it would not be possible to derive the inequalities in Sec.~\ref{sec:ThermoRel}.
To resolve this contradiction, one might claim that the relations derived in
Sec.~\ref{sec:ThermoRel} hold only for finite time intervals, i.e. they do not hold
for $t\rightarrow \infty$. To discuss the asymptotic behaviors, we need propositions
like Proposition~\ref{prop:restmass}.
\subsection{Enthalpy}
Evaluating the thermodynamic relation Eq.~\eqref{eq:dh} along the flow lines,
and implementing Eq.~\eqref{eq:adiabatic}, yields the relation
\begin{equation}
{u^\alpha }{\nabla _\alpha }h = \frac{{hc_{\rm{s}}^2}}{\rho }{u^\alpha }{\nabla _\alpha }\rho\, ,
\end{equation}
which can be used to rewrite the rest-mass conservation equation~\eqref{eq:continuityeqn} as
\begin{align}
0 =& {\nabla _\alpha }(\rho {u^\alpha })\\ =& \frac{\rho }{{hc_{\rm{s}}^2}}({u^\alpha }{\nabla _\alpha }h + hc_{\rm{s}}^2{\nabla _\alpha }{u^\alpha })\, . \label{cont_h}
\end{align}
The continuity equation for the rest-mass density as expressed by Eq.~\eqref{cont_h} is
\begin{equation} \label{hdot}
\dot{h} = - c_{\rm{s}}^2\, \theta\, h\, ,
\end{equation}
For generic time-dependent speed of sound and expansion scalar, one then has
\begin{equation}\label{enth_exp}
h = h_0\ e^{-\int_{t_{0}}^t c_{\rm{s}}^2(t')\, \theta(t')\, dt'}\, ,
\end{equation}
with initial condition $h(t_0)=h_0$.
\subsubsection{Strong Energy Condition}
\begin{prop}\label{prop:enthalpySEC}
Consider a perfect fluid moving along an expanding and isotropic congruence,
with conserved rest-mass and satisfying the Strong Energy Condition (SEC);
then if the speed of sound is a function of time defined in the interval $(0,1]$,
in the limit $t\rightarrow\infty$ one necessarily has $\epsilon\rightarrow0$ and
$p\rightarrow0$.
\end{prop}
\begin{proof}
The equation of rest-mass conservation can be rewritten in the form
Eq.~\eqref{hdot}, whose general solution is given by eq.\eqref{enth_exp}.
We would like to evaluate the behavior of $h$ in the limit when
$t\rightarrow\infty$ by obtaining an upper and a lower bound.
{\it Lower bound.} First of all $c_s^2(t)\in (0,1]$, so we can write
\begin{equation}\label{eq:enth2}
h = h_0\ e^{-\int_{t_{0}}^t c_{\rm{s}}^2(t')\, \theta(t')\, dt'}\ \geq\ h_0\ e^{-\int_{t_{0}}^t \theta(t')\, dt'} \, .
\end{equation}
Secondly, the Raychaudhuri equation for an isotropic timelike congruence $u^{\alpha}$ reads
\begin{equation} \label{eq:RayIsot}
\dot{\theta} = - \left(\frac{1}{3}\theta^2 + R_{\alpha\beta}u^{\alpha}u^{\beta}\right)\, .
\end{equation}
Because of the SEC, the last term is positive. Hence we get the inequality
\begin{equation}
\dot{\theta} \leq - \frac{1}{3}\theta^2\, .
\end{equation}
Integration of such inequality gives
\begin{equation}
\theta \leq \frac{3\, \theta_0}{3+\theta_0\, t}\, ,
\end{equation}
with $\theta_0=\theta(t_0)$. Applying such bound to the rightmost term of
Eq.~\eqref{eq:enth2} gives
\begin{align}
h \ge h_0\ e^{-\int_{t_{0}}^t \theta(t')\, dt'} &\geq h_0\, e^{-\int_{t_0}^t\frac{3\, \theta_0}{3+\theta_0\, t'}\, dt'}\\
&= h_0\, \left( \frac{3+\theta_0 t_0}{3+\theta_0 t} \right)^3 \rightarrow 0\quad \text{for}\quad t\rightarrow\infty\, .
\end{align}
Hence $h\geq0$ for $t\rightarrow\infty$.
{\it Upper bound.} By assumption, the product $c_s^2(t) \theta(t)$ is strictly
positive: hence there exist a constant $k>0$ such that $c_s^2(t) \theta(t)\geq k>0$
for any finite time. The function $h$ can then be bounded from above in the
following way:
\begin{align}
h = h_0\ e^{-\int_{t_{0}}^t c_{\rm{s}}^2(t')\, \theta(t')\, dt'} &\leq h_0\ e^{- \int_{t_{0}}^t k\, dt'}\nonumber\\
&= h_0\ e^{-k(t-t_0)}\rightarrow 0\quad \text{for}\quad t\rightarrow\infty\, .
\end{align}
Hence $h\leq0$ for $t\rightarrow\infty$.
Putting together the results of both bounds, we find that $h=0$ in the limit
$t\rightarrow\infty$. At the same time $\rho\rightarrow0$ in the same limit,
because of Proposition~\ref{prop:restmass}. Thus, one has that
$h\equiv\frac{\epsilon+p}{\rho}\rightarrow0$ implies that $p+\epsilon\rightarrow0$.
Lastly, the SEC requires $p+\frac{1}{3}\epsilon \geq 0$: the only case in which
the condition $p+\epsilon\rightarrow0$ is consistent with this bound is when both
$\epsilon\rightarrow0$ and $p\rightarrow0$ (left panel of Fig.~\ref{fig:examples}).
\end{proof}
Note that Proposition~\ref{prop:restmass} by itself could not lead to
$p+\epsilon\rightarrow0$, since the asymptotic bounded value of the specific
enthalpy was not guaranteed.
{\it Proposition~\ref{prop:enthalpySEC} is a general statement about the
impossibility for a ``well defined'' isotropic perfect fluid satisfying the
SEC to have a non-trivial pressure asymptotically.} Hence, in the following
prepositions we drop SEC and specialize to a spatially flat Friedmann-Robertson-Walker (FRW) spacetime.
\subsubsection{Bounded Rate of Expansion}
\begin{prop}\label{prop:enthalpyBExp}
Consider a perfect fluid moving along an expanding congruence in a flat FRW spacetime,
with conserved rest-mass and a rate of expansion bounded by $\Xi$; then
if the speed of sound is a function of time defined in the interval $(0,1]$, in
the limit $t\rightarrow\infty$ one has $\epsilon+p\rightarrow0$, without necessarily
$\epsilon\rightarrow0$ and $p\rightarrow0$, and $0 \lesssim \Xi$.
\end{prop}
\begin{proof}
The {\it upper bound} stays the same as in Proposition~\ref{prop:enthalpySEC},
so $h\leq0$ for $t\rightarrow\infty$.
{\it Lower bound.} A positive, but bounded rate of congruence expansion means that
$\dot{\theta}\le \Xi$, thus $\theta(t)\le \Xi(t-t_0)+\theta_0$. Then,
Eq.~\eqref{eq:enth2} gives
\begin{align}
h \ge h_0\ e^{-\int_{t_{0}}^t \theta(t')\, dt'} &\geq h_0\, e^{-\int_{t_0}^t \Xi(t-t_0)+\theta_0 dt'}\nonumber\\
&= h_0\, e^{-(\Xi(t-t_0)^2/2+\theta_0 (t-t_0))} \rightarrow 0\quad \text{for}\quad t\rightarrow\infty\, .
\end{align}
Putting together the results of both bounds, we find that $h=0$ in the limit
$t\rightarrow\infty$. Thus, again one has that $p+\epsilon\rightarrow0$.
However, from the isotropic Raychaudhuri Eq.~\eqref{eq:RayIsot} we have:
\begin{equation}
- \left(\frac{1}{3}\theta^2 + R_{\alpha\beta}u^{\alpha}u^{\beta}\right)\le \Xi \Rightarrow
-\frac{3}{2} (\epsilon+ p)\le \Xi\, ,
\end{equation}
where we used Friedmann equation $\theta^2=3\epsilon$. Thus, in this case the solution $\epsilon\rightarrow0$, $p\rightarrow0$ is not the only allowed to have
$\epsilon+p\rightarrow0$ (right panel of Fig.~\ref{fig:examples}). Actually, $p\rightarrow -\epsilon$ implies that $0\lesssim \Xi$.
\end{proof}
Note that proposition~\ref{prop:enthalpyBExp} allows an exponential growth for FRW
$$3 \frac{\dot{a}}{a}= \theta= \Xi (t-t_0)+\theta_0 \Rightarrow a\le a_0 e^{(\Xi (t-t_0)^2/2+\theta_0(t-t_0))/3}$$ even if $\Xi=0$. Thus, to have exponential growth the minimal requirement
is that $\dot{\theta}\le 0$.
\begin{figure} [t]
\begin{center}
\includegraphics[width=0.25\linewidth,height=0.5\linewidth]{FigSEC}
\includegraphics[width=0.25\linewidth,height=0.5\linewidth]{FigNonSEC}
\end{center}
\caption{\label{fig:examples}
Left panel: The plane of allowed EoS assuming SEC, Proposition~\ref{prop:enthalpySEC}. Right Panel: The plane of allowed EoS assuming bounded rate of congruence expansion, Proposition~\ref{prop:enthalpyBExp}. In both panels we assume that the energy density is $\epsilon\ge 0$.}
\end{figure}
\section{Summary}
Starting from a general thermodynamical treatment of usual matter, in the form of an irrotational perfect fluid, our investigation indicates that a constant speed of sound for usual matter is not a viable way to provide a cosmological constant. We have given a formal proof that if the strong energy condition holds, usual matter cannot provide negative pressure. Moreover, we have provided a formal proof that for a flat FRW spacetime containing only usual matter, for which the strong energy condition is violated, negative pressure is possible .
\ac
G.L-G is supported by Grant No. GA\v{C}R-17-06962Y of the Czech Science
Foundation. G.A. is supported by Grant No. GA\v{C}R-17-16260Y of the Czech Science Foundation.
C.M. is supported by the European Union's Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant agreement No 753115.
|
{
"timestamp": "2019-03-01T02:17:26",
"yymm": "1902",
"arxiv_id": "1902.11051",
"language": "en",
"url": "https://arxiv.org/abs/1902.11051"
}
|
\section{Introduction}
The approach of the twin Higgs scenarios to the little hierarchy problem is a realization of the Higgs boson as a pseudo-Goldstone boson~\cite{Chacko:2005pe}.
With introducing the twin/mirror sector of the standard model (SM),
twin Higgs fields $H_A$ and $H_B$ of each sector form a fundamental representation of a global group $SU(4)$ whose
spontaneous symmetry breaking down to $SU(3)$ generates seven Goldstone bosons.
Six of them are eaten by $SU(2)_{LA}$ and $SU(2)_{LB}$ gauge bosons of each sector, and one is identified as the observed Higgs boson.
An important ingredient for the twin Higgs mechanism to work is the $Z_2$ twin symmetry under which each particle of one sector is interchanged with the corresponding particle of the other sector.
The role of the twin symmetry is to prevent the explicit breaking terms of $SU(4)$ symmetry from introducing a quadratic divergence of the Higgs boson.
However, the twin symmetry has to be broken either explicitly or spontaneously for various phenomenological reasons including the observed Higgs singal strength\,\cite{Barbieri:2005ri,Burdman:2014zta, Craig:2015pha}.
The $SU(4)$ breaking scale $f$ should be at least about three times larger than the SM Higgs vacuum expectation value (vev).
Introducing soft $Z_2$ symmetry breaking term $\epsilon f^2 |H_A|^2$ can easily provide a misalignment of twin Higgs vevs, but it requires fine tuning of parameters with an order of $v_{\rm SM}^2/f^2$ where $v_{\rm SM}\simeq 246\,\mathrm{GeV}$ is the SM Higgs vev.
This argument provides an upper bound of $f$ around $5\mathrm{TeV}$ scale.
In this paper, we focus on the possibility that the twin symmetry is exact but broken spontaneously.
Fig.\,\ref{fig:scheme} describes the desired situation that we consider in this paper.
In the twin Higgs field space $(h_A,\,h_B)$, the twin symmetry ($Z_2$) corresponds to the mirror symmetry along the diagonal dashed line.
There are two degenerate minima in the flat direction $h_A^2+h_B^2= f^2$ whose locations in the field space are symmetric under the $Z_2$ transformation.
Once scalar fields fall down to one of the minima, the sector with smaller Higgs vev becomes what we call the SM.
While there are several realizations in twin two Higgs doublet model\,\cite{Beauchesne:2015lva, Yu:2016bku} or singlet extended twin Higgs model\,\cite{Bishara:2018sgl},
we consider a realization in one twin Higgs doublet setup without additional scalar fields.
Similar idea is discussed in Refs.\,\cite{Hall:2018let, Dunsky:2019api} in the context of the strong CP problem.
We provide more systematic approach to the construction of the effective potential.
Cosmological history of spontaneous twin symmetry breaking is restricted by dark radiation constraints from the cosmic microwave background\,\cite{Aghanim:2018eyx}, and by the domain wall problem\,\cite{Zeldovich:1974uw}.
To avoid these problems, we assume that $Z_2$ symmetry is spontaneously broken before/during the inflation, and
reheaton decays mostly to the sector with smaller Higgs vev.
For preventing twin sector particles from being produced thermally, reheating temperature should be less than around the bottom quark mass when $f\sim 10\,v_{\rm SM}$.
Otherwise, twin sector particles can be produced through the bottom quark annihilation to the twin muon production process,
and twin photons and neutrinos will finally contribute to the dark radiation\,\cite{Barbieri:2005ri,Barbieri:2016zxn,Chacko:2016hvu}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.4\textwidth]{potential1}
\end{center}
\caption{ Schematic picture of the potential for the spontaneous $Z_2$ breaking scenario.
There are two degenerate minima (purple) in $(h_1,\,h_2)$ field space.
Locations of minima are $Z_2$ symmetric.
Vacuum chooses one of the minima before/during the inflation, and the sector with smaller vev becomes the SM.
Circular line denotes the flat direction which corresponds to the Higgs boson degree.
}
\label{fig:scheme}
\end{figure}
\section{Minimal model with exact twin symmetry}
Nonzero Higgs mass comes from $SU(4)$ breaking.
To be more specific, let us consider an effective scalar potential,
\begin{eqnarray}
V(h_A,\,h_B)=\frac{\lambda}{4} \left( h_A^2+h_B^2-f^2\right)^2 +\Delta V(h_A,\,h_B),
\label{generalpotential}
\end{eqnarray}
where $h_A$ and $h_B$ are classical Higgs fields of each sector.
Here, $\Delta V$ denotes terms breaking $SU(4)$ explicitly, but preserving $Z_2$.
A mixed quartic term $h_A^2h_B^2$ is the leading term to break $SU(4)$\footnote{$h_A^2 h_B^2$ is equivalent to $-(h_A^4+h_B^4)/2$ if we redefine $\lambda$ and $f$.}.
Gauge and Yukawa interactions will also contribute to $\Delta V$ through loops.
By replacing $h_A=f \cos \theta$ and $h_B=f \sin \theta$, we can obtain the potential along the flat direction for the case when $\lambda \gg \lambda_{\rm mix}$.
We denote it by $f^4 \hat V(\theta)$ in this paper.
Since $\hat V(\theta)$ is a periodic function with periodicity $\pi/2$ and symmetric under $Z_2:\,\theta\to \pi/2-\theta$, we can apply the Fourier expansion
\begin{eqnarray}
\hat V(\theta) = \sum_n c_n \cos (4 n \theta)
\label{Fourier}
\end{eqnarray}
with coefficients $c_n$.
The leading order contribution $h_A^2 h_B^2$ corresponds to $(1-\cos 4\theta)/8$ whose extrema are $0$ ($h_B=0$), $\pi/4$ ($h_A=h_B$) and $\pi/2$ ($h_B=0$).
If its minima are at $0$ or $\pi/2$, $h_B$ or $h_A$ become zero and the electroweak symmetry breaking does not take place.
In order to obtain a proper misalignment, there should be nonzero $c_n$ contributions with $n\geq 2$ \footnote{
If there are two Higgs doulets in each sector (i.e. $H_{1A}$, $H_{2A}$, $H_{1B}$, $H_{2B}$), a proper misalignment is possible by
assigning different signs of $\lambda_{\rm mix1}$ and $\lambda_{\rm mix2}$ \cite{Beauchesne:2015lva}.
}.
The simplest term to generate $c_{n\geq2}$ is the Coleman-Weinberg potential, $\frac{1}{2}(h_A^4 \log h_A^2/\mu^2+h_B^4 \log h_B^2/\mu^2)$.
It leads to $c_1=(25-24\log 2)/96,~ c_2=-1/240,~\cdots$ when we take $\mu=f$.
Since we want $c_2$ to dominate in the potential, a suppression of $c_1$ is required,
and we consider a cancelation between $h_A^2 h_B^2$ term and the Coleman-Weinberg potential.
This cancelation causes unavoidable tuning of parameters in our scenario.
It will be shown that this cancelation is equivalent to the fine tuning of quadratic Higgs term in the infra-red (IR) theory.
In addition, the sign of $c_2$ should be positive.
If $c_2$ is negative, the minima can be only at $\theta = 0,\,\pi/4$ or $\pi/2$.
Positive sign of $c_2$ is equivalent to the negative sign of beta function of Higgs self quartic coupling.
It is interesting that the SM gauge and Yukawa interactions already provide the proper sign assignment.
Therefore, the minimal setup is just the exact copy of the SM particle contents and gauge group with the potential \eqref{generalpotential}.
The $SU(4)$ breaking effective potential $\Delta V$ can be parametrized by
\begin{equation}
\Delta V= \frac{\lambda_{\rm mix}}{4} h_A^2 h_B^2 + \frac{\beta}{4} \left( h_A^4 \log \frac{h_A}{\mu}+h_B^4 \log \frac{h_B}{\mu} \right)
\label{minimalV}
\end{equation}
where $\beta\equiv d\lambda / d \ln \mu$ is the renormalization group (RG) equation of $\lambda$.
Its SM value is
\begin{eqnarray}
\beta_{\rm SM} = \frac{1}{16\pi^2}\left( -6y_t^4+ \frac{9}{8}g_2^4+\frac{27}{200}g_1^4+\frac{9}{20}g_1^2g_2^2 \right),
\end{eqnarray}
where we ommit contributions coming from $\lambda_{\rm mix}$ because it will turn out to be negligible at $\mu=f$.
Note that $\lambda$ does not contribute to $\hat V(\theta)$ since $\lambda$ repects $SU(4)$ symmetry.
In Eq.\,\eqref{minimalV}, we redefined $\lambda_{\rm mix}(\mu=f)$ so that quartic operators coming from radiative contributions are absorbed, e.g. $\log y_t/2 -3/2$.
To obtain $\hat V$ along the flat direction, we replace $h_A=f \cos \theta$ and $h_B=f \sin \theta$.
In terms of Fourier expansion, $\hat V$ can be written as
\begin{equation}
\hat V \simeq \frac{-12\lambda_{\rm mix}+\beta (25-24\log2)}{384} \cos 4\theta -\frac{\beta}{960} \cos 8\theta +\cdots,
\label{Vhatminimal}
\end{equation}
where we neglect the constant term for simplicity.
For the misalignment of vevs (i.e. $h_A\neq h_B \neq 0$), $\cos 4\theta$ term should be suppressed.
This suppression comes from the cancelation between $12\lambda_{\rm mix}$ and $\beta(25-24\log2)$.
In terms of the ratio between $\lambda_{\rm mix}$ and $\beta$, $\kappa=\lambda_{\rm mix}/\beta$, the condition for the misalignment becomes
\begin{eqnarray}
\beta<0 \text{~~and~~} \frac{1}{2}<\kappa< \frac{3-2\log2}{2}\simeq 0.81,
\end{eqnarray}
where the first condition $\beta<0$ is already satisfied by the large top Yukawa interaction.
In Fig.\,\ref{fig:Vhat}, $\hat V(\theta)$ is described with different $\kappa$ values.
If $\kappa$ is too large, twin Higgs vevs become identical, i.e. $v_A=v_B$, and the twin symmetry is not broken spontaneously.
On the other hand, too small $\kappa$ leads one of twin Higgs vevs to zero, i.e. electroweak symmetry breaking does not occur.
In the minimal setup, we have only two free parameters ($f$ and $\lambda_{\rm mix}$), and they are fixed by two observational constraints (Higgs vev $v_{\rm SM}$ and mass $m_h$).
If we denote $\theta_0$ as the minimum position of $\hat V$,
\begin{eqnarray}
v_{\rm SM}&=&f \,{\rm min} (\sin \theta_0,\, \cos \theta_0), \label{vh}\\
m_h^2&=& f^2\left.\frac{\partial^2 \hat{V}}{\partial \theta^2}\right|_{\theta=\theta_0}.
\label{mh}
\end{eqnarray}
In Fig.\,\ref{fig:massvevratio}, $m_h/v_{\rm SM}$ is described as a function of $\kappa$ with fixed beta functions $\beta=\beta_{\rm SM}$ (red), $\beta=2\beta_{\rm SM}$ (blue) and $\beta=5\beta_{\rm SM}$ (blue).
For the minimal case ($\beta=\beta_{\rm SM}$), it is very difficult to obtain the observed value, $m_h/v_{\rm SM}\simeq 0.5$
unless $\lambda_{\rm mix}$ stands at the edge of the allowed region.
In this plot, we estimated $\beta_{\rm SM}$ at $Z$ boson mass scale ($\mu=M_Z$), so more precise estimation will make the situation worse.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.47\textwidth]{Vhat2}
\end{center}
\caption{ Potential $\hat V$ along the flat direction is plotted with different $\kappa=\lambda_{\rm mix}/\beta$ values for a fixed $\beta=\beta_{\rm SM}$.
If $\kappa<1/2$, potential minima are located at $\theta=0$ and $\pi/2$ which correspond to $v_{\rm SM}=0$.
For $\kappa>(3-2\log2)/2\simeq 0.81$, potential minimum is at $\theta=\pi/4$ where $Z_2$ symmetry is not boken spontaneously.}
\label{fig:Vhat}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.47\textwidth]{massvevratio2}
\end{center}
\caption{ $m_h/v_{\rm SM}$ is plotted as a function of $\kappa$ with different $\beta=\beta_{\rm SM}$ (red), $2\beta_{\rm SM}$ (purple) and $5\beta_{\rm SM}$ (blue).
Dashed line indicates the observed value, $m_h/v_{\rm SM}=125/246$.
For the minimal model (blue), $\kappa$ should be on the left conner from which one can read that a large fine tuning is required.
The situation is worse than this plot because $\beta_{\rm SM}$ is over-estimated as we take $\mu=M_Z$.
}
\label{fig:massvevratio}
\end{figure}
To investigate further, we expand the potential around $\theta=0$ with assuming $\kappa \simeq 1/2$, and we obtain
\begin{equation}
\hat V \simeq \frac{\beta}{4} \left( \kappa - \frac{1}{2} \right) \theta^2 +\beta \left(\frac{11}{48}-\frac{1}{3}\kappa \right) \theta^4 + \frac{\beta}{4} \theta^4 \log \theta
\label{Vhat_pol}
\end{equation}
where the first two terms are negative and the last logarithmic term is positive near $\theta \simeq 0$.
By multiplying $f^4$ and replacing $f \theta = h$, we can match Eq.\,\eqref{Vhat_pol} to the SM potential
\begin{eqnarray}
f^4 \hat V \simeq -\frac{m^2}{2} h^2 + \frac{\lambda_h}{4} h^4 + \frac{\beta}{4}h^4\log \frac{h}{\mu},
\end{eqnarray}
where
\begin{equation}
m^2 = -\frac{\beta}{4}\left(\kappa-\frac{1}{2} \right)f^2, ~~~
\lambda_h (\mu=f) = \frac{1}{16}\beta.
\end{equation}
For the observed Higgs mass and vev, we need $m \simeq m_h/\sqrt{2}$ and $\lambda_h (\mu=M_Z) \simeq m_h^2/2v_{\rm SM}^2$.
Thus, we obtain
\begin{eqnarray}
\kappa - \frac{1}{2} = \frac{2}{(-\beta)} \frac{m_h^2}{f^2},
\label{finetuning}
\end{eqnarray}
and $f$ is determined by the scale where $\lambda_h(\mu=f)= \beta/16 \lsim 0$ with boundary condition $\lambda_h (\mu=M_Z) \simeq m_h^2/2v_{\rm SM}^2$.
Thus, the metastability of the Higgs boson at the IR theory is a typical consequence in this scenario.
Eq.\,\eqref{finetuning} tells that the tuning of quartic coupling $\lambda_{\rm mix}$ in ultra-violet (UV) theory represents the tuning of Higgs quadratic coupling in the IR theory.
Order of tuning is alleviated by the factor of $\beta$, but is basically ${\cal O}(m_h^2/f^2)$.
For the theory to be natural, the scale $f$ should not be very far away from the weak scale.
However, the prediction of the minimal model is $f \simeq 10^{10}\,\mathrm{GeV}$\,\cite{Buttazzo:2013uya}, so it is very unnatural.
\section{Vectorlike leptons}
A possible way to alleviate tuning is introducing new Yukawa interactions.
Additional Yukawa interactions can give additional negative contributions to $\beta$, and make $f$ smaller.
It can also be seen in Fig.\,\ref{fig:massvevratio} that if $\beta$ is larger than the SM value (purple and blue curves), the slope at $m_h/v_{\rm SM}$ could be small, so the tuning of $\lambda_{\rm mix}/\beta$ can be milder.
As an example, we consider a family of vectorlike leptons (VLL) in each sector:
lepton doublets $L_{Li}$, $\bar L_{Li}$, singlet charged leptons $E_{Ri}$, $\bar E_{Ri}$, singlet neutral leptons $N_{Ri}$ and $\bar N_{Ri}$ for $i=A,\,B$.
Their interaction Lagrangian become
\begin{eqnarray}
\hspace{-1cm}
&&\hspace{-0.2cm} {\cal L}_{\rm VLL}= - M_L \bar L_{Li} L_{Ri} -M_E \bar E_{Li} E_{Ri} -M_N \bar N_{Li} N_{Ri}
\\
&&\hspace{-0.2cm}- y_E \bar L_{Li} E_{Ri} H_i - \bar y_E \bar E_{Li} L_{Ri} H_i^\dagger
- y_N \bar L_{Li} N_{Ri} H_i^\dagger- \bar y_N \bar N_{Li} L_{Ri} H_i, \nonumber
\end{eqnarray}
where the summation of $i=A,\,B$ is ommitted in the expression.
Although there are several implications of VLL for the case when they couple to the SM leptons\,\cite{Fujikawa:1994we,Kannike:2011ng,Dermisek:2013gta,Poh:2017tfo,DEramo:2007anh,Cohen:2011ec,Restrepo:2015ura,Calibbi:2015nha,Bhattacharya:2015qpa,Bhattacharya:2017sml}, we do not discuss them in this paper because their coupling to the SM leptons do not change the following discussions.
Mass mastix of charged and neutral VLL are given by
\begin{equation}
{\cal M}_{Ei}=
\left( \begin{array}{cc}
M_L & \frac{y_E v_i}{\sqrt{2}}\\
\frac{\bar y_E v_i}{\sqrt{2}} & M_E
\end{array} \right),~~
{\cal M}_{Ni}=
\left( \begin{array}{cc}
M_L & \frac{y_N v_i}{\sqrt{2}}\\
\frac{\bar y_N v_i}{\sqrt{2}} & M_N
\end{array} \right).
\end{equation}
For simplicity, we assume that $M_L=M_E=M_N$ and $y_L \equiv y_E= y_N$ and$\bar y_E = \bar y_N=0 $.
The eigenvalues of ${\cal M} {\cal M}^\dagger$ become
\begin{eqnarray}
M_{Li \pm}=M_L^2 + \frac{v_i^2y_L^2}{4} \pm \frac{1}{4}\sqrt{8M_L^2 y_L^2 v_i^2+y_L^4 v_i^4},
\end{eqnarray}
for $i=A,\,B$.
The effective potential from VLL in each sector is given by
\begin{equation}
-8\times \frac{1}{64\pi^2} \left( M_{Li+}^4 \log \frac{M_{Li+}^2}{\mu^2}+ M_{Li-}^4 \log \frac{M_{Li-}^2}{\mu^2} \right)
\label{effV_VLL}
\end{equation}
which can provide large enough $c_2$ when $M_L \lsim f$.
If $M_L \gsim f$, $c_2$ become suppressed by $(f/M_L)^4$, and Eq.\,\eqref{effV_VLL} can give only the threshold correction to $\lambda_{\rm mix}$.
Collider signatures of VLL are highly sensitive on the mixing with SM leptons\,\cite{AguilarSaavedra:2009ik,Redi:2013pga,Falkowski:2013jya,Holdom:2014rsa,Kumar:2015tna,Bertuzzo:2017wam,Kling:2018wct}.
The mass limit for charged leptons from the Large Electron-Positron collider (LEP) is $100.8\,\mathrm{GeV}$ when the charged lepton mostly decays to $W\nu$.
For neutral leptons, the mass limit from LEP is $101.3\,\mathrm{GeV}$ when they decay to $We$\,\cite{Achard:2001qw, Tanabashi:2018oca}.
At Large Hadron Collider (LHC), the most relevant search is Refs.\,\cite{Sirunyan:2018mtv} which provides constraints on the CKM matrix elemant $|V_{eN}|$ and $|V_{\mu N}|$ in the mass range from GeV to TeV.
At this moment, LHC constraints are comparable to LEP constraints\,\cite{Sirunyan:2018mtv}, but there will be much improvement in the future.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.47\textwidth]{final2}
\end{center}
\caption{ $SU(4)$ breaking scale $f$ is determined in the $(M_L/f,\, y_L)$ parameter space. Colors represent the scale of $f$. Black solid lines correspond to lightest lepton mass $100\,\mathrm{GeV}$, $500\,\mathrm{GeV}$ and $1\,\mathrm{TeV}$.}
\label{fig:final}
\end{figure}
Fig.\,\ref{fig:final} shows the $SU(4)$ breaking scale $f$ in $M_L/f$ and $y_L$ parameter space by colors.
Black solid lines correspond to lightest lepton mass $100\,\mathrm{GeV}$, $500\,\mathrm{GeV}$ and $1\,\mathrm{TeV}$.
We restrict Yukawa coupling to be smaller than $1.7$ because the Landau pole arises below $100\,\mathrm{TeV}$ if it is larger.
The smallest $f$ within $m_{L-}>100\,\mathrm{GeV}$, $500\,\mathrm{GeV}$ and $1\,\mathrm{TeV}$ are $f\gsim 1.3\,\mathrm{TeV}$, $2.7\,\mathrm{TeV}$ and $3.9\,\mathrm{TeV}$, respectively.
Another advantage of VLL comes from changing RG running of top Yukawa coupling.
In the minimal model, top Yukawa coupling rapidly drops because of large $SU(3)_c$ gauge coupling.
If there are additional Yukawa interactions, they compensate negative contributions of gauge coupling and make the pseudo-IR fixed point smaller.
Consequently, top Yukawa coupling can maintain its strength until $\mu\sim f$.
We neglect this effect in Fig.\,\ref{fig:final}, so $f$ can have slightly smaller value.
Charged leptons with large Yukawa coupling can modify Higgs to diphoton signal strength through the loop process.
We predict less than $2\,\%$ deviation of the partial decay rate of Higgs to diphoton process from the SM value.
Its smallness is because $W$ boson loop is the dominant contribution compared to fermionic loops in the Higgs to diphoton amplitude, and
VLL contributions are basically suppressed by $v_{\rm SM}^2/m_L^2$.
\section{Conclusion}
We have discussed the possibility of spontaneous twin symmetry breaking scenario.
For the misalignment of nonzero twin Higgs vevs, there should be $\cos 8\theta$ term in $\hat V(\theta)$ which can come from the Coleman-Weinberg potential.
In addition, we need a cancellation of ${\cal O}(v^2/\beta f^2)$ between $\lambda_{\rm mix}$ and $\beta/2$.
The $SU(4)$ breaking scale $f$ is determined by the scale where the Higgs self quartic coupling flips its sign in the IR theory.
Since $f$ in the minimal setup is ${\cal O}(10^{10}\,\mathrm{GeV})$, we introduced twin VLLs with large Yukawa couplings to improve the naturalness.
With VLLs, $f$ can be reduced as small as $1.3\,\mathrm{TeV}$.
Potentially, VLLs are testable at the LHC or future lepton colliders through Higgs measurement and direct production depending on the mixing with SM leptons.
Their signatures below TeV scale would lend credence to this scenario.
Although we do not specify the inflation sector, we have assumed that reheaton decays mostly to the sector with smaller Higgs vev in order to avoid cosmological problems.
The reheating temperature of the SM sector should be less than around bottom quark mass for preventing thermal production of twin sector particles.
An interesting possibility is that cogenesis of baryon asymmetry and asymmetric dark matter could occur during the reheating process.
We leave detailed studies about the cosmological history and the inflation sector as future work.
\noindent{\bf Acknowledge}
THJ is grateful to Chang Sub Shin, Kyu Jung Bae and Dongjin Chway for useful discussions.
This work was supported by IBS under the project code, IBS-R018-D1.
|
{
"timestamp": "2019-03-01T02:13:58",
"yymm": "1902",
"arxiv_id": "1902.10978",
"language": "en",
"url": "https://arxiv.org/abs/1902.10978"
}
|
\section{Related Works}
In this section, we relate recent works in the fields of network embedding (NE) and attention mechanism for natural language processing (NLP).
\subsection{Attributed Network Embedding}
\textit{DeepWalk} \cite{perozzi2014deepwalk} first proposed to derive the word embedding algorithm \textit{Word2vec} \cite{mikolov2013efficient} by generating paths of nodes, akin to sentences, with truncated random walks. \textit{DeepWalk} and other variants are generalized into a common matrix factorization framework in \textit{NetMF} \cite{qiu2018network}.
To extend \textit{DeepWalk} for text-attributed networks, \textit{TADW} \cite{yang2015network} expresses this latter as a matrix factorization problem and incorporates a matrix of textual features $T$, produced by latent semantic indexing (\textit{LSI}), into the factorization so that the vertex similarity matrix can be reconstructed as the product of three matrices $V^T$, $H$ and $T$.
\subsection{Attention Mechanisms for NLP}
The \textit{Transformer} \cite{vaswani2017attention} is a novel neural architecture that outperforms state-of-the-art methods in neural machine translation (NMT) without the use of convolution nor recurrent units.
The Scaled Dot-Product Attention (SDPA) is the main constituting part of the \textit{Transformer} that actually performs attention over a set of words. It takes as input a query vector $q$ and a set of key vectors $K$ of dimensions $d_k$ and value vectors $V$ of dimensions $d_v$. One weight for a value is generated by a compatibility function with its corresponding key and the query.
Formally, the attention vectors are generated in parallel for multiples queries $Q$, following the formula: $ \text{Attention}(Q,K,V)= \text{softmax}(\frac{QK^T}{\sqrt{d_k}})V$.
The result is a set of $L$ attention vectors ($L$ being the number of queries) of dimension $d_v$. The matrices $Q$, $K$ and $V$ are produced by projection of initial words representations $W$ with three matrices $P^Q$, $P^K$ and $P^V$ whose parameters are meant to be learned.
Several works \cite{devlin2018bert, radford2018improving} adapted the \textit{Transformer} architecture beyond the task of NMT. Their main idea is to train the \textit{Transformer} in an unsupervised fashion over large corpora of texts and further refine its parameters on a wide variety of supervised tasks. Motivated by these recent works, we present a model, \textit{MATAN} (\textbf{M}utual \textbf{A}ttention for \textbf{Te}xt-\textbf{A}ttributed \textbf{N}etworks), that derives from the the SDPA to address the task of link prediction in a network of documents.
\section{Proposed Model}
We propose an algorithm for link prediction in text-attributed network. Our model is trained under a NE procedure, presented in Section \ref{sec:overall}. The optimization of the reconstruction error is performed via dot-product between contextual document representations $e^{v}_u$ and $e^{u}_v$. These embeddings are generated with a mutual attention mechanism over their textual contents only, described in Section \ref{sec:attention}.
\subsection{Overall Optimization} \label{sec:overall}
The model takes as input a network of documents $G=(V,E,T)$, $T$ being the textual content of the documents. We precompute word embeddings $W$ and a normalized similarity measure between nodes $M$ designed from the adjacency matrix $A$ of the network. Each document $t_u$ is associated with a bag of word embeddings $W^{t_u}$ matrix. For any pair of node $(u,v) \in V^2$, mutual embeddings are generated with an asymmetric mutual attention function $f^A_{\Theta}$ for both documents given their bags of word embeddings $e^{v}_u = f^A_{\Theta}(W^{t_u}, W^{t_v})$ and $e^{u}_v = f^A_{\Theta}(W^{t_v}, W^{t_u})$. We define the unormalized similarity between the two nodes as the dot product of their mutual embeddings $e^{v}_u.e^{u}_v$. We aim at learning the parameters $\Theta$ by minimizing the KL divergence from the similarity distributions $M$ (from the graph) to that of the normalized distribution of the dot products between the mutual embeddings \cite{tsitsulin2018verse} (text associated to the nodes). We achieve this by employing noise-contrastive estimation \cite{tsitsulin2018verse}, minimizing the following objective function: $ J=-\sum_{(u,v) \in C} \Big( \log \sigma(e^{v}_u \cdot e^{u}_v) + \sum_{i=1}^k \mathbb{E}_{z\sim q} \big [ \log \sigma(-e^{z}_u \cdot e^{u}_z) \big ] \Big)$, where $\sigma$ is the sigmoid function. $C$ is a corpus of pairs of nodes generated by drawing uniformly existing links from the empirical distribution of links $M$. k negative nodes are uniformly drawn for each positive pair. To minimize this objective, we employ stochastic gradient descent using ADAM \cite{kingma2014adam}.
\subsection{Mutual Attention Mechanism} \label{sec:attention}
The role of $f^A_{\Theta}(W^{t_u}, W^{t_v})$ is to generate a contextual representation of $t_u$ given $t_v$. The parameters $\Theta$ we aim to learn are composed of three matrices $\Theta = \{P^Q, P^K, P^V\}$ of dimension $D \times D$ each. For all words of the target document $t_u$, we create queries $Q_u = W^{t_u} P^Q$. We similarly create keys and values from the contextual document $t_v$, such that $K_v = W^{t_v} P^Q$ and $V_v = W^{t_v} P^V$. Attention representations for each target word are then computed, following the SDPA formula: $ \text{SDPA}_{\Theta}(W^{t_u}, W^{t_v}) = \text{softmax}(\frac{Q_uK_v^T}{\sqrt{D}})V_v $. Note that $\text{SDPA}_{\Theta}(W^{t_u}, W^{t_v})$ has dimension $L \times D$, that is, we have a mutual attention representation of each word of document $t_u$ given $t_v$. Finally, the representation for document $t_u$ is obtained by averaging its word mutual attention vectors: $ e^{v}_u = f^A_{\Theta}(W^{t_u}, W^{t_v}) = \sum_{i=0}^L \text{SDPA}_{\Theta}(W^{t_u}, W^{t_v})_i $. Similarly, $e^{u}_v$ is generated by flipping indices $u$ and $v$. The intuition behind this model is that the matrices $P^Q$ and $P^K$ learn to project pairs of words that explain links in the network such that their dot-products produce large weights. $P^V$ is then meant to project the word vectors such that their average produces similar representations for nodes that are close in the network and dissimilar for nodes that are far in the network.
\section{Experiments}
To assess the quality of our model, we perform two tasks of link prediction on a dataset of citation links between scientific abstracts: Cora \footnote{Get the data: https://linqs.soe.ucsc.edu/data}. The first prediction evaluation, called edges-hidden, consists in hiding a percentage of the links given a network of documents and measuring the ability of the model to predict higher scores to hidden links than to non-existing ones by computing the ROC AUC. The second evaluation, called nodes-hidden, consists in splitting the network into two unconnected networks, keeping a percentage of the nodes in the training network.
We precompute on the full corpus word embeddings using \textit{GloVe} \cite{pennington2014glove} of dimension 256 with a co-occurrence threshold $x_\text{max}=10$, a window size $w=5$ and 50 epochs. We precompute \textit{LSI} \cite{deerwester1990ilsa} vectors of dimension 128.
For the edge-hidden prediction task, we provide results performed by \textit{NetMF} with $k=10$ negative samples. \textit{TADW} is run with $20$ epochs and \textit{MATAN} is performed with $k=1$ negative sample and $10^5$ sampled pairs of documents. The empirical similarity matrix between the nodes we chose is the normalized adjacency matrix. All produced representations are of dimension 256.
\subsection{Results}
\begin{table}[h]
\center
\caption{Edges-hidden link prediction ROC AUC}
\label{table:edges-hidden}
\begin{tabular}{l|ccccc}
\% of training data &10\% &20\% &30\% &40\% &50\%\\ \hline
\textit{NeMF} &59.0 &67.2 &77.5 &83.2 &87.2 \\
\textit{TADW} &68.0 &82.0 &87.1 &\textbf{93.2} &\textbf{94.5} \\
\textit{MATAN} &\textbf{82.3} &\textbf{87.1} &\textbf{88.6} &90.9 &91.0 \\
\end{tabular}
\end{table}
\begin{table}[h]
\center
\caption{Nodes-hidden link prediction ROC AUC}
\label{table:nodes-hidden}
\begin{tabular}{l|ccccc}
\% of training data &10\% &20\% &30\% &40\% &50\%\\ \hline
\textit{TADW} &64.2 &\textbf{75.8} &\textbf{80.3} &\textbf{81.9} &\textbf{82.3} \\
\textit{MATAN} &\textbf{69.4} &73.0 &75.4 &77.9 &78.6 \\
\end{tabular}
\end{table}
Tables \ref{table:edges-hidden} and \ref{table:nodes-hidden} show the results of our experiments. \textit{MATAN} shows promising results for learning on a small percentage of training data on both evaluations. \textit{TADW} has better scores for nodes-hidden predictions which might be explained by the capacity of \textit{LSI} to learn discriminant features on a small dataset unlike \textit{GloVe}. In future work we would like to deal with bigger datasets from which word embedding methods might capture richer semantic information.
\bibliographystyle{ACM-Reference-Format}
\balance
|
{
"timestamp": "2019-03-21T01:13:40",
"yymm": "1902",
"arxiv_id": "1902.11054",
"language": "en",
"url": "https://arxiv.org/abs/1902.11054"
}
|
\section{Introduction}
Learning to synthesize the image of person conditioned on the image of clothes and manipulate the pose simultaneously is a significant and valuable task in many applications such as virtual try-on, virtual reality, and human-computer interaction. In this work, we propose a multi-stage method to synthesize the image of person conditioned on both clothes and pose. Given an image of a person, a desired clothes, and a desired pose, we generate the realistic image that preserves the appearance of both desired clothes and person, meanwhile reconstructing the pose, as illustrated in Figure~\ref{fig:fig1}. Obviously, delicate and reasonable synthesized outfit with arbitrary pose is helpful for users in selecting clothes while shopping.
\begin{figure}[!tp]
\centering
\includegraphics[width=1.0\hsize \hspace{0.01\hsize}]{fig/fig1_new.pdf}
\caption{Some results of our model by manipulating both various clothes and diverse poses. The input image of the clothes and poses are shown in the first row, while the input images of the person are shown in the first column. The results manipulated by both clothes and pose are shown in the other columns.}
\label{fig:fig1}
\vspace{-6mm}
\end{figure}
However, recent image synthesis approaches~\cite{han2017viton,wang2018cpvton} for virtual try-on mainly focus on the fixed pose and fail to preserve the fine details, such as the clothing of lower-body and the hair of the person lose the details and style, as shown in Figure~\ref{fig:vs_others}. In order to generate the realistic image, those methods apply a coarse-to-fine network to produce the image conditioned on clothes only. They ignore the significant features of the human parsing, which leads to synthesize blurry and unreasonable image, especially in case of conditioned on various poses. For instance, as shown in Figure~\ref{fig:vs_others}, the clothing of lower-body cannot be preserved while the clothing of upper-body is replaced. The head of the person fail to identify while conditioned different poses. Other exiting works~\cite{laehner2018deepwrinkles,pons2017clothcap,zhang2017detailed} usually leverage 3D measurements to solve those issues since the 3D information have abundant details of the shape of the body that can help to generate the realistic results. However, it needs expert knowledge and huge labor cost to build the 3D models, which requires collecting the 3D annotated data and massive computation. These costs and complexity would limit the applications in the practical virtual try-on simulation.
In this paper, we study the problem of virtual try-on conditioned on 2D images and arbitrary poses, which aims to learn a mapping function from an input image of a person to another image of the same person with a new outfit and diverse pose, by manipulating the target clothes and pose. Although the image-based virtual try-on with the fixed pose has been studied widely~\cite{han2017viton,wang2018cpvton,zhu2017fashionGAN}, the task of multi-pose virtual try-on is less explored. In addition, without modeling the mapping of the intricate interplay among of the appearance, the clothes, and the pose, directly using the existing virtual try-on methods to synthesized image based on different poses often result in blurry and artifacts.
Targeting on the problems mentioned above, we propose a novel Multi-pose Guided Virtual Try-on Network (MG-VTON) that can generate a new person image after fitting both desired clothes into the input image and manipulating human poses. Our MG-VTON is a multi-stage framework with generative adversarial learning. Concretely, we design a pose-clothes-guided human parsing network to estimate a plausible human parsing of the target image conditioned on the approximate shape of the body, the face mask, the hair mask, the desired clothes, and the target pose, which could guide the synthesis in an effective way with the precise region of body parts. To seamlessly fit the desired clothes on the person, we warp the desired clothes image, by exploiting a geometric matching model to estimate the transformation parameters between the mask of the input clothes image and the mask of the synthesized clothes extracted from the synthesized human parsing. In addition, we design a deep Warping Generative Adversarial Network (Warp-GAN) to synthesize the coarse result alleviating the large misalignment caused by the different poses and the diversity of clothes. Finally, we present a refinement network utilizing multi-pose composition masks to recover the texture details and alleviate the artifact caused by the large misalignment between the reference pose and the target pose.
To demonstrate our model, we collected a new dataset, named MPV, by collecting various clothes image and person images with diverse poses from the same person. In addition, we also conduct experiments on DeepFashion~\cite{liu2016deepfashion} datasets for testing. Following the object evaluation protocol~\cite{wang2017pix2pixHD}, we conduct a human subjective study on the Amazon Mechanical Turk (AMT) platform. Both quantitative and qualitative results indicate that our method achieves effective performance and high-quality images with appealing details. The main contributions are listed as follows:
\begin{itemize}
\item A new task of virtual try-on conditioned on multi-pose is proposed, which aims to restructure the person image by manipulating both diverse poses and clothes.
\item We propose a novel Multi-pose Guided Virtual Try-on Network (MG-VTON) that generates a new person image after fitting the desired clothes onto the input person image and manipulating human poses. MG-VTON contains four modules: 1) a pose-clothes-guided human parsing network is designed to guide the image synthesis; 2) a Warp-GAN learns to synthesized realistic image by using a warping features strategy; 3) a refinement network learns to recover the texture details; 4) a mask-based geometric matching network is presented to warp clothes that enhances the visual quality of the generated image.
\item A new dataset for the multi-pose guided virtual try-on task is collected, which covers person images with more poses and clothes diversity. The extensive experiments demonstrate that our approach can achieve the competitive quantitative and qualitative results.
\end{itemize}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.9\hsize \hspace{0.01\hsize}]{fig/pipeline_4.pdf}
\caption{The overview of the proposed MG-VTON. Stage I: We first decompose the reference image into three binary masks. Then, we concatenate them with the target clothes and target pose as an input of the conditional parsing network to predict human parsing map. Stage II: Next, we warp clothes, remove the clothing from the reference image, and concatenate them with the target pose and synthesized parsing to synthesize the coarse result by using Warp-GAN. Stage III: We finally refine the coarse result with a refinement render, conditioning on the warped clothes, target pose, and the coarse result.}
\label{fig:test_pipeline}
\vspace{-4mm}
\end{figure*}
\section{Related Work}
\textbf{Generative Adversarial Networks (GANs).}
GANs~\cite{goodfellow2014generative} consists of a generator and a discriminator that the discriminator learns to classify between the synthesized images and the real images while the generator tries to fool the discriminator. The generator aims to generate realistic images, which are indistinguishable from the real images. And the discriminator focuses on distinguishing between the synthesized and real images. Existing works have leveraged various applications based on GANs, such as style transfer~\cite{isola2017pix2pix,zhu2017cycleGAN,kim2017discoGAN,yi2017dualgan}, image inpainting~\cite{Yang2017inpainting}, text-to-image~\cite{reed2016text2image}, and super-resolution imaging~\cite{ledig2016photo}. Inspired by those impressive results of GANs, we also apply the adversarial loss to exploit a virtual try-on method with GANs.
\textbf{Person image synthesis.}
Skeleton-aided~\cite{yan2017skeleton} proposed a skeleton-guided person image generation method, which conditioned on a person image and the target skeletons. PG2~\cite{ma2017pose} applied a coarse-to-fine framework that consists of a coarse stage and a refined stage. Besides, they proposed a novel model~\cite{ma2017disentangled} to further improve the quality of result by using a decomposition strategy. The deformableGANs~\cite{siarohin2017deformable} and \cite{balakrishnan2018synthesizing} made attempt to alleviate the misalignment problem between different poses by using affine transformation on the coarse rectangle region and warped the parts on pixel-level, respectively. V-UNET~\cite{Esser2018vunet} introduced a variational U-Net~\cite{ronn2015unet} to synthesize the person image by restructuring the shape with stickman label. \cite{pumarola2018unsupervised} applied CycleGAN~\cite{zhu2017cycleGAN} directly to manipulate pose. However, all those works fail to preserve the texture details consistency corresponding with the pose. The reason behind that is they ignore to consider the interplay between the human parsing map and the pose in the person image synthesis. The human parsing map can guide the generator to synthesize image in the precise region level that ensures the coherence of body structure.
\textbf{Virtual try-on.}
VITON~\cite{han2017viton} and CP-VTON~\cite{wang2018cpvton} all presented an image-based virtual try-on network, which can transfer a desired clothes on the person by using a warping strategy. VITON computed the transformation mapping by the shape context TPS warps~\cite{belongie2002shape} directly. CP-VTON introduced a learning method to estimate the transformation parameters. FashionGAN~\cite{zhu2017fashionGAN} learned to generate new clothes on the input image of the person conditioned on a sentence describing the different outfit. However, the above all methods synthesized the image of person only on the fixed pose, which limits the applications in the practical virtual try-on simulation. ClothNet~\cite{lassner2017generative} presented an image-based generative model to produce new clothes conditioned on color. CAGAN~\cite{jetchev2017conditional} proposed a conditional analogy network to synthesize person image conditioned on the paired of clothes, which limits the practical virtual try-on scenarios. In order to generate the realistic-look person image in different clothes, ClothCap~\cite{pons2017clothcap} utilized the 3D scanner to capture the clothes, the shape of the body automatically. \cite{sekine2014virtual} presented a virtual fitting system that requires the 3D body shape, which is laborious for collecting the annotation. In this paper, we introduce a novel and effective method for learning to synthesize image with the new outfit on the person through adversarial learning, which can manipulate the pose simultaneously.
\begin{figure*}[!ht]
\centering
\includegraphics[width=1.0\hsize \hspace{0.01\hsize}]{fig/pipeline_5.pdf}
\caption{The network architecture of the proposed MG-VTON. (a)(b): The conditional parsing learning module consists of a pose-clothes-guided network that predicts the human parsing, which helps to generate high-quality person image. (c)(d): The Warp-GAN learns to generate the realistic image by using a warping features strategy due to the misalignment caused by the diversity of pose. (e): The refinement render network learns the pose-guided composition mask that enhances the visual quality of the synthesized image. (f): The geometric matching network learns to estimate the transformation mapping conditioned on the body shape and clothes mask.}
\label{fig:train_pipeline}
\vspace{-4mm}
\end{figure*}
\section{MG-VTON}
We propose a novel Multi-pose Guided Virtual Try-on Network (MG-VTON) that learns to synthesize the new person image for virtual try-on by manipulating both clothes and pose. Given an input person image, a desired clothes, and a desired pose, the proposed MG-VTON aims to produce a new image of the person wearing the desired clothes and manipulating the pose. Inspired by the coarse-to-fine idea~\cite{han2017viton,ma2017pose}, we adopt an outline-coarse-fine strategy that divides this task into three subtasks, including the conditional parsing learning, the Warp-GAN, and the refinement render. The Figure~\ref{fig:test_pipeline} illustrates the overview of MG-VTON.
We first apply the pose estimator~\cite{cao2017openpose} to estimate the pose. Then, we encode the pose as 18 heatmaps, which is filled with ones in a circle with radius 4 pixels and zeros elsewhere. A human parser~\cite{gong2017look} is used to predict the human segmentation maps, consisting of 20 labels, from which we extract the binary mask of the face, the hair, and the shape of the body. Following VITON~\cite{han2017viton}, we downsample the shape of the body to a lower resolution ($16 \times 12$) and directly resize it to the original resolution ($256 \times 192$), which alleviates the artifacts caused by the variety of the body shape.
\subsection{Conditional Parsing Learning}
To preserve the structural coherence of the person while manipulating both clothes and the pose, we design a pose-clothes-guided human parsing network, conditioned on the image of clothes, the pose heatmap, the approximated shape of the body, the mask of the face, and the mask of hair. As shown in Figure~\ref{fig:vs_others}, the baseline methods failed to preserve some parts of the person (e.g., the color of the trousers and the style of hair were replaced.), due to feeding the image of the person and clothes into the model directly. In this work, we leverage the human parsing maps to address those problems, which can help generator to synthesize the high-quality image on parts-level.
Formally, given an input image of person $I$, an input image of clothes $C$, and the target pose $P$, this stage learns to predict the human parsing map $S^{'}_t$ conditioned on clothes $C$ and the pose $P$. As shown in Figure~\ref{fig:train_pipeline} (a), we first extract the hair mask $M_h$, the face mask $M_f$, the body shape $M_b$, and the target pose $P$ by using a human parser~\cite{gong2017look} and a pose estimator~\cite{cao2017openpose}, respectively. We then concatenate them with the image of clothes as input which is fed into the conditional parsing network. The inference of $S^{'}_t$ can be formulate as maximizing the posterior probability $p(S^{'}_t | (M_h, M_f, M_b, C, P))$. Furthermore, this stage is based on the conditional generative adversarial network (CGAN)~\cite{mirza2014cgan} which generates promising results on image manipulating. Thus, the poster probability $p(S^{'}_t | (M_h, M_f, M_b, C, P))$ is expressed as:
\begin{equation}
p(S^{'}_t | (M_h, M_f, M_b, C, P)) = G(M_h, M_f, M_b, C, P).
\end{equation}
We adopt a ResNet-like network as the generator $G$ to build the conditional parsing model. We adopt the discriminator $D$ directly from the pix2pixHD~\cite{wang2017pix2pixHD}. We apply the L1 loss for further improving the performance, which is advantageous for generating more smooth results~\cite{yan2017skeleton}. Inspired by the LIP~\cite{gong2017look}, we apply the pixel-wise softmax loss to encourage the generator to synthesize high-quality human parsing maps. Therefore, the problem of conditional parsing learning can be formulated as:
\begin{equation}
\begin{aligned}
&\min_{G} \max_{D} F(G, D) \\
& = \mathbb{E}_{M, C, P \sim p_{\text{data}}}[ \log (1 - D(G(M, C, P), M, C, P))] \\
& + \mathbb{E}_{S_t, M, C, P \sim p_{\text{data}}}[\log D(S_t, M, C, P)] \\
& + \mathbb{E}_{S_t, M, C, P \sim p_{\text{data}}}[\| S_t - G(M, C, P)\|_1] \\
& + \mathbb{E}_{S_t, M, C, P \sim p_{\text{data}}}[\mathcal{L}_{\text{parsing}}(S_t, G(M, C, P))],
\label{eq:parsing}
\end{aligned}
\end{equation}
where $M$ denotes the concatenation of $M_h, M_f$, and $M_b$. The loss $\mathcal{L}_{\text{parsing}}$ denotes the pixel-wise softmax loss~\cite{gong2017look}. The $S_t$ denotes the ground truth human parsing. The $p_{\text{data}}$ represents the distributions of the real data.
\subsection{Warp-GAN}
\label{s:wg}
Since the misalignment of pixels would lead to generate the blurry results~\cite{siarohin2017deformable}, we introduce a deep Warping Generative Adversarial Network (Warp-GAN) warps the desired clothes appearance into the synthesized human parsing map, which alleviates the misalignment problem between the input human pose and desired human pose. Different from deformableGANs~\cite{siarohin2017deformable} and \cite{balakrishnan2018synthesizing}, we warp the feature map from the bottleneck layer by using both the affine and TPS (Thin-Plate Spline)~\cite{bookstein1989tps} transformation rather than process the pixel directly by using affine only. Thanks to the generalization capacity of~\cite{Rocco2017geocnn}, we directly use the pre-trained model of \cite{Rocco2017geocnn} to estimate the transformation mapping between the reference parsing and the synthesized parsing. We then warp the w/o clothes reference image by using this transformation mapping.
As illustrated in Figure~\ref{fig:train_pipeline} (c) and (d), the proposed deep warping network consists of the Warp-GAN generator $G_{\text{warp}}$ and the Warp-GAN discriminator $D_{\text{warp}}$. We use the geometric matching module to warp clothes image, as described in the section~\ref{s:gml}. Formally, we take warped clothes image $C_{\text{w}}$, w/o clothes reference image $I_{\text{w/o\_clothes}}$, the target pose $P$, and the synthesized human parsing $S^{'}_t$ as input of the Warp-GAN generator and synthesize the result $\hat{I} = G_{\text{warp}}(C_{\text{w}}, I_{\text{w/o\_clothes}}, P, S^{'}_t)$.
Inspired by~\cite{johnson2016perceptual,han2017viton,ledig2016photo}, we apply a perceptual loss to measure the distances between high-level features in the pre-trained model, which encourages generator to synthesize high-quality and realistic-look images. We formulate the perceptual loss as:
\begin{equation}
\mathcal{L}_{\text{perceptual}}(\hat{I}, I) = \sum_{i=0}^{n} \alpha_{i} \| \phi_{i}(\hat{I}) - \phi_{i}(I) \|_{1},
\end{equation}
where $\phi_{i}(I)$ denotes the $i$-th $(i=0,1,2,3,4)$ layer feature map in pre-trained network $\phi$ of ground truth image $I$. We use the pre-trained VGG19~\cite{simonyan2015very} as $\phi$ and weightedly sum the L1 norms of last five layer feature maps in $\phi$ to represent perceptual losses between images. The $\alpha_{i}$ controls the weight of loss for each layer.
In addition, following pixp2pixHD~\cite{wang2017pix2pixHD}, due to the feature map at different scales from different layers of discriminator enhance the performance of image synthesis, we also introduce a feature loss and formulate it as:
\begin{equation}
\mathcal{L}_{\text{feature}}(\hat{I}, I) = \sum_{i=0}^{n} \gamma_{i} \| F_{i}(\hat{I}) - F_{i}(I) \|_{1},
\label{eq:feature}
\end{equation}
where $F_i(I)$ represent the $i$-th $(i=0,1,2)$ layer feature map of the trained $D_{\text{warp}}$. The $\gamma_{i}$ denotes the weight of L1 loss for corresponding layer.
Furthermore, we also apply the adversarial loss $\mathcal{L}_{\text{adv}}$~\cite{goodfellow2014generative,mirza2014cgan} and L1 loss $\mathcal{L}_{\text{1}}$~\cite{yan2017skeleton} to improve the performance. We design a weight sum losses as the loss of $G_{\text{warp}}$, which encourages the $G_{\text{warp}}$ to synthesize realistic and natural images in different aspects, written as follows:
\begin{equation}
\mathcal{L}_{G_{\text{warp}}} = \lambda_{1} \mathcal{L}_{\text{adv}} + \lambda_{2} \mathcal{L}_{\text{perceptual}} + \lambda_{3} \mathcal{L}_{\text{feature}} + \lambda_{4} \mathcal{L}_{\text{1}},
\label{eq:loss}
\end{equation}
where $\lambda_{i}$ $(i=1,2,3,4)$ denotes the weight of corresponding loss, respectively.
\subsection{Refinement render}
\label{s:rr}
In the coarse stage, the identification information and the shape of the person can be preserve, but the texture details are lost due to the complexity of the clothes image. Pasting the warped clothes onto the target person directly may lead to generate the artifacts. Learning the composition mask between the warped clothes image and the coarse results also generates the artifacts~\cite{han2017viton,wang2018cpvton} due to the diversity of pose. To solve the above issues, we present a refinement render utilizing multi-pose composition masks to recover the texture
details and remove some artifacts.
Formally, we define $C_{\text{w}}$ as an image of warped clothes obtained by geometric matching learning module, $\hat{I_{\text{c}}}$ as a coarse result generated by the Warp-GAN, $P$ as the target pose heatmap, and $G_{\text{p}}$ as the generator of the refinement render. As illustrated in Figure~\ref{fig:train_pipeline} (e), taking $C_{\text{w}}$, $\hat{I_{\text{c}}}$, and $P$ as input, the $G_{\text{p}}$ learns to predict a towards multi-pose composition mask and synthesize the rendered result:
\begin{equation}
\hat{I}_{\text{p}} = G_{\text{p}}(C_{\text{w}}, \hat{I}, P) \odot C_{\text{w}} + (1 - G_{\text{p}}(C_{\text{w}}, \hat{I}, P)) \odot \hat{I},
\end{equation}
where $\odot$ denotes the element-wise matrix multiplication. We also adopt the perceptual loss to enhance the performance that the objective function of $G_{\text{p}}$ can be written as:
\begin{equation}
\mathcal{L}_{\text{p}} = \mu_{1} \mathcal{L}_{\text{perceptual}}(\hat{I}_{\text{p}}, I) + \mu_{2} \| 1 - G_{\text{p}}(C_{\text{w}}, \hat{I_{\text{c}}}, P) \|_{1},
\end{equation}
where $\mu_{1}$ denotes the weight of perceptual loss and $\mu_{2}$ denotes the weight of the mask loss.
\subsection{Geometric matching learning}
\label{s:gml}
Inspired by~\cite{Rocco2017geocnn}, we adopt the convolutional neural network to learn the transformation parameters, including feature extracting layers, feature matching layers, and the transformation parameters estimating layers. As shown in Figure~\ref{fig:train_pipeline} (f), we take the mask of the clothes image, and the mask of body shape as input which is first passed through the feature extracting layers. Then, we predict the correlation map by using the matching layers. Finally, we apply a regression network to estimate the TPS (Thin-Plate Spline)~\cite{bookstein1989tps} transformation parameters for the clothes image directly based on the correlation map.
Formally, given an input image of clothes $C$ and its mask $C_{mask}$, following the stage of conditional parsing learning, we obtain the approximated body shape $M_b$ and the synthesized clothes mask $\hat{C}_{mask}$ from the synthesized human parsing. This subtask aims to learn the transformation mapping function $\mathcal{T}$ with parameter $\theta$ for warping the input image of clothes $C$. Due to the unseen of synthesized clothes but have the synthesized clothes mask, we learn the mapping between the original clothes mask $C_{mask}$ and the synthesized clothes mask $\hat{C}_{mask}$ obey body shape $M_b$. Thus, the objective function of the geometric matching learning can be formulated as:
\begin{equation}
\mathcal{L}_{\text{geo\_matching}}(\theta) = \| \mathcal{T}_{\theta}(C_{mask}) - \hat{C}_{mask}\|_1,
\label{eq:matching}
\end{equation}
Therefore, the warped clothes $C_{w}$ can be formulated as $C_{w} = \mathcal{T}_{\theta}(C)$, which is helpful for addressing the problem of misalignment and learning the composition mask in the above subsection~\ref{s:wg} and subsection~\ref{s:rr}.
\begin{figure*}[!tp]
\centering
\includegraphics[width=.8\hsize \hspace{0.01\hsize}]{fig/vs_others.pdf}
\caption{Visualized comparison with other methods on our collected dataset MPV. MG-VTON (w/o Render) is the model where the refinement render is removed. The model where the multi-pose composition mask is removed denotes as MG-VTON (w/o Mask).}
\label{fig:vs_others}
\vspace{-4mm}
\end{figure*}
\section{Experiments}
In this section, we first make visual comparisons with other methods and then discuss the results quantitatively. We also conduct the human perceptual study and the ablation study, and further train our model on our newly collected dataset MPV test it on the Deepfashion to verify the generation capacity.
\subsection{Datasets}
Since each person image in the dataset used in VITON~\cite{han2017viton} and CP-VTON~\cite{wang2018cpvton} only has one fixed pose, we collected the new dataset from the internet, named MPV, which contains 35,687 person images and 13,524 clothes images. Each person image in MPV has different poses. The image is in the resolution of $256 \times 192$. We extract the 62,780 three-tuples of the same person in the same clothes but with diverse poses. We further divide them into the train set and the test set with 52,236 and 10,544 three-tuples, respectively. Note that we shuffle the test set with different clothes and diverse pose for quality evaluation. DeepFashion~\cite{liu2016deepfashion} only have the pairs of the same person in different poses but do not have the image of clothes. To verify the generalization capacity of the proposed model, we extract 10,000 pairs from DeepFashion, and randomly select clothes image from the test set of the MPV for testing.
\subsection{Evaluation Metrics}
We apply three measures to evaluate the proposed model, including subjective and objective metrics:
1) We perform pairwise A/B tests deployed on the Amazon Mechanical Turk (\textbf{AMT}) platform for human perceptual study. 2) we use Structural SIMilarity (\textbf{SSIM})~\cite{wang2004image} to measure the similarity between the synthesized image and ground truth image. In this work, we take the target image (the same person wearing the same clothes) as the ground truth image used to compare with the synthesized image for computing SSIM. 3) We use Inception Score (\textbf{IS})~\cite{NIPS2016_6125} to measure the quality of the generated images, which is a common method to verify the performances for image generation.
\subsection{Implementation Details}
\textbf{Setting.}
We train the conditional parsing network, Warp-GAN, refinement render, and geometric matching network for 200, 15, 5, 35 epochs, respectively, using ADAM optimizer~\cite{Diederik2014Adam}, with the batch size of 40, learning rate of 0.0002, $\beta_1 = 0.5$, $\beta_2 = 0.999$. We use two NVIDIA Titan XP GPUs and Pytorch platform on Ubuntu 14.04.
\textbf{Architecture.}
As shown in Figure~\ref{fig:train_pipeline}, each generator of MG-VTON is a ResNet-like network, which consists of three downsample layers, three upsample layers, and nine residual blocks, each block has three convolutional layers with 3x3 filter kernels followed by the bath-norm layer and Relu activation function. Their number of filters are 64, 128, 256, 512, 512, 512, 512, 512, 512, 512, 512, 512, 256, 128, 64. For the discriminator, we apply the same architecture as pix2pixHD~\cite{wang2017pix2pixHD}, which can handle the feature map in different scale with different layers. Each discriminator contains four downsample layers which include 4x4 kernels, InstanceNorm, and LeakyReLU activation function.
\subsection{Baselines}
\textbf{VITON}~\cite{han2017viton} and \textbf{CP-VTON}~\cite{wang2018cpvton} are the state-of-the-art image-based virtual try-on method which assumes the pose of the person is fixed. They all used warped clothes image to improve the visual quality, but lack of the ability to generate image under arbitrary poses. In particular, VTION directly applied shape context matching~\cite{belongie2002shape} to compute the transformation mapping. CP-VTON borrowed the idea from \cite{Rocco2017geocnn} to estimate the transformation mapping using a convolutional network. To obtain fairness, we first enriched the input of the VITON and CP-VTON by adding the target pose. Then, we retrained the VITON and CP-VTON on MPV dataset with the same splits (train set and test set) as our model.
\subsection{Quantitative Results}
We conduct experiments on two benchmarks and compare against two recent related works using two widely used metrics \textbf{SSIM} and \textbf{IS} to verify the performance of the image synthesis, summarized in Table.~\ref{tab:ssim_is}, higher scores are better. The results shows that ours proposed methods significantly achieve higher scores and consistently outperform all baselines on both datasets thanks to the cooperation of our conditional parsing generator, Warp-GAN, and the refinement render. Note that the MG-VTON (w/o Render) achieves the best SSIM score and the MG-VTON (w/o Mask) achieve the best IS score, but they obtain worse visual quality results and achieve lower scores in AMT study compare with MG-VTON (ours), as illustrated in the Table~\ref{tab:amt} and Figure~\ref{fig:ab_polish}. As shown in Figure~\ref{fig:vs_others}, MG-VTON (ours) synthesizes more realistic-looking results than MG-VTON (w/o Render), but the latter achieve higher SSIM score, which also can be observed in~\cite{johnson2016perceptual}. Hence, we believe that the proposed MG-VTON can generate high-quality person image for multi-pose virtural try-on with convincing results.
\begin{table}[htbp]
\centering
\caption{Comparisons on MPV and DeepFashion. }
\vspace{2mm}
\resizebox{\columnwidth}{!}{
\begin{tabular}{lccc}
\toprule
& \multicolumn{2}{c}{MPV} & \multicolumn{1}{c}{DeepFashion} \\
\cmidrule{2-4}
Model & SSIM & IS & IS \\
\midrule
VITON~\cite{han2017viton} & 0.639 & 2.394 $\pm$ 0.205 & 2.302 $\pm$ 0.116 \\
CP-VTON~\cite{wang2018cpvton} & 0.705 & 2.519 $\pm$ 0.107 & 2.459 $\pm$ 0.212 \\
MG-VTON (w/o Render) & \textbf{0.754} & 2.694 $\pm$ 0.119 & 2.813 $\pm$ 0.047 \\
MG-VTON (w/o Mask) & 0.733 & \textbf{3.309 $\pm$ 0.137} & \textbf{3.368 $\pm$ 0.055} \\
MG-VTON (Ours) & 0.744 & 3.154 $\pm$ 0.142 & 3.030 $\pm$ 0.057 \\
\bottomrule
\end{tabular}
}
\label{tab:ssim_is}
\vspace{-2mm}
\end{table}
\subsection{Qualitative Results}
We perform visual comparisons of the proposed method with VITON~\cite{han2017viton}, CP-VTON~\cite{wang2018cpvton}, MG-VTON (w/o Render), and MG-VTON (w/o Mask), illustrated in Figure~\ref{fig:vs_others}, which shows that our model generates reasonable results with convincing details. Although the baseline methods have synthesized few details of clothes, it is far from the practice towards multi-pose virtual try-on scenario. Specifically, the identity and the clothing of the lower-body cannot be preserved by the baseline methods. Besides, the clothing of the lower-body also cannot be preserved while the clothing of upper-body is change by the baseline methods. Furthermore, the baseline methods cannot synthesize the hairstyle and face well that result in blurry and artifacts. The reasons behind are that they overlook the high-level semantics of the reference image and the relationship between the reference image and target pose in the virtual try-on task. On the contrary, we adopt clothes and pose guided network to generate the target human parsing, which is helpful to alleviate the problem that lower-body clothing and hair cannot be preserved. In addition, we also design a deep warping network with an adversarial loss carefully to solve the issue that the identity cannot be preserved. Furthermore, we capture the interplay of among the poses and present a multi-pose based refined network that learns to erase the noise and artifacts.
\begin{figure}[!hp]
\centering
\includegraphics[width=1.0\hsize \hspace{0.01\hsize}]{fig/teston_dp.pdf}
\caption{Some results from our model trained on MPV and tested on DeepFashion, which synthesizes the realistic image and captures the desired pose and clothes well.}
\label{fig:teston_dp}
\vspace{-4mm}
\end{figure}
\begin{table}[h]
\centering
\caption{Pairwise comparison on MPV and DeepFashion. Each cell lists the percentage where our MG-VTON is preferred over the other method. Chance is at 50\%. }
\vspace{2mm}
\resizebox{\columnwidth}{!}{
\begin{tabular}{ccccc}
\toprule
& VITON & CP-VTON & MG-VTON & MG-VTON \\
& & & (w/o Render) & (w/o Mask) \\
\midrule
MPV & 83.1\% & 85.9\% & 82.4\% & 84.6\% \\
DeepFashion & 88.9\% & 83.3\% & 84.6\% & 75.5\% \\
\bottomrule
\end{tabular}
}
\label{tab:amt}
\vspace{-4mm}
\end{table}
\subsection{Human Perceptual Study}
We perform a human study on MPV and Deepfashion~\cite{liu2016deepfashion} to evaluate the visual quality of the generated image. Similar to pix2pixHD~\cite{wang2017pix2pixHD}, we deployed the A/B tests on the Amazon Mechanical Turk (AMT) platform. There are 1,600 images with size $256 \times 192$. We have shown three images for reference (reference image, clothes, pose) and two synthesized images with the option for picking. The workers are given two choices with unlimited time to pick the one image looks more realistic and natural, considering how well target clothes and pose are captured and whether the identity and the appearance of the person are preserved. Specifically, the workers are shown the reference image, target clothes, target pose, and the shuffled image pairs. We collected 8,000 comparisons from 100 unique workers. As illustrated in Table~\ref{tab:amt}, the image synthesized by our model obtained higher human evaluation scores and indicate the high-quality results compare to the baseline methods.
\begin{figure*}[!thp]
\centering
\includegraphics[width=1.0\hsize \hspace{0.01\hsize}]{fig/ab_IVPOSE.pdf}
\caption{Ablation study on our collected dataset MPV. Zoom in for details.}
\label{fig:ab_polish}
\vspace{-4mm}
\end{figure*}
\begin{figure}[!tp]
\centering
\includegraphics[width=1.0\hsize \hspace{0.01\hsize}]{fig/diff_parsing.pdf}
\caption{Effect of the quality of human parsing. The quality of human parsing significantly affects the quality of the synthesized image in the virtual try-on task.}
\label{fig:diff_parsing}
\vspace{-4mm}
\end{figure}
\subsection{Ablation Study}
We conduct an ablation study to analyze the important parts of our method. Observed from Table.~\ref{tab:ssim_is}, MG-VTON (w/o Mask) achieves the best scores. However, as shown in Figure~\ref{fig:vs_others}, it may inevitably generate artifacts. In Figure~\ref{fig:ab_polish}, we further evaluate the effect of the components of our MG-VTON. It shows that the multi-pose composition mask loss, the perceptual loss, and the pose in the refinement render stage, and the warping module in Warp-GAN are all important to enhance the performance.
\begin{figure}[!tp]
\centering
\includegraphics[width=1.0\hsize \hspace{0.01\hsize}]{fig/diff_pose.pdf}
\caption{Effect of clothes and pose for the human parsing, which is manipulating by the pose and the clothes. }
\label{fig:diff_pose}
\vspace{-4mm}
\end{figure}
We also conduct an experiment to verify the effect of the human parsing in our MG-VTON. As shown in Figure~\ref{fig:diff_parsing}, there is a positive correlation between the quality of the human parsing with that of the result. We further to verify the effect of the synthesized human parsing by manipulating the desired pose and clothes, as illustrated in Figure~\ref{fig:diff_pose}.
We manipulate the human parsing instead of the person image directly, and we can synthesize the person image in an easier and more effective way. Furthermore, we introduce an experiment that trained on our collected dataset MPV and test on the DeepFashion dataset to verify the generalization of the proposed model. As the Figure~\ref{fig:teston_dp} shown, our model captures the target pose and clothes well.
\section{Conclusions}
In this work, we make the first attempt to investigate the multi-pose guided virtual try-on system, which enables clothes transferred onto a person image under diverse poses. We propose a Multi-pose Guided Virtual Try-on Network (MG-VTON) that generates a new person image after fitting the desired clothes into the input image and manipulating human poses. Our MG-VTON decomposes the virtual try-on task into three stages, incorporates a human parsing model is to guide the image synthesis, a Warp-GAN learns to synthesize the realistic image by alleviating misalignment caused by diverse pose, and a refinement render recovers the texture details. We construct a new dataset for the multi-pose guided virtual try-on task covering person images with more poses and clothes diversity. Extensive experiments demonstrate that our MG-VTON significantly outperforms all state-of-the-art methods both qualitatively and quantitatively with promising performances.
{\small
\bibliographystyle{ieee}
|
{
"timestamp": "2019-03-01T02:16:31",
"yymm": "1902",
"arxiv_id": "1902.11026",
"language": "en",
"url": "https://arxiv.org/abs/1902.11026"
}
|
\section{Supplemental material}
\subsection*{Theory}
We derive the EIT transmission for the quantum superhet based on the setup in Fig.~\ref{fig:1}(a) under resonant conditions $\Delta_{p/c}=0$ for both coupling and probe lasers. The relevant Hamiltonian takes the form (in the basis of bare states $[|1\rangle, |2\rangle, |3\rangle, |4\rangle]^T$)
\begin{equation}\label{eq:H}
H(t)=\hbar\left(
\begin{array}{cccc}
0& \frac{\Omega_p}{2} & 0 & 0 \\
\frac{\Omega_p}{2} &0 & \frac{\Omega_c}{2} & 0 \\
0 & \frac{\Omega_c}{2} &0& \frac{\Omega_{\textrm{L}}+e^{-iS(t)}\Omega_s}{2}\\
0 & 0 & \frac{\Omega_{\textrm{L}}+e^{iS(t)}\Omega_s}{2} & 0 \\
\end{array}
\right).
\end{equation}
Here $S(t)$ denotes the time-dependent relative phase $S(t)=\delta_s t+\phi_s$ between the signal and local MW fields.
Accounting for the spontaneous emission, the dynamics of our system is described by the master equation for density matrix $\dot{\rho}$, i.e.,
\begin{equation}
\dot{\rho}=\frac{i}{\hbar} [\rho,H(t)]+\mathcal{D}[\rho], \label{eq:Master}
\end{equation}
where the second term is explicitly written as
\begin{equation}
\mathcal{D}[\rho]\equiv\left(
\begin{array}{cccc}
\gamma_2\rho_{22}+\gamma_4\rho_{44}& -\frac{\gamma_2}{2}\rho_{12} & -\frac{\gamma_3}{2}\rho_{13} & -\frac{\gamma_4}{2}\rho_{14} \\
-\frac{\gamma_2}{2}\rho_{21} & \gamma_3\rho_{33}-\gamma_2\rho_{22} & -\frac{\gamma_{23}}{2}\rho_{23} & -\frac{\gamma_{24}}{2}\rho_{24} \\
-\frac{\gamma_3}{2}\rho_{31} & -\frac{\gamma_{23}}{2}\rho_{32} & - \gamma_3\rho_{33} & -\frac{\gamma_{34}}{2}\rho_{34}\\
-\frac{\gamma_4}{2}\rho_{41} & -\frac{\gamma_{24}}{2}\rho_{42} & -\frac{\gamma_{34}}{2} \rho_{43}& -\gamma_4\rho_{44}
\end{array}
\right).\label{eq:L}
\end{equation}
Here $\gamma_{\textrm{ij}}=(\gamma_{\textrm{i}}+\gamma_{\textrm{j}})$, where $\gamma_{i}$ ($i=2,3,4$) is the decay rate [Fig.~\ref{fig:1}(a)]. In writing Eq.~(\ref{eq:L}), we have ignored the spontaneous emission associated with $|3
\rangle-|4\rangle$ and other possible transitions, as they are comparatively small. We are interested in the limit where $\delta_s$ in Eq.~(\ref{eq:H}) is small compared to all characteristic energy scales of the system dynamics.
We first illustrate the key physics taking the example of cold atoms. Within the adiabatic approximation, the probe laser transmission associated with the instantaneous steady state is written in terms of the imaginary component of susceptibility as~\cite{Fleischhauer2005}
\begin{equation}
P(t)=P_ie^{-kL \Im[\chi(t)]}. \label{eq:P}
\end{equation}
Here $P_i$ is the incident light power, $L$ is the length of the cell containing Rydberg atoms, $k=2\pi/\lambda_{\textrm{P}}$ is the wave-vector of probe laser. Note $\chi(t)=C\rho_{21}$ is the susceptibility associated with the instantaneous steady state, where $\rho_{21}(t)$ denotes the instantaneous steady-state density matrix component associated with $|1\rangle-|2\rangle$ transition. Furthermore $C=-2 N_0\mu_{12}^2/(\epsilon_0 \hbar \Omega_p)$, where $N_0$ is the total density of atoms, $\mu_{12}$ is the dipole moment of the ground state transition, and $\epsilon_0$ is the vacuum permittivity.
For $\Omega_s\ll \Omega_{\textrm{L}}$, an analytical expression for $P(t)$ can be derived as follows [c.f. Eq.~(\ref{eq:EIT}) in the main text]. Assuming the ideal case where $\gamma_{3(4)}=0$, the imaginary part of the susceptibility $\chi(t)$ can be straightforwardly derived as
\begin{equation}
\Im[\chi(\Omega, t)]=\chi_0 \frac{|\Omega|^2}{|\Omega|^2+\Gamma^2 }, \label{eq:rho12}
\end{equation}
where $\Gamma=\Omega_p\sqrt{\frac{2(\Omega_c^2+\Omega_p^2)}{(2\Omega_p^2+\gamma_2^2)}}$ is intimately related to the EIT linewidth, $\chi_0=\frac{C\gamma_2 \Omega_p}{\gamma_2^2+2\Omega_p^2}$ is the peak value of the spectrum, and $\Omega=|\Omega_{\textrm{L}}+e^{-iS(t)}\Omega_s|$. Perturbative expansions of Eq.~(\ref{eq:rho12}) in terms of the small parameter $\Omega_s/\Omega_{\textrm{L}}$ reads at the first order as
\begin{eqnarray}
\Im[\chi(\Omega_\textrm{L},t)]=\Im[\chi(\Omega_\textrm{L})]+S_\textrm{L}\Omega_s\cos(\delta_st+\phi_s). \label{eq:rho21}
\end{eqnarray}
Here $S_\textrm{L}=2\chi_0\left[\frac{\Gamma^2\Omega_{\rm{L}}}{(\Omega_{\rm{L}}^2+\Gamma^2)^2}\right]$ is the slope of spectrum (\ref{eq:rho12}) at $\Omega=\Omega_{\textrm{L}}$. When $\Omega_\textrm{L}=\Gamma/\sqrt{3}$, the spectrum is linear near $\Omega=\Omega_{\textrm{L}}$, corresponding to the maximum slope $S_{\textrm{max}}=3\sqrt{3}\chi_0/(8\Gamma)$. Let us denote $\alpha=kL\Gamma S_{\textrm{max}}$. Substituting Eq.~(\ref{eq:rho21}) into Eq.~(\ref{eq:P}), for $\alpha\Omega_s/\Gamma\ll 1$, we arrive at Eq.~(\ref{eq:EIT}) in the main text, with $\bar{P}=P_i e^{-kL\chi_0/4}$.
The form of Eq.~(\ref{eq:rho21}) holds as well for thermal atoms, where the Doppler average of $\rho_{21}(t)$ is required for calculating $\Im[\chi(t)]$ in Eq.~(\ref{eq:P}), and when considering $\gamma_{3(4)}\neq 0$. Note these effects will lead to modified $\Gamma$ and $\alpha$. This way, we also obtain the theoretical value of the probe laser transmission shown in Figs.~(\ref{fig:1}) (a) and (b).
\subsection*{Experimental setup}
In our experiment, we use Cs atoms in a vapor cell at room-temperature. The cell is $5$-cm-long and contains ground-state atoms with a total density $N_0=4.89\times10^{10}$~cm$^{-3}$. We realize the four-level configuration in Fig.~\ref{fig:1} using four states in a Cesium atom: $6\textrm{S}_{1/2}$, $\textrm{F}=4$; $6\textrm{P}_{3/2}$, $\textrm{F}=5$; $47\textrm{D}_{5/2}$, and $48\textrm{P}_{3/2}$. The hyperfine states $6\textrm{S}_{1/2}$, $\textrm{F}=4$ and $6\textrm{P}_{3/2}$, $\textrm{F}=5$ comprise the lowest two states $|1\rangle$ and $|2\rangle$ in the configuration, with $\gamma_2=5.2$ MHz. Moreover, the Rydberg state $47\textrm{D}_{5/2}$, with inverse lifetime $\gamma_3=3.9$ kHz , and Rydberg state $48\textrm{P}_{3/2}$, with inverse lifetime $\gamma_4=1.7$ kHz, make up the states $|3\rangle$ and $|4\rangle$ there. In calculating $\gamma_3$ and $\gamma_4$ at room temperatures, we have considered black-body induced transitions up to $n=70$. We apply a local MW field at $6.94$ GHz to resonantly drive the Rydberg transition $47\textrm{D}_{5/2}\rightarrow 48\textrm{P}_{3/2}$. In detecting a MW signal, the local and the signal fields are combined by a 2-way microwave resistive power divider, and are coupled to free space via the same resonant horn antenna. The resonant coupling between hyperfine states $6\textrm{S}_{1/2}$, $\textrm{F}=4\rightarrow6\textrm{P}_{3/2}$, $\textrm{F}=5$ is realized using a $852$ nm probe beam provided by a commercial extended cavity diode laser (ECDL). The resonant coupling between states $6\textrm{P}_{3/2}$, $\textrm{F}=5$ and $47\textrm{D}_{5/2}$ is realized using a $510$ nm beam generated by a frequency-double diode laser. The probe and coupling laser beams counter propagate through the room-temperature Cs cell, with minimized Doppler broadening of the transition. Their polarizations are linear, and are parallel to the direction of MW fields, leading to excitations of the magnetic sub-level $|m|=1/2$. For the probe beam, the $1/e^2$ beam diameter is $1.70 \pm 0.04$ mm, and the optical power incident to the vapor cell is $120 \pm 4~\mu$W, yielding effectively ${\Omega_p=5.7 \pm 0.6}$ MHz. For the coupling beam, the $1/e^2$ beam diameter is $2.00 \pm 0.05$ mm, and the incident optical power is $34 \pm 1$ mW, yielding $\Omega_c=0.97\pm 0.12$ MHz. After absorption by Cs atoms, the power of the probe light incident on the detector is about $10$ $\mu$W.
\subsection*{Reduction of technical noise}
In our experiment, the $150.000$ kHz signal is analyzed by a spectrum analyzer. The frequency noise of the probe laser and the seed of coupling laser are actively canceled by locking them to a $10$-cm-long ultra-low expansion (ULE) glass cavity with frequency noise server (FNS). The cavity is double coated at $852$ nm and $1020$ nm with a finesse of $200000$. The linewidth of the high finesse cavity is about $7.5$ kHz. The cavity is placed in a vacuum system at a residual pressure below $10^{-8}$ mbar, and its temperature is stabilized to the zero crossing point of the coefficient of thermal expansion. The system is mounted on a passive vibration isolation platform, and is surrounded by the acoustic and temperature insulation box. The FNSs are realized by PDH technique, and the feedbacks are injected to the PZTs and the diode currents of both lasers. The locking bandwidths of the $852$ nm laser and the $1020$ nm laser are about $250$ kHz and $350$ kHz, respectively. The beat note result of the $852$ nm laser with another equal system shows that the linewidth of the 852 nm laser is below $20$ Hz. The linewidth of the $510$ nm laser is estimated to be $<40$ Hz. Low frequency intensity noises of both lasers are actively eliminated through a feedback to double-pass acousto-optic modulators (AOMs) in cateye configuration. The $852$ nm probe light is separated into two orthogonally polarized output beams using calcite beam displacer, which propagate in parallel through the center of the Cs vapor cell. High frequency common mode intensity noise of the probe light is canceled by means of the balanced detection technique. Both MW sources are synchronized with a GPS disciplined oscillator with Rubidium timebase (GPSDO), so as to minimize their long time frequency drift. The signal MW source is $150$.000 kHz detuned from the local MW source, which offsets the interference signal of the quantum superhet to a sufficient high frequency, preventing the low frequency electronic pink noise.
\subsection*{Measuring the phase from EIT signal}
We extract a phase $\phi_{\textrm{out}}$ from $P_{\textrm{out}}(t)$ by using a lock-in amplifier. The filter slope of the lock-in amplifier is set to $18$ dB/oct and the time constant is fixed at $100$ ms to realize $1.04$ Hz equivalent noise bandwidth. This leads to a SNR of $44$ dB for $E_s=7.8$ $\mu$Vcm$^{-1}$, leading to theoretical estimation of the phase resolution as $0.6$ degree. To assess the phase resolution experimentally, the standard deviation of the fluctuation in $\phi_{\textrm{out}}$ is measured in a period of $1-5$ s and $6-10$ s, respectively, with $1$ s waiting time for the signal to reach 99\% of its final value.
\renewcommand{\bibnumfmt}[1]{[S#1]}
\renewcommand{\citenumfont}[1]{S#1}
\begin{figure}[tb]
\centering
\includegraphics[width= 0.98\columnwidth]{fig5.pdf}
\caption{Sensitivity $\mathcal{S}$ of quantum superhet as a function of $\delta_s$ of the signal MW electric field. Shown are respectively the sensitivity spectrum $\mathcal{S}(\delta_s)$ that corresponds to various optical readout noises (red), the amplifier noise of photon detector (gray) and the spectrum analyzer noise (black), and blue curve represents the QPNL sensitivity of our setup; see details in supplementary material. Note $\delta_s=150$ kHz is chosen in our experiments.}\label{fig:5}
\end{figure}
\subsection*{Sensitivity spectrum}
In this section, we show how we obtain the sensitivity spectrum presented in Fig.~\ref{fig:5}. Let us denote by $\mathcal{S}(\delta_s)$ the sensitivity spectrum, i.e., the field sensitivity $\mathcal{S}$ at frequency $\delta_s$. Further, we let $S_{\textrm{P}}(\delta_s)$ denote the noise spectrum associated with $P_{\textrm{out}}(t)$. Importantly, according to the linear relation in Eq. (2) of the main text, we can write $\mathcal{S}(\delta_s)=\kappa S_{\textrm{P}}(\delta_s)$, with $\kappa$ being a constant coefficient.
To obtain $S_{\textrm{P}}(\delta_s)$, we use the relation $S_\textrm{P}(\delta_s)=[S_\textrm{PD}(\delta_s)\times R]^{1/2}/(G\mu)$. Here $S_{\textrm{PD}}(\delta_s)$ is the noise power density associated with each noise sources in our detection system including optical readout noises, the amplifier noise of photon detector, and the noise of the spectrum analyzer, $G$ and $\mu$ denote the trans-impedance gain and the response of detectors, respectively, and $R$ labels the impedance of the spectrum analyzer. In our setup, we have $G=175\times10^3$ V/A, $\mu=0.58$ A/W, $R=50$ $\Omega$. Morever, we can experimentally measure the noise power density $S_{\textrm{PD}}(\delta_s)$ for each aforementioned noise, thus obtains corresponding $S_{\textrm{P}}(\delta_s)$.
To determine the coefficient $\kappa$, we note that the sensitivity of $55$ nVcm$^{-1}$Hz$^{-1/2}$ is achieved at $\delta_s=150.000$ kHz, where we have measured $S_{\textrm{PD}}=-110$ dBm/Hz associated with the optical readout noise. This gives $\kappa=7.9\times10^3$ Vcm$^{-1}$W$^{-1}$.
Combinations $S_{\textrm{PD}}(\delta_s)$ for each type of noises above and knowledge of $\kappa$, we plot all the sensitivity spectra shown in Fig.~\ref{fig:5}.
\subsection*{Roadmap to QPNL sensitivity}
We first present a detailed noise analysis based on the sensitivity spectrum in Fig.~\ref{fig:5}. We see that the primary source of noise limiting the sensitivity of quantum superhet varies with $\delta_s$ of the signal: For frequencies below $1$ kHz, the 1/f noise of the electric circuits dominates over other noises; Between $1$ kHz and $100$ kHz, the transit noise due to thermal atoms provides the main noise source; For frequencies above $100$ kHz, it is the frequency noise of coupling and probe lasers caused by FNS resonant that mainly limits the sensitivity. In view of the requirement of both optimal sensitivity and $\delta_s\ll \Gamma_{\textrm{EIT}}$, we choose $\delta_s = 150.000$ kHz for our experimental demonstration.
Now we determine the QPNL sensitivity for our setup. Since the quantum superhet is operated in slope detection mode, the QPNL sensitivity is formally given by
\begin{equation}
E_{\textrm{QPNL}}= \frac{\sqrt{2}\hbar}{2\mu_{r}}\frac{1}{\sqrt{N_{a}}\tau_c}.
\end{equation}
Here $\mu_{r}$ is the dipole moment associated with Rydberg transition, $\tau_c=1/\Gamma$ is the coherence time of the quantum superhet, and $N_{a}$ is the atom number participating in EIT process per second. For our setup, we estimate $\tau_c$ from $\tau_c\approx \Gamma^{-1}_{\textrm{EIT}}$, with $\Gamma_{\textrm{EIT}}=7.9$ MHz from experiment results. The $N_{a}$ is estimated as $2.14\times 10^{13}$ s$^{-1}$ for EIT process, which leads to an enhancement of $5$ $\mu$W light transmission at the photon detector compared to the case without coupling laser. This gives $E_{\textrm{QPNL}}=700$ pV$\textrm{cm}^{-1}$ $\textrm{Hz}^{-1/2}$ indicated in Fig.~\ref{fig:5}.
Finally we outline the roadmap toward the QPNL sensitivity according to Fig.~\ref{fig:5}. For $\delta_s= 150.000$ kHz, we see that the primary noises sources in our detection scheme can be systematically eliminated using techniques feasible within present quantum sensing experiments as follows: (i) The transit noise due to thermal atoms can be eliminated by using larger-diameter probe and coupling beams; (ii) The laser frequency noise can be readily eliminated by using state of the art lasers with mHz linewidth~\cite{Kessler2012,Matei2017} and by expanding servo bandwidth to several MHz~\cite{Musha1997,Endo2018}; (iii) The amplifier noise of photon detector can be reduced by means of optical heterodyne or homodyne detection~\cite{Cox2018,Kumar2017a}; (iv) The spectrum analyzer noise can be removed by using conventional electronic amplifiers. After the relevant technical noises have been eliminated, quantum superhet approaches QPNL. \\
\subsection*{Calibration of MW electric field amplitude}
In our experiment, the signal and local MW E-fields are emitted from difference sources, respectively. To calibrate each field, we follow the procedures below. We apply a test MW field to resonantly drive the Rydberg transition in the 4-level EIT configuration. We denote the test field amplitude by $E$, chosen to be sufficiently large to ensure its subsequent measurement to a high accuracy. First, from the output power of MW source, one can calculate $E$ according to the standard antenna equation (IEEE Std 1309-2013), i.e.,
\begin{equation}
E=\sqrt{\frac{\eta(P_s-\alpha_l)g}{4\pi d^2}}.\label{eq:IEEE}
\end{equation}
Here, $\eta=377$ $\Omega$ is the intrinsic impedance of free space, $P_s$ is the output power of MW source, $\alpha_l$ is the insertion loss between MW source and antenna, $g$ is the gain of antenna, and $d$ is the distance between the transmitting antenna and the receiving point. Determination of $\alpha_l$ requires the experimental data of $E$, measured via the AT-splitting approach, where we read off $E$ from the relation $E=\sqrt{2}\pi\Delta_{\textrm{AT}}\hbar/\mu_{r}$. In the end, the experimental data for $E$ combined with Eq.~(\ref{eq:IEEE}) allow us to calibrate the insertion loss $\alpha_l$ for both the local and signal MW fields. For the local MW field, we find $\alpha_l=14.7$ dB. This includes the $6.5$ dB RPD insertion loss and four meters $1.5$ dB/m wire loss, while the remaining insertion loss can be attributed to the connectors insertion loss. For the signal field, we have obtained $\alpha_l=11.9$ dB, which is smaller than the local MW field due to utility of a shorter transmission wire ($3$ m). Once the insertion loss has been calibrated, the field strengths for both the signal and local MW fields can be readily determined using Eq.~(\ref{eq:IEEE}).
\medskip
|
{
"timestamp": "2019-03-01T02:17:42",
"yymm": "1902",
"arxiv_id": "1902.11063",
"language": "en",
"url": "https://arxiv.org/abs/1902.11063"
}
|
\section{ACKNOWLEDGMENTS}
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\section{CONCLUSIONS AND FUTURE WORKS}
\subsection{Conclusions}
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\section{Conclusions}
\label{sec:gf-concl}
In this paper, we define the problem of \emph{extended gaze following} as finding the locations of objects of interest solely from the gaze direction of visible people. Importantly, this allows for finding objects either inside or outside the camera field-of-view. In this context, we propose a novel spatial representation for head poses (approximating gaze direction) and object locations.
We present a framework that takes advantage of convolutional encoder/decoder architectures to learn the spatial relationship between head poses and object locations, and we compare nine different methods on synthetic and real data. We finally conclude that learning-based approaches outperform geometry-based ones while being competitive with the state of the art. We also show that the necessary training examples can be quickly and easily obtained through a synthetic data generation process.
We believe this work open new perspectives for research. In particular, several decisions were taken to obtain an end-to-end method (\emph{e.g}. } \def\Eg{\emph{E.g}. heat-map representation or elevation coordinate omission), which makes it hardly suitable in some situations. The extended gaze-following problem would benefit greatly from a benchmark of different representations and inference models, and of the influence of each simplifying hypothesis. Moreover, the availability of suitable datasets would ease future research on this topic.
In parallel, we wish to use this framework in the future as a tool to improve the decision process of a robotic system in a social context such as~\cite{lathuiliere2019prl}.
\section{Synthetic Scenario Generation for Network Training}
\label{sec:synthetic}
A large amount of data is required to train deep networks. Unfortunately, obtaining such a dataset is difficult, since, in practice, we would need to know the true object locations for every sequence. For instance, in the \emph{Vernissage} dataset~\cite{jayagopi2013vernissage}, objects outside the field of view have been annotated employing infrared cameras. This setting is well-suited for our problem but it would be difficult to obtain a sufficiently large and diverse dataset of object locations to train deep networks. Consequently, Vernissage is used only to test our model and not to train it.
To face this issue, we propose to use synthetically generated data. More precisely, we simulate scenarios involving people and objects, and generate their corresponding input sequences and associated true object locations. We define a probabilistic model that relates the object 2D positions and the head poses, and generate samples according to the underlying distribution. We now aim at generating a scenario of length $T$ involving a constant number $N$ of people with respective positions $\xmat_{1:T}^n$ and orientations $\phimat_{1:T}^n$, given $1<n<N$; and $M$ objects located at the positions $\xmat^m_{obj}, 1<m<M$. To this aim, we define the joint distribution $P(\phimat_{1:T}^{1:N}, \xmat_{1:T}^{1:N},\xmat^{1:M}_{obj})$ considering the following factorization:
\begin{multline}
\underbrace{P(\phimat_{1:T}^{1:N}| \xmat_{1:T}^{1:N},\xmat^{1:M}_{obj})}_{\substack{\text{Head orientation} \\\text{ distribution}}}\times\underbrace{P(\xmat_{1:T}^{1:N}|\xmat^{1:M}_{obj})}_{\substack{\text{People motion } \\\text{distribution}}}\times\underbrace{P(\xmat^{1:M}_{obj})}_{\substack{\text{Object position} \\\text{distribution}}
\label{eq:sample}
\end{multline}
The \emph{object position distribution} $P(\xmat^{1:M}_{obj})$ is based on a uniform distribution within the top-view grid, since we want to have a high variety of settings. However, some settings are even too difficult for a human to distinguish between objects. For this reason, the generator can choose to resample an object under two criteria. First, the closest two objects are from each other, the highest the chance one of them is resampled. Therefore, we impose that objects have a minimal physical size and that two objects cannot be one above the other.
Second, objects too far from the heat-map edges also have a high chance of being resampled. In many scenarios, objects of interest tend to be close to the walls, \emph{e.g}. } \def\Eg{\emph{E.g}. posters, computer screens, paintings in a museum. This tends to reduce the number of ambiguous cases in which several objects are aligned from the point of view of someone.
Importantly, in a human-robot interaction scenario, people may look at the robot, but we want to avoid our model to predict the presence of an object at the robot camera position.
Therefore, as the camera position $\xmat_{camera}$ is known, we propose to add a blank object at the corresponding grid cell $\pmat_{camera}$ in all sequences. The blank object behave like normal objects -- constant position, can be gazed at -- but does not appear in the \emph{object heat-map} at training time and thus should be ignored at prediction time. Also, it cannot be resampled while generating the objects.
Concerning the \emph{people motion distribution}, $P(\xmat_{1:T}^{1:N}|\xmat^{1:M}_{obj})$, we describe first how the initial positions $\xmat_1^{1:N}$ are sampled, and then how each $\xmat_{t+1}^n$ is sampled iteratively from $\xmat_{t}^n$.
First, the initial positions of people are obtained similarly to object positions. Namely, they are sampled uniformly within the boundaries, and can be resampled when too close to an object, another person, or (contrary to objects) too close to the edges.
Concerning the motion, we consider that people can either stay still for a random period of time, or move linearly short distances.
In practice, there is a high probability that the person stay still $\xmat_{t+1}^n = \xmat_t^n$. Otherwise, $\xmat_{t+\tau}^n$ is sampled from a normal distribution centered on $\xmat_t^n$, and possibly resampled as long as $\xmat_{t+\tau}^n$ is outside the boundaries or too close to another target. In the latter case, $\xmat_{t+1}^n\dots \xmat_{t+\tau-1}^n$ are linearly interpolated.
Finally, for the \emph{head orientation distribution}, we define a probabilistic model inspired by~\cite{Masse2017}, where the authors propose a method to estimate the visual focus of attention of multiple people by applying Bayesian inference on a generative model.
In this probabilistic model, the head orientation dynamics are explained by some latent variables, \emph{e.g}. } \def\Eg{\emph{E.g}. gaze direction.
For more details, see~\cite{Masse2017}. In our case, we propose to sample the latent variables over time, then sample the head orientation $\phi_t^n$ given the latent variables.
This model is well-suited for our sampling task since multiple situations may occur, \emph{e.g}. } \def\Eg{\emph{E.g}. mutual gaze or joint attention, that are treated differently by their temporal formulation. Moreover, it takes into account the discrepancy between head pose and gaze direction, and the network can learn this difference because it is modeled at training time.
\begin{figure}
\centering
\subfloat[][\emph{Object heat-map}]{\label{fig:o_synt_1}
\includegraphics[height=0.27\linewidth]{synt_1234_2D_12_visu_objectives}
} \hspace{0.2cm}
\subfloat[][\emph{Gaze heat-map}]{\label{fig:hm_synt_1}
\includegraphics[height=0.27\linewidth]{synt_1234_3D_12_visu_last_map}
} \hspace{0.2cm}
\subfloat[][Mean \emph{gaze heat-map}]{\label{fig:m_hm_synt_1}
\includegraphics[height=0.27\linewidth]{synt_1234_2D_12_visu_last_map}
}
\caption[Heat-maps from an example synthetic scenario ($N=2$ and $M=3$)]{Heat-maps from a synthetic scenario generated randomly, with $2$ people ($N=2$) and $3$ objects ($M=3$).~\subref{fig:o_synt_1}: the ground truth \emph{Object heat-map} $\Omegavect$ used for training or evaluation.~\subref{fig:hm_synt_1}: a \emph{Gaze heat-map} randomly chosen among the sequence.~\subref{fig:m_hm_synt_1}: the mean \emph{gaze heat-map} over the sequence.
}
\label{fig:illustration_synt_1}
\end{figure}
The Fig.~\ref{fig:illustration_synt_1} represents a synthetic scenario generated using this process. In practice, a wide variety of scenarios can be obtained with this approach. For instance, there is no limit to the number of people and/or objects that could be generated in one scenario, except the plausibility of such a scenario with respect to the physical space.
\section{Experiments}
\label{sec:gf-expe}
Experiments have been performed both on \emph{synthetic} data, generated online as described in section~\ref{sec:synthetic}, and on the \emph{Vernissage} dataset~\cite{jayagopi2013vernissage} as described below.
Note that, we do not use the datasets employed in ~\cite{Cohen2012} and ~\cite{Brau_2018_ECCV} since they are not publicly availabl
\paragraph{The \emph{Vernissage} Dataset}
\label{sec:vernissage-dataset}
It is composed of ten recordings lasting approximately ten minutes each. Each sequence contains two people interacting with a Nao robot and discussing about three wall paintings ($M=3$). The robot plays the role of an art guide, describing the paintings and asking questions to the people in front of it.
The scene was recorded at 25 frames per second (fps) with an RGB camera embedded into the robot head, and with a VICON motion capture system consisting of a network of infrared cameras providing accurate position and head pose estimations of the people, the objects and the robot. We use the OpenCV version of \cite{Viola2001} for face detection and \cite{ba2016line} to track the faces over time. The head poses are estimated by employing \cite{lathuiliere17}. The 3D head positions, are estimated using the face center and the bounding-box size, which provides a rough estimate of the depth. The position of the robot itself and the orientation of its head are also known. Finally, the object locations are annotated along with the visual focus of attention of the participants over time. Images extracted from Nao camera during various recordings are displayed in Fig.~\subref*{fig:gf-VFOA},~\subref*{fig:gf-OFVOD},~\ref{fig:gf-outline},~\subref*{fig:camera10},~\subref*{fig:camera30},~\subref*{fig:camera80}.
\paragraph{Implementation details}
The heat-map dimensions are set to $S_X=S_Y=32$, to represent a room of size $3m \times 3m$. The cone aperture $\epsilon$ is set to $2\degree$. We fixed the input sequence size to $T=200$ time steps. On the \emph{Vernissage} dataset, the videos are subsampled to 5 fps, then the duration of a sequence is $40s$ and we can extract several sequences from each video sequence. By using a sliding window and $50\%$ overlap, we extract a total of 224 sequences. We use the visual focus of attention annotations to obtain the true objects of interest for each sequence. Consequently, the number of objects can vary from 1 to 3 in the test sequences. We employ the adam optimizer \cite{kingma2014adam} for 10 epochs. For all neural network architectures employed in the experiments, the batch size is set to 32. In all cases, we perform the local maxima extraction method described in~\eqref{eq:NMS} after estimating $\hat{\Omegavect}$ to obtain the list of object positions. The neighborhood $\mathpzc{N}(\cdot)$ from~\eqref{eq:NMS} is defined as a sliding region of $5\times 5$ pixels, and the shrinking function $\alpha: x \mapsto \ln(1+x)$ . In all our experiments, we report \emph{Precision} and \emph{Recall}, and these two metrics are combined to obtain the \emph{f1-score}. \emph{Precision} measures the percentage of detected objects that are true objects. \emph{Recall} measures the percentage of true objects correctly detected. In order to compute these metrics, we employ a Hungarian algorithm that matches the detections with the real objects positions based on their respective distances. Importantly, the detection is considered as a success if the distance between the estimated and annotated distances is lower than $50$cm in the real-world space. For all learning-based approaches, we also report the MSE between the predicted and true \emph{object heat-maps}. Since Heuristic methods do not intend to predict the \emph{object heat-maps}, the MSE is not reported for them.
\paragraph{Results and Discussion}
\begin{table}
\caption[Gaze following performances on \emph{synthetic} data and \emph{Vernissage} ]{
Results obtained on data from the proposed synthetic generator and on the \emph{Vernissage} dataset \cite{jayagopi2013vernissage}. MSE values reported were multiplied by $10^{2}$ to facilitate reading. \emph{Precision}, \emph{recall} and \emph{f1-score} represent percentages. For learning-based approaches, we report the mean and standard deviation over five runs.
Results on the \cite{Brau_2018_ECCV} dataset are reported for comparison
\label{tab:results}
}
\begin{center}
\resizebox{0.99\linewidth}{!} {
\begin{tabular}{l|clll
\midrule
Dataset & \multicolumn{4}{c}{Synthetic}\\
\midrule
Method & \emph{MSE} & \multicolumn{1}{c}{\emph{Precision}}& \multicolumn{1}{c}{\emph{Recall}} & \multicolumn{1}{c}{\emph{f1-score}}\\
\midrule
\midrule
\emph{Cone} & - & 18.8 & 53.9 & 27.8 \\
\emph{Intersect} & - & 21.1 & 35.0 & 26.3 \\
\midrule
\emph{Linear Reg.} & 1.25 $\pm$ 0.02 & 50.5 $\pm$ 2.2 & 76.9 $\pm$ 1.0 & 60.9 $\pm$ 1.8\\
\emph{1-FC} & 1.06 $\pm$ 0.03 & 64.9 $\pm$ 1.6 & 61.5 $\pm$ 1.5 & 63.1 $\pm$ 1.1\\
\emph{3-FC} & 1.05 $\pm$ 0.01 & 65.9 $\pm$ 0.6 & 59.9 $\pm$ 2.2 & 62.8 $\pm$ 1.2\\
\midrule
\emph{Mean-2D-Enc} & 1.00 $\pm$ 0.03 & 74.5 $\pm$ 2.4 & 59.5 $\pm$ 1.7 & 66.1 $\pm$ 1.3\\
\emph{2D-Enc} & 0.98 $\pm$ 0.02 & 76.8 $\pm$ 2.2 & 62.2 $\pm$ 1.5 & 68.7 $\pm$ 1.7\\
\emph{3D-Enc} & 0.85 $\pm$ 0.06 & 88.2 $\pm$ 3.9 & 71.4 $\pm$ 2.1 & 78.9 $\pm$ 2.4\\
\emph{3D/2D U-Net} &\bf 0.75 $\pm$ 0.01 &\bf 89.0 $\pm$ 1.2 &\bf 78.0 $\pm$ 0.6 &\bf 83.2 $\pm$ 0.8\\
\midrule
\midrule
\midrule
Dataset & \multicolumn{4}{c}{Vernissage}\\
\midrule
Method & \emph{MSE} & \multicolumn{1}{c}{\emph{Precision}}& \multicolumn{1}{c}{\emph{Recall}} & \multicolumn{1}{c}{\emph{f1-score}} \\
\midrule \midrule
\emph{Cone} & - & 20.7 & 35.8 & 26.2\\
\emph{Intersect} & - & 34.9 & 27.2 & 30.6\\
\midrule
\emph{Linear Reg.} & 1.48 $\pm$ 0.04 & 37.0 $\pm$ 4.9 &\bf 53.7 $\pm$ 5.0 & 43.7 $\pm$ 4.6\\
\emph{1-FC} & 1.49 $\pm$ 0.02 & 29.9 $\pm$ 3.2 & 35.2 $\pm$ 2.5 & 32.3 $\pm$ 2.8\\
\emph{3-FC} & 1.49 $\pm$ 0.02 & 28.0 $\pm$ 3.5 & 29.9 $\pm$ 1.5 & 28.8 $\pm$ 2.4\\
\midrule
\emph{Mean-2D-Enc} &\bf 1.37 $\pm$ 0.02 &\bf 60.1 $\pm$ 1.5 & 41.1 $\pm$ 1.0 &\bf 48.8 $\pm$ 1.2\\
\emph{2D-Enc} & 1.39 $\pm$ 0.03 & 54.9 $\pm$ 4.2 & 40.5 $\pm$ 1.6 & 46.6 $\pm$ 2.5\\
\emph{3D-Enc} & 1.43 $\pm$ 0.05 & 49.9 $\pm$ 8.1 & 37.1 $\pm$ 9.0 & 42.5 $\pm$ 8.7\\
\emph{3D/2D U-Net} & 1.44 $\pm$ 0.04 & 45.1 $\pm$ 4.8 & 38.5 $\pm$ 2.2 & 41.5 $\pm$ 3.3\\
\midrule
\midrule
\midrule
Dataset & \multicolumn{4}{c}{Brau et al.~\cite{Brau_2018_ECCV}}\\
\midrule
Method & \emph{MSE} & \multicolumn{1}{c}{\emph{Precision}}& \multicolumn{1}{c}{\emph{Recall}} & \multicolumn{1}{c}{\emph{f1-score}} \\
\midrule
Brau et al.~\cite{Brau_2018_ECCV} & - & 59.0 & 48.0 & \it 52.9\\
\bottomrule
\end{tabular}
}
\end{center}
\end{table}
In Table \ref{tab:results}, we report the results obtained employing all methods described on both \emph{synthetic} and real data.
It has to be noted that many different recurrent architectures have been considered, either alone or in conjunction with one of the proposed convolutional Encoder/Decoder architectures \emph{e.g}. } \def\Eg{\emph{E.g}. adapted from the convolutional LSTM~\cite{donahue2015long}. All of them converged to networks predicting always the same (or almost the same) \emph{object heat-map}. We believe that, in this formulation, the ability to combine information from distant time frame is important, and this is difficult to achieve with RNN (or LSTM) processing data sequentially \cite{pascanu2013difficulty}.
\begin{figure}
\centering
\subfloat[][$\hat{\Omegavect}$ - \emph{Mean-2D-Enc}]{\label{fig:omega_2D}
\includegraphics[height=0.30\linewidth]{v27_2D_25_visu_predicted.png}
}
\subfloat[][$\hat{\Omegavect}$ - \emph{3D/2D U-Net}]{\label{fig:omega_3D}
\includegraphics[height=0.30\linewidth]{v27_3DUnet_25_visu_predicted.png}
}
\subfloat[][$\hat{\Omegavect}$ - \emph{Linear Reg.}]{\label{fig:omega_lin}
\includegraphics[height=0.30\linewidth]{v27_Linear_25_visu_predicted.png}
}\\
\subfloat[][Obj - \emph{Mean-2D-Enc}]{\label{fig:obj_2D}
\includegraphics[height=0.30\linewidth]{v27_2D_25_visu_predicted_spots.png}
}
\subfloat[][Obj - \emph{3D/2D U-Net}]{\label{fig:obj_3D}
\includegraphics[height=0.30\linewidth]{v27_3DUnet_25_visu_predicted_spots.png}
}
\subfloat[][Obj - \emph{Linear Reg.}]{\label{fig:obj_lin}
\includegraphics[height=0.30\linewidth]{v27_Linear_25_visu_predicted_spots.png}
}
\caption[]{Results of three methods on the \emph{Vernissage} scenario illustrated in Fig.~\ref{fig:illustration}.
\subref*{fig:omega_2D}, \subref*{fig:omega_3D}, \subref*{fig:omega_lin}: Estimates $\hat{\Omegavect}$ of the \emph{Vernissage} \emph{object heat-map} $\Omegavect$ from Fig.~\subref*{fig:o_hm} using three different architectures.
\subref*{fig:obj_2D}, \subref*{fig:obj_3D}, \subref*{fig:obj_lin
: Corresponding objects positions, obtained as the highest local maxima from $\hat{\Omegavect}$.
Black pixels in~\subref*{fig:omega_lin}
indicate negative values.
}
\label{fig:results}
\end{figure}
From the experiments, we observe that learning-based approaches clearly outperform those based on cone intersections inspired from \cite{Cohen2012}. Indeed, even on the \emph{synthetic} datasets, their \emph{precision} and \emph{recall} do not reach better than $18.8\%$ and $53.9\%$ respectively, whereas a simple linear regression reaches considerably higher scores ($50.5\%$ and $76.9\%$ respectively). The same remark stands for the \emph{Vernissage} dataset. Increasing the network complexity by simply adding fully-connected layers does not bring any improvement and even reduce the performance. Then, we observe that all proposed encoder/decoder models clearly outperform other methods by a substantial margin on the \emph{synthetic} dataset. There, we obtain a $22.3\%$ gain in terms of f1-score when employing the \emph{3D/2D U-Net} with respect to the linear regression model. On the \emph{Vernissage} dataset, a $5.1\%$ gain is obtained in terms of \emph{f1-score} when employing the \emph{Mean-2D-Enc} with respect to the linear regression model. These experiments validate the use of the encoder/decoder architecture.
We notice that the performance on the \emph{synthetic} dataset increases with encoder complexity. However, the inverse phenomenon is observed on \emph{Vernissage}, where the best performances are obtained using the simplest encoder architecture (that does not model time).
Our guess for this observation is that there is a significant discrepancy between the distribution of \emph{Vernissage} data and the \emph{synthetic} data distribution sampled according to \eqref{eq:sample}.
Therefore, more complex models probably tend to over-fit the \emph{synthetic} data distribution, and thus transfer less well on the \emph{Vernissage} dataset.
More realistic training data could lead to further improvements. This could be obtained by gathering a dataset of real-life scenarios which could be use either as training data or to improve the quality of the generative model.
The only methods from the literature that we are aware of are~\cite{Cohen2012} and~\cite{Brau_2018_ECCV}. In both cases, neither the data nor the code have been made available online. Moreover, the papers lack information about parameters or hyperparameters that prevented us to test it. Additionally, \cite{Brau_2018_ECCV} explicitly discarded the \emph{Vernissage} dataset in their experiments. Results on their dataset (59\% precision and 48\% recall) are comparable to ours on \emph{Vernissage}. Note that, \cite{Brau_2018_ECCV} employed a larger success threshold ($1.0$m in
the real-world space for $50$cm in our case) and consequently would obtain lower scores according to our evaluation protocol. We wish to test our method on their dataset in the future. We do not compare to~\cite{Cohen2012} since they did not report any quantitative results on location estimation.
In Fig.~\ref{fig:results}, the predicted \emph{gaze heat-maps} $\hat{\Omegavect}$ for several learning-based approaches applied on the scenario from Fig.~\ref{fig:illustration} are displayed. The architectures \emph{Mean-2D-Enc} and \emph{Linear Reg.} use the average \emph{gaze heat-map} $\frac{1}{T} \sum_{t=1}^T \Gammavect_t$ as input, whereas \emph{3D/2D U-Net} takes the whole concatenated sequence $\Gammavect_{1:T}$. All three approaches are approximately able to predict the positions of two objects of interest. The third object is probably not targeted enough during the sequence to be found. The black pixels in the \emph{Linear Regression} indicate negative values. All other approaches end with a sigmoid activation so each pixel value is homogeneous to a probability. The lower number of falsely proposed object positions for the \emph{Mean-2D-Enc} is consistent with the higher mean precision reported.
For comparison, experiments on the \emph{synthetic} scenario from Fig.~\ref{fig:illustration_synt_1} are available in the supplementary materials.
\section{Introduction}
\label{sec:gf-intro}
\emph{Gaze following} is the ability to intuit the region of space that an observer is looking at. Humans learn this skill during infancy
\cite{baldwin1995understanding}, and use it very frequently in many social activities~\cite{land2009looking}. An accurate estimation of where one or several persons look has an enormous potential in order to determine which are the objects of interest in a scene, predict the actions and movements of the participants and, in general terms, advance towards a better visual scene understanding. It has applications in various fields such as human-robot interaction~\cite{schauerte2014look, domhof2015multimodal, lathuiliere2019prl}, or action recognition~\cite{Wei2018Where}. However, automatically estimating the visual region of attention remains an open challenge, particularly when the gaze target is not visible within the field of view.
This paper addresses the detection of visual regions of attention, which are expected to contain objects of interest. People in a video generally either look at other people or at an object of interest. Such an object can be indistinctly located inside or outside the current image. In the standard \emph{gaze-following} problem, addressed \emph{e.g}. } \def\Eg{\emph{E.g}. in \cite{Recasens2015}, both the observer and the targeted object are within the same image. An example is provided on Fig.~\subref*{fig:gf-gazeF}.
This is related to -- but significantly different from -- estimating the \emph{visual focus of attention} \emph{i.e}. } \def\Ie{\emph{I.e}. whom or what a person is looking at~\cite{Masse2017}. In this case, object locations are known, but potentially non-visible (occluded or outside the field of view, see Fig.~\subref*{fig:gf-VFOA}). However, in a general setting, an object may not be visible within the image, and its location is most probably unknown.
All the more in a social interaction, an object is not ``of interest'' until people actually start paying attention to it. In this paper, we deal with \emph{extended gaze following} in videos, see Fig.~\subref*{fig:gf-OFVOD}, meaning that we tackle the more general problem of predicting the location of objects of interest whose number and locations are not known a priori, and that are not necessarily visible.
\begin{figure*}
\centering
\subfloat[Gaze following~\cite{Recasens2015}]{\label{fig:gf-gazeF}\includegraphics[height=0.215\linewidth]{gazefollow}}
\subfloat[Visual Focus of Attention~\cite{Masse2017}]{\label{fig:gf-VFOA}\includegraphics[height=0.175\linewidth]{VFOA.pdf}}
\subfloat[Extended Gaze following]{\label{fig:gf-OFVOD}\includegraphics[height=0.175\linewidth]{OFVOD.pdf}}
\caption[A comparison of gaze-related computer vision problems]{
A comparison of gaze-related computer vision problems. In the standard formulation of gaze following (a), the problem consists in localizing the objects that people are likely to be looking at (and both observer and objects are visible in the input image). Visual focus of attention estimation (b) consists in associating which person is looking at what object at a certain moment (considering that the objects locations are known). In extended gaze following (or visual regions of attention detection) (c), we aim at localizing objects of interest even if they are not visible in the video image. }
\label{fig:gf-gaze-cv-problems}
\end{figure*}
Our method takes as input a video sequence containing a group of people, and outputs a set of estimated locations for the objects of interest. This work makes the assumption that objects do not move across the video sequence. As in~\cite{Masse2017, mukherjee2015deep}, we propose to use the head orientation as a strong cue for the gaze direction.
The pipeline is illustrated in Fig.~\ref{fig:gf-outline}.
The contribution of this paper is threefold. First, we propose a novel formalism for embedding the spatial representation of directions of interest and regions of attention. They are modeled as a top-view heat-map, \emph{i.e}. } \def\Ie{\emph{I.e}. a discrete grid of spatial regions from a top-view perspective. Contrary to the majority of previous work, this formalism is not limited to representing locations within the field of view. Second, we propose several convolutional encoder/decoder neural architectures that learn to predict object locations from head poses in our proposed embedding, and we compare them with several baselines inspired from earlier work. Third, since a large amount of data are required to train a deep neural network, we propose an algorithm based on a generative probabilistic framework that can sample an unlimited number of synthetic conversational scenarios, involving people and objects of interest. The method has been tested both on synthetic data and on a publicly available dataset.
The remainder of this paper is organized as follows. The state of the art is presented in Section~\ref{sec:gf-related}. Then, the details of the proposed heat-map representations and neural network architectures are respectively given in Sections~\ref{sec:HM} and~\ref{sec:gf-inference}. The synthetic data generation process is described in Section~\ref{sec:synthetic}. The Section~\ref{sec:gf-expe} is dedicated to experimental results, both on synthetic and real data. To conclude, Section~\ref{sec:gf-concl} discuss the perspectives and limitations of this work.
\section{Deep Learning for Extended Gaze Following}
We note $N_t$ the number of persons at time $t \in \{1\dots T\}$. For each person, we suppose that we can estimate its corresponding 3D head location $[x^{n}_t,y^{n}_t,z^{n}_t]^\top$, and head orientation $[\phi_{t}^n, \theta_{t}^n]^\top$ for person $n \in \{1\dots N_t\}$ in a common scene-centered coordinate frame. However, we additionally choose to drop the z-coordinate (the height) and the head tilt angle as in \cite{Cohen2012}, projecting every object and every person in the same horizontal plane. As we will see later, this simplification drastically reduces the complexity of the model while still representing plausible scenarios. In addition, the tilt angle is commonly the one estimated with the largest mean absolute error \cite{lathuiliere17}.
In the remaining of the paper, the term \emph{position} refers to 2D coordinates $\xmat_t^n = [x_t^n, y_t^n]^\top$ in the horizontal plane (top-view perspective), and \emph{head orientation} refers to the head pan angle $\phi_t^n$.
As mentioned before, we decided to use heat-map embeddings. The reasons for this are multiple.
First, the exact number of people and objects is not known a priori and may vary within and between video sequences. Heat-map structures are independent of the number of participants (people and objects of interest). Additionally, the problem addressed is fundamentally geometric, and heat-maps intrinsically encode the geometry of the scene. Moreover, convolutional neural networks are able to efficiently extract this structured information in order to obtain a descriptive input representation. A drawback of the heat-map representation is the difficulty to predict an object outside the modeled area. Nevertheless, for indoor scenarios, the area containing the objects is bounded. It is then possible to adapt the heat-map size for the current setup and train the model using scaled simulated scenarios (see Section \ref{sec:synthetic}). For all these reasons, we employ heat-map embeddings to model the geometry of the scene.
\subsection{Heat-Map Representation}
\label{sec:HM}
We propose several heat-map representations of the scene from a top-view perspective. The scene is discretized into a 2D grid of dimension $S_U\times S_V$.
Each position in the scene $\xmat=(x,y)$ is associated to a grid cell $\pmat=(u,v) \in \{1\dots S_U\}\times \{1\dots S_V\}$. As stated previously, $\xmat$ is bounded in both dimensions: $x \in [x_{min}, x_{max}]$ and $y \in [y_{min}, y_{max}]$. With these notations, $\pmat=(u,v)$ is obtained from $\xmat$ as
\begin{equation}
\begin{cases}
u &= \lceil S_U \times \frac{x-x_{min}}{x_{max}-x_{min}} \rceil\\
v &= \lceil S_V \times \frac{y-y_{min}}{y_{max}-y_{min}} \rceil\\
\end{cases}
\end{equation}
where $\lceil \cdot \rceil$ is the \emph{ceiling} function.
The grid cell associated to $\xmat^n_t=(x^{n}_t,y^{n}_t)$, the position of a person $n$ at time $t$, is $\pmat^n_t$.
In this formalism, a heat-map $\Lambdavect$ is a 2D map of $S_U \times S_V$ elements that attaches to each cell $\pmat$ of the grid a value $\Lambdavect(\pmat)$ between $0$ and $1$. The meaning of this value depends on what the heat-map represents. In this paper, there are two different categories of heat-map. First, a \emph{gaze heat-map} $\Gammavect$ is an embedding for head pose information. A value close to one indicates a region of space consistently situated in front of someone's head. Second, an \emph{object heat-map} $\Omegavect$ embeds the likelihood for each region to contain an object of interest.
\begin{figure}
\centering
\subfloat[][Camera Image at $t=10$]{\label{fig:camera10}
\includegraphics[height=0.23\linewidth]{camera_21633}
} \hspace{0.2cm}
\subfloat[][\emph{Gaze heat-map} $\Gammavect_{10}$]{\label{fig:hm10}
\includegraphics[height=0.23\linewidth]{roa_map_21633}
} \hspace{0.4cm}
\subfloat[][Mean \emph{gaze heat-map} $\Gammavect$]{\label{fig:mean_hm}
\includegraphics[height=0.23\linewidth]{cumulative_25}
} \\
\subfloat[][Camera Image at $t=30$]{\label{fig:camera30}
\includegraphics[height=0.23\linewidth]{camera_21713}
} \hspace{0.2cm}
\subfloat[][\emph{Gaze heat-map} $\Gammavect_{30}$]{\label{fig:hm30}
\includegraphics[height=0.23\linewidth]{roa_map_21713}
} \hspace{0.4cm}
\subfloat[][Object positions]{\label{fig:objects}
\includegraphics[height=0.23\linewidth]{expected_21633}
} \\
\subfloat[][Camera Image at $t=80$]{\label{fig:camera80}
\includegraphics[height=0.23\linewidth]{camera_21958}
} \hspace{0.2cm}
\subfloat[][\emph{Gaze heat-map} $\Gammavect_{80}$]{\label{fig:hm80}
\includegraphics[height=0.23\linewidth]{roa_map_21958}
} \hspace{0.4cm}
\subfloat[][\emph{Object heat-map} $\Omega$]{\label{fig:o_hm}
\includegraphics[height=0.23\linewidth]{expected_21633_blur}
}
\caption[Illustration of the heat-map representations on a \emph{Vernissage} sequence]{
Illustration of the heat-map representations using a sequence extracted from the \emph{Vernissage} dataset~\cite{jayagopi2013vernissage}. The camera is located close to the bottom left corner of the \emph{gaze heat-maps}. Heat-map colors range from blue to red to indicate number from $0$ to $1$.
\subref{fig:camera10}, \subref{fig:camera30}, \subref{fig:camera80}: camera images.
\subref{fig:hm10}, \subref{fig:hm30}, \subref{fig:hm80}: corresponding \emph{gaze heat-maps}. Cone origins in the \emph{gaze heat-maps} indicate people positions; cone axes represent head orientations.
\subref{fig:mean_hm}: mean \emph{Gaze heat-maps} over the sequence.
The object ground truth is represented in the heat-map coordinate frame \subref{fig:objects}. This provides the ground truth \emph{Object heat-map} \subref{fig:o_hm} used for training and \emph{MSE} evaluation.
}
\label{fig:illustration}
\end{figure}
\paragraph{\emph{Gaze heat-map} representation $\Gammavect$}
Motivated by the use of cones for modeling the dependency between head pose and gaze \cite{Marin-Jimenez2014}, we compute a heat-map $\Gammavect^{n}_t \in [0,1]^{S_U\times S_V}$ for each person $n \in \{1\dots N_t\}$ by considering a cone whose axis is the direction spanned by the head pan angle $\phi_t^n$. Formally, the value of $\Gammavect^{n}_t$ at any grid cell $\pmat$ is given by:
\begin{equation}
\label{eq:gazemap}
\Gammavect^{n}_t(\pmat) =
\begin{cases}
1 &\quad\text{if }|\phi(\pmat)-\phi^{n}_t|<\epsilon\\
0 &\quad\text{otherwise} \\
\end{cases}
\end{equation}
where $\phi(\pmat)$ is the angle corresponding to the direction of vector $\overrightarrow{\pmat^n_t\pmat}$. The parameter $\epsilon$ controls the aperture of the cone.
We obtain the \emph{gaze heat-map} illustrated in Fig.~\subref*{fig:hm10}, \subref*{fig:hm30} and \subref*{fig:hm80}:
\begin{equation}
\Gammavect_t= \frac{1}{N_t} \sum_{n=1}^{N_t} \Gammavect^{n}_t.
\end{equation}
It is sometimes useful to aggregate the \emph{gaze heat-maps} through time into a mean \emph{gaze heat-map} (see Fig.~\subref*{fig:mean_hm}) to have an compact representation of the scenario:
\begin{equation}
\Gammavect= \frac{1}{T} \sum_{t=1}^T \Gammavect_t.
\end{equation}
\paragraph{\emph{Object heat-map} $\Omegavect$}
Considering a scenario with $M$ objects (\emph{e.g}. } \def\Eg{\emph{E.g}. Fig.~\subref*{fig:objects}), we compute a heat-map $\Omegavect \in [0,1]^{S_U\times S_V}$ (Fig.~\subref*{fig:o_hm}) whose value at grid cell $\pmat$ is given by:
\begin{equation}
\Omegavect(\pmat) =\max_{1\leq m\leq M}\exp\left(-\frac{||\pmat-\pmat^m_{obj}||_2^2}{2\sigma_\Omega^2}\right)
\end{equation}
where $\pmat^m_{obj}$ is the grid cell corresponding to the scene position of the $m^{th}$ object. The variance $\sigma_\Omega$ controls the spread of the peaks. As objects do not move, $\Omegavect$ remains constant during a scenario.
Now, let us suppose we have been able to obtain an estimate $\hat{\Omegavect}$ of $\Omegavect$ from $\Gammavect_1 \dots \Gammavect_T$.
Finally, to obtain an actual list of object positions, we extract the local maxima from $\hat{\Omegavect}$ and discard local maxima that are too low compared to the global maximum. More precisely, given a candidate position $\pmat_C$, a neighborhood of this position $\mathpzc{N}(\pmat_C)$ and a shrinking function $\alpha(\cdot)$ such that $\alpha(x)\leq x$, we consider that $\pmat_C$ contains an object if
\begin{equation}
\label{eq:NMS}
\pmat_C = \argmax_{\pmat \in \mathpzc{N}(\pmat_C)} \hat{\Omegavect}(\pmat) \qquad \text{and} \qquad \hat{\Omegavect}(\pmat_C) \ge \alpha \left( \max_{\pmat} \hat{\Omegavect}(\pmat) \right).\\
\end{equation}
The section~\ref{sec:gf-inference} below is dedicated to propose a neural network that learns to predict an estimate $\hat{\Omegavect}$ of the \emph{object heat-map} from the set of \emph{gaze heat-maps} $\Gammavect_1\dots\Gammavect_T$.
\subsection{Object heat-map inference}
\label{sec:gf-inference}
Now, we address the problem of estimating $\hat{\Omegavect}$, on which the local maxima detection algorithm can be run. We propose several baselines with justification for their relevance. Then, we present our architectures based on convolutional encoder/decoder.
\paragraph{Heuristics without learning}
First, we propose two heuristics with no training. The local maxima detection is performed directly on a combination of gaze heat-maps. Indeed, the regions that are activated (close to one) in multiple gaze heat-maps are consistently in front of someone's head and have a high chance of containing an object. Previous works~\cite{Cohen2012, Marin-Jimenez2014} already used geometric features based on cone intersections. The heuristics are as follow.
\begin{itemize}
\item \emph{Cone}: The local maxima extraction is performed directly on the \emph{mean gaze heat-map} $\Gammavect = \frac{1}{T}\sum_{t=1}^{T}\Gammavect_t$.
\item \emph{Intersect}: We define a \emph{gaze intersection heat-map} $\Gammavect_t^{inter}$ per time frame, by setting regions to one only if they are at the intersection of multiple cones. More formally,
\begin{equation}
\Gammavect_t^{inter} (\pmat) =
\begin{cases}
1 & \text{if} \sum_{n=1}^{N_t} \Gammavect_t^n (\pmat) \geq 2 \\
0 & \text{otherwise}
\end{cases}
\end{equation}
The local maxima extraction is performed on $\Gammavect^{inter} = \frac{1}{T} \sum_{t=1}^T \Gammavect_t^{inter}$.
\end{itemize}
\paragraph{Learning-based Baselines}
We define some simple regression models. They learn a regression from the \emph{mean gaze heat-map} $\Gammavect = \frac{1}{T}\sum_{t=1}^{T}\Gammavect_t$ to the \emph{Object heat-map} $\hat{\Omegavect}$. These models consider the input and output as flattened vectors of $S_U\times S_V$ components.
\begin{itemize}
\item \emph{Linear Reg.}: We learn a linear regression model from $\Gammavect$ to $\hat{\Omegavect}$. Interestingly, the output of a linear regression is not constrained to lie between $0$ and $1$, contrary to the definition of $\Omegavect$. The local maxima extraction is performed after $\hat{\Omegavect}$ has been rescaled in $[0,1]$.
\item \emph{d-FC}: The regression is performed on $\Gammavect$ by a network composed of $d\in\{1,3\}$ fully connected hidden layers of $S_U\times S_V$ units, with ReLU activations. The last hidden layer is fully connected to the output \emph{object heat-map} with sigmoid activations.
\end{itemize}
\paragraph{Encoder/Decoder Architectures} They have been used for many computer vision tasks where the goal is to perform a regression between high dimensional spaces \cite{isola2017image,badrinarayanan2017segnet}. Such architectures are composed of two sub-networks, where the first reduces the spatial resolution of the input to obtain a compact description of it, and the second alternates between up-sampling and fully-connected layers until recovering a high dimensional output. In our particular problem, we use convolutional layers instead of fully-connected layers to model the spatial connections. Moreover, as the input is a sequence, several encoder architectures can be employed. We propose to use a decoder composed of three successive up-sampling and convolutional layers with $3 \times 3$ kernels. The last convolution layer of the decoder employs sigmoid activations. The whole network is trained employing the Mean Squared Error (MSE) loss. We propose the four following architectures that represent a progressively increasing complexity. Graphical representations of the proposed networks are given in the supplementary material\footnote{see \textit{https://team.inria.fr/perception/research/extended-gaze-following}}.
\begin{itemize}
\item\emph{Mean-2D-Enc}:
This is the simplest model. We use the \emph{mean gaze heat-maps} $\Gammavect$ as in the baselines. It is fed to a standard 2D convolutional encoder composed of three successive convolutional and down-sampling layers.
\item\emph{2D-Enc}:
In this model, we consider that time plays the role of the color-axis in standard 2D convolutions. $\Gammavect_1\dots\Gammavect_T$ are concatenated along the third dimension to obtain the \emph{sequence gaze heat-map} $\Gammavect_{1:T}$. Therefore, the first layer kernels have as dimension $3\times3\times T$ instead of $3\times3\times1$ like in \emph{Mean-2D-Enc}.
\item\emph{3D-Enc}:
Inspired by \cite{ji20133d}, that shows that 3D convolutions are able to extract reliable features from both the spatial and the temporal dimensions, we propose a 3D-Encoder network on $\Gammavect_{1:T}$. By performing 3D convolutions, the model can capture orientation changes and people motion in successive frames. The time dimension is reduced, from $T$ to $1$ after three convolutional and max-pooling layers, before feeding it to the 2D-Decoder.
\item\emph{3D/2D U-Net}:
This variant of the \emph{3D-Enc} architecture is inspired from the U-Net architecture \cite{ronneberger2015u}. In our specific case, since we have a 3D encoder, we need to squeeze the time dimension. To do so, we combine over time the feature maps of the encoder with max-pooling, before concatenation to the decoder.
\end{itemize}
\section{Related work}
\label{sec:gf-related}
\emph{Gaze following}, or more generally any problem based on the visual attention of a person within an image, is intrinsically based on estimating the gaze direction. In practice, estimating the gaze direction is a complicated problem that still requires to compromise between being precise and non-invasive. When precision is crucial, a head-mounted system, \emph{e.g}. } \def\Eg{\emph{E.g}. ~\cite{Hong2012}, can provide very accurate gaze direction. However, it cannot be used in a natural scenario since it requires a specific setup. Since the head-mounted system is visible to all participants, it may bias what would be the nature of social behaviour. Another issue is that the head-mounted system can hardly be used to annotate training data since the system would appear in the images recorded by external cameras and, therefore, real environment images that are recorded without head-mounted system would differ from the training set images~\cite{Fischer_2018_ECCV}.
On the other hand, estimating gaze direction from remote camera images is a difficult task, with non-frontal faces, or eyelid occlusions~\cite{Hansen2010}. Moreover, since it is difficult to obtain gaze annotations in scenarios where people can move freely, most learning-based methods are trained on extremely simplified setups. For instance, in~\cite{Krafka2016} and~\cite{zhang2015appearance}, subjects were asked to fixate a region on the screen of a camera-equipped device. Alternatively, in unconstrained settings, the head pose is highly correlated with the gaze direction, and the former can be used as an approximation for the latter~\cite{mukherjee2015deep, Masse2017}.
Finding objects of interest generally requires to analyze the visual field of view and look for highly contrasting regions. Indeed, an object or a person is likely to look different from the background, thus highly contrasting regions have higher chance of containing something interesting. This approach, similar to the human brain pipeline~\cite{treue2003visual}, is known in computer vision as \emph{saliency}~\cite{itti2001computational,rudoy2013learning,parks2015augmented,wang2018deep}, where a salient region is one that attracts the visual attention of an observer.
In the context of gaze following, the goal is to find regions that are salient, \emph{i.e}. } \def\Ie{\emph{I.e}. that attracts gaze, from another point of view. However, a salient region is most likely salient from most points of view. Based on this remark,~\cite{Recasens2015} combines a saliency model with a gaze direction model to find salient objects at the intersection of the image and the person's field of view. The attention predictor in~\cite{Wei2018Where} also uses both saliency and gaze. By combining multiple gaze directions, \cite{fan2018inferring} estimates shared attention of multiple people, but still within the image. In~\cite{recasens2017following}, the authors further investigate this problem based on the idea that the gaze target of a person inside a video may be visible in another video frame. Their method still relies on a saliency model. Recently, \cite{Chong_2018_ECCV} uses a similar combination of gaze and saliency but is also able to predict whether the object of attention lies within the image or not.
Finally, \cite{schauerte2014look, domhof2015multimodal} merge the problems of saliency and gaze following in the context of human-robot interaction. Indeed, the robot is both an active member of the scenario, and an observer behind the camera. Both papers are based on saliency and gaze direction, as well as additional data such as pointing gesture and speech.
However, all works based on saliency require that the object of interest lies within the field of view. By contrast, we wish to be able to locate out-of-view objects; therefore, we cannot rely on this category of methods.
\begin{figure*}[t!]
\centering
\includegraphics[width=\linewidth]{full_model_sec.pdf}
\caption[Outline of the proposed model]{
Outline of the proposed model. For every frame and detected face, orientation and 3D location are estimated, and both sources of information are combined to obtain a top-view representation of the scene encoded in a heat-map. The sequence of heat-maps is then given to a neural network with an encoder/decoder architecture. The network outputs a heat-map that predicts the position of the objects of interest in the top-view domain.
}
\label{fig:gf-outline}
\end{figure*}
Apart from saliency-based gaze following, a few other methods have been published, addressing the gaze-following problem in the 3D space instead of the 2D image plane.
\cite{soo2015social} proposes to estimate 3D regions of attention using only the location of people. They model social group structures that constrain the set of candidate locations. In this framework, they learn to locate regions of attention independently of visual saliency. Their method only needs people locations and can work in adversarial scenarios, using only spatial data from first person cameras. However, it fails when some people are undetected and the group structures are wrongly estimated, or when a person is isolated and should not be integrated into a group structure.
By contrast, both~\cite{Cohen2012} and~\cite{Brau_2018_ECCV} independently propose to use 3D intersection of gazes in a probabilistic framework to estimate locations of objects of interest, possibly outside the camera field of view. The methods achieve good levels of performance -- even though~\cite{Cohen2012} lacks quantitative evaluation.
In both cases, no training data have been used. Each method is designed with strong geometric assumptions so that location inference can be performed without any prior learning phase. At the time this article was written, the data on which the methods have been tested were not released yet for comparison.
In this paper, we combine a learning-based model with a geometric formulation to address the gaze-following problem, without the restriction of being limited to the image plane. Only very few works exist in this direction~\cite{soo2015social, Cohen2012, Brau_2018_ECCV}, and all employ strong social or geometric assumptions.
|
{
"timestamp": "2019-03-01T02:12:14",
"yymm": "1902",
"arxiv_id": "1902.10953",
"language": "en",
"url": "https://arxiv.org/abs/1902.10953"
}
|
\section{Introduction}
One notable feature of human learners is that we are able to
carry out counter-factual reasoning over unrealized events. That is,
we contemplate potential answers to questions of the form, ``What
would I do in situations $X$, $Y$, and $Z$?'' A related, and perhaps
more pertinent, form of question is, ``Are there situations $X$, $Y$, and $Z$,
such that in these situations I would select actions $A$, $B$, and
$C$?'' In this case, the actions $A$, $B$, etc., might be
actions that are likely to result in particularly good or bad
outcomes, and answers $X$, $Y$, etc., can be useful, especially when
they are of an unexpected nature, since they reveal potential failures
of robustness (in the case of bad examples) or potential strengths (in
the case of good examples). In this paper, we describe a novel
approach to answering and utilizing the answers to questions of this
form when asked not of a human agent, but of a reinforcement learning
agent. Our approach is not based solely on the deployment of
techniques from the typical machine learning toolbox, as we make
crucial use of SMT-solving, which is more familiar to researchers in
the field of formal methods. In the theoretical development, we
capture our use of SMT-solving technology via the abstraction of what
we are calling \emph{introspection oracles}: oracles that may give us
direct access to sets of (state, action) pairs satisfying fixed
constraints with respect to the policy network.
By querying the oracle during training it is possible to generate
(state, action)-pairs capturing failures/strengths of the agent with
respect to properties of interest. For instance, if there are certain
``obviously wrong'' actions that the agent should never take (e.g.,
selecting a steering angle that would cause the automobile controlled
by the policy network to drive off of the road when there are no
obstacles or other dangers present), we query the oracle as to whether
there exists states in which the agent would select such actions. Our
algorithm then uses this data to train so as to improve the safety of
the agent and without requiring that such potentially dangerous or
costly situations be encountered in real life. It is true that such
(state, action) pairs are potentially discoverable in
simulation/testing, but when the set of such pairs is known beforehand
we save time and improve policy robustness by generating them analytically.
In this paper, we introduce a new algorithm for reinforcement learning, which we
call the \emph{Introspection Learning Algorithm}, that exploits
introspection oracles to improve the training and robustness of reinforcement learning (RL) agents versus baseline training algorithms. This algorithm
involves modifying the underlying MDP structure and we derive
theoretical results that justify these modifications. Finally, we
discuss several experimental results that clearly showcase the
benefits to both performance and robustness of this approach. In
particular, in the case of robustness, we evaluated our results by
querying the weights after training to determine numbers of Sat
(examples found), Unsat (examples mathematically impossible) and
Timeout (ran out of time to find or refute existence of examples)
results.
The paper is organized as follows. In Section~\ref{oracles} we
introduce the mathematical abstraction of introspection oracles and
discuss briefly their embodiment as SMT-solvers. Section
\ref{introspection} details our Introspection Learning
Algorithm. Finally, Section \ref{methods} captures our
empirical results. The Appendix (Section \ref{appendix}) includes the proof of
a basic result that justifies the modification of MDPs made in our algorithms.
\subsection{Related work}
Previously, Linear Programming, which is itself is a constraint
solving technique, has been employed in reinforcement
learning to constrain the exploration space for the agent’s policy to
improve both the speed of convergence and the quality of the policy
converged to \cite{Amos2017} or as a replacement for
more traditional Dynamic Programming methods in Q-Learning to solve
for equilibria policies in zero-sum multi-agent Markov game MDPs
\cite{Littman,Greenwald}. Previous work has also been done on
incorporating Quadratic Program solvers to restrict agent exploration to ``safe''
trajectories by constraining the output of a neural network policy
\cite{Pham2017,Amos2017}. Introspection Learning is fundamentally
different from these approaches as rather than restricting the action space, or
replacing our Q function, we are instead shaping our agents in policy space by
asking our policy for state batches where it would satisfy stated
constraints, without needing the agent to actually experience these
states.
Exciting recent work on verification of neural networks
(e.g., \cite{Reluplex,Lomuscio}) is closely related the work described
here. In addition to the similarity of the techniques, we are indeed
capturing verification results as a robustness measure (see
below). One practical distinction is that we are using the dReal
solver \cite{dReal}, which is able to handle
networks with general non-linear activations, but as a trade-off (not
made in other SMT-solvers) admits the possibility of
``false-positive'' $\delta$-satisfiable instances. In principle, our
algorithm can be used with any compatible combination of SMT-solvers and neural
network architectures.
\section{Introspection Oracles}\label{oracles}
In order to set the appropriate theoretical stage, we will first
introduce some notation and terminology.
\begin{definition}\label{definition:pre_mdp}
A \emph{pre-Markov decision process} (pre-MDP) consists of a set $S$ of
\emph{states}, a set $A$ of \emph{actions}, and \emph{transition
probabilities} $p(s,a,s')$ in $[0,1]$ for $s,s'\in S$ and $a\in A$
such that $\sum_{s'}p(s,a,s')=1$.
\end{definition}
Intuitively, the value $p(s,a,s')$ is the
probability $\mathbb{P}(s'|s,a)$ transitioning from state $s$ to state
$s'$ on taking action $a$.
\begin{definition}\label{definition:policy}
Given a \emph{pre-Markov Decision Process} (pre-MDP) $D=(S,A,p)$, a \emph{policy for $D$} assigns to each state $s$
a probability distribution $\pi(s)$ over the set $A$.
\end{definition}
A pre-MDP is called a MDP$\backslash$R in, e.g., \cite{Abbeel}.
Often we are concerned with cases where $A$
is finite and the policies $\pi$ under consideration are \emph{deterministic} in the sense that,
for each state $s$, $\pi(s)(a)=0$ for all but a single element $a$ of
$A$. When $p(s,a,s')=1$ we write ${a\colon s\to s'}$. Given a pre-MDP
$D$, we denote by $\Pi(D)$ the set of all
policies for $D$.
\begin{definition}\label{def:mdp}
A \emph{Markov decision process} (MDP) consists of a pre-MDP
$(S,A,p)$ together with a \emph{reward} function ${r\colon
S\times A\to \mathbb{R}}$ which is bounded, a subset $T\subseteq
S\times A$
of \emph{terminal (state, action)-pairs}, and a new state $s_{t}$
not in $S$ such that:
\begin{itemize}
\item For any $(s,a)\in T$, $a\colon s\to s_{t}$;
\item For any $a\in A$, $a\colon s_{t}\to s_{t}$; and
\item For any $a\in A$, $r(s_{t},a)=0$.
\end{itemize}
\end{definition}
One non-standard feature of Definition \ref{def:mdp} is that we consider terminal
pairs $(s,a)\in S\times A$ rather than terminal states. This will be
technically useful below. We also follow \cite{SuttonBarto} in that
the provision of terminal pairs modifies the pre-MDP
structure in adding a dummy stable state $s_{t}$ to which all terminal
states canonically transition such that subsequent transitions from
$s_{t}$ have no reward. This is a
technical convenience which streamlines some of the theory.
We denote by $\mathcal{M}(D)$ the set of all Markov decision processes over
the pre-MDP $D$ and by $\Pi(D)$ the set of all policies over
$D$. Given an MDP $M$ in $\mathcal{M}(D)$, we denote by $\textnormal{Opt}(M)$ the
subset of $\Pi(D)$ consisting of those policies that are optimal
for $M$. In broad strokes, inverse reinforcement learning \cite{Ng} is
concerned with, given a policy $\pi$ in $\Pi(D)$ (or, more
often, a set of its
trajectories), determining an element $M$ of $\mathcal{M}(D)$ such that
$\pi$ is in $\textnormal{Opt}(M)$. We are concerned with a closely related
problem.
One difference between our approach and that of inverse reinforcement
learning is that instead of assuming access to a target policy
$\pi$ or its trajectories, we assume that we have access to certain
\emph{properties} that target policies \emph{ought} to have. In the
simplest case, such a property is given by a subset of the set
$S\times A$ of (state, action) pairs.\footnote{In the more general
case, the relevant properties should be (non-empty) subsets of space
$(S\times A)^{*}$ of finite sequences of (state, action) pairs that
are compatible with the underlying transition probabilities of $D$.
In this paper, we restrict attention to the more elementary notion.}
We refer to policies with the required properties as \emph{good}
policies. There is considerable flexibility in the notion of goodness
here, but in many cases it will be associated with safety and
robustness. E.g., a good policy for driving a car would not make
unexpected sharp turns when the road ahead is straight and clear of
obstacles. Much of our focus is on these kinds of examples, but it is
worth emphasizing that goodness could instead be associated with
performance rather than safety.
In order to make the problem tractable, it is necessary to restrict to
sufficiently well-behaved subsets of $S\times A$. For us, the
well-behaved subsets are those definable in the first-order theory of
real arithmetic with common non-linear function symbols (e.g., $\sin$,
$\log$, $\max$, $\tanh$, etc.).\footnote{In the experimental
results captured in this paper, we restricted further to
semialgebraic subsets. I.e., those describable as finite unions of
sets defined by finitely many polynomial equations and inequations.}
Denote by $\mathcal{P}_{\!\!r}(X)$ the set of all such
subsets of $X\subset\mathbb{R}^{n}$. With this notation in place, we
arrive the definition of introspection oracle.
\begin{definition}\label{def:introspection_oracle}
Given policy $\pi$ in $\Pi(D)$, an \emph{introspection oracle
for $\pi$} is a map $\omega_{\pi}\colon\mathcal{P}_{\!\!r}(S\times
A)\to\{\perp\}+S$ such that if $\omega_{\pi}(U)\neq\perp$, then
$(\omega_{\pi}(U),\pi(\omega_{\pi}(U)))$ is in $U$. An
introspection oracle is \emph{non-trivial} when there exists $U$ in
$\mathcal{P}_{\!\!r}(S\times A)$ such that $\omega_{\pi}(U)\neq\perp$.
\end{definition}
Intuitively, an introspection oracle $\omega_{\pi}$ for
$\pi$ attempts to answer questions of the form: ``Are there inputs
that give rise via $\pi$ to a (state, action) pair with property $U$?''
Here $\perp$ is an error signal which can be provided with several
possible semantics. Here it is best understood as indicating
that the oracle was unable to find an element of $U$ in a reasonable amount
of time.
Before turning to describe our use of introspection oracles in
reinforcement learning, we observe that non-trivial introspection
oracles do indeed exist:
\begin{observation}
For policy functions $\pi$ definable in the language of first-order
real arithmetic with non-linear function symbols ($\sin$, $\cos$,
$\log$, $\tanh$, etc.) there exist non-trivial introspection oracles.
\end{observation}
The existence of such introspection oracles which are moreover
practically useful in the sense of returning outputs $\neq\perp$ in
a wide range of feasible cases is guaranteed by the $\delta$-decision
procedure of Gao, Avigad and Clarke \cite{Gao}, which is implemented
in the dReal non-linear SMT-solver. The novelty of dReal is that it
overcomes the undecidability of real arithmetic with non-linear
function symbols by accepting a compromise: whereas unsatisfiable
(Unsat) results are genuine, satisfiable (Sat) results may be
false-positives. Note that, unlike in many of the other applications
of SMT-solving to verification of neural networks such as
\cite{Reluplex,Lomuscio}, dReal is able to handle all common non-linear
activations. In terms of our abstraction, spurious Sat results, which
are easily detected by a forward pass of the network, can be regarded
as instances where $\omega_{\pi}(U)=\perp$.
\section{The Introspection Learning Algorithm}\label{introspection}
We now describe the Introspection Learning Algorithm in detail,
starting with its inputs. First, this algorithm assumes given an
off-policy reinforcement learning algorithm (OPRL) and corresponding policy
function $\pi$. It is furthermore assumed that $\pi$ is describable
in the language of real arithmetic with non-linear function symbols.
Additionally assume given a family $(U_{i})_{i}$ of
subsets $U_{i}\in\mathcal{P}_{\!\!r}(S\times A)$, which will be used when we query
the oracle $\omega$. Having a sufficiently rich family
$(U_{i})_{i}$ will provide a mechanism
for generating more useful examples and the design
of these properties is one of the main engineering challenges
involved in utilizing the algorithm effectively. Pairs $(s,\pi(s))$
obtained from the oracle as $s=\omega(U_{i})$ are added to the OPRL agent's
replay buffer.
\begin{algorithm}\label{algorithm}
\DontPrintSemicolon
\KwData{Off-policy RL algorithm OPRL, policy function
$\pi$, family of queries $(U_{i})_{i}$, a schedule
$\sigma$, a reward cutoff $R$}
Initialize OPRL policy $\pi$ with random weights $\vartheta$
and replay buffer $D$\;
\For{episode $e\in \{1, \ldots, M\}$} {
Train OPRL as specified\;
\If{moving average reward $< R$ and $e\in\sigma$} {
For each $i$, query
$\omega_{\pi}(U_{i})$
and add examples
$\omega_{\pi}(U_{i})\in S$ to $D$ as terminal
}
}
\caption{{\sc Introspection Learning}}
\label{algo:introspection}
\end{algorithm}
Finally, we assume given a schedule determining when during training to perform
queries and updates. For simplicity in describing the algorithm we
assume that the schedule is controlled by two factors. First, a
simple set $\sigma$ of training indices. Second, a bound $R$ on
moving average reward such that once moving average reward is greater
than or equal to $R$ we no longer perform queries or updates on gathered
examples.
In summary, given the aforementioned inputs, the Introspection
Learning Algorithm \ref{algorithm} proceeds by training $\pi$ as usual
according to the OPRL except that, when episode indices $e$ in $\sigma$
are arrived at and the moving average reward remains below $R$, the
oracle is queried with the specified family of pairs, examples are
gathered (when possible) and inserted into the replay buffer as
terminal.
Mathematically, this algorithm effectively produces a modified MDP structure
$M^{\dagger}$ by altering the terminal pairs and the reward structure. In the
Appendix (Section \ref{appendix}), we show (Theorem \ref{theorem:equiv}) that,
under reasonable hypotheses, the sets of optimal policies for the
original MDP $M$ and the modified MDP $M^{\dagger}$ coincide.
There are several parameters and variations of this algorithm
possible, of which we now mention several. First, in some cases it
may be necessary or useful to post-process the gathered
state batches (e.g., to ensure sufficient balance/symmetry
properties). Here consideration should be paid to the bias introduced by state
batches which are in one sense ``on policy'' (if the agent were in a state
returned by the SMT-solver it would have taken the specified action with high
probability), but are not guaranteed to be ``on trajectory'' as we have no
guarantee the state would be reachable by policy $\pi$. In practice,
we have found such processing to be unnecessary provided that
suitable $(U_{i})_{i}$ are selected and a reasonable schedule is followed.
In addition to varying the schedule, it is also possible to consider
a range of options for the behavior of the replay buffer and
how to train on the examples contained therein. We have found it to
usually be sufficient to train on these as terminal
states with high-negative or high-positive reward, however
other approaches can also be considered. It should be noted that treating these
states as terminal will alter the optimal policy, which may or may not be
desired, and alternatively one could query the training environment with the
state batches and specified actions to recover the reward signal and next state
from the environment in order to reduce the change in the optimal policy. Our
intention was to take a na\"ive approach as we are interested in applications
where acquiring experience is potentially risky or expensive.
\section{Experimental Environments and Results}\label{methods}
Our experiments were conducted with the
Double Deep Q Network algorithm DDQN \cite{VanHasselt2015} with Prioritized
Experience Replay\cite{Schaul2015} and the OpenAI Gym
``Lunar Lander'' environment \cite{OpenAI}, OpenAI Gym ``Cliff Walk'' environment
\cite{OpenAI} and the DeepMind AI Safety Gridworld ``Absent Supervisor''
environment \cite{SafetyGridworlds}. Prioritized Experience Replay augments the
selection of experience tuples from the DDQN replay buffer by preferentially
selecting experience with high TD error and simultaneously correcting for the
bias this introduces by scaling the loss in the neural network update
proportionally to the size of the TD error.
In the ``Lunar Lander'' environment the objective is to safely land a spacecraft
on the surface of the moon by controlling four discrete actions for each of its
three engines.
The state space is eight dimensional with six continuous variables representing
location in two-dimensional cartesian-coordinates, linear velocity,
angle and angular velocity, and two boolean variables indicating
whether or not contact is being made with the ground by each of the
lander's two legs. The reward signal positively reinforces movement
toward the landing pad, as well as bonus for making leg contact with the ground.
Negative reward is given for moving away from the
landing pad or losing contact with the ground. The environment is
considered solved when the agent achieves a 100 episode moving
average reward of at least 200.
In the ``Cliff Walk'' gridworld environment (Figure \ref{fig:cliffwalk}) the objective is to reach the goal
state while avoiding the row of terminal ``cliff'' states along the bottom edge
by controlling four discrete actions up, down, left, right. The state is
encoded as a binary vector. The environment provides the agent a reward of -1
at each step and a reward of -100 for entering the cliff. The goal provides no
reward and terminates the episode. In our experimentation the environment was
considered solved when the agent achieved a 100 episode moving average reward
of at least -30.
\begin{figure}[h]
\centering
\includegraphics[width=85mm]{cliffwalk_gridworld}
\caption{Cliff Walk Gridworld\label{fig:cliffwalk}}
\end{figure}
In the ``Absent Supervisor'' gridworld environment (Figure
\ref{fig:absent_supervisor}) the objective is to reach the
goal state by controlling four discrete actions up, down, left, right. The four
center squares are impassable. For each episode a supervisor is absent or
present with uniform probability. The state is encoded as a binary vector. The
environment provides the agent a reward of -1 at each time step and a reward of
+50 for entering the goal. When the supervisor is present the orange state,
located immediately above the goal state, highlighted in Figure
\ref{fig:absent_supervisor} provides a large negative reward (-30) but no such
reward when the supervisor is absent. We would like the agent to never pass
through the orange punishment state. The intent of the environment is to
demonstrate that when provided the opportunity to cheat by passing through the
orange state when the supervisor is absent traditional deep reinforcement
learning algorithms will do so.
\begin{figure}[h]
\centering
\includegraphics[width=40mm]{absent_supervisor_gridworld}
\caption{Absent Supervisor Gridworld\label{fig:absent_supervisor}}
\end{figure}
\begin{figure}[h]
\includegraphics{figure_1_20}
\caption{Episodes until ``Lunar Lander'' solved for DDQN (baseline) with
and without
Introspection Learning\label{fig:results}}
\end{figure}
In each case, the policy $\pi$ considered was a neural network with two hidden layers each
having 32 nodes and hyperbolic tangent activations. The output
activation was linear with one node for each action. DDQN with soft
target network updates \cite{Lillicrap2015}, the proportional variant of
Prioritized Experience Replay\cite{Schaul2015}, and an $\epsilon$ greedy
exploration policy were employed to train the agent
with the hyperparameters summarized in Table
\ref{tab:hypers}.
\begin{table}[h]
\begin{center}
\begin{tabular}{l|l}
\textbf{Hyperparameter} & \textbf{Value}\\
\hline
experience replay every $n$ timesteps & 2\\
replay buffer size & 1e5\\
batch size & 64\\
$\gamma$ (Discount factor) & 0.99\\
$\alpha$ (Learning rate) & 1e-3\\
$\tau$ (Soft target network update rate) & 1e-2\\
PER $\alpha$ (TD error prioritization) & 0.6\\
PER $\beta$ (Bias correction) & 0.6\\
\end{tabular}
\caption{DDQN hyperparameters used during training\label{tab:hypers}
}
\end{center}
\end{table}
In the ``Lunar Lander'' environment, the Introspection Learning parameters were
set as follows. For the query schedule,
we determine at what interval batches will be searched for and when
searching for batches will cease and training will proceed as normal.
We experimented with solving for state batches at a predetermined interval
(every 100 episodes) and ceasing when the 100 episode moving average reward
crossed a predetermined threshold.
For training on state batches, states found were treated as terminal
states with high negative reward (-100)
as determined by the rules of the environment for terminal states.
We have generally found that
incorporating the state batches into the replay buffer
is beneficial early in the learning process when the policy is poor, as it
introduces bias (cf. \cite{Schaul2015}).The query
constraints in both cases were to look for states whose $x$-coordinates were
outside of the landing zone ($x<-0.25$ or $x> 0.25$), such that the
agent favors selecting an action that would result in it moving
further away from from the landing zone.\footnote{Note
that alternative choices of query constraints are also possible
including, e.g., querying for those states that move the agent in the
correct direction, which could be given extra reward. Our approach
here is based on trying to minimize the number of obviously risky
actions the agent is likely to carry out during training, while
allowing the agent freedom to explore reasonable actions.}
This region of the state-space was divided into boxes using a
simple quantization scheme that
ignored regions of state space where examples satisfying the query
constraints would be impossible to find. In general, such
quantization schemes should be
sufficiently fine-grained to allow generation of many and diverse
examples. Twenty training runs
with a set of twenty random seeds were run with and without our approach for a
maximum of 500,000 timesteps. Results averaged over the training runs are
summarized in Figure \ref{fig:results}. DDQN with Introspection Learning
solved the environment in a mean of 893 episodes while DDQN without
Introspection Learning (baseline) failed to successfully solve the environment
on average within 500,000 timesteps.
In addition to observing performance benefits, we also evaluated the
agents trained with Introspection Learning for robustness benefits.
In particular, we periodically stored the weights of both the
Introspection Learning agent and the baseline agent during training
for each of the twenty runs. We then recorded, for different regions
of state space, statistics regarding the Sat, Unsat and Timeout results obtained
when querying the SMT-solver on these agents across training. To
recall, in this case, a Sat result indicates that there exists a state
$s$ in the specified region $U$ of state space such that an undesirable action
$\pi(s)$ (in this case, moving away from the landing zone) is selected by the
agent. Likewise, an Unsat result indicates that there
is a mathematical proof that there exists no state $s$ in $U$ such
that $\pi(s)$ is undesirable. We gathered Sat, Unsat and Timeout data
across a number of different selections of $U$. Tables
\ref{tab:small_baseline} and \ref{tab:small_il} record the percentages
of each kind of result across all twenty test runs that were captured
at four points during training. The selection of $(U_{i})_{i}$
queried here were a subset of the subsets of (state,action)-space
queried during the actual Introspection Learning training and the
results show a clear improvement of robustness over the baseline.
Timeouts during training were set to five seconds and to ten seconds
during evaluation.
One interesting point that we noticed in analyzing the robustness
evaluation data is that larger numbers of Unsat results for the
Introspection Learning agents were obtained at the beginning of
training than the end. This is illustrated, for a typical example
(the run with ID number 480951) in Figure \ref{fig:unsats}. This
is likely due to the schedule employed as part of the introspection
learning algorithm and highlights the more general fact that
reinforcement learning agents are sometimes subject to ``forgetting''
important learned behavior at later stages of training. Since the
agents at the end of training were typically very good at solving the
task, the regions of state space in which this forgetfulness would
manifest themselves were likely off-trajectory (i.e., unreachable by
the current policy).
In order to emphasize that this improvement is very much a function of
the specific $(U_{i})_{i}$ used during training, and tested at
evaluation time, we include for comparison in Table \ref{tab:large_percentages} the average
percentages for an alternative selection of $(U_{i})_{i}$ used at
evaluation time. Here the improvements are more modest.
\begin{figure}[h]
\includegraphics[scale=0.5]{robustness_results}
\caption{Total number of unsat instances as a function of
time for baseline (DDQN) and IL.\label{fig:unsats}
}
\end{figure}
\begin{table}[h]
\centering
\begin{tabular}[h]{l|ccc}
\textbf{Run ID} & \textbf{Unsat} & \textbf{Sat} & \textbf{Timeout}\\
\hline
34001 & 0\% & 62.5\% & 37.5\%\\
390797 & 0\% & 100\% & 0\%\\
747524 & 0\% & 75\% & 25\%\\
480621 & 25\% & 50\% & 25\%\\
475982 & 50\% & 25\% & 25\%\\
319324 & 25\% & 62.5\% & 12.5\%\\
449374 & 0\% & 50\% & 50\%\\
491386 & 0\% & 50\% & 50\%\\
532333 & 0\% & 50\% & 50\%\\
55487 & 0\% & 75\% & 25\%\\
4211 & 0\% & 50\% & 50\%\\
480951 & 0\% & 100\% & 0\%\\
219015 & 0\% & 87.5\% & 12.5\%\\
481614 & 0\% & 75\% & 25\%\\
367249 & 25\% & 50\% & 25\%\\
508732 & 0\% & 100\% & 0\%\\
521233 & 0\% & 50\% & 50\%\\
543696 & 0\% & 75\% & 25\%\\
998982 & 0\% & 100\% & 0\%\\
36067 & 0\% & 75\% & 25\%\\
\hline
Average & 6.250\% & 68.125\% & 25.625\%
\end{tabular}
\caption{Percentages of Sat, Unsat and Timeout instances
for Baseline DDQN at four points during training.\label{tab:small_baseline}}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}[h]{l|ccc}
\textbf{Run ID} & \textbf{Unsat} & \textbf{Sat} & \textbf{Timeout} \\
\hline
34001 & 50\% & 25\% & 25\%\\
390797 & 25\% & 25\% & 50\%\\
747524 & 0\% & 50\% & 50\%\\
480621 & 25\% & 0\% & 75\%\\
475982 & 25\% & 50\% & 25\%\\
319324 & 0\% & 75\% & 25\%\\
449374 & 0\% & 50\% & 50\%\\
491386 & 0\% & 45.8333\% & 54.1667\%\\
532333 & 25\% & 25\% & 50\%\\
55487 & 25\% & 50\% & 25\%\\
4211 & 50\% & 25\% & 25\%\\
480951 & 0\% & 75\% & 25\%\\
219015 & 25\% & 37.50\% & 37.50\%\\
481614 & 25\% & 75\% & 0\%\\
367249 & 25\% & 75\% & 0\%\\
508732 & 0\% & 75\% & 25\%\\
521233 & 50\% & 50\% & 0\%\\
543696 & 79.1667\% & 0\% & 20.8333\%\\
998982 & 25\% & 33.3333\% & 41.6667\%\\
36067 & 25\% & 0\% & 75\%\\
\hline
Average & 23.958\% & 42.083\% & 33.958\%
\end{tabular}
\caption{Percentages of Sat, Unsat and Timeout instances
for Introspection Learning at four points during training.\label{tab:small_il}}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}[h]{l|ccc}
\textbf{Run ID} & \textbf{Unsat} & \textbf{Sat} & \textbf{Timeout}\\
\hline
\textbf{Baseline} & 83.3\% & 1.4\% & 15.3\%\\
\textbf{Introspection} & 85.3\% & 0.6\% & 14.2\%
\end{tabular}
\caption{Average percentages of Sat, Unsat and Timeout instances
for Baseline DDQN versus Introspection Learning for the
full batch of all twenty runs on another selection of query
subsets $(U_{i})_{i}$. For this choice of subsets, the
gains in robustness are more modest.\label{tab:large_percentages}}
\end{table}
In the ``Absent Supervisor'' environment the Introspection Learning parameters
were set as follows. Solving for state batches is unnecessary as in this
discrete state environment we are only concerned with the agent choosing to
enter the orange punishment state from the state directly above it. For the
query schedule solving for this specific behavior is performed at every
timestep and during training this transition is treated as terminal with high
negative reward (-100). Results for DDQN with and without Introspection
Learning are provided in Figures \ref{fig:absent_supervisor_introspection} and
\ref{fig:absent_supervisor_baseline} respectively. One interesting
point about the ``Absent Supervisor'' environment is that, for the
evident notion of good policy, one of the hypotheses (the ``Strong
Compatiblity'' assumption) of our Theorem \ref{theorem:equiv} is violated.
\begin{figure}[h]
\centering
\includegraphics[width=40mm]{as_final_policy_oracleTrue}
\caption{Final DDQN policy with Introspection Learning does not select to
enter the orange punishment state when the supervisor is absent.\label{fig:absent_supervisor_introspection}}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=40mm]{as_final_policy_oracleFalse}
\caption{Final DDQN policy without Introspection Learning (baseline)
selects to cheat and enter the orange punishment state when the
supervisor is absent.\label{fig:absent_supervisor_baseline}}
\end{figure}
In the ``Cliff Walk'' environment the Introspection Learning parameters
were set as follows. Solving for state batches is unnecessary as in this
discrete state environment we are only concerned with the agent choosing to
enter the cliff states which can only be done from the state directly above
each cliff state respectively. For the
query schedule solving for these specific behaviors is performed at every
timestep and during training this transition is treated as terminal with high
negative reward (-100). It should be noted that in this particular case the
environment already treats these transitions as terminal with high negative
reward (-100) and thus Introspection Learning will not alter the optimal
policy (in particular, the hypotheses of Theorem \ref{theorem:equiv}
are satisfied).
In this experiment, five training runs with a set of five random seeds were run with and without
our approach until the environment was solved. During training, at each
timestep, a running count was kept of the number of states from which the agent
would
select to enter the cliff states ``lemming''. During training the policies were
found to lemming on average 112 times with Introspection Learning and 29,501
times without. It was experimentally found that an agent with Introspection
Learning would rarely learn a policy during training that would enter the cliff
after the first training episode while it was routine for an agent without
Introspection Learning. Representative policies learned by DDQN with and
without Introspection Learning after 30 training episodes are provided in
Figures \ref{fig:cliffwalk_introspection_30} and
\ref{fig:cliffwalk_baseline_30} respectively. Additionally, agents with
Introspection Learning enjoyed a small performance benefit solving the
environment in 208 episodes on average over the five training runs
while agents without Introspection Learning averaged 229 episodes to solve the
environment.
\begin{figure}[h]
\centering
\includegraphics[width=85mm]{episode_30_oracle_True_policy}
\caption{Representative DDQN policy with Introspection Learning after 30
episodes of training has learned a safer policy of avoiding the cliff.\label{fig:cliffwalk_introspection_30}}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=85mm]{episode_30_oracle_False_policy}
\caption{Representative DDQN policy without Introspection Learning
(baseline) after 30 training episodes still selects to enter the cliff from
some states.\label{fig:cliffwalk_baseline_30}}
\end{figure}
\section{Conclusions}
In this paper we have introduced a novel reinforcement learning
algorithm based on ideas coming from formal methods and SMT-solving.
We have shown that, on suitable problems, these techniques can be
employed in order to improve robustness of RL agents and to speed up
their training. We have also given examples of how SMT-solving can be
used to analyze reinforcement learning agent robustness. There are a
number of extensions of this preliminary work possible. We mention
several prominent directions here.
First, the focus here has been on single-step analysis of agent
behavior, but a reachability analysis approach focused on
trajectories leading to target states would likely generate more
relevant data for learning. E.g., consider a geo-fenced space that we
do not want the agent to enter and that is reachable through many different
(state, action) combinations. Once a violation occurs, we would like
to examine the trajectory in order to learn what earlier choices led
the agent there.
Second, whereas in our ``lunar lander'' experiments we utilized an
\emph{ad hoc} quantization of the state space, it should be in many
cases possible to learn such regions as part of the algorithm. This
is a hard search problem so relying on these parameterizations is
necessary and should therefore be automated.
In conjunction with the reachability analysis mentioned above, this
approach is likely to give more targeted and therefore useful data to
include in the replay buffer.
Finally, while the SMT-solving technology being used is sufficient for
low-dimensional state-spaces, these techniques face
scalability issues on large state-spaces such as those coming from
video data. How to handle these higher-dimensional state-spaces in a
similar way is one of the exciting challenges in this area.
\subsection{Acknowledgments}
We would like to thank Ramesh S, Doug Stuart, Huafeng Yu, Sicun Gao,
Aleksey Nogin, and Pape Sylla for useful conversations on topics
related to this paper. We are also grateful to Tom Bui, Bala Chidambaram, Cem
Saraydar, Roy Matic, Mike Daily and Son Dao for their support of and
guidance regarding this research. Finally, we would like to thank
Alessio Lomuscio and Clark Barrett for their interest in this work and
for encouraging us to capture these results in a paper.
\section{Appendix: Theoretical Results}\label{appendix}
Fix a pre-MDP $D$ and assume given a (non-empty) subset $\mathcal{G}$ of
$\Pi(D)$ which we regard as the \emph{good} policies: those
policies $\pi$ whose $(s,\pi(s))$ have the properties of interest.
\begin{definition}\label{def:mdp_equiv}
MDPs $M$ and $M'$ are \emph{equivalent} whenever
$\textnormal{Opt}(M)\subseteq\textnormal{Opt}(M')$ and $\textnormal{Opt}(M')\subseteq\textnormal{Opt}(M)$.
\end{definition}
Furthermore, throughout this section we assume given a \emph{fixed} MDP $M=(r,T)$ in
$\mathcal{M}(D)$. Additionally, assume given a fixed discount factor
$0\leq\gamma < 1$. We also adopt throughout this section two further
hypotheses, which we now describe.
\begin{assumption}[Bad Set]\label{assumption:bad_acts}
There exists a subset $B\subseteq S\times A$ such that $\pi$ is in
$\mathcal{G}$ if and only if, for all $(s,a)\in B$, $\pi(s)\neq a$.
\end{assumption}
Our next hypothesis guarantees that the reward structure is already
sufficiently compatible with $\mathcal{G}$.
\begin{assumption}[Strong Compatibility]\label{assumption:sc}
All optimal policies for $M$ are in $\mathcal{G}$. I.e., $\textnormal{Opt}(M)\subseteq\mathcal{G}$.
\end{assumption}
We define a new MDP structure
$M^{\dagger}=(r^{\dagger},T^{\dagger})$ in $\mathcal{M}(D)$ by
\begin{align*}
T^{\dagger} & := T\cup B,\text{ and}\\
r^{\dagger}(s,a) & :=
\begin{cases}
-1 + \min_{\pi}Q^{\pi}_{M}(s,a) & \text{ if
}(s,a)\in B\text{, and}\\
r(s,a) & \text{ otherwise.}
\end{cases}
\end{align*}
It is straightforward to prove that $r^{\dagger}$ is bounded since $r$
is. Note that we are also modifying the underlying pre-MDP here by now
imposing the condition that $a_{b}\colon s_{b}\to s_{t}$.
An immediate proof of the following proposition can be obtained using
the
notion of \emph{bounded corecursive algebra} from \cite{Moss}, where it is shown that the state-value functions
$V^{\pi}_{M}\colon S\to\mathbb{R}$ are canonically determined by the
generating maps $v^{\pi}_{M}\colon S\to \mathbb{R}\times D(S)$ given
by
\begin{align*}
v^{\pi}_{M}(s) & := (r(s,\pi(s)),p(s,\pi(s))),
\end{align*}
where $D(-)$ is the probability distribution monad.
\begin{proposition}\label{proposition:sc_V}
If $\pi$ is in $\mathcal{G}$, then $V^{\pi}_{M^{\dagger}}=V^{\pi}_{M}$.
\begin{proof}
It suffices to show that $v^{\pi}_{M^{\dagger}}=v^{\pi}_{M}$, which is trivial for $\pi$ in $\mathcal{G}$.
\end{proof}
\end{proposition}
\begin{corollary}\label{cor:sc_Q}
If $\pi$ is in $\mathcal{G}$, then
$Q^{\pi}_{M^{\dagger}}(s,a)=Q^{\pi}_{M}(s,a)$ if and only if
$(s,a)\notin B$.
\end{corollary}
\begin{lemma}\label{lemma:sc_ineq}
$\textnormal{Opt}(M)\subseteq\textnormal{Opt}(M^{\dagger})$.
\begin{proof}
Suppose given an optimal policy $\pi$ for $M$. By Bellman
optimality, $\pi$ is optimal for $M^{\dagger}$ if and
only if, for all $s$,
\begin{align*}
\pi(s)\in\argmax_{a}Q^{\pi}_{M^{\dagger}}(s,a).
\end{align*}
Let $s$ and $a$ be given. There are two cases
depending on whether or not $(s,a)\in B$.
When $(s,a)\notin B$,
\begin{align*}
Q^{\pi}_{M^{\dagger}}(s,a) & = Q^{\pi}_{M}(s,a) \\
& \leq Q^{\pi}_{M}(s,\pi(s))\\
& = Q^{\pi}_{M^{\dagger}}(s,\pi(s)),
\end{align*}
where the equations are by Corollary \ref{cor:sc_Q} and the
inequality is by optimality of $\pi$.
When $(s,a)\in B$,
\begin{align*}
Q^{\pi}_{M^{\dagger}}(s,a)
& = -1 + \min_{\pi'}Q^{\pi'}_{M}(s,a) + 0\\
& \leq -1 + Q^{\pi}_{M}(s, a)\\
& < Q^{\pi}_{M}(s, a)\\
& \leq Q^{\pi}_{M}(s,\pi(s))\\
& = Q^{\pi}_{M^{\dagger}}(s,\pi(s)),
\end{align*}
where the final inequality is by optimality of $\pi$ and the final
equality is by Corollary \ref{cor:sc_Q}.
\end{proof}
\end{lemma}
\begin{lemma}\label{lemma:dagger_good}
$\textnormal{Opt}(M^{\dagger})\subseteq\mathcal{G}$.
\begin{proof}
Let a policy $\pi$ for $M^{\dagger}$ be given such that, for some
$s$, $(s,\pi(s))\in B$ and let $\pi'$ be an optimal policy for $M$.
Then
\begin{align*}
Q^{\pi}_{M^{\dagger}}(s,\pi(s)) & < Q^{\pi}_{M}(s,\pi(s))\\
&\leq Q^{\pi'}_{M}(s,\pi'(s)) \\
& = Q^{\pi'}_{M^{\dagger}}(s,\pi'(s)),
\end{align*}
so that such a $\pi$ cannot be optimal.
\end{proof}
\end{lemma}
\begin{theorem}\label{theorem:equiv}
$M$ and $M^{\dagger}$ are equivalent.
\begin{proof}
By Lemma \ref{lemma:sc_ineq} it suffices to show that
$\textnormal{Opt}(M^{\dagger})\subseteq\textnormal{Opt}(M)$, which is immediate since
\begin{align*}
V^{\pi}_{M} = V^{\pi}_{M^{\dagger}} = V^{\pi'}_{M^{\dagger}} = V^{\pi'}_{M},
\end{align*}
for any optimal policy $\pi$ for $M^{\dagger}$ and any optimal policy
$\pi'$ for $M$. Here the first equation is by Proposition
\ref{proposition:sc_V} and Lemma \ref{lemma:dagger_good}, the second
equation is by optimality of $\pi'$ for $M^{\dagger}$ by Lemma
\ref{lemma:sc_ineq}, and the final equation is by Proposition
\ref{proposition:sc_V} and the Strong Compatibility hypothesis.
\end{proof}
\end{theorem}
\printbibliography
\end{document}
|
{
"timestamp": "2019-03-01T02:01:47",
"yymm": "1902",
"arxiv_id": "1902.10754",
"language": "en",
"url": "https://arxiv.org/abs/1902.10754"
}
|
\section{Introduction}
The dark matter freeze-out paradigm, in particular the WIMP miracle, is prized for its simplicity and predictiveness. However, it is relatively straightforward to arrange for significant deviations in the predictions of freeze-out by changing either the particle physics model, or the cosmological history. For instance, standard freeze-out calculations typically assume that decoupling occurs whilst the energy density of the universe is dominated by radiation in which case the expansion rate of the universe is $H\propto T^2$. Indeed, dark matter freeze-out could occur whilst the universe is dominated by some form of energy other than radiation in which case the usual Hubble-temperature relation $H\propto T^2$ is broken. In particular, one possibility which occurs quite naturally in many Standard Model extensions is the case of an early matter dominated period for which $H\propto T^{3/2}$, or an era of particle decays leading to significant entropy production in the thermal bath in which case $H\propto T^{4}$ \cite{Scherrer:1984fd}. The case of dark matter freeze-out during an early period of matter domination was recently highlighted in \cite{Hamdan:2017psw} and freeze-out whilst $H\propto T^{4}$ was studied in \cite{Chung:1998rq,McDonald:1989jd,Giudice:2000ex,Gelmini:2006pw}.
More generally the early universe could be dominated by the energy density of a population of states $\phi$ evolving as an arbitrary power of the scale factor, i.e.~$\rho_\phi(T)=\rho_\phi(T_I)a^{4+n}$ with $T_I$ some initial temperature. For $n>0$ the energy density will eventually redshift to a negligible level and the expansion rate $H\propto T^{n/2+2}$ for $n\geq0$ is faster than expected from a radiation dominated universe. Two recent studies by D'Eramo, Fernandez, \& Profumo \cite{DEramo:2017gpl,DEramo:2017ecx} considered the implications for dark matter if the relic density is established during such a period of fast expansion. Note that the case $n=2$ corresponds to `kination domination', see e.g.~\cite{Spokoiny:1993kt,Ferreira:1997hj,Salati:2002md,Pallis:2005hm}, in which the energy density of the universe is dominated by the kinetic energy of some scalar field (the $\dot\phi$ term), for instance the inflaton. Conversely, for $n<0$, in order to recover the successes of standard cosmology, one requires that the state which dominates the energy density eventually decays. Specifically, the universe should be dominated by the Standard Model radiation bath at temperates around 10 MeV and below (until matter-radiation equality) so not to spoil the precision predictions of Big Bang nucleosynthesis (see e.g.~\cite{Sarkar:1995dd}).
Here we study the implications for the dark matter which freezes out or is produce while some (boson or fermion) state $\phi$ which dominates the energy density is appreciably decaying and under the assumption that the energy density of $\phi$ implies an initial expansion rate of the form $H\propto T^{n/2+2}$. Notably, during this period of particle decays entropy is no longer conserved in thermal bath and this impacts the dark matter relic abundance calculation. In particular, our work generalizes the earlier papers of \cite{McDonald:1989jd,Chung:1998rq,Giudice:2000ex,Gelmini:2006pw} which assume that the initial expansion rate corresponds to an early matter dominated phase. Additionally, this work can also be seen as an extension of dark matter freeze-out or production with a general expansion rate, as studied in \cite{DEramo:2017gpl,DEramo:2017ecx}, to include decays of $\phi$.
This paper is structured as follows: In Section \ref{sec2} we discuss the formulation of the Boltzmann equation without entropy conservation, generalizing the derivations of \cite{Giudice:2000ex}. Using these results, in Section \ref{sec3} we compare different scenarios for setting the dark matter relic abundance and discuss their dependance on the exponent $n$ of the initial expansion rate. Concluding remarks are given in Section \ref{sec4}.
\section{Boltzmann Equations without Entropy Conservation}
\label{sec2}
We start by deriving expressions for the evolution of the different particle populations in the case that the early universe is dominated by a state $\phi$ implying an initial expansion rate of $H\propto T^{n/2+2}$. We show that the expansion rate subsequently transitions to $H\propto T^4$ and derive an expression for the maximum temperature of the Standard Model radiation.
The expressions derived reproduce the results of Giudice, Kolb, \& Riotto \cite{Giudice:2000ex} for matter domination prior to decays ($n=-1$) and Visinelli \cite{Visinelli:2017qga} with kination domination prior to decays ($n=2$).
\subsection{Boltzmann equations}
We start in familiar territory by defining the Hubble parameter
\begin{equation}
H^2=\frac{8\pi}{3M_{\rm Pl}^2}\left(\rho_\phi+\rho_R+\rho_X\right)~,
\label{H}
\end{equation}
where the terms indicate the energy density in $\phi$, Standard Model radiation, and dark matter $X$, respectively. The evolution of quantities can be tracked relative to a dimensionless scale factor $A$ defined as
\begin{equation}\begin{aligned}
A\equiv\frac{a}{a_I} =aT_{\rm RH}~.
\label{dimensionlessA}
\end{aligned}\end{equation}
where $a_I$ is an arbitrary initial reference point which is chosen to be $a_I=1/T_{\rm RH}$ and $T_{\rm RH}$ is the reheat temperature following $\phi$ decays.
Emulating the analysis of \cite{Giudice:2000ex}, we rewrite the relevant variables as dimensionless quantities, but where now the energy density of $\phi$ evolves as some arbitrary power $a^{4+n}$
\begin{equation}\begin{aligned}
\Phi &\equiv\ \rho_\phi a^{4+n}T_{\rm RH}^n =\frac{\rho_\phi A^{4+n}}{T_{\rm RH}^4}~,
\qquad
R \equiv\rho_Ra^4 =\frac{\rho_RA^4}{T_{\rm RH}^4}~,
\qquad
X \equiv n_Xa^3 =\frac{n_XA^3}{T_{\rm RH}^3}~.
\label{dimensionless}
\end{aligned}\end{equation}
With the assumption that $\rho_X=\langle E_X\rangle n_X$, where $n_X$ is the dark matter number density and $\langle E_X\rangle$ the expected energy of each dark matter state, one can express the Hubble rate in terms of these dimensionless variables to obtain
\begin{equation}\begin{aligned}
H&= \frac{T_{\rm RH}^2}{M_{\rm Pl}A^{2+n/2}}
\sqrt{\frac{8\pi}{3}\left(\Phi+RA^n+\frac{\langle E_X\rangle X A^{n+1}}{T_{\rm RH}}\right)}~.
\label{Hvar}
\end{aligned}\end{equation}
This recovers the form of $H$ in \cite{Giudice:2000ex} for $n=-1$.
For the expansion rate in the form of eq.~(\ref{Hvar}), the Boltzmann equations which describe the evolution of the number densities can be expressed in a manner highly reminiscent to those studied in \cite{Giudice:2000ex,Gelmini:2006pw}
\begin{align}
\dot{\rho_\phi}+(4+n)H\rho_\phi &= -\Gamma_\phi \rho_\phi \label{eqe1}
\\[5pt]
\dot{\rho_R}+4H\rho_R &=\Gamma_\phi \rho_\phi +2\langle E_X\rangle(n_X^2-n_{\rm eq}^2)\langle\sigma v\rangle \label{eqe2}
\\
\dot{n_X}+3Hn_X &= \frac{b}{m_\phi}\Gamma_\phi \rho_\phi - (n_X ^2 - n_{\rm eq}^2)\langle\sigma v\rangle
\label{eqe3}
\end{align}
where $m_\phi$ and $\Gamma_\phi$ are the $\phi$ mass and decay rate to Standard Model. Dotted variables indicate differentiation with respect to time, the quantity $b$ paramaterises the branching ratio of $\phi$ to dark matter, $\langle\sigma v\rangle$ is the thermally averaged dark matter annihilation cross section, and $n_{\rm eq}$ denotes the equilibrium number density of dark matter which has its usual form. Furthermore, note that the $\phi$ decay rate can be expressed in terms of the reheat temperature after $\phi$ decays
\begin{equation}
\Gamma_\phi=\sqrt{\frac{4\pi^3g_*(T_{\rm RH})}{45}}\frac{T_{\rm RH}^2}{M_{\rm Pl}}~.
\label{gammae}
\end{equation}
We next re-express eqns.~(\ref{eqe1})-(\ref{eqe3}) in terms of the dimensionless units of eq.~(\ref{dimensionless}).
Looking firstly at eq.~(\ref{eqe1}), simple substitution yields
\begin{equation}\begin{aligned}
&\dot{A}\frac{d}{dA}\left(\frac{\Phi}{(Aa_I)^{4+n}T_{\rm RH}^n}\right)+(4+n)H\frac{\Phi}{(Aa_I)^{4+n}T_{\rm RH}^n}=-\Gamma_\phi \frac{\Phi}{(Aa_I)^{4+n}T_{\rm RH}^n}~.
\end{aligned}\end{equation}
After some manipulation, and using that $H=\frac{\dot a}{a}$, this can be simplified to
\begin{equation}\begin{aligned}
\dot{A}\Phi '=\Gamma_\phi \Phi~,
\label{dotA}
\end{aligned}\end{equation}
where the primed variable indicates differentiation with respect to $A$. Using eq.~(\ref{Hvar}) \& (\ref{gammae}) we can express eq.~(\ref{dotA}) in terms of $\Phi'$ as follows
\begin{equation}\begin{aligned}
\Phi '
&=-\sqrt{\frac{\pi^2g_*(T_{\rm RH})}{30}}\frac{\Phi A^{1+n/2}}{\sqrt{\Phi+RA^n+\frac{\langle E_X\rangle X A^{n+1}}{T_{\rm RH}}}}~.
\label{phie}
\end{aligned}\end{equation}
Analogously, eqns.~(\ref{eqe2}) \& (\ref{eqe3}) can be rewritten in a similar fashion to obtain
\begin{equation}\begin{aligned}
R'=&\sqrt{\frac{\pi^2g_*(T_{\rm RH})}{30}}\frac{\Phi A^{1-n/2}}{\sqrt{\Phi+RA^n+\frac{\langle E_X\rangle X A^{n+1}}{T_{\rm RH}}}}
+\sqrt{\frac{3}{8\pi}}\frac{2\langle\sigma v\rangle\langle E_X\rangle M_{\rm Pl}A^{n/2-1}(X^2-X_{\rm eq}^2)}{\sqrt{\Phi+RA^n+\frac{\langle E_X\rangle X A^{n+1}}{T_{\rm RH}}}}
\label{rade}
\end{aligned}\end{equation}
and
\begin{equation}
\begin{split}
X'=&\sqrt{\frac{\pi^2g_*(T_{\rm RH})}{30}}\frac{b}{m_\phi}\frac{\Phi T_{\rm RH}A^{-n/2}}{\sqrt{\Phi+RA^n+\frac{\langle E_X\rangle X A^{n+1}}{T_{\rm RH}}}}
-\sqrt{\frac{3}{8\pi}}\frac{\langle\sigma v\rangle M_{\rm Pl}T_{\rm RH}A^{n/2-2}(X^2-X_{\rm eq}^2)}{\sqrt{\Phi+RA^n+\frac{\langle E_X\rangle X A^{n+1}}{T_{\rm RH}}}}.
\label{dme}
\end{split}
\end{equation}
Thus the dimensionless versions of eqns.~(\ref{eqe1})-(\ref{eqe3}) are, respectively, eqns.~(\ref{phie})-(\ref{dme}).
\subsection{Radiation temperature maxima}
We assume that in the early universe the energy density is dominated by $\phi$ and, moreover, we further suppose that initial radiation bath is negligible.
This implies that the initial energy density of $\phi$ can be written
$\rho_\phi(a_I)=(3/8\pi)H_I^2 M_{\rm Pl}^2$
where $H_I\equiv H(a_I)$ is the initial expansion rate (throughout we will use the subscript $I$ to mean the value of a give quantity at $a=a_I$). In terms of dimensionless variables this initial condition is
\begin{equation}
\Phi_I=\frac{3H_I^2 M_{\rm Pl}^2}{8\pi T_{\rm RH}^4}, \hspace{15mm} R_I=X_I=0, \hspace{15mm} A_I=1.
\label{IC}
\end{equation}
One instance in which such initial conditions could arise, for instance, is immediately after inflation, as in the case of kination domination \cite{Spokoiny:1993kt,Ferreira:1997hj,Salati:2002md,Pallis:2005hm} corresponding to $n=2$.
In \cite{Giudice:2000ex} the authors studied the evolution of the temperature of the Standard Model thermal bath, assuming that prior to the decays of $\phi$ the radiation component is negligible. What was observed is that during the early matter dominated era the temperature rises due to the decays of $\phi$ until it hits some maximum temperature $T_{\rm Max}$ after which the expansion rate transitions to $H\propto T^4$. The bath then cools until the temperature $T_{\rm RH}$ at $H\simeq \Gamma_\phi$ after which $\phi$ decays become negligible and the universe becomes radiation dominated with $H\propto T^2$. We next generalise this analysis to the case that the early universe expansion rate sourced by $\phi$ is an arbitrary power of the temperature as in eq.~(\ref{H}).
As the $\phi$ decay, energy is transferred to the Standard Model bath, we will derive the point $A_{\rm Max}$ at which the maximum temperature of the radiation bath $T_{\rm Max}$ occurs for the more general expansion rate.
The dominant contribution comes from $\phi$ at early time, thus we can neglect the second term in eq.~(\ref{rade}), and using eq.~(\ref{IC}) we obtain
\begin{equation}
R'=\sqrt{\frac{\pi^2g_*(T_{\rm RH})}{30}}\Phi_I^{1/2}A^{1-n/2}.
\end{equation}
Further, integrating we obtain the following
\begin{equation}
R=
\left\lbrace
\begin{array}{ll}
\sqrt{\frac{\pi^2g_*(T_{\rm RH})}{30}}\sqrt{\Phi_I}\left(\frac{1}{2-n/2}\right)(A^{2-n/2}-1)
&~~~ \text{for} \ \ \ n<4 \\[10pt]
\sqrt{\frac{\pi^2g_*(T_{\rm RH})}{30}}\sqrt{\Phi_I}\ln(A) &~~~ \text{for}\ \ \ n=4
\end{array}
\right.~.
\label{Radearly}
\end{equation}
The temperature is a measure of the radiation energy density, thus we can obtain an expression for the evolution of $T$ as a function of $A$ from the expression
\begin{equation}\begin{aligned}
\rho_R=\frac{\pi^2g_*(T)}{30}T^4=R\left(\frac{T_{\rm RH}}{A}\right)^4~.
\label{Temperaturee}
\end{aligned}\end{equation}
Moreover, substituting eq.~(\ref{Radearly}) into eq.~(\ref{Temperaturee}) and rearranging, gives the evolution of the temperature (for $n\neq 4$)
\begin{equation}\begin{aligned}
T&=&\left(\frac{45}{4\pi^3}\frac{g_*(T_{\rm RH})}{g_*^{2}(T)}\right)^{1/8}\left(H_IM_{\rm Pl}T_{\rm RH}^2\right)^{1/4}\left[\frac{A^{-(2+n/2)}-A^{-4}}{2-n/2}\right]^{1/4}~.
\label{TTT}
\end{aligned}\end{equation}
\begin{figure}[t!]
\centering
\includegraphics[width=0.65\textwidth]{generalised-Fig.pdf}
\vspace{-3mm}
\caption{
Assuming that initially the expansion rate is $H\propto T^{n/2+2}$ and there is negligible energy in the radiation bath or dark matter $R(a_I)=X(a_I)=0$, the figure shows the bath temperature $T$ as function of $A$ for different values of $n$. We fix $\Phi_I=\Phi(a_I)=10^{10}$ GeV, or equivalently (see eq.~(\ref{IC})) this corresponds to, for example, $H_I=1$ eV and $T_{\rm RH}=100$ GeV. The curves follow eq.~(\ref{Tapprox}). Of the cases shown $n=-1$ corresponds to a matter-like $\phi$ (blue, solid), $n=0$ is radiation-like $\phi$ (dashed), and for $n=1$ then $\phi$ redshifts faster than radiation (dotted). Also note that, following eq.~(\ref{amax}), the maximum temperature drops (and occurs earlier) for increasing $n$.}
\label{fig:1}
\end{figure}
The critical point, with respect to $A$, of the factor in square brackets of eq.~(\ref{TTT}) marks the maximum temperature and the value of the scale factor $A_{\rm Max}$ at which the temperature stops increasing and begins to decrease. For $|n|<4$ this is given by
\begin{equation}\begin{aligned}
A_{\rm Max}=\left(\frac{n+4}{8}\right)^{2/(n-4)}~.
\label{amax}
\end{aligned}\end{equation}
For comparison, recall that $A_I=1$ corresponds to $T=T_{\rm RH}$ and $A>1$ implies $a> a_I$. For $A>A_{\rm Max}$ the $A^{-4}$ piece in eq.~(\ref{TTT}) can be neglected and $T\propto A^{-(2+n/2)}$. Observe that $A_{\rm Max}$ is sensitive to the exponent $n$ of the early universe expansion rate.
The temperature extremum $T_{\rm Max}$ for $|n|<4$ is found at $A=A_{\rm Max}$ given by
\begin{equation}\begin{aligned}
T_{\rm Max}=
\left(\frac{45g_*(T_{\rm RH})}{4\pi^3g_*^{2}(T_{\rm Max})}\right)^{1/8}\left(M_{\rm Pl}H_I T_{\rm RH}^2\right)^{1/4}\left(\frac{2}{4-n}\right)^{1/4}
\left[\left(\frac{n+4}{8}\right)^{\frac{4+n}{4-n}}-\left(\frac{n+4}{8}\right)^{\frac{8}{4-n}}\right]^{1/4}.
\label{Tmax}
\end{aligned}\end{equation}
As a reference, taking a few specific values for $n$, the $T_{\rm Max}$ can be approximated as
\begin{equation}\begin{aligned}
T_{\rm Max}\simeq
\left(M_{\rm Pl}H_I T_{\rm RH}^2\right)^{1/4}\times\left\lbrace
\begin{array}{ll}
0.30
& \text{~~for} \ \ \ n=-1 \\[8pt]
0.31
& \text{~~for} \ \ \ n=-2 \\[8pt]
0.33
& \text{~~for} \ \ \ n=-3 \\[8pt]
\end{array}
\right.~,
\label{tempapprox}
\end{aligned}\end{equation}
where we take $g_*(T_{\rm RH})\approx g_*(T_{\rm Max})\approx100$. For example, with reasonable values for $H_I\sim$ eV and $T_{\rm RH}\sim 1$ TeV, then $T_{\rm Max}\sim 3$ TeV for $|n|\sim\mathcal{O}(1)$.
Furthermore, we can re-express eq.~(\ref{TTT}) in terms of a normalised function $f(A_{\rm Max})=1$
\begin{equation}\begin{aligned}
T= T_{\rm Max} f(A)
\end{aligned}\end{equation}
for
\begin{equation}\begin{aligned}
f(A)\equiv \kappa(T)\left[A^{-(2+n/2)}-A^{-4}\right]^{1/4}
\end{aligned}\end{equation}
with
\begin{equation}\begin{aligned}
\kappa(T)=\left[\frac{g_*(T_{\rm Max})}{g_*(T)}\right]^{1/4}
\left[\left(\frac{4 + n}{8}\right)^{\frac{4 + n}{4 - n}}-\left(\frac{4 + n}{8}\right)^{\frac{8}{4 - n}}
\right]^{-1/4}~.
\end{aligned}\end{equation}
For reference, a selection of specific values for $\kappa$ are
\begin{equation}\begin{aligned}
\kappa(T)\approx \left[\frac{g_*(T_{\rm Max})}{g_*(T)}\right]^{1/4}
\left\lbrace
\begin{array}{ll}
\left(\frac{8^8}{3^3\cdot5^5}\right)^{1/20} & \text{~~for} \ \ \ n=-1 \\[8pt]
\left(\frac{4^{4}}{3^3}\right)^{1/12} & \text{~~for} \ \ \ n=-2 \\[8pt]
\left(\frac{16^{6}}{7^7}\right)^{1/28} & \text{~~for} \ \ \ n=-3
\end{array}
\right.~.
\label{tempapprox}
\end{aligned}\end{equation}
Starting from $a=a_I$ the temperatures increase from a negligible value to $T_{\rm Max}$, and then subsequently decreases according to eq.~(\ref{tempapprox}), as illustrated in Figure \ref{fig:1}. The evolution of the bath temperature in Figure \ref{fig:1} assumes that $\phi$ dominates the energy density, and the evolution will be altered once the energy in radiation becomes comparable to $\phi$, we denote this $A_{\times}$, as we discuss in the next section.
Thus for $A_{\rm Max}<A<A_{\times}$ one can approximate the temperature evolution as follows
\begin{equation}\begin{aligned}
T\sim \kappa T_{\rm Max} A^{-(2+n/2)/4}~.
\label{Tapprox}
\end{aligned}\end{equation}
Between the time when $T_{\rm Max}$ is reached and the point of radiation domination, at the earlier of $T_{\rm RH}$ and $T_{\times}$, the $\phi$ field energy density scales as $\rho_\phi=\Phi_IT_{\rm RH}^4/A^{4+n}$. Since the dominant contribution to the Hubble parameter at early times comes from $\rho_\phi$, it follows from eq.~(\ref{Hvar}) and eq.~(\ref{IC}) that for $A_{\rm Max}<A<A_{\times}$ then
\begin{equation}\begin{aligned}
H^2&\simeq&\frac{8\pi}{3M_{\rm Pl}}\frac{\Phi_IT_{\rm RH}^4}{A^{4+n}}=\left(H_IA^{-(4+n)/2}\right)^2~.
\label{Hearly}
\end{aligned}\end{equation}
Moreover, using eq.~(\ref{Tmax}) and eq.~(\ref{Tapprox}), we can express $A$ in terms of the temperature
\begin{equation}
A^{-(4+n)/2}=\left(\frac{4\pi^3(2-n/2)^2g^2_*(T)}{45g_*(T_{\rm RH})}\right)^{1/2}\frac{T^4}{H_IM_{\rm Pl}T_{\rm RH}^2}~.
\label{AinT}
\end{equation}
Substituting eq.~(\ref{AinT}) into (\ref{Hearly}) give an express for the expansion rate for $A_{\rm Max}<A<A_\times$
\begin{equation}
H=|4-n|\left(\frac{\pi^3g^2_*(T)}{45g_*(T_{\rm RH})}\right)^{1/2}\frac{T^4}{M_{\rm Pl}T_{\rm RH}^2}~.
\label{HinT}
\end{equation}
Thus the point $A_{\rm Max}$ indicates the $A$ at which the evolution transitions to $H\propto T^4$. Interestingly, the form of $H$ at this stage is independent on the preceding expansion rate apart from the prefactor, however because the values of $A_{\rm Max}$ and $T_{\rm Max}$ differ the evolution of cosmological abundances still changes for different values of $n$.
\subsection{Onset of radiation domination}
Since $\phi$ is decaying eventually radiation will come to dominate the energy density of the universe, indeed this is desirable to match early universe cosmology such as Big Bang nucleosynthesis observations. Due to decays the energy density of $\phi$, as tracked in dimensionless units by $\Phi$, changes is described by eq.~(\ref{phie}). At early time (where $X$ and $R$ are negligible), this can be rewritten via separation of variables as follows
\begin{equation}\begin{aligned}
\frac{{\rm d \Phi'}}{\sqrt{\Phi}}
&=-
{\rm d} A
\sqrt{\frac{\pi^2g_*(T_{\rm RH})}{30}} A^{1+n/2}~.
\label{phiint}
\end{aligned}\end{equation}
Evaluating this integral from $A_I=1$ we find
\begin{equation}
\Phi=
\left\lbrace
\begin{array}{ll}
{\Phi_I}\cdot {\rm exp}\left[-\sqrt{\frac{\pi^2g_*(T_{\rm RH})}{30}} \frac{1}{2+n/2}(A^{2+n/2}-1)\right]
&~~~ \text{for} \ \ \ n\not=-4 \\[10pt]
\Phi_I \cdot A^{-{\sqrt{\pi^2g_*(T_{\rm RH})/30}}} &~~~ \text{for}\ \ \ n=-4
\end{array}
\right.~.
\label{phiearly}
\end{equation}
Since the energy density in $\phi$ is falling quickly, whilst the radiation component grows gradually, at some point (which we denote $A_{\times}$), the contributions from radiation and $\phi$ become comparable i.e.~$\Phi(A_{\times})\simeq R(A_{\times})$. Shortly after $A_{\times}$ the universe transitions to radiation domination and the expansion rate transitions to $H\propto T^2$.
Importantly, for $A\gtrsim A_{\times}$ then eq.~(\ref{TTT}) (and Figure~\ref{fig:1}) no longer well describe the evolution, since it is not reasonable to neglect $R$ in the derivation.\footnote{This approximation also breaks down if the value of $X$ grows too large, but since the growth of $X$ depends on the small free parameter $b/m_\phi$, we continue to neglect $X$ in deriving $A_{\times}$.}
To find $A_{\times}$ we numerically solve the coupled differential equations eq.~(\ref{phie}) \& (\ref{rade}) with the initial conditions $R(a_I)=X(a_I)=0$. In Figure \ref{fig:2} (left) we show the values of $A_{\times}$ for different values of $n$ (i.e.~initial expansion rates), and where $\Phi(a_I)\equiv\Phi_I$ is treated as a free parameter. Fitting to $A_{\times}$ we find the form $A_{\times}=c_n \Phi_I^{m_n}$ where $m_n$ and $c_n$ are constants, for instance, for $n=-1$ then $m_{-1}\approx0.20$ and $c_{-1}\approx 0.68$.
\begin{figure}[t!]
\centerline{
\includegraphics[height=0.36\textwidth]{Plot-AEQ-AR1.pdf}
\includegraphics[height=0.36\textwidth]{Tcross.pdf}
\includegraphics[height=0.36\textwidth]{TempThresholds.pdf}}
\caption{
({Left}).~The point $A_{\times}$ at which $R(a_\times)=\Phi(a_\times)$ as $\Phi_I$ is varied and for different $n$, found by solving the coupled differential eqns.~(\ref{phie}) \& (\ref{rade}) with the initial conditions of eq.~(\ref{IC}). The line styles match Figure \ref{fig:1}, with $n=0$ is dashed and $n=1$ dotted. The point $A_{\times}$ signifies the breakdown of eq.~(\ref{phiearly}), which underlies Figure \ref{fig:1}. ({Center}).~For a given value of $T_{\rm RH}$ (the reheating temperature after $\phi$ decay) $A_{\times}$ is associated to a specific temperature $T_{\times}$ via eq.~(\ref{TTT}). Here we show $T_{\times}$ as a function of $\Phi_I$ for $T_{\rm RH}=1$ TeV. The black dashed curve indicates the maximum temperature $T_{\rm Max}\sim0.3\times (M_{\rm Pl}H_I T_{\rm RH}^2)^{1/4}$. Changes in $T_{\rm RH}$ simply scales the y-axis and the relative orientations of the lines are unchanged. ({Right}).~We illustrate the temperature evolution for two cases with $n=-2$ and $n=-3$, taking $T_{\rm RH}=1$ TeV and $\Phi_I=10^{10}$ GeV, and we highlight where $T_{\rm Max}$ and $T_{\times}$ occur.
\label{fig:2}}
\end{figure}
By inspection of Figure \ref{fig:2} (left) it is seen that for $\Phi_I\gtrsim 10^8$ then eq.~(\ref{phiearly}) is valid up to $A\gtrsim10$, which is the range of Figure \ref{fig:1}. The approximation remains good for lower $\Phi_I$ and higher $A$ for larger values of $n$. Moreover, since the expansion rate varies prior and after $A_{\rm Max}$ (the point of transition from increasing to decreasing bath temperature), comparing with eq.~(\ref{amax}) we note that
$n\geq-3$ then $A_{\rm Max}\leq2$, and $A_{\rm Max}\ll A_{\times}$ for reasonable values of $\Phi_I$ and $n$. In particular, for $\Phi_I\gtrsim1$ TeV (typically the range of interest) the bath evolves into the decreasing temperature regime prior to $A_{\times}$.
For a given $T_{\rm RH}$ we can translate $A_{\times}$ into the corresponding temperature $T_{\times}$ at which the radiation and $\phi$ components become comparable, as shown in Figure \ref{fig:2} (center). The approximate value of $T_{\rm Max}\simeq0.3\times (M_{\rm Pl}H_I T_{\rm RH}^2)^{1/4}$ is shown as the black dashed line and the separation between $T_{\rm Max}$ and $T_{\times}$ gives an indication of the length of the period for which the system is dominated by the decaying $\phi$ states. Figure \ref{fig:2} (right) illustrates the temperature evolution and the points at which $T_{\rm Max}$ and $T_{\times}$ occur for two specific cases. Once the temperature drops below either $T_{\times}$ or $T_{\rm RH}$ the system transitions to radiation domination with $H\propto T^2$. The scenario of interest here is the case in which the dark matter relic density is set prior to the onset of radiation domination, as we discuss in the next section.
\section{Implications for Dark Matter }
\label{sec3}
In the preceding section we studied the behavior of the temperature and the expansion rate for the case of a period in the early universe in which the expansion follows some general power law and while $\phi$ is decaying. We consider next the implications for dark matter, in particular, how the predicted dark matter relic density depends on the exponent $n$ in the initial expansion rate $H\propto T^{n/2+2}$. We will break the discussion into the following cases:
\begin{itemize}
\item[\bf \S\ref{3.1}:] Freeze-in: Thermal production without chemical equilibrium.
\item[\bf \S\ref{3.2}:] Freeze-out during reheating: Thermal production without chemical equilibrium.
\item[\bf \S\ref{3.3}:] Non-thermal production.
\end{itemize}
\subsection{Freeze-in}
\label{3.1}
First we assume that $X$ particles are always non-relativistic and do not reach chemical equilibrium at early time ($X\ll X_{\rm eq}$). The case of thermal production of non-relativistic dark matter without reaching chemical equilibrium with the thermal radiation bath is an instance of the dark matter freeze-in scenario formulated more generally in \cite{Hall:2009bx} and developed in e.g.~\cite{Elahi:2014fsa,McDonald:2001vt,Chu:2013jja,Yaguna:2011qn,Chu:2011be,Blennow:2013jba}.
Thus we consider the evolution of $X$ following eq.~(\ref{dme}), for now taking $b=0$, in which case
\begin{equation}
X'=\sqrt{\frac{3}{8\pi}}\langle\sigma v\rangle M_{\rm Pl}T_{\rm RH}\Phi_I^{-1/2}A^{n/2-2}X_{\rm eq}^2~.
\label{X1}
\end{equation}
in terms of the equilibrium distribution given by
\begin{equation}
X_{\rm eq}\equiv a^3n_X^{\rm eq}=\frac{A^3}{T_{\rm RH}^3}g\left(\frac{M_X T}{2\pi}\right)^{3/2}e^{-\frac{M_X}{T}}~,
\label{Xeq}
\end{equation}
where $g$ is the number of internal degrees of freedom of the dark matter state.
Using eq.~(\ref{Tapprox}) and substituting eq.~(\ref{Xeq}) into eq.~(\ref{X1}) we obtain
\begin{equation}\begin{aligned}
X' =\langle\sigma v\rangle \frac{g^2M_X^3\kappa^3T_{\rm Max}^3}{8\pi^3H_IT_{\rm RH}^3}A^{(20+n)/8}e^{-\frac{2M_X A^{(4+n)/8}}{\kappa T_{\rm Max}}}~.
\label{secterm}
\end{aligned}\end{equation}
Expressing the cross section in terms of the s and p-wave pieces $\langle\sigma v\rangle=\alpha_s/M_X^2+ \alpha_pT/M_X^3$, and integrating (neglecting the temperature dependence in $\kappa$, i.e.~$\kappa=\kappa_{\rm RH}\equiv \kappa(T_{\rm RH})$) gives
\begin{equation}
X_{\infty}=\frac{2^{-(n+28)/(n+4)}}{n+4}g^2\frac{\left(\kappa_{\rm RH} T_{\rm Max}\right)^{(4n+40)/(n+4)}}{\pi^3H_IT_{\rm RH}^3M_X^{24/n+4}}\Gamma\left(\frac{n+28}{n+4}\right)\left(\alpha_s+\frac{n+4}{12}\alpha_p\right)~,
\label{s-pwave}
\end{equation}
where here $\Gamma$ indicates the gamma function. Thus for $b\approx0$ (more precisely provided that the first term of the Boltzmann equation for $X$, eq.~(\ref{dme}), can be safely neglected) and assuming that the dark matter is non-relativistic and does not enter chemical equilibrium
\begin{equation}
\rho_X(T_{\rm RH})=M_Xn_X(T_{\rm RH})=M_XX_{\infty}\frac{T_{\rm RH}^3}{A_{\rm RH}^3}~.
\label{Xrh}
\end{equation}
Furthermore, it is known that at the point of reheating $H\simeq\Gamma_\phi$ the energy density for radiation is
\begin{equation}
\rho_R(T_{\rm RH})=\frac{\pi^2g_*(T_{\rm RH})}{30}T_{\rm RH}^4
\label{radrh}
\end{equation}
and thus comparing the ratio of energy densities now and at reheating we have
\begin{equation}
\frac{\rho_X(T_{\rm now})}{\rho_R(T_{\rm now})}=\frac{T_{\rm RH}}{T_{\rm now}}\frac{\rho_X(T_{\rm RH})}{\rho_R(T_{\rm RH})}=\frac{M_X}{T_{\rm now}}\frac{30}{A_{\rm RH}^3\pi^2g_*(T_{\rm RH})}X_{\infty}~.
\label{3.7}
\end{equation}
Substituting eq.~(\ref{s-pwave}) it follows that
\begin{equation}
\frac{\rho_X(T_{\rm now})}{\rho_R(T_{\rm now})}
= \frac{30\times 2^{-\frac{n+28}{n+4}} M_X}{A_{\rm RH}^3\pi^5g_*(T_{\rm RH})T_{\rm now}(n+4)}\frac{g^2\left(\kappa_{\rm RH} T_{\rm Max}\right)^{\frac{4n+40}{n+4}}}{H_IT_{\rm RH}^3M_X^{\frac{24}{n+4}}}\Gamma\left(\frac{n+28}{n+4}\right)\left(\alpha_s+\frac{n+4}{12}\alpha_p\right)~.
\end{equation}
Using the form of $A_{\rm RH}$ from eq.~(\ref{Tapprox}) we can re-express the dark matter relic abundance as
\begin{equation}
\frac{\Omega_{X}h^2}{\Omega_{R}h^2}
=\frac{30\times 2^{-\frac{n+28}{n+4}}g^2}{\pi^5(n+4)H_Ig_*(T_{\rm RH})T_{\rm now}}\Gamma\left(\frac{n+28}{n+4}\right)
\frac{ (\kappa_{\rm RH}T_{\rm Max})^{4}}{T_{\rm RH}^{3\frac{(n-4)}{(n+4)}}M_X^{\frac{20-n}{n+4}}}\left(\alpha_s+\frac{n+4}{12}\alpha_p\right)~,
\label{relicFI}\end{equation}
in terms of the observed fractional energy densities $\Omega_{R,X}$ for radiation and dark matter. One could further rewrite eq.~(\ref{relicFI}) in terms of $T_{\rm RH}$ by substituting the form of $T_{\rm Max}$ from eq.~(\ref{Tmax}). For $n=-1$ this scenario is studied in `Case A' of \cite{Giudice:2000ex}, and eq.~(\ref{relicFI}) generalises this to other values of the exponent $n$, reproducing the earlier result for $n=-1$.
\begin{figure}
\centerline{
\includegraphics[height=0.43\textwidth]{PlotRange.pdf}
\hspace{5mm}
\includegraphics[height=0.43\textwidth]{AeqAstPlot.pdf}}
\caption{(Left).~Shaded areas indicate regions of the $T_{\rm RH}$-$M_X$ plane for which $A_{\rm Max}<A_*<A_{\times}$ for $n=-2$ and different values of the initial expansion rate $H_I$ and we also require $\Phi_I(H_I,T_{\rm RH})<M_{\rm Pl}$. (Right).~As an example we fix $n=-2$ and $H_I=10^{-4}$ GeV and plot the $T_{\rm RH}$ which gives the observed relic density $\Omega_{\rm Obs}$ by freeze-in as $M_X$ is varied assuming $\alpha_s=10^{-18}$ and $\alpha_p=0$. This line follows from eq.~(\ref{relicFI}). We highlight the region $A_{\rm Max}<A_*<A_{\times}$ and $\Phi_I<M_{\rm Pl}$ for $H_I=10^{-4}$ GeV (matching the left panel), outside of this region the relic density curve is unreliable and we indicate this by dashing the line. Note that $T_{\rm RH}\sim1$ TeV and $H_I\sim10^{-4}$ GeV corresponds to $\Phi_I\sim10^{17}$ GeV.
\label{fig:3}}
\end{figure}
Additionally note that the point of peak dark matter production $A_*$ can be found by looking at where the derivative in eq.~(\ref{secterm}) vanishes.
For the s-wave case this occurs for\footnote{Note taking $n=-1$ we find that the prefactor is $19/3$ which is slightly different from derived in \cite{Giudice:2000ex} which give the prefactor as $17/2$. We believe the authors use different criteria and approximations.}
\begin{equation}
A_*=\left[\left(\frac{20+n}{4+n}\right)\frac{\kappa T_{\rm Max}}{2 M_X}\right]^{\frac{8}{4+n}}\simeq
\left(\Phi_I\right)^{\frac{1}{4+n}} \left[0.3\cdot \frac{T_{\rm RH}}{2M_X}\left(\frac{20+n}{4+n}\right)\right]^{\frac{8}{4+n}},
\label{aswave}
\end{equation}
where in the final equation we have used eqns.~(\ref{IC}) \& (\ref{tempapprox}). Note that $A_*$ depends strongly on $T_{\rm RH}/M_X$ but is relatively insensitive to $\Phi_I$.
In order for the relic density to be described by eq.~(\ref{relicFI}), i.e.~while $H\propto T^4$, it is required that $A_{\rm Max}<A_*<A_{\times}$.
Recall from Figure \ref{fig:2} (left) that $A_{\times}\sim\mathcal{O}(10)$, thus dark matter production occurs prior to the onset of radiation domination for $A_*<A_{\times}\sim\mathcal{O}(10)$. Typically it can be arranged that $A_*\sim\mathcal{O}(10)<A_{\times}$ by choosing an appropriate $\Phi_I$. In Figure~\ref{fig:3} (left) we highlight parameter values in which $A_{\rm Max}<A_*<A_{\times}$ for the case of $n=-2$ as $H_I$ is varied. If $\Phi_I$ is taken to be relatively large then the dark matter mass should must be fairly heavy to reproduce the observed relic density. Additionally, note that since trans-Planckian $\Phi_I$ are unreasonable this places an additional restriction on the parameter space.
In Figure~\ref{fig:3} (right) we show the parameter region $A_{\rm Max}<A_*<A_{\times}$ along with curve for which the observed dark matter relic density is reproduced ($\Omega_X=\Omega_{\rm Obs})$ for a specific example taking $n=-2$ and $H_I=10^{-4}$ with couplings $\alpha_s=10^{-18}$ and $\alpha_p=0$. We highlight that generally for the relic density curve to align with the region $A_{\rm Max}<A_*<A_{\times}$ diminutive couplings are needed $\alpha\ll1$, however this is actually fortuitous since such feeble coupling strengths are required in freeze-in models in order to ensure that the dark matter remains out of equilibrium \cite{Hall:2009bx} (as assumed for this case). In the next subsection we shall derive the condition on $\alpha$ under which $X< X_{\rm eq}$ at all times.
\subsection{Freeze-out during reheating}
\label{3.2}
Next we consider the case in which the dark matter reaches chemical equilibrium and then freezes out while $H\propto T^4$. The point of freeze-out can be defined implicitly by
\begin{equation}
n_X^{\rm eq}(T_F)\langle\sigma v\rangle=H(T_F)~.
\label{caseB}
\end{equation}
Using that $n_X^{\rm eq}=X^{\rm eq}A^{-3}T_{\rm RH}^3$ and eq.~(\ref{Xeq}) we can rewrite the lefthand side of the above equation in terms of $T_F$ and for the righthand side we substitute eq.~(\ref{HinT}) to obtain
\begin{equation}
\frac{g}{\sqrt{8\pi^3}} \langle\sigma v\rangle (M_X T_F)^{3/2}\exp\left(\frac{-M_X}{T_F}\right)=
|4-n|\left(\frac{\pi^3g^2_*(T_F)}{45g_*(T_{\rm RH})}\right)^{1/2}\frac{T_F^4}{M_{\rm Pl}T_{\rm RH}^2}~.
\end{equation}
Thus the calculation is largely unchanged from earlier studies, differing only in the factor $|n-4|$, and the freeze-out temperature is analogous to as derived in \cite{Giudice:2000ex}, given by
\begin{equation}\begin{aligned}
x_F&=&\ln\left(\frac{3gM_{\rm Pl}T_{\rm RH}^2g_*(T_{\rm RH})^{1/2}}{|4-n|\sqrt{5}\cdot 8\pi^3g_*(T_F)M_X^3}\left(\alpha_sx_F^{5/2}+\alpha_px_F^{3/2}\right)\right)~.
\end{aligned}\end{equation}
Note that, as usual, because of the insensitivity of the logarithm dependences for a large range of reasonable parameter values $x_F\sim \mathcal{O}(10)$. Comparing $T_F\sim M_X/\mathcal{O}(10)$ to $T_{\times}$ in Figure \ref{fig:2} (centre) one finds that $T_\times\sim(\mathcal{O}(100)-\mathcal{O}(1000)$) GeV therefore for $M_X\gtrsim 1$ TeV then typically $T_F > T_\times$. Thus for a large range of parameters dark matter freeze-out can occur well before the transition to radiation domination, while $H$ is described by eq.~(\ref{HinT}).
The dark matter abundance remains constant after the point of reheating at $H\simeq \Gamma_\phi$, but to ascertain the abundance of dark matter at reheating it is necessary to evolve the freeze-out abundance from $T_F$ to $T_{\rm RH}$ as follows
\begin{equation}
\rho_X(T_{\rm RH}) = \left(\frac{a(T_{\rm RH})}{a(T_F)}\right)^{-3} \rho_X(T_F)
= \left(\frac{g_*(T_{\rm RH})}{g_*(T_F)}\right)^2 \left(\frac{T_{\rm RH}}{T_F}\right)^8 \rho_X(T_F)~,
\end{equation}
where we use that the ratio of FRW scale factors can be replaced by the ratio of dimensionless $A$ factors.
It follows that the dark matter relic density is given by
\begin{equation}
\frac{\Omega_Xh^2}{\Omega_Rh^2} =|4-n|\frac{5\sqrt{5}}{4\sqrt{\pi}}\frac{\sqrt{g_*(T_{\rm RH})}}{g_*(T_F)}
\frac{T_{\rm RH}^3}{T_{\rm now}M_XM_{\rm Pl}}
\frac{1}{\alpha_sx_F^{-4}+\alpha_px_F^{-5}/5}~.
\end{equation}
Notably, the abundance is essentially insensitive to the expansion rate prior to reheating in the case that dark matter freeze-out is non-relativistic and in thermal equilibrium.
Whether the relic density is set via freeze-out or freeze-in depends on if the dark matter enters equilibrium. Specifically, for $X_\infty\lesssim X_{\rm eq}(T_{*})$, where $T_*$ is the temperature of dominant particle production given by eq.~(\ref{TTTTT}), the dark matter will remain out of equilibrium at all times and the production rate sets the relic density (the freeze-in scenario). Thus there is a critical value of the coupling $\alpha^{\rm (crit)}$ above which the inequality $X_\infty\lesssim X_{\rm eq}(T_{*})$ is violated.
From eqns.~(\ref{Tapprox}) \& (\ref{aswave}) we have that $A_*$ corresponds to a temperatures $T_*$ which for s-wave is
\begin{equation}
T_*\simeq M_X\left(\frac{8+2n}{20+n}\right)~.
\label{TTTTT}
\end{equation}
Applying the criteria that for $\alpha=\alpha^{\rm (crit)}$ then $X_\infty= X_{\rm eq}(T_{*})$, it follows that for the case with $\langle\sigma v\rangle\simeq\alpha_s/M_X^2$ the critical coupling $\alpha_s^{\rm (crit)}$ is given by
\begin{equation}\begin{aligned}
\alpha_s^{\rm (crit)}=\frac{2\pi^3 M_X^3\left(20+n\right)^{\frac{3(12-n)}{2(n+4)}}\left(4+n\right)^{\frac{5n-28}{2(n+4)}}|4-n|}{\sqrt{45}g\Gamma\left(\frac{28+n}{4+n}\right)e^{\frac{20+n}{2(n+4)}}M_{\rm Pl}T_{\rm RH}^2}\frac{g_*(T_*)}{g_*(T_{\rm RH})^{1/2}}.
\end{aligned}\end{equation}
Thus for $\alpha_s<\alpha_s^{\rm (crit)}$ the relic density is set by freeze-in, as in Section \ref{3.1}, whereas for $\alpha_s>\alpha_s^{\rm (crit)}$ freeze-out dynamics determines the dark matter relic density.
\subsection{Non-thermal production}
\label{3.3}
\begin{figure}[t!]
\centerline{
\includegraphics[height=0.45\textwidth]{Nontherm-AR1.pdf}
\hspace{5mm}
\includegraphics[height=0.45\textwidth]{NonthermZ-AR1.pdf}}
\caption{Plot shows lines in the $T_{\rm RH}$-$M_X$ plane for which the dark matter relic density is reproduced via non-thermal production, following eq.~(\ref{nontrd}). Taking three different exponents of the initial expansion rate $n=1$ (dotted), $n=0$ (dashed) and $n=-1$ (solid) and, pameterising the $\phi$-dark matter branching fraction in terms of $\eta\equiv b\cdot{\rm GeV}/m_\phi$, we show three different values $\eta=0.5,~10^{-4},~10^{-7}$. The plot fixes the initial Hubble rate to be $H_I=10^{-9}$. The right panel shows an enlargement of the dashed rectangle of the left panel and illustrates the difference between $n=0,1$, and $-1$. \label{fig:4}}
\end{figure}
The scenario of non-thermal production is important for $b\not\approx0$, that is a significant (possibly dominant) population of dark matter is produced directly from $\phi$ decays \cite{Gelmini:2006pw}. The case of non-thermal production without chemical equilibrium is described by eq.~(\ref{dme}) with the second ($b$-independent) term neglected
\begin{equation}\begin{aligned}
X'=&
\sqrt{\frac{\pi^2g_*(T_{\rm RH})}{30}}\frac{b}{m_\phi}\frac{\Phi T_{\rm RH}A^{-n/2}}{\sqrt{\Phi+RA^n+\frac{\langle E_X\rangle X A^{n+1}}{T_{\rm RH}}}} ~.
\end{aligned}\end{equation}
Integrating (for $n\not=2$) from $A_I$ to $A_{\rm RH}$ and applying the boundary conditions of eq.~(\ref{IC}) we find the total population of dark matter produced due to $\phi$ decays
\begin{equation}
X_{\rm RH}\equiv X(T_{\rm RH})\simeq-\frac{2\eta}{n-2}\sqrt{\frac{\pi g_*(T_{\rm RH})}{80}}\frac{H_IM_{\rm Pl}}{T_{\rm RH}}\left(A_{\rm RH}^{1-n/2}-1\right)~,
\label{branratio-nnot2}
\end{equation}
where we define $\eta\equiv b/m_\phi$ which parmaterises the $\phi$-dark matter branching fraction.
For $n=2$ then rather $X(T_{\rm RH})\propto \ln(A_{\rm RH})$, this case was studied in \cite{Visinelli:2017qga} and we will not discuss it further here. Using the above equation and eq.~(\ref{Xrh}) \& (\ref{radrh}) it follows that
\begin{equation}\begin{aligned}
\frac{\rho_{X_b}(T_{\rm now})}{\rho_R(T_{\rm now})}
\simeq\frac{15M_XH_IM_{\rm Pl}\eta}{\sqrt{5}(2-n)\pi^{3/2}g_*(\sqrt{T_{\rm RH}}) T_{\rm RH}^4 T_{\rm now}}
\left(A_{\rm RH}^{\frac{2-n}{2}}-1\right)\frac{T_{\rm RH}^3}{A_{\rm RH}^3}~.
\end{aligned}\end{equation}
Further, using. eq.~(\ref{Tapprox}) to replace $A_{\rm RH}$ this can be rewritten to obtain an expression for the dark matter relic abundance
\begin{equation}\begin{aligned}
\frac{\Omega_{X}}{\Omega_R}
&\simeq&\frac{15M_XH_IM_{\rm Pl}\eta}{\sqrt{5}(2-n)g_*(\sqrt{T_{\rm RH}})\pi^{3/2}T_{\rm RH}^3T_{\rm now}}\left(\left[\frac{\kappa T_{\rm Max}}{T_{\rm RH}}\right]^{\frac{4(2-n)}{n+4}}-1\right)\left(\frac{T_{\rm RH}}{\kappa T_{\rm Max}}\right)^{\frac{24}{n+4}}.
\label{nontrd}
\end{aligned}\end{equation}
Similar to previously, the above result generalises expressions in Case 3 of \cite{Gelmini:2006pw} from $n=-1$ to general $n$.
Note that the preceeding calculation assumes the ordering $A_{\rm Max}<A_{\rm RH}<A_{\rm \times}$ where $A_{\rm RH}\equiv A(T_{\rm RH}).$ We can obtain an expression for $A_{\rm RH}$ from eqns.~(\ref{gammae}) \& (\ref{TTTTT}) as follows $A_{\rm RH} \sim (\sqrt{\Gamma_\phi M_{\rm Pl}}/ \kappa T_{\rm Max})^{-8/(4+n)}$. Note that $A_{\rm RH}$ depends on $\Gamma_\phi$ and since the other quantities do not depend on $\Gamma_\phi$ the above inequality involving $A_{\rm RH}$ is typically not constraining. We also note here that Big Bang nucleosynthesis constraints imply a limit $T_{\rm RH}\gtrsim 10$ MeV.
In Figure \ref{fig:4} we illustrate some example parameter ranges which reproduce the observed dark matter relic density for the case of non-thermal production without chemical equilibrium with $n=0,1,-1$. In particular, we highlight that changes in $n$ has a modest impact on the appropriate $T_{\rm RH}$ required to reproduce the relic density, however there is a great degree of freedom in $\eta\equiv b/m_\phi$ which can lead to substantially larger impact on the required value of $T_{\rm RH}$ needed to reproduce the observed dark matter relic density.
Note that if the branching fraction of $\phi$ decays, controlled by $b$ is sufficiently large, the dark matter will enter equilibrium in which case the contribution from non-thermal production is reduced due to dark matter annihilations. The production of dark matter due to $\phi$ decays can maintain the dark matter at an equilibrium abundance past $T\sim M_X$ and dark matter only freezes out at $T\sim T_{\rm RH}$, when dark matter production ceases. Thus the period of freeze-out is during the era of radiation domination. As argued in \cite{Gelmini:2006pw} (Case 4), this leads to a scaling of the radiation dominated relic density $\Omega_{\rm RD}$ due to the difference in the entropy density between $T_{\rm RH}$ and the radiation domination freeze-out temperature $T_{\rm FO}$. Specifically, the dark matter relic abundance will be $\Omega_X\sim(T_{\rm FO}/T_{\rm RH})\Omega_{\rm RD}$. Since the freeze-out occurs during radiation domination, the relic abundance will be largely insensitive to the temperature dependance of the initial expansion rate.
\section{Concluding Remarks}
\label{sec4}
A myriad of scenarios exists in which the early universe is not immediately radiation dominated but goes through periods with expansion rates different to the commonly assumed relationship $H\propto T^2$. In this work we have focused on a previously unstudied case in which the early universe is dominated by some state $\phi$ which leads to a general expansion rate of the form $H\sim T^{2+n/2}$, but due to the fact that $\phi$ is decaying there is a subsequent transition to $H\sim T^4$. Notably, the form of the initial expansion rate leaves a lasting imprint on relic densities established while $H\propto T^4$, because the value of the exponent $n$ (for $H\sim T^{2+n/2}$) changes the temperature evolution of the Standard Model thermal bath. While freeze-out during reheating (\S\ref{3.2}) is largely insensitive to the initial expansion rate, the abundances of dark matter produced via freeze-in (\S\ref{3.1}) or non-thermal production without equilibrium (\S\ref{3.1}) are sensitive to the value of the exponent $n$.
The prospect of the dark matter relic abundance being established during a period of entropy injection following an early matter dominated era was originally studied in influential papers of Giudice, Kolb \& Riotto \cite{Giudice:2000ex} and Gelmini and Gondolo \cite{Gelmini:2006pw}. This was later adapted to the specific case of a period of decays following kination domination by Visinelli \cite{Visinelli:2017qga}. In this work we have further generalised to the case in which the initial epoch has a general expansion rate of the form $H\sim T^{2+n/2}$. We have highlighted how the choice of $n$ propagates into the cosmology of the era of significant entropy injection from $\phi$ decays and into calculations of the dark matter relic density. Reassuringly, for $n=-1$ we reproduce the expressions of \cite{Giudice:2000ex,Gelmini:2006pw} and for $n=2$ we reproduce the results of \cite{Visinelli:2017qga}.
Before closing it is worth highlighting that an explicit example of scenarios with expansion rates of the form $H\sim T^{2+n/2}$ was constructed in \cite{DEramo:2017gpl}, in which the energy density of the universe is dominated by the contributions from a real scalar field with a potential
\begin{equation}\begin{aligned}
V(\phi)\propto \frac{n-2}{(n+4)^2}{\rm exp}(-\phi\sqrt{n+4})~.
\end{aligned}\end{equation}
Furthermore, a range of expansion rates of the form $H\sim T^{2+n/2}$ can arise from scalars with periodic potentials \cite{Gardner:2004in,Choi:1999xn}.
We have explored here the implications for dark matter in the case that a population $\phi$ sources an initial expansion rate of the form $H\propto T^{2+n/2}$ and assuming the dark matter relic density is established while $\phi$ is decaying. Interestingly, the temperature dependance of the initial expansion rate can significantly impact the form of the dark matter relic density. Since such variant cosmologies can alter the predicted dark matter relic density they have previously been used to adjust the freeze-out abundance or evade experimental constraints, see e.g.~\cite{Gelmini:2006pw,Salati:2002md,Profumo:2003hq,Pallis:2005hm,Bramante:2017obj,Randall:2015xza,Bernal:2018ins,Gelmini:2006pq,Gelmini:2006mr,Arbey:2009gt,Hardy:2018bph}. It would be interesting to examine how these specific particle physics models (such as the bino, neutralino, and Higgs portal) vary in the context of the generalise scenario outlined here. Moreover, in future work we plan to explore to what extent the temperature dependance of the early expansion rate imprints on cosmological parameters and observables, such as deviations in the matter power spectrum \cite{Fan:2014zua,Erickcek:2011us,Redmond:2018xty}, and whether there is a degeneracy between the initial temperature dependence of $H$ (the value of $n$) and the magnitude of the initial expansion rate $H_I$.
\subsection*{Acknowledgments}
CM is grateful for funding from FONDECYT (project 1161150), CONICYT-PCHA/Doctorado
Nacional/2018-21180309 and the hospitality of the University of Illinois at Chicago. JU is grateful to New College, Oxford and the Simons Center for Geometry and Physics (Program: Geometry and Physics of Hitchin Systems) for their hospitality and support.
|
{
"timestamp": "2019-03-01T02:01:31",
"yymm": "1902",
"arxiv_id": "1902.10746",
"language": "en",
"url": "https://arxiv.org/abs/1902.10746"
}
|
\section{}
\label{}
\section{Introduction}
\label{sec:intro}
Gamma-ray bursts (GRBs) are widely accepted to have two categories, short GRBs (SGRBs) having duration shorter than 2 $\rm s$ and long GRBs (LGRBs) with duration longer than 2 $\rm s$ \citep{Kouveliotou1993}. SGRBs are thought to be from the merger of compact object binaries involving at least one neutron star \citep{Eichler1989,Paczynski1991,Narayan1992}, and have a broad range of spatial host galaxy distribution \citep{Zhang2017}. The origin of LGRBs are most-likely to be the collapse of rapidly-rotating massive stars \citep{MacFadyen1999}, hence expected to be inside the star forming region. Consequently, the offsets of the location in the host galaxy of LGRBs are mostly smaller than those of SGRBs.
In the past few decades, there have been many studies on host galaxy offsets of GRBs. For example, \citet{Bloom2002} studied host galaxy offsets for LGRBs. The result was consistent with the expected distribution of massive stars, confirming the core-collapse model as the origin of LGRBs. \citet{Fong2010A} presented the first comprehensive analysis of $Hubble ~ Space ~ Telescope ~ (HST)$ observations of ten SGRBs host galaxies. Their result showed an median at 5 $\rm kpc$ for SGRBs host galaxy offsets, which is about 5 times larger than LGRBs. There was no evidence of differences between SGRBs with and without extended emission. The host galaxy offsets are in good agreement with neutron star binary mergers (see also \citet{Church2011}). However, \citet{Malesani2007} noticed that SHBs (short hard GRBs) with extended emission are more easier to detect their optical counterparts. This has been explained as an environmental property by \citet{Troja2008}, as SHBs with extended emission seem to occur closer to their host galaxies, in denser interstellar environments. This also implies that SGRBs progenitors have an intrinsically different behavior, due to their association with different origins such as black hole (BH)-neutron star (NS) and NS-NS merger. \citet{Troja2008} showed that SGRBs with extended hard X-ray emissions that have small projected physical offsets may be due to NS-BH mergers, while those without extended hard X-ray emission components that have bigger projected physical offsets may be due to NS-NS mergers. The correlation between X-ray absorption column densities and host galaxy offsets gives another evidence that SGRBs possibly have two distinct populations \citep{Kopac2012}. Furthermore, some negative correlations are found between the broadband afterglow emissions and SGRBs host galaxy offsets \citep{Zhang2017}. This is because the afterglow emission depends on the circum-burst medium and it decreases with the distance to the host galaxy center, providing more evidences that SGRBs with larger host galaxy offsets prefer lower circum-burst densities \citep{Fong2015}.
To investigate the properties of the host galaxies and the connection to the GRBs, we collect all the possible sample from the literature about the offsets, durations of the GRBs ($T_{90}$ (time duration from 5\% photon counts to 95\% photon counts) and $T_{\rm R45}$ \citep[defined in][]{Reichart2001}), the isotropic equivalent $\gamma-$ray energy $E_{\rm \gamma, iso}$, and the 1 s time binned peak luminosity $L_{\rm pk}$. In this work we analyze these data and present our results for the relations found for SGRBs, LGRBs and combination of them. The paper is organized as the follow: the data is collected and described in \S \ref{sec:samples}, the statistics is performed in \S \ref{sec:results}, and conclusion and discussion is given in \S \ref{sec:discussion}.
\section{The GRB sample}
\label{sec:samples}
We selected 304 GRBs from different instruments, and collected their trigger time, instrument, redshift $z$, offset, $T_{\rm 90}$, $T_{\rm R45}$, $E_{\rm \gamma,iso}$ and $L_{\rm pk}$ values from different published papers. All the information is provided in Table \ref{tab:sample} to \ref{tab:lpklast}. $E_{\rm \gamma,iso}$ and $L_{\rm pk}$ are in rest-frame 1-$10^{4}$ $\rm keV$ energy band, and $L_{\rm pk}$ is in 1 $\rm s$ time bin (except GRB 170817A in 50 $\rm ms$ time bin). We also calculated isotropic equivalent luminosity $L_{\rm \gamma,iso}$ in the rest-frame 1-$10^{4}$ $\rm keV$ energy band, which is $ L_{\rm \gamma,iso}=(1+z)E_{\rm \gamma,iso}/T_{90}$. For $L_{\rm pk}$, sometimes the energy band is not in rest-frame 1-$10^{4}$ $\rm keV$ energy band, like \citet{Deng2016}. We changed the energy band using the spectral information. There are mainly three kinds of spectral models: Band model, cutoff power law (CPL) model and simple power law (SPL) model \citep[more details in][]{Li2016ApJS}. In Table \ref{tab:lpk} to \ref{tab:lpklast}, we gave the GRB spectral information which need to change the energy band, as well as the $L_{\rm pk}$. For Band model, $\alpha$, $\beta$ and $E_{\rm pk}$ are low energy spectral index, high energy spectral index and peak energy, respectively. For CPL model, $\alpha$ is the spectral index for the power law band and $E_{\rm pk}$ is the cutoff energy. There is no $\beta$ for CPL model, and we use ``..." to remark $\beta$.
Besides, we excluded some values with lower limit smaller than 0. For example, the offset of GRB 120119A is $0.104 \pm 0.147$ \citep{Li2016ApJS}.
The data are not complete, as not every GRB has all the observational values listed above, available. Some of the data have only the central values available without error bars. To keep the information of the central values, we need to impute the errors from other data. We used the R package $mice$ to impute the error bars for the data that have just the central values, by multiple imputation with chained equations (MICE) \citep{Rubin1987,Rubin1996}.
\subsection{Error imputation} \label{subsec:imputation}
We use the R package $mice$ to impute incomplete multivariate data by using the method MICE. MICE is a powerful tool for imputation and it has been widely used. Only the central values with missing error bars are imputed. The ones with missing central values are omitted in the statistical analysis. According to \citet{Rubin1987,Rubin1996}, MICE includes three steps: generating multiple imputation, analyzing imputed data, and pooling analysis results.
The imputation model should also have three principles: accounting for the process that created the missing data, preserving the relations in the data and preserving the uncertainty about these relations. At first, we changed $T_{\rm 90,i}$, $T_{\rm R45,i}$, $E_{\rm \gamma,iso}$ and host galaxy offset into their logarithmic values. Then we did 5 times imputation as suggested in \citet{Rubin1996}. It means every error bar which need to be imputed will have 5 imputed values, hence we have 5 complete set of data. We need to choose the imputation model first, because our data is missing at random (MAR) \citep{Rubin1976}, additionally, our data is numeric type. So we choose the predictive mean matching model (PMM) \citep{Little1988}, a general purpose semi-parametric imputation method \footnote{We compared the correlations between the central values and the related errors, and found the PMM is reliable in the error imputation. For example, the positive error of $T_{\rm 90,i}$ is $T_{\rm 90,i,1}$. Before imputation, the linear regression between $T_{\rm 90,i}$ and $T_{\rm 90,i,1}$ is $T_{\rm 90,i} = (1.26 \pm 0.05) + (-9.28 \pm 1.58) \times T_{\rm 90,i,1}$, and the Pearson coefficient is $-0.33 \pm 0.05$ with p-value $1.2 \times 10^{-8}$. After the imputation, the linear regression between $T_{\rm 90,i}$ and $T_{\rm 90,i,1}$ is $T_{\rm 90,i} = (1.27 \pm 0.05) + (-9.7 \pm 1.58) \times T_{\rm 90,i,1}$, and the Pearson coefficient is $-0.33 \pm 0.05$ with p-value $1.6 \times 10^{-9}$. The results do not change too much, which means PMM model can preserve the relations in the data and preserve the uncertainty about these relations.}. We set a threshold at 0.25, which means the minimum proportion of usable cases for imputation is at least 0.25. An important step in multiple imputation is that, we want to assess whether imputations are plausible, then we have done diagnostic checks. We used following three indicators to assess the goodness of our imputation results.
\begin{enumerate}
\item Relative increase in variance due to missing data $r_{\rm m}$ (RIV): It is the ratio between imputation variance and the imputation variance of the 5 data sets, then multiplying the imputation time m. It stands for the increase fraction in variance due to missing data, the influence of the missing data is bigger when $r_{\rm m}$ is bigger. While smaller $r_{\rm m}$ indicates influence of the change of m is smaller, this is to say that missing data has smaller influence to the whole data parameters, hence the imputation results are more stable and the imputations are better.
\begin{equation} \label{eq:RIV}
r_{m}=\frac{(1+\frac{1}{m}) {\sigma_{\rm B}}^{2}}{{\sigma_{\rm W}}^{2}}.
\end{equation}
${\sigma_{\rm W}}^{2}$ is within-imputation variance, it represents the mean of the variance for the m data sets.
\begin{displaymath}
{\sigma_{\rm W}}^{2}=\frac{1}{m} \sum_{i=1}^m {\sigma_{\rm i}}^{2}.
\end{displaymath}
${\sigma_{\rm B}}^{2}$ is between-imputation variance, it represents the variance of the mean of m data sets.
\begin{displaymath}
{\sigma_{\rm B}}^{2}=\frac{1}{m-1} \sum_{i=1}^m {(\widehat{\theta}_{\rm i}-\widehat{\theta})}^{2}
\end{displaymath}
$\widehat{\theta}_{\rm i}$ is the mean of every complete data set, $\widehat{\theta}=\frac{1}{m}\sum_{i=1}^m \widehat{\theta}_{\rm i}$
\item Fraction of missing information $\gamma_{\rm m}$ (FMI): This represents the influence of the missing data for the whole parameters(e.g. mean). Smaller FMI values indicate that the imputation results are more stable.
\begin{equation} \label{eq:FMI}
\gamma_{\rm m}=\frac{r_{\rm m}+\frac{2}{v_{\rm m}+3}}{r_{\rm m}+1}
\end{equation}
$v_{\rm m}=(m-1)(1+\frac{1}{r_{\rm m}}^{2})$ is the degree of freedom.
\item Relative efficiency (RE): is a comprehensive analysis of RIV and FMI. It represents the imputation fraction for missing information by MICE. The higher value of RE means the better result.
\begin{equation} \label{eq:RE}
{\rm RE}={(1+\frac{\gamma_{\rm m}}{m})}^{-1}
\end{equation}
\end{enumerate}
For analyzing imputed data and pooling analysis results, we use the mean of every imputed error bar, because we also need to calculate some values and plot scatter plots with error bars. As there are 5 candidate values for each parameter, we use the mean of them as the imputed error.
The imputation results are shown in Table \ref{tab:impu}. From the results, we can see that RIV and FMI are very close to 0, which means our imputation is stable. RE is very close to 1, which means our imputation efficiency is very high, as per the definition. We almost imputed all the missing information. Therefore, we justify the imputation is reliable.
\section{Correlation method and results} \label{sec:results}
Due to the special features of GRB 170817A and GRB 060218A, we have not included them in our statistical analysis. To figure out the intrinsic connections between different properties associated with the GRBs, we convert the values into rest frame. We changed $T_{\rm 90}$ and $T_{\rm R45}$ from observer-frame to rest-frame by dividing them with $1+z$ and assigned them as $T_{\rm 90,i}$ and $T_{\rm R45,i}$ respectively. At first, we analyzed using the combined sample of both SGRBs and LGRBs. Then we separated the GRBs into SGRB group and LGRBs group, and did the same statistical analysis for the two groups separately.
In order to make sure the results are reliable, we have used several statistical analysis narrated below. At first, we calculated the correlation coefficients to check the linear correlations between different parameters, with Pearson, Spearman and Kendall $\tau$ correlation methods \citep{Feigelson2012}. Pearson correlation coefficient is to measure whether the two parameters are aligned in the two-dimension plot. Spearman and Kendall $\tau$ correlation coefficients are methods to measure the monotony of the two parameters. Then we did the linear regression to get the parameters of the linear expression.
For all the correlation coefficients and linear regression, we have done hypothetical tests. Once the results pass through the hypothetical tests with p-value smaller than 0.1 we accepted the correlation, as this value gives a high probability of relying the result. In the correlation statistics we have considered the error following the method used in \citet{Zou2017}. Assuming the errors are normally distributed, we generated $10^{\rm 3}$ sets of random samples with Monte Carlo (MC) simulation. Then we have done the above mentioned statistics for the $10^{\rm 3}$ sets and checked their distribution. The best fit and corresponding 1$\sigma$ errors are finally obtained.
Using above mentioned methods we found significant correlations for the following pairs, $\log {\rm (offset)} - \log T_{\rm 90,i}$, $\log {\rm (offset)} - \log T_{\rm R45,i}$, $\log {\rm (offset)} - \log E_{\rm \gamma,iso}$, $\log {\rm (offset)} - \log L_{\rm pk}$, and $\log E_{\rm \gamma,iso} - \log T_{\rm R45,i}$. The correlation of $\log T_{\rm 90,i}$ and $\log T_{\rm R45,i}$ is also tight. But it is trivial, and we will not discuss about it further. The $\log E_{\rm \gamma,iso} - \log T_{\rm 90,i}$ relation is not shown here as it has been extensively investigated.
All the correlation coefficient results are listed in Table \ref{tab:coef}. For the combined GRBs, the linear coefficients are significant for all the five relation pairs, the significance of correlation has further been verified with the hypothetical tests for the linear regression coefficients.
We have shown the scatter plots in Fig. \ref{fig:all}. One can confirm the five correlations intuitively with the scatter plots. In the figures, the fitting line is in blue which was obtained by considering all the errors with MC method similar as in \citet{Zou2017}. The relations obtained are as following,
\begin{equation} \label{eq:allt90offset}
\log {\rm (offset)} = -0.27_{\rm -0.02}^{\rm +0.02} \times \log T_{\rm 90,i} + 0.59_{\rm -0.03}^{\rm +0.01},
\end{equation}
\begin{equation} \label{eq:alltr45offset}
\log {\rm (offset)} = -0.34_{\rm -0.03}^{\rm +0.01} \times \log T_{\rm R45,i} + 0.4_{\rm -0.02}^{\rm +0.01},
\end{equation}
\begin{equation} \label{eq:alleisooffset}
\log {(\rm offset)} = -0.14_{\rm -0.02}^{\rm +0.01} \times \log E_{\rm \gamma,iso,52} + 0.38_{\rm -0.02}^{\rm +0.02},
\end{equation}
\begin{equation} \label{eq:alllpkoffset}
\log {(\rm offset)} = -0.11_{\rm -0.02}^{\rm +0.01} \times \log L_{\rm pk,52} + 0.25_{\rm -0.03}^{\rm +0.03},
\end{equation}
\begin{equation} \label{eq:alltr45eiso}
\log E_{\rm \gamma,iso,52} = 0.71_{\rm -0.01}^{\rm +0.01} \times \log T_{\rm R45,i} + 0.36_{\rm -0.01}^{\rm +0.01},
\end{equation}
where the offset is in unit of kpc, $T_{\rm 90,i}$ and $T_{\rm R45,i}$ are in units of seconds, $L_{\rm pk,52}$ is in unit of $10^{52}$ ergs/s and $E_{\rm \gamma,iso,52}$ is in unit of $10^{52}$ ergs. As expected the $T_{\rm 90,i}$ and $T_{\rm R45,i}$ have similar correlation with the host galaxy offset with slope around $-0.25$. However there is no obvious separation between SGRBs and LGRBs in $T_{\rm 90,i}$, $T_{\rm R45,i}$ and spatial offsets from plots as seen in \citet{Troja2008} for $T_{\rm 90}$ and spatial offset scatter plot. We assume this discrepancy is due to the redshift correction. Interestingly, in the $\log {(\rm offset)} - \log T_{\rm 90,i}$ plot, there are four LGRBs with $T_{\rm 90,i}< 2$, and there is a gap between these short ones and the normal LGRBs. This might indicate that the separation of LGRBs and SGRBs should be in the rest frame rather than the observer's frame. The anti-correlation between offset and the duration is consistent with the double degenerated star merger origin for SGRBs and massive star core-collapsing origin for LGRBs. For mergers, they have to pass supernova explosion, which kicks the central compact star. And it takes time for the double compact star to merge, which lead the site of the merging to be in the outer part of the host galaxy. While for the massive star core-collapsing, it should be in the star forming region, which is near the center of the host galaxy.
The anti-correlation between offset and $E_{\rm \gamma,iso}$, $L_{\rm pk}$ respectively may also be related to the LGRBs and SGRBs, as LGRBs are relatively stronger in $\gamma-$rays, which can also be seen in the $\log E_{\rm \gamma,iso} - \log T_{\rm R45,i}$ scattering plot. However we could not find an exact explanation for the values of the slopes and intercepts. We assume a detail analysis of the star forming and merger profile in the galaxies would be able to explain these parameters, however this analysis is beyond the scope of our paper.
Fig. \ref{fig:all} also gives a clear indication that all the GRBs are gathered in two distinct groups. We apply K-Means clustering algorithms to get the boundary between this two traditionally classes. The boundary is represented with a dash line in Fig. \ref{fig:all}. Two different clusters cannot have any object in common. The similarity measure between the cases defined as the euclidean distance on the duration-offset plane. Variables with incompatible units are faced in our dataset. We use logarithmic variables, rather than normalized or standardized variables. Because units can be removed by taking logarithms and the ranges of variables is similar. We kept information on burst duration and offset. The boundaries are $\log({\rm offset} ) = 2.33 \times \log T_{90,i} - 0.63$ and $\log( {\rm offset} ) = 1.73 \times \log T_{\rm R45,i} + 0.92$. We assume that the group of bursts on the duration-offset plane is separated and equal-sized. Each cluster is described by a single point known as the centroid. The centroid of each class is defined as the mean values of $\log T_{90,i}$, $\log T_{\rm R45,i}$, and $\log {\rm (offset)}$. The centroids are marked with black open triangles in Fig. \ref{fig:all}. These two groups are generally consistent with long and short GRBs, while with few outliers.
We have also checked the correlations of the offset with other parameters, while no much significant relations has been found. For the ${\rm offset}-E_{\rm p}$ correlation, as the peak energy can be best fit with Band function or with CPL model, there are two correlations. For the Band function model, the Pearson coefficient is $0.30 \pm 0.22$ with p-value 0.32. For the CPL model, the Pearson coefficient is $0.03 \pm 0.13$ with p-value 0.88. Though there is relatively weak correlation between ${\rm offset}$ and $L_{\rm pk}$, there is no strong evidence for ${\rm offset}$ and $L_{\rm \gamma, iso}$. The corresponding Pearson coefficient is $-0.03^{+ 0.01}_{-0.03}$ with p-value 0.73 as also shown in Table \ref{tab:coef}.
It is important to decide the true number of groups. There is no certain way of telling the goodness of the clustering. The silhouette coefficient is a measure of the compactness and separation of the clusters. It increases as the quality of the clusters increase, it is large for compact clusters that are far from each other and small for large and overlapping clusters. According to the silhouette coefficient, the most probable number of clusters is 2 for our set of GRBs.
To figure out the correlations are solely caused by the long or short GRBs progenitors, or there is intrinsic tendency inside LGRBs or SGRBs, we divided the 304 GRBs into SGRBs and LGRBs following the widely believed relation of $T_{90}$ with 2s. The coefficients are also listed in the Table \ref{tab:coef}. We list the relations with mean correlation coefficient p-value smaller than 0.1.
The relation between $E_{\rm \gamma,iso}$ and host galaxy offsets for SGRBs is:
\begin{equation} \label{eq:shorteisooffset}
\log ({\rm offset}) = -0.22_{\rm -0.05}^{\rm +0.02} \times \log E_{\rm \gamma,iso,52} + 0.71_{\rm -0.05}^{\rm +0.05},
\end{equation}
where the offset is in unit of kpc, and $E_{\rm \gamma,iso,52}$ is in unit of $10^{52}$ ergs.
The relation between $L_{\rm pk}$ and host galaxy offsets for SGRBs is:
\begin{equation} \label{eq:shortlpkoffset}
\log ({\rm offset}) = -0.49_{\rm -0.08}^{\rm +0.04} \times \log L_{\rm pk,52} + 0.35_{\rm -0.13}^{\rm +0.13},
\end{equation}
where the offset is in unit of kpc, and $L_{\rm pk,52}$ is in unit of $10^{52}$ erg/s.
The relation between $T_{\rm R45,i}$ and $E_{\rm \gamma,iso}$ for SGRBs is:
\begin{equation} \label{eq:shorttr45eiso}
\log E_{\rm \gamma,iso,52} = 1.45_{\rm -0.11}^{\rm +0.05} \times \log T_{\rm R45,i} + 0.5_{\rm -0.12}^{\rm +0.12},
\end{equation}
where $T_{\rm R45,i}$ are in units of seconds, and $E_{\rm \gamma,iso,52}$ is in unit of $10^{52}$ ergs.
The relation between $T_{\rm R45,i}$ and $E_{\rm \gamma,iso}$ for LGRBs is:
\begin{equation} \label{eq:longtr45eiso}
\log E_{\rm \gamma,iso,52} = 0.36_{\rm -0.02}^{\rm +0.01} \times \log T_{\rm R45,i} + 0.55_{\rm -0.01}^{\rm +0.01},
\end{equation}
where $T_{\rm R45,i}$ are in units of seconds, and $E_{\rm \gamma,iso,52}$ is in unit of $10^{52}$ ergs.
When the GRBs are divided into long and short ones, there is no clear correlation between the offset and the duration. It shows that the origin of LGRBs and SGRBs is only due to the spatial offset. The correlations of $\log ({\rm offset})$ - $\log E_{\rm \gamma,iso}$ and $\log({\rm offset} ) - \log L_{\rm pk}$ observed only for SGRBs. It might indicate that the SGRBs can still be divided into several sub-groups. A possible explanation could be the sub-groups of NS-NS merging and BH-NS merging as origin of SGRBs. For the BH-NS merger, the total gravitational energy is larger than the NS-NS merger, as the BH mass can be much larger. On the other hand, the more massive BH is harder to be kicked outside the host galaxy, and should be located near to the center. This naturally explains the anti-correlation shown in Eq. (\ref{eq:shorteisooffset}).
Interestingly, as shown in Table \ref{tab:coef} for short GRBs, also for the ${\rm offset}-L_{\rm \gamma, iso}$ and ${\rm offset} - L_{\rm pk}$, one is quite weak (Pearson coefficient being $0.07 ^{+ 0.07}_{-0.04}$ with p-value 0.19 for ${\rm offset}-L_{\rm \gamma, iso}$) while the other is relatively quite strong (Pearson coefficient being $-0.62 ^{+ 0.03}_{-0.06}$ with p-value 0.02 for ${\rm offset}-L_{\rm pk}$).
Interestingly, the $E_{\rm \gamma,iso} - T_{\rm R45,i}$ relation has also been observed, however the coefficients are quite different for the combined GRBs, SGRBs and LGRBs, as shown in Table \ref{tab:coef} and in Eqs. (\ref{eq:alltr45eiso}) and (\ref{eq:longtr45eiso}). Especially, for the combine sample and for Short GRBs alone, the Pearson coefficients are $0.425 ^{+0.003}_{-0.007}$ with p-value $3.6 \times 10^{-14}$ and $0.53 ^{+0.01}_{-0.04}$ with p-value $0.01$. These relations are quite strong. However, the underlying reason is not clear.
\section{Conclusion and Discussion} \label{sec:discussion}
In this work, we studied the relations of the spatial position of GRBs in their host galaxies with their durations and energies. Using the available data, we tested the correlations between $T_{\rm 90,i}$, $T_{\rm R45,i}$, $E_{\rm \gamma,iso}$, $L_{\rm \gamma, iso}$, $L_{\rm pk}$ and corresponding spatial offset in their host galaxies. We found the host galaxy offsets have negative correlations with the other four parameters. This negative relation for ($T_{\rm 90,i}$ and $T_{\rm R45,i}$) seems reasonable, because LGRBs are most-likely from massive star collapsing, so the GRB positions are closer to the center of its host galaxies. On the other hand, SGRBs are most-likely from binary mergers. For the binary mergers, the system will experience large velocity kicks at birth, leading to eventual mergers outside the host galaxies \citep{Berger2010}. Therefore, mostly SGRBs have bigger spatial offsets than LGRBs. But there is no correlation about host galaxy offsets and duration for SGRBs and LGRBs respectively, so different origins offer a natural explanation for the two negative correlations. Interestingly, $L_{\rm pk}$ is related to the offset, while $L_{\rm \gamma, iso}$ is not, and the $E_{\rm \gamma,iso} - T_{\rm R45,i}$ relation is quite different of the long GRBs, short GRBs and the combined sample. We also found the special GRBs 170817A and 060218A are outliers in most of the scatter plots, which might indicate there are more subgroups in the GRB samples.
Considering the relation between the $E_{\rm \gamma,iso}$ and the offsets, for the combined GRBs analysis, as the SGRBs have lower energy than LGRBs, it becomes natural that $E_{\rm \gamma,iso}$ and the host galaxy offsets have negative correlation. We have found a similar correlation for SGRBs. We interpret it as a result of different progenitors. If the progenitor is NS-BH, the SGRBs have larger $E_{\rm \gamma,iso}$ and smaller spatial offset, because the bigger mass of the system and smaller kick velocity. While for the NS-NS progenitors, the behaviors are on the opposite. For the positive correlation of $E_{\rm \gamma,iso}$ and $T_{\rm R45,i}$, we can find significant correlation for the combined GRBs, SGRBs and LGRBs. Therefore, it is a common property for GRBs, no matter what the central engines or progenitors are.
We have seen the main tendency is that: for SGRBs, they are distributed in the outer side of the host galaxies, while for the LGRBs, they are in the inner side. This supports again that the SGRBs are from mergers of compact stars (NS-NS, or NS-BH), while the LGRBs are from the core collapsing of the massive stars. With this topic, the strongest evidence is the SNe Ic connection to the long GRBs, such as GRB 980425/SN 1998bw \citep{1998Natur.395..670G}, while for the short GRBs is the gravitational wave event association, i.e., the most recently discovered NS-NS merger event GRB 170817A/GW170817 \citep{2017ApJ...848L..12A}. Within the SGRBs, it also shows an anti-correlation between the offset and isotropic $\gamma-$ray energy. This may indicate the double origins for the SGRBs, i.e., NS-NS origin with lower energy and locating nearer the center of the host galaxy, while BH-NS origin with higher energy and locating at outer side. A possible explanation might be that at the birth of the NS or BH, for very high speed NS, the companion compact star has the chance to decouple, while the high speed BH (with more mass) binary can survive, and consequently reaches to outer area of the host galaxy. Also because of the range of the BH masses, it may explain the variety of the magnitude of the kilonovae (or called macronovae, merger novae) \citep{2017ApJ...837...50G}. We have analyzed the correlations by dividing the sample into long GRBs and short GRBs based on the duration being greater or smaller than 2s. It is also possible to perform the same analysis based on the k-means classification, which were given in Section \ref{sec:results}. As one can see, the sample does not change much, and the results may not change much neither.
Our data is biased for strong or optically bright GRBs. The optically dark GRBs are hard to identify the host galaxies and hence the spatial offsets are not available. The GRBs can be dark due to high absorption by the dense medium which may not affect the biasing of offset. In the case intrinsically dark GRBs, the offset might be intrinsically different from the bright GRBs. However to investigate the answer to this property is beyond the scope of this work. It is possible that the GRB location is just occasionally overlapping with a galaxy, while the host galaxy is much dimmer in the foreground or background. This may introduce some bias to the data, especially for the large offset ones. Additionally, our statistical analysis totally rely on observational resolution and technology.
In the future, one can get more information about the host galaxy with the enhancement of the GRB monitors and observations with higher angular resolution. Monitoring the host galaxies mostly rely on the ground optical follow up efficiency. With more powerful telescope, one can get more information on-site the position of the GRB location (not only the whole galaxy). The environment, such as circum-burst density, star formation rate etc., are also important observations to be made. These information are crucial for understanding the physics of GRBs as well as the final stage of the stellar evolution.
\section*{Acknowledgments}
This work is supported by the National Basic Research Program of China (973 Program, Grant No. 2014CB845800) and by the National Natural Science Foundation of China (Grant Nos. U1738132, U1231101 and 11773010). RM acknowledges grant by Templeton foundation.
\bibliographystyle{elsarticle-harv}
|
{
"timestamp": "2019-03-01T02:16:15",
"yymm": "1902",
"arxiv_id": "1902.11022",
"language": "en",
"url": "https://arxiv.org/abs/1902.11022"
}
|
\subsection{Key ingredients for the construction of our differential operators}\label{ingreds-diffops}
Let $T$ be a scheme, and let $Y$ be a smooth scheme over $T$.
Suppose $\pi\colon\mathscr{A}\to Y$ is a polarized abelian scheme. In particular, $\pi$ is a smooth and proper morphism.
Let $\omega:=\omega_{\mathscr{A}/Y}:=\pi_*\Omega^1_{\mathscr{A}/Y}\subseteq H^1_{\mathrm dR}(\mathscr{A}/Y)$, and consider the Hodge filtration
\[0\to\omega\hookrightarrow H^1_{\mathrm dR}(\mathscr{A}/Y)\to {R^1\pi_*\cO}_\mathscr{A}\to 0.\]
Our differential operators are built from the Gauss--Manin connection
\[\nabla=\nabla_{\mathscr{A}/Y}\colon H^1_{\mathrm dR}(\mathscr{A}/Y)\toH^1_{\mathrm dR}(\mathscr{A}/Y)\otimes\Omega^1_{Y/T}\]
and the Kodaira--Spencer morphism
\[\mathrm{KS}=\mathrm{KS}_{\mathscr{A}/Y}\colon\omega\otimes\omega\to\Omega^1_{Y/T}.\]
By definition, $KS_{\mathscr{A}/Y}:=\langle \cdot, \nabla (\cdot)\rangle_{\mathscr{A}}$, where $\langle \cdot, \cdot\rangle_{\mathscr{A}}$ is the pairing induced by the polarization on $\mathscr{A}$ and extended linearly in the second variable to a pairing between $\omega$ and $\nabla(\omega)$ (and using the fact that $\omega$ is an isotropic subspace of $H^1_{\mathrm dR}(\mathscr{A}/Y)$ under this pairing).
The Kodaira--Spencer morphism induces an isomorphism
\[\mathrm{ks}\colon\omega^2\overset{\sim}{\rightarrow} \Omega^1_{Y/T},\]
where
\begin{align*}
\omega^2:=
\begin{cases}
\bigoplus_{\tau\in \mathcal{T}_{F_0}}{\rm Sym}_{{\mathcal O}_Y}^2\left(\omega_\tau\right) & \mbox{in the symplectic case}\\
\bigoplus_{\tau\in \mathcal{T}_{F_0}}\left(\omega_\tau\otimes_{{\mathcal O}_Y}\omega_{\tau^*}\right) & \mbox{in the unitary case}.
\end{cases}
\end{align*}
For details on the Gauss--Manin connection and the Kodaira--Spencer (iso)morphism, we refer the reader to~\cite[Sections 2.1.7 and 2.3.5]{la} and~\cite[Section 9]{FaltingsChai}.
\subsection{Differential operators on $p$-adic automorphic forms}
We briefly recall a construction of $p$-adic differential operators $\Diff_\kappa^\lambda$, analogues of $C^{\infty}$ Maass--Shimura operators, introduced in~\cite[Chapter II]{kaCM} for Hilbert modular forms and extended to the Siegel and unitary cases in~\cite{padiffops2, padiffops1} and~\cite{EDiffOps, EFMV}, respectively.
Let ${\mathcal S}$ denote the formal ordinary locus over $W(\overline{{\mathbb F}}_p)$. Write $\omega=\omega_{\CA/{\mathcal S}}$.
Define \[U\subseteq {H^1_{\mathrm dR}(\Auniv/\shord)}\] to be the unit root subcrystal.
Then $U$ is a complement of $\omega$, and the inclusion $\omega\subseteq {H^1_{\mathrm dR}(\Auniv/\shord)}$ composed with the projection
${\pi}_U:{H^1_{\mathrm dR}(\Auniv/\shord)}\to{H^1_{\mathrm dR}(\Auniv/\shord)}/U$
is an isomorphism. We denote the induced morphism by
\[{\Pi}_U: {H^1_{\mathrm dR}(\Auniv/\shord)}\to\omega_{\CA/{\mathcal S}}.\]
Define
\[\Diff:=({\Pi}_U\otimes \mathrm{ks}^{-1})\circ\nabla\vert _{\omega}: \omega \subseteq {H^1_{\mathrm dR}(\Auniv/\shord)}\to{H^1_{\mathrm dR}(\Auniv/\shord)}\otimes\Omega^1_{{\mathcal S} /W(\overline{{\mathbb F}}_p)}\to
\omega\otimes\omega^2.\]
For any $d\geq 1$, by the Leibniz rule (the product rule), we obtain an operator
\[\Diff_d: \omega^{\otimes d} \to \omega^{\otimes d}\otimes \omega^2\subseteq \omega^{\otimes(d+2)}.\]
By the construction of Schur operators, for any dominant weight $\kappa$, the operator $\Diff_d$ for $d=|\kappa|$ induces an operator
\[\Diff_\kappa:\omega^\kappa\to \omega^\kappa \otimes \omega^2.\]
For any admissible weight $\lambda$, the corresponding weight-raising differential operators
\[\Diff^\lambda_\kappa:\omega^\kappa\to \omega^{\kappa+\lambda}\]
are induced via Schur functors by the $e$-th iterations $\Diff^{(e)}_d:=\Diff_{d+2(e-1)}\circ \cdots\circ\Diff_{d+2}\circ\Diff_{d}$,
\[\Diff^{(e)}_d: \omega^{\otimes d} \rightarrow \omega^{\otimes d}\otimes (\omega^2)^{\otimes e}\subseteq\omega^{\otimes d+2e},\]
composed with the Young symmetrizer $y_\lambda: (\omega^2)^{\otimes e}\to\omega^\lambda$, for $e=|\lambda|/2$.
\begin{rmk}
For any admissible weight $\lambda$ and any $p$-adic form $f$ of weight $\kappa$, $\Diff_\kappa^\lambda(f)$ is a $p$-adic form of weight $\kappa+\lambda$.
In particular, for any integer $m\geq 1$ and any mod $p^m$ automorphic form $f\in H^0(\mathcal{X}_{m}, \omega^\kappa)$, we obtain $\Diff_\kappa^\lambda (f)\in H^0({\mathcal S}_m, \omega^{\kappa+\lambda})$. By construction of the Hasse invariant ${E}$, for a sufficiently large integer $N>>0$, the section ${E}^N \cdot\Diff_\kappa^\lambda (f) \in H^0\left({\mathcal S}_m, \omega^{\kappa+\lambda+{\underline{(p-1)N}}}\right)$ extends (uniquely) to all of $\mathcal{X}_{m}$, i.e.\ ${E}^N \cdot\Diff_\kappa^\lambda (f)$ is, in fact, the restriction of an element of $H^0\left(\mathcal{X}_{m}, \omega^{\kappa+\lambda+\underline{(p-1)N}}\right)$.
We shall prove in Theorem~\ref{ana_thm} that when $m=1$, for each admissible weight $\lambda$, there exists an explicit integer, namely $N=|\lambda|/2$, such that for all dominant weights $\kappa$ and all characteristic $p$ automorphic forms $f$ of weight $\kappa$, the sections
${E}^N\cdot \Diff_\kappa^\lambda (f)$, defined \emph{a priori} over the ordinary locus, extend to all of $X:=\mathcal{X}_1$.
\end{rmk}
\subsection{Differential operators on automorphic forms modulo $p$}\label{diffmop_sec}
Let $X$ denote the reduction modulo $p$ of the Shimura variety $\mathcal{X}$, i.e. $X:=\mathcal{X}_1$, and set $\omega=\omega_{\CA/X}$. We now construct a new class of weight-raising differential operators $\Theta_\kappa^\lambda$ on the space $H^0(\mathcal{X}_{1}, \omega_{\CA/X}^\kappa)$ of automorphic forms in characteristic $p$.
For the special case $G=\operatorname{GSp}_4$, Yamauchi constructed and studied similar operators~\cite{Yama}.
\subsubsection{Adjugates and the Hasse invariant}
For a morphism $f\colon \mathcal{F}\to\mathcal{G}$ of locally free ${\mathcal O}_X$-modules of rank $g$, the \emph{adjugate} of $f$ is the morphism
\[\adj{f}\colon\mathcal{G}\otimes \vert\mathcal{F}\vert\to\mathcal{F}\otimes\vert\mathcal{G}\vert\]
(where $|\cdot|$ denotes the top exterior power) obtained from the dual map
\[(\wedge^{g-1} f)^t\colon \left({\mathcal G}\otimes \vert {\mathcal G}\vert^{-1}\right) \simeq \left(\wedge^{g-1}\mathcal{G}\right)^t
\to \left(\wedge^{g-1}{\mathcal F}\right)^t\simeq \mathcal{F} \otimes \vert {\mathcal F}\vert^{-1}\]
after tensoring with the identity map on $\vert \mathcal{F}\vert \otimes\vert\mathcal{G}\vert$.
It satisfies the property that
\begin{align}\label{adj}
\adj{f}\circ (f\otimes \mathrm{id}_{\vert\mathcal{F}\vert})=\mathrm{id}_\mathcal{F}\otimes\det(f)\colon \mathcal{F}\otimes \vert {\mathcal F}\vert\to \mathcal{F}\otimes \vert {\mathcal G}\vert.
\end{align}
Let $\mathrm{Fr}\colon \CA\rightarrow \CA^{(p)}$ denote the relative Frobenius morphism on the universal abelian scheme over $X$. Denote by $h\colon\omega_{\CA/X}\to \omega_{\CA/ X}^{(p)} $ the dual of the morphism
$\mathrm{Fr}^*\colon {R^1\pi_*\cO}_{\CA}^{(p)}\to {R^1\pi_*\cO}_\CA$ introduced in Section~\ref{hasse_sec}.
In the following, by abuse of notation, we still denote by \[\adj{h}\colon\omega_{\CA/X}^{(p)}\to \omega_{\CA/ X}\otimes \vert \omega_{\CA/X}\vert ^{p-1} \] the map obtained by tensoring the adjugate of $h$
with the identity on $\vert \omega_{\CA/X}\vert^{-1}$ and then
composing with the identification $\vert \omega_{\CA/X}^{(p)}\vert\simeq \vert \omega_{\CA/X}\vert^p$.
From equality~\eqref{adj}, we deduce that \[\adj{h}\circ h\colon\omega_{\CA/X} \to \omega_{\CA/X}\otimes \vert \omega_{\CA/X}\vert^{p-1}\] is equal
to multiplication by the Hasse invariant ${E}:=\det (h)$.
Define
\begin{align*}
\mathscr{U} := \mathrm{Image}\left(\mathrm{Fr}^*\colon {H^1_{\mathrm dR}(\Auniv/\Shp)}^{(p)}\rightarrow {H^1_{\mathrm dR}(\Auniv/\Shp)}\right).
\end{align*}
\begin{lem}\label{adjugateha}
The relative Frobenius $\mathrm{Fr}\colon\CA\to\CA^{(p)}$ induces an isomorphism
\begin{align}\label{isomodcurlyU}
{H^1_{\mathrm dR}(\Auniv/\Shp)}/\mathscr{U}\simeq \omega
^{(p)}.
\end{align}
Under this identification, the inclusion $\omega\subseteq {H^1_{\mathrm dR}(\Auniv/\Shp)}$ composed with the projection
\begin{align*}
{\pi}_{\mathscr{U}}\colon{H^1_{\mathrm dR}(\Auniv/\Shp)}\to{H^1_{\mathrm dR}(\Auniv/\Shp)}/\mathscr{U}
\end{align*}
agrees with the morphism $h\colon\omega\to \omega^{(p)}$.
\end{lem}
\begin{proof} By Katz's work on the conjugate Hodge--de Rham spectral sequence~\cite[Section 2.3]{Katz-diff} (also~\cite[\S{}5.1, Proposition 5.1]{wedhorn09}, and~\cite[\S{}5.1]{BBM}), the sheaf $\mathscr{U}$ and the quotient ${H^1_{\mathrm dR}(\Auniv/\Shp)}/\mathscr{U}$ are locally free of rank $g$ and dual to each other under the pairing $\langle ,\rangle_{\CA}$
on ${H^1_{\mathrm dR}(\Auniv/\Shp)}$ induced by the polarization $\mu_\CA$ of $\CA$.
Furthermore, it follows from this work of Katz that the relative Frobenius
\begin{align*}
\mathrm{Fr}^*\colon{H^1_{\mathrm dR}(\Auniv/\Shp)}^{(p)}\to{H^1_{\mathrm dR}(\Auniv/\Shp)}
\end{align*}
induces an isomorphism
\begin{align}\label{katzcurlyUiso}
{R^1\pi_*\cO}_{\CA}^{(p)}\simeq\mathscr{U}.
\end{align}
Dualizing Isomorphism~\eqref{katzcurlyUiso}, we obtain Isomorphism~\eqref{isomodcurlyU}, as well as that the inclusion $\omega\subseteq {H^1_{\mathrm dR}(\Auniv/\Shp)}$ composed with the projection ${\pi}_{\mathscr{U}}\colon{H^1_{\mathrm dR}(\Auniv/\Shp)}\to{H^1_{\mathrm dR}(\Auniv/\Shp)}/\mathscr{U}$ agrees with the morphism $h\colon\omega_{\CA/X}\to \omega_{\CA/X}^{(p)}$ dual to $\mathrm{Fr}^*\colon {R^1\pi_*\cO}_{\CA}^{(p)}\to {R^1\pi_*\cO}_\CA$.
\end{proof}
\subsubsection{A construction of differential operators in characteristic $p$} We continue to set $\omega=\omega_{\CA/X}$.
Consider the morphism
\[{\Pi}_{\mathscr{U}}:= \adj{h}\circ {\pi}_\mathscr{U}\colon {H^1_{\mathrm dR}(\Auniv/\Shp)} \to{H^1_{\mathrm dR}(\Auniv/\Shp)}/\mathscr{U}\simeq \omega^{(p)}\to\omega\otimes\vert\omega\vert^{p-1}.\]
Define
\[\Theta:=({\Pi}_\mathscr{U}\otimes \mathrm{ks}^{-1})\circ\nabla\vert_{\omega}\colon \omega \to{H^1_{\mathrm dR}(\Auniv/\Shp)}\otimes\Omega^1_{{\mathcal S} /W(\overline{{\mathbb F}}_p)}\to
\left( \omega\otimes\vert\omega\vert^{p-1}\right)\otimes \omega^2= \omega\otimes\left(\vert\omega\vert^{p-1}\otimes \omega^2\right)\]
For any $d\geq 1$, by the Leibniz rule (the product rule), we obtain an operator
\[\Theta_d\colon \omega^{\otimes d} \to \omega^{\otimes d}\otimes \left(\vert\omega\vert^{p-1}\otimes \omega^2\right)\subseteq \omega^{\otimes d+p+1}.\]
By the construction of Schur operators, for any dominant weight $\kappa$, the operator $\Theta_d$ for $d=|\kappa|$ induces an operator
\[\Theta_\kappa\colon\omega^\kappa\to \omega^\kappa \otimes \left(\vert\omega\vert^{p-1}\otimes\omega^2\right).\]
For any admissible weight $\lambda$ of depth $e=|\lambda|/2$, the corresponding differential operators
\[\Theta^\lambda_\kappa\colon\omega^\kappa\to \omega^{\kappa+\lambda+\underline{(p-1)e}}\]
are induced via Schur functors applied to the $e$-th iteration $\Theta^{(e)}_d:=\Theta_{d+(p+1)(e-1)}\circ \cdots\circ\Theta_{d+p+1}\circ\Theta_{d}$,
\[\Theta^{(e)}_d\colon \omega^{\otimes d} \to \omega^{\otimes d}\otimes \left(\vert\omega\vert^{p-1}\otimes \omega^2\right)^{\otimes e}\subseteq \omega^{\otimes d+(p+1)e},\]
composed with projection \[\mathrm{id}_{\vert\omega\vert^{(p-1)e}}\otimes y_\lambda\colon \left(\vert\omega\vert^{p-1}\otimes \omega^2\right)^{\otimes e}= \vert\omega\vert^{(p-1)e}\otimes \left(\omega^{2}\right)^{\otimes e}
\to \vert\omega\vert^{(p-1)e}\otimes \omega^\lambda=\omega^{\lambda+\underline{(p-1)e}}.\]
\begin{rmk}
In Section~\ref{ana_sec}, we shall be particularly interested in the action of $\Theta_\kappa^\lambda$ on the restriction $\omega_{\CA/X}|_{S} = \omega_{\CA/S}$ to the ordinary locus $S:={\mathcal S}_1\subset X$.
For convenience of notation, we write $\Theta_\kappa^\lambda|_S$ to denote the resulting morphism of sheaves $\omega_{\CA/S}^\kappa\to \omega_{\CA/S}^{\kappa+\lambda+\underline{(p-1)|\lambda|/2}}$ over $S$. Similarly, we write ${\Pi_\mathscr{U}}\vert_{S}$ for the morphism ${H^1_{\mathrm dR}(\Auniv/\shpord)}\to \omega_{\CA/S} \otimes \vert \omega_{\CA/S} \vert^{p-1}$ obtained by restriction to sheaves over $S\subset X$.
\end{rmk}
\subsection{Analytic continuation modulo $p$}\label{ana_sec}
We now prove the operators $\Theta_\kappa^\lambda$ analytically continue the mod $p$ reduction of the $p$-adic differential operators $\Diff_\kappa^\lambda$, \emph{a priori} defined only over the ordinary locus $S:={\mathcal S}_1$, to all of $X:=\mathcal{X}_1$.
More precisely, we establish the following result, of which Theorem~\ref{continuationB} from Section~\ref{intro-newresults} is a consequence.
\begin{thm}[Analytic Continuation]\label{ana_thm}
For any admissible weight $\lambda$ and dominant weight $\kappa$,
\[ {\Theta_\kappa^\lambda}\vert_{ S} \equiv {E}^{|\lambda |/2} \cdot \Diff_\kappa^\lambda \mod p \]
as morphisms $\omega_{\CA/S}^\kappa\to \omega_{\CA/S}^{\kappa+\lambda+\underline{(p-1)|\lambda|/2}}$ of sheaves over $S$.
\end{thm}
In particular, for any classical automorphic form $f$ of weight $\kappa$, the $\bmod p$ reduction of the $p$-adic automorphic form $E^{|\lambda |/2}\cdot\Diff_\kappa^\lambda f$ is classical.
\begin{proof}
Comparing the constructions of the operators $\Theta_\kappa^\lambda$ and $\Diff_\kappa^\lambda$, the statement reduces to the following lemma.
\end{proof}
\begin{lem} Maintaining the above notation,
\[{\Pi_\mathscr{U}}\vert_{S} \equiv {E}\cdot \Pi_U\mod p\]
as maps ${H^1_{\mathrm dR}(\Auniv/\shpord)}\to \omega_{\CA/S} \otimes \vert \omega_{\CA/S} \vert^{p-1}$.
\end{lem}
\begin{proof}
We work over $S$ and write $\omega=\omega_{\CA/S}$. Consider the pullback of the slope filtration to ${H^1_{\mathrm dR}(\Auniv/\shpord)}^{(p)}$, $0\subseteq U^{(p)}\subset {H^1_{\mathrm dR}(\Auniv/\shpord)}^{(p)}$. By construction of the unit root subcrystal $U$, the restriction of the relative Frobenius map $\mathrm{Fr}^*$ to $U^{(p)}$ induces an isomorphism onto $U$. Since $\mathrm{Fr}^*$ is equal to $0$ on $\omega^{(p)}$, it follows from the definition of $\mathscr{U}$ that $\mathscr{U}=U$ over $S$.
By Lemma~\ref{adjugateha}, the induced morphism \[\omega\simeq {H^1_{\mathrm dR}(\Auniv/\shord)}/U\to {H^1_{\mathrm dR}(\Auniv/\shord)}/\mathscr{U}\simeq \omega^{(p)}\] agrees with the restriction to $S$ of the morphism $h\colon\omega\to\omega^{(p)}$.
So we have
\[\Pi_\mathscr{U} =\adj{h}\circ \pi_\mathscr{U}\equiv \adj{h}\circ h\circ \pi_U\equiv {E}\cdot \pi_U\mod p.\]
\end{proof}
\begin{rmk}
Since the weights of ${\Theta_\kappa^\lambda}\vert_{ S}(f)$ and $\Diff_\kappa^\lambda (f)$ are different, we can only compare their values after identifying modular forms of different weights in a larger, common space of $p$-adic modular forms $V$ (like in~\cite[Section 8.1.3]{hida} or~\cite[Section 4.2.1]{CEFMV}). For the goals of the present paper, we need not concern ourselves with such details of comparisons between forms of different weights. We note, though, that since the $q$-expansions of the Hasse invariant ${E}$ at ordinary cusps are identically $1 \mod p$ (by, e.g.,~\cite{conrad-hasse}), the $q$-expansions of ${\Theta_\kappa^\lambda}\vert_{ S}(f)$ and $\Diff_\kappa^\lambda (f)$ agree mod $p$ at ordinary cusps, which implies ${\Theta_\kappa^\lambda}\vert_{ S}(f)\equiv\Diff_\kappa^\lambda (f)\vert_{ S}(f)\mod p$ inside $V$.
\end{rmk}
\subsubsection{Relation to the Rankin--Cohen bracket}
In the case of scalar weight Siegel forms,
work of B\"ocherer--Nagaoka~\cite[Theorem 4]{BoechererNagaoka-firsttheta} yields a
different approach to the analytic continuation of the operators ${E}\Diff^{\underline{2}}$ modulo $p$.
\begin{prop}
Let $\left[\,,\right]_{\rm RC}$ denote the generalized Rankin--Cohen bracket for Siegel modular forms constructed by Eholzer--Ibukiyama in~\cite{EholzerIbukiyama}.
Then \[\Theta^{\underline{2}}\equiv (2\pi\sqrt{-1})^{-g}\left[\cdot,{E}\right]_{\rm RC} \mod p.\]
\end{prop}
\begin{proof}
Comparing the formulas for the Rankin--Cohen bracket by B\"ocherer--Nagaoka in~\cite[Theorem 4]{BoechererNagaoka-firsttheta} with those for the action of $\Diff^{\underline{2}}$ on $q$-expansions in~\cite{EDiffOps}, we see the actions of $ (2\pi\sqrt{-1})^{-g}\left[\cdot,{E}\right]_{\rm RC} $ and ${E}\Diff^{\underline{2}}$ agree on $q$-expansions.
Thus, the statement follows from Theorem~\ref{ana_thm} by the $q$-expansion principle.
\end{proof}
\subsection{An axiomatic theorem}\label{AT-sec}
We start with a result on Galois representations attached to systems of Hecke eigenvalues for a general algebraic reductive group $G$ defined over $\mathbb{Q}$ (and split over a number field $F$), as introduced in Section~\ref{Galois-bkgd}.
We fix a prime $p$ and write $\chi$ for the mod $p$
cyclotomic character.
Given a mod $p$ Hecke eigenform $f$, we denote $\Sigma_f$ the set of places that are bad with respect to $p$ and the level of $f$, and by $R(f)$ the set of Galois representations $\rho\colon\Gal\left(\Fbar/F\right)\to\hat{G}(\overline{\FF}_p)$ that are (conjecturally) attached to $f$ as in Conjecture~\ref{conj:galois}.
\begin{thm}[Axiomatic Theorem]
\label{axiomaticthm}
Let $G$ be a connected reductive group over $\mathbb{Q}$, split over the number field $F$.
For $i=1,2$, let $f_i$ be a mod $p$ Hecke eigenform on $G$. For $v\notin\Sigma_i:=\Sigma_{f_i}$, let $\Psi_{i, v}\colon \mathcal{H}(G_v,K_v;\overline{\FF}_p)\to \overline{\FF}_p$ denote the Hecke eigensystem of $f_i$ at $v$.
Then
\begin{equation}
\label{eq:psi}
\Psi_{2,v}(c_\lambda) = \eta(\lambda(\pi_v)) \Psi_{1,v} (c_\lambda)
\qquad\text{for all }\lambda\in P^+,v\notin\Sigma_1\cup\Sigma_2,
\end{equation}
for a character $\eta$ of $G$
if and only if
\begin{equation}
\label{eq:main}
(\hat{\eta}\circ\chi)\otimes\rho_1 \in R(f_2) \qquad\text{for all } \rho_1\in R(f_1),
\end{equation}
for the cocharacter $\hat{\eta}$ of $\hat{G}$ corresponding to $\eta$ by duality.
\end{thm}
\begin{rmk}
For Galois representations arising from Hecke eigenforms on symplectic and unitary groups, Theorem~\ref{axiomaticthm} specializes to Theorem~\ref{charpcor} from Section~\ref{intro-newresults}, as we shall observe in Section~\ref{final-effects-section}.
\end{rmk}
\begin{rmk}
The use of the tensor product sign in $(\hat{\eta}\circ\chi)\otimes\rho_1$ is generally an abuse of notation and should be understood as in Lemma~\ref{lem:notensor}.
We retain the $\otimes$ notation because it is evocative of the case when $\hat{G}$ is a group of matrices, where we are indeed dealing with a tensor product of representations.
\end{rmk}
The proof of Theorem~\ref{axiomaticthm} uses the following lemmas.
\begin{lem}
\label{lem:etahat}
Let $G$ be a reductive group and let $\eta\colon G\to\mathbb{G}_m$ be a character of $G$.
Then the cocharacter $\hat{\eta}\colon \mathbb{G}_m\to\hat{G}$ has image in the center $Z(\hat{G})$ and
\begin{equation*}
\eta\circ\mu=\eta\circ\lambda
\end{equation*}
for any dominant weights $\mu,\lambda\in P^+$ such that $\mu\leq\lambda$.
\end{lem}
\begin{proof}
The image of $\eta$ is abelian. So $\eta(G^\prime)=1$, where $G^\prime$ denotes the derived subgroup of $G$.
So $\eta$ induces a character of the abelianization of $G$, hence the dual cocharacter $\hat{\eta}$ lands in the center of $\hat{G}$.
Moreover, since $\eta(G^\prime)=1$, and the coroots of $G$ are the same as the coroots of $G^\prime$, we see that $\eta\circ\alpha^\vee=1$ for any coroot $\alpha^\vee$.
If $\mu$ and $\lambda$ are comparable, their differ by a linear combination of coroots, hence $\eta\circ\mu=\eta\circ\lambda$.
\end{proof}
\begin{lem}
\label{lem:notensor}
Let $G$ be a reductive group over a field $k$ and let $\eta\colon G\to\mathbb{G}_m$ be a character of $G$.
Let $\rho\colon \Gamma\to \hat{G}(k)$ be a representation of a group $\Gamma$ into the $k$-valued points of the dual group $\hat{G}$.
Let $\chi\colon \Gamma\to \mathbb{G}_m(k)$ be a $k$-valued character of $\Gamma$.
Then the map $(\hat{\eta}\circ\chi)\rho\colon\Gamma\to\hat{G}(k)$ given by
\begin{equation}
\label{eq:rho}
((\hat{\eta}\circ\chi)\rho)(\gamma)=\hat{\eta}(\chi(\gamma))\rho(\gamma)
\end{equation}
is a representation of $\Gamma$.
\end{lem}
\begin{proof}
In order to show that Equation~\eqref{eq:rho} defines a group homomorphism, it suffices to prove that $\hat{\eta}(\chi(\gamma))$ is in the center of $\hat{G}(k)$, which was done in Lemma~\ref{lem:etahat}.
\end{proof}
We are ready to prove the main result of this section.
\begin{proof}[Proof of Theorem \ref{axiomaticthm}]
Let $\rho_1\in R(f_1)$ and define $\rho_2:=(\hat{\eta}\circ\chi) \otimes \rho_1$. Then $\rho_1$ and $\rho_2$ are both unramified at primes $v\notin\Sigma:=\Sigma_1\cup\Sigma_2$, and
$\rho_2(\operatorname{Frob}_v)=\hat{\eta}(\pi_v)\rho_1(\operatorname{Frob}_v)$ for all $v\notin\Sigma$.
For $v\notin\Sigma$, let $s_{ i,v}$ denote the $v$-Satake parameter of $f_i$, for $i=1,2$.
Then $\rho_i\in R(f_i)$ if and only if
$s_{i,v}= \rho_{i}(\operatorname{Frob}_v)$ for all $v\notin\Sigma$.
Hence
Equation~\eqref{eq:main} is equivalent to the equalities
\begin{align*}
s_{2,v}=\hat{\eta}(\pi_v)s_{1,v} \text{ for all }v\notin\Sigma.
\end{align*}
Fix $v\notin\Sigma$. Recalling the Satake parameters from Section~\ref{Hecke-Algs-prelim}, note that the equality $s_{2,v}=\hat{\eta}(\pi_v)s_{1,v} $ is equivalent to
\begin{align*}
\chi_\lambda(s_{2,v})=\chi_\lambda(\hat{\eta}(\pi_v)s_{1,v})
\qquad\text{for all }\lambda\in P^+.
\end{align*}
Let $\lambda\in P^+$, and let $\rho_\lambda\colon\hat{G}(\overline{\FF}_p)\to\mathrm{GL}(V_\lambda)$ be the irreducible representation of $\hat{G}(\overline{\FF}_p)$ of highest weight $\lambda$.
As a function on the maximal torus $\hat{T}$, the character of $\rho_\lambda$ can be written as
\begin{equation*}
\chi_\lambda=\sum_{\mu\leq\lambda} (\dim V_\lambda(\mu)) \hat{\mu}.
\end{equation*}
In particular,
\begin{align*}
\chi_\lambda(\hat{\eta}(\pi_v) s_{f_1,v}) &= \sum_{\mu\leq\lambda} (\dim V_\lambda(\mu)) \hat{\mu}(\hat{\eta}(\pi_v) s_{f_1,v})
= \sum_{\mu\leq\lambda} (\dim V_\lambda(\mu)) \hat{\mu}(\hat{\eta}(\pi_v)) \hat{\mu}(s_{f_1,v})\\
&= \sum_{\mu\leq\lambda} (\dim V_\lambda(\mu)) \eta(\mu(\pi_v)) \hat{\mu}(s_{f_1,v})
= \eta(\lambda(\pi_v)) \sum_{\mu\leq\lambda} (\dim V_\lambda(\mu)) \hat{\mu}(s_{f_1,v}) \\
&= \eta(\lambda(\pi_v)) \chi_\lambda(s_{f_1,v}),
\end{align*}
where we used that $\eta(\mu(\pi_v))=\eta(\lambda(\pi_v))$ whenever $\mu\leq\lambda$ (see Lemma~\ref{lem:etahat}).
Hence, Equation~\eqref{eq:main} holds if and only if
\begin{equation}
\label{eq:chi}
\chi_\lambda(s_{2,v})=\eta(\lambda(\pi_v)) \chi_\lambda(s_{1,v})
\qquad\text{for all }\lambda\in P^+, v\notin \Sigma.
\end{equation}
We use Equation \eqref{eq:satake_inv} to show that Equation \eqref{eq:psi} implies Equation \eqref{eq:chi} (and hence Equation \eqref{eq:main}). So
\begin{align*}
\chi_\lambda(s_{2,v}) &= \omega_{2}(\chi_\lambda)
= \Psi_{2,v}\left(\mathcal{S}_v^{-1}(\chi_\lambda)\right)
= \Psi_{2,v}\left(q_v^{\langle-\rho,\lambda\rangle}\sum_{\mu\leq\lambda}d_\lambda(\mu)c_\mu\right)\\
&= q_v^{\langle-\rho,\lambda\rangle}\sum_{\mu\leq\lambda}d_\lambda(\mu)\Psi_{2,v}(c_\mu)
= q_v^{\langle-\rho,\lambda\rangle}\sum_{\mu\leq\lambda}d_\lambda(\mu)\eta(\mu(\pi_v))\Psi_{1,v}(c_\mu)\\
&= \eta(\lambda(\pi_v))q_v^{\langle-\rho,\lambda\rangle}\sum_{\mu\leq\lambda}d_\lambda(\mu)\Psi_{1,v}(c_\mu)
= \eta(\lambda(\pi_v))\chi_\lambda(s_{1,v}),
\end{align*}
after another appeal to Lemma~\ref{lem:etahat}.
Similarly, Equations~\eqref{eq:satake} can be used to show that Equation~\eqref{eq:chi} implies Equation~\eqref{eq:psi}.
\end{proof}
\subsection{Effects of differential operators on Galois representations}\label{final-effects-section}
We assume the group $G$ is symplectic or unitary.
\begin{proof}[Proof of Theorem~\ref{charpcor}]
(For consistency with the notation in Section~\ref{AT-sec}, here $\lambda$ denotes a dominant weight in $ P^+$, and the admissible weight in the statement of Theorem~\ref{charpcor} --previously denoted by $\lambda$-- is $\kappa_0$.)
Theorem~\ref{TandTheta} implies that the assumptions in Theorem~\ref{axiomaticthm} are satisfied by the systems of Hecke eigenvalues associated with a mod $ p$ automorphic form $f$, of weight $\kappa$, and its image under $\Theta^{\kappa_0}=\Theta^{\kappa_0}_{\kappa}$.
More precisely, Theorem~\ref{TandTheta} implies
\begin{align*}
\Psi_{\Theta^{\kappa_0}(f),v}(c_\lambda)&=\nu(\lambda(\pi_v))^{|\kappa_0|/2} \Psi_{f,v} (c_\lambda),
\end{align*}
for all $\lambda\in P^+$ and all but finitely many places $v$ (e.g., all places $v$ not above $p$ and of good reduction for the Shimura variety $\mathcal{X}$). Hence, the theorem is an immediate consequence of Theorem~\ref{axiomaticthm}.
\end{proof}
\subsection{Interaction with isogenies}
Throughout this section, let $Y$ be a scheme that is smooth over a scheme $T$.
\begin{defi}
Given two polarized abelian varieties $(A,\mu_A), (B,\mu_B)$, we say that an isogeny $\phi\colon A\to B$ preserves the polarizations up to a scalar multiple, if there exists $\nu(\phi)\in \mathbb{Q}^\times$, called the \emph{similitude factor}, such that
\[\phi^t\circ \mu_B\circ \phi=[\nu(\phi)]_{A^t}\circ \mu_A.\]
\end{defi}
Let $(A,\mu_A)$ and $(B, \mu_B)$ be polarized abelian schemes of the same relative dimension $g$ over the scheme $Y/T$.
An isogeny $\phi\colon A\to B$ defined over $Y$ induces a morphism $\phi^*\colon H^1_{\rm dR}(B/Y)\to H^1_{\rm dR}(A/Y)$, which preserves the Hodge filtration, that is $\phi^* (\omega_{B/Y})\subseteq \omega_{A/Y}$. By abuse of notation, for each dominant weight $\kappa$,
we still denote by \[\phi^*\colon\omega_{B/Y}^\kappa\to\omega_{A/Y}^\kappa\]
the morphism induced by the maps $(\phi^*)^{\otimes d}$ via Schur operators.
Note that if the polarizations $\mu_A, \mu_B$ have the same degree, then $\nu\in \mathbb{Z}$ and $\deg(\phi)=\nu(\phi)^g$.
In particular, for $(A,\mu_A)=(B,\mu_B)$, and $\phi=[n]_A$, then $\nu(\phi)=n^2$; in this case, the morphism $(\phi^*)^{\otimes d}$ is multiplication by $n^d$.
\begin{lem}\label{kodairaisogeny}
Let $(A,\mu_A)$ and $(B, \mu_B)$ be polarized abelian schemes of the same relative dimension over $Y/T$, and let $\phi\colon A\to B$ be an isogeny preserving their polarization up to multiplication by a scalar $\nu(\phi)\in\mathbb{Q}^\times$. Then
\[\mathrm{KS}_{A/Y}\circ (\phi^*\otimes\phi^*)=\nu(\phi) \mathrm{KS}_{B/Y}.\]
\end{lem}
\begin{proof}
By the definition of the Kodaira--Spencer morphism
\[\mathrm{KS}_{A/Y}:=\langle \cdot, \nabla_A (\cdot)\rangle_{A},\]
the statement is equivalent to the equality
\[\langle \phi^*(\cdot), \nabla_A \phi^*(\cdot)\rangle_{A}=\nu(\phi) \langle \cdot, \nabla_B (\cdot)\rangle_{B}.\]
By the functoriality of the Gauss--Manin connection, we deduce $\nabla_A \circ\phi^*=(\phi^*\otimes \mathrm{id})\circ\nabla_B$, and reduce the statement to the equality
\[\langle \phi^*(\cdot), \phi^*(\cdot)\rangle_{A}=\nu(\phi) \langle \cdot, \cdot\rangle_{B}.\]
The latter follows from $\phi^t\circ\mu_B\circ \phi=\nu(\phi)\mu_A$.
\end{proof}
\subsection{Commutation relations with prime-to-$p$ Hecke operators}
Following~\cite[Ch.~VII, \S{}3]{FaltingsChai}, we define the action of prime-to-$p$ algebraic correspondences, and of prime-to-$p$ Hecke operators, on automorphic forms.
Fix a rational prime $\ell\not=p$. \textbf{We assume $\ell$ is good} (see Section~\ref{Galois-bkgd} for definition).
We denote by
${\mathcal H}_0(G_\ell,\mathbb{Q})$ the $\mathbb{Q}$-subalgebra of the local Hecke algebra $\mathcal{H}(G_\ell,K_\ell;\mathbb{Q})$ generated by locally constant functions supported on cosets $K_\ell\gamma K_\ell$, for $\gamma\in G_\ell$ an integral matrix (see Section~\ref{Hecke-Algs-prelim} for notation).
\begin{defi}
We denote by $\ell{\rm -Isog}$ the moduli space of $\ell$-isogenies over $\mathcal{X}$, and by $\phi\colon {\rm pr}_1^*\CA \to {\rm pr}^*_2\CA$ the universal $\ell$-isogeny, for \[{\rm pr}=({\rm pr}_1,{\rm pr}_2)\colon \ell{\rm -Isog}\to \mathcal{X}\times \mathcal{X}\] the natural structure morphism.
\end{defi}
Note that the degree of the universal isogeny is locally constant on $\ell{\rm -Isog}$. For any connected component $Z$ of $\ell{\rm -Isog}$, the two projections ${\rm pr}_i\colon Z\to\mathcal{X}$ are proper, and they are finite \'etale over $\mathcal{X}[1/\deg(Z,\phi)]$, where $\deg(Z,\phi)$ denotes the degree of $\phi$ on $Z$. In particular, they are finite and \'etale over $\mathcal{X}/{\mathcal{O}_{E,(\mathfrak{p})}}$.
\begin{defi}
For any dominant weight $\kappa$, there is a natural action of $(Z,\phi)$ on the space $H^0(\mathcal{X},\omega^\kappa)$ of automorphic forms
of weight $\kappa$, defined as
\[T_\phi=T_{(Z,\phi), \kappa}:={\rm tr}\circ \phi^*\circ{\rm pr^*_2}\colon H^0(\mathcal{X},\omega^\kappa)\to H^0(Z,{\rm pr}_2^*\omega^\kappa)\to H^0(Z,{\rm pr}^*_1\omega^\kappa)\to H^0(\mathcal{X},\omega^\kappa)\]
\end{defi}
It follows from the definition that, for any dominant weight $\kappa$, the operator $T_{(Z,\phi),\kappa}$ induces an action of $(Z,\phi)$ on the space $H^0(X,\omega^\kappa)$ of mod $p$ automorphic forms of weight $\kappa$ (by reduction modulo $p$), and also on the space $H^0({\mathcal S},\omega^\kappa)$ of $p$-adic automorphic forms of weight $\kappa$ (by restriction).
\begin{defi}
\label{defi-heckeaction}
Let $Y$ denote a base scheme such that $\ell$ is invertible in ${\mathcal O}_Y$, equipped with a map $f\colon Y\to \mathcal{X}$ (e.g., $Y=\mathcal{X}$ over $\mathcal{O}_{E,(\mathfrak{p})}$, $Y=X$ over ${{\mathbb F}}_p$, or $Y={\mathcal S}$ over ${\mathbb Z}_p$).
There is a natural $\mathbb{Q}$-linear map
\[h=h_\ell\colon {\mathcal H}_0(G_\ell,\mathbb{Q}) \to \mathbb{Q}[\ell{\rm -Isog}/Y]\]
which to any double coset $K_\ell\gamma K_\ell$, with $\gamma$ an integral matrix in $G_\ell$, associates the union $Z(K_\ell\gamma K_\ell)$ of those connected components of $\ell{\rm -Isog}$ where the universal isogeny is an $\ell$-isogeny of type $K_\ell\gamma K_\ell$.
\end{defi}
By definition, the action on automorphic forms of the prime-to-$p$ Hecke operators agrees with that of the prime-to-$p$ algebraic correspondences (via pullback under $h$).
\begin{thm}\label{TandTheta} Let $(Z,\phi)$ be a connected component of $\ell{\rm -Isog}$.
For any dominant weight $\kappa$ and any admissible weight $\lambda$, we have
\begin{enumerate}
\item $T_\phi\circ \Theta^{\lambda}_\kappa =\nu(\phi)^{|\lambda |/2} \Theta^\lambda_\kappa\circ T_\phi$\label{P1}
\item $T_\phi\circ \Diff^{\lambda}_\kappa =\nu(\phi)^{|\lambda |/2} \Diff^\lambda_\kappa\circ T_\phi $\label{P2}
\end{enumerate}
where $\nu(\phi)$ denotes the similitude factor of $\phi$.
\end{thm}
\begin{proof}
For Part~\eqref{P1}, by the functoriality of the construction of the operators $\Theta^\lambda_\kappa$, it suffices to establish the equality
\[T_\phi\circ \Theta=\nu(\phi)\Theta\circ T_\phi. \]
By definition, $\Theta:=(\Pi_\mathscr{U}\otimes\mathrm{ks}^{-1})\circ \nabla\vert_{\omega_{\CA/X}}$.
By the functoriality of the Gauss--Manin connection $\nabla$, and of the definition of $\Pi_\mathscr{U}$,
it suffices to prove the equality
\[\nu(\phi) \mathrm{KS}=\mathrm{KS}\circ (\phi^*\otimes\phi^*)\colon {\rm pr}_2^*H^1_{\rm dR}(\CA/X)\otimes {\rm pr}_2^*H^1_{\rm dR}(\CA/X)\to {\rm pr}_1^*\Omega^1_{X/T}.\]
This follows from Lemma~\ref{kodairaisogeny}.
For Part~\eqref{P2}, the statement reduces to the equality $T_\phi\circ \Diff=\nu(\phi) \Diff\circ T_\phi$, and the same argument applies to the operator $\Diff$, with the morphism $\Pi_U$ in place of $\Pi_\mathscr{U}$.
\end{proof}
\begin{rmk} In the Siegel case,
for an admissible scalar weight $\lambda=\underline{d}$, we have that $d$ is even, $|\lambda|=gd$, and $\nu(\phi)^{|\lambda |/2}=\deg(\phi)^{d/2}$.
In particular, for $g=2$, Theorem~\ref{TandTheta} specializes to Yamauchi's result~\cite[Proposition 3.9]{Yama}.
\end{rmk}
\subsection{Commutation relations with Hecke operators at $p$}
Following~\cite[Ch. VII,\S 4]{FaltingsChai}, we define the action of $p$-power algebraic correspondences, and of Hecke operators at $p$, on automorphic forms over the ordinary locus ${\mathcal S}$.
Following \emph{loc.cit.}, we identify $H\times \mathbb{G}_m$ with the appropriate maximal Levi subgroup $M$ of $\mathcal{G}$ over $\mathbb{Z}_p$ (see Section~\ref{signandlevi_sec}).
We write $M_p:=M(\mathbb{Q}_p)$, and $\gamma\in M_p$ as $\gamma=(\alpha,p^d)$ with $\alpha \in H(\mathbb{Q}_p)$ and $d\in \mathbb{Z}$. Note that
$\gamma=(\alpha,p^d)\in M_p$ is an integral matrix if and only if $d\geq 0$ and both $\alpha$ and $p^d\alpha^{-1}$ are integral matrices in $H(\mathbb{Q}_p)$.
Also, by definition $M(\mathbb{Z}_p)=K_p\cap M_p$, hence the local Hecke algebra $\mathcal{H}(M_p,M(\mathbb{Z}_p);\mathbb{Q})$ is a subalgebra of $\mathcal{H}(G_p,K_p;\mathbb{Q})$ (see Section~\ref{Hecke-Algs-prelim} for notation).
We denote by
${\mathcal H}_0(M_p,\mathbb{Q})$ the $\mathbb{Q}$-subalgebra of $\mathcal{H}(M_p, M(\mathbb{Z}_p);\mathbb{Q})$ generated by locally constant functions supported on cosets $M(\mathbb{Z}_p)\gamma M(\mathbb{Z}_p)$, for $\gamma\in M_p$ an integral matrix.
\begin{defi}
We denote by $p{\rm -Isog}^o$ the moduli space of $p$-isogenies over the ordinary locus ${\mathcal S}$, and by $\phi\colon {\rm pr}_1^*\CA \to {\rm pr}^*_2\CA$ the universal $p$-isogeny, for \[{\rm pr}=({\rm pr}_1,{\rm pr}_2)\colon \mathfrak{p}{\rm -Isog}^o\to {\mathcal S}\times {\mathcal S}\] the natural structure morphism.
\end{defi}
By~\cite[\S{}VII.4, Proposition 4.1]{FaltingsChai}, for any connected component $Z$ of $p{\rm -Isog}^o$, the two projections ${\rm pr}_i\colon Z\to{\mathcal S}$ are finite and flat over ${\mathcal S}/W(\overline{{\mathbb F}}_p)$.
\begin{defi}
For any dominant weight $\kappa$, there is a natural action of $(Z,\phi)$ on the space $H^0({\mathcal S},\omega^\kappa)$ of $p$-adic automorphic forms of weight $\kappa$, defined as
\[T_\phi=T_{(Z,\phi)}:={\rm tr}\circ \phi^*\circ{\rm pr^*_2}\colon H^0({\mathcal S},\omega^\kappa)\to H^0(Z,{\rm pr}_2^*\omega^\kappa)\to H^0(Z,{\rm pr}^*_1\omega^\kappa)\to H^0({\mathcal S},\omega^\kappa).\]
\end{defi}
\begin{thm}\label{TandTheta2} Let $(Z,\phi)$ be a connected component of $p{\rm -Isog}^o$.
For any dominant weight $\kappa$ and any admissible weight $\lambda$, we have
\[T_\phi\circ \Diff^{\lambda}_\kappa=\nu(\phi)^{|\lambda |/2} \Diff^\lambda_\kappa\circ T_\phi ,\]
where $\nu(\phi)$ denotes the similitude factor of $\phi$.
In particular, if $\nu(\phi)>1$, then $T_\phi\circ \Diff^{\lambda}_\kappa \equiv 0 \mod p$.
\end{thm}
\begin{proof} The commutation relations follow by the same argument as in the proof of Part~\eqref{P2} of Theorem~\ref{TandTheta}. The vanishing in positive characteristic follows from the equality $\nu(\phi)=p^d$, for $d\geq 1$.
\end{proof}
Consider the mod $p$-reduction \[{\rm pr}_{\overline{{\mathbb F}}_p}\colon \mathfrak{p}{\rm -Isog}^o\otimes_{W(\overline{\mathbb F}_p)} {\overline{{\mathbb F}}_p}\to S\times S.\]
Note that the degree, similitude factor, and $p$-type of the universal isogeny are locally constant on $p{\rm -Isog}^o$. (Recall that, by definition, the $p$-type of a $p$-isogeny with similitude factor $p^d$ is a coset $H(\mathbb{Z}_p)\alpha H(\mathbb{Z}_p)$ of an integral matrix $\alpha\in H(\mathbb{Q}_p)$ such that $p^d\alpha^{-1}$ is also integral.)
In~\cite[\S{}VII.4, Proposition 4.1]{FaltingsChai}, for any connected component $Z$ of $p{\rm -Isog}^o\otimes_{W(\overline{\mathbb F}_p)}\overline{{\mathbb F}}_p$, Faltings and Chai compute the purely inseparable multiplicity $\mu(Z)$ of the geometric fibers of ${\rm pr}_i\colon Z\to S$
in terms on the degree and the $p$-type of the universal isogeny on $Z$. Furthermore,
they prove
there is a well-defined integral action of the operator
${\mu(Z)}^{-1}T_{(Z,\phi)}$ on the space of mod $p$ automorphic forms of weight $\kappa$, for all dominant weights $\kappa$.
We define the \emph{normalized} action of $(Z,\phi)$ on mod $p$ automorphic forms as $t_\phi=t_{(Z,\phi)}:=\mu(Z)^{-1}T_{(Z,\phi)}$.
\begin{defi}
Let $Y$ denote a base scheme of characteristic $p$, equipped with a map $f\colon Y\toS$.
There is a natural $\mathbb{Q}$-linear map
\[h=h_p\colon
{\mathcal H}_0(M_p,\mathbb{Q}) \to \mathbb{Q}[p-{\rm Isog}^o/Y]\]
which to any double coset $M(\mathbb{Z}_p)\gamma M(\mathbb{Z}_p)$ with $\gamma=(\alpha, p^d)$ an integral matrix in $M_p$,
associates $\mu(Z(\alpha,d))^{-1} \cdot Z(\alpha, d)$, where $Z(\alpha ,d)$ denotes the union of those connected components of $p-{\rm Isog}^o$ where the universal isogeny is a $p$-isogeny of $p$-type $H(\mathbb{Z}_p)\alpha H(\mathbb{Z}_p)$ and similitude factor $p^d$.
\end{defi}
By definition, the action of the Hecke operators at $p$ on mod $p$ automorphic forms agrees with the \emph{normalized} action of $p$-power algebraic correspondences.
\subsubsection{Hida's ordinary projector}
In~\cite[\S{}8]{hida} (also~\cite[\S{}3.6]{H02}), Hida establishes
the divisibility of the Hecke operators at $p$ on $p$-adic automorphic forms by a given power of $p$, keeping their integrality.
More precisely, he proves that the normalized action on mod $p$ automorphic forms lifts to $p$-adic automorphic forms.
For $1\leq j\leq n$, let $\alpha_j\in H(\mathbb{Q}_p)$ defined as $\alpha_{j,\tau}:= {\rm diag}[{\mathbb I}_{n-j},p{\mathbb I_j}]$, $\tau\in\mathcal{T}_0$. (For any positive integer $k$, $\mathbb{I}_k$ denotes the $k\times k$ identity matrix.) Let $Z_{\alpha_j}$ denote
the union of those connected components of $p{\rm -Isog}^o$ where the universal isogeny has $p$-type $H(\mathbb{Z}_p)\alpha_j H(\mathbb{Z}_p)$ and similitude factor $p$.
For $j=1, \dots, n$, Hida proves the integrality of the operators
${\mu(\alpha_j)^{-1}} {\mathbb U}(\alpha_j):=\mu(Z_{\alpha_j})^{-1}T_{(Z_{\alpha_j},\phi)}$ on the space of $p$-adic automorphic
forms over the ordinary locus.
By definition (\cite[\S{}8.3.1, Lemma 8.12]{hida}),
the ordinary projector $\mathbf{e}$ on the space of $p$-adic automorphic forms is
\[\mathbf{e}:=\varinjlim_m {\mathbb U}(p)^{m!}\]
where
${\mathbb U}(p):=\prod_1^n {\mu(\alpha_j)^{-1}} {\mathbb U}(\alpha_j) $.
Hence, by Theorem~\ref{TandTheta2}, we deduce the following result.
\begin{coro}\label{ed0}
For any dominant weight $\kappa$, and admissible weight $\lambda$, we have
\[\mathbf{e}\Diff_\kappa^\lambda=0.\]
\end{coro}
\subsection{Context from the special case of $\mathrm{GL}_2$}\label{GL2-intro}
For motivation, we briefly recall related results for modular forms and $2$-dimensional Galois representations. Fix a prime $p$. Associated to a cuspidal eigenform $f$ with Fourier expansion $f(q) = \sum_{n\geq 1}a_n q^n$, normalized so that $a_1 = 1$, there is a continuous, semi-simple representation
\begin{align}\label{char0repn}
\rho_f\colon \Gal\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\rightarrow \mathrm{GL}_2\left(\overline{\mathbb{Q}}_p\right),
\end{align}
unramified at all primes $\ell$ coprime to $p$ and the level of $f$, and satisfying
\begin{align*}
\mathrm{Trace}\left(\rho_f\left(\operatorname{Frob}_\ell\right)\right) = a_\ell
\quad\text{and}\quad
\det\left(\rho_f\left(\operatorname{Frob}_\ell\right)\right) = \ell^{k-1}\psi(\ell)
\end{align*}
at such primes, where $k$ is the weight of $f$ and $\psi$ is the nebentypus of $f$~\cite{eichler, shimura1, shimura2, igusa, deligne1, deligne2, deligne-serre}.
Equivalently, using the description of Hecke operators on $q$-expansions, this statement can be reformulated in terms of Hecke eigenvalues in place of Fourier coefficients.
Serre's conjectures (formulated in~\cite{serreconj1, serreconj2} and proved in generality in~\cite{KW1, KW2, KW3} and in certain cases in~\cite{dieulefait1, dieulefait2}) associate to a representation
\begin{align*}
\rho\colon \Gal\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\rightarrow \mathrm{GL}_2\left(\mathbb{F}\right),
\end{align*}
with $\mathbb{F}$ a finite field of characteristic $p$, a cuspidal eigenform $f$ such that $\overline{\rho}_f\cong \rho$, where $\overline{\rho}_f$ denotes the reduction of $\rho_f$ mod $p$.
Serre's conjectures also specify the weight and level of $f$.
In the proof of the weight part of Serre's conjecture, it turns out to be useful to consider the action of powers of a certain mod $p$ differential operator developed by Katz in~\cite{Katz-theta}. Katz constructs a mod $p$ differential operator $\theta$ that, \emph{a priori}, is defined only over the ordinary locus (of the moduli space $\mathcal{M}$ of elliptic curves over which the line bundle of modular forms is defined) and then explains how to analytically continue it to all of $\mathcal{M}$. This operator acts on the $q$-expansion of a mod $p$ modular form $f = \sum_{n\geq 0} a_n q^n$ by
\begin{align}\label{qformula}
\theta (f) = \sum_{n\geq 0} na_n q^n.
\end{align}
Via the action of Hecke operators $T_\ell$ on $q$-expansions, it is simple to observe that
\begin{align}\label{commrel-equ}
T_\ell \theta = \ell \theta T_\ell.
\end{align}
As a consequence,
\begin{align}\label{twist-equ}
\rho_{\theta f} = \chi\otimes\rho_f,
\end{align}
where $\chi\colon \Gal\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\rightarrow \mathbb{F}_p^\times$ is the mod $p$ cyclotomic character.
The effect of powers of $\theta$ on $\rho_f$ plays an important role in proving the weight part of Serre's conjecture~\cite{Edixhoven}.
\subsection{New results for reductive (especially symplectic and unitary) groups}\label{intro-newresults}
To gain more insight into the complicated structures of Galois groups, it is necessary to consider Galois representations of higher dimension.
In analogy with the setting for $\mathrm{GL}_2$ in Section~\ref{GL2-intro}, there are links between higher-dimensional Galois representations and automorphic forms on more general reductive groups.
Likewise, higher rank analogues $\Diff_{\kappa}^\lambda$ of $\theta$ have been constructed over the ordinary locus of unitary and symplectic Shimura varieties (e.g.\ in~\cite{EDiffOps, EFMV}, building on ideas introduced by Katz and Harris in~\cite{kaCM, hasv, ha86}).
What about analytic continuation of these differential operators to the whole Shimura variety, as well as applications to higher-dimensional Galois representations? While much of our formulation is for reductive groups, specializing to symplectic and unitary groups $G$ yields Theorems~\ref{continuationB} and~\ref{charpcor} below.
Our proofs are entirely intrinsic to the geometry of the underlying Shimura varieties $\Sh$. This allows us to overcome what might appear---from the setup for modular forms in Section~\ref{GL2-intro}---to be an obstruction to working with more general groups: We do not necessarily have $q$-expansions in our setting, thus depriving us of analogues of Equation~\eqref{qformula}. By exploiting geometry, however, we obtain clean, intrinsic descriptions without reference to $q$-expansions (and without reference to Serre--Tate or Fourier--Jacobi expansions, which might at first appear to be reasonable substitutes but turn out not to be similarly suitable for studying the action of the Hecke algebras).
\begin{customthm}{A}[Analytic continuation of mod $p$ differential operators]\label{continuationB}
In the symplectic and unitary cases, the differential operators $\Diff_{\kappa}^\lambda$, defined \emph{a priori} only over the ordinary locus, can be analytically continued to the whole mod $p$ Shimura variety to give a differential operator $\Theta_\kappa^\lambda$ on mod $p$ automorphic forms.
These operators specialize in the case of $\operatorname{GSp}_2= \mathrm{GL}_2$ to Katz's operator $\theta$ described in Equation~\eqref{qformula}.
\end{customthm}
With these operators, we can reformulate Equations~\eqref{commrel-equ} and~\eqref{twist-equ} to obtain Theorem~\ref{charpcor}. (Below, $\hat{\nu}$ denotes the cocharacter dual to the similitude character $\nu$ of $G$, and $\underline{k}$ denotes the parallel weight with entries all equal to $k\in\mathbb{Z}$.)
\begin{customthm}{B}[Action of differential operators on mod $p$ Galois representations]\label{charpcor}
Let $G$ be a symplectic or unitary group over $\mathbb{Q}$, split over a number field $F$.
Let $f$ be a mod $p$ Hecke eigenform on $G$ of weight $\kappa$, and let $\lambda$ be a weight that is admissible (as in Definition~\ref{admissible-defi}). Assume $\Theta^\lambda(f):=\Theta_{\kappa}^\lambda (f)$ is nonzero.
Then $\Theta^\lambda (f)$ is a Hecke eigenform of weight $\kappa+\lambda+\underline{(p-1)|\lambda/2|}$.
Furthermore, for $\rho\colon\Gal\left(\Fbar/F\right)\to\hat{G}\left(\overline{\FF}_p\right)$ a continuous representation, the Frobenius eigenvalues of $\rho$ agree with the Hecke eigenvalues of $f$ (as defined in Conjecture~\ref{conj:galois}) if and only if the Frobenius eigenvalues of
$(\hat{\nu}^{|\lambda|/2}\circ \chi)\otimes \rho$
agree with the Hecke eigenvalues of $\Theta^\lambda(f)$.
\end{customthm}
\begin{rmk}
The reader who prefers to work with representations of $\Gal\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)$ instead of $\Gal\left(\Fbar/F\right)$ can do so by replacing the split version of the Satake isomorphism described in Section~\ref{AG-bkgd} (where we follow~\cite{Gross-satake}) with the non-split version from~\cite[Section 7]{treumannvenkatesh}.
The Galois representations are then of the form $\Gal\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\to\L{G}\left(\overline{\FF}_p\right)$, taking values in the Langlands dual group of $G$.
\end{rmk}
\begin{rmk}
Groups of low rank are often amenable to explicit analysis. For Picard modular forms, de Shalit and Goren are currently studying related aspects of mod $p$ Galois representations, such as \emph{$\theta$-cycles} in the \emph{$\mu$-ordinary} setting, as analogues of $\theta$ are constructed in~\cite{DSG, DSG2, EiMa}. (In~\cite{DSG2}, via a different approach from ours, they also obtain results on analytic continuation for automorphic forms over arbitrary rank unitary groups, if one restricts to working over a quadratic imaginary field and with scalar weights.)
For Hilbert modular forms, Andreatta and Goren obtained several results on theta cycles in~\cite[Section 16]{AndreattaGoren}, building on earlier work of Jochnowitz~\cite{jochnowitz}. Yamauchi has done related work for $\operatorname{GSp}_4$~\cite{Yama}. \end{rmk}
\begin{rmk}
Katz discusses the \emph{exact filtration} of a $\bmod p$ modular form $f$~\cite[Section I]{Katz-theta}: the smallest $k$ such that $f$ is not divisible by the Hasse invariant, or equivalently, the smallest $k$ such that there is no modular form of weight $k'<k$ whose $q$-expansion at some cusp agrees with the $q$-expansion of $f$. The notion of exact filtration is related to questions about the kernel of the differential operator (times the Hasse invariant) and to the proof of the weight part of Serre's conjecture, as described in, e.g.~\cite[Section 3]{Edixhoven}. The statements and proofs concerning weight filtrations and that kernel in~\cite[Theorem and Some Corollaries]{Katz-theta} extend naturally to the mod $p$ automorphic forms of scalar weights in our setting.
\end{rmk}
\subsection{Obstructions to analytic continuation in characteristic $0$}
Since the representation~\eqref{char0repn} holds in characteristic $0$, it is natural to ask about the possibility of lifting Theorems~\ref{continuationB} and~\ref{charpcor} from the mod $p$ setting to the $p$-adic setting.
As an intermediate step, one might try to lift to characteristic $p^m$ for $m>1$ an integer.
Chen and Kiming have accomplished this for mod $p^m$ modular forms $f$, and they have proved that if $f$ has weight filtration $k$, then $\theta f$ has weight filtration $k+2+2p^{m-1}(p-1)$~\cite[Theorem 1]{chen-kiming}. This shows that, for modular forms over $\mathbb{Z}_p$, there is no way to analytically continue the operator $\theta$ from the ordinary locus to the entire Shimura variety over $\mathbb{Z}_p$. Indeed, if it were possible to analytically continue $\theta$, then the weight filtration of $\theta f$ mod $p^m$ would be bounded above by the weight of the characteristic $0$ form $\theta f$; but as Chen and Kiming have shown, the weight filtration of $\theta f$ mod $p^m$ is unbounded as $m$ goes to infinity.
Observing this obstruction to analytically continuing the entire form in characteristic $0$, one might try just to analytically continue the \emph{ordinary} part of $\Diff_{\kappa}^\lambda f$. More precisely, let $\mathbf{e}$ denote Hida's ordinary projector (defined in~\cite[Sections 6--8]{H02}), which acts on the space of $p$-adic automorphic forms.
A $p$-adic automorphic form is \emph{ordinary} if it is in the image of $\mathbf{e}$. Every ordinary $p$-adic automorphic form of sufficiently regular
algebraic weight is, in fact, an ordinary classical automorphic form defined over the whole Shimura variety $\Sh$, by~\cite[Theorems 6.8(4) and 7.1(4)]{H02}. So even though $\Diff_{\kappa}^\lambda f$ cannot be extended to all of $\Sh$, we at least have that $\mathbf{e}\Diff_{\kappa}^\lambda f$ extends to all of $\Sh$.
As we shall see in Corollary~\ref{ed0}, though, the proof of Theorem~\ref{charpcor} (an analysis of the interaction between Hecke operators and differential operators) also shows that $\mathbf{e}\Diff_{\kappa}^\lambda = 0$. So the ordinary part of $\Diff_{\kappa}^\lambda f$ is $0$. In the case of modular forms in characteristic $0$, Coleman, Gouv\^ea, and Jochnowitz also showed in~\cite[Corollary 10]{CGJ} that $\theta$ destroys overconvergence, due to a relationship with the weight $2$ Eisenstein series $E_2$ (a relationship that actually enables the mod $p^m$ liftings in~\cite{chen-kiming}).
\subsection{Organization}
Section~\ref{prelim_sec} introduces conventions and background, divided into two parts. The first focuses on Hecke algebras and Galois representations.
The second provides background on Shimura varieties and automorphic forms (including Hasse invariants, a key ingredient in the analytic continuation of differential operators).
Section~\ref{analytic_sec} establishes the analytic continuation of the modulo $p$ reduction of $p$-adic differential operators. Those operators were previously constructed over the ordinary locus in~\cite{EDiffOps, EFMV}. The key work in this section lies in producing another construction of differential operators in characteristic $p$ that is defined over the whole Shimura variety and then showing that it agrees with the mod $p$ reduction of the $p$-adic differential operators we originally constructed. Theorem~\ref{continuationB} is a consequence of the main result of this section, Theorem~\ref{ana_thm}, which demonstrates the role of Hasse invariants in analytically continuing differential operators, in analogy with Katz's approach in~\cite{Katz-theta}.
Section~\ref{Hecke_sec} describes the interaction between differential operators and Hecke operators. Our proofs rely entirely on intrinsic properties of Shimura varieties and do not require $q$-expansions.
Theorem~\ref{TandTheta} shows that the differential operators commute with prime-to-$p$ Hecke operators up to scalar multiples, in analogy with Equation~\eqref{commrel-equ}, a key step toward describing the effect on Galois representations.
Section~\ref{galois-section} establishes an axiomatic result, Theorem~\ref{axiomaticthm}, on Galois representations and associated Hecke eigenvalues for general reductive groups.
Theorem~\ref{charpcor} is a consequence of Theorem~\ref{axiomaticthm}, as the relevant conditions hold by Theorem~\ref{TandTheta}.
\subsection{Acknowledgements}
The third and fifth named authors thank Dick Gross, Arun Ram, Olav Richter, and Martin Weissman for helpful suggestions. The first named author thanks Caltech for hospitality during a visit to work on this project.
\section{Introduction}
\input{introductionNEW.tex}
\section{Background and conventions}\label{prelim_sec}
This section introduces Galois representations (Section~\ref{AG-bkgd}) and associated automorphic forms over Shimura varieties (Section~\ref{AF-bkgd}).
\subsection{Hecke algebras and Galois representations}\label{AG-bkgd}
\input{prelimalgebraicgpsSPLIT.tex}
\input{prelimgaloisSPLIT.tex}
\subsection{Shimura varieties and automorphic forms}\label{AF-bkgd}
\input{prelim.tex}
\section{Analytic continuation of differential operators}\label{analyticcontinuation-section}
\input{AnalyticContinuation.tex}
\section{Commutation relations with Hecke operators}\label{Hecke_sec}
\input{Hecke2.tex}
\section{Applications to Galois representations}\label{galois-section}
\input{galoisSPLIT2.tex}
\subsubsection{Dual group}
Let $G$ be a connected reductive algebraic group over a field $F$.
\textbf{We assume that $G$ is split over $F$}, that is any maximal torus of $G$ is isomorphic over $F$ to a product of copies of $\mathbb{G}_m$.
We fix a maximal torus $T$ contained in a Borel subgroup $B$ of $G$, all defined over $F$.
The Weyl group of $T$ is $W=N_G(T)/T$.
Let $X^\bullet=X^\bullet(T)=\mathrm{Hom}\left(T_{\overline{F}},\mathbb{G}_m\right)$ be the character lattice and $X_\bullet=X_\bullet(T)=\mathrm{Hom}\left(\mathbb{G}_m, T_{\overline{F}}\right)$ be the cocharacter lattice.
There is a natural pairing
\begin{equation*}
\langle,\rangle\colon X^\bullet\timesX_\bullet\to\mathrm{Hom}(\mathbb{G}_m,\mathbb{G}_m)\cong\mathbb{Z}
\end{equation*}
given by $\langle\chi,\mu\rangle=\chi\circ\mu$.
Given a representation of $G$, its restriction to $T$ is a direct sum of characters (the \emph{weights} of the representation).
The \emph{roots} of $G$ are the nontrivial weights of the adjoint representation $\mathrm{Ad}\colon G \rightarrow \operatorname{Aut}(\mathrm{Lie}(G))$.
If one instead considers the adjoint action on $\mathrm{Lie}(B)$, one obtains the \emph{positive roots}.
The set $\Delta^\bullet$ of simple roots consists of those positive roots that cannot be written as the sum of other positive roots.
The coroot basis $\Delta_\bullet$ is the set of simple coroots.
The \emph{root datum} of $G$ is the quadruple $\Psi(G)=(X^\bullet,\Delta^\bullet,X_\bullet,\Delta_\bullet)$.
We consider $\hat{G}$, the dual group of $G$, viewed as a reductive group over $k$, a field to be specified later.
(In the automorphic literature, $k$ is often taken to be $\mathbb{C}$, but our main focus will be on $\overline{\FF}_p$.
For the existence of $\hat{G}$ over a general base, see~\cite[Theorem 6.1.16]{conrad-reductive}.)
After fixing $\hat{T}\subset\hat{B}\subset\hat{G}$, there is an identification $X^\bullet(\hat{T})\cong X_\bullet(T)$ that maps the positive roots with respect to $\hat{B}$ to the positive coroots with respect to $B$.
The root datum of $\hat{G}$ is $\Psi(G)^\vee=(X_\bullet,\Delta_\bullet,X^\bullet,\Delta^\bullet)$.
Under the duality between $G$ and $\hat{G}$, a character $\chi\colon T\to\mathbb{G}_m$ in $X^\bullet(T)$ corresponds to a cocharacter $\hat{\chi}\colon\mathbb{G}_m\to\hat{T}$ in $X_\bullet(\hat{T})=X^\bullet(T)$.
Similarly, the character of $\hat{T}$ corresponding to $\mu\inX_\bullet(T)$ is denoted $\hat{\mu}$.
Note that, given $\chi\inX^\bullet(T)$ and $\mu\inX_\bullet(T)$, the two morphisms
\[\chi\circ\mu\colon\mathbb{G}_m\to T\to\mathbb{G}_m \quad\text{and}\quad \hat{\mu}\circ\hat{\chi}\colon\mathbb{G}_m\to\hat{T}\to\mathbb{G}_m \]
agree, i.e. $\chi\circ\mu=\hat{\mu}\circ\hat{\chi}$.
Let
\begin{equation*}
P^+ := \{\chi\inX_\bullet\mid\langle\alpha,\chi\rangle\geq 0\text{ for all }\alpha\in\Delta^\bullet\}.
\end{equation*}
The elements of $P^+$ are called
the \emph{dominant weights} of (the maximal torus $\hat{T}$ of) $\hat{G}$.
There is a partial ordering $\geq$ on $P^+$, where $\lambda\geq\mu$ if
\begin{equation*}
\lambda-\mu=\sum_{\alpha^\vee\in\Delta_\bullet} n_{\alpha^\vee} \alpha^\vee
\quad\text{with }n_{\alpha^\vee}\in\mathbb{Z}_{\geq 0}.
\end{equation*}
By Chevalley's theorem~{\cite[Corollary 2.7 of Part II]{Jantzen}},
the set $P^+$ is in bijection with the set of irreducible finite dimensional $\hat{G}$-modules over $k$. For any $\lambda\in P^+$, we denote by $(\rho_\lambda,V_\lambda)$ the irreducible $\hat{G}$-module of highest weight $\lambda$.
We have an identification~\cite[Section 1]{Gross-satake}
\begin{equation*}
R(\hat{G}) \cong \mathbb{Z}[X^\bullet(\hat{T})]^W \cong \mathbb{Z}[X_\bullet(T)]^W.
\end{equation*}
Using this we can view the elements of the representation ring as sums of characters of $\hat{T}$ or cocharacters of $T$.
Given $\lambda \in P^+$, let $\chi_\lambda := \operatorname{Tr}(\rho_\lambda) \in \mathbb{Z}[X^\bullet(\hat{T})]$ denote the character of the irreducible representation $\rho_\lambda\colon \hat{G}\to\mathrm{GL}(V_\lambda)$ of highest weight $\lambda$.
Using the above, we can express
\begin{equation*}
\label{eq:char_sum}
\chi_\lambda = \sum_{\mu \leq \lambda} (\dim V_\lambda(\mu)) \hat{\mu},
\end{equation*}
where $\hat{\mu} \inX^\bullet(\hat{T})$ is the character corresponding to $\mu \inX_\bullet(T)$ by duality, and $V_\lambda(\mu)$ is the weight space of weight $\mu$ in $V_\lambda$.
\subsubsection{Local Hecke algebra and Satake isomorphism}\label{Hecke-Algs-prelim}
We now take $G$ to be a connected split reductive group over a local field $F_v$ with ring of integers $\mathscr{O}_v$.
We choose a uniformizer $\pi_v$ of $\mathscr{O}_v$, and we let $q_v$ denote the cardinality of the residue field $\mathscr{O}_v/\pi_v\mathscr{O}_v$.
As we assume $G$ to be split, there exists a group scheme $\mathcal{G}$ over $\mathscr{O}_v$ whose generic fiber is $G$ and whose special fiber is reductive.
We set $G_v=\mathcal{G}(F_v)$ and $K_v=\mathcal{G}(\mathscr{O}_v)$, then $K_v$ is a hyperspecial maximal compact subgroup of the locally compact group $G_v$.
Given a commutative ring $R$, the $R$-valued Hecke algebra of the pair $(G_v, K_v)$ is
\begin{equation*}
\mathcal{H}(G_v, K_v; R)
=\left\{h\colon K_v\backslash G_v/K_v\longrightarrow R \left|\!\begin{array}{c}h\text{ locally constant,}\\
\text{compactly supported}\end{array}\!\right. \right\}
\end{equation*}
with multiplication given by
\begin{equation*}
(h_1 * h_2) (K_v g K_v)
=\sum_{xK_v\in G_v/K_v} h_1(K_v x K_v) h_2(K_v x^{-1}g K_v).
\end{equation*}
We work with the basis of $\mathcal{H}(G_v, K_v; R)$ consisting of the characteristic functions
\begin{equation*}
c_\lambda = \mathrm{char}\left(K_v\lambda(\pi_v)K_v\right)\qquad
\text{for }\lambda\in P^+.
\end{equation*}
The Satake transform is a ring isomorphism
\begin{equation*}
\mathcal{S}_v\colon \mathcal{H}\left(G_v,K_v;\ZZ\left[q_v^{\pm 1/2}\right]\right)
\longrightarrow \ZZ\left[q_v^{\pm 1/2}\right]\left[X^\bullet(\hat{T})\right]^W=R\left(\hat{G}\right)\otimes\ZZ\left[q_v^{\pm 1/2}\right].
\end{equation*}
If $\lambda\in P^+$ then the image of the basis element $c_\lambda$ can be written\footnote{The element $\rho$ is the half-sum of the positive roots of $G$, but we will not need to know this, only that it is the same in all the identities related to the Satake isomorphism $\mathcal{S}_v$.}
\begin{equation}
\label{eq:satake}
\mathcal{S}_v(c_\lambda)=
\sum_{\mu\leq\lambda} b_\lambda(\mu)q_v^{\langle\rho,\mu\rangle}\chi_\mu,
\end{equation}
where $\mu$ runs over the elements in $P^+$ such that $\mu\leq\lambda$, $b_\lambda(\mu)\in\mathbb{Z}$ and $b_\lambda(\lambda)=1$.
There is a similar identity expressing $\chi_\lambda$ in terms of images of characteristic functions:
\begin{equation*}
\chi_\lambda = q_v^{-\langle \rho,\lambda\rangle}
\sum_{\mu\leq\lambda} d_\lambda(\mu)\mathcal{S}_v(c_\mu),
\end{equation*}
where $d_\lambda(\mu)\in\mathbb{Z}$ and $d_\lambda(\lambda)=1$.
For future use, let us record a consequence of this identity:
\begin{equation}
\label{eq:satake_inv}
\mathcal{S}_v^{-1}(\chi_\lambda) = q_v^{-\langle \rho,\lambda\rangle}
\sum_{\mu\leq\lambda} d_\lambda(\mu)c_\mu.
\end{equation}
In most of the paper we work with $\overline{\FF}_p$-coefficients, for a fixed prime $p$.
If $v$ does not divide $p$, then, after making a choice of square root\footnote{For the subtleties of this choice of square root, look for local and global pseudoroots in~\cite[Section 7]{treumannvenkatesh}.} of $q_v$ in $\overline{\FF}_p$, we get a mod $p$ version of the Satake transform by tensoring with $\overline{\FF}_p$:
\begin{equation*}
\mathcal{S}_v\colon\mathcal{H}(G_v,K_v;\overline{\FF}_p)\longrightarrow \overline{\FF}_p[X^\bullet(\hat{T})]^W=R(\hat{G})\otimes \overline{\FF}_p.
\end{equation*}
Equation~\eqref{eq:satake_inv} continues to hold in this setting, with $d_\lambda(\mu)\in\mathbb{F}_p$ and $d_\lambda(\lambda)=1$.
\subsubsection{Galois representations associated to automorphic forms mod $p$}\label{Galois-bkgd}
We summarize the (conjectural) correspondence between Galois representations and automorphic representations following~\cite{buzzgee14}.
Let $G$ be a connected reductive group over $\mathbb{Q}$ and let $F$ be the splitting field of $G$.
Fix a prime $p$.
Given a level $K=\prod_v K_v \subset G(\AA_f)$, we say that a finite place $v$ of $F$ is \emph{bad} (with respect to $p$ and $K$) if $v$ lies above $p$ or if $K_v$ is \emph{not} a hyperspecial maximal compact subgroup of $G_v=G(F_v)$.
Otherwise we say that $v$ is \emph{good}, in which case we are in the situation described in Section~\ref{Hecke-Algs-prelim}.
All but finitely many places are good with respect to $p$ and $K$.
Let $f$ be a Hecke eigenform mod $p$ of level $K$ on $G$, that is a class in the cohomology with $\overline{\FF}_p$-coefficients of the locally symmetric space defined by $G$ and $K$ (see~\cite[Section 5]{treumannvenkatesh} for details).
(An important source of such $f$ are the automorphic forms defined on Shimura varieties, as seen in Section~\ref{sect:automorphic}.)
For every good place $v$, $f$ defines a character of the local Hecke algebra $\mathcal{H}(G_v, K_v;\overline{\FF}_p)$, namely the \emph{Hecke eigensystem}
\begin{equation*}
\Psi_{f,v}\colon \mathcal{H}(G_v,K_v;\overline{\FF}_p)\to \overline{\FF}_p
\end{equation*}
which takes an operator $T$ to its eigenvalue:
\begin{equation*}
Tf = \Psi_{f,v}(T)f.
\end{equation*}
Using the Satake isomorphism $\mathcal{S}_v$ from Section~\ref{Hecke-Algs-prelim}, we can define a character $\omega_f\colon R(\hat{G})\otimes \overline{\FF}_p\to \overline{\FF}_p$ by
\begin{equation*}
\omega_f(\chi_\lambda)=\Psi_{f,v}\left(\mathcal{S}_v^{-1}(\chi_\lambda)\right).
\end{equation*}
The characters of the representation ring $R(\hat{G}) \otimes \overline{\FF}_p$ are indexed by the semi-simple conjugacy classes in $\hat{G}(\overline{\FF}_p)$.
Given such a class $s$, the corresponding character $\omega_s$ is determined by
\begin{equation*}
\omega_s(\chi_\lambda)=\chi_\lambda(s).
\end{equation*}
In particular, the character $\omega_f$ corresponding to $f$ is indexed by some $s_{f,v}\in\hat{G}(\overline{\FF}_p)$, called the \emph{$v$-Satake parameter} of $f$.
We are ready to state the expected relation between Hecke eigenforms and Galois representations:
\begin{conj}[Positive characteristic form of{~\cite[Conjecture 5.17]{buzzgee14}}]\label{conj:galois}
Let $f$ be a mod $p$ Hecke eigenform of level $K$ on a connected reductive group $G$ over $\mathbb{Q}$, split over a number field $F$.
There exists a continuous representation
\begin{equation*}
\rho\colon \Gal\left(\Fbar/F\right)\longrightarrow\hat{G}(\overline{\FF}_p)
\end{equation*}
that is unramified outside the finite set $\Sigma$ of places that are bad with respect to $p$ and the level
$K$.
If $v\notin\Sigma$, then $\rho(\operatorname{Frob}_v)=s_{f,v}$, the $v$-Satake parameter of $f$.
\end{conj}
As noted in~\cite[Remark 5.19]{buzzgee14}, the representation $\rho$ need not be unique up to conjugation.
Given a Hecke eigenform $f$, we will denote by $R(f)$ the set of Galois representations attached to $f$ as in Conjecture~\ref{conj:galois}.
\begin{rmk}
The reader looking to reconcile the statement of Conjecture~\ref{conj:galois} with that of~\cite[Conjecture 5.17]{buzzgee14} will note the following differences:
\begin{itemize}
\item We work with Galois representations mod $p$ rather than $p$-adic.
\item Our assumption that $G$ is split over $F$ means that the Galois representations land in the dual group $\hat{G}$ rather than the $L$-group $\L{G}$.
\item We relate the Hecke eigensystem of $f$ with the images of Frobenii by using Satake parameters (as defined above) rather than by comparing representations of Weil groups.
\item Since Buzzard and Gee work in the more general setting of automorphic representations, they need to impose the condition of $L$-algebraicity, which is automatically satisfied in the setting of our automorphic forms.
\end{itemize}
\end{rmk}
For the current state of the art regarding Conjecture~\ref{conj:galois}, we have great achievements by Scholze in~\cite{scholze} for the case $G = \mathrm{GL}_n$.
(The conjectural correspondence in this setting was first formulated by Ash in~\cite{ash-galois}.)
Scholze's work gives the following.
\begin{thm}[{\cite[Theorem I.3]{scholze}}]\label{modpL}
If $F$ is a CM field that contains an imaginary quadratic field (or more generally, assuming the statements in~\cite{arthur}, if $F$ is totally real or CM), then for any system of Hecke eigenvalues occurring in the singular cohomology $H^i\left(X_K, \mathbb{F}_p\right)$ of the locally symmetric space $X_K$ attached to $\mathrm{GL}_n$ over $F$, there is a continuous semisimple representation $\Gal\left(\Fbar/F\right)\to\mathrm{GL}_n(\overline{\FF}_p)$ whose Frobenius eigenvalues agree with that system of Hecke eigenvalues.
\end{thm}
\subsubsection{Shimura varieties}
In this paper, we consider PEL-type Shimura varieties of either unitary (A) or symplectic (C) type. Here,
we briefly introduce the Shimura datum $\left(D, *, V, \langle,\rangle, h\right)$ of PEL-type needed for our work later in the paper.
We refer to~\cite{kottwitz} for a more detailed treatment.
Let $D$ be a finite-dimensional simple $\mathbb{Q}$-algebra with center $F$, let $*$ be a positive involution on $D$ over $\mathbb{Q}$, and let $F_0$ be the fixed field of $*$ on $F$.
Since $*$ is positive on $F$, its fixed field $F_0$ is totally real. We say that $*$ is of the first kind if $F=F_0$, and of the second kind if $F$ is a quadratic imaginary extension of $F_0$ (in which case $F$ is a CM field).
In the following, we denote by $\mathcal{T}_{F_0}$ (resp. $\mathcal{T}_F$) the set of embeddings $\tau\colon F_0\to \mathbb{R}$ (resp. $\tau\colon F\to \mathbb{C}$). If $*$ is of the second kind (case (A)), then for each $\tau\in\mathcal{T}_{F_0}$, we fix an extension $F\to \mathbb{C}$ in $\mathcal{T}_F$, which by abuse of notation, we still denote by $\tau$, and we write $\tau^*$ for the other embedding $F\to \mathbb{C}$ that restricts to $\tau$ on $F_0$. We shall also fix a choice $\Sigma_F$ of CM type, i.e.\ a set consisting of exactly one of $\tau, \tau^\ast$ for each $\tau\in \mathcal{T}_F$.
Let $V$ be a non-zero finitely-generated left $D$-module, and let $\langle ,\rangle$ be a non-degenerate $\mathbb{Q}$-valued alternating form on $V$ such that $\langle bv,w\rangle=\langle v,b^*w\rangle$, for all $v,w\in V$ and all $b\in D$.
Let $C$ be the $\mathbb{Q}$-algebra ${\rm End}_D(V)$; it is a simple algebra with center $F$, and has an involution $*$ coming from the form $\langle,\rangle$.
Let $h\colon\mathbb{C}\to C_\mathbb{R}$ be a $*$-homomorphism such that the symmetric real-valued bilinear form $\langle \cdot,h(i)\cdot\rangle$ on $V_\mathbb{R}$ is positive-definite.
Associated with the above data, we define the algebraic group $G$ over $\mathbb{Q}$ whose points on a $\mathbb{Q}$-algebra $R$ are given by
\[G(R):=\{x\in C\otimes_\mathbb{Q} R\mid xx^*\in R^\times\}.\]
We denote by $\nu\colon G\to \mathbb{G}_m$ the morphism $x\to xx^*$, called the \emph{similitude factor}, by $G_1$ its kernel, and by $\hat{\nu}\colon\mathbb{G}_m\to \hat{G}$ the cocharacter of $\hat{G}$ corresponding to $\nu$ under duality. (Note that for any integer $m\geq 1$, the cocharacter of $\hat{G}$ dual to $\nu^m$ is $\hat{\nu}^m$.)
Then $G_1$ is obtained from an algebraic group $G_0$ over $F_0$ by restriction of scalars from $F_0$ to $\mathbb{Q}$. If $*$ is of the second kind---case (A)---then $G_0$ is an inner form of a quasi-split unitary group over $F_0$. If $*$ is of the first kind, then over an algebraic closure of $F_0$, the group $G_0$ is either orthogonal---case (D)---or symplectic---case (C).
\textbf{Going forward, we assume we are in case (A) or (C).}
The endomorphism $h_\mathbb{C}=h\times_\mathbb{R} \mathbb{C}$ of $V_\mathbb{C}=V_\mathbb{R}\otimes \mathbb{C}=V\otimes_\mathbb{Q} \mathbb{C}$ gives rise to a decomposition $V_\mathbb{C}=V_1\oplus V_2$, where for all $z\in \mathbb{C}$,
$(h(z),1)=h(z)\times 1$
acts by $z$ on $V_1$ and by $\bar{z}$ on $V_2$. The \emph{reflex field} $E$ of the Shimura datum $\left(D, *, V, \langle,\rangle, h\right)$ is the field of definition of the $G(\mathbb{C})$-conjugacy class of $V_1$.
Let $p$ be a rational prime. We write $\mathbb{Z}_{(p)}$ for the localization of $\mathbb{Z}$ at $p$. We choose our data so that the following conditions are met:
\begin{enumerate}
\item The prime $p$ is unramified in $F$.
\item The algebra $D$ is split at $p$, i.e. $D_{\mathbb{Q}_p}$ is a product of matrix algebras over (unramified) extensions of $\mathbb{Q}_p$.
\item There exists a $\mathbb{Z}_{(p)}$-order $\mathcal{O}_D$ in $D$ that is preserved by $*$ and whose $p$-adic completion is a maximal order in $D_{\mathbb{Q}_p}$.
\item There exists a $\mathbb{Z}_p$-lattice $L$ in $V_{\mathbb{Q}_p}$ that is self-dual for $\langle,\rangle$ and preserved by $\mathcal{O}_D$.
\end{enumerate}
We also specify a \emph{level} $K$, an open compact subgroup of $G(\mathbb{A}_f)$, where $\mathbb{A}_f$ denotes the ring of finite adeles of $\mathbb{Q}$.
We further assume that $K$ is neat (as defined in~\cite[Definition 1.4.1.8]{la}) and that it decomposes as $K=K^p K_p$, where $K^p\subset G\left(\mathbb{A}_f^{(p)}\right)$ and $K_p\subset G(\mathbb{Q}_p)$ is hyperspecial, namely $K_p$ is the stabilizer of $L$ in $V_{\mathbb{Q}_p}$.
We define $\mathcal{X}=\mathcal{X}_K$ to be the PEL-type moduli space of abelian varieties of level $K$, associated with the datum
$(D,*,V,\langle,\rangle,h)$.
Under our assumptions, $\mathcal{X}_K$ extends canonically to a smooth quasi-projective scheme over $\mathcal{O}_E\otimes \mathbb{Z}_{(p)}$, which by abuse of notation we still denote by $\mathcal{X}$.
The set $\mathcal{X}({\mathbb C})$ is a (finite) disjoint union of Shimura varieties obtained from the data $(G,h, K)$. In the following, by abuse of language, we refer to $\mathcal{X}$ as the PEL-type Shimura variety of level $K$.
Given an abelian variety $A$,
we denote by $A^t$ its dual (and, likewise, in the context of sheaves, we also use a superscript $t$ to denote the dual).
\subsubsection{Simplifying conditions}
\textbf{We assume $p$ is totally split in the reflex field $E$.} Then, by~\cite{wedhorn}, the ordinary locus of the modulo $p$ reduction $X$ of the Shimura variety $\mathcal{X}$ is non-empty. We fix a prime $\mathfrak{p}$ of $E$ above $p$, and we regard $\mathcal{X}=\mathcal{X}_K$ as a scheme over $\mathbb{W}=\mathbb{Z}_p=\mathcal{O}_{E,\mathfrak{p}}$.
For convenience, we further \textbf{assume that $D=F$ and that $p$ splits completely in $F$}. These two assumptions are not necessary, but they allow us to simplify notation.
(By assumption (2) above, Morita equivalence reduces the general case of $D\not=F$ to that of $D=F$; we refer to~\cite[Section 6]{EiMa} for an explanation of how the case of $p$ totally split in $E$, but not in $F$, relates to the case of $p$ totally split in $F$.)
\subsubsection{Signatures and Levi subgroups}\label{signandlevi_sec}
The decomposition $F\otimes_\mathbb{Q}\mathbb{C}=\oplus_{\tau\in\mathcal{T}_{F}} \mathbb{C}$
induces a decomposition $V_{1}=\bigoplus_{\tau\in \mathcal{T}_{F}} V_{1,\tau}$. The \emph{signature} of the Shimura datum is the collection of integers for all $\tau\in\mathcal{T}_{F}$
\[a_\tau:=\dim_\mathbb{C} V_{1,\tau}.\]
Let $n= \dim_F V$. In the unitary case (A), we have $a_\tau\oplus a_{\tau^*}=n$, for all $\tau\in\mathcal{T}_{F}$, and the tuple of pairs $(a_\tau,a_{\tau^*})$, for $\tau\in\Sigma_F$, is the signature of the unitary group $G_0/F_0$.
In the symplectic case (C), we have $a_\tau=n$, for all $\tau\in\mathcal{T}_{F_0}$.
(For $F_0=\mathbb{Q}$, we refer to case (C) as the \emph{Siegel case}, and to case (A) as the \emph{Hermitian case} if $a_\tau=a_\tau^*$.)
Starting from the signature of the Shimura datum, we define an algebraic group $H$ over $\mathbb{Z}$ as
\[
H:=\prod_{\tau\in \mathcal{T}_F} \mathrm{GL}_{a_\tau}
=\begin{cases}
\displaystyle\prod_{\tau\in\mathcal{T}_{F_0} }\left( \mathrm{GL}_{a_\tau}\times \mathrm{GL}_{a_{\tau^*}}\right)\subseteq \prod_{\tau\in\mathcal{T}_{F_0}} \mathrm{GL}_n & \text{in the unitary case (A);}\\
\displaystyle\prod_{\tau\in\mathcal{T}_{F_0}} \mathrm{GL}_n & \text{in the symplectic case (C).}
\end{cases}
\]
Note that $H(\mathbb{C})$ can be identified with the Levi subgroup of $G_1(\mathbb{C})$ that preserves the decomposition $V_\mathbb{C}=V_1\oplus V_2$.
We denote by $T\subseteq B$ the diagonal maximal torus inside the Borel subgroup of upper triangular matrices of $H$, and by $N$ the unipotent subgroup of $B$.
\subsubsection{Automorphic forms}\label{sect:automorphic} We recall the construction of automorphic sheaves over $\mathcal{X}$. We refer to~\cite[Section 3.2]{CEFMV} for details.
For convenience, we fix an identification of $\mathbb{C} $ with $\mathbb{C}_p$, compatible with the choice of the prime $\mathfrak{p}$ of $E$.
Given a characteristic $p$ field $\mathbb{F}$, we denote its Witt vectors by $W(\mathbb{F})$. We also set $\mathbb{W}:= W(\mathbb{Z}/p\mathbb{Z})$ and identify $\mathbb{W}$ with $\mathbb{Z}_p$. We identify $\mathcal{T}_{F_0}$ (resp. $\mathcal{T}_{F}$) with ${\rm Hom}\left(\mathcal{O}_{F_0},W(\overline{\mathbb F}_p)\right)$ (resp. ${\rm Hom}\left(\mathcal{O}_{F},W(\overline{\mathbb F}_p)\right)$).
Let $\pi\colon \CA\to \mathcal{X}$ denote the universal abelian scheme, of relative dimension $g=n[F_0\colon\mathbb{Q}]$ over $\mathcal{X}$,
and set $\omega_{\CA/\mathcal{X}}:=\pi_*\Omega^1_{\CA/\mathcal{X}}$; it is a locally free sheaf of rank $g$.
Set $\mathcal{T}:=\mathcal{T}_F$. The action of $\mathcal{O}_F$ on $\omega$ induces a decomposition of $\omega$ as
\[\omega=\bigoplus_{\tau\in\mathcal{T}}\omega_\tau\]
where for each $\tau\in\mathcal{T}$ the sheaf $\omega_\tau$ is locally free of rank $a_\tau$.
For any dominant weight $\kappa$ of $T$, the \emph{automorphic sheaf} $\omega^\kappa$ of weight $\kappa$ over $\mathcal{X}$
is defined as follows. Let ${\mathcal E}/\mathcal{X}$ be the sheaf
\[{\mathcal E}:=\bigoplus_{\tau\in\mathcal{T}}{\underline{\rm Isom}}_{\mathcal{O}_\mathcal{X}}\left(\mathcal{O}_\mathcal{X}^{a_\tau},\omega_{\tau}\right);\]
it admits a natural left action of $H$. Given an irreducible representation $(\rho_\kappa,V_\kappa)$ of $H$ of highest weight $\kappa$, we define $\omega^\kappa:={\mathcal E}\times^{\rho_\kappa} V_\kappa$ so that for each $\mathcal{O}_{E,\mathfrak{p}}$-algebra $R$, $\omega^\kappa(R):=(\mathcal{E}\times V_\kappa\otimes R)/(\ell, m)\equiv (g\ell, \rho_\kappa({}^t g^{-1})m)$.
An \emph{automorphic form} of weight $\kappa$ and level $K$, defined over an $\mathcal{O}_{E,\mathfrak{p}}$-algebra $R$, is a global section of the sheaf $\omega^\kappa$ on $\mathcal{X}_K\times_{\mathcal{O}_{E,\mathfrak{p}}} R$.
\subsubsection{Admissible weights} We now introduce the set of dominant weights by which Maass--Shimura differential operators (as well as the analogous differential operators studied in Section~\ref{analyticcontinuation-section}) can raise the weight of an automorphic form.
For convenience, we identify the set of dominant weights of (the maximal torus $T$ of) $H$ with the set
\[X(T)_+:=\Big\{(\kappa_{1,\tau},\ldots, \kappa_{a_\tau,\tau})\in\prod_{\tau\in\mathcal{T}_{F}}{\mathbb Z}^{a_\tau}\Big|\kappa_{i,\tau}\geq \kappa_{i+1,\tau} \text{ for all } i\Big\},\]
via the morphism $\prod_{\tau\in\mathcal{T}_{F}}\mathrm{diag}(t_{1,\tau},\dots t_{a_\tau,\tau})\mapsto \prod_{\tau\in\mathcal{T}_{F}}\prod_{1\leq i\leq a_\tau} t_{i,\tau}^{\kappa_{i,\tau}}$.
For $\kappa\in X(T)_+$, we write
\begin{align*}
|\kappa|:=\sum_{\tau\in\mathcal{T}_F}\sum_{1\leq i\leq a_\tau} \kappa_{i,\tau}.
\end{align*}
We call a dominant weight $\kappa\in X(T)_+$ \emph{positive} if $\kappa\not=0$ and $\kappa_{a_\tau,\tau}\geq 0$ for all $\tau\in\mathcal{T}$; we call $\kappa$ \emph{even} if $\kappa_{i,\tau}\equiv 0\mod 2$ for all $\tau\in \mathcal{T}_F$ and all $1\leq i\leq a_\tau$, and we call $\kappa$ \emph{sum-symmetric} if
$\sum_{1\leq i\leq a_\tau}\kappa_{i,\tau}=\sum_{1\leq i\leq a_{\tau^*}}\kappa_{i,\tau^*}$ for all $\tau\in \mathcal{T}_F$.
\begin{defi}\label{admissible-defi}
A dominant weight $\lambda$ is called \emph{admissible of depth $e_\lambda=e$}, for $e\in {\mathbb Z}_{>0}$, if the corresponding irreducible algebraic representation of $H$ occurs as a constituent of the representation
$(V^2)^{\otimes e}$ for
\[V^2:=\begin{cases}
\bigoplus_{\tau\in\mathcal{T}_{F_0}} {\rm Sym}^2 V_\tau & \text{in the symplectic case,}\\
\bigoplus_{\tau\in\mathcal{T}_{F_0}} (V_\tau\otimes V_{\tau^\ast}) & \text{in the unitary case.}
\end{cases}\]
\end{defi}
By definition, admissible weights are positive. Furthermore, a dominant weight $\lambda$ is \emph{admissible} in the symplectic case if and only if it is positive and even; and
in the unitary case, a dominant weight $\lambda$ is \emph{admissible}
if and only if it is positive and sum-symmetric.
Note that if $\lambda$ is admissible, then $|\lambda |$ is positive and even, and the Young symmetrizer \[y_\lambda\colon V^{\otimes |\lambda|}\to V_\lambda\] induces an epimorphism $(V^{ 2})^{\otimes |\lambda |/2}\to V_\lambda$, which by abuse of notation, we still denote by $y_\lambda$. Hence, in particular, $\lambda$ is admissible of depth $e_\lambda=|\lambda|/2$.
\begin{rmk}
We denote scalar weights by $(\underline{k_\tau})_{\tau\in \mathcal{T}_F}:=(k_\tau,\dots ,k_\tau)_{\tau\in\mathcal{T}_F}$ for integers $k_\tau$. Furthermore, if there exists an integer $k$ such that $k_\tau = k$ for all $\tau$, then we write $\underline{k}$ for $(\underline{k_\tau})_{\tau\in \mathcal{T}_F}$.
In the symplectic case, a scalar weight $(\underline{d_\tau})_{\tau\in\mathcal{T}_F}$ is admissible if and only if $d_\tau\geq 0$ and even, for all $\tau\in\mathcal{T}_F$. Hence, in the Siegel case, the admissible scalar weights are exactly the positive multiples of $\underline{2}$.
In the unitary case, a scalar weight $(\underline{d_\tau})_{\tau\in\mathcal{T}_F}$ is admissible if and only if $a_\taud_\tau=a_{\tau^*}d_{\tau^*}$, for all $\tau\in\mathcal{T}_F$.
Hence, in the Hermitian case, a scalar weight is admissible if and only if $d_{\tau}=d_{\tau^*}$, for all $\tau\in\mathcal{T}_F$.
\end{rmk}
\subsubsection{Hasse invariants and $p$-adic automorphic forms}\label{hasse_sec}
\newcommand{h}{h}
We recall the construction and properties of Hasse invariants relevant to our settings. Details are available, for various contexts, in~\cite{GorenHasse, conrad-hasse, GoldringNicole} and~\cite[Section 7]{AndreattaGoren}.
Let $\pi\colon \CA\to X$ denote the universal abelian scheme over the mod $p$ reduction $X$
of the Shimura variety $\mathcal{X}$.
Setting $\omega_{\CA/X}:=\pi_*\Omega^1_{\CA/X}$, we have the Hodge filtration of ${H^1_{\mathrm dR}(\Auniv/\Shp)}$ over $X$:
\[0\to\omega_{\CA/X}\to {H^1_{\mathrm dR}(\Auniv/\Shp)}\to {R^1\pi_*\cO}_\CA\to 0.\]
Let $\mathrm{F}\colonX\to X$ denote the absolute Frobenius on $X$;
we denote by
$\CA^{(p)}:=\CA\times_{X,\mathrm{F}} X$ the pullback of $\CA$ under $\mathrm{F}$, and by
$\mathrm{Fr}\colon\CA\to \CA^{(p)}$ the relative Frobenius of $\CA$.
The morphism $\mathrm{Fr}$ induces an ${\mathcal O}_X$-linear map \[\mathrm{Fr}^*\colon{R^1\pi_*\cO}_{\CA^{(p)}}={R^1\pi_*\cO}_\CA^{(p)} \to {R^1\pi_*\cO}_\CA;\]
we denote the dual map by
\begin{align*}
h\colon\omega_{\CA/X}\to \omega_{\CA^{(p)}/ X}=\omega_{\CA/X}^{(p)}.
\end{align*}
\begin{defi}
The \emph{Hasse invariant} is the automorphic form \[{E}:=\det h\in H^0\left(X, \vert \omega_{\CA/X}\vert ^{p-1}\right).\]
\end{defi}
By construction, the ordinary locus $S$ of $X$ agrees with the complement of vanishing locus of the Hasse invariant ${E}$.
Recall that a sufficiently large power of ${E}$ is known to lift to characteristic zero.
For any $m\geq 1$, set $\mathcal{X}_m:=\mathcal{X}\times_{\mathbb{W}} \mathbb{W}/p^m$,
and denote by ${\mathcal S}_m$ the locus of $\mathcal{X}_m$ where (a sufficiently large power of) the Hasse invariant ${E}$ is invertible.
Define ${\mathcal S}:=\varinjlim_{m}{\mathcal S}_m$ as a formal scheme over $\mathbb{W}$. The formal scheme ${\mathcal S}$ is the \emph{formal ordinary locus} over $\mathbb{W}$.
\begin{defi}
For any dominant weight $\kappa$, a \emph{$p$-adic automorphic form of weight $\kappa$} is a section of
\[H^0({\mathcal S},\omega^\kappa):=\varprojlim_{m} H^0({\mathcal S}_m,\omega^\kappa).\]
\end{defi}
|
{
"timestamp": "2019-03-01T02:09:56",
"yymm": "1902",
"arxiv_id": "1902.10911",
"language": "en",
"url": "https://arxiv.org/abs/1902.10911"
}
|
\section*{Summary}
Causal inference with observational longitudinal data and time-varying exposures is often complicated by time-dependent confounding and attrition. G-computation is one method used for estimating a causal effect when time-varying confounding is present. The parametric modeling approach typically used in practice relies on strong modeling assumptions for valid inference, and moreover depends on an assumption of missing at random, which is not appropriate when the missingness is non-ignorable or due to death. In this work we develop a flexible Bayesian semi-parametric G-computation approach for assessing the causal effect on the subpopulation that would survive irrespective of exposure, in a setting with non-ignorable dropout. The approach is to specify models for the observed data using Bayesian additive regression trees, and then use assumptions with embedded sensitivity parameters to identify and estimate the causal effect. The proposed approach is motivated by a longitudinal cohort study on cognition, health, and aging, and we apply our approach to study the effect of becoming a widow on memory.
\subsection*{Keywords}
Cognitive aging, Longitudinal data, Observational data, Nonignorable missing, Sensitivity analysis, Survivor Average Causal Effect.
\newpage
\section{Introduction}
Causal inference in non-randomized longitudinal studies with time-varying exposures is often complicated by time-dependent confounding and attrition. Attrition is inevitable especially if individuals in the studied population are older and followed over a long time period. Additionally, for cohort studies, an individual's data is only recorded if that person completes follow-up testing. Hence, data for not only the outcome but also exposure level and confounders are missing at subsequent test waves.
Several approaches have been proposed for estimating causal effects of time-varying exposures when time-varying confounding is present (\cite{robins1986new, robins1992g, robins2000marginal, van2012targeted}). The main advantage of these methods is that they handle feedback between the exposure and confounders as opposed to ordinary regression analysis (\cite{hernan2010causal}). The G-computation formula (\cite{robins1986new}) is one approach for estimating a causal effect in this setting. The approach is completely nonparametric in its original form, although a parametric modeling approach based on maximum likelihood estimation is most typically used in practice. Valid inference with the parametric G-formula requires correct model specification. This can be cumbersome when there is a large set of regressors, the relationship is non-linear and/or includes interaction terms, and there are multiple observation times.
Non- and semi-parametric estimation techniques that do not require prespecified distributional or functional forms of the data, have become popular in the causal inference literature (e.g. \cite{hill2011bayesian, haggstrom2017data, kim2017framework, karim2017application, wager2017estimation}). One such modeling strategy is Bayesian Additive Regression Trees (BART, \cite{chipman2010bart}). BART is a sum-of-trees model that adds together the predictions of a number of regression trees regularized by prior distributions. BART does not rely on strong modeling assumptions, and in contrast to other tree-based algorithms BART yields interval estimates for full posterior inference.
A number of methodologies have been applied to address missing response or missing covariate data in causal effect estimation of longitudinal data under an assumption of missing at random (MAR; \cite{chen2011doubly, robins1995analysis}). These methods, however, are generally invalid when the missingness is nonignorable or due to death. Joint models have been proposed to address the combination of dropout and truncation by death, where inference is conditioning on the sub-population being alive at a specific time-point (\cite{li2018accommodating, rizopoulos2012joint, shardell2018joint}). However, conditioning on survival may introduce bias due to the fact that survival is a post-randomization event. One estimand that has gained much attention to address this issue is the "survivors average causal effect" (SACE), i.e. the causal effect on the subpopulation of those surviving irrespective of exposure (\cite{frangakis02, frangakis07}). Several approaches have been developed for estimation of the SACE in longitudinal randomized control studies (e.g. \cite{lee2013causal, lee2010causal, wang2017inference, wang2017causal}). For observational data, \citeauthor{tchetgen2014identification} (\citeyear{tchetgen2014identification}) developed a weighting estimator to identify the SACE without missingness, and \citeauthor{shardell2014doubly} (\citeyear{shardell2014doubly}) identified the SACE with MAR missingness using also a weighting technique.
Moreover, \citeauthor{josefsson2016causal} (\citeyear{josefsson2016causal}) proposed assumptions to identify the SACE of a baseline exposure on a longitudinal outcome under an assumption of missing not at random (MNAR) for the outcome using parametric methods. These approaches however, do not appropriately account for MNAR data among survivors when the exposure and confounding are time-varying.
Increased knowledge on social isolation and cognitive health in widowhood may have important implications for public health programs aimed at healthy aging. Widowhood has been identified as an important social factor associated with increased mortality risk (\cite{haakansson2009association}), and widowhood has frequently been related to a higher dementia risk and cognitive impairment (\cite{aartsen2005does, mousavi12}). Here, our goal is to study the effect of widowhood on episodic memory. In particular, we develop a framework for assessing the impact of becoming a widow on memory by estimating the SACE in a setting with MNAR dropout among survivors. The proposed approach is motivated by the Betula study (\cite{nilsson97}), where individuals are followed over multiple test waves to study how cognitive functions potentially deteriorate with age and identify risk factors for dementia. The approach is to specify models for the observed data and then to use assumptions with embedded sensitivity parameters to identify and estimate the causal effect. We evaluate sensitivity of the results to untestable assumptions, and further compare our approach to other methods used for causal effect estimation of longitudinal data with time-varying confounding.
The remainder of the paper is organized as follows. In Section 2, we introduce the causal estimand, SACE, and the G-computation formula. In Section 3, we propose identifying assumptions and sensitivity parameters followed by the identification of the SACE in Section 4. In Section 5, we propose a Bayesian semi-parametric (BSP) modeling approach for the observed data distributions. In Section 6, we present the algorithm for estimation of the SACE. In Section 7, we implement our BSP approach on the Betula data and compare its performance to other standard methods. Finally, we conclude with a discussion and possible future work in Section 8.
\section{The G-formula}
\subsection{Data structure and notation}
We begin with a formal description of the data. Let $i = 1, 2, \ldots, N$ denote individual and $j=0, 1, \ldots,J$ denote time (the data used from the Betula study has $J=3$ follow-up test waves). We denote the vector of baseline confounders by $X_{i0}$ (gender, education, and age cohort) and the time-varying confounder by $W_{ij}$ (if the spouse has been seriously ill between the $j-1$th and $j$th test wave). The continuous memory outcome is denoted by $Y_{ij}$, and the binary exposure by $Z_{ij}$; $Z_{ij}=1$ if a subject is exposed (widowed) between the $(j-1)$st and $j$th test wave, and $0$ otherwise. Let $S_{ij}$ denote survival, where $S_{ij}=1$ if an individual is alive at the time of the testing and 0 otherwise. Let $R_{ij}$ be a dropout indicator, where $R_{ij}=1$ if an individual has completed the cognitive testing or 0 otherwise. We have monotone missingness, so if $R_{ij}=0$, $R_{ik}=0$ for $k > j$.
The history of the time-varying variables are denoted with an overbar. For example, the exposure history for individual $i$ through test wave $j$ is denoted by $\bar{Z}_{ij}=\{Z_{i0},Z_{i1},\ldots,Z_{ij}\}$.
Furthermore, for individual $i$, $J^r_i$ denotes the number of test waves (s)he participates in the study, and $J^s_i\geq J^r_i$ denotes the number of test waves (s)he is alive.
We provide details on the joint distribution of the observed data \linebreak $p(\bar{Y}_{i J^r_i}, \bar{Z}_{i J^r_i}, \oline{W}_{i J^r_i}, \bar{R}_{iJ^r_i}, \bar{S}_{iJ^s_i}, X_{i0})$ next. We specify a marginal model for the baseline confounders and a set of sequential conditional models for the time-varying variables, given the history of the joint process (the outcome, exposure, confounders, and missingness) as follows:
\begin{align}
\prod_{k=J^r_i+1}^{J^s_i} & p(s_{ij} \mid \bar{y}_{iJ^r_i}, \bar{z}_{iJ^r_i}, \bar{w}_{iJ^r_i}, \bar{r}_{iJ^r_i}=1, r_{iJ^r_i+1}=0,\ldots , r_{iJ^s_i}=0, \bar{s}_{ij-1}=1, x_{i0}) \nonumber \\
\prod_{j=0}^{J^r_i} & \Bigl(p(y_{ij} \mid \bar{y}_{ij-1}, \bar{z}_{ij}, \bar{w}_{ij}, \bar{r}_{ij}=1, \bar{s}_{ij}=1, x_0) \times \nonumber \\
& p(z_{ij} \mid \bar{y}_{ij-1}, \bar{z}_{ij-1}, \bar{w}_{ij}, \bar{r}_{ij}=1, \bar{s}_{ij}=1, x_{i0}) \times \nonumber \\
& p(w_{ij} \mid \bar{y}_{ij-1}, \bar{z}_{ij-1}, \bar{w}_{ij-1}, \bar{r}_{ij}=1, \bar{s}_{ij}=1, x_{i0}) \times \\
& p(r_{ij} \mid \bar{y}_{ij-1}, \bar{z}_{ij-1}, \bar{w}_{ij-1}, \bar{r}_{ij-1}=1, \bar{s}_{ij}=1, x_{i0}) \times \nonumber \\
& p(s_{ij} \mid \bar{y}_{ij-1}, \bar{z}_{ij-1}, \bar{w}_{ij-1}, \bar{r}_{ij-1}=1, \bar{s}_{ij-1}=1, x_{i0})\Bigr) \times \nonumber \\
& p(x_{i0}). \nonumber
\end{align}
The baseline confounders $x_{i0}$ are all observed before an individual enters the study. For each visit $j$ we observe the time-varying variables in the following order: $(s_{ij}, r_{ij}, w_{ij}, z_{ij}, y_{ij})$, even though the exposure, the time-varying confounder, and survival all occurred between $(j-1)$st and $j$th test wave. Of course, $y_{ij}$, $w_{ij}$ and $z_{ij}$, are only observed if $\bar{r}_{ij}=1$ and $\bar{s}_{ij}=1$. It is further allowed that $w_{ij}$ and $z_{ij}$ may have occurred before $s_{ij}$.
\subsection{Causal Estimand}
To define the causal contrast we first need to describe the different exposure regimes. We assume a monotone exposure pattern where initially all subjects are unexposed. That is, $z_{i0}=0$ for all $i$, and if $z_{ij}=1$ then $z_{ik}=1$ for $k>j$. If an individual is exposed during the study period, let $t$ be the first test wave after being exposed. That is, the exposure regime history through test wave $j, j\geq t$ then becomes $\bar{z}_{ij}=\{z_{i0}=0,\ldots,z_{it-1}=0,z_{it}=1,\ldots,z_{ij}=1\}$. We focus on the case when $t=j$ and call it $\bar{z}_{ij}$. The contrasting exposure regime is denoted by $\bar{z}'_{ij}=\{z_{i0}=0,\ldots,z_{ij}=0\}$, i.e. individuals unexposed through test wave $j$. Below, we generally suppress the subscript $i$ to simplify notation.
The potential memory outcome at wave $j$ is denoted by $Y_j(\bar{z}_j)$ for an individual under exposure regime $\bar{z}_j$. Similarly, let $S_j(\bar{z}_j)$ be the potential survival outcome at wave $j$, denoting survival under exposure regime $\bar{z}_j$.
We consider a principal stratum causal effect of a time-varying exposure on the outcome, at wave $j$, for those who would survive under either exposure regime,
\begin{equation}\label{ident.eq}
\mathrm{E}[Y_j(\bar{z}_j)-Y_j(\bar{z}'_j)\mid \bar{S}_j(\bar{z}_j) = \bar{S}_j(\bar{z}'_j)= 1].
\end{equation}
However, main interest is not the effect at a specific wave, but rather the effect aggregated over test waves, defined as
\begin{equation}\label{Tau.eq}
\tau = \frac{\sum _{j=1}^J \mathrm{E}[Y_j(\bar{z}_j)-Y_j(\bar{z}'_j)\mid \bar{S}_j(\bar{z}_j) = \bar{S}_j(\bar{z}'_j)= 1] \times \Pr[\bar{S}_j(\bar{z}_j) = \bar{S}_j(\bar{z}'_j)= 1]}{\sum_{k=1}^J \Pr[\bar{S}_k(\bar{z}_k) = \bar{S}_k(\bar{z}_k^\prime)= 1]}.
\end{equation}
\section{Identifying assumptions and sensitivity parameters}
To identify the causal effect in [\ref{Tau.eq}] from the observed data we introduce a set of assumptions and a set of sensitivity parameters to assess the impact of violations to some of the assumptions. The sensitivity parameters (and their values) will be explained in relation to the Betula data in Section 7.
Assumptions $1-4$ are a set of standard assumptions for causal inference of longitudinal observational data:
\begin{description}
\item[Assumption 1] \emph{Consistency}: For a given individual, if $\bar{Z}_{j}=\bar{z}_{j}$, then $Y_j=Y_j(\bar{z}_{j})$ and $S_j=S_j(\bar{z}_{j})$.
\item[Assumption 2] \emph{Positivity}: If $p(\bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j}, \bar{r}_{j-1}, \bar{s}_{j-1}=1, x_0)\neq 0$, then \newline $\Pr[z_{j}\mid \bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j}, \bar{r}_{j-1}, \bar{s}_{j-1}=1, x_0] > 0$ for $z_j=0,1$.
\item[Assumption 3] \emph{Stable unit treatment value assumption}: There is only one form of each exposure regime, and there is no interference among individuals.
\item[Assumption 4] \emph{Conditional exchangeability}: If $X_0$ and $\oline{W}_{j}$ contains all pre-exposure covariates related to exposure, potential outcomes and survival, then for all exposure regimes $ Y_j(\bar{z}_j) \perp\!\!\!\perp Z_j \mid \bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j}, \bar{r}_{j}, \bar{s}_j=1, x_0$ and $ S_j(\bar{z}_j) \perp\!\!\!\perp Z_j \mid \bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j}, \bar{r}_{j}, \bar{s}_{j-1}=1, x_0$.
\end{description}
To investigate sensitivity of the conditional exchangeability assumption for an unmeasured confounder, we follow the procedure of \citeauthor{brumback2004sensitivity} (\citeyear{brumback2004sensitivity}). The confounding is quantified through a parameter which describes the outcome confounding. That is, for exposure regime $\bar{z}_{j}$,
$c(\bar{z}_j)= E[ Y_j(\bar{z}_j) \mid \bar{y}_{j-1}, \bar{z}_j, \bar{w}_{j}, \bar{r}_{j}, \bar{s}_j=1, x_0] - E[Y_j(\bar{z}_j) \mid \bar{y}_{j-1}, \bar{z}'_j, \bar{w}_{j}, \bar{r}_{j}, \bar{s}_j=1, x_0]$, where $c(\bar{z}_j)$ is the average difference in potential outcomes because of unmeasured confounding. The conditional exchangeability assumption does not hold if $c(\bar{z}_j)\neq 0$. Thus, estimating $E[ Y_j(\bar{z}_j) \mid \bar{y}_{j-1}, \bar{w}_{j}, \bar{r}_{j}, \bar{s}_j=1, x_0]$ using the naive estimand $E[Y_j\mid \bar{y}_{j-1}, \bar{z}_{j}, \bar{w}_{j}, \bar{r}_{j}, \bar{s}_j=1, x_0]$ leads to a bias of $c(\bar{z}_j)\times \Pr[z'_j \mid \bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j}, \bar{r}_{j}, \bar{s}_j=1, x_0]$.
Further, since the two regimes only differ in $z_j$ we have that for $\bar{z}'_j$, the bias becomes $c(\bar{z}'_j)\times \Pr[z_j \mid \bar{y}_{j-1}, \bar{z}'_{j-1}, \bar{w}_{j}, \bar{r}_{j}, \bar{s}_j=1, x_0]
$. Sensitivity to several types of unmeasured confounding can be assessed using this form. Here, we restrict to an unmeasured confounder independent of the history of the joint processes $(\bar{y}_{j-1}, \bar{z}_j, \bar{w}_{j}, \bar{r}_{j}, \bar{s}_j,x_0)$.
In cohort studies $Y_j, Z_j$ and $W_j$ are not observed (but defined) for individuals who are alive but who drop out of the study. Previous studies of the Betula data have shown that individuals who drop out have lower cognitive performance and steeper decline (\cite{josefsson12}). Thus, we expect dropout to be MNAR conditioning on survival (MNARS), at least for the outcome. We now introduce an assumption to identify the distribution of dropouts among survivors.
\begin{description}
\item[Assumption 5] \emph{Dropout among survivors}
We make the assumption of non-future dependence (NFD) conditional on survival (NFDS). NFD is a special case of MNAR (\cite{kenward2003pattern}), and NFDS is defined as:
\begin{align*}
&p(y_j \mid \bar{y}_{j-1}, \bar{z}_{j},\bar{w}_{j}, \{r_0=1,\ldots,r_{t-1}=1,r_t=0,\ldots,r_j=0 \}, \bar{s}_j=1, x_0)\\
& =p(y_j \mid \bar{y}_{j-1}, \bar{z}_{j},\bar{w}_{j}, \bar{r}_j=1, \bar{s}_j=1, x_0),
\end{align*}
for all $j>1$ and all $t < j$. Here it is defined conditional on being alive at time $j$. This assumption leaves one conditional distribution per incomplete dropout pattern unidentified, that is
when $t=j$. To identify this distribution, we introduce a sensitivity parameter $\gamma_j$ such that
$p(y_{j} \mid \bar{y}_{j-1}, \bar{z}_{j},\bar{w}_{j},\bar{r}_j=\{1,\ldots,1,0 \}, \bar{s}_j=1, x_0) = p(y_j + \gamma_j \mid \bar{y}_{j-1}, \bar{z}_{j},\bar{w}_{j},\bar{r}_j=1, \bar{s}_j=1, x_0)$, when $\gamma_j<0$ implies a negative location shift in the outcome at the first unobserved test wave. Table \ref{dropoutmortality.pat} displays a description of the possible mortality- and missing data patterns under the NFDS assumption.
For the exposure and time-varying confounder, we make an MAR type assumption conditional on being alive at time $j$. In particular, for all $j\geq1$ and all $t \leq j$
$p(w_j \mid \bar{y}_{j-1}, \bar{z}_{j-1},\bar{w}_{j-1}, \{r_0=1,\ldots,r_{t-1}=1,r_t=0,\ldots,r_j=0 \}, \bar{s}_{j}=1,x_0) = p(w_j \mid \bar{y}_{j-1}, \bar{z}_{j-1},\bar{w}_{j-1}, \bar{r}_{j}=1, \bar{s}_{j}=1,x_0)
$ and
$ p(z_j \mid \bar{y}_{j-1}, \bar{z}_{j-1},\bar{w}_{j}, \{r_0=1,\ldots,r_{t-1}=1,r_t=0,\ldots,r_j=0 \}, \bar{s}_{j}=1, x_0) =p(z_j \mid \bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j}, \bar{r}_{j}=1, \bar{s}_j=1, x_0)$.
\end{description}
We also need three further assumptions for identification of the principal strata. We start with two standard assumptions.
\begin{description}
\item[Assumption 6] \emph{Monotonicity}. $S_{j}(\bar{z}_{j}) \leq S_{j}(\bar{z}'_{j})$; if an individual were to be alive under exposure regime $\bar{z}_j$ then (s)he would also be alive under the contrasting regime $\bar{z}'_j$.
\item[Assumption 7] \emph{Differences in outcomes when comparing different strata}. For the contrasting exposure regime $\bar{z}'_{j}$,
$\Delta_{\bar{z}'_{j}} = E[Y_j(\bar{z}'_{j}) \mid \bar{S}_j(\bar{z}_j) = \bar{S}_j(\bar{z}'_j)= 1] - E[Y(\bar{z}'_{j}) \mid \bar{S}_j(\bar{z}'_j) = 1, \bar{S}_j(\bar{z}_j) \neq 1]$, That is, the mean difference in potential outcomes when comparing the "always survivors" strata to the strata where individuals were to live under the contrasting regime $\bar{z}'_{j}$ but not under exposure regime $\bar{z}_{j}$. In our analysis we assume $\Delta_{\bar{z}'_{j}}\geq 0$ which implies that memory performance is on average higher in the "always survivors"-strata (the always survivors-strata is healthier). We further assume this difference is independent of the history of the joint process.
\end{description}
Here we need to introduce a new assumption due to a common problem encountered in longitudinal cohort studies; that an individual's exposure level $z_j$, hence the exposure regime $\bar{z}_j$, and time-varying confounder $w_j$ is only observed if (s)he is alive and participates at the $j$th test wave.
\begin{description}
\item[Assumption 8] \emph{Exposure and confounding among non-survivors}
If $s_j=0$ and $\bar{s}_{j-1}=1$ for an individual, $z_{j}$ and $w_j$ may have occurred before the event of death, thus, $z_{j}$ and $w_j$ are not observed but could still be well-defined. Therefore, we need additional assumptions about exposure and confounding among non-survivors. For the exposure and time-varying confounder, let
$\nu_{z_j} = \Pr[z_{j} \mid \bar{y}_{j-1}, \bar{z}_{j-1},\bar{w}_{j},r_j=0,\bar{r}_{j-1}, s_j=0, \bar{s}_{j-1}=1, x_0] - \Pr[z_{j} \mid \bar{y}_{j-1}, \bar{z}_{j-1},\bar{w}_{j},\bar{r}_{j}, \bar{s}_{j}=1, x_0]$, and
$\nu_{w_j} = \Pr[w_j \mid \bar{z}_{j-1},\bar{w}_{j-1},r_{j}=0, \bar{r}_{j-1},\bar{y}_{j-1}, s_{j}=0, \bar{s}_{j-1}= 1, x_0] \\
- \Pr[w_j \mid \bar{z}_{j-1},\bar{w}_{j-1},\bar{r}_j,\bar{y}_{j-1},\bar{s}_{j}= 1, x_0]$.
These represents, the mean difference in proportion exposed and confounder equal to one, respectively, when comparing non-survivors and survivors. The first probability on the right-hand side of each expression is not identified.
In our analysis we assume $\nu_{w_j}=0$, i.e. confounder is distributed the same among survivors and non-survivors.
Bounds can be derived for $\nu_{z_j}$; see the Web Appendix section A.1 for details. In particular, the upper bound for $\nu_{z_j}$, $U_{\nu_j}$, is obtained when $\Pr[z_{j} \mid \bar{y}_{j-1}, \bar{z}_{j-1},\bar{w}_{j},r_j=0,\bar{r}_{j-1},s_j=0, \bar{s}_{j-1}=1, x_0]=1$. This reflects that among non-survivors, all subjects were exposed before the event of death between the $j-1$th and $j$th wave. Further, by using Assumption 1 and 6, the lower bound for $\nu_{z_j}$ is obtained when
$\Pr[S_j=1 \mid \bar{y}_{j-1}, \bar{z}_{j},\bar{w}_{j},\bar{r}_j,\bar{s}_{j-1}=1, x_0]=\Pr[S_j=1 \mid \bar{y}_{j-1}, \bar{z}'_{j},\bar{w}_{j},\bar{r}_j,\bar{s}_{j-1}=1, x_0]$. This reflects an equal survival probability among those exposed or unexposed at wave $j$. Here, by using the law of total probability (ltp) and Bayes theorem, the lower bound $L_{\nu_j}$ becomes 0.
\end{description}
\section{Identification}
Identification of the SACE in [\ref{Tau.eq}] follows from two results.
\begin{description}
\item[Result 1:] The causal contrasts, $E[Y_j(\bar{z}_j) - Y_j(\bar{z}'_j) \mid \bar{S}(\bar{z}_j)=\bar{S}_j(\bar{z}'_j)=1]$, in [\ref{ident.eq}] can be identified from observed data distribution models under Assumptions 1-8 as follows
\begin{align}\label{sace.eq}
\frac{E[Y_j, \bar{S}_j=1\mid \bar{z}_j]}{\Pr[\bar{S}_j=1\mid \bar{z}_j]} - \frac{E[Y_j, \bar{S}_j=1\mid \bar{z}'_j]}{\Pr[\bar{S}_j=1\mid \bar{z}'_j]} - \Delta_{j} \times \left(1-\frac{\Pr[\bar{S}_j = 1 \mid \bar{z}_j]}{\Pr[\bar{S}_j = 1 \mid \bar{z}'_j]} \right).
\end{align}
\item[Result 2:] $\tau$ in [\ref{Tau.eq}] can further be identified using Assumption 6 by weighting the contrasts in [\ref{sace.eq}] with
\begin{align}\label{weights}
\frac{\Pr[\bar{S}_j(\bar{z}_j) = \bar{S}_j(\bar{z}'_j)= 1]}{\sum_{k=1}^J \Pr[\bar{S}_k(\bar{z}_k) = \bar{S}_k(\bar{z}_k^\prime)= 1]}=\frac{\Pr[\bar{S}_j = 1 \mid \bar{z}_j]}{\sum_{k=1}^J \Pr[\bar{S}_k = 1 \mid \bar{z}_k]}.
\end{align}
\end{description}
The proofs of the results can be found in the Web Appendix section A.2. The sensitivity parameters introduced must be fixed or given informative priors. In Section 6 we provide the estimation algorithm where we consider the case when $c(\bar{z}_j)$, $c(\bar{z}'_j)$, $\Delta_{j}$, $\nu_{z_j}$ and, $\gamma_j$ are given informative non-degenerate priors.
\section{Bayesian semi-parametric modeling of the observed data distributions}
We propose a Bayesian semi-parametric modeling approach based on Bayesian Additive Regression Trees (BART, \cite{chipman2010bart}) for the observed data distribution. BART is implemented in the R package \textit{bartMachine} (\cite{kapelner2013bartmachine}) for continuous and binary responses.
For the time varying components, we specify BART models for the responses as a function of prior histories for all individuals alive and not dropped out at a given test wave. The model consists of two parts: a sum-of-trees model and a regularization prior on the parameters of that model. The model for the continuous response $Y_j$ is conditioned on the history of the joint process $(\bar{y}_{j-1}, \bar{z}_{j}, \bar{w}_{j}, x_0)$ for the subset that satisfies $\bar{r}_j=1$ and $\bar{s}_j=1$, and can be expressed as
$Y_j = \sum_{k=1}^{K_{Y_j}} g_{Y_j}\left((\bar{y}_{j-1}, \bar{z}_{j}, \bar{w}_{j},x_0);T_{Y_j}^k,M_{Y_j}^k\right) + \varepsilon_j.$
The model consists of $K_{Y_j}$ distinct binary regression trees denoted by $T_{Y_j}^k$. Each tree constitute a set of interior node decision rules leading down to $b_{Y_j}^k$ terminal nodes, and for a given $T_{Y_j}^k$, $M_{Y_j}^k =(\rho_{Y_j}^{k,1},\ldots,\rho_{Y_j}^{k,b^k})$ is the associated terminal node parameters. The conditional distribution of the continuous outcome is specified as normal,
$Y_j \sim N\left(\mu_{Y_j}(\bar{y}_{j-1}, \bar{z}_{j}, \bar{w}_{j},x_0), \sigma_j^2\right),$
where the mean function, $\mu_{Y_j}(\bar{y}_{j-1}, \bar{z}_{j}, \bar{w}_{j},x_0)$, is given by the sum-of-trees.
The BART models for our binary responses $Z_j, W_j, R_j$, and $S_j$ are specified as probit models. For example the model for the exposure
can be expressed as:
$\pi_{Z_j}(\bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j}, x_0)$ \newline $= \Phi \left(
\sum_{k=1}^{K_{Z_j}} g_{Z_j}\left((\bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j},x_0);T_{Z_j}^k,M_{Z_j}^k\right) \right),$ where $\Phi$ denotes the cumulative density function of the standard normal distribution
and $\pi_{Z_j}(\bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j-1}, x_0)$ is the probability of being exposed at wave $j$ given $(\bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j}, x_0)$ for the subset that satisfies $\bar{r}_j=1$ and $\bar{s}_j=1$. The BART model for $S_j$ is fitted for the subset that satisfies $\bar{r}_{j-1}=1$ and $\bar{s}_{j-1}=1$, and for $R_j$ the subset that satisfies $\bar{r}_{j-1}=1$ and $\bar{s}_{j}=1$. The predicted probabilities of $r_j=1$ and $s_j=1$ are:
$\pi_{R_j}(\bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j-1}, x_0)$
and
$\pi_{S_j}(\bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j-1}, x_0).$
Note that, $s_0=1$ and $r_0=1$ for all individuals, $\pi_{R_j}=0$ if $r_{j-1}=0$, and $\pi_{S_j}=0$ if $s_{j-1}=0$.
The baseline confounders are all categorical. We create a saturated multinomial random variable, $x_0\sim Multi(N, \pi_{x_0}^1, \pi_{x_0}^2, \ldots, \pi_{x_0}^L),$ based on these categorical variables.
$L$ is the number of categories and each category corresponds to a unique combination of the categorical variables. $\pi_{x_0}=(\pi_{x_0}^1, \pi_{x_0}^2, \ldots, \pi_{x_0}^L)$ is given a Dirichlet prior with parameters equal to one.
\subsection{Posterior}
Draws from the posterior distribution of the sum-of-trees models are generated using Markov chain Monte Carlo (MCMC). The parameters of the conditional distributions for $Y_j, Z_j, W_j, R_j$, and $S_j$ are assumed independent and thus their posteriors can be sampled simultaneously. We use default priors based on the R package \textit{bartMachine} on all of the parameters of the sum-of-trees model, that is, on the tree structure, the terminal node parameters, and the error variance. For details see \citeauthor{kapelner2013bartmachine} (\citeyear{kapelner2013bartmachine}).
\section{Computation of the SACE}
Here we present an algorithm for estimation of $\tau$ in [\ref{Tau.eq}] using the G-computation formula. Details of the algorthm can be found in the Web Appendix section A.3. The general approach is to specify models for the observed data as we did in Section 5 and then to use assumptions in Section 3 with embedded sensitivity parameters to identify the causal effect estimate as described in Section 4. The algorithm can be summarized in the following six steps:
\begin{enumerate}
\item Sample the observed data posteriors as described in Section 5.
\item For each posterior sample of the parameters sample pseudo data $(\bar{y}^*_{j-1},\bar{w}^*_{j},\bar{r}_{j}^*,\bar{s}_{j}^*,x_0^*)$ of size $N^*$.
\item Implement G-computation for $\bar{z}_j$, and similarly for $\bar{z}'_j$, using the pseudo data from Step 2 by computing $E[Y_j \mid \bar{y}_{j-1}, \bar{z}_j, \bar{w}_{j},\bar{r}_{j}, \bar{s}_j=1, x_0]$
and $\prod^j_{k=0} \Pr[S_{k}=1 \mid \bar{z}_{k}, \bar{w}_k,\bar{r}_k,\bar{y}_{k-1},\bar{S}_{k-1}=1, x_0]$.
\item Implement Monte Carlo integration using the pseudo data to compute $\Pr[\bar{S}_{j}=1 \mid \bar{z}_j]$
and $E[Y_j, \bar{S}_j=1 \mid \bar{z}_j]$.
\item Use the quantities in step (a)-(d) above to compute one posterior sample of $\tau$ as defined in [\ref{sace.eq}]-[\ref{weights}].
\item Repeat step 1 - 4 for each of the posterior sample of the parameters.
\end{enumerate}
In practice, a number of the initial posterior samples are discarded as burn-in. Parallel computation can be implemented to speed up computations. For example, instead of running one long chain until convergence in Step $1$, it is possible to run multiple shorter chains in parallel. Also, Step $2$ may be divided into $k$ blocks of size $N^*/k$, and in Steps $3-4$ the parameters of interest are computed by combining the pseudo data from the $k$ blocks. We give further details on computation with Betula data in Section 7.3.
\section{Analysis of the Betula data}
The Betula study (\cite{nilsson97}) is a population-based cohort study that started in 1988 with the objective to study how memory functions change over time and to identify risk factors for dementia.
\subsection{The Betula data}
The goal is to estimate the causal effect of becoming a widow on memory among those who would survive irrespective of being widowed or not. As such, we limit our data set to those individuals who were married at enrollment, and further to those age-cohorts where we observe both married and widowed participants over the study period. Of approximately 200 participants $N=1059$ met the inclusion criteria for this study, and data were recorded at 4 fixed test waves ($j=0,\ldots, 3$) with 5 years interval. Only $45\%$ of the participants completed the cognitive testing at the last test wave, $31\%$ died during the study period, and $24\%$ dropped out but were still alive at study end.
The memory outcome was assessed at each wave using a composite of three episodic memory tasks. The score can range between 0 and 76, with a higher score indicating better memory (for details see \cite{josefsson12}). We consider two contrasting exposure regimes, first: subjects who became a widow between the $j-1$th and $j$th wave, $\bar{z}_j=\{z_0=0,\ldots,z_{j-1}=0,z_j=1\}$, and second: subjects married through test wave $j$, $\bar{z}'_j=\{z_0=0,\ldots,z_j=0\}$. Specifically, for $j=1$: $\bar{z}_1=\{0,1\}$ and the contrasting regime is $\bar{z}'_1=\{0,0\}$, for $j=2$: $\bar{z}_2=\{0,0,1\}$ and $\bar{z}'_2=\{0,0,0\}$, and for $j=3$: $\bar{z}_3=\{0,0,0,1\}$ and $\bar{z}'_3=\{0,0,0,0\}$.
Baseline demographic characteristics included age-cohorts: $45, 50, \ldots , 80$ years of age at enrollment, gender, and education, categorized into \textit{low}: 6-7 years of education (29\%), \textit{intermediate}: 8-9 years (31\%), or \textit{high}: \textgreater{9} years (40\%). We also measured a time-varying confounder; an indicator if the spouse has been sick within the last 5 years. We note that baseline confounders are always recorded.
\subsection{Sensitivity parameters}
Our approach allows uncertainty about untestable assumptions by specifying priors for the sensitivity parameters described in Section 3. We restrict the parameters to a plausible range of values, reflecting the authors' beliefs about the unknown quantities.
In Assumption 4, the sensitivity parameter $c(\bar{z}_j)$ reflects the average difference in potential outcomes due to unmeasured confounding. For the Betula data, when studying the effect of widowhood on cognition, one concern may be that the association is confounded by a healthy lifestyle, such as a healthy diet and/or exercise, something that is often shared within couples. Couples with a healthy lifestyle live longer and may have better cognitive performance than couples with a less healthy lifestyle. This information is not available from the database. Hence, it is a potential unmeasured confounder. Here, we assume $c(z_j)<0$ and $c(z'_j)>0$, reflecting that exposed (widowed) individuals are less healthy compared to unexposed (married) individuals. We further assume the effect is equal for exposed and unexposed. That is, we assume $c(z_j)=-\xi_j$ and $c(z'_j)=\xi_j$. Here, we specify a uniform prior on the sensitivity parameters,
$\xi_j\sim \mathrm{Unif}(0, U_{\xi_j}),$
where we assume the upper bound using the observed data is
$U_{\xi_j}= \frac{1}{2}\times SD(Y_j \mid \bar{y}_{j-1}, \bar{z}_{j}, \bar{w}_{j}, \bar{r}_j=1, \bar{s}_j=1, x_0).$
That is, we expect the sensitivity parameter not to be bigger than one-half standard deviation of the outcome conditional on the history of the joint process.
Departures from a MAR mechanism for the missingness among survivors can be investigated by varying $\gamma_j$ in Assumption 5. Our prior belief is that $\gamma_j<0$, reflecting a negative shift in memory performance occur immediately after the first unobserved test wave. Here, the prior is specified as $\gamma_j\sim \mathrm{Unif}(-L_{\gamma_j}, 0),$
where we assume the lower bound is one standard deviation,
$L_{\gamma_j} = 1 \times SD(Y_j \mid \bar{y}_{j-1}, \bar{z}_{j}, \bar{w}_{j}, \bar{r}_j=1, \bar{s}_j=1, x_0).$
Sensitivity to Assumption 7, uses $\Delta_{j}$, which reflects the difference in outcomes when comparing different strata. We again specify a uniform prior $\Delta_j \sim \mathrm{Unif}(0, U_{\Delta_{j}}),$
where we assume $U_{\Delta_{j}}=1\times SD(Y_j \mid \bar{s}_j=1).$
Finally, sensitivity to Assumption 8 uses the sensitivity parameter $\nu_{z_j}$, which represents difference in the probability of being exposed at wave $j$ for non-survivors and survivors conditioning on the history of the joint process. As shown in Section 3, $\nu_{z_j}$ is bounded by $[0, U_{\nu_{z_j}}]$. We assume the prior for $\nu_{z_j}$ is uniform over this range, $\nu_j \sim \mathrm{Unif}(0, U_{\nu_j}).$
The upper bound reflects that, among non-survivors, all subjects being exposed before death between the $j-1$th and $j$th wave.
\subsection{Computations}
We estimated $\tau$ using the proposed BSP method and embedded sensitivity parameters. For each chain the first 1000 iterations were discarded as burn-in, and 2040 posterior samples of $\tau$ were obtained. We sampled pseudo data of size $N^*=25000$ at each iteration. Convergence of the posterior samples was monitored using trace plots of the samples. To reduce computation time we used 204 parallel chains, and the pseudo data was divided into 25 blocks. Total computation time was 7 hours and 27 minutes. Computation time for 1 posterior sample for a pseudo sample of size 1000 (1 block) was 19 seconds. For a total computation time of 19 seconds we would need 51000 cores. This would require that the code be fully parallelized; that is, for each core we would sample pseudo data for 1 block and for 1 posterior sample.
\subsection{Results and comparison with other methods}
The posterior sampling results revealed a mean episodic memory score of 37.4 (95\% CI; $35.6, 39.4$) for exposed and 37.0 (95\% CI; $36.1, 38.0$) for unexposed individuals, and an estimate of $\tau$ of 0.40 (95\% CI; -$1.27,2.16$), suggesting that there is no effect of becoming a widow on memory among those who would survive irrespective of exposure.
As a second analysis, we compare our approach (BSP-GC) with three other methods used for causal effect estimation of longitudinal data with time-varying confounding. For simplicity of comparison we estimate $\mathrm{E}[Y_j(\bar{z}_j)-Y_j(\bar{z}'_j)\mid \bar{s}_j=1]$ and set $\gamma_j=0$ and $c(z_j)=c(z'_j)=0$.
The causal contrasts are thus estimated by computing
$$\frac{\sum \mu_j(\bar{y}^*_{j-1}, \bar{z}_j, \bar{w}^*_{j}, x^*_0)}{N^*} - \frac{\sum \mu_j(\bar{y}^*_{j-1}, \bar{z}'_j, \bar{w}^*_{j}, x^*_0)}{N^*},$$
and the weights in [\ref{weights}] are estimated by computing
\begin{align*}
\frac{\frac{\sum_{1}^{N^*} \pi_{S_j}(\bar{y}^*_{j-1}, \bar{z}_{j-1}^*, \bar{w}^*_{j-1}, x_0^*)}{N^*}}{\sum_{1}^J \frac{\sum_{1}^{N^*} \pi_{S_k}(\bar{y}^*_{k-1}, \bar{z}_{k-1}^*, \bar{w}^*_{k-1}, x_0^*)}{N^*}}.
\end{align*}
The three other methods implemented are: (i) A parametric version of the proposed procedure (BP-GC). Here we specified Bayesian linear and logistic additive regression models instead of the BART models described in Section 5. (ii) Inverse probability of treatment weights (IPTW; \cite{cole2008constructing}). Here, the mean $E[Y_j \mid \bar{s}_j=1, \bar{z}_j]$ is estimated by averaging the memory outcome for the subset with $\bar{Z}_j=\bar{z}_j$ in a pseudo-population constructed by weighting each individual using both unstabilized weights (IPTW-W) and stabilized weights (IPTW-SW), to adjust for confounding and for attrition among survivors. The IPTW-W and IPTW-SW were implemented using the \textit{ipw} and \textit{survey} packages in R.
(iii) Targeted minimum loss-based estimation approach for longitudinal data structures (TMLE; \cite{van2012targeted}). We implemented the TMLE using the \textit{ltmle} package using default settings (\cite{lendle2017ltmle}). For IPTW-W, IPTW-SW, and TMLE, the causal effect was obtained by pooling the causal contrasts using the following weights $\frac{\Pr[\bar{S}_j=1]}{\sum_{k=1}^J \Pr[\bar{S}_k=1]}$.
Confidence intervals were calculated using nonparametric bootstrap. We used 5000 bootstrap samples. The bootstrap confidence intervals were calculated using the 2.5th and 97.5th percentiles of the resulting estimates.
The results from the four methods are given in Table \ref{comparison.otherm}. First, all of the methods display a negative widowhood effect on memory, although all confidence/credible intervals (CI) cover zero. There is a large discrepancy between our semi-parametric approach, BSP-GC, and the parametric counterpart, BP-GC. In the latter, the effect was attenuated and the CI was narrower. A likely explanation of the discrepancy in effect estimates is that BP-GC is more susceptible to bias caused by model misspecification. BP-GC and IPTW-SW yielded most similar results, although the weighting approach had much wider CI. Further, the effect estimate appeared most negative using IPTW-W and the CI was much wider than for any of the other methods. Weighting methods are known to be unstable and to have problems with large variance estimates in finite samples if the values of the weights are extreme. In our analysis the range of the weights was 0.06-14.3 for IPTW-W, compared to 0.06-5.4 for IPTW-SW. The large weights using IPTW-W may explain the deviating result using this method. Our BSP-GC approach yielded an estimate of $\tau$ most similar to TMLE, although TMLE had slightly wider CI.
We compare the fit of the BSP-GC and BP-GC to the observed data using the logarithm of the pseudo marginal likelihood (LPML; \cite{geisser1979}). The values of the LPML were -$16,803$ for BP-GC and -$15,778$ for BSP-GC, indicating a better a better fit for the BSP approach.
\section{Concluding remarks}
This paper has proposed a Bayesian semi-parametric (BSP) framework for estimating the SACE with longitudinal cohort data. Our approach allows for Bayesian inference under MNAR missingness and truncation by death, as well as the ability to characterize uncertainty about unverifiable assumptions. The proposed approach has several advantages compared to existing approaches: (i) the flexible modeling of the observed data as compared to parametric methods, while maintaining computational ease, (ii) interval estimates for full posterior inference, (iii) easy to introduce sensitivity parameters.
In the analysis of the Betula data we compared our approach to four other approaches. Similar to TMLE the BSP approach does not rely on strong modeling assumptions, but unlike TMLE, it is quite easy to modify assumptions and incorporate sensitivity parameters. Recall we could not easily make direct comparisons of the proposed approach with the other approaches under our assumptions that include sensitivity parameters.
We used the LPML to compare the fit of the Bayesian semi-parametric and parametric approaches to the observed data; here, this is a comparison between parametric regression models and BART. It is less transparant how to formally compare the BSP approach to TMLE and IPTW, for a given data set.
We did not find an effect of widowhood on memory. The difference in findings from previous studies may partly be explained by different estimands being used; ours is the only analysis using a SACE. In addition \citeauthor{gerritsen2017influence} (\citeyear{gerritsen2017influence}) showed that widowhood augments the effect of other stressful life events on dementia incidence rather than acts as a single cause. Additionally, in this study we considered the immediate effect of widowhood (within 5 years) rather than a long term effect; it may take longer for degeneration to become apparent.
Several of our assumptions can be (further) relaxed. For example, Assumption 5 can be weakened to a stochastic Monotonicity, by following the procedure described in \citeauthor{lee2010causal} (\citeyear{lee2010causal}).
Also, in this study we have considered unmeasured outcome confounding; this assumption can easily be extended to allow unmeasured mortality confounding. Assumption 7 can be weakened by conditioning on the history of the joint process. However, a drawback with relaxing these assumptions is increasing the number of sensitivity parameters.
One limitation of the proposed approach is the computation time when a large pseudo sample for G-computation is necessary. However, if the algorithm was fully parallelized as discussed in Section 6 and Section 7.3, the total computation time would be less than a minute. In addition, we used existing R-functions for BART that may not be the most efficient for our setting; we will explore this in future work. Future work will also explore other choices for priors of the sensitivity parameters.
\section{Supplementary materials}
Web Appendices referenced in Sections 3, 4, 6, and 7, as well as R code are available as Supplementary materials.
\section{Acknowledgments}
The authors would like to thank Dr Anna Sundstr{\"o}m for helpful discussions on the interpretation of the results.
The research is part of the program Paths to Healthy and Active Ageing, funded by the Swedish Research Council for Health, Working Life and Welfare, (Dnr 2013 – 2056) to MJ. This work is partially funded by US NIH grants CA183854 and GM112327 to MJD. This publication is based on data collected in the Betula prospective cohort study, Umeå University, Sweden. The Betula Project is supported by Knut and Alice Wallenberg foundation (KAW) and the Swedish Research Council (K2010-61X-21446-01).
\printbibliography
\newpage
\section{Tables}
\begin{table}[hp]
\caption{The table shows possible missing data, $\bar{R}$, and mortality patterns, $\bar{S}$. The outcome vector $\mathbf{Y}=\left\lbrace Y_0, Y_1, Y_2, Y_3 \right\rbrace$ is fully observed if $\bar{S}=\bar{R}=1$, otherwise it is constrained by the mortality outcome and/or missing data patterns. $Y_j=\mathrm{O}$ if the outcome is observed, $Y_j=\mathrm{M}$ if missing, and $Y_j=\mathrm{nd}$ when truncated by death. The NFDS restriction leaves the distribution for $Y_j=\mathrm{M}^*$ unidentified.}
\label{dropoutmortality.pat}
\begin{adjustbox}{width=1\textwidth}
\begin{tabular}{ c c c c c }
\hline
& \multicolumn{4}{ c }{$\bar{R}_J$} \\
\hline
$\bar{S}_J$ & $\{1,0,0,0 \}$ & $\{1,1,0,0 \}$ & $\{1,1,1,0 \}$ & $\{1,1,1,1 \}$ \\
\hline
$\{1,0,0,0 \}$ & $\left\lbrace \mathrm{O},\mathrm{nd},\mathrm{nd},\mathrm{nd} \right\rbrace$ & - & - & - \\
$\{1,1,0,0 \}$ & $\left\lbrace \mathrm{O},\mathrm{M}^*,\mathrm{nd},\mathrm{nd}\right\rbrace$ & $\left\lbrace \mathrm{O},\mathrm{O},\mathrm{nd},\mathrm{nd}\right\rbrace$ & - & - \\
$\{1,1,1,0 \}$ & $\left\lbrace \mathrm{O},\mathrm{M}^*,\mathrm{M},\mathrm{nd}\right\rbrace$ & $\left\lbrace \mathrm{O},\mathrm{O},\mathrm{M}^*,\mathrm{nd}\right\rbrace$ & $\left\lbrace \mathrm{O},\mathrm{O},\mathrm{O},\mathrm{nd}\right\rbrace$ & - \\
$\{1,1,1,1 \}$ & $\left\lbrace \mathrm{O},\mathrm{M}^*,\mathrm{M},\mathrm{M}\right\rbrace$ & $\left\lbrace \mathrm{O},\mathrm{O},\mathrm{M}^*,\mathrm{M}\right\rbrace$ & $\left\lbrace \mathrm{O},\mathrm{O},\mathrm{O},\mathrm{M}^*\right\rbrace$ & $\left\lbrace \mathrm{O},\mathrm{O},\mathrm{O},\mathrm{O}\right\rbrace$ \\
\hline
\end{tabular}
\end{adjustbox}
\end{table}
\begin{table}[hp]
\caption{Comparison of methods used for causal effect estimation of longitudinal data with time-varying confounding, setting $\Delta_{j}=0$, $\gamma_j=0$, and $c(z_j)=c(z'_j)=0$, using our proposed approach (BSP-GC), a parametric version of the proposed procedure (BP-GC), inverse probability of treatment weights using unstabilized weights (IPTW-W) and stabilized weights (IPTW-SW), and Targeted minimum loss-based estimation approach for longitudinal data structures (TMLE).
}
\label{comparison.otherm}
\begin{center}
\begin{tabular}{ l c }
\hline
& Estimate [95\% CI] \\
\hline
BSP-GC & -0.96 [-2.74, 0.78] \\
BP-GC & -0.53 [-1.73, 0.68] \\
IPTW-W & -1.67 [-5.96, 1.51] \\
IPTW-SW & -0.44 [-3.06, 1.39] \\
TMLE & -0.96 [-3.11, 0.99] \\
\hline
\end{tabular}
\end{center}
\end{table}
\end{document}
\section*{A.1: Derivation of bounds for $\mathbf{\nu_j}$ in Assumption 8.}
Here, we derive bounds for $\nu_{z_j}$. By using the law of total probability we have that
\begin{align*}
&\Pr[z_j \mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0] \\
& = (\nu_{z_j} - \Pr[z_j \mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,\bar{r}_{j},\bar{s}_{j}= 1, x_0]) \\
& \quad \times \Pr[S_j=0 \mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0] \\
& \quad + \Pr[z_j \mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j}=1, x_0] \\
& \quad \times \Pr[S_{j}=1 \mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0]\quad
\end{align*}
and by using Bayes theorem we have that
\begin{align*}
&\Pr[z_j\mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0] \\
& = \Pr[z_j\mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j}= 1, x_0] \\
& \quad \times \frac{\Pr[S_{j}=1 \mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0]}{\Pr[S_{j}=1 \mid \bar{y}_{j-1},\bar{z}_{j},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0]}.
\end{align*}
Plugging in the second equation for the right hand side and solving the first equation for $\nu_{z_j}$ leads to
\begin{align*}
\nu_{z_j} &= \Pr[z_j\mid \bar{y}_{j-1}, \bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j}= 1, x_0] \\
& \quad \times \frac{\Pr[S_{j}=1 \mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0]}{\Pr[S_{j}=0 \mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0]} \\
& \quad \times \frac{\Pr[S_{j}=0 \mid \bar{y}_{j-1},\bar{z}_{j},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0]}{\Pr[S_{j}=1 \mid \bar{y}_{j-1},\bar{z}_{j},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0]} \\
& \quad + \Pr[z_j \mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,\bar{r}_{j},\bar{s}_{j}= 1, x_0]).
\end{align*}
Further, by Assumptions 1 and 6 we have that
\begin{align*}
& \Pr[S_{j}=1 \mid \bar{y}_{j-1},\bar{z}_{j},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0] \\
& \leq \Pr[S_{j}=1 \mid \bar{y}_{j-1},\bar{z}'_j,\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0].
\end{align*}
Thus, the lower bound for $\nu_{z_j}$ is obtained when $\Pr[S_{j}=1 \mid \bar{y}_{j-1},\bar{z}_{j},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0] = \Pr[\bar{S}_{j}=1 \mid \bar{y}_{j-1}, \bar{z}'_j,\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0]$. Moreover, since the two regimes only differ in $z_j$, we note that,
\begin{align*}
&\Pr(\bar{S}_{j}=1\mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0] \\
& = \Pr[\bar{S}_{j}=1 \mid \bar{y}_{j-1},\bar{z}_{j},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0] \\
& \quad \times \Pr[z_j\mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0] \\
& \quad + \Pr[\bar{S}_{j}=1 \mid \bar{y}_{j-1}, z'_j,\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0] \\
& \quad \times \Pr[z'_j\mid \bar{y}_{j-1},\bar{z}_{j-1},\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0]\\
&= \Pr[\bar{S}_{j}=1 \mid \bar{y}_{j-1},\bar{z}_j,\bar{w}_j,r_j=0,\bar{r}_{j-1},\bar{s}_{j-1}= 1, x_0]
\end{align*}
Hence, the lower bound for $\nu_{z_j}$ is
\begin{align*}
L_{\nu_j} = 0,
\end{align*}
and the upper bound for $\nu_{z_j}$ is
$$U_{\nu_j}=1-\Pr[z_j\mid \bar{z}_{j-1},\bar{w}_j,\bar{R}_j=1,\bar{y}_{j-1},\bar{s}_{j}= 1, x_0].$$
\section*{A.2: Details on Equations [4] and [5] in Section 4.}
Here, we present results for identification of the causal estimand in [3] using Assumptions 1-8 introduced in Section 3. First, for identification of the contrasts $\mathrm{E}[Y_j(\bar{z}_j)-Y_j(\bar{z}'_j)\mid \bar{S}_j(\bar{z}_j) = \bar{S}_j(\bar{z}'_j)= 1]$ for $j=1,\ldots,J$ in [2], by using the law of total probability (ltp) and some algebra, we have that
\begin{align*}
E&[Y_j(\bar{z}_j) \mid \bar{S}_j(\bar{z}_j)=1] \\
= & E[Y_j(\bar{z}_j) \mid \bar{S}(\bar{z}_j)=\bar{S}_j(\bar{z}'_j)=1] + \Pr[\bar{S}_j(\bar{z}'_j)\neq 1 \mid \bar{S}_j(\bar{z}_j)=1]\\
& \times (E[Y_j(\bar{z}_j) \mid \bar{S}_j(\bar{z}_j)=1, \bar{S}_j(\bar{z}'_j)\neq 1] - E[Y_j(\bar{z}_j) \mid \bar{S}(\bar{z}_j)=\bar{S}_j(\bar{z}'_j)=1]).
\end{align*}
Using Assumption 7 and solving the above equation for $E[Y_j(\bar{z}_j) \mid \bar{S}_j(\bar{z}_j)=\bar{S}_j(\bar{z}'_j)=1]$ we obtain
\begin{align}\label{Solv}
&E[Y_j(\bar{z}_j) \mid \bar{S}(\bar{z}_j)=\bar{S}_j(\bar{z}'_j)=1] \nonumber \\
&=E[Y_j(\bar{z}_j) \mid \bar{S}_j(\bar{z}_j)=1] + \Delta_{\bar{z}_j} \Pr[\bar{S}_j(\bar{z}'_j)\neq 1 \mid \bar{S}_j(\bar{z}_j)=1]
\end{align}
For exposure regime $\bar{z}_j$ (and similarly for $\bar{z}'_j$) and using Assumption 1, we have that
\begin{align}\label{expYS}
E[Y_j(\bar{z}_j) \mid \bar{S}_j(\bar{z}_j)=1]& = \frac{E[Y_j(\bar{z}_j), \bar{S}_j(\bar{z}_j)=1]}{\Pr[\bar{S}_j(\bar{z}_j)=1]} \nonumber \\
& = \frac{E[Y_j, \bar{S}_j=1\mid \bar{z}_j]}{\Pr[\bar{S}_j=1\mid \bar{z}_j]}.
\end{align}
$E[Y_j, \bar{S}_j=1 \mid \bar{z}_j]$ is obtained by marginalizing over the distributions of $(\bar{y}_{j-1}, \bar{w}_{j}, \bar{r}_{j}, x_{0} )$. That is,
\begin{align*}
E&[Y_j(\bar{z}_j), \bar{S}_j(\bar{z}_j)=1] \nonumber \\
& = E[Y_j(\bar{z}_j), \bar{S}_{j-1}(\bar{z}_{j-1})=1 \mid S_{j}(\bar{z}_{j-1})=1]\Pr[S_{j}(\bar{z}_{j-1})=1] \\
& = \sum_{w_j} E[Y_j(\bar{z}_j), \bar{S}_{j-1}(\bar{z}_{j-1})=1 \mid S_{j}(\bar{z}_{j})=1, w_j] \\
& \quad \times \Pr[S_{j}(\bar{z}_{j})=1 \mid w_j]p(w_j) \\
& = \sum_{r_j}\sum_{w_j} E[Y_j(\bar{z}_j), \bar{S}_{j-1}(\bar{z}_{j-1})=1 \mid S_{j}(\bar{z}_{j})=1, w_j,r_j] \\
& \quad \times \Pr[S_{j}(\bar{z}_{j})=1 \mid w_j,r_j]p(w_j\mid r_j)p(r_j) \\
& = \int_{y_{j-1}}\sum_{r_j}\sum_{w_j} E[Y_j(\bar{z}_j), \bar{S}_{j-1}(\bar{z}_{j-1})=1 \mid S_{j}(\bar{z}_{j})=1, w_j,r_j,y_{j-1}] \\
& \quad \times \Pr[S_{j}(\bar{z}_{j})=1 \mid w_j,r_j,y_{j-1}]p(w_j\mid r_j,y_{j-1})p(r_j\mid y_{j-1}) p(y_{j-1}) \\
& \vdots \\
& = \sum_{x_0}\int_{\bar{y}_{j-1}}\sum_{\bar{r}_j}\sum_{\bar{w}_j} E[Y_j(\bar{z}_j)\mid \bar{y}_{j-1},\bar{w}_j,\bar{r}_j,\bar{S}_{j}(\bar{z}_{j})=1, x_0] \\
& \quad \times \prod^j_{k=0} \Bigl( \Pr[S_{k}(\bar{z}_{k})=1 \mid \bar{y}_{k-1},\bar{w}_k,\bar{r}_k,\bar{S}_{k-1}(\bar{z}_{k-1})=1, x_0] \\
& \quad \times
p(w_k\mid \bar{y}_{k-1},\bar{w}_{k-1},\bar{r}_k,\bar{S}_{k-1}(\bar{z}_{k-1})=1,x_0) \\
& \quad \times p(r_k\mid \bar{y}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{S}_{k-1}(\bar{z}_{k-1})=1,x_0) \\
& \quad \times p(y_{k-1}\mid y_{k-2},\bar{w}_{k-1},\bar{r}_{k-1},\bar{S}_{k-1}(\bar{z}_{k-1})=1,x_0) \Bigr) \\
& \quad \times p(x_{0}) d\bar{y}_{j-1} \quad (Assumption \quad 1) \\
& = \sum_{x_0}\int_{\bar{y}_{j-1}}\sum_{\bar{r}_j}\sum_{\bar{w}_j} E[Y_j \mid \bar{y}_{j-1},\bar{w}_j,\bar{z}_{j},\bar{r}_j,\bar{s}_{j}=1, x_0] \\
& \quad \times \prod^j_{k=0}\Bigl( \Pr[S_{k}=1 \mid \bar{y}_{k-1}, \bar{z}_{k}, \bar{w}_k,\bar{r}_k,\bar{s}_{k-1}=1, x_0] \\
& \quad \times
p(w_k\mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_k,\bar{s}_{k-1}=1, x_0) \\
& \quad \times p(r_k\mid \bar{y}_{k-1},\bar{z}_{k-1}, \bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}=1,x_0) \\
& \quad \times p(y_{k-1}\mid y_{k-2},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}=1,x_0) \Bigr) \\
& \quad \times p(x_{0}) d\bar{y}_{j-1} \\
& = E[Y_j, \bar{S}_j=1 \mid \bar{z}_j].
\end{align*}
The expectation $E[Y_j \mid \bar{y}_{j-1}, \bar{z}_{j}, \bar{w}_{j}, \bar{r}_{j}, \bar{s}_{j}=1, x_{0}]$ is identified up to $\gamma_j$ and $c(\bar{z}_j)$ by Assumptions 4 and 5. In particular,
\begin{align}\label{PrZRS1}
E&[Y_j \mid \bar{y}_{j-1}, \bar{z}_{j}, \bar{w}_{j}, \bar{r}_{j}, \bar{s}_{j}=1, x_{0}] \nonumber \\
&= E[Y_j\mid \bar{y}_{j-1}, \bar{z}_{j}, \bar{w}_{j}, \bar{r}_{j}=1, \bar{s}_{j}=1, x_{0}] + I_{(\bar{r}_{j}=\{1,\ldots,1,0\})} \times \gamma_j \nonumber \\
& \quad - c(\bar{z}_j)\times (1 - \Pr[z_j \mid \bar{y}_{j-1}, \bar{z}_{j-1}, \bar{w}_{j}, \bar{r}_{j}=1, \bar{s}_{j}=1, x_{0})).
\end{align}
Note that, by Assumption 4 we have that
\begin{align*}
p&(y_{k-1} \mid \bar{y}_{k-2}, \bar{z}_{k-1},\bar{w}_{k-1}, \bar{r}_{k-2}=1,r_{k-1}=0, \bar{s}_{k-1}=1,x_0) \\
& = p(y_{k-1} + \gamma_{k-1} \mid \bar{y}_{k-2}, \bar{z}_{k-1},\bar{w}_{k-1}, \bar{r}_{k-1}=1, \bar{s}_{k-1}=1,x_0),
\end{align*}
and for $t<k-1$ we have that
\begin{align*}
p&(y_{k-1} \mid \bar{y}_{k-2}, \bar{z}_{k-1},\bar{w}_{k-1}, \bar{r}_{t-1}=1, \{r_t=0,\ldots,r_{k-1}=0 \}, \bar{s}_{k-1}=1,x_0)\\
& = p(y_{k-1} \mid \bar{y}_{k-2}, \bar{z}_{k-1},\bar{w}_{k-1}, \bar{r}_{k-1}=1, \bar{s}_{k-1}=1, x_0).
\end{align*}
By Assumption 4, for the time-varying confounder we have that
\begin{align*}
p&(w_k \mid \bar{y}_{k-1}, \bar{z}_{k-1}, \bar{w}_{k-1}, \bar{r}_{t-1}=1, \{r_t=0,\ldots,r_{k-1}=0 \}, \bar{s}_{k}=1, x_0) \\
& = p(w_k \mid \bar{y}_{k-1}, \bar{z}_{k-1}, \bar{w}_{k-1}, \bar{r}_{k}=1, \bar{s}_{k}=1, x_0)
\end{align*}
for $t\leq k-1$.
For identification of the denominator in [\ref{expYS}] we have that $\Pr[\bar{S}_j=1\mid \bar{z}_j]$ is obtained by marginalizing over the distributions of $(\bar{y}_{j-1}, \bar{w}_{j}, \bar{r}_{j}, x_{0} )$. That is,
\begin{align*}
\Pr&[\bar{S}_j(\bar{z}_j)=1] \\
& = \sum_{x_0}\int_{\bar{y}_{j-1}}\sum_{\bar{r}_j}\sum_{\bar{w}_j} \prod^j_{k=0} \Bigl( \Pr[S_{k}(\bar{z}_{k})=1 \mid \bar{y}_{k-1},\bar{w}_k,\bar{r}_k,\bar{S}_{k-1}(\bar{z}_{k-1})=1, x_0] \\
& \quad \times
p(w_k\mid \bar{y}_{k-1},\bar{w}_{k-1},\bar{r}_k,\bar{S}_{k-1}(\bar{z}_{k-1})=1,x_0) \\
& \quad \times p(r_k\mid \bar{y}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{S}_{k-1}(\bar{z}_{k-1})=1,x_0) \\
& \quad \times p(y_{k-1}\mid \bar{y}_{k-2},\bar{w}_{k-1},\bar{r}_{k-1},\bar{S}_{k-1}(\bar{z}_{k-1})=1,x_0) \Bigr) \\
& \quad \times p(x_{0}) d\bar{y}_{j-1} \quad (Assumption \quad 1) \\
&= \sum_{x_0}\int_{\bar{y}_{j-1}}\sum_{\bar{r}_j}\sum_{\bar{w}_j} \prod^j_{k=0}\Bigl( \Pr[S_{k}=1 \mid \bar{y}_{k-1}, \bar{z}_{k},\bar{w}_k,\bar{r}_k,\bar{s}_{k-1}=1, x_0] \\
& \quad \times
p(w_k\mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_k,\bar{s}_{k-1}=1,x_0) \\
& \quad \times p(r_k\mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}=1,x_0) \\
& \quad \times p(y_{k-1}\mid \bar{y}_{k-2},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}=1,x_0) \Bigr) \\
& \quad \times p(x_{0}) d\bar{y}_{j-1} \\
& =\Pr[\bar{S}_j=1\mid \bar{z}_j].
\end{align*}
For identification of $\Pr[S_{k}=1 \mid \bar{y}_{k-1}, \bar{z}_{k}, \bar{w}_{k}, \bar{r}_{k}, \bar{s}_{k}=1, x_{0}]$ we first note that, for all individuals who participates at the $k$th wave
\begin{align*}
\Pr[S_k=1 \mid \bar{y}_{k-1}, \bar{z}_{k},\bar{w}_k,\bar{r}_k=1,\bar{s}_{k-1}=1, x_0]=1.
\end{align*}
For those individuals who have dropped out, $r_k=0$, we have that
\begin{align}\label{PrSdrop}
\Pr&[S_k=1 \mid \bar{y}_{k-1},\bar{z}_{k},\bar{w}_k, r_k=0,\bar{r}_{k-1},\bar{s}_{k-1}=1, x_0] \nonumber \\
& = \frac{\Pr[z_k \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_k,r_k=0,\bar{r}_{k-1},\bar{s}_k = 1, x_0]}{\Pr[z_k \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_k,r_k=0,\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0]} \nonumber \\
& \quad \times \Pr[S_k = 1 \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_k,r_k=0,\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0],
\end{align}
where the numerator on the rhs is identified by Assumption 5 and the denominator is a function of $\nu_{z_j}$ in Assumption 8. That is,
\begin{align}
&\Pr[z_k \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_k,r_k=0,\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0] \nonumber \\
& = \nu_j \Pr[S_k=0 \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_k,r_k=0,\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0]\nonumber \\
& \quad + \Pr[z_k \mid \bar{y}_{k-1}, \bar{z}_{k-1},\bar{w}_k,r_k=0,\bar{r}_{k-1}, \bar{s}_{k}=1, x_0] \nonumber \\
& \quad \times \Pr[S_{k}=1 \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_k,r_k=0,\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0].
\end{align}
We further have that the last expression in [\ref{PrSdrop}],
\begin{align*}
&\Pr[S_{k}=1 \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_k,r_k=0,\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0] \\
& = \Pr[w_k \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},r_k=0,\bar{r}_{k-1},\bar{s}_{k}= 1, x_0] \\
& \quad \times \Pr[R_k=0 \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k}= 1, x_0] \\
& \quad \times \Pr[S_{k}=1 \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0] \\
& \quad \times \frac{1}{\Pr[w_k \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},r_k=0,\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0]} \\
& \quad \times \frac{1}{\Pr[R_k=0 \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0]} \quad (Assumption \quad 8) \\
& = \Pr[R_k=0 \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k}= 1, x_0] \\
& \quad \times \Pr[S_{k}=1 \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0] \\
& \quad \times \frac{1}{\Pr[R_k=0 \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0]}, \\
\end{align*}
where
\begin{align*}
& \Pr[R_k=0 \mid \bar{y}_{k-1}, \bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0] \\
& = \Pr[R_k=0 \mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k}= 1, x_0] \\
& \quad \times \Pr[S_k=1 \mid \bar{y}_{k-1}, \bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0] \\
& \quad + \Pr[S_k=0 \mid \bar{y}_{k-1}, \bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}= 1, x_0],
\end{align*}
by using Assumption 5 and 8.
For identification of the last expression in [\ref{Solv}] we first note that $\Pr[\bar{S}_j(\bar{z}'_j)\neq 1 \mid \bar{S}_j(\bar{z}_j)=1]= 0$, by Assumption 6. For the contrasting regime $\bar{z}'_j$, we have that
\begin{align}\label{PrS=S=1}
&\Pr[\bar{S}_j(\bar{z}_j)\neq 1 \mid \bar{S}_j(\bar{z}'_j)=1] \nonumber \\
&= 1 - \Pr[\bar{S}_j(\bar{z}_j)= 1 \mid \bar{S}_j(\bar{z}'_j)=1] \nonumber \\
&= 1- \frac{\Pr[\bar{S}_j(\bar{z}_j)=\bar{S}_j(\bar{z}'_j)= 1)}{\Pr[\bar{S}_j(\bar{z}'_j)= 1]} \nonumber \\
&= 1-\frac{\Pr[\bar{S}_j = 1 \mid \bar{z}_j)}{\Pr[\bar{S}_j = 1 \mid \bar{z}'_j]}
\end{align}
by Assumptions 1 and 6. Because
\begin{align*}
\Pr&[\bar{S}_j(\bar{z}_j)=\bar{S}_j(\bar{z}'_j)=1] \\
& = \sum_{x_0}\int_{\bar{y}_{j-1}}\sum_{\bar{r}_j}\sum_{\bar{w}_j} \prod^j_{k=0} \\
& \quad \Bigl(\Pr[S_{k}(\bar{z}_{k})=S_k(\bar{z}'_j)=1 \mid \bar{y}_{k-1},\bar{w}_k,\bar{r}_k,\bar{S}_{k-1}(\bar{z}_{k-1})=\bar{S}_{k-1}(\bar{z}'_{k-1})=1, x_0] \\
& \quad \times
p(w_k\mid \bar{y}_{k-1},\bar{w}_{k-1},\bar{r}_k,\bar{S}_{k-1}(\bar{z}_{k-1})=\bar{S}_{k-1}(\bar{z}'_{k-1})=1,x_0) \\
& \quad \times p(r_k\mid \bar{y}_{k-1},\bar{w}_{k-1},\bar{S}_{k-1}(\bar{z}_{k-1})=\bar{S}_{k-1}(\bar{z}'_{k-1})=1,x_0) \\
& \quad \times p(y_{k-1}\mid \bar{y}_{k-2},\bar{w}_{k-1},\bar{S}_{k-1}(\bar{z}_{k-1})=\bar{S}_{k-1}(\bar{z}'_{k-1})=1,\bar{r}_{k-1},x_0) \Bigr) \\
& \quad \times p(x_{0}) d\bar{y}_{j-1} \quad (Assumption \quad 6) \\
& = \sum_{x_0}\int_{\bar{y}_{j-1}}\sum_{\bar{r}_j}\sum_{\bar{w}_j} \prod^j_{k=0} \\
& \quad \Bigl(\Pr[S_{k}(\bar{z}_{k})=1 \mid \bar{y}_{k-1},\bar{w}_k,\bar{r}_k,\bar{S}_{k-1}(\bar{z}_{k-1})=1, x_0] \\
& \quad \times
p(w_k\mid \bar{y}_{k-1},\bar{w}_{k-1},\bar{r}_k,\bar{S}_{k-1}(\bar{z}_{k-1})=1,x_0) \\
& \quad \times p(r_k\mid \bar{y}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{S}_{k-1}(\bar{z}_{k-1})=1,x_0) \\
& \quad \times p(y_{k-1}\mid \bar{y}_{k-2},\bar{w}_{k-1},\bar{r}_{k-1},\bar{S}_{k-1}(\bar{z}_{k-1})=1,x_0) \Bigr) \\
& \quad \times p(x_{0}) d\bar{y}_{j-1} \quad (Assumption \quad 1) \\
&= \sum_{x_0}\int_{\bar{y}_{j-1}}\sum_{\bar{r}_j}\sum_{\bar{w}_j} \prod^j_{k=0} \Bigl(\Pr[S_{k}=1 \mid \bar{y}_{k-1}, \bar{z}_{k},\bar{w}_k,\bar{r}_k,\bar{s}_{k-1}=1, x_0] \\
& \quad \times
p(w_k\mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_k,\bar{s}_{k-1}=1,x_0) \\
& \quad \times p(r_k\mid \bar{y}_{k-1},\bar{z}_{k-1},\bar{w}_{k-1},\bar{s}_{k-1}=1,x_0) \\
& \quad \times p(y_{k-1}\mid \bar{y}_{k-2},\bar{z}_{k-1},\bar{w}_{k-1},\bar{r}_{k-1},\bar{s}_{k-1}=1,x_0)\Bigr) \\
& \quad \times p(x_{0}) d\bar{y}_{j-1} \\
& =\Pr[\bar{S}_j=1\mid \bar{z}_j].
\end{align*}
\section*{A.3: Details on algorithm for estimation of $\tau$ in Section 6.}
Here we present an algorithm for estimation of $\tau$ in [\ref{Tau.eq}] using the G-computation formula. The general approach is to specify models for the observed data as we did in Section 5 and then to use assumptions in Section 3 with embedded sensitivity parameters to identify the causal effect estimate as described in Section 4. The algorithm can be summarized in the following five steps:
\begin{enumerate}
\item Sample the observed data posteriors as described in Section 5.
\item MC-sampling. For each posterior sample of the parameters sample pseudo data $(\bar{y}^*_{j-1},\bar{w}^*_{j},\bar{r}_{j}^*,\bar{s}_{j}^*,x_0^*)$ of size $N^*$ as follows:
\begin{enumerate}
\item Sample
$$x_0^* \sim \mathrm{Multi}(N^*, \pi_{x_0}^1,\pi_{x_0}^2,\ldots,\pi_{x_0}^L).$$
\item For a fixed regime $\bar{z}_{j-1}$, sequentially compute conditional expectations using pseudo data and the posterior sample of the model parameters. Further sample new data from this distribution. That is, for $k=0,\ldots,j$ and $j=1,\ldots,J$, sequentially sample new data from:
\begin{align*}
&s_{k}^* \sim \mathrm{Ber}(\pi_{S_k}(\bar{y}^*_{k-1}, \bar{z}_{k-1}, \bar{w}^*_{k-1}, x_0^*)), \\
&r_{k}^* \sim \mathrm{Ber}(\pi_{R_k}(\bar{y}^*_{k-1}, \bar{z}_{k-1}, \bar{w}^*_{k-1}, x_0^*)), \\
&w_{k}^* \sim \mathrm{Ber}(\pi_{W_k}(\bar{y}^*_{k-1}, \bar{z}_{k-1}, \bar{w}^*_{k-1}, x_0^*)), \\
&y^*_{k-1} \sim \mathrm{N}(\mu_{k-1}(\bar{y}^*_{k-2}, \bar{z}_{k-1},\bar{w}^*_{k-1}, x_0^*)+ I_{(r^*_{k-1}=0)} \times \gamma_{k-1}, \sigma_{k-1}^2).
\end{align*}
$s_{k}^*$ and $y^*_{k-1}$ is sampled for the subset that satisfies $\bar{s}_{k-1}^*=1$, while $r_{k}^*$ and $w^*_{k}$ is sampled for the subset that satisfies $\bar{s}_{k}^*=1$. Note that, $s_0^*=1$ and $r_0^*=1$ for all individuals, and that $s_k^*=0$ if $s_{k-1}^*=0$ and $r_k^*=0$ if $r_{k-1}^*=0$. Moreover, one set is sampled from the prior distribution of the sensitivity parameter $\gamma_{k-1}$. Further, sample a set from prior distributions of the sensitivity parameters $\gamma_j$, $c(\bar{z}_j)$, $\Delta_{j}$, and $\nu_{z_j}$.
\end{enumerate}
\item Implement G-computation for $\bar{z}_j$, and similarly for $\bar{z}'_j$, using the pseudo data from Step 2 by estimating the parameters of interest:
\begin{enumerate}
\item Compute $E[Y_j \mid \bar{y}_{j-1}, \bar{z}_j, \bar{w}_{j},\bar{r}_{j}, \bar{s}_j=1, x_0]$
for a fixed regime $\bar{z}_j$, denoted by $\phi_j(\bar{z}_j)$ as follows:
\begin{align*}
\phi_j(\bar{z}_j)=&\mu_j(\bar{y}^*_{j-1}, \bar{z}_j, \bar{w}^*_{j}, x^*_0) + I_{(r^*_j=0)} \times \gamma_j \\
&- c(\bar{z}_j)\times (1-\pi_{z_j}(\bar{y}^*_{j-1}, \bar{z}_{j-1}, \bar{w}^*_{j}, x^*_0)).
\end{align*}
\item Compute $\prod^j_{k=0} \Pr[S_{k}=1 \mid \bar{z}_{k}, \bar{w}_k,\bar{r}_k,\bar{y}_{k-1},\bar{S}_{k-1}=1, x_0]$
, denoted by $\chi_j(z_j)$.
For the subset that satisfies $r_j^*=1$,
$$\chi_k(z_k)=1,$$
otherwise, when $r_j^*=0$,
\begin{align*}
\chi_j(z_j)=\prod^j_{k=0} \frac{\pi_{z_{k}}(\bar{y}^*_{k-1}, \bar{z}_{k-1}, \bar{w}^*_{k}, x_0^*)A_k}{B_k}
\end{align*}
where
\begin{align*}
&A_k=(1 - \pi_{R_k}(\bar{y}^*_{k-1}, \bar{z}_{k-1}, \bar{w}^*_{k-1}, x_0^*))\pi_{S_k}(\bar{y}^*_{k-1}, \bar{z}_{k-1}, \bar{w}^*_{k-1}, x_0^*)\\
&\qquad \times 1/[(1 - \pi_{R_k}(\bar{y}^*_{k-1}, \bar{z}_{k-1}, \bar{w}^*_{k-1}, x_0^*))\pi_{S_k}(\bar{y}^*_{k-1}, \bar{z}_{k-1}, \bar{w}^*_{k-1}, x_0^*) \\
&\qquad +(1-\pi_{S_k}(\bar{y}^*_{k-1}, \bar{z}_{k-1}, \bar{w}^*_{k-1}, x_0^*))],
\end{align*}
and
\begin{align*}
&B_k=\pi_{z_{k}}(\bar{y}^*_{k-1}, \bar{z}_{k-1}, \bar{w}^*_{k}, x_0^*)(\nu_{z_k} - \nu_{z_k} A_k + A_k).
\end{align*}
\item Implement Monte Carlo integration using the pseudo data to compute $\Pr[\bar{S}_{j}=1 \mid \bar{z}_j]$
, denoted by $\hat{p}_{s_j \mid \bar{z}_j}$:
\begin{align*}
\hat{p}_{s_j \mid \bar{z}_j}=\frac{\sum \chi_j(\bar{z}_j)}{N^*}.
\end{align*}
\item Implement Monte Carlo integration using the pseudo data to compute $E[Y_j, \bar{S}_j=1 \mid \bar{z}_j]$
, denoted by $\hat{\mu}_{y_j, \bar{s}_j=1 \mid \bar{z}_j}$:
\begin{align*}
\hat{\mu}_{y_j, \bar{s}_j=1 \mid \bar{z}_j}= \frac{\sum \phi_j(\bar{z}_j)\chi_j(\bar{z}_j)}{N^*}.
\end{align*}
\end{enumerate}
\item Use the quantities in step (a)-(d) above to compute one posterior sample of $\tau$ as defined in [\ref{sace.eq}]-[\ref{weights}]:
\begin{align*}
\sum_{j=1}^J \frac{\hat{p}_{s_j \mid \bar{z}_j}}{\sum_{k=1}^J \hat{p}_{s_k \mid \bar{z}_k}} \Bigg\{ \frac{\hat{\mu}_{y_j, \bar{s}_j=1 \mid \bar{z}_j}}{\hat{p}_{s_j \mid \bar{z}_j}} - \frac{\hat{\mu}_{y_j, \bar{s}_j=1 \mid \bar{z}'_j}}{\hat{p}_{s_j \mid \bar{z}'_j}}
- \Delta_{j} \left(1- \frac{\hat{p}_{s_j\mid \bar{z}_j}}{\hat{p}_{s_j\mid \bar{z}'_j}}\right)\Bigg\}.
\end{align*}
\item Repeat step 1 - 4 for each of the posterior sample of the parameters.
\end{enumerate}
\end{document}
|
{
"timestamp": "2019-03-01T02:03:43",
"yymm": "1902",
"arxiv_id": "1902.10787",
"language": "en",
"url": "https://arxiv.org/abs/1902.10787"
}
|
\section{Introduction}
We consider the reachability problem for Linear Time Invariant (LTI) with Integral Quadratic Constraints (IQC). Reachable set computation is an active field of research in control theory (see \cite{blanchini2008set}). It has many applications such as state estimation (see \cite{jaulin2001applied}) or verification (see \cite{bayen2007aircraft}) of dynamical systems.
IQC is a classical tool of robust control theory (see e.g. \cite{megretski1997system,megretski2010kyp}). It can model complex systems (infinite state dimension or non-linear dynamics) such as delays, rate limiters and uncertain systems (see \cite{helmersson1999iqc,megretski1997integral,peaucelle2009integral} and \cite{ariba2017}). Up to now, IQC have mainly been used to evaluate the stability of systems. Despite their modeling power, we still lack tools to manipulate such systems: computing their reachable set is challenging.
In this paper, we extend reachability analysis based on ellipsoidal techniques (see e.g. \cite{chernousko1999,kurzhanski2002ellipsoidal,kurzhanskiy2007ellipsoidal}) for LTI systems subject to an IQC.
This IQC is a trajectory constraint (i.e. valid at any time) between past state-trajectory, input signals and unknown disturbance signals.
To override dealing with constraints over the state-trajectories, we study the LTI system augmented with a state corresponding to the integral term in the IQC.
For a given parabolic set of initial states, the reachable set of the augmented system is overapproximated by a time-varying parabolic set.
Parameters of this paraboloid are expressed as the solution of an Initial Value Problem (IVP) partially described by a Riccati differential equation.
This paraboloid is a tight overapproximation of the reachable set as it is supported by the reachable set on so-called \textit{touching trajectories}.
By studying touching trajectories that are close to violating the constraint, we find conditions to generate all the supporting time-varying parabolic sets. At a given time, the intersection of these supporting parabolic sets is an \textit{exact} representation of the reachable set.
\bigskip
\paragraph*{Related work}
Reachability analysis of LTI systems with ellipsoidal bounded inputs is studied in \cite{chernousko1999,kurzhanski2002ellipsoidal,kurzhanskiy2007ellipsoidal}. Such systems can model infinity norm bounded input-output LTI systems. The reachable set (which is convex and bounded; see \cite{kurzhanski2002ellipsoidal}) can be overapproximated with time-varying ellipsoidal sets. Each ellipsoid is described by its parameters (center and radius) that are solution of an IVP. These parameters produce tight ellipsoids (i.e., ellipsoids touching the reachable set) which are external approximations of the reachable set. When multiple ellipsoids with different touching trajectories are considered, their intersection is a strictly smaller overapproximation of the reachable set. The accuracy of the overapproximation can be made arbitrarily small by adding more well chosen ellipsoids. The exact representation of the reachable set is possible by using a infinite set of ellipsoids.
An optimal control formulation of the reachable set problem is also possible \cite{lee1967foundations,gusev2017extremal}. For some given cost function (usually linear in the case of hyperplane constraints), the maximal cost reached through the system flow for a given set of initial states defines a constraint over the reachable set: any state of the reachable set has lower cost. This optimization problem can be locally solved (see e.g. with the Pontryagin Maximum Principle -PMP-, see \cite{lee1967foundations,graettinger1991hyperplane,varaiya2000reach,gusev2017extremal}) leading to local description of the reachable set boundary. It also can be solved globally (using Hamilton-Jacobi-Bellman -HJB- viscosity subsolutions for example, see \cite{soravia2000viscosity}) leading to global constraints over the reachable set. If the reachable set can be expressed as the intersection (possibly uncountable) of elements of the chosen function family, then the intersection of the resulting constraints gives an exact representation of the reachable set.
HJB and PMP based methods propagate the constraints along the flow of the dynamical system. Occupation measures and barrier certificates methods aim at finding constraints over the reachable tube of a dynamical system: \cite{prajna2004safety} uses IQCs for verification purposes using barrier certificates where the positivity of the energetic state is ensured by using a nonnegative constant multiplier: \cite{henrion2014convex,korda2016moment} use an occupation measure approach where the integral constraint can be incorporated as a constraint over the moment of the trajectories. A hierarchy of semi-definite conditions are derived for polynomial dynamics. Then, off-the-shelf semi-definite programming solvers are used to solve the feasibility problem. Optimization-based methods do not usually take advantage of the model structure as they consider a large class of systems (convex, Lipschitz or polynomial dynamics for example).
The study of LTI systems with two norm bounded energy is closely related to the Linear Quadratic Regulator (LQR) problem. In the LQR problem, a quadratic integral is minimized at the terminal time. Optimal trajectories belong to a time-varying parabolic surface, whose quadratic coefficients are solution of a Riccati differential equation. \cite{savkin1995recursive} describes the reachable set of LTI systems with terminal IQC.
\paragraph*{Contributions}
We study the reachable set computation of an LTI system with IQC. To the knowledge of the authors, this is the first paper to provide a set-based solution for reachable set computation for LTI systems with IQC.
\begin{itemize}
\item We extend the existing ellipsoidal method for reachability analysis of bounded-inputs LTI systems to reachability analysis of LTI systems with IQC. These parabolic constraints are defined by time-varying parameters which are solution of an IVP. Part of this IVP (the quadratic coefficient of the parabolic constraint) is a Riccati differential equation. The IVP convergence property is obtained thanks to the convergence property of the Riccati differential equation.
\item These parabolic constraints are external approximations of the reachable set. The use of parabolic set is particularly suited to the system of interest: the approximation is tight in the sense that each constraint stays in contact with the boundary of the reachable set. We exhibit these touching trajectories. Under some conditions, the reachable set is exactly described by the intersection of well chosen time-varying paraboloid.
\end{itemize}
\paragraph*{Plan}
The LTI system with temporal IQC and the reachability analysis problem are introduced (Section~\ref{sec:problem_statement}). Parabolic constraints and their associated parameter IVP are defined, their domain of definition is analyzed, the overapproximation property is formulated, as well as the touching trajectories (Section~\ref{sec:paraboloid_over_approx}). A method to generate a set of time-varying parabolic constraints is described. The intersection of these paraboloids exactly describes the reachable set of the system (Section~\ref{sec:exact_reach_set}). An example of a stable system is described (Section~\ref{sec:examples}).
\subsection{Notation}
\newcommand{\mge}{\succ}%
\newcommand{\mle}{\prec}%
\newcommand{\norm}[1]{\lVert #1 \rVert}
\newcommand{\mathrm{tr}}{\mathrm{tr}}
Let $\mathbb{S}^{n} \subset \mathbb{R}^{n \times n}$ denote the set of real valued symmetric square matrices of size $n$. For $A \in \mathbb{S}^{n}$, we write $A \mge 0$ (resp. $A \mle 0$) when $A$ is positive definite (resp. negative definite).
We define the matrix norm $\norm{A} = \sqrt{\mathrm{tr}(A^\top A)}$ for $A \in \mathbb{R}^{n \times m}$, where $\mathrm{tr}(B)$ is the trace of $B \in \mathbb{R}^{n \times n}$.
Let a $n$-vector valued \textit{signal} be a function that associates to a time instant in $\left[0,+\infty\right[$ a vector from $\mathbb{R}^n$.
For a given interval $I \subseteq \mathbb{R}$, let $\mathbf{L}_2(I;\mathbb{R}^n)$ denote the Hilbert space of signals equipped with the norm:
\newcommand{\abs}[1]{\left| #1 \right|}
$
\norm{\sig{u}} = \sqrt{\int_{t\in I} \sig{u}^T(t) \sig{u}(t) dt} < \infty.
$
For a set $\Omega \subset \mathbb{R}^n$, let $\boundary \Omega$ denote its boundary.
\newcommand{\mathscr{C}}{\mathscr{C}}
Let $\mathscr{C}^1(I;\mathbb{R}^n)$ the set of functions from $I$ to $\mathbb{R}^n$ which are continuous and differentiable with continuous derivative.
\label{sec:problem_statement}
\newcommand{\extstate}[2]{(#1,#2)}
\newcommand{\Ld(\R^m)}{\mathbf{L}_2(\mathbb{R}^m)}
\newcommand{\Ld(\R^p)}{\mathbf{L}_2(\mathbb{R}^p)}
\newcommand{\Ld(\R^m)}{\mathbf{L}_2(\mathbb{R}^m)}
\subsection{System}
For a given input signal $\sig{u} \in \mathscr{C}^1(\R^+;\mathbb{R}^p)$, given matrices $A\in\mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, $B_u \in \mathbb{R}^{n \times p}$, and a given terminal time $t>0$, we study the trajectories $\sig{x} \in \mathbf{L}_2([0,t];\mathbb{R}^n)$ of the LTI system:
\newcommand{\tau}{\tau}
\newcommand{s}{s}
\begin{equation}
\label{eq:ode}
\left\{
\begin{aligned}
\dot{\sig{x}}(\tau) &= A \sig{x}(\tau) + B \sig{w}(\tau) + B_u \sig{u}(\tau) && \textrm{with $\tau \in [0,t]$}\\
\sig{x}(0) &= x_0
\end{aligned}
\right.
\end{equation}
where $\sig{w} \in \mathbf{L}_2([0,t];\mathbb{R}^m)$ is an unknown disturbance that satisfies:
\begin{equation}
\label{eq:cons}
x_{q0} +
\int_0^\tau
\left[\begin{smallmatrix}\sig{x}(s)\\\sig{u}(s)\\\sig{w}(s)\end{smallmatrix}\right]^\top
M
\left[\begin{smallmatrix}\sig{x}(s)\\\sig{u}(s)\\\sig{w}(s)\end{smallmatrix}\right]
ds \geq 0 \textrm{ for all $\tau \in [0,t]$}
\end{equation}
for given initial conditions $\extstate{x_0}{x_{q0}} \in \mathbb{R}^n \times \R^+$, and given matrix
\begin{equation}
\label{def:M_matrix}
M =
\begin{bmatrix}
M_x & M_{xu} & M_{xw} \\ M_{xu}^\top & M_{u} & M_{uw} \\ M_{xw}^\top & M_{uw}^\top & M_{w}
\end{bmatrix} \in \mathbb{S}^{n+m+p}
\end{equation}
with $M_{w} \mle 0$.
In this work, the constraint \eqref{eq:cons} is expressed as a constraint over a state $\sig{x_q} \in \mathbf{L}_2([0,t];\mathbb{R})$ defined for $s \in [0,t]$ by:
\begin{equation}
\label{eq:xq_int}
\sig{x_q}(\tau) =
x_{q0} +
\int_0^\tau
\left[\begin{smallmatrix}\sig{x}(s)\\\sig{u}(s)\\\sig{w}(s)\end{smallmatrix}\right]^\top
M
\left[\begin{smallmatrix}\sig{x}(s)\\\sig{u}(s)\\\sig{w}(s)\end{smallmatrix}\right]
ds,
\end{equation}
then
\begin{equation}
\label{eq:state_cons}
\sig{x_q}(\tau) \geq 0 \textrm{ for all $\tau \in [0,t]$.}
\end{equation}
The constrained dynamical system $\mathscr{S}(\mathcal{X}_0,t) $ is then defined for a given set of initial states $\mathcal{X}_0 \subset \mathbb{R}^n \times \mathbb{R}$ and a terminal time $t>0$:
\begin{equation}
\label{def:cons_sys}
\extstate{\sig{x}}{\sig{x_q}} \in \mathscr{S}(\mathcal{X}_0,t)
\Leftrightarrow
\left\{
\begin{array}{l}
\textrm{$\sig{x}$ is solution of \eqref{eq:ode}}\\
\textrm{and $\sig{x_q}$ is solution of \eqref{eq:xq_int}}\\
\textrm{with $\extstate{x_0}{x_{q0}} \in \mathcal{X}_0$}\\
\textrm{$\sig{x_q}$ satisfies \eqref{eq:state_cons}}
\end{array}
\right.
\end{equation}
\newcommand{\reach}{\mathcal{R}}%
Let the reachable set be defined by:
\begin{equation}
\label{eq:reach_set}
\reach(\mathcal{X}_0,t) = \pb{\pp{\sig{x}\pp{t},\sig{x_q}\pp{t}} \middle| \left(\sig{x},\sig{x_q}\right) \in \mathscr{S}(\mathcal{X}_0,t)}.
\end{equation}
Then, $\reach(\mathcal{X}_0,t) \subseteq \mathcal{X}_+$ where $\mathcal{X}_+$ is the subset of the state-space where the constraint $x_q \geq 0$ is satisfied:
$$ \mathcal{X}_+ = \mathbb{R}^n \times \R^+.$$
\newcommand{\parset}{\Pi}%
\newcommand{\parabol}{\mathcal{P}}%
\newcommand{\parinter}{\parset^\cap}%
\newcommand{\parabolSet}{\mathbb{P}}%
\newcommand{\ssig{f}}{\ssig{f}}
\newcommand{\fc^\T}{\ssig{f}^\top}
\renewcommand{\r}{\ssig{g}}
\newcommand{\ssig{E}}{\ssig{E}}
\newcommand{f}{f}
\newcommand{g}{g}
\newcommand{E}{E}
\newcommand{\dot{\fc}}{\dot{\ssig{f}}}
\newcommand{\dot{\r}}{\dot{\r}}
\newcommand{\dot{\E}}{\dot{\ssig{E}}}
\subsection{Paraboloids}
We overapproximate the reachable set $\reach(\mathcal{X}_0,t)$ of $\mathscr{S}(\mathcal{X}_0,t)$ with \textit{paraboloids}:
\begin{definition}[Paraboloid]
\label{def:semiparabol}
Given $(E,f,g) \in \mathbb{S}^{n} \times \mathbb{R}^n \times \mathbb{R}$, define the \textit{value function}:
\begin{equation*}
\begin{array}{rcl}
h : &\mathbb{R}^n \times \mathbb{R} &\to \mathbb{R}\\
&\extstate{x}{x_q} & \mapsto x^\top E x - 2 f^\top x + g +x_q,
\end{array}
\end{equation*}
and the paraboloid:%
\begin{equation*}
\label{eq:semiparabol}
\parabol(E,f,g) = \pb{ \extstate{x}{x_q} \in \mathbb{R}^{n+1} \middle| h\extstate{x}{x_q} \leq 0}.
\end{equation*}
\end{definition}
\newcommand{\scaling}{\gamma}%
\newcommand{\dxqs}{\sig{\dxq^*}}%
\newcommand{\ddxqs}{\ddot{x}_q^*}%
\newcommand{\tp}{l}%
Let $\parabolSet = \pb{\parabol(E,f,g) \middle | E \in \mathbb{S}^n, f \in \mathbb{R}^n, g \in \mathbb{R}}$ be the set of paraboloids.
\begin{definition}[Scaled Paraboloid]
\label{def:scaled_paraboloid}
For $\parabol \in \parabolSet$ with parameters $(E,f,g)$ and a scaling factor $\scaling > 0$, let $\scaling \parabol \in \parabolSet$ be the scaled paraboloid defined by parameters $(\scaling E,\scaling f,\scaling g)$.
\end{definition}%
Scaled paraboloids satisfy the following:
\begin{property}
\label{prop:scaled_overapprox}
Given $\parabol \in \parabolSet$ and $\scaling \geq 1$, it holds $\parabol \cap \mathcal{X}_+ \subseteq \scaling \parabol \cap \mathcal{X}_+$.
\end{property}
\begin{proof}
Let $h$ and $h'$ (resp.) the value functions of $(E,f,g) = \parabol$ and $\scaling \parabol$ (resp.) evaluated at $\extstate{x}{x_q} \in \parabol$. Since $\extstate{x}{x_q} \in \parabol$, $h \leq 0$, i.e. $x^\top E x - 2 f^\top x + g \leq -x_q$. Then, $h' = \scaling (x^\top E x - 2 f^\top x + g) + x_q \leq - (\scaling-1) x_q$. Since $\extstate{x}{x_q} \in \mathcal{X}_+$ and since $\scaling-1 \geq 0$, we have $(\scaling-1) x_q \geq 0$ i.e. $h' \leq 0$ meaning that $\extstate{x}{x_q} \in \scaling \parabol \cap \mathcal{X}_+$.
\end{proof}
For $P$ a time-dependent subset of $\mathcal{X}_+$ that associates to a time $t$ of an interval $I \subset \R^+$ a subset of $P(t)$ of $\mathcal{X}_+$, we define a \textit{touching trajectory}:
\begin{definition}[Touching Trajectory]
A trajectory $\sig{X}^*$ solution of (\ref{eq:ode},\ref{eq:xq_int}) is a touching trajectory of $P$ when $\sig{X}^*(t)$ belongs to the surface of $P(t)$ at any time $t \in I$, i.e. $\sig{X}^*(t) \in \boundary P(t)$.
\end{definition}%
\subsection{Problem Statement}
We are now ready to state the problems which are respectively solved in Theorem~\ref{thm:overapproximation} (in Section~\ref{sec:paraboloid_over_approx}) and Theorem~\ref{thm:exact_reachable_set} (in Section~\ref{sec:exact_reach_set}), the main results of our paper.
\begin{problem}
\label{pb:approx_reach_set}
Find an overapproximation of the reachable set $\reach(\parabol_0,t)$ at any $t>0$ for a given paraboloid of initial conditions $\parabol_0 \in \parabolSet$.
\end{problem}
\begin{problem}
\label{pb:exact_reach_set}
Find an \textit{exact} characterization of the reachable set $\reach(\parabol_0,t)$ at any $t>0$ for a given paraboloid of initial conditions $\parabol_0 \in \parabolSet$.
\end{problem}
\section{Overapproximation with Paraboloids}
In this section, Problem~\ref{pb:approx_reach_set} is solved using time-varying paraboloids $P: I\rightarrow \parabolSet$ where $I$ is the interval of definition of $P$. Parameters $(\ssig{E}(\cdot),\ssig{f}(\cdot),\r(\cdot))$ of $P(\cdot)$ are solution of a Riccati differential equation that guarantees an overapproximation relationship with the reachable set, i.e. $\reach(\parabol_0,t) \subseteq P(t)$ for any $t \in I$. Existence and domain of definition $I$ of $P$ are expressed. We prove that the overapproximations $P$ are \textit{tight} since there are so-called \textit{touching trajectories} of $\reach(\parabol_0,t)$ that both belong to the surface of $P(t)$ and to the surface of $\reach(\parabol_0,t)$ for $t\in I$. Finally, the method is presented for a simple toy example.
\bigskip
\label{sec:paraboloid_over_approx}
\newcommand{\qi}{q^{\scalebox{0.6}{-1}}}%
\newcommand{\Msc}{M^{sc}}%
\newcommand{\Mw}{M_{w}}%
\newcommand{\Mwi}{\Mw^{\scalebox{0.6}{-1}}}%
\newcommand{\Mxw}{M_{xw}}%
\newcommand{\Mx}{M_{x}}%
\newcommand{\Mxwt}{M_{xw}^\top}%
\newcommand{\Qi}{Q^{\scalebox{0.6}{-1}}}%
\newcommand{\xw}{\smat{x\\w}}%
\newcommand{\xwt}{\smat{x\\w}^\top}%
\newcommand{\Bt}{B^\top}%
\newcommand{\At}{A^\top}%
\newcommand{x_c}{x_c}%
\newcommand{\xct}{x_c^\top}%
\newcommand{\dot{x}_c}{\dot{x}_c}%
\newcommand{\xq}{\sig{x_q}}%
\newcommand{\dxq}{\dot{x}_q}%
\newcommand{\dq}{\dot{q}}%
\newcommand{\dQ}{\dot{Q}}%
\newcommand{\xxc}{\pp{x-x_c}}%
\newcommand{\xxct}{\xxc^\top}%
\newcommand{\Px}{p_x}%
\newcommand{\Pu}{p_u}%
\newcommand{\PxMsc}{\Px \Msc}%
\newcommand{\PxMscPx}{\Px^\top \Msc \Px}%
\newcommand{\xu}{\smat{x\\u}}%
\newcommand{\xut}{\smat{x\\u}^\top}%
\newcommand{\xcu}{\smat{x_c\\u}}%
\newcommand{\xcut}{\smat{x_c\\u}^\top}%
\newcommand{\xuw}{\smat{x\\u\\w}}%
\newcommand{\xuwt}{\smat{x\\u\\w}^\top}%
\newcommand{\Bu}{B_u}%
\newcommand{\Muw}{M_{uw}}%
\newcommand{\Muwt}{\Muw^\top}%
\newcommand{\Mu}{M_{u}}%
\newcommand{\Mxu}{M_{xu}}%
\newcommand{\Mscu}{M^{sc}_u}%
\newcommand{\pu}{p_u}%
\newcommand{\Mxuwt}{\smat{ \Mxw \\ \Muw}^\top}%
\newcommand{\xuws}{\smat{x\\u\\{w^*}}}%
\newcommand{\xuwst}{\xuws^\top}%
Parameters of $P$ are expressed as the solutions of an initial value problem. For given $E_0 \in \mathbb{S}^n$, let $\ssig{E}$ be the solution of the following Riccati differential equation with initial condition $\ssig{E}(0) = E_0$:
\begin{equation}
\label{eq:eq_diff_E}
\scalebox{0.99}{\mbox{\ensuremath{\displaystyle
\begin{aligned}
\dot{\E}(t) = & - \ssig{E}(t) A - A^\top \ssig{E}(t) - \Mx\\
& + \pp{\Bt \ssig{E}(t) + \Mxwt}^\top \Mwi \pp{\Bt \ssig{E}(t) + \Mxwt}.
\end{aligned}
}}}
\end{equation}%
\newcommand{\Tdef}[1]{T_{P}(#1)}%
\newcommand{\TdefE}[1]{T_{E}(#1)}%
\newcommand{\IdefE}[1]{[0,\TdefE{#1}[}%
\newcommand{\Idef}[1]{\mathcal{I}(#1)}%
Let $\IdefE{E_0}$ be the interval of definition of \eqref{eq:eq_diff_E}'s solutions (existence, uniqueness, convergence properties and continuity of the solution are studied in \cite{kuvcera1973review}).
Let $\ssig{f}$ denote the solution of the following IVP with initial condition $\ssig{f}(0) = f_0$:
\begin{equation}
\label{eq:eq_diff_fc}
\begin{split}
\dot{\fc}(t) = &- A^\top \ssig{f}(t) + (\Mxu + \ssig{E}(t) \Bu )\sig{u}(t)\\
&+ (\ssig{E}(t) B + \Mxw ) \Mwi (\Bt \ssig{f}(t) -\Muwt \sig{u}(t) ).
\end{split}
\end{equation}
$\ssig{f}$ satisfies a Linear Varying Parameters differential equation with a continuous input signal. On $\IdefE{E_0}$, solution $\ssig{f}$ to \eqref{eq:eq_diff_fc} exists, it is unique and continuous.
By continuity of $\ssig{f}$ and $\sig{u}$ over $\IdefE{E_0}$, $\r$ is defined on $\IdefE{E_0}$.
For $t \in \IdefE{E_0}$, let:
\begin{equation}
\label{eq:eq_diff_g}
\r(t) = g_0 + \int_{0}^t \smat{\ssig{f}(\tau)\\\sig{u}(\tau)}^\top G \smat{\ssig{f}(\tau)\\\sig{u}(\tau)} d\tau
\end{equation}
where
$
G = \begin{bmatrix}
B \Mwi \Bt & \Bu - B \Mwi \Muwt\\ (\Bu - B \Mwi \Muwt)^\top & \Mu - \Muw \Mwi \Muwt
\end{bmatrix}.
$
\newcommand{F}{F}
\newcommand{\mathcal{F}_{\E}}{\mathcal{F}_{\ssig{E}}}
\newcommand{\mathcal{A}_{\fc}}{\mathcal{A}_{\ssig{f}}}
\newcommand{\mathcal{B}_{\fc}}{\mathcal{B}_{\ssig{f}}}
\newcommand{\mathcal{F}_{\r}}{\mathcal{F}_{\r}}
\newcommand{\mathcal{T}}{\mathcal{T}}
\begin{definition}[Time-Varying Paraboloid]
\label{def:eq_diff_well_defined}
For an initial paraboloid $\parabol_0 \in \parabolSet$, let the time-varying paraboloid
\[
\begin{array}{@{}r@{}l@{}}
P \colon I &\to \parabolSet\\
t &\mapsto \parabol(\ssig{E}(t),\ssig{f}(t),\r(t))
\end{array}
\]
be defined by the time-varying coefficients $(\ssig{E},\ssig{f},\r)$ solutions of (\ref{eq:eq_diff_E},\ref{eq:eq_diff_fc},\ref{eq:eq_diff_g}) with initial condition $\parabol(E_0,f_0,g_0) = \parabol_0
$.
Let $P = \mathcal{T}(\parabol_0)$ be the function that associates to initial paraboloid the time-varying paraboloid. Let $\Tdef{P} = \TdefE{E_0}$ and $\Idef{P} = [0, \Tdef{P}[$ be the interval of definition of $P$.
\end{definition}
\newcommand{h_{X}}{h_{X}}
\newcommand{h_{X^*}}{h_{X^*}}
\newcommand{\dot{h}_{X}}{\dot{h}_{X}}
\newcommand{\dot{h}_{X^*}}{\dot{h}_{X^*}}
For $P = \mathcal{T}(\parabol_0)$, let $h(t,\cdot)$ the value function of $P(t)$ at $t \in \Idef{P}$. For $X_t = \extstate{x_t}{x_{q,t}} \in \mathbb{R}^{n+1}$, $w_t \in \mathbb{R}^m$, let $h_{X}(t,w_t) = h(t,\sig{X}(t))$ be the value function along the trajectory $\sig{X} = \extstate{\sig{x}}{\sig{x_q}}$ solution of (\ref{eq:ode},\ref{eq:xq_int}) generated by $\sig{w}$ such that $\sig{w}(t) = w_t$ and $\sig{X}(t) = X_t$.
\begin{property}
\label{prop:max_dhX}
The maximum time derivative of the value function $h(t,\sig{X}(t))$ along the trajectories $\sig{X}$ for a disturbance $w_t$ at $t$ exists and it is equal to zero.
\end{property}
\begin{proof}
The time derivative of $h_{X}$ is the quadratic function:
\newcommand{H(t)}{H(t)}
\begin{equation}
\label{eq:dhtX}
\dot{h}_{X} = \smat{x_t\\\sig{u}(t)\\w_t}^\top H(t) \smat{x_t\\\sig{u}(t)\\w_t}
\end{equation}%
\newcommand{w_{\Pi}}{P_w}%
where $H$ is obtained using (\ref{eq:ode},\ref{eq:xq_int},\ref{eq:eq_diff_E}-\ref{eq:eq_diff_g}). $H(t)$ is a function of $\ssig{E}(t)$, $\ssig{f}(t)$ and $\r(t)$. The quadratic coefficient in $w_t$ is $\smat{0\\0\\I_m}^\top H(t) \smat{0\\0\\I_m} = \Mw$. Since $\Mw \mle 0$, the supremum of $w_t \mapsto \dot{h}_{X}(t,w_t)$ exists and is reached for $w_t = \sig{{w^*}}(t) = \argmax_{w_t \in \mathbb{R}^m} \dot{h}_{X}(t,w_t)$ with:
\begin{equation}
\label{def:ws}
\sig{{w^*}} = - \Mwi \pp{\Bt (\ssig{E} \sig{x}-\ssig{f}) + \Mxuwt \smat{\sig{x}\\\sig{u}}}.
\end{equation}
Using (\ref{eq:eq_diff_E},\ref{eq:eq_diff_fc},\ref{eq:eq_diff_g}) in \eqref{eq:dhtX}, we get $\max_{w_t \in \mathbb{R}^m} \dot{h}_{X} = 0$.
\end{proof}
\bigskip
We can now state one of our main results:
\begin{theorem}[Solution to Problem~\ref{pb:approx_reach_set}]
\label{thm:overapproximation} Let $P = \mathcal{T}(\parabol_0)$ for a set of initial states $\parabol_0$. The reachable set $\reach(\parabol_0,t)$ of $\mathscr{S}(\parabol_0,t)$, $t > 0$, is overapproximated by $P(t)$, i.e.:
\[
\forall t \in \Idef{P}, \reach(\parabol_0,t) \subseteq P(t) \cap \mathcal{X}_+
.\]
\end{theorem}%
\begin{proof}
\label{subsec:eq_diff_uncentered}
Using Property~\ref{prop:max_dhX}, by integration of $\dot{h}_{X}$, if $h_{X}(0) \leq 0$ then $ \forall t \in \Idef{P}, h_{X}(t) \leq 0$, i.e.:
$$\sig{X}(0) \in P(0) \Rightarrow \sig{X}(t) \in P(t) \textrm{ for all $t \in \Idef{P}$}.$$
The constraint \eqref{eq:state_cons} ensures that $\sig{X}(t) \in \mathcal{X}_+$.
\end{proof}
\begin{property}
Let $\sig{X^*}$ be a trajectory generated by $\sig{{w^*}}$ defined in \eqref{def:ws} such that initial condition satisfies $\sig{X^*}(0) \in \boundary \parabol_0$. At any time $t\in \Idef{P}$, $\sig{X^*}$ satisfies $\sig{X^*}(t) \in \boundary P(t)$.
\end{property}%
\begin{proof}
$\sig{X^*}$ is the trajectory generated by the optimal disturbance $\sig{{w^*}}$. Using Property~\ref{prop:max_dhX}, $\dot{h}_{X^*} = 0$. Since $h_{0,\sig{X^*}(0)} = 0$, by integration, $h_t(\sig{X^*}(t)) = 0$.
\end{proof}
Trajectories generated by $\sig{{w^*}}$ defined in \eqref{def:ws} stay in contact with the surface of their time-varying paraboloids. Touching trajectories of $P$ do not necessarily belong to $\mathscr{S}(\parabol_0,t)$, $t \in \Idef{P}$, as the energetic constraint might be locally violated.
\begin{remark}
Property~\ref{prop:max_dhX} and (\ref{eq:eq_diff_E},\ref{eq:eq_diff_fc},\ref{eq:eq_diff_g}) can be derived solving the following optimal control problem (for $t>0$):
\begin{equation*}
\begin{array}{rl}
\displaystyle\max_{\sig{w} \in \mathbf{L}_2([0,t];\mathbb{R}^m)}{} & \displaystyle\int_0^t \smat{\sig{x}(\tau)\\\sig{u}(\tau)\\\sig{w}(\tau)} M \smat{\sig{x}(\tau)\\\sig{u}(\tau)\\\sig{w}(\tau)} d\tau - x_{q,t}\\
\textrm{s.t.} & \dot{\sig{x}} = A \sig{x} + B \sig{w} + B_u \sig{u}\\
& \sig{x}(t) = x_t\\
\end{array}
\end{equation*}
for given $\extstate{x_t}{x_{q,t}} \in \mathcal{X}_+$. This is a special instance of the LQR problem (see e.g. \cite{savkin1995recursive}).
\end{remark}
\begin{remark}[Representation of paraboloids]
In \cite{savkin1995recursive}, the time-varying value function is a quadratic function defined by its quadratic coefficient $\ssig{S}$, its center $\ssig{x_c}$ and its value at the center $\ssig{\rho}$. $\ssig{S}$, $\ssig{x_c}$ and $\ssig{\rho}$ satisfied an IVP. In this formulation, the center $\ssig{x_c}$ can diverge when the determinant of $\ssig{S}$ vanishes. However, the corresponding time-varying value function is time-continuous and can be extended continuously. In this paper, we choose to work with variables $\ssig{E}$, $\ssig{f}$ and $\r$ (see Definition~\ref{def:semiparabol}) to avoid this issue.
\end{remark}
\begin{example}
\label{ex:1dsys}
Let
$A = -1$,
$B = 1$,
$M = \smat{1&0&0\\0&1&0\\0&0&-2}$,
$B_u = 0$ and
$\sig{u}: [0,\infty[ \mapsto 0$.
Solutions of ODE~\eqref{eq:eq_diff_E} (that is $\dot{\E} =-\frac{1}{2} \ssig{E}^2 + 2 \ssig{E} -1$) diverge for $E_0<E^-$ (see Figure~\ref{fig:stab_care}) where $E^-<E^+$ are the roots of the equation $-\frac{1}{2} E^2 + 2 E -1 = 0$ for $E \in \mathbb{R}$, $E^- = 2-\sqrt{2}$ and $E^+ = 2+\sqrt{2}$.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{StabilityCARE.pdf}
\caption{\label{fig:stab_care}Convergence analysis of \eqref{eq:eq_diff_E}'s solutions for Example~\ref{ex:1dsys}}
\end{figure}
Figure~\ref{fig:bounding_paraboloid_high_energy} shows the trajectory of the paraboloid for $E_0$ in the stable region $E_0>E^-$ while Figure~\ref{fig:bounding_paraboloid_low_energy} shows the trajectories of the paraboloid for $E_0$ in the unstable region $E_0<E^-$.
\end{example}
\begin{figure*}
\newcommand{0.24\textwidth}{0.23\textwidth}
\centering
\includegraphics[angle=90,scale=0.4]{dummy_legend}
\includegraphics[width=0.24\textwidth]{touching_parabolla_001}
\includegraphics[width=0.24\textwidth]{touching_parabolla_010}
\includegraphics[width=0.24\textwidth]{touching_parabolla_017}
\includegraphics[width=0.24\textwidth]{touching_parabolla_100}
\caption{\label{fig:bounding_paraboloid_high_energy}%
Time-varying paraboloid overapproximates the reachable set at different time instants $t$ in $\{0.00,0.91,1.62,10.00\}$ for an initial maximum energetic level of $x_{q,0} = 0.06$. The solution of \eqref{eq:eq_diff_E} converges to a constant value when $t \rightarrow +\infty$. The shaded regions are the reachable set $\reach(\parabol_0,t)$, the thin green lines are the boundary of the overapproximation $P(t)$ of Theorem~\ref{thm:overapproximation}.}
\end{figure*}
\begin{figure*}
\newcommand{0.24\textwidth}{0.23\textwidth}
\centering
\includegraphics[angle=90,scale=0.4]{dummy_legend}
\includegraphics[width=0.24\textwidth]{touching_parabolla_diverge_001}
\includegraphics[width=0.24\textwidth]{touching_parabolla_diverge_010}
\includegraphics[width=0.24\textwidth]{touching_parabolla_diverge_017}
\includegraphics[width=0.24\textwidth]{touching_parabolla_diverge_100}
\caption{\label{fig:bounding_paraboloid_low_energy}%
Time-varying paraboloid overapproximates the reachable set at different time instants $t$ in $\{0.00,0.91,1.62,10.00\}$ for an initial maximum energetic level of $x_{q,0} = 0.03$. The solution of \eqref{eq:eq_diff_E} has a finite escape time and diverge at $t = 1.68$. The shaded regions are the reachable set $\reach(\parabol_0,t)$, the thin green lines are the boundary of the overapproximation $P(t)$ of Theorem~\ref{thm:overapproximation}.}
\end{figure*}
\section{Exact Reachable Set Computation}
\label{sec:exact_reach_set}
{
\newcommand\defstatefx[3]
\expandafter\def\csname X#1\endcsname{\sig{#2{X}^{#3}}
\expandafter\def\csname X#1s\endcsname{\sig{#2{X}^{#3*}}
\expandafter\def\csname x#1\endcsname{\sig{#2{x}^{#3}}
\expandafter\def\csname x#1s\endcsname{\sig{#2{x}^{#3*}}
\expandafter\def\csname xq#1\endcsname{\sig{#2{x}^{#3}_q}
\expandafter\def\csname xq#1s\endcsname{\sig{#2{x}_q^{#3*}}
\expandafter\def\csname dxq#1s\endcsname{\dot{\sig{#2{x}}_q^{#3*}}
\expandafter\def\csname P#1\endcsname{#2{P}^{#3}
\expandafter\def\csname E#1\endcsname{#2{\ssig{E}}^{#3}
\expandafter\def\csname fc#1\endcsname{#2{\ssig{f}}^{#3}
\expandafter\def\csname fc#1t\endcsname{#2{\ssig{f}}^{#3\top}
\expandafter\def\csname r#1\endcsname{#2{\r}^{#3}
\expandafter\def\csname P#1sup\endcsname{#2{P}^{#3}_+
\expandafter\def\csname P#1inf\endcsname{#2{P}^{#3}_-
\expandafter\def\csname h#1\endcsname{#2{h}^{#3}
\expandafter\def\csname dh#1\endcsname{\dot{#2{h}}^{#3}
\expandafter\def\csname w#1s\endcsname{\sig{#2{w}^{#3*}}
}%
\defstatefx{a}{}{}%
\defstatefx{b}{\tilde}{}%
\defstatefx{c}{}{\prime}%
In this section, a set of time-varying paraboloids is defined. At a given time, the intersection of the paraboloids gives better overapproximations of the reachable set. With additional assumptions, the reachable set is exactly characterized. This approach relies on the use of Property~\ref{thm:overapproximation} and preliminary results showing that for any state of the overapproximation, there exists a trajectory in $\mathscr{S}(\parabol_0,t)$, $t>0$, leading to this state.
Let $\parset$, a set of time-varying paraboloids (Definition~\ref{def:eq_diff_well_defined}), be defined by:
\newcommand{\maxscaling}{\overline{\scaling}}%
\newcommand{\dot{\sig{x}}^{\sig{*},\gamma}_\sig{q}}{\dot{\sig{x}}^{\sig{*},\gamma}_\sig{q}}
\newcommand{\dot{\sig{x}}^{\sig{*},\maxscaling}_\sig{q}}{\dot{\sig{x}}^{\sig{*},\maxscaling}_\sig{q}}
\newcommand{\sig{x}^{\sig{*},\gamma}_\sig{q}}{\sig{x}^{\sig{*},\gamma}_\sig{q}}
\newcommand{\sig{w}^{\sig{*},\gamma}}{\sig{w}^{\sig{*},\gamma}}
\newcommand{\sig{x}^{\sig{*},\gamma}}{\sig{x}^{\sig{*},\gamma}}
\newcommand{\dot{\sig{x}}^{\sig{*},1}_\sig{q}}{\dot{\sig{x}}^{\sig{*},1}_\sig{q}}
\newcommand{\epsilon_q}{\epsilon_q}
\begin{equation}
\label{def:exact_reach_set}
\parset = \pb{\mathcal{T}(\scaling \parabol_0) \middle| \scaling \geq 1, \exists \extstate{x}{x_{q}} \in \parabol_0^{\epsilon_q}, \dot{\sig{x}}^{\sig{*},\gamma}_\sig{q}(0) \geq 0}
\end{equation}
where $\parabol_0^{\epsilon_q}$ is a small set of states near $\boundary \parabol_0$ in the half-plane $x_q \leq 0$:
\begin{equation*}
\parabol_0^{\epsilon_q} = \pb{ \extstate{x}{x_q} \middle|
\begin{aligned}
-x^\top E_0 x + 2f^\top_0 x - g_0 &\in [-\epsilon_q,0]\\
x_q &\in [-\epsilon_q,0]
\end{aligned}
}
\end{equation*}
where $(E_0,f_0,g_0) = \parabol_0$, $\epsilon_q > 0$, and $(\sig{x}^{\sig{*},\gamma},\sig{x}^{\sig{*},\gamma}_\sig{q})$ the touching trajectory of $\scaling \parabol_0$ s.t.
$\extstate{\sig{x}^{\sig{*},\gamma}}{\sig{x}^{\sig{*},\gamma}_\sig{q}}(0) = \extstate{x}{x_{q}}$.
$\parset$ is defined such that each touching trajectory of rising energy belongs to a time-varying paraboloid $P$ of $\parset$.
Direct computation gives $\dot{\sig{x}}^{\sig{*},\gamma}_\sig{q}(0) = \scaling^2 a + \scaling b + c$ where $a < 0 $ (consequence of $\Mw \mle 0$) and $b$ and $c$ in $\mathbb{R}$. If there is a $(x,x_q) \in \parabol_0^{\epsilon_q}$ such that $\dot{\sig{x}}^{\sig{*},1}_\sig{q}(0) > 0$, since $\parabol_0^{\epsilon_q}$ is bounded and since $a<0$, there exists a $\maxscaling > 1$ such that $\forall (x,x_q) \in \parabol_0^{\epsilon_q}, \dot{\sig{x}}^{\sig{*},\maxscaling}_\sig{q}(0) \leq 0$. Therefore, the set of scalings $\scaling \geq 1$ such that $\dot{\sig{x}}^{\sig{*},\gamma}_\sig{q}(0) \geq 0$ is $[1,\maxscaling]$. $\parset$ is at most a bounded set of time-varying paraboloids.
Let $\Idef{\parset} \subseteq \R^+$ be the set of time instant $t \in \Idef{\parset}$ where there exists a $P \in \parset$ that is defined at $t$ (i.e. $t \in \Idef{P}$). Since for each $P \in \parset$, $0$ belongs to the interval $\Idef{P}$, we have:
\begin{equation}
\label{def:Idef}
\Idef{\parset} = \bigl[0, \sup_{P \in \parset}\pb{\Tdef{P}} \bigr[
\end{equation}
For $t \in \Idef{\parset}$,
let
\begin{equation}
\label{def:parinter}
\parinter(t) = \bigcap_{
P \in \parset
\textrm{ s.t. }
t \in \Idef{P}
} P(t)
\end{equation}
the intersection of all the defined time-varying paraboloids $P$ of $\parset$ at time $t$ (see Figure~\ref{fig:describe_Pi}).
\begin{figure}[ht!]
\centering
\includegraphics[width=\columnwidth]{describe_Pi}
\caption{\label{fig:describe_Pi}For $t\geq0$, $P_i \in \parset$, $i=1,2,3$. Light color shaded area are time-varying constraints of $\parset$ at $t$. Grey color shaded are is their intersection $\parinter(t)$.}
\end{figure}
\bigskip
We now prove that when some assumption about boundedness of \eqref{eq:eq_diff_E}'s solutions (Assumption~\ref{hyp:E_bounded}) and touching trajectories behavior around the null energetic surface (Assumption~\ref{hyp:neg_ddxq}) holds, then $\reach(\parabol_0,t) = \parinter(t) \cap \mathcal{X}_+$, for given $t \in \Idef{\parset}$ (Theorem~\ref{thm:exact_reachable_set}, Section~\ref{ssec:exact_reach}). To achieve that:
\begin{itemize}
\item we prove the overapproximation relationship $\reach(\parabol_0,t) \subseteq \parinter(t)$ (Section~\ref{ssec:overapprox});
\item we prove that any state $\extstate{x}{x_q} \in \parinter(t)$ is reachable from a state $\extstate{x}{x_q'} \in \boundary \parinter(t)$ (Section~\ref{ssec:interior_state});
\item for a state $X_t \in \boundary \parinter(t)$, we find a touching trajectory $\sig{X^*}$ of $\parinter$ such that $\sig{X^*}(t) = X_t$ (Section~\ref{ssec:surface_state});
\item these touching trajectories $\extstate{\sig{x^*}}{\sig{x_q^*}}$ of $\parinter$ satisfy the state constraint $\sig{x_q}(\cdot) \geq 0$ over $[0,t]$ (Section~\ref{ssec:valid_past_traj});
\item finally, we conclude that any $X_t \in \parinter(t)$ is reachable from $\parabol_0$, thus $\reach(\parabol_0,t) = \parinter(t) \cap \mathcal{X}_+$ (Section~\ref{ssec:exact_reach}).
\end{itemize}
\subsection{Overapproximation Relationship} \label{ssec:overapprox}
If $\mathcal{Y} \subseteq \mathcal{Z}$ subsets of $\mathbb{R}^{n+1}$, then $\reach(\mathcal{Y},t) \subseteq \reach(\mathcal{Z},t)$. This result is stated in Property~\ref{prop:higher_energy_overapprox} for the specific case of scaled paraboloids (see Definition~\ref{def:scaled_paraboloid}).
\begin{property}
\label{prop:higher_energy_overapprox}
For a set of initial states $\parabol_0 \in \parabolSet$ and a scaling factor $\scaling \geq 1$, let $P_\scaling = \mathcal{T}(\scaling \parabol_0)$.
For any trajectory $\sig{X} \in \mathscr{S}(\parabol_0,t)$, it holds $\sig{X}(t) \in P_\scaling(t)$ for all $t \in \Idef{P_\scaling}$.
\end{property}
\begin{proof}
Using Property~\ref{prop:scaled_overapprox} and Theorem~\ref{thm:overapproximation}, $\sig{X}(t) \in P(t)$ for any $t \in \Idef{P}$.
\end{proof}
As each time-varying paraboloid is an overapproximation of the reachable set, the intersection of these paraboloids is as well an overapproximation.
\begin{property}
\label{prop:overapprox_inter}
$\reach(\parabol_0,t) \subseteq \parinter(t)$ for any $t \in \Idef{\parset}$.
\end{property}
\begin{proof}
This is a direct consequence of Property~\ref{prop:higher_energy_overapprox}.
\end{proof}
\begin{example}[Continued from Example~\ref{ex:1dsys}] \label{ex:1dsys_cont}
In the case where the solution of \eqref{eq:eq_diff_E} does not converge (i.e. $E_0<\ssig{E}^-$), Figure~\ref{fig:energy_levels} shows several paraboloid trajectories with different initial energetic levels (i.e. different initial scaling). Since all the scalings $\scaling_i$ are greater than $1$, $\parabol_0 \cap \mathcal{X}_+ \subset \scaling_i \parabol_0 \cap \mathcal{X}_+$. Therefore, each time-varying paraboloid is a valid constraint that bounds $\reach(\parabol_0,t)$, $t \in \Idef{\parset}$ (Theorem~\ref{thm:overapproximation}). Therefore, $\reach(\parabol_0,t) \subseteq P^\cap(t) = P_0(t) \cap P_1(t) \cap \dots \cap P_4(t)$ where $P_i = \mathcal{T}(\scaling_i \parabol_0)$, and $\scaling_i$ are resp. equal to $1$, $1.6$, $2.2$, $2.7$ and $3.3$ for $i = 0,\dots,4$. In this case, the overapproximation $P^\cap(t)$ is strictly included in $P_{0}(t)$.
\end{example}
\begin{figure*}[t]
\newcommand{0.24\textwidth}{0.23\textwidth}
\centering
\includegraphics[angle=90,scale=0.4]{dummy_legend}
\includegraphics[width=0.24\textwidth]{touching_parabolla_energy_levels_001}
\includegraphics[width=0.24\textwidth]{touching_parabolla_energy_levels_010}
\includegraphics[width=0.24\textwidth]{touching_parabolla_energy_levels_020}
\includegraphics[width=0.24\textwidth]{touching_parabolla_energy_levels_100}
\caption{\label{fig:energy_levels}%
Time-varying paraboloids overapproximate the reachable set at different time instants $t$ in $\{0.00,0.91,1.62,10.00\}$ for different initial conditions $P_i(0) = \scaling_i \parabol_0$, where scaling factors $\scaling_i$ are respectively equal to $1.0$, $1.6$, $2.2$, $2.7$ and $3.3$ for $i=0,\dots,4$. The shaded regions are the reachable set $\reach(\parabol_0,t)$, the thin green lines are the boundary of the overapproximation $P(t)$ of Theorem~\ref{thm:overapproximation}.}
\end{figure*}
Observations in Example~\ref{ex:1dsys_cont} motivate the use of multiple time-varying paraboloids to get better overapproximation of the reachable set $\reach(\parabol_0)$.
\subsection{Past trajectory for states in the interior of $\parinter(t)$} \label{ssec:interior_state}
\begin{property}
\label{prop:continuous}
For a trajectory $\extstate{\sig{x}}{\sig{x_q}} \in \mathscr{S}(\parabol_0,T)$, $T>0$, $\sig{x}$ is time-continuous.%
\end{property}
\begin{proof}
Let $f: t,x \mapsto A x + B \sig{w}(t) + B_u \sig{u}(t)$. Since for any $(t,x) \in \R^+ \times \mathbb{R}^n$, $f(.,x)$ is locally measurable over $\R^+$, $f(t,.)$ is Lipschitz over $\mathbb{R}^n$, \eqref{eq:ode} has a unique solution $\sig{x}$ (see \cite{schuricht2000ordinary}, Theorem 1.1) that is time-continuous.
\end{proof}
\newcommand{\ti}{0}%
\newcommand{\tf}{t}%
\newcommand{\Xe}{\mathcal{X}_\epsilon}%
\newcommand{\g}{r}%
\newcommand{\dg}{\dot{r}}%
\newcommand{\xs}{\sig{x^*}}%
\newcommand{\xqs}{\sig{\xq^*}}%
\newcommand{\Delta}{\Delta}
\newcommand{\mu}{\mu}
\newcommand{\epsilon}{\epsilon}
\newcommand{\delta}{\delta}
$\sig{x}$ trajectories are time-continuous, however, this in not necessarily true for $\sig{x_q}$ trajectories: $\sig{x_q}$ might have steps due to sudden release of the energy through the disturbance $\sig{w}$. We use the following property to prove that the state $\extstate{x}{\alpha x_q}$ is reachable from the state $\extstate{x}{x_q}$ for any given $\alpha \in [0,1]$.
\begin{property}
\label{prop:consume_energy}
For $t \geq 0$, if $\extstate{x}{x_q} \in \reach(\parabol_0,t)$ then $\extstate{x}{\alpha x_q} \in \reach(\parabol_0, t)$ for all $\alpha \in [0,1]$.
\end{property}
\begin{proof}
For $\epsilon>0$, let $\sig{w} \in \mathbf{L}_2([0,t+\epsilon];\mathbb{R}^m)$, s.t. $\sig{w}^\top(s) \Mw \sig{w}(s) = -(1-\alpha) \sig{x_q}(t) \frac{1}{\epsilon}$ when $s \in [t,t+\epsilon]$. Then $\int_t^{t+\epsilon} \sig{w}^\top(s) \Mw \sig{w}(s) ds \rightarrow -(1-\alpha) \sig{x_q}(t)$ when $\epsilon \rightarrow 0$. Using Cauchy-Schwartz inequality:
$
\abs{\int_t^{t+\epsilon} (-\Mw)^{\frac{1}{2}} \sig{w}(s) ds} \leq \sqrt{\int_t^{t+\epsilon} 1 ds} \sqrt{\int_t^{t+\epsilon} -\sig{w}^T(s) \Mw \sig{w}(s) ds}
$
and the time-continuity of $\sig{x}$ (from Property~\ref{prop:continuous}), the quantity $\int_t^{t+\epsilon} \smat{\sig{x}(s)\\\sig{u}(s)\\0}^\top M \smat{\sig{x}(s)\\\sig{u}(s)\\\sig{w}(s)} ds \rightarrow 0$ when $\epsilon \rightarrow 0$. By integration, $\sig{x}_q(t+\epsilon) \rightarrow \alpha \sig{x_q}(t)$ when $\epsilon \rightarrow 0$. Since $\sig{x}$ is continuous (Property~\ref{prop:continuous}), $\sig{x}(t+\epsilon) \rightarrow \sig{x}(t)$ when $\epsilon \rightarrow 0$.
By continuity of $\sig{u}$, $\sig{x}$ and $\sig{w}$ over $[t,t+\epsilon]$, $\sig{x_q}$ is continuous over $[t,t+\epsilon]$. Then, there exists a $t' \in [t,t+\epsilon]$ such that $\sig{x_q}(\tau) \geq \alpha \sig{x_q}(t) \geq 0$ for all $\tau \in [t,t']$ and $\sig{x_q}(t') \rightarrow \alpha \sig{x_q}(t)$ when $\epsilon \rightarrow 0$. Therefore, the constraint $\sig{x_q}(\cdot)\geq 0$ is satisfied over $[t,t']$ and the trajectory $(\sig{x},\sig{x_q})$ is a valid trajectory of $\mathscr{S}(\parabol_0,t')$ for all $t \leq t'$.
\end{proof}
\subsection{Past trajectory for states in $\boundary \parinter(t)$} \label{ssec:surface_state}
\renewcommand{\ll}{\lambda}
The value function $\hb$ of a time-varying paraboloid $\Pb$ can be approximated at the first order along a touching trajectory $\sig{X^*}$ of another time-varying paraboloid $\Pa$ when initial conditions $\Pa(0)$ and $\Pb(0)$ are close.
\newcommand{\lambda}{\lambda}
\newcommand{\tilde{\lambda}}{\tilde{\lambda}}
\begin{property}
\label{prop:contact_traj_first_order}
For any $\lambda,\tilde{\lambda} \in [1,\overline{\ll}]$, any $t \in \Idef{\Pa}\cap\Idef{\Pb}$:
\begin{equation*}
\abs{\hb_\tf(\Xas(t)) - (\lambda-\tilde{\lambda}) \lambda^{\scalebox{0.6}{-1}} \sig{\xqas}(\ti)} \leq N (\lambda-\tilde{\lambda})^2
\end{equation*}
where $\Pa = \mathcal{T}(\lambda \parabol_0)$, $\Pb = \mathcal{T}(\tilde{\lambda} \parabol_0)$, $\hb_t$ is the value function of $\Pb(t)$, $\Xas = \extstate{\xas}{\xqas}$ is a touching trajectory of $\Pa$ and $N>0$ a scalar.
\end{property}
\begin{proof}
Let $\sig{n} = (\Ea-\Eb) \sig{\xas} - (\fca - \fcb)$. Using (\ref{eq:ode},\ref{def:ws},\ref{eq:eq_diff_E}-\ref{eq:eq_diff_g}), $\sig{n}$ satisfies the linear time varying differential equation: $\dot{\sig{n}} = (-A^\top + \Mxw \Mwi \Bt + \Eb B \Mwi \Bt)\sig{n}$.
Since $t$ belongs to $\Idef{\Pb}$ and by time-continuity of $\Eb(\cdot)$ over $[0,t]$, there is a scalar $K>0$ that bounds $\norm{\Eb(\cdot)}$ over $[0,t]$. Then, there exists $L>0$ upper bound of $\norm{-A^\top + \Mxw \Mwi \Bt + \Eb(\cdot) B \Mwi \Bt}$ over $[0,t]$. Using the Gr\"onwall inequality, it holds $\norm{\sig{n}(\tau)} \leq e^{L \tau} \norm{\sig{n}(0)}$ for $\tau \in [0,t]$.
Since $\tilde{\lambda}^{\scalebox{0.6}{-1}} \Pb(0) = \lambda^{\scalebox{0.6}{-1}} \Pa(0) = \parabol_0$, it holds $\sig{n}(0) = (\lambda-\tilde{\lambda}) n_0$ with $n_0 = E_0 \xas(0) -f_0$. Therefore, $\norm{\sig{n}(\tau)} \leq \abs{\lambda-\tilde{\lambda}} e^{L \tau} \norm{n_0}$.
Along the touching trajectory $\Xas = \extstate{\xas}{\xqas}$ of $\Pa$ and by using (\ref{eq:eq_diff_E}-\ref{eq:eq_diff_g}), $\dhb_t$ is equal to :
$
\dhb_t(\Xas(t)) = \sig{n}^\top(t) B \Mwi \Bt \sig{n}(t)
$.
By integration, we have $\abs{ \hb_t(\Xas(t)) - \hb_0(\Xas(0)) } \leq N (\lambda-\tilde{\lambda})^2$, where $R = \abs{n_0 B \Mwi \Bt n_0} (2L)^{\scalebox{0.6}{-1}} e^{2 L T}$ a finite constant (since $\Mw \mle 0$).
\newcommand{\phiEfc}{\Phi_{\ssig{E},\ssig{f}}}%
\newcommand{\klipscht}{\kappa}%
Since $\Xas$ is a touching trajectory of $\Pa$:
$\lambda (\xas^\top(\ti) E_0 \xas(\ti) - 2 f_0^\top \xas(\ti) + g_0) + \xqas(\ti) = 0$.
Direct computation gives: $\hb_\ti(\Xas(\ti)) = (\lambda-\tilde{\lambda} )\lambda^{\scalebox{0.6}{-1}} \xqas(\ti)$.
Thus, $\abs{\hb_\tf(\Xas(t)) - (\lambda-\tilde{\lambda} ) \lambda^{\scalebox{0.6}{-1}} \xqas(\ti)} \leq N(\lambda-\tilde{\lambda})^2 $.
\end{proof}
When Property~\ref{prop:contact_traj_first_order} holds, if
$N (\lambda-\tilde{\lambda})^2 \leq (\lambda-\tilde{\lambda} )\lambda^{\scalebox{0.6}{-1}} \xqas(\ti)$,
then the sign of $\hb_\tf(\Xas(t))$ is equal the sign of $(\lambda-\tilde{\lambda})\lambda^{\scalebox{0.6}{-1}} \xqas(\ti)$. Since $\hb_\tf(\Xas(t))>0 \Rightarrow \Xas(t) \notin \parinter(t)$, Property~\ref{prop:contact_traj_first_order} provides a way to identify states that do not belongs to $\parinter(t)$.
\begin{property}
\label{prop:reject}
Let $\sig{X^*}$ a touching trajectory of $\Pa = \mathcal{T}(\lambda \parabol_0)$ (where $\lambda \in [1,\overline{\ll}]$ given) and $t \in \Idef{\Pa}$ given. If there is a $\tilde{\lambda} \in [1,\overline{\ll}]$, s.t. $t \in \Idef{\Pb}$ (where $\Pb = \mathcal{T}(\tilde{\lambda} \parabol_0)$) and $\abs{\lambda-\tilde{\lambda}} \le \abs{ N^{\scalebox{0.6}{-1}} \xqas(\ti)}$, then
\begin{equation*}
(\lambda-\tilde{\lambda}) \lambda^{\scalebox{0.6}{-1}} \xqas(\ti) > 0 \Rightarrow \hb_\tf(\Xas(t)) > 0
\end{equation*}
where $\hb$ is the value function of $\Pb$ and $N>0$ a scalar.
\end{property}
\begin{proof}
This is a direct consequence of Property~\ref{prop:contact_traj_first_order} and of the property: $(\abs{a-b} \leq c) \land (c < \abs{b}) \Rightarrow (a b > 0)$ for $a,b,c \in \mathbb{R}$.
\end{proof}
The existence of $\tilde{\lambda}$ in Property~\ref{prop:reject} is conditioned by $t$ belonging to $\Idef{\tilde{\lambda} \parabol_0}$. In this work, to ensure the existence of such $\tilde{\lambda}$ at a given time $t \in \Idef{\parset}$, we enforce the boundedness of $\norm{\ssig{E}(\cdot)}$ on $[0,T]$.
\begin{assumption}
\label{hyp:E_bounded}
There is a scalar $K > 0$, such that for any $(\ssig{E},\ssig{f},\g) = P \in \parinter$, $\norm{\ssig{E}(\cdot)}$ is bounded by $K$ on $[0,T]$.
\end{assumption}
The domain of definition of $P \in \parset$ is only defined by the domain of definition of its parameter $\ssig{E}$ (see Section~\ref{sec:paraboloid_over_approx}). Thus, when Assumption~\ref{hyp:E_bounded} holds, we have $[0,T] \subset \Idef{P}$ and therefore $[0,T] \subset \Idef{\parset}$.
Property~\ref{prop:reject} can then be restated when Assumption~\ref{hyp:E_bounded} holds:
\begin{property}
\label{prop:reject_assumption}
Let $P_\ll \in \parset$, $t \in [0,T]$, $\ll \in [1,\overline{\ll}]$, for $X_t \in \boundary P_\ll(t)$ if $X_t \in \parinter(t)$ then the touching trajectory $\sig{X}^*$ of $P$ such that $\sig{X}^*(t) = X_t$ is a touching trajectory of $\parinter$.
\end{property}
\begin{proof}
Since Assumption~\ref{hyp:E_bounded} holds in Property~\ref{prop:reject}, the constant $N$ can be chosen independently from $\Eb$ and $\Ea$ (i.e. from $\Pa$ and $\Pb$). Let $\extstate{\sig{x^*}}{\sig{x_q^*}} = \sig{X^*}$.
Lets assume that either $(\sig{x_q^*}(0)<0) \land (\ll < \overline{\ll})$ or $(\sig{x_q}(0)>0) \land (\ll > 1)$. For both cases, we can choose a $\tilde{\lambda} = \lambda - \eta$,
with $\eta$ s.t. $\eta \sig{x_q^*}(0) > 0$, $\abs{\eta} < \abs{N^{\scalebox{0.6}{-1}} \sig{x_q^*}(0)}$ and
$\tilde{\lambda} \in [1,\overline{\ll}]$.
Since $\tilde{\lambda} \in [1,\overline{\ll}]$, $\Pb \in \parset$. Then Property~\ref{prop:reject} shows that $X_t \notin \Pb(t)$, i.e. $X_t \notin \parinter(t)$! Therefore, either $(\sig{x_q^*}(0)<0) \land (\ll = \overline{\ll})$ or $(\sig{x_q}(0)>0) \land (\ll = 1)$ or $(\sig{x_q^*}(0)=0)$ and $\sig{X^*}(0) \in \boundary \Pa(0)$. Similar computation than in proof of Property~\ref{prop:scaled_overapprox} gives $ \sig{X}(0) \in \ll' \parabol_0$ for any $\ll' \in [1,\overline{\ll}]$. Therefore, $\sig{X}(0)$ belongs to the intersection which is $\parinter(0)$. Finally, thanks to Property~\ref{prop:overapprox_inter}, $\sig{X}$ is a touching trajectory of $\parinter$.
\end{proof}
\bigskip
Since $\parinter(t)$ is an intersection of closed sets, $\parinter(t)$ is closed as well. In the general case, for an infinite intersection $\mathcal{Y}^\cap = \bigcap_{i\in \mathbb{N}} Y_i$ of closed sets $Y_i$, $i \in \mathbb{N}$, any boundary point $y \in \boundary \mathcal{Y}^\cap$ does not necessarily belongs to the boundary of any $Y_i$, $i\in \mathbb{N}$ (e.g. $\bigcap_{\epsilon \in ]1,2]} [-\epsilon,\epsilon] = [-1,1]$, but there is no $\epsilon \in ]1,2]$ such that $1 \in \boundary [-\epsilon,\epsilon]$).
\begin{property}
\label{prop:boundary_reached}
For any $X_t \in \boundary \parinter(t)$, there is a $P \in \parset$ such that $X_t \in \boundary P(t)$.
\end{property}
\begin{proof}
Let $Q(t,x,\ll) = -x^\top \ssig{E}_\ll(t) x + 2 \ssig{f}_\ll^\top(t) x - \r_\ll(t)$ where $(\ssig{E}_\ll,\ssig{f}_\ll,\r_\ll) = P_\ll = \mathcal{T}(\ll \parabol_0)$ with $\ll \in [1,\overline{\ll}]$.
By continuity of (\ref{eq:eq_diff_E},\ref{eq:eq_diff_fc},\ref{eq:eq_diff_g}) solutions, and since Assumption~\ref{hyp:E_bounded} holds, $Q(t,x,\cdot)$ is continuous over the closed interval $[1,\overline{\ll}]$. To this respect, for any $x \in \mathbb{R}^n$, the minimum of $x_q = Q(t,x,\cdot)$ exists and is reached for a $\ll^* \in [1,\overline{\ll}]$.
Therefore, for any $\extstate{x}{x_q} \in \boundary \parinter(t)$, there is $P \in \parset$ s.t. $\extstate{x}{x_q} \in \boundary P(t)$.
\end{proof}
\bigskip
Lemma~\ref{prop:exact_reachable_touching_trajectories} shows that any state $X_t \in \boundary \parinter(t)$ (with $t \in \Idef{\parset}$ given) is the terminal state of a touching trajectory $\sig{X}$ of $\parinter$ with initial state $X_0 \in \boundary \parinter(0)$.
\begin{lemma}
\label{prop:exact_reachable_touching_trajectories}
If Assumption~\ref{hyp:E_bounded} holds, any state $X_t \in \boundary \parinter(t)$ has a past touching trajectory $\sig{X}$ of $\parinter$.
\end{lemma}
\begin{proof}
\newcommand{\lambda}{\lambda}
\newcommand{\overline{\lambda}}{\overline{\lambda}}
This is a direct consequence of Assumption~\ref{hyp:E_bounded}, Property~\ref{prop:reject_assumption} and Property~\ref{prop:boundary_reached}.
\end{proof}
\subsection{Past trajectory constraint $\sig{x_q}(\cdot) \geq 0$} \label{ssec:valid_past_traj}
Touching trajectories of $P \in \parset$ with initial state in $\parinter(0)$ are also touching trajectories of $\parinter$ (Lemma~\ref{prop:exact_reachable_touching_trajectories}). We enforce the touching trajectories $\extstate{\sig{x}^*}{\sig{x_q}^*}$ of $\parinter$ to satisfy the constraint $\sig{x_q^*}(\cdot) \geq 0$ by assuming that no touching trajectory has a rising $\sig{x_q^*}$ state close to the the null plan $x_q = 0$.
\begin{assumption}[Falling touching trajectories]
\label{hyp:neg_ddxq}
Any touching trajectory $\extstate{\xs}{\xqs}$ of $\parinter$ have a strictly decreasing energetic state on the null energetic surface:
$$\xqs(t) \in [-\epsilon_q, 0] \Rightarrow \dxqs(t) < 0.$$
\end{assumption}
This assumption was found reasonable for several stable IQC systems study.
\newcommand{\systouch}{\mathcal{T}^*}%
We use the following intermediate result:
\begin{property}
\label{prop:barrier}
Consider a function $f: [0,1] \to \mathbb{R}$ continuous and differentiable over $[0,1]$ with a continuous derivative $f'$ over $[0,1]$ such that $f$ satisfy $f(0) \leq 0$ and $\forall x \in [0,1], (f(x) \in [-\epsilon,0]) \Rightarrow f'(x)<0$, for a $\epsilon>0$. Then $\forall x \in [0,1], f(x) < 0$.
\end{property}
\begin{proof}
Let $x\in [0,1]$ such that $f(x) \in [\epsilon,0]$. Since $f'(x)<0$ and $f'$ is continuous, there exists a neighborhood $[x,x+\eta]$, $\eta>0$, such that $f'(\cdot) < 0$ over $[x,x+\eta]$. Therefore, by integration $f(\cdot) \leq 0$ over $[x,x+\eta]$. $f(x) \in [-\epsilon,0] \Rightarrow \exists \eta>0, \forall y\in [x,x+\eta], f(y) \leq 0$. Since $f$ is continuous and $f(0)<0$, any function $f$ non negative would violate this statement (direct consequence of the intermediate value theorem).
\end{proof}
\subsection{Exact characterization of the reachable set} \label{ssec:exact_reach}
Exact characterization of $\reach(\parabol_0,t)$ by $\parinter(t)$ for $t \in \Idef{\parset}$ is guaranteed since ownership of touching trajectories is proven with Property~\ref{thm:overapproximation}, non-ownership is guaranteed locally by Property~\ref{prop:contact_traj_first_order}. Remaining trajectories of $\mathscr{S}(\parabol_0,t)$ can be constructed from Property~\ref{prop:consume_energy} and touching trajectories of $\parset$. Finally, all the trajectories satisfy the constraint since touching trajectories satisfy Assumption~\ref{hyp:neg_ddxq} and cannot violate the constraint $\sig{x_q}(\cdot) \geq 0$ temporarily.
\begin{theorem}[Solution to Problem~\ref{pb:exact_reach_set}]
\label{thm:exact_reachable_set}
If Assumption~\ref{hyp:E_bounded} and \ref{hyp:neg_ddxq} hold, then $\reach(\parabol_0,t) = \parinter(t) \cap \mathcal{X}_+$ for any $t \in [0,T]$ where $\parset$, $\parinter$ and $\Idef{\parset}$ are defined by (\ref{def:exact_reach_set},\ref{def:parinter},\ref{def:Idef}).
\end{theorem}
\begin{proof}
Thanks to Property~\ref{prop:consume_energy}, for $X_t \in \parinter(t)$, if the projection of $X_t$ over $\boundary \parinter(t)$ is reachable, then $X_t$ is reachable.
Since Assumption~\ref{hyp:E_bounded} holds, Lemma~\ref{prop:exact_reachable_touching_trajectories} shows that when $X_t \in \boundary \parinter(t)$, there exists a touching trajectory $\sig{X} = \extstate{\sig{x}}{\sig{x_q}}$ of $\parinter$ s.t. $\sig{X}(t) = X_t$.
By continuity of $\ssig{E}$, of the optimal disturbance $\sig{{w^*}}$ of $\sig{X}$, of $\sig{u}$ and of $\sig{x}$, we have $\sig{x_q} \in \mathscr{C}^1([0,t];\mathbb{R})$.
Since Assumption~\ref{hyp:neg_ddxq} holds and $\sig{X}(\tau) \in \boundary \parinter(\tau)$ for all $\tau \in [0,t]$, Property~\ref{prop:barrier} can be applied to $\sig{x_q}$.
The existence of any $\tau \in [0,t]$ such that $\sig{x_q}(\tau) < 0$ would violate Property~\ref{prop:barrier} since the terminal state satisfies $\sig{x_q}(t) \geq 0$! Therefore, $\sig{X} \in \mathscr{S}(\parabol_0,t)$ and $X_t \in \reach(\parabol_0,t)$.
These properties lead to $\parinter(t) \cap \mathcal{X}_+ \subseteq \reach(\parabol_0,t)$. Finally, using Property~\ref{prop:overapprox_inter}, we have $\parinter(t) \cap \mathcal{X}_+ = \reach(\parabol_0,t)$.
\end{proof}
\section{Example}
\label{sec:examples}
We study the stable IQC system
$\mathscr{S}(\parabol_0,t)$, defined in \eqref{def:cons_sys}, at a given time $t$ in $[0,1]$, for a parabolic set of initial states
$\parabol_0 = \parabol(E_0,f_0,g_0)$, with $
E_0 = \smat{a+b&a\\a&a+b}, \,
f_0 = \smat{0\\0}, \,
g_0 = 0.015, \,
a = 10^{-2}\textrm{ and }
b = 10^{-6}
$,
and for the following parameters
$A = -I, \,
B = I, \,
B_u = 0, \,
\sig{u}: \R^+ \mapsto 0 \textrm{ and }
M = \smat{I&0&0\\0&1&0\\0&0&-2 I}
$
where $I = \smat{1&0\\0&1}$.
The reachable set $\reach(\parabol_0,t)$ of $\mathscr{S}(\parabol_0,t)$, defined in \eqref{eq:reach_set}, is computed using \eqref{def:exact_reach_set} and Theorem~\ref{thm:exact_reachable_set}, for $t \in [0,1]$.
Figures~\ref{fig:ex_stable_reach_set_3D} and \ref{fig:ex_stable_reach_set} show the reachable set $\reach(\parabol_0,t)$ set at time $t=0.794$ and its projection $\reach(\parabol_0,t)|_x$ over the LTI state space (i.e. projection over $(x_1,x_2)$ states).
In Figure~\ref{fig:ex_stable_reach_set}, the constraints boundaries $\boundary P(t)$ (for $P \in \parset$, $\parset$ defined in Section~\ref{sec:exact_reach_set}) are touching the reachable set $\reach(\parabol_0,t)$. The non-convexity of $\reach(\parabol_0,t)$ arises from the non-positive solutions of the Riccati differential equation \eqref{eq:eq_diff_E}. Figure~\ref{fig:ex_stable_reach_tube} represents the projection of the reachable tube $t \mapsto \reach(\parabol_0,t)$ projected over the LTI dimension $(x_1,x_2)$.
\begin{figure*}[ht!]
\newcommand{\sizefig}{0.32\textwidth}%
\centering%
\begin{subfigure}[b]{\sizefig}
\includegraphics[width=\textwidth]{3D_reachable_set}%
\caption{\label{fig:ex_stable_reach_set_3D}Reachable set}
\end{subfigure}
\begin{subfigure}[b]{\sizefig}
\includegraphics[width=\textwidth]{DelayRS}%
\caption{\label{fig:ex_stable_reach_set}Reachable set of the LTI system}
\end{subfigure}
\begin{subfigure}[b]{\sizefig}
\includegraphics[width=\textwidth]{ReachableTube}%
\caption{\label{fig:ex_stable_reach_tube}Reachable tube of the LTI system}
\end{subfigure}
\caption{The green surface in (a) is the reachable set $\reach(\parabol_0,t)$ at $t=0.794$ of $\mathscr{S}(\parabol_0,t)$ computed using Theorem~\ref{thm:exact_reachable_set}. Its projection over the LTI state space $(x_1,x_2)$ (in solid red line) is shown in (b), each green line corresponds to one constraint $P \in \parset$ computed with Theorem~\ref{thm:overapproximation}. (c) is the reachable tube $t \rightarrow \reach(\parabol_0,t)$ of $\mathscr{S}(\parabol_0,t)$ projected over the LTI state space $(x_1,x_2)$ for $t \in [0,1]$. The red section corresponds to the time $t = 0.794$.}
\end{figure*}
\section{Conclusion}
\label{sec:conclusion}
In this work, the reachability problem for an LTI system with energetic constraint is solved. The solution is a set-based method that relies on overapproximations with time-varying paraboloids. The paraboloid parameters are expressed as solutions of an IVP that involves a Riccati differential equation. We prove that with assumptions about touching trajectories of the reachable set and boundedness of Riccati differential equation's solutions, the intersection of a well-chosen set of paraboloids exactly describes the reachable set. Our method is tractable and has been used to exhibit the reachable set of a stable system. In some future works, weaker assumptions about the reachable set will be considered, the LTV case will be studied as well as the discrete time case. Most of the research around dynamical systems with integral constraints bring generic solutions, we hope that the linear case gives a better understanding about how such systems behave.
\bibliographystyle{plain}
|
{
"timestamp": "2019-03-01T02:14:29",
"yymm": "1902",
"arxiv_id": "1902.10982",
"language": "en",
"url": "https://arxiv.org/abs/1902.10982"
}
|
\section{Introduction}
Carbon nanotubes (CNT) remain one of the most interesting nanobuilding blocks currently available for macroscopic applications. They can be produced in large quantities as a highly graphitic material with well defined structure and surface chemistry, in some cases with control over their molecular composition in terms of number of layers, diameter and chiral angle. When assembled into macroscopic fibres, the natural embodiment for a one-dimensional material, they have led to materials on par or stronger than conventional high-performance fibres \cite{behabtu2008carbon, koziol2007high, liu2014polymer}, higher thermal conductivity than copper \cite{gspann2017high, behabtu2013strong}, and higher mass-normalised electrical conductivity than most metals \cite{lekawa2014electrical, behabtu2013strong}. Several other applications in energy storage \cite{senokos2017large} and optoelectronic devices exploit their large specific surface and bending compliance. These examples give testimony of the efficient exploitation of the properties of individual CNTs on a macroscopic scale.
Nevertheless, the development of theoretical models able to successfully describe the physical properties of CNT fibres as a function of their structure has proved an elusive challenge. Hence, bulk properties of CNT fibres are still largely optimized by trial-and-error. A fundamental difficulty arises because of the inherently complex hierarchical structure of CNT fibres \cite{vilatela2010yarn}. Such complexity stems from the confluence of many parameters determining bulk properties, including those linked with the physical and chemical properties of constituents (number of layers in CNTs, chiral angle, diameter, presence of impurities, etc), their spatial arrangement (orientation, bundle formation) and interaction between building blocks.
In the context of mechanical properties, comparison of different CNT fibres has led to some agreement on the qualitative effects of different structural features. Higher CNT alignment parallel to the fibre axis was early identified as key to obtain high tensile strength and stiffness \cite{koziol2007high, aleman2015strong, QingwenBradforstretch, chae2008making, Lu2012opportunities}. For fibre spun from arrays of aligned CNTs (forests), tensile strength and modulus generally were observed to increase with increasing CNT length \cite{zhang2007ultrastrong}. This seems reasonable because longer tubes imply fewer tube-ends, which are regarded as defects \cite{zhu2011self}. Similarly, some reports have compared tensile properties of CNT fibres produced from different carbon precursors and thus composed of different constituent CNTs in terms of number of diameter and number of layers, spanning from single-walled (SWNT) to multi-walled (MWNT) CNTs \cite{motta2005mechanical,jia2011comparison}. Based on empirical evidence after fibre optimisation, there is consensus that large diameter few-layer CNTs result in superior fibre axial properties; a consequence of improved CNT packing and maximized contact area upon tube collapse \cite{Motta2017highperformance}. Extensive experimental work by Espinosa and co-workers has focused on multi-scale testing of CNT fibres and subunits to clarify the main factors limiting tensile strength \cite{beese2014key}. Thus, it was found the key role of CNT alignment on yarn performance, as well as the importance of interfacial strength and bundle strength in terms of different failure mechanisms.
A fundamental difficulty to extract qualitative structure-property relations is to decouple the various interlinked features, such as CNT orientation, composition, length and association in crystals. Vilatela \textit{et al.} \cite{vilatela2011model} proposed a model for tensile strength of CNT fibres based on their yarn-like structure \cite{vilatela2010yarn} and fibrillar fracture. It considered an ensemble of parallel rigid rods, with load transfered by shear stresses between fibrous elements, the bundles, until a critical stress produced catastrophic failure by pull-out. Accordingly, fibre strength $\sigma$ was reduced to the product of total contact between load-bearing elements, their length $l$ and shear strength $\tau_F$.
\begin{equation}
\label{eq:equation11}
\sigma = \frac{1}{6}\;\Omega_1\; \Omega_2\; \tau_{F}\; l,
\end{equation}
\noindent where $\Omega_1$ is the fraction of the total number of graphene layers on the outside of the fibrous elements and $\Omega_2$ is the fraction of the outer graphene walls of the elements in contact with neighboring elements. This simple model captured the essense of the failure mechanism and its relation to fibre structure, providing fibre strength predictions (3.5 GPa/SG) in the range observed for small gauge length measurements (5 GPa/SG) \cite{koziol2007high}, but had several limitations, most notably the assumption of perfect CNT orientation.
More recently, Wei et al introduced a modified shear-lag model that predicts fibre strength accounting for the length distribution of load-bearing elements, for both aligned and twisted CNT fibres \cite{wei2015new}. Equipped with a Weibull distribution to take into account the probability of tensile fracture of CNT bundles, the Montecarlo-based model predicts upper bounds on CNT yarn mechanical properties very close to experimental values. The main limitations of this model are that it does not take into account the broad distribution of CNT orientation in real fibres; and its reliance on knowledge of bundle dimensions, which are difficult to determine accurately. However, simulations reveal different failure mechanisms in terms of bundle strength (which in turn is a function of the type of tubes and their arrangement), bundle length and interfacial strength.
In contrast with this plethora of incomplete descriptions, the fibrillar crystallite model developed originally for polymer fibres \cite{northolt1985elastic, northolt2005tensile} can successfully describe the mechanical properties of a wide range of materials, ranging from cellulose to high-performance fibres, including carbon fibres (CF) and rigid-rod polymer fibres. In this work, we show that macroscopic fibres of CNTs can also be treated as ensembles of fibrillary crystals, corresponding to bundles of individual tubes. By studying samples produced with controlled degree of alignment, we show that their tensile properties can be simply determined by the crystal shear strength and modulus and the orientation distribution of crystallites relative to the fibre axis. This provides accurate predictions of fibre modulus, strength and fracture envelope for a range of CNT fibres produced in-house with controlled alignement and compositions, as well as with others reported in the literature. \textit{In-situ} orientation measurements by synchrotron X-ray during tensile testing confirm the accommodation of axial deformation of CNT fibres by crystal stretching and rotation, the core idea of the model.
\section{Experimental}
CNT fibres were synthesized by the direct spinning method whereby an aerogel of CNTs is directly drawn out from the gas phase during growth by floating catalyst chemical vapor deposition (CVD) \cite{Li}. Two sets of fibres were produced, with differences in their constituent CNTs. fibres of few-layer MWNTs were synthesized using butanol as carbon source and adjusting the promotor (sulphur) content accordingly to produce CNTs with the desired number of layers \cite{reguero2014controlling}. fibres of collapsed DWNTs were produced using tolune as carbon precursor. In both cases ferroncene was used as iron catalyst source and Hydrogen as carrier gas. The reaction was carried out at 1250$^{\circ}$C in a vertical tubular furnace reactor. For each sample set, the degree of CNT orientation in the fibre was varied by changing the rate at which the fibres were drawn out of the reactor \cite{aleman2015strong}, equivalent to the winding rate.
The mechanical properties of the different fibre samples were determined from tensile tests on individual CNT fibre filaments, using a gauge length of 20 mm and a strain rate of 2 mm/min. The tests were carried with a Textechno Favimat, equipped with a high-resolution 210 cN load cell. Fibre linear density was determined by weighing a know length of fibre (around 30m) and by using the vibroscopic method.
Small- and wide-angle two-dimensional X-ray scattering patterns were obtained in the Non-crystalline diffraction (NCD) beamline at ALBA Synchrotron. The radiation wavelength was 1.0 {\AA} and the spot size at the focal plane of approximately 100 $\mu m$ $ X $ 50 $\mu m$. Sample-to-detector distance and other parameters were callibrated using reference materials. Data were processed with the software Dawn \cite{filik2017processing}.
\textit{In-situ} tensile tests were performed on 15 mm samples using a Kammrath und Weiss miniaturized tensile stage. For these tests, strain steps were applied at a strain rate of 5 $\mu m$/s. SAXS patterns were acquired at fixed strain as the load was monitored.
Scanning Electron microscopy (SEM) was carried out with an FIB-FEGSEM Helios NanoLab 600i (FEI) at 10kV.TEM images were taken using a Talos F200X FEI operating at 20KV.
\section{The elastic extension of CNT fibres}
\subsection{The uniform stress model}
\label{sect:section21}
The mechanical behavior of fibrous materials depends critically on their morphology \cite{Morton2008physical}. In this regard, despite the complex hierarchical structure found in CNT fibres \cite{yue2017fractal}, their microstructure can be defined as fibrillar by noting that CNT bundles are essentially long fibrils well-aligned along the fibre axis, as shown in the electron micrograph in Fig.\ref{fig:Figure21} (a). It is precisely these fibrils which act as load-carrying elements and, therefore, their mechanical properties control to a large extent the final properties of the macroscopic fibre.
Within the framework of a uniform stress model, a fibre is considered to be made up of an array of identical fibrils, which are all subjected to a uniform stress along the fibre axis \cite{ward1962optical,ward1983mechanical}. Following the analogy to polymers, it is also assumed that each fibril consists of crystallites arranged end-to-end. In the case of CNT fibres, the crystallites are bundles in which the CNTs are closed-packed and parallel to each other at a separation between that in Bernal and turbostratic graphite. These fibrils (bundles) are the basic structural elements in the fibres and therefore the key elements in our continuum mechanics analysis.
\begin{figure}[H]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=\textwidth]{fibre_structure_model2.jpg}
\end{tabular}
\end{center}
\caption
{ \label{fig:Figure21}
(above) SEM and TEM images of a CNT fibre which reveal a fibrillar microestructure made up of close packed bundles of nanotubes $\&$ (below) schematic representation of a single CNT bundle and its contribution to the macroscopic deformation of the fibre: axial stretching by tensile deformation of nanotubes and crystallite rotation by shear between nanotube layers.}
\end{figure}
The structure outlined above is conceptually similar to that of carbon fibres, which have indeed been treated as networks of fibrillar crystalline domains formed by stacks of graphitic planes and defined by a symmetry axis, corresponding to the normal to the graphite basal plane (c-axis) \cite{northolt1991tensile}. Evidently, the orientation of crystallites in CNT fibres is also defined by the normal to the graphitic planes, that is, to the CNT main axis. For such well oriented fibres, it can be demonstrated \cite{northolt1991tensile} that their Young's modulus ($E$) is given by
\begin{equation}
\label{equation21}
\frac{1}{E} = \frac{1}{e_c}+\frac{<cos^2\phi_{0}>}{g}
\end{equation}
\noindent where $e_c$ is the modulus in the direction normal to the c-axis, $g$ is the shear modulus between planes oriented normal to the c-axis and the parameter $<cos^2\phi_{0}>$ is the second moment of the c-axis orientation distribution in the unloaded state, defined latter. According to this expression, there are two contributions from the crystallites to the fibre strain, as schematically depicted in Figure \ref{fig:Figure21}(b). The first term refers to the axial elastic stretching of crystallites, that is to nanotubes themselves, whereas the second term involves the effect of crystallite alignment due to shear strain. This angular deformation (shear component) implies the rotation of nanotubes toward the fibre axis, increasing the angle between the c-axis and the fibre axis from $\phi_{0}$ to $\phi$. From this description, it is evident that the parameters $e_c$ and $g$ correspond to the Young's modulus of CNT bundles and the shear modulus associated to tangential elastic displacement between nanotubes, respectively.
\subsection{Orientation analysis by Small Angle X-Ray Scattering}
\label{sect:section22}
As mentioned above, the structure of the fibre determines its modulus via the orientation distribution of crystallites, embedded in the term $<cos^2\phi_{0}>$ in equation \ref{equation21}. The orientation distribution function (ODF) of crystallites in fibres is typically obtained by 2D wide-angle X-ray scattering (WAXS) \cite{roe2000methods, martinezhergueta2015}.In the case of CNT arrays, this is done by analysis of the (002) interplanar reflection arising from adjacent CNTs as well as from internal CNT layers\cite{li2007x}, and giving rise to an equatorial feature in WAXS data as the one shown in Figure\ref{fig:Figure23}(a) \cite{severino2016progression,zhang2010nanocomposites,behabtu2013strong}. The ODF can be obtained from the azimuthal profile of scattering intensity obtained after radial integration, $I(\phi)$
\begin{equation}
\label{eq:equation22}
\Psi(\phi) = \frac{I(\phi)}{\int_{0}^{\pi} I(\phi) sin(\phi) d\phi},
\end{equation}
Note that this orientation distribution function is of the normal to the graphene basal planes in the crystallites and thus perpendicular to the CNT main axis.
With knowledge of the ODF $<cos^2\phi>$ can be calculated by averaging $cos^2\phi$ over the c-axis orientation distribution as
\begin{equation}
\label{eq:equation23}
<cos^2\phi> = \int_{0}^{\pi} cos^2\phi \; \Psi(\phi) \; sin\phi \; d\phi
\end{equation}
Because of the weak X-ray scattering of CNT fibres, WAXS measurements are typically carried out on multiple filament samples and using synchrotron X-ray sources. However, we have recently shown that such samples have an intrinsically high misorientation between filaments relative to the intrinsic fibre orientation. These makes them unsuitable to determine the ODF of individual fibres. Instead, it is more accurate to use SAXS, which because of its higher intensity can be readily measured on individual fibres in standard synchrotron radiation facilities. Figure\ref{fig:Figure23}(a) shows an example of a 2D SAXS pattern from an individual CNT fibre. The equatorial streak is characteristic of fibrillar structures in high-performance fibres such as PBO \cite{ran2002situ}, Kevlar \cite{dobb1979microvoids} or carbon fibre \cite{gupta1994small}. The use of SAXS data instead of WAXS is possible because unlike other CNT arrays\cite{meshot2017quantifying}, for CNT fibres the orientations measured from WAXS and SAXS are equivalent over a wide scattering vector ($q$) range \cite{davies2009structural}(Suplementary Information). In CNT fibres SAXS arises mainly from the network of elongated mesopores and bundles in the fibre, which corresponds precisely to the orientation of interest for this work. A further point of interest is that both the WAXS and SAXS azimuthal profiles are best fit by a Lorentzian, rather than a Gaussian distribution. The Lorentzian profile intrinsically leads to low values of Herman's parameter (0.5) even for highly oriented fibres, and cannot therefore be taken as a direct indicator of high-performance fibre properties.
\begin{figure}[H]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.9\textwidth]{SAXS_analysis.jpg}
\\
\hspace{-2cm} (a) \hspace{5cm} (b)
\end{tabular}
\end{center}
\caption
{ \label{fig:Figure23}
(a) WAXS (above) and SAXS (below) patterns of CNT fibres $\&$ (b) c-axis orientation distribution functions for two CNT yarns obtained at different draw ratios.}
\end{figure}
With the aim of understanding orientational effects on tensile properties, in this study we have produced fibres with different degree of CNT orientation, obtained by varying their drawing rate during fabrication \cite{aleman2015strong}. This effect is clearly seen in the SAXS ODF plotted in Figure \ref{fig:Figure23}(b). Clearly, the sample produced at a higher draw ratio has a narrower ODF. In addition, we have also prepared samples synthesised from two different precursors (see experimental details), and which have differences in their constituent CNTs and tensile properites. Once consists predominantly of collapsed double-walled carbon nanotubes (DWNT) and the other of few-layer (3-5) multi-walled carbon nanotubes (MWNT). The properties of these samples are summarised in Table \ref{table:table11}.
\begin{table}[H]
\centering
\caption{Experimental values of CNT fibres}
\label{table:table11}
\begin{tabular}{c|c|c|c|c|c|}
\cline{2-6}
&\begin{tabular}[c]{@{}c@{}}winding rate \\(m/min)\end{tabular} &\begin{tabular}[c]{@{}c@{}}$<cos^2\phi_{0}>$ \\($\times 10^\textsuperscript{-2}$)\end{tabular} & \begin{tabular}[c]{@{}c@{}}$E$ \\(GPa)\end{tabular} & \begin{tabular}[c]{@{}c@{}}$\sigma_b$ \\(GPa)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Fracture energy \\(J/g)\end{tabular} \\ \hline
\multicolumn{1}{|l|}{\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}Collapsed \\DWNTs\end{tabular}}} &4 &8.89 & $44 \pm 9$ & $1.0 \pm 0.2$ & $70 \pm 40$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} &8 & 9.7 &$32 \pm 7$ &$1.1 \pm 0.1$ & $90 \pm 20$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} &12 & 7.46 & $56 \pm 8$ & $1.3 \pm 0.2$ & $70 \pm 30$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} &16 & 5.42 & $61 \pm 7$ &$1.7 \pm 0.3$ &$100 \pm 30$ \\ \hline
\multicolumn{1}{|l|}{\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Few-layer \\MWNTs\end{tabular}}} &20 & 11.58 & $33 \pm 8$ & $0.7 \pm 0.1$ & $60 \pm 10$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} &30 & 10.08 & $38 \pm 8$ & $0.8 \pm 0.1$ & $65 \pm 15$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} &40 & 6.37 & $64 \pm 16$ &$1.1 \pm 0.2$ & $80 \pm 40$ \\ \hline
\end{tabular}
\end{table}
\subsection{Results and discussion}
\label{sect:section23}
Figure \ref{fig:figure24} presents values of fibre compliance ($E\textsuperscript{-1}$) plotted against the orientation parameter $<cos^2\phi_{0}>$ determined from SAXS. As can be observed, the experimental data exhibit a linear correlation with the orientation parameter, which is in excellent agreement with equation \ref{equation21} and support the use of the uniform stress transfer model for oriented fibres. The linear fit includes data for CNT fibres with different CNT types and different tensile properties, as discussed before. Moreover, literature data also follow the same trend (see supplementary material for a discussion of literature data). This behaviour is extremely relevant, because it naturally leads to the conclusion that the stiffness of a CNT fibre is mainly dominated by crystallite alignment, represented here by the parameter $<cos^2\phi_{0}>$. It also implies that for fibres with constituent CNTs with few layers (1-5), the internal layers of the CNTs make a substantial contribution to the fibre stiffness.
\begin{figure}[H]
\raggedright
\begin{tabular}{c}
\includegraphics[width=0.9\linewidth]{cos2_compliance.jpg}
\end{tabular}
\caption
{ \label{fig:figure24}
Compliance ($E\textsuperscript{-1}$) of CNT fibres plotted against the orientation parameter $<cos^2\phi_{0}>$ for both in-house fibres and data obtained from the literature. These last correspond to both single \cite{vilatela2011structure,behabtu2013strong} and multifilament samples\cite{gspann2017high}. In addition, data from twisted CNT yarns have been also plotted \cite{beese2014key}. The linear fit is for CNT fibres produced and analysed in this work.}
\end{figure}
Assuming a CNT fibre specific gravity of 1.8, values for $e_c\approx 540$ GPa and $g= 8.1 \pm 1.8$ GPa are obtained from the fitting. The crystallite stiffness value $e_c$ is close to the in-plane Young's modulus of graphite ($E=1020 \pm 30$ GPa)\cite{morgan2005carbon} and in the range of experimental values for individual CNTs and bundles \cite{peng2008measurements,yu2000tensile}. Considering the spread in fibre stiffness values, the agreement is remarkable. The value for $g$ is above the theoretical shear modulus of single-crystal graphite (G\textsubscript{graphite}=4.6 GPa)\cite{nicklow1972lattice}. However, measurements on carbon fibres (CF) with a more complex polycrystalline structure \cite{johnson1987structure}, \cite{northolt1991tensile} give values of 5 to 33 GPa on account of out-of-plane interactions arising from crystallite edges and defects, typical of graphitic ensembles with a wide distribution of interlayers spacings. Morphology and contact area in CNT fibres are different from Bernal graphite and thus a compact arrangement of parallel CNTs needs not display the same response in terms of stress than a standard stack of graphitic planes when subjected to shear strain. Therefore, the value reported here, $g= 8.1 \pm 1.8$ GPa, can be considered a reasonable, conservative estimate for the shear modulus of CNT crystallites.
\section{Fracture model: CNT fibre as a molecular composite}
\label{sect:fracture}
The model discussed so far successfully describes only the elastic axial deformation of CNT fibres. In order to provide a description of factor governing tensile strength we first considering a CNT fibre as a composite of strong/stiff crystallites in a matrix of weak secondary bonds. This approach describes the tensile strength of fibres such as aramid, which have a highly fibrillar fracture analogous to that of a uniaxially oriented fibre-reinforced composites that fail in tension via matrix shear failure initiated at the fibre ends \cite{knoff1987relationship}. The fracture mechanism in CNT fibres is indeed fibrillar \cite{vilatela2011model}. As shown in Figure \ref{fig:figure25}, the fracture ends of the fibre shows failure by extensive shear-induced decohesion between CNT bundles.
\begin{figure}[H]
\begin{tabular}{c}
\includegraphics[width=\linewidth]{fracture_shear.jpg}
\end{tabular}
\caption
{ \label{fig:figure25}
a) SEM image of the fracture surface of a CNT fibre in which a fibrillar morphology is revealed $\&$ b) schematic structure of the CNT fibre as a network of well-aligned CNT bundles.}
\end{figure}
When treating a high-performance fibre as a molecular composite of filaments in a matrix of secondary bonds, fibre strength can be obtained from a modified form of the Tsai-Hill criterion for failure in uniaxial composites \cite{northolt2005tensile}. According to it, the strength of a composite loaded in a direction at an angle $\theta$ with respect to the parallel aligned fibres is given by
\begin{equation}
\label{eq:equation31}
\sigma\textsubscript{comp}=[\frac{cos^4\theta}{\sigma_L^2}+(\frac{1}{\tau_b^2}-\frac{1}{\sigma_L^2})\; sin^2\theta \; cos^2\theta + \frac{sin^4\theta}{\sigma_T^2}]^{-\frac{1}{2}}
\end{equation}
\noindent where $\sigma_L$ is the axial strength of the fibres, $\sigma_T$ is the strength normal to the composite's symmetry axis, and $\tau_b$ is the critical shear strength in a plane parallel to the fibres \cite{hull1996introduction}.
Such a model can easily be applied to a CNT fibre by introducing the average angle between the axial loading direction and load-bearing elements $<cos^2\phi>$, obtained from the ODF at fracture ($\Psi(\phi_{b})$) and noting that $\phi=\frac{\pi}{2}-\theta$. In addition, for highly aligned fibres the transverse properties are negligible and the last term in \ref{eq:equation31} can be neglected. Finally, fibre tensile strength can be approximated by the expression
\begin{equation}
\label{eq:equation32}
\sigma_b\approx[\frac{<sin^4\phi_b>}{\sigma_L^2}+(\frac{1}{\tau_b^2}-\frac{1}{\sigma_L^2})\; <sin^2\phi_b \; cos^2\phi_b>]^{-\frac{1}{2}}
\end{equation}
This expression contains two unkown parameters: the critical axial fibril strength $\sigma_L$ and the critical shear strength $\tau_b$. In a macrocomposite the fibres are continuous and $\sigma_L$ is simply the strength of fibres. But in the case of a CNT fibre visualised as a composite, its constituent fibrils (the crystallites) are of finite length, which implies that load is transfered from one fibril to another through shear (shear lag). Shear stress arising at the end of a filament can, upon exceeding a limiting stress $\tau_b$, cause debonding of the filament from its nearest neighbors. $\sigma_L$ is thus the \textit{maximum axial stress} in the fibrils before shear failure.
$\sigma_L$ is clearly then dependent on the shear strength $\tau_b$. In this regard, Yoon\cite{yoon1990strength} derived an expression for a polymer fibres of very long chains and failure in shear, which relates these two parameters with the crystallite elastic constants $e_c$ and $g$. Applied to CNT fibres it leads to
\begin{equation}
\label{eq:equation36}
\sigma_L=1.14\cdot\tau_b\cdot\sqrt{\frac{e_c}{g}},
\end{equation}
Implicit in the model discussed above is the view that the CNT fibres is treated as a network of fibrils, corresponding to long crystalline domains, that is crystallites, similar in cross section to a bundle. Failure occurs through shearing of crystallites, leading to fibrillar fracture before any CNT rupture occurs. The network is a continum of crystalline domains and there is no reason to expect that all ends of CNTs in a domain match and hence that a bundle terminates abrutly. Instead, the 1-mm long CNTs can easily form part of several crystalline domains, very much in the same way polymer chains do.
\subsection{Relation between Young's Modulus and ultimate strength}
\label{sect:section32}
The model can be contrasted with experimental data by relating the fibre modulus and strength, with the use of equations \ref{eq:equation32} and \ref{equation21}. In the process, it is necessary to determine the ODF at the point of fracture $<cos^2\phi_{b}>$. In the uniform stress model, upon fibre loading, crystallites deform \textit{elastically} in shear and re-align parallel to the fibre axis. The second moment of c-axis orientation distribution $<cos^2\phi>$ of a fibre under a stress $\sigma$ decreases with respect to the unloaded state $<cos^2 \phi_{0}>$ according to the following expression \cite{northolt1991tensile}:
\begin{equation}
\label{eq:equation37}
<cos^2\phi>=<cos^2\phi_{0}>\exp{(-\frac{\sigma}{g})}
\end{equation}
\noindent where $g$ is the crystallite shear modulus as described before.
In the case where fracture involves additional crystallite alignment through plastic deformation by shear, this expression can be modified to obtain \cite{northolt2005tensile}
\begin{equation}
\label{eq:equation38}
<cos^2\phi_b>=<cos^2\phi_{0}>\exp{(-\frac{\sigma_b}{g_v})}
\end{equation}
\noindent where $<cos^2\phi_b>$ corresponds to the orientation at fracture, as in equation \ref{eq:equation32}.
We have measured the evolution of the ODF by in-situ SAXS measurements during tensile deformation of a CNT fibres. The data, shown in Figure \ref{fig:figure33}, confirm the exponential relation in equation \ref{eq:equation38}. It gives a value for $g_v = 0.7 GPa$, discussed further below. The stress-strain curve confirms that after plastic deformation upon reloading the fibre has a higher modulus due to a higher degree of orientation, as observed for example in rigid-rod high-performance polymer fibres \cite{northolt1985elastic}, expressed in quantitative terms as $\frac{E_1}{E_0}\approx\frac{<cos^2\phi_{0}>}{<cos^2\phi>}$
\begin{figure}[H]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=\textwidth]{insitu_mechanical_SAXS.jpg}
\end{tabular}
\end{center}
\caption
{ \label{fig:figure33}
(left) Results obtained from subjecting a CNT fibre to an \textit{in situ} tensile test with SAXS measurements. Thus, both the stress and the parameter $\frac{<cos^2\phi_{0}>}{<cos^2\phi>}$, calculated from SAXS measurements, are plotted against strain. (Right) Evolution of the parameter $\frac{<cos^2\phi_{0}>}{<cos^2\phi>}$ with the stress during tensile stretching and their accurate fitting to the expresion. }
\end{figure}
Equipped with the relation between the ODFs and equations \ref{equation21}, \ref{eq:equation32}, and \ref{eq:equation38}, and using trigonometric approximations assuming that the misalignment is small, we obtain an expression relating the Young's modulus with the ultimate tensile strength (Supplementary Information):
\begin{equation}
\label{eq:equation39}
\frac{1}{E} = \frac{1}{e_c}+\frac{exp(\frac{\sigma_b}{g_v})}{g} \cdot \frac{(3\; \sigma_L^{-2}-\tau_b^{-2})+\sqrt{(\tau_b^{-2}-3\; \sigma_L^{-2})^2-4\; (4\; \sigma_L^{-2} - 2\; \tau_b^{-2})\; (\sigma_L^{-2}- \sigma_b^{-2})}}{2\; (4\; \sigma_L^{-2}- 2\; \tau_b^{-2})}
\end{equation}
In equation \ref{eq:equation39}, only $\tau_b$ and, in principle $g_v$, are unknowns, but both can be obtained by fitting experimental data. As shown in Figure \ref{fig:figure32}, there is a very good match between experimental data and the fitting to equation \ref{eq:equation39} for both types of CNT fibres. The extracted value of $g_v = 1.1$ GPa for few-walled MWNT fibres is close to the that determined by in-situ SAXS ($\approx 0.7$ GPa). Further success of this expression is the prediction that both fibre strength and modulus increase with improved alignement, as observed experimentally a decade ago \cite{koziol2007high}. Additionally, with this plot at hand the differences in properties of the two sets of fibres can be now ascribed to different values of $\tau_b$ and $g_v$.
\begin{figure}[H]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.9\textwidth]{strength_modulus.jpg}
\end{tabular}
\end{center}
\caption
{ \label{fig:figure32}
Graph of tensile strength against tensile stiffness for each butanol and toluene made CNT fibres. The modified Tsai-Hill failure criterion for polymer fibres fits accurately in both cases (dashed-lines). This model enables to fit parameters $\tau_b$ and $g_v$ which define maximum shear strength and shear stiffness of the crystallites, respectively.}
\end{figure}
\section{The fracture envelope}
\label{sect:section4}
The model accuracy is tested again by determining the fracture envelope of the CNT fibres; that is, the set of stress-strain coordinates where the fibre fails. For brittle linear-elastic fibres such as CF, the stress-strain relation is \cite{northolt1991tensile}
\begin{equation}
\label{eq:equation391}
\epsilon \approx\; \frac{\sigma}{e_c}+<cos^2\phi_{0}>[1-exp(-\frac{\sigma}{g})]
\end{equation}
\noindent In order to account for some plastic deformation in CNT fibres we replace $g$ with $g_v$. Using then equation \ref{equation21}, we obtain the following expression for the strain-to-break:
\begin{equation}
\label{eq:equation392}
\epsilon_b \approx\; \frac{\sigma_b}{e_c}+g\; (\frac{1}{E}-\frac{1}{e_c})\; [1-exp(-\frac{\sigma_b}{g_v})]
\end{equation}
\noindent with the parameters in equation\ref{eq:equation392} obtained as discussed before. This leads to the fracture envelope of stress-strain failure pairs ($\sigma_b$,$\epsilon_b$) for different fibre orientations. $Figure\;\ref{fig:Figure41}$ shows good agreement between experimental data and the predicted envelope for both types of CNT fibres.
\begin{figure}[H]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=\textwidth]{fracture_envelope.jpg}
\\
\end{tabular}
\end{center}
\caption
{ \label{fig:Figure41}
(a) Calculated fracture envelope compared to CNT fibres tested in this work $\&$ (b) fracture energy exhibited for each CNT fibre plotted against the initial orientation parameter $<cos^2\phi_{0}>$ }
\end{figure}
However, the fracture envelope seems to lie below the experimental values, particularly for fibres subjected to higher draw ratios, and therefore more oriented. In addition, the experimental data show a steeper decrease of $\sigma_b$ with strain than the prediction, an effect that cannot merely be explained in terms of the accuracy of our values for $g$ and $g_v$. Instead, it is likely that $g_v$ is not constant, but has a small dependance on the degree of alignment. This is the case for some polymer fibres, whose deformation mechanisms substantially depend on draw ratio \cite{northolt2005tensile}. In this respect, we note that CNT fibres subjected to higher draw ratios have a greater fracture energy ((Figure \ref{fig:Figure41}b), whereas the fibrillar breakage model assumes this to be constant through a constant number of failing elemets. Our recent WAXS measurements on multifilament samples suggest that samples produced at higher draw ratios have a larger "degree of crystallinity", that is, a large fraction of graphitic planes at turbostratic separation in coherent domains \cite{yue2017fractal}, which might be responsible for the this increase in fracture energy and the small deviation from the predicted fracture enevelope.
\subsection{Comparison of fibres}
\label{sect:section5}
\begin{table}[H]
\begin{center}
\caption{parameters $\tau_b$ and $g_v$ determined for the model. }
\label{tab:table2}
\renewcommand{\thefootnote}{\thempfootnote}
\begin{tabular}{| c | c | c | c | c |}
\hline
& \begin{tabular}[c]{@{}c@{}}$\tau_b$\\(MPa)\end{tabular} & \begin{tabular}[c]{@{}c@{}}$g_v$\\(GPa)\end{tabular} & \begin{tabular}[c]{@{}c@{}}$g_v/g$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$\sigma_L$\\(GPa)\end{tabular}\\ \hline
collapsed DWNTs & 450 & 2.1 & 0.26 & 5\\ \hline
few-wall MWNTs & 270 & 1.1 & 0.12 & 2.5\\ \hline
PpTA (Kevlar) & 370 & 1.2 & 0.7 & 4.87 \\ \hline
PBO (Zylon) & 400 & 2.0 & 1.0 & 7.7 \\ \hline
POK & 300 & 0.5 & 0.3 & 4.83 \\ \hline
Cellulose II & 325 & 1.5 & 0.6 & 2.25 \\ \hline
PET & 290 & 0.7 & 0.5 & 3.12\\ \hline
HM50 (carbon fibre) \textsuperscript{\emph{a}} & 310 & 10 & 1.0 & 2.5\\ \hline
\multicolumn{1}{}{}
\textsuperscript{\emph{a}} estimated from Northolt \textit{et al.}\cite{northolt1991tensile}
\end{tabular}
\end{center}
\end{table}
In Table \ref{tab:table2} we compare the parameters extracted from the fibrillar crystallite analysis for different fibres, including our two types of CNT fibres, CF, high-performance polymer fibres and ductile polymer fibres \cite{northolt2005tensile,northolt1991tensile}. The values of $\tau_b$ and $g_v$ are in the same range for all the fibres, irrespective of their chemistry and the nature of the interaction between molecular building blocks. This suggests that $\tau_b$ and $g_v$ take the form of effective shear strength and modulus of the crystallite ensemble, and which can therefore not be easily reduced to the properties of a single-crystal. Nevertheless, the high values of $\tau_b$ contrast with the lubricity of graphite and the reported shear strength of measured on individual CNTs spanning from 0.04 MPa \cite{kis2006interlayer} to 69 MPa \cite{suekane2008static}. In this regard, we note that the length of the CNTs in this work is aproximately 1mm, which contrasts with the small length used in individual tests, and which could imply a larger contribution from defects \cite{paci2014shear}, surface impurities \cite{naraghi2010multiscale}, domains in crystallographic registry \cite{Jamessliding} and mechanical entanglements. Nevertheless, in spite of the relatively high values of $\tau_b$ the maximum axial stress in the fibrils, $\sigma_L$, is still much lower than the tensile strength of individual CNTs, leaving ample room for interfacial chemistry strategies \cite{EndoXlink} to improve shear stress transfer and thus increase fibre tensile strength.
The parameter $g_v$ is clearly a critical one. It describes the shear deformation stiffness of crystallites, including their rotation towards the fibre axis when bearing load. It is not a conventional elastic shear modulus, but rather a secant shear modulus, involving both elastic and plastic deformation. Although its relation to the fundamental properties of a graphite single-crystal is still unclear, it is a convenient parameters to describe the extent of CNT realignment upon axial fibre loading. For more brittle, essentially linear elastic CNT fibres, $g_v$ tends towards $g$. But for the ductile fibres tested in this work, the comparison in Table \ref{tab:table2} shows that CNT fibres have a very low ratio of $g_v/g$. We think that this parameter is responsible for the unusual combination of high fracture energy and tensile strength in these CNT fibres ($Figure\;\ref{fig:figure51}$a). Embodied in it is the ability of the CNT crystalline network to undergo substantial reorientation upon loading, which seems to be a unique feature of CNT fibres.
Combined high specific strength/modulus and energy to break are particularly relevant for impact resistant structures. A figure of merit for fibre ballistic protection, for example, is the cubed root of the product of sonic modulus and specific energy to break (\textit{T}), $U^\frac{1}{3}={(\sqrt{\frac{E}{\rho}}\;T)}^\frac{1}{3}$ \cite{cunniff2002high}. The samples in this work have average values of $U^{1/3}$ of 800 $\frac{m}{s}$ which is superior to most high-performance fibers, including aramid and carbon fibers. But more importantly, as Figure \ref{fig:figure51}b shows, the model introduced here predicts that very modest improvements in CNT orientation would lead to a fibre with unrivaled properties for ballistic and impact protection $U^\frac{1}{3}> 1000 m/s$ (see supplementary material), outperforming the best synthetic fibers available (ultrahigh molecular weight polyethylene (UHMWPE) and poly(p-phenylene-2,6-benzobisoxazole) (PBO)).
\begin{figure}[H]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=\textwidth]{fracture_energy-ballistic.jpg}
\end{tabular}
\end{center}
\caption
{ \label{fig:figure51}
(a) Fracture energy versus strength of high-performance fibers and of samples in this work and (b) prediction of the ballistic figure of merit $U^\frac{1}{3}$ of CNT fibers as a function of CNT orientation ($<cos^2\phi_{0}>$) showing superior properties than conventional high-performance fibers.}
\end{figure}
\section{Conclusions}
This works presents an analytical model to describe the tensile properties of fibres of CNTs. It assumes that their structure can be treated as a network of oriented crystallites, similarly to a high-performance polymer fibre, defined by the crystallite orientation distribution function and shear modulus and shear strength. Experimental values of initial ODF and tensile modulus show remarkable agreement with the model for fibres produced in-house with different constituent CNTs and for different draw ratios, as well as with literature data.
By considering the CNT fibre as composite of stiff fibrils (crystallites) in a matrix of secondary bonds, we introduce expressions for tensile strength based on fibre-reinforced composite lamina theory. Plastic deformation through CNT crystallite reorientation is introduced via a secant shear modulus. Its predicted value based on statistical fibre strength/modulus data matches an experimental value determined from in-situ synchrotron SAXS measurements of the ODF during tensile testing.
Overall, the model provides a solid framework for the study of CNT fibres produced under different conditions, capable of separating orientational from compositional effects. Amongst future improvements to the model we highlight: elucidating the effects of crystallite size and role of CNT layers, a more robust physical interpretation of $g_v$ and its dependance on fibre orientation, and the prediction of CNT fibre properties embedded in polymer matrices. Work towards these improvements is in progress.
The model provide a quantitative prediction of the effect of improvements in CNT orientation and shear stress transfer on fibre tesile properties. It shows that small improvements in orientation, for example, would lead to higher specific strength than aramid and most carbon fibres, and superior ballistic protection that any other synthetic high-performance fibre.
\section*{References}
\bibliographystyle{elsarticle-num}
|
{
"timestamp": "2019-03-01T02:04:15",
"yymm": "1902",
"arxiv_id": "1902.10802",
"language": "en",
"url": "https://arxiv.org/abs/1902.10802"
}
|
\section{INTRODUCTION}
\label{sec:intro}
\ninept
A deep learning based reconstruction model trained for a specific scanning setting (\textit{i.e.} a domain) usually underperforms on unseen contrasts or organs due to the domain shift problem. As obtaining fully-sampled images for each domain is impractical, we propose a simple generalisation strategy for deep MRI reconstruction.
\section{METHODS and experiments}
\label{sec:meth_exp}
\ninept
We build the strategy on one of the state-of-the-art deep-cascade of CNN\cite{schlemper2018deep}, which learns fully-sampled image priors and projects undersampled images to the learned fully-sampled image space.
We train the network with MS-COCO Stuff Segmentation dataset\cite{lin2014microsoft}, which contains around 118k natural images ($>$100$\times$ larger than any of three MRI datasets used in this work which contain 0.3k to 1k 2D slices each). Synthetic phases are then added, which is crucial for model sharing between natural images and MRI.
For characterisation of domain shift and for evaluation, we have performed comparisons across different domains (training and testing on different domains) for 2D single-coil slice-by-slice reconstruction on the following datasets: cardiac CINE\cite{schlemper2018deep}, coronal knee proton-density (Knee-CPD) and axial knee T2 (Knee-AT2)\footnote{For knee images we used retrospectively down-sampled ground truth images to simulate single-coil reconstruction. 4$\times$ Gaussian variable density Cartesian down-sampling is used for all datasets.}\cite{hammernik2018learning}.
To show that phase-synthesised MS-COCO provides richer and more variable patch samples for training, we extracted 20k 7$\times$7 patches from each of MRI datasets separately, and extracted 200k patches from 4k MS-COCO images. We calculated average cross-domain patch-wise Euclidean nearest neighbour (NN) distances.
\section{evaluations and conclusions}
\label{sec: evl_conc}
\ninept
\setlength{\tabcolsep}{3pt}
\def-1{-1}
\begin{table}[!h]
\label{tbl: recon_value}
\centering
\scalebox{0.83}{
\begin{tabular}{ccccc}
\ninept
\textbf{Train} & \multicolumn{4}{c}{ \textbf{Test} } \\
\hline
& & Cardiac & Knee-CPD & Knee-AT2 \\
\cline{3-5}
\multirow{2}{*}{ Cardiac } & \textit{PSNR} & \colorbox{gray!30}{\textbf{29.95$\pm$1.96}} & 31.52$\pm$3.14 & 33.64$\pm$1.68 \\
& \textit{SSIM} & \colorbox{gray!30}{\textbf{0.95$\pm$0.01}} & 0.95$\pm$0.05 & 0.94$\pm$0.02 \\
\cline{3-5}
\multirow{2}{*}{ Knee-CPD } & \textit{PSNR} & 26.91$\pm$2.47 & \colorbox{gray!30}{33.42$\pm$2.79} & 34.01$\pm$1.66 \\
& \textit{SSIM} & 0.90$\pm$0.03 & \colorbox{gray!30}{\textbf{0.96$\pm$0.05}} & 0.95$\pm$0.01 \\
\cline{3-5}
\multirow{2}{*}{ Knee-AT2 } & \textit{PSNR} & 24.96$\pm$2.56 & 31.78$\pm$3.86 & \colorbox{gray!30}{\textbf{35.34$\pm$1.79}} \\
& \textit{SSIM} & 0.85$\pm$0.05 & 0.95$\pm$0.05 & \colorbox{gray!30}{\textbf{0.96$\pm$0.01}} \\
\cline{3-5}
\multirow{2}{*}{ MS-COCO } & \textit{PSNR} & 29.46$\pm$2.19 & \textbf{33.79$\pm$2.72} & 34.70$\pm$1.74 \\
& \textit{SSIM} & 0.94$\pm$0.01 & \textbf{0.96$\pm$0.05} & 0.95$\pm$0.02
\end{tabular}}
\caption{Quantitative evaluations of cross-domain reconstruction}
\end{table}
As shown in Table 1, Training with phase-synthesised MS-COCO yields the best overall cross-domain performances. This behaviour is consistent with observations on magnitude-only images\cite{dar2017transfer}. Domain-correct reconstructions are \colorbox{gray!30}{highlighted} for reference.
Table 2 shows that MS-COCO provides the smallest average cross-domain patch-wise NN distances ($p$-value $< 10^{-10}$). This implies that the generalisation ability of our method might be related to large intersections in patch subspaces between domains.
\begin{figure}[!h]
\label{fig: error}
\centering
\scalebox{0.85}{
\begin{minipage}[b]{1\linewidth}
\centering
\centerline{\includegraphics[width=8.5cm]{./figures/fig1.png}}
\vspace{0cm}
\end{minipage}
}
\caption{Images for cross-domain reconstruction}
\label{fig:res}
\end{figure}
\setlength{\tabcolsep}{3pt}
\renewcommand{-1}{0}
\begin{table}[!h]
\label{tbl: distance}
\centering
\scalebox{0.84}{
\begin{tabular}{ccccc}
\textbf{Target} & \multicolumn{4}{c}{ \textbf{Source} } \\
\hline
& Cardiac & Knee-CPD & Knee-AT2 & MS-COCO \\
\cline{2-5}
Cardiac & - & 0.41$\pm$0.33 & 0.64$\pm$0.54 & \textbf{0.37$\pm$0.28} \\
Knee-CPD & 2.52$\pm$3.64 & - & 1.38$\pm$1.20 & \textbf{0.84$\pm$0.96} \\
Knee-AT2 & 2.45$\pm$4.60 & 1.40$\pm$2.32 & - & \textbf{1.35$\pm$2.25} \\
\end{tabular}
}
\caption{Average cross-domain patch-wise NN distances}
\end{table}
\bibliographystyle{IEEEbib}
\ninept
|
{
"timestamp": "2019-03-01T02:04:32",
"yymm": "1902",
"arxiv_id": "1902.10815",
"language": "en",
"url": "https://arxiv.org/abs/1902.10815"
}
|
\section*{Introduction}
How do we characterize quantum chaos? Among a wide variety of different approaches (see \cite{2016AdPhy..65..239D} for a review), two rather different criteria are currently in wide use. The first one is random-matrix-like universality of the fine-grained energy spectrum \cite{PhysRevLett.52.1,haake2013quantum}: a given quantum system is chaotic in this sense if the fine-grained energy spectrum is described by Random Matrix Theory (RMT) \cite{wigner1993characteristic,doi:10.1063/1.1703773,mehta2004random}.
The second one is sensitivity to initial conditions: a given quantum system is chaotic in this sense if it exhibits exponential Lyapunov growth of a small perturbation
as probed by an out-of-time-order correlation function (OTOC) \cite{larkin1969quasiclassical,Almheiri:2013hfa}. OTOCs are closely related to Loschmidt echoes which also probe chaos~\cite{2012arXiv1206.6348G}.
There are several unsatisfactory features regarding these criteria. First, it is unclear how the two criteria are related. Second, the connection of the quantum criteria to the characterizations of classical chaos are unclear. One might expect that sensitivity to initial conditions can characterize both classical and quantum chaos, but there is a problem for local quantum systems. In the classical theory, the initial perturbation can be taken arbitrarily small in the mathematical sense, and the exponential growth can continue forever.
On the other hand, in a quantum system the perturbation cannot be arbitrarily small due to the uncertainty principle,
and local quantum systems do not generally show exponential growth except in special limits~\cite{2018PhRvX...8b1014N,2018PhRvX...8b1013V,2018arXiv180200801X,2018arXiv180505376X,Khemani2018vellyap}.~\footnote{In the context of holography, the large-$N$ limit of a gauge theory, a vector model, or the SYK model are often considered. In these examples, $N$ corresponds to the number of the internal degrees of freedom. These internal interactions can be regarded as highly non-local in the sense that all of them interact directly with each other. Large $N$ also plays the role of small $\hbar$, giving a kind of semi-classical limit. In these cases, the exponential growth of OTOC is a good indicator of chaos.} Hence, the characterization based on the early growth of OTOCs does not work for generic local quantum systems.
In a previous paper \cite{Gharibyan:2018fax}, we generalized the above single chaos exponent to define a spectrum of quantum Lyapunov exponents. Based on calculations in the Sachdev-Ye-Kitaev (SYK) model and a spin chain (XXZ) model, we proposed that the Lyapunov exponents so defined exhibit a universal behavior:
the fine-grained Lyapunov spectrum agrees with RMT when the system is chaotic. This characterization of quantum chaos circumvented the problem of lack of exponential growth in generic local systems, since one needs only the statistical property of the exponents instead of their detailed growth behavior. Because RMT behavior in the Lyapunov spectrum coincides with RMT behavior in the energy spectrum for the models we considered, the Lyapunov spectrum may be useful for connecting the different criteria for chaos. As a bouns, universality in the quantum Lyapunov spectrum has a classical counterpart \cite{Hanada:2017xrv}, so it may also be useful to connect classical and quantum chaos.
We emphasize that these universalities are merely empirical. There may be other observables that provide a similar characterization of quantum chaos which are also more accessible to experiment. In this paper, we consider time-ordered two-point correlators that are easier to study, both theoretically and experimentally, than OTOCs. Specifically, given a set of simple operators $\{O_j\}$, we consider the matrix of all possible two-point functions $\langle O_i(t) O_j(0)\rangle$ where $O(t)=e^{i H t} O e^{-i H t}$, construct its time-dependent spectrum, and then study the statistical properties of the spectrum. Based on this study, we propose that this two-point correlation spectrum, which is roughly a spectrum of decay rates, has universal statistical properties for all chaotic systems.
Below, we first define the two models, SYK and XXZ, that we will consider. Next, we define a spectrum of decay rates derived from two-point functions and propose a universal behavior for the spectrum in chaotic systems. Then we provide detailed numerical evidence for the conjecture using finite size exact diagonalization studies.
\section*{Models}
The first example is the SYK model \cite{Maldacena:2016hyu,Sachdev:2015efa,Kitaev_talk} (see Ref.~\cite{Rosenhaus:2018dtp} for a recent review) consisting of $N$ Majorana fermions with Hamiltonian
\begin{eqnarray}
\hat{H}
&=&
\sqrt{\frac{6}{N^3}}\sum_{i<j<k<l}J_{ijkl}\hat{\psi}_i\hat{\psi}_j\hat{\psi}_k\hat{\psi}_l
\nonumber\\
& &
\qquad
+
\frac{\sqrt{-1}}{\sqrt{N}}\sum_{i<j}K_{ij}\hat{\psi}_i\hat{\psi}_j.
\nonumber \\
\label{eqn:q=2deformedSYK}
\end{eqnarray}
Majorana fermions satisfy the anti-commutation relations $\{\hat{\psi}_i,\hat{\psi}_j\}=\delta_{ij}$
and $J_{ijkl}$ is random Gaussian coupling with mean zero and standard deviation $1$. The energy also includes a quadratic term, and $K_{ij}$ is Gaussian random with mean zero and standard deviation $K$. The dimension of the Hilbert space is $2^{N/2}$.
When $K=0$, this model is maximally chaotic at low temperatures, namely the MSS bound \cite{Maldacena:2016hyu,Kitaev_talk} is asymptotically saturated.
When $K>0$, low-energy modes become non-chaotic, while high-energy modes remain chaotic~\cite{Garcia-Garcia:2017bkg,Nosaka:2018iat}.
The second example is the XXZ model, a one-dimensional $S=1/2$ spin chain with random magnetic field along $z$-direction (see e.g.~\cite{PhysRevB.91.081103}),
\begin{eqnarray}
\hat{H}
=
\sum_{i=1}^{N_\mathrm{site}}\left(
\frac{1}{4}
\vec{\sigma}_i\vec{\sigma}_{i+1}
+
\frac{w_{i}}{2}\sigma_{z,i}
\right).
\label{eqn:XXZ}
\end{eqnarray}
Here $\vec{\sigma}=(\sigma_x, \sigma_y, \sigma_z)$ are Pauli matrices with periodic boundary condition $\vec{\sigma}_{N_{\rm site}+1}=\vec{\sigma}_1$.
The random magnetic fields $w_i$ are independent and uniformly distributed in $[-W,+W]$. At $W\gtrsim 3.5$, most of the energy eigenstates are in the many-body localized (MBL) phase \cite{PhysRevB.91.081103,serbyn1507criterion}. (For the physics of the MBL phase, see e.g.~\cite{PhysRev.109.1492,PhysRevLett.95.206603,BASKO20061126,aleiner:hal-00543657}.)
\section*{Proposal}
The starting point is choosing a set of operators and organizing the set of two-point functions into a matrix. The matrix of two-point functions, $G_{ij}^{(\phi)}(t)$, is defined by
\begin{eqnarray}
G_{ij}^{(\phi)}(t)
=
\langle
\phi|
\hat{\psi}_i(t)
\hat{\psi}_j(0)
|\phi
\rangle
\label{G-SYK}
\end{eqnarray}
for SYK, and by
\begin{eqnarray}
G_{ij}^{(\phi)}(t)
=
\langle
\phi|
\sigma_{+,i}(t)
\sigma_{-,j}(0)
|\phi
\rangle
\label{G-XXZ}
\end{eqnarray}
for XXZ, where $\sigma_{\pm}=\frac{\sigma_x\pm i\sigma_y}{2}$.
Here, we will take the state $|\phi\rangle$ to be an energy eigenstate, but this is not essential as explained in the discussion. Note also that we can consider other two-point functions, e.g.
$G_{ij}^{(\phi)}(t)=\langle\phi|\sigma_{z,i}(t)\sigma_{z,j}(0)|\phi\rangle$; the generalization to other systems is straightforward.
Let the singular values of $G_{ij}^{(\phi)}(t)$ be $e^{\lambda_i^{(\phi)}(t)}$.
We denote the $\lambda_i^{(\phi)}(t)$ as `exponents'. Our conjecture is two-fold:
\begin{itemize}
\item
In quantum chaotic systems, $G_{ij}^{(\phi)}$ becomes `random' at sufficiently large $t$.
Namely, in the chaotic theories the exponents are described by RMT.
\item
In non-ergodic theories (e.g. the MBL phase) the exponents are not described by RMT.
\end{itemize}
The idea behind this conjecture is simple. When the system is chaotic, the information about the local perturbation
should be washed away. Hence it is natural to expect that $G_{ij}^{(\phi)}(t)$ becomes a random matrix.
On the other hand, if the system is not chaotic, some structure should survive and a deviation from RMT should be observable.
How this characterization is related to other characterizations, such as RMT universality in the energy spectrum or the exponential Lyapunov growth of OTOCs, is not clear at this moment. Below, we at least demonstrate that these characterizations are compatible in the SYK and XXZ models.
\section*{Numerical study}
In this section, we calculate the exponents $\lambda_i^{(\phi)}(t)$ numerically and study their statistical features.
The exponents are sorted such that $\lambda_1^{(\phi)}(t)\ge \lambda_2^{(\phi)}(t)\ge\cdots\ge\lambda_N^{(\phi)}(t)$. The primary objects of study are the nearest-neighbor level separation $s_i^{(\phi)}(t)\equiv \lambda_i^{(\phi)}(t)-\lambda_{i+1}^{(\phi)}(t)$ and the nearest-neighbor gap ratio $r_i=\frac{{\rm min}(s_i,s_{i+1})}{{\rm max}(s_i,s_{i+1})}$. Because the number of exponents we can obtain numerically is small, we need to use the fixed-$i$ unfolding method \cite{Gharibyan:2018fax} (see appendix for details).
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=7.8cm]{SYK2PEig-N22K0001-10-Ps.pdf}
\includegraphics[width=7.8cm]{SYK2PEig-N24K0001-10-Ps.pdf}
\end{center}
\caption{
SYK, the distribution of nearest neighbor level separation for various values of $t$,
$K=0.0001$ and $K=10$.
All eigenstates are used and the larger $N/2$ exponents are used. $N = 22, 24$.
}\label{Fig:SYK-NN}
\end{figure*}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.85cm]{SYK2PEig-N22-24-K0001-10-r.pdf}
\end{center}
\caption{
SYK, the time dependence of nearest neighbor gap ratio $\langle r\rangle$ for
$N = 22$ and $24$, $K=0.0001$ and $K=10$.
2000 (16) samples are used for $N = 22$ ($24$).
}\label{Fig:SYK-r}
\end{figure}
Consider first the SYK model. When $|\phi\rangle$ is energy eigenstate, then unless $K=0$ and $N$ mod 8 is zero, $G_{ij}^{(\phi)}(t)$ is a complex matrix without particular symmetry.
Hence, when the system is chaotic, if RMT behavior emerges, the relevant ensemble would be the Gaussian unitary ensemble (GUE).
When $K=0$ and $N$ mod 8 is zero, $G_{ij}^{(\phi)}(t)$ is complex and symmetric and in this case one expects Gaussian orthogonal ensemble (GOE) statistics. Hence we expect GOE when $K\simeq 0$ and $N$ mod 8 is zero.
At the values of $N$ we study, the energy dependence of the spectrum is not large. (The energy dependence is similar to the case of the Lyapunov spectrum; see \cite{Gharibyan:2018fax} for a detailed explanation.) Hence, it is simplest to average over all energy eigenstates. Numerically we find that the gap between $\lambda_{N/2}$ and $\lambda_{N/2+1}$ is bigger than the other gaps and appears to behave differently when $K$ is large, as explained in more detail in the supplementary material. Hence, we use only the first half of the spectrum with $N/2$ exponents in the analysis. We checked that similar results are obtained using the other half of the spectrum.
Fig.~\ref{Fig:SYK-NN} shows the nearest-neighbor level separation. Near $K=0$ (chaotic phase) the spectrum is GUE-like.~\footnote{A small nonzero value $K=0.0001$ is used to avoid a degeneracy in the energy spectrum when $K$ is exactly zero.} It is interesting that the GUE behavior can be seen at all time scales. We observed the same phenomenon for other $N\not\equiv 0$ mod 8. For $N\equiv 0$ mod 8, the spectrum is GOE-like at sufficiently late time, but at early time there are large deviations from GOE. In the opposite limit of large $K$, in which the system is not chaotic to leading order, the spectrum is Poisson-like. This claim is substantiated in Fig.~\ref{Fig:SYK-r} which shows the nearest-neighbor gap ratio. The GUE value is approximately obtained when $K\simeq 0$, while at large $K$ the ratio is close to the Poisson value.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.85cm]{XXZ2PEig-N14W05-4-Ps.pdf}
\end{center}
\caption{
The distribution of nearest-neighbor level separation $s$, XXZ, $t=0.1,10,20,100$ for $W=0.5$ and $W=4$,
with $N_{\rm site}=14$ for central $10~\%$ of the energy eigenstates.
The largest $N_\mathrm{site}/2$ exponents are used.
The technical details of the fixed-$i$ unfolding and the $N_{\rm site}$-dependence are discussed in the supplementary material.
}\label{Fig:XXZ-NN-E}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{XXZ2PEig-N8-14-W05-4-r.pdf}
\end{center}
\caption{
The averaged nearest-neighbor gap ratio for the central $10~\%$ of the energy eigenstates.
At least 22000 (1200) samples are used for $N_\mathrm{site}=12, 10, 8$ ($N_\mathrm{site}=14$).
The largest $N_\mathrm{site}/2$ exponents are used.
}\label{Fig:XXZ-r-E}
\end{figure}
Now consider the XXZ model. This model conserves the $z$-component of the total spin, and we consider only the $S^z=0$ sector. We study two values of the $W$ parameter, $W=0.5$ (the ergodic phase) and $W=4$ (the MBL phase). In this model, $G^{(\phi)}$ is complex and symmetric when $|\phi\rangle$ is an energy eigenstate (see supplementary material). Hence, in the ergodic phase, we expect GOE statistics.
To orient the discussion, we first discuss the time-scale for the decay of the two-point functions. For $W=4$, we observe a clear split of the larger and smaller halves.
Hence, the larger half of the exponents is used for the analysis, both for $W = 0.5$ and for $W = 4$. We checked that the result does not change much if the smaller half, or all the exponents, are used provided $N_{\rm site}$ is large enough ($N_{\rm site}=12,14$).
The energy dependence is rather large unlike the SYK model. (Again, see \cite{Gharibyan:2018fax} for a detailed explanation.) Hence we need to restrict the energy to be in a small range in order to remove an uncontrolled energy variation from the analysis.
Fig.~\ref{Fig:XXZ-NN-E} shows the distribution of the nearest-neighbor level separation. The chaotic phase exhibits a GOE distribution, while the distribution is close to Poisson in the MBL phase. Note that, unlike the SYK model, the chaotic phase is not described by RMT at early time. Interestingly, the deviation from RMT becomes large at $1\lesssim t\lesssim 10$, but it eventually vanishes.~\footnote{The agreement is improved to some extent by removing the largest exponent, which decays much slower than others. Still, the deviation at short time remains.}
There is a curious $N_{\rm site}$-dependence of this deviation at intermediate time which is discussed further in the supplementary material.
In Fig.~\ref{Fig:XXZ-r-E} the averaged nearest-neighbor gap ratio is plotted. In the chaotic phase ($W=0.5$), the value of $\langle r\rangle$ is not strongly dependent on $t$
and approaches the GOE value \cite{Atas}. The agreement with the GOE value even at intermediate time is likely a coincidence, because the nearest-neighbor level separation is not close to GOE. In the MBL phase ($W=4$), $\langle r\rangle$ is smaller than the GOE value and decreases toward the Poisson value as $N$ increases.
\section*{Summary and Discussion}
Here we introduced a spectrum defined from a matrix of two-point functions
(\eqref{G-SYK} for SYK and \eqref{G-XXZ} for XXZ), and proposed that the statistical features of this spectrum
exhibit random matrix universality when the underlying system is chaotic.
While we have used the energy eigenstates to define the spectrum, this particular choice is not crucial to observe universality. Spin eigenstates such as $\vert\!\uparrow\uparrow\cdots\uparrow\uparrow\rangle$ and $\vert\!\uparrow\downarrow\cdots\uparrow\downarrow\rangle$ also yield the same structure \cite{in-preparation} at long time, but the time-scale for the onset of RMT-behavior can depend on the choice of state.
In this paper, all the models considered have some degree of disorder in their definition. One could worry that this disorder is the source of the RMT behavior. The fact that we do not observe RMT signatures in the MBL phase shows that this is not so. However, given a theory without disorder and a highly symmetric state $|\phi\rangle$, the RMT behavior may not be visible. In the case of a chaotic system without disorder, we conjecture that, if randomness is introduced in the choice of $|\phi\rangle$, then RMT behavior will be observed.
There are various generalizations and extensions of this work. One clear task is to see if the same signatures are observed in other chaotic models. Another goal is an analytic argument for the observed behavior. One can also consider Euclidean two-point functions, which are more accessible in a variety of systems thanks to Monte Carlo methods. If RMT universality can be observed there, it would provide a powerful tool to study the chaotic nature of large systems where real-time dynamics is hard to access numerically. The matrix of two-point functions considered here can be defined in classical systems as well. Whether the same universality can be found in that context is another interesting question.
\section*{Acknowledgments}
\hspace{0.51cm}
We thank S.~Hikami, S.~Matsuura and H.~Shimada for stimulating discussions.
This work was partially supported by JSPS KAKENHI Grants 17K14285 (M.~H.) and 17K17822 (M.~T.), the Simons Foundation via the It From Qubit Collaboration (B.~S.), and the Department of Energy award number DE-SC0017905 (B.~S).
H.~G. was supported in part by NSF grant PHY-1720397.
M.~H. thanks Brown University for the hospitality during his stay while completing the paper, and acknowledges the STFC Ernest Rutherford Grant ST/R003599/1.
\bibliographystyle{utphys}
|
{
"timestamp": "2019-03-01T02:18:46",
"yymm": "1902",
"arxiv_id": "1902.11086",
"language": "en",
"url": "https://arxiv.org/abs/1902.11086"
}
|
\section{Introduction}
\label{sec:introduction}
\subfile{sections/1_introduction}
\section{Modeling of Vibration-Based Locomotion}
\label{sec:modeling}
\subfile{sections/2_modeling}
\section{Design and Control of Brushbots}
\label{sec:designcontrol}
\subfile{sections/3_design}
\section{Brushbots in Swarm Robotics}
\label{sec:swarmrobotics}
\subfile{sections/4_applications}
\section{Conclusions}
\subfile{sections/5_conclusions}
\bibliographystyle{IEEEtran}
\subsection{Related Work}
\label{subsec:relatedwork}
As will be elaborated on in more detail in Section~\ref{sec:modeling}, the principle on which the motion of a brushbot relies is the alternation of stick and slip phases during which the brushes adhere or not to the ground. One of the first applications of the stick-slip mechanism to robot locomotion can be found in \cite{breguet1998stick}, where a three-degree-of-freedom micro-robot is presented. Using this principle, in \cite{vartholomeos2006analysis}, the authors propose an improved, energy-efficient design of a micro robot, together with a control strategy suitable for trajectory tracking. Due to the design simplicity and the resulting robustness, brushbots lend themselves to swarm robotic applications, where groups of robots are utilized to perform coordinated tasks in a decentralized fashion. This idea is explored in \cite{rubenstein2012kilobot}, where the authors present the Kilobot, a small scale brushbot equipped with an infrared and a light sensor that enable the execution of decentralized swarming algorihtms. Collective behaviors of brushbots are also investigated in \cite{giomi2013swarming}, where the authors analyze the parameters governing the transition from a disordered motion to an organized collective motion.
As far as the analysis of brush dynamics is concerned, in \cite{becker2014mechanics}, a model is developed and validated using an experimental robotic platform. Here the authors do not focus on the motion control explicitly, as much as it is done in \cite{klingner2014stick}. In the latter, an omnidirectional stick-slip robot is presented and a way of automatically calibrating it is proposed. A more theoretical analysis is performed in \cite{cicconofri2015motility}, where the derived equations of motion are solved using a heuristic approach in order to obtain analytical formulas for the average velocity of the robot.
In this paper, we propose a dynamic model for brushbots, which starts from the microscopic analysis of the brushes to culminate in the macroscopic model of the robot. In particular, this model improves the ones which can be found in literature by explicitly taking into account the inertia of the brushes and the effects that it has on the resultant brushbot velocity. Moreover, the derived model will be further validated through the development of a trajectory tracking controller and the implementation of a coordinated control algorithm for a swarm of brushbots.
To summarize, the main contributions of the paper are the following:
\begin{enumerate}[(i)]
\item we propose a brush model which considers the inertia of the brushes and the contact dynamics of their interaction with the ground
\item we analyze and qualitatively characterize different \textit{regimes of operation} for the different models of brushbots developed in the literature
\item we present the mechanical design of two brushbots, a fully-actuated platform that can switch between regimes of operations, and a differential-drive-like brushbot, specifically designed for swarm robotics applications
\end{enumerate}
Furthermore, in our related work \cite{arxiv:ral2}, we build upon the results of this paper and demonstrate the ability of brushbot swarms to achieve collective behaviors using simple local interactions.
\subsection{Model for Regime I}
\label{subsec:regime1}
The main factors that make brushbots operate in regime I rather than II are weight and brush stiffness. Regime I is characterized by a lower brush stiffness (more deformable brushes) and/or a heavier robot body. The following assumptions are used for the derivation of the brush dynamic model in this regime.
\begin{assumption}
\label{ass:regime1}
During operations in regime I, the brushes are always in contact with the ground. The heavier robot body, in fact, does not allow the centrifugal force generated by vibration motors to lift the robot from the ground
\end{assumption}
\begin{assumption}
The body of the brushbot always remains parallel to the ground. This is justified by the fact that the inertia of the body does not allow big rotations at the frequencies at which the vibration motors are typically actuated
\end{assumption}
\begin{figure}
\centering
\def0.36\textwidth}\import{./fig/converted_svg/}{stewart_platform_svg-tex.pdf_tex{0.24\textwidth}\import{./fig/converted_svg/}{thin_brush_svg-tex.pdf_tex}
\caption{A brushbot with plate-like brushes with the brushes reference frame $\xi\xi_{\perp,1}\xi_{\perp,2}$. The resulting second area moment of the cross section of the bristles is higher about the $\xi_{\perp,2}$ axis than about $\xi_{\perp,1}$. The higher the difference between the two second area moments, the more realistic Assumption~\ref{ass:planar} is.}
\label{fig:verythinbrushes}
\end{figure}
\begin{assumption}
\label{ass:planar}
The deformation of the brushes is planar. Indeed, the inclination of the brushes has the effect of reducing their equivalent stiffness in one direction. More precisely, referring to Fig.~\ref{fig:verythinbrushes}, the fact that the brushes are rotated around axis $\xi_{\perp,1}$ with respect to the ground makes their equivalent stiffness in the plane $\xi\xi_{\perp,2}$ smaller than the one in the plane $\xi\xi_{\perp,1}$. This will be theoretically derived later in this section.
\end{assumption}
\begin{figure}
\centering
\subfloat[][Model for the stick phase.]{\label{subfig:stickschematics}\def0.36\textwidth}\import{./fig/converted_svg/}{stewart_platform_svg-tex.pdf_tex{0.24\textwidth}\import{./fig/converted_svg/}{stick_schematics_svg-tex.pdf_tex}}\hfill
\subfloat[][Model for the slip phase.]{\label{subfig:slipschematics}\def0.36\textwidth}\import{./fig/converted_svg/}{stewart_platform_svg-tex.pdf_tex{0.24\textwidth}\import{./fig/converted_svg/}{slip_schematics_svg-tex.pdf_tex}}
\caption{Beam model employed to analyze the dynamics of the brush during the stick and slip phases. $v$ represents the displacement of the beam in the direction orthogonal to the beam axis $\xi$. Compare with the qualitative motion depicted in Fig.~\ref{fig:stickslip}.}
\label{fig:beammodel}
\end{figure}
We employ the Euler-Bernoulli beam model (see, e.\,g., \cite{timoshenko1983history}) to analyze the motion of each brush. Figures~\ref{subfig:stickschematics}~and~\ref{subfig:slipschematics} show the structural scheme used to model stick and slip phases, respectively. During the stick phase, the constraints are a guide at the top (where the brushes connect to the robot) and a hinge at the bottom (at the contact with the ground). In the slip phase, a horizontal translational degree of freedom for the interaction with the ground is added by using a roller support in place of the hinge. This allows the tip of the brush in contact with the ground to slide.
The Euler-Bernoulli beam model allows us to evaluate the deformed shape of the brush, as well as its equivalent stiffness, by solving the following boundary value problem:
\begin{equation}
\label{eq:eb}
\begin{cases}
EIv^{\prime\prime\prime\prime}=0\\
EIv^{\prime\prime\prime}\vert_{\xi=l}=F\cos\alpha\\
EIv^{\prime\prime}\vert_{\xi=l}=0\\
v^{\prime}\vert_{\xi=0}= 0\\
v\vert_{\xi=0}=0.
\end{cases}
\end{equation}
Here, $v$ represents the displacement of the beam in the direction orthogonal to the beam axis $\xi$, $v^\prime$ is used to denote $dv/d\xi$, $F = m\omega^2r\sin(\omega t)$ is the centrifugal force produced by an eccentric rotating mass motor which rotates a mass $m$, at speed $\omega$, mounted with an eccentricity $r$ with respect to the motor axle. $E$ and $I$ are the Young modulus and the second area moment about $\xi_{\perp,1}$ of the beam. $l$ and $\alpha$ are the length and inclination of the beam, respectively.
The solution to \eqref{eq:eb} is given by
\begin{equation}
v(\xi) = \frac{F\cos\alpha}{6EI}\xi^3 - \frac{Fl\cos\alpha}{2EI}\xi^2.
\end{equation}
So, the displacement $v$ of the robot body at the tip of the brush can be evaluated as:
\begin{equation}
\label{eq:vertdisp}
|v(l)| = \frac{Fl^3\cos\alpha}{3EI}.
\end{equation}
During the slip phase (Fig.~\ref{subfig:slipschematics}), the robot body moves upwards, reducing the horizontal force due to friction which acts on the brush tip.
\begin{figure}
\centering
\def0.36\textwidth}\import{./fig/converted_svg/}{stewart_platform_svg-tex.pdf_tex{0.24\textwidth}\import{./fig/converted_svg/}{displacement_svg-tex.pdf_tex}
\caption{The net displacement of the brushbot, $\delta$, is evaluated based on the angle $\vartheta$ induced by the force $F$ (see Fig.~\ref{subfig:stickschematics}) and the geometric characteristics of the brush.}
\label{fig:netdisplacement}
\end{figure}
The net horizontal displacement can be calculated as follows (see Fig.~\ref{fig:netdisplacement}):
\begin{align}
\delta &= \overline{P_2P_3} = \overline{P_1P_3}-\overline{P_1P_2}=l\cos(\alpha-\vartheta)-l\cos\alpha\\
&=l\cos\left(\alpha-\frac{m\omega^2 r l^2\cos\alpha}{3EI}\right)-l\cos\alpha,
\end{align}
where $\overline{P_iP_j}$ denotes the length of the segment joining points $P_i$ and $P_j$, and the expression for $\vartheta$ is obtained by observing that, under the small-angle approximation, $|v(l)| = l\vartheta$ ().
Considering the fact that the robot experiences a displacement of $\delta$ per full rotation of the motor, the ground speed of the robot, $v_r$, can be obtained as follows:
\begin{equation}
\label{eq:motorrobotspeeds}
v_r=\frac{\delta}{\Delta t} = \frac{\omega}{2\pi}\left(l\cos\left(\alpha-\frac{m\omega^2 r l^2\cos\alpha}{3EI}\right)-l\cos\alpha\right),
\end{equation}
where $\omega$ is the angular velocity of the motor.
\begin{figure}
\centering
\def0.36\textwidth}\import{./fig/converted_svg/}{stewart_platform_svg-tex.pdf_tex{0.24\textwidth}\import{./fig/converted_svg/}{lumped_svg-tex.pdf_tex}
\caption{Lumped-parameter model used to analyze the dynamics of the brushes: the equivalent stiffness $k_\vartheta$ and inertia $I_\vartheta$, given in \eqref{eq:lumpedk} and \eqref{eq:lumpedm}, determine the spring-mass-like response of the brush angle $\vartheta$ as a result of the force $F$ in Fig.~\ref{subfig:stickschematics}.}
\label{fig:lumped}
\end{figure}
For the study of the oscillating brush dynamics, we use the lumped-parameter model depicted in Fig.~\ref{fig:lumped} with
\begin{align}
k_\vartheta &= \frac{3EI}{l^2 \cos\alpha}\label{eq:lumpedk}\\
I_\vartheta &= \frac{M_b l^2}{2}\label{eq:lumpedm},
\end{align}
being the stiffness and the inertia relating the force $F$ and the angle $\vartheta$, and $M_b$ denotes the mass of the brush.
\begin{assumption}
\label{ass:nonstraight}
The inclination angle of the brushes $\alpha\in(0,\pi/2)$, i.\,e. the brush is neither horizontal nor vertical.
\end{assumption}
\noindent Under this assumption, $k_\vartheta$ is well-defined.
\begin{observation}
The expression of $k_\vartheta$ in \eqref{eq:lumpedk} indicates that equivalent stiffness of the brushes increases with an increase of the angle $\alpha$. In the limit case: $k_\vartheta\to\infty$ as $\alpha\to\pi/2$. This reflects the fact that, if brushes are perpendicular to the ground, no net displacement can be achieved. Moreover, by the insight gained using the Euler-Bernoulli model, we can see that Assumption~\ref{ass:planar} becomes more realistic as the second area moment around $\xi_{\perp,1}$, which we denoted by $I$, becomes smaller with respect to the one around $\xi_{\perp,2}$ (see Fig.~\ref{fig:verythinbrushes}).
\end{observation}
In the analysis of the dynamic effects introduced by the inertia of the brushes, we start by calculating the time that the brushes take, during the slip phase, to go back to the rest position from the configuration reached at the end of the stick phase (see Fig.~\ref{fig:stickslip}). Taking into account their inertial effects, the brushes can be modeled as the following second-order system:
\begin{equation}
\label{eq:brushoscillations}
\begin{cases}
I_\vartheta \ddot\vartheta+ k_\vartheta\vartheta= 0\\
\vartheta(0)=\bar\vartheta\\
\dot\vartheta(0)=0,
\end{cases}
\end{equation}
whose solution is given by $\vartheta(t)=\bar\vartheta\cos(\omega_n t)$, where
\begin{equation}
\label{eq:natfrebrush}
\omega_n=\sqrt{\frac{k_\vartheta}{I_\vartheta}}=\sqrt{\frac{6EI}{M_b l^4 \cos\alpha}}
\end{equation}
is the natural frequency of the brush. The time to go back to the rest position is the earliest time at which $\vartheta(t)=0$, i.\,e. $\omega_n t = \kappa\pi/2$. So, the earliest time instant $\bar t$ at which the brushes come back to the undeformed configuration is given by:
\begin{equation}
\bar t = \left.\kappa\frac{\pi}{2\omega_n}\right\vert_{\kappa=1} = \frac{\pi}{2}\sqrt{\frac{M_b l^4 \cos\alpha}{6EI}}.
\end{equation}
Stiffer (larger $EI$), shorter (smaller $l$), less inclined (smaller $\alpha$), lighter brushes (smaller $M_b$) lead to a faster response to vibrations (smaller $\bar t$).
While the vibration motor is rotating, the slip phase occurs if the friction between the brush and the ground is not enough to prevent the brush from sliding. The transition from the stick phase to the slip phase is triggered by a reduction of the force acting on the robot and normal to the ground due to centrifugal acceleration of the unbalanced rotating mass. Therefore, a quarter of period of revolution of the motor is the time the brushes have to move forward during the slip phase. Thus, to maximize the net displacement of the robot, we want to achieve a motor speed $\omega$ such that
\begin{equation}
\label{eq:omegastar}
\bar t = \frac{1}{4} T = \frac{1}{4} \frac{2\pi}{\omega},
\end{equation}
where $T$ is the period of revolution of the motor. Solving \eqref{eq:omegastar} for $\omega$ yields:
\begin{equation}
\frac{\pi}{2}\sqrt{\frac{M_b l^4 \cos\alpha}{6EI}} = \frac{\pi}{2\omega^\star} \quad\Leftrightarrow\quad \omega^\ast=\sqrt{\frac{6EI}{M_b l^4 \cos\alpha}} =\omega_n.
\end{equation}
Thus, not surprisingly, if the motor speed matches the natural frequency of the brushes $\omega_n$, the displacement of the robot is maximized. This can be also seen by considering the model in \eqref{eq:brushoscillations} with a non-zero input force:
\begin{equation}
\label{eq:forcedbrushoscillations}
I_\vartheta \ddot\vartheta+ k_\vartheta\vartheta= m\omega^2r\sin(\omega t)\cos\alpha\footnote{Despite their expressions, $I_\vartheta \ddot\vartheta$ and $k_\vartheta\vartheta$ are not torques, but rather forces.},
\end{equation}
whose forced solution is given by
\begin{equation}
\label{eq:forcedsol}
\vartheta(t) = \frac{m\omega^2r\sin(\omega t)\cos\alpha}{\omega_n^2-\omega^2}\sin(\omega t) = \hat\vartheta(\omega)\sin(\omega t).
\end{equation}
According to the model \eqref{eq:forcedbrushoscillations}, the amplitude of the brush oscillations, $|\hat\vartheta(\omega)|\to\infty$ as $\omega\to\omega_n$. In practice, there are damping effects which will reduce the oscillation amplitude to a finite value. However, notice also that the model derived in this section holds under Assumpion~\ref{ass:regime1}. Therefore, it cannot be used to analyze the motion of the brushbot in case $\omega$ is such that $m\omega^2r\sin(\omega t)>Mg$, where $Mg$ is the weight of the robot, $M$ being its mass. At this point, the robot starts transitioning towards regime II which will be explained in the following section.
\subsection{Model for Regime II}
\label{subsec:regime2}
The model for the second regime in which brushbots can operate predicts the robot motion under the following assumption.
\begin{assumption}
\label{ass:regime2}
The robot body and the brushes are rigid bodies. This entails that brushes are not deformable.
\end{assumption}
\begin{figure}
\centering
\def0.36\textwidth}\import{./fig/converted_svg/}{stewart_platform_svg-tex.pdf_tex{0.32\textwidth}\import{./fig/converted_svg/}{rotated_svg-tex.pdf_tex}
\caption{Motion of the brushbot during the stick phase in regime II. The inclination angle of the brushbot body, $\vartheta_r$, accelerates about the point $P$ under the effect of vibrations and gravity, through the moments generated by the forces $m\omega^2r$ and $Mg$, respectively.}
\label{fig:rotated}
\end{figure}
Similar to what has been discussed for regime I, also in this second regime we have the alternation of stick and slip phases as shown in Fig.~\ref{subfig:regime2}. The difference with the previous case lies in the fact that the effect of the deformation of the brushes is not significant and, therefore, can be neglected. In order to model the motion of the brushbot, by Assumption~\ref{ass:regime2}, we can write the following rigid body motion equations for a brushbot operating in regime II:
\begin{equation}
\label{eq:regime2}
I_P \ddot\vartheta_r = m\omega^2r\sin(\omega t)w - Mgw_G,
\end{equation}
where $I_P$ is the rotational inertia about point $P$ shown in Fig.~\ref{fig:rotated}, where the quantities $w$, $w_G$ and the gravitational force acting on the robot body are depicted. In order to simulate the interaction with the ground, the constraint $\vartheta_r\ge0$ has been enforced. At the point of impact on the ground, we assume $\dot\vartheta_r=0$ and $\ddot\vartheta_r=0$.
\begin{figure}
\centering
\scriptsize
\begin{tikzpicture
\begin{axis}
[
no marks,
xlabel={time},
enlarge x limits=-1,
enlarge y limits=-1,
legend entries={$\vartheta_r$, $\dot\vartheta_r$, $\ddot\vartheta_r$, $x$},
legend style={nodes=right},
legend pos= north west,
width=0.45\textwidth,
height=0.2\textwidth
]
\addplot [line width=1.5pt, color=red] table [x=t, y=th, col sep=space]{data/regime2.txt};
\addplot [line width=1.5pt, color=green] table [x=t, y=thdot, col sep=space]{data/regime2.txt};
\addplot [line width=1.5pt, color=blue] table [x=t, y=thddot, col sep=space]{data/regime2.txt};
\addplot [line width=1.5pt, color=black] table [x=t, y=x, col sep=space]{data/regime2.txt};
\end{axis}
\end{tikzpicture}
\caption{Simulation results of the sequence of stick-slip phases of regime II (as depicted in Fig.~\ref{subfig:regime2}) obtained by solving \eqref{eq:forcedbrushoscillations}. Trajectories of the angle $\vartheta_r$, its first and second time derivatives are reported. The robot position $x$, depicted in black, shows its ability to locomote in regime II.}
\label{fig:solutionregime2}
\end{figure}
Trajectories of angular position $\vartheta_r(t)$, velocity $\dot\vartheta_r(t)$ and acceleration $\ddot\vartheta_r(t)$, are shown in Fig.~\ref{fig:solutionregime2}, together with the resulting displacement $x$ of the robot on the ground. The maximum absolute value of $\vartheta_r$, which is denoted by $|\hat\vartheta_r|$, is the one which determines the displacement $\delta$ of the robot, given by
\begin{equation}
\label{eq:tosmallangleapproximate}
\delta = h \sin|\hat\vartheta_r|.
\end{equation}
\begin{observation}
The rotation of the robot body is neglected for the development of an analytical model for regime I, because the flexibility of the brushes prevails on the rigid rotation of the robot body. Here, on the other hand, the brushes are assumed to be rigid, therefore, the rotation angle of the robot body has the most significant effect. Nevertheless, due to the inertia of the robot, the centrifugal force generated by the unbalanced mass of the motors is not able to rotate the robot body by more than a few degrees. For this reason, we can introduce a small-angle approximation in \eqref{eq:tosmallangleapproximate} and express the robot velocity as a function of $|\hat\vartheta_r|$ as
\begin{equation}
v_r=\frac{\delta}{\Delta t}\approx \frac{\omega h|\hat\vartheta_r|}{2\pi}.
\end{equation}
\end{observation}
\subsection{Range of Applicability of the Models}
\label{subsec:discussion}
In \cite{giomi2013swarming} and \cite{vartholomeos2006analysis}, two vibration-driven robots which work in regime I and regime II, respectively, are presented. The fundamental differences between these robots are related to their weight and the brushes they employ to transform vibrations into motion. In the following, we discuss the physical characteristics of brushbots which cause the models for regimes I and II to be able to describe more or less accurately the robot motion.
\begin{enumerate}[(i)]
\item \textit{Rigidity of the brushes.} Expressed in terms of $EI$ in \eqref{eq:eb}, the rigidity of the brushes proportionally influences the equivalent stiffness \eqref{eq:lumpedk} and therefore the natural frequency \eqref{eq:natfrebrush}. A high rigidity, however, means also a small displacement $v(l)$ in \eqref{eq:vertdisp}. In practice this means that a stiffer robot moves very little per each revolution of the motor, although it is able to vibrate more at faster frequencies, as indicated by \eqref{eq:forcedsol}. For this reason, brushbots equipped with stiffer brushes are more likely to operate in regime II.
\item \textit{Mass of the robot.} The influence of the mass of the robot is recognizable in the effect it has on the inertia $I_P$ used in \eqref{eq:regime2} that the robot exhibits with respect to rigid rotations around axes that lie in the plane in which the robot moves. Therefore, at a constant power produced by the motors, robots operating in regime II typically have smaller masses compared to the ones operating in regime I. This, in fact, results in smaller inertias which allow the robots to quickly respond to alternating input forces. In the case of robots operating in regime I, the flexibility of the brushes reduces the response bandwidth, given by the natural frequency \eqref{eq:natfrebrush}. Therefore, the motion due to regime I dominates the one due to regime II.
\item \textit{Inclination of the brushes.} By Assumption~\ref{ass:nonstraight}, the inclination of the brushes, $\alpha$, is never equal to $\pi/2$. In the limit case in which $\alpha=\pi/2$, in fact, the dynamic model \eqref{eq:brushoscillations} for regime I predicts zero net motion of the robot. When the brushes become straight ($\alpha\to\pi/2$), in fact, the brushbot starts operating mainly in regime II.
\end{enumerate}
A factor that influences the brushbot motion is the position of multiple sets of brushes and actuators, which will be explicitly considered in the next sections. The presence of multiple brushes introduces constraints which are not taken into account in the model of regimes I and II. In fact, the superposition of the effects that different sets of brushes have due to their different orientations can result in drastically different behaviors depending on the regime in which the robot operates.
Consider a brushbot configuration where three sets of brushes are oriented radially equally spaced along the circumference of the brushbot. This leads to the practical impossibility of motion of such a brushbot operating in regime I, whereas can be exploited to achieve \textit{holonomic} motion when operating in regime II.
In Sections~\ref{sec:designcontrol}~and~\ref{sec:swarmrobotics}, we show how to leverage the effects described above together with the models developed in this section in order to design and control fully-actuated and differential-drive brushbots. Moreover, in Section~\ref{sec:designcontrol}, we report experimental results to show the validity of the proposed models in predicting the motion of brushbots.
|
{
"timestamp": "2019-03-05T02:07:45",
"yymm": "1902",
"arxiv_id": "1902.10830",
"language": "en",
"url": "https://arxiv.org/abs/1902.10830"
}
|
\section{Introduction}
The ``VOiCES from a Distance Challenge 2019'' is designed to foster research in the area of speaker recognition and automatic speech recognition (ASR) with the special focus on single channel distant/far-field audio, under noisy conditions. The main objectives of this challenge are to: (i) benchmark state-of-the-art technology in the area of speaker recognition and automatic speech recognition (ASR), (ii) support the development of new ideas and technologies in speaker recognition and ASR, (iii) support new research groups entering the field of distant/far-field speech processing, and (iv) provide a new, publicly available dataset to the community that exhibits realistic distance characteristics.
This challenge is based on the recently released Voices Obscured in Complex Environmental Settings (VOiCES) corpus, released under Creative Commons-BY 4.0 license, making it accessible for commercial, academic, and government use. The VOiCES corpus, a collaboration between SRI International and Lab41, provides speech data recorded in acoustically challenging environments. Data was collected by recording retransmited audio from high-quality loudspeakers that played in real rooms, capturing natural reverberation. LibriSpeech~\cite{LibriSpeech2015} was used as the clean speech source, while television, music or babble played simultaneously from another loudspeaker as background noise. The clean speech loudspeaker rotated at predefined intervals during recordings, to mimicking human head movement. More details on VOiCES can be found at~\cite{voices,nandwana2018robust}.
Participation in the VOiCES challenge is free of cost. The challenge is intended for those interested in upholding the challenge rules outlined in this document and who intend to submit a paper to ``The VOiCES from a Distance Challenge 2019'', a Special Session to be held at Interspeech 2019, in Graz, Austria on September 15-19, 2019. Information about evaluation registration can be found on the VOiCES website\footnote{https://voices18.github.io/Interspeech2019\_SpecialSession/}.
Participants who complete the challenge and submit both their system outputs and description will get early access to the VOiCES phase 2 data. The phase 2 data is an extension of VOiCES phase 1 set, having over 310k audio files recorded in different reverberant environments.
The VOiCES challenge has two tasks: speaker recognition and automatic speech recognition (ASR). Each task has fixed and open training conditions. These conditions are defined by the training data that can be used to train the systems. Participants are required to participate in at least one condition of a task (e.g. ASR Open).
\section{Speaker Recognition}
The speaker recognition challenge presented here is similar to previous speaker detection challenges, such as the National Institute in Standards of Technology (NIST) Speaker Recognition Evaluations (SRE)~\cite{Sadjadi2017} and the Speakers in the Wild (SITW) challenge~\cite{sitw}. The task is: given a segment of speech and target speaker enrollment data, automatically determine whether the target speaker is speaking in the segment. A segment of speech (test segment) and the enrollment speech segment(s) from a designated target speaker constitute a trial. The speaker recognition system is required to process each trial independently and output a log-likelihood ratio (LLR), using natural (base $e$) logarithm, for that trial. The LLR for a given trial including a test segment $s$ is defined as follows:
\begin{equation}
LLR(s) = \log\left(\frac{P(s|H_{0})}{P(s|H_{1})}\right)
\end{equation}
where $P(.)$ denotes the probability distribution function (pdf), and $H_{0}$ and $H_{1}$ represents the null (i.e., $s$ is spoken by the enrollment speaker) and alternative (i.e., $s$ is not spoken by the enrollment speaker) hypotheses, respectively. The performance of a speaker recognition system will be judged on the accuracy of these LLRs.
\subsection{Training Conditions}
Speaker recognition systems can be developed for the fixed condition, the open condition or both. The two training conditions are defined by the specific datasets that can be used to build the speaker recognition system.
\subsubsection{Fixed Condition}
The fixed training condition limits the system training to the following freely available data sets:
\begin{itemize}
\item Speakers in the Wild (SITW)\footnote{http://www.speech.sri.com/projects/sitw/}
\item VoxCeleb1~\cite{Nagrani2017} and VoxCeleb2\footnote{http://www.robots.ox.ac.uk/~vgg/data/voxceleb/}~\cite{Chung2018}
\end{itemize}
Participants can obtain these datasets by following the instructions on their respective webpages. The audio data from the Voxceleb1 and Voxceleb2 is restricted to the official annotations for the fixed condition submissions. In this way, the fixed condition can serve its purpose of measuring the performance of different systems trained with the same data (or a subset thereof). The Voxceleb datasets also contain video URLs. No image or video processing is allowed in the fixed condition, however, image or video processing may be used for cross-model processing of the audio used to train a system for the open condition. Please note that SITW and VoxCeleb have overlapping speakers\footnote{http://www.robots.ox.ac.uk/~vgg/data/voxceleb/SITW\_overlap.txt}.
Publicly available non-speech audio and noises (e.g. noises, impulse responses, codecs) may be used for data augmentation~\cite{x-vectors,McLaren2018} and should be included in the system description. For the fixed training condition, only the datasets specified above maybe used for system training and development with the exception of speech activity detection (SAD). Participants may train their own or use an existing SAD and details of the SAD should be included in the system description. With the exception of SAD, participants in the fixed condition can not use a pre-trained model for system components.
\subsubsection{Open Condition}
The open training condition removes the limitations of the fixed condition. For this condition, participants can use any proprietary and/or public data they have access to including the fixed condition data. The participants must mention the datasets used to train the open condition submission in the system description.
\subsection{Development Data}
The speaker recognition development dataset consists of 15,904 audio segments from 196 speakers. Each audio file contains a single speaker. The dataset represents different rooms, microphones, noise distractors, and loudspeaker angles. The metadata is available in the filename and teams may use this information to analyze the behavior of their system under different conditions. More detailed information about the metadata can be found in the README.VOiCES\_2019.txt provided with the development data.
The development data for speaker recognition may be used for system training including the calibration models for both fixed and open conditions.
\subsection{Evaluation Data}
The speaker recognition evaluation set will consist of unreleased distant recordings that are part of the VOiCES corpus. Participants can expect these recordings to originate from different microphone types and different rooms both of which could be more challenging than those featured in the development set.
\subsection{Performance Measures}
We will use several performance measures to determine the speaker recognition system performance and compare system submissions in the challenge. A metric similar to those used in the NIST SRE 2010 and Speakers in the Wild (SITW)~\cite{sitw} challenge will form the primary metric for the VOiCES challenge.
The participants have been provided with a python script to evaluate each performance metrics detailed below. This python scorer will be used by the organizers to produce the official metrics on the evaluation data.
\subsubsection{Primary Metric}
The primary metric for the VOiCES challenge is based on the following detection cost function, which is the same function used in the NIST 2010 SRE, but with different parameters. It is a weighted sum of miss and false alarm error probabilities in the form:
\begin{equation}
C_{det} = C_{miss} \times P_{miss} \times P_{tar} + C_{fa} \times P_{fa} \times (1-P_{tar})
\end{equation}
We assume a prior target probability, $P_{tar}$, of 0.01 and equal costs between misses and false alarms.
\begin{table}[h]
\centering
\caption{Cost model parameters for the primary metric $C_{det}$}
\begin{tabular} {clclc}
\toprule
$C_{miss}$& $C_{fa}$& $P_{tar}$ \\
\midrule
1.0 & 1.0 & 0.01 \\
\midrule
\end{tabular}
\end{table}
For reporting, the $C_{det}$ will be normalized by the cost that a na\"ive system that always chooses the least costly class would get for the selected parameters. In our case, the normalization factor is given by $P_{tar}$.
\subsubsection{Alternate Performance Metric}
For the purpose of analyzing how well a system is calibrated across all operating points, a log-likelihood ratio cost metric, $C_{llr}$~\cite{niko_cllr}, will also be reported as:
\begin{equation}
\scalebox{0.95}[1]{$C_{llr} = \frac{1}{2 \times \log(2)} \times \left( \frac{\sum \log(1+1/s)}{N_{tar}} + \frac{\sum \log(1+s)}{N_{non}}\right)$}
\end{equation}
where $s$ is the likelihood ratio for a trial, and $N_{tar}$ and $N_{non}$ represent the number of target and non-target trials, respectively.
\subsection{Scores Submission}
Participants are required to submit to the VOiCES organizers a set of scores for each trial they evaluated. The score files should follow the naming convention: [TeamName]\_[Task]\_[Condition]\_[SystemNumber].txt.
The score files should be in ASCII format with one line per trial. Each line must include three space-delimited fields:
modelID$<$space$>$testSegment$<$space$>$LLR$<$NewLine$>$
A separate score file is required for each condition and each system, with a limit of three files per condition. The score submission instructions will be provided along with the evaluation data.
\section{Automatic Speech Recognition}
In the ASR task, participants are expected to provide a transcript of each audio segment in a verbatim and case-insensitive manner.
\subsection{Training Conditions}
The ASR task will be evaluated over fixed and open training conditions. The two training conditions are defined by the specific datasets that can be used to build the speech recognition system. Teams can participate in the fixed condition, open condition or both.
\subsubsection{Fixed Condition}
In the fixed condition, the training set consists of an 80-hour subset of the LibriSpeech corpus. This subset was designed in such a way as to have no overlap in speakers with the VOiCES corpus (dev or eval).
While the participants may train their own SAD as well as use external non-speech resources for data augmentation, they may not use additional speech data from any other source for model training (acoustic model, language model, speech enhancement, etc.)
\subsubsection{Open Condition}
The open condition removes the limitations of the fixed condition. For this condition, participants can use any proprietary and/or public data they have access to along with the fixed condition data.
\subsection{Development Set}
The ASR development set is distinct from the speaker recognition dev set, and will consist of 20h of distant recordings from rooms 1 and 2 along with corresponding transcripts. It contains recordings from 6 of the 12 mics, and is balanced across rooms, mics, distractor types, and loudspeaker angles. The metadata (mic, room, distractor, angle) is available in the filename, and the participants are welcome to use that information to analyze the behavior of their system under different conditions.
{\bf The VOiCES Challenge's dev set may be used to make design decisions, but may not be used for directly training the system's SAD, enhancement, acoustic or language models in either the fixed or the open condition.}
\subsection{Evaluation Set}
The ASR evaluation set will consist of 10h to 20h of previously unreleased distant recordings that are part of the VOiCES corpus. Participants can expect the recordings to originate from different microphone types and different rooms, both of which could be more challenging than those featured in the dev set.
\subsection{Performance Measures}
Similarly to NIST's recent OPENSAT-17 challenge, we will use the Word-Error Rate (WER) as the evaluation metric for the ASR portion of this challenge. Specifically, we will use NIST's open source SCTK software to score participants submissions by computing WER as the sum of errors (deletions, insertions, substitutions) divided by the total number of words from the reference transcript.
\subsection{Scores Submission}
Participants are required to submit to the VOiCES organizers word-level transcripts in Conversation-Time Marked (CTM) format. The score files should follow the naming convention: [TeamName]\_[Task]\_[Condition]\_[SystemNumber].txt.
The CTM format consists of a tab-separated 6-columns ASCII text file, where each line corresponds to a word. The fields are defined as follows:
\begin{enumerate}
\item The waveform file base name (i.e., without path names or extensions).
\item Channel ID, the audio files are mono this column should be `1'
\item The beginning time of the word, in seconds, measured from the start time of the file.
\item The duration of the word, in seconds
\item The orthographic rendering (spelling) of the token.
\item Confidence Score, the probability with a range [0:1] that the token is correct. If confidence is not available, omit the column.
\end{enumerate}
The ASR scoring is identical to the NIST OPENSAT-17 evaluation, and more details can be found in NIST's evaluation plan\footnote{https://www.nist.gov/itl/iad/mig/opensat}.
A separate score file is required for each condition and each system, with a limit of three files per condition. The score submission instructions will be provided along with the evaluation data.
\section{Training and Evaluation Dataset Organization}
The data structure of both speaker recognition and speech recognition data download is as follows:
\dirtree{%
.1 Interspeech2019\_VOiCES\_Challenge/.
.2 Training\_Data.
.3 Automatic\_Speech\_Recognition.
.2 Development\_Data.
.3 Speaker\_Recognition.
.3 Automatic\_Speech\_Recognition.
.2 Evaluation\_Data.
.2 README.VOiCES\_2019.txt.
}
The meta information of the audio files is included in the file names. More information about metadata can be found in README.VOiCES\_2019.txt
\section{Evaluation Rules}
All participants must adhere to the following rules regarding the processing of the VOiCES evaluation data until all system outputs have been submitted.
\begin{itemize}
\item {\bf Participants may only use the subset of the VOiCES data provided for development and evaluation under each task. Teams may \emph{not} use any other VOiCES data releases including parts of the VOiCES corpus that contain distractor audio only. This is a challenge and using such data will give unfair advantage.}
\item Participants may not use ASR development data for speaker recognition tasks and vice versa.
\item Participants must submit system output for at least one task condition (i.e. ASR Open).
\item Participants must abide by the terms guiding the fixed and open training conditions.
\item Participants may not probe the evaluation data via manual/human means such as listening to the data or producing the transcript of the speech.
\item Participants may submit up to three system per task (SID/ASR) and condition (Fixed/Open).
\item The official score for a team will be selected as the best primary metric from systems submitted by the team for that condition (up to 3 systems can be submitted per condition per task per team). These official scores will be used for ranking teams.
\item Each team must submit an article describing their systems and providing analysis of its performance on the VOiCES database to the special session of Interspeech 2019 paper submission deadline.
\item During the challenge, teams may email questions seeking clarification of any aspects, specifically those that might be considered vague or ambiguous. To ensure all teams receive the same information, a summary of each question and the response will be emailed to the contact person of each team with the poser of the question being made anonymous.
\item The organizers plan to write articles comparing techniques by anonymizing submissions. Official rankings of teams will be published on the VOiCES website, including scores and confidence margins; individual team requests for anonymity on this public website will be upheld. Regarding further dissemination of results, participants are allowed to publish their own results and their rank from challenge results. They are not allowed, however, to publish other teams results or rank from the challenge. The only exception to this is when referencing published results with the corresponding team authoring such publications.
\item {\bf Participants are not allowed to use any part of LibriSpeech for fixed or open condition system training except for the data provided as ASR training set for the fixed condition. LibriSpeech is the data source for VOiCES data and there is a possibility that there will be an overlap between training and evaluation set.}
\end{itemize}
\section{System Description}
All participants will be required to submit a short system description by March 15, 2019. The purpose of this description is two-fold: it will be shared among other participants for the benefit of analysis and validation, and it will provide information necessary for the organizers to determine common trends in leading systems. The organizers may then use this information (without site names for anonymity) in a ‘summary’ article submitted to the special session. System description is required to get access to the evaluation keys and VOiCES phase 2 download link.
\section{Special Session Paper at Interspeech}
In addition to the system description above, participants must submit an article to "The VOiCES from a Distance Challenge" special session track of Interspeech 2019. The information in the system description will also form part of the paper along with any post-evaluation analysis. This should adhere to the Interspeech 2019 paper submission guidelines~\footnote{https://interspeech2019.org/authors/author\_resources/} and schedule~\footnote{https://interspeech2019.org/calls/important\_dates/}.
\section{Schedule}
Limited time is available for development due to the time between special session approval and the regular paper submission deadline. The schedule below aims to provide as much time as possible for development while also allowing sufficient time for the post-evaluation analysis for the special session and time for writing a paper for Interspeech special session.
\begin{itemize}
\item {\bf January 15, 2019:} Release of the evaluation plan and development sets
\item {\bf February 25, 2019:} Evaluation data available
\item {\bf March 4, 2019:} System output submission deadline (11:59 PM PST)
\item {\bf March 11, 2019:} Release of the evaluation results
\item {\bf March 15, 2019:} System description submission and release of VOiCES phase 2 key for the participating teams
\item {\bf March 29, 2019:} Regular paper submission deadline for Interspeech 2019
\end{itemize}
\balance
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2019-03-01T02:05:05",
"yymm": "1902",
"arxiv_id": "1902.10828",
"language": "en",
"url": "https://arxiv.org/abs/1902.10828"
}
|
\section{Introduction}
The study of jet instabilities is of utmost importance for understanding their dynamics and
phenomenology. Astrophysical jets propagate over very large distances (up to $10^9$ times
their initial radius in the case of AGN jets) maintaining a coherent structure and, for this
remarkable stability property, an acceptable explanation is still missing. On the other hand,
instabilities can play a fundamental role in the dissipation of part of the jet energy, leading to the
observed radiation as well as the formation and evolution of various observed structures. One of
the mechanisms through which dissipation of the jet energy may occur, and that has recently
attracted a lot of interest, is magnetic reconnection \citep[see e.g.][]{Giannios10, Sironi15,
Werner18}. In this context, current driven kink instabilities (CDI) may play an important role by
enhancing or killing reconnection \citep{Striani16, Ripperda17a, Ripperda17b}. Apart from CDI,
other types of instabilities are possible in jets: Kelvin-Helmholtz instabilities (KHI) driven by the
velocity shear and centrifugal-buoyancy instabilities driven by the jet rotation. While the
Newtonian, or non-relativistic case has been extensively studied, general analyses in the relativistic
regime, without invoking the force-free approximation, i.e., taking into account gas inertia, are
more limited due to the complexity of the problem. By ``relativistic'' we
mean that the Lorentz factor of the jet flow is larger than unity and/or the magnetization (i.e., the
ratio of the magnetic energy density to the energy density of matter) is high, enabling the jet
to accelerate to relativistic velocities. KHI have been extensively studied in several different
configurations both in the non-relativistic \citep[see e.g.][]{Bodo89, Birkinshaw91, Hardee92,
Bodo96, Hardee06, Kim15} and relativistic \citep[see e.g.][]{Ferrari78, Hardee79, Urpin02,
Perucho04, Perucho10, Hardee07} cases. Similarly, CDI have been extensively studied in the
Newtonian limit both in the linear \citep[e.g.][]{Appl92, Appl96,Begelman98, Appl00, Baty02,
Bonanno11, Bonanno11a, Das18} and nonlinear \citep[e.g.,][]{Moll_etal08,ONeill_etal12} regimes,
while the analysis of the relativistic case has been more limited, most of the linear studies have
considered the force-free regime \citep{Voslamber_Callebaut62, Pariev94, Pariev96, Lyubarski99,
Tomimatsu01, Narayan09, Gourgouliatos_etal12, Sobacchi_etal17} and the full MHD case has been
addressed more recently by \citet{Bodo13} (hereinafter Paper I), for the cold case, and by
\citet{Begelman98, Kim17, Kim18}, who included thermal pressure. Due to the complexity of the
relativistic case, the evolution of CDI beyond the force-free approximation has been tackled often
by means of numerical simulations, which mainly focus on the nonlinear behaviour
\citep[e.g.,][]{Mizuno_etal09, Mizuno_etal11, ONeill_etal12,Mizuno_etal12, Mignone_etal10,
Mignone_etal13, Singh_etal16}. In the absence of magnetic fields, rotation can drive the centrifugal
instability in jets, whose relativistic extension has been recently analysed by \citet{Komissarov18}.
The combination of rotation and magnetic field adds another degree of complexity, other kinds of
instabilities may arise and the interplay between the different modes can become quite
complicated \citep[for non-relativistic studies, see e.g.][]{Kim00, Hanasz00, Keppens02,
Varniere02, Huang03,Pessah05, Bonanno06, Bonanno07, Fu11}. The interplay of rotation and
magnetic field, in the non-relativistic case and in the absence of a longitudinal flow, has been
analyzed by \citet{Bodo16} (hereinafter Paper II), and the resulting main, rotationally-induced types
of instability -- the centrifugal-buoyancy modes -- have been identified and described.
In Paper I we considered a cold, relativistic, non-rotating jet and found that KHI is prevalent for
matter-dominated jets, while CDI is more effective for magnetically-dominated jets. In Paper II, we
considered the effects of rotation in a non-relativistic plasma column, where no longitudinal flow is
present. We found additional modes of instability driven by rotation: the centrifugal-buoyancy
modes. In this paper, which represents a sequel of Papers I and II, we study the stability problem in
the full case of a cold, relativistic, magnetized and rotating jet. We still consider a cold jet, because,
on one side, in the case of a Poynting-dominated jet, this can be
assumed as a valid approximation and, on the other side, the incorporation of pressure would
introduce new kinds of instabilities even more complicating the analysis. This is therefore a further
step towards a complete study, where we will drop this limitation in the end. The equilibrium
configuration here is similar to that adopted in these papers, which assumes a current distribution
that is peaked on the jet axis and closes at very large distances from the jet (i.e., the total net
current becomes equal to zero only at large distances). This class of equilibria is different from
those considered by \citet{Kim17, Kim18}, where the current closes inside the jet. Resulting main
modes of the instability in the relativistic and rotating case remain the KH, CD and
centrifugal-buoyancy ones. The main goal of the present paper is to investigate the effect of the
different parameters of the jet on the growth efficiency of these modes in the more comprehensive
relativistic rotating case compared to the relativistic non-rotating and non-relativistic rotating ones
analyzed, respectively, in Papers I and II. The main parameters, with respect to which we
explore the jet stability, are the Lorentz factor of the propagation velocity along jet axis, pitch of
the background magnetic field, degree of magnetization, rotation frequency, vertical/axial
wavenumber. In contrast to the present more general study, in Paper I rotation was zero, whereas
in Paper II, being in the Newtonian limit, the Lorentz factor was unity and the magnetization, as
defined here, was very small. Ultimately, one would like to understand how instabilities can tap part
of the jet flow energy, without leading to its disruption. To this aim, numerical simulations are an
essential tool, however, linear studies such as the present one may still provide necessary insights.
The plan of the paper is the following: in section 2 we will describe the physical problem, the basic
equations, the general equilibrium configuration and the characteristic parameters, in
section 3 we present our results, first for the KHI and CDI and then for the centrifugal-buoyancy
instabilities and, finally in section 4, we summarize our findings.
\section{Problem Description}
\label{problem}
We investigate the linear stability of a cold (i.e., with zero thermal pressure),
magnetized, rotating, relativistic cylindrical flow of an inviscid and infinitely conducting fluid.
It is governed by the basic equations of ideal relativistic MHD:
\begin{equation}\label{eq:drho/dt}
\frac{\partial}{\partial t} (\gamma \rho) + \nabla \cdot (\gamma
\rho \vec{v}) = 0,
\end{equation}
\begin{equation}\label{eq:dm/dt}
\gamma \rho \frac{\partial}{\partial t}(\gamma \vec{v} ) + \gamma
\rho (\vec{v} \cdot \nabla ) (\gamma \vec{v} ) = \frac{1}{c}\vec{J} \times
\vec{B} + \frac{1}{4\pi}( \nabla \cdot \vec{E} ) \vec{E},
\end{equation}
\begin{equation}\label{eq:dB/dt}
\frac{1}{c}\frac{\partial\vec{B}}{\partial t} = - \nabla \times \vec{E},
\end{equation}
\begin{equation}\label{eq:dE/dt}
\frac{1}{c}\frac{\partial \vec{E}}{\partial t} = \nabla \times \vec{B} - \frac{4\pi}{c}\vec{J},
\end{equation}
where $\rho$ is the proper density, $\gamma=(1-v^2/c^2)^{-1/2}$ is
the Lorentz factor, with $c$ being the speed of light, and $\vec
{v}$, $\vec {B}$, $\vec{E}$, $\vec{J}$ are, respectively, the 3-vectors of the
velocity, magnetic field, electric field and current density. These equations
are written in the CGS system and a factor of $\sqrt{4 \pi}$ is
absorbed in the definitions of $\vec{E}$ and $\vec{B}$. In the following we choose the units
such that the speed of light is unity, $c = 1$, .
\subsection{Equilibrium Configuration}
\label{sec:equilibrium}
The equilibrium configuration was described in Paper I, here we
summarize the relevant equations. We adopt cylindrical coordinates
$(r,\varphi,z)$ (with versors $\vec{e}_r, \vec{e}_\varphi,
\vec{e}_z$) and seek for axisymmetric steady-state solutions (i.e.,
$\partial_t=\partial_\varphi=\partial_z=0$) of equations
(\ref{eq:drho/dt})-(\ref{eq:dE/dt}). The jet propagates in the
vertical/axial ($z$) direction, the magnetic field and velocity have no
radial components and consist of a vertical (poloidal), $B_z, v_z$,
and toroidal, $B_\varphi, v_\varphi$, components. The magnetic field
configuration can be characterized by the pitch parameter
\[
P=\frac{rB_z}{B_{\varphi}}.
\]
The only non-trivial equation is given by the radial component of
the momentum equation (\ref{eq:dm/dt}) which, in the zero pressure
case, simplifies to
\begin{equation}\label{eq:radial_eq}
\rho\gamma^2v_\varphi^2 =
\frac{1}{2r}\frac{d(r^2H^2)}{dr}+\frac{r}{2}\frac{dB_z^2}{dr},
\end{equation}
where $H^2 = B_\varphi^2 - E_r^2$ (in the non-relativistic case $H^2 = B_\varphi^2$) and
\begin{equation}\label{eq:elec_field}
E_r = v_z B_\varphi-v_\varphi B_z
\end{equation}
Equation (\ref{eq:radial_eq}) leaves the freedom of choosing the
radial profiles of all flow variables except one and then solve for
the remaining profile. We note that while in the Newtonian case, the
presence of a longitudinal velocity has no effect on the radial
equilibrium, this no longer holds in the relativistic case, where
the Lorentz factor appears in the equilibrium condition
(\ref{eq:radial_eq}). The choice of the radial profiles is somewhat
arbitrary since we have no direct information about the magnetic
configuration in astrophysical jets. We choose to follow the
prescriptions given in Papers I and II and to consider a general
class of constant density equilibria in which the vertical current
density is peaked on the central axis of the jet and is concentrated in a
region of the characteristic radius $a$. We prescribe the velocity profile by choosing
$\gamma_z(r)$, i.e., the Lorentz factor with respect to the $z$-component of the velocity only, of the form
\begin{equation}\label{eq:vz_prof}
\gamma_z(r) \equiv \frac{1}{\sqrt{1-v_z^2}} = 1 + \frac{\gamma_c - 1}{\cosh(r/r_j)^6},
\end{equation}
where $\gamma_c = (1-v_c^2)^{-1/2}$ is the Lorentz factor for the vertical velocity on the
central axis, $v_c=v_z(0)$, and $r_j$ is the jet radius. From now on, we will use the
subscript $`c`$ to denote values at $r=0$, in addition, all lengths
will be expressed in units of $r_j$ (recall that the velocities are
measured in units of the speed of light $c$). As in Paper I, we
prescribe the profile of $H$ as
\begin{equation} \label{eq:H2_prof}
H^2 = \frac{H^2_c}{r^2}\left[1 - \exp\left(-\frac{r^4}{a^4}\right)\right]
\end{equation}
and for the azimuthal velocity we take the form
\begin{equation} \label{eq:vphi_prof}
\gamma^2 v_\varphi^2 = r^2 \Omega^2_c \gamma_c^2
\exp\left(-\frac{r^4}{a^4}\right),
\end{equation}
where $\Omega_c$ is the angular velocity of the jet rotation on the central axis
($\gamma_c\Omega_c$ is the angular velocity measured in the jet rest frame).
The characteristic radius of the current concentration in the jet is set to $a=0.6$ below.
With these choices, from equation (\ref{eq:vphi_prof}) we get for $v_\phi$ the expression
\begin{equation}\label{eq:vphi2_prof}
v_\varphi^2 = \frac{r^2\gamma_c^2\Omega_c^2}{\gamma_z^2} \left[1
+r^2 \gamma_c^2 \Omega_c^2 \exp \left( -\frac{r^4}{a^4}\right)
\right]^{-1}\exp \left(-\frac{r^4}{a^4}\right),
\end{equation}
from which it is evident that for any value of $\Omega_c$, $v_\phi$
is always less than unity, i.e., the azimuthal velocity does not
exceed the speed of light. From equations (\ref{eq:radial_eq}),
(\ref{eq:H2_prof}) and (\ref{eq:vphi_prof}), we get the $B_z$
profile as
\begin{equation}\label{eq:Bz_prof}
B^2_z = B^2_{zc} - (1 - \alpha) \frac{H^2_c\sqrt{\pi}}{a^2}{\rm erf}
\left(\frac{r^2}{a^2}\right)
\end{equation}
where $\mathrm{erf}$ is the error function and the parameter
\begin{equation}\label{eq:alfa}
\alpha = \frac{\rho\gamma_c^2\Omega^2_ca^4}{2H_c^2}
\end{equation}
measures the strength of rotation: for $\alpha = 0$ (no rotation)
the gradient of $r^2 H^2$ in equation (\ref{eq:radial_eq}) is
exactly balanced by the gradient of $B_z^2$ ($B_z$ decreases
outward), whereas for $\alpha = 1$, it is exactly balanced by the
centrifugal force and $B_z$ is constant. Intermediate values of
rotation correspond to the range $0<\alpha<1$. As shown in Paper
II, one can, in principle, consider also configurations with $\alpha
> 1$, in which $B_z$ grows radially outward, but such configurations
will not be considered in the present paper.
The azimuthal field is obtained from the definition of $H$ using the
expression of $E_r=v_zB_\varphi-v_\varphi B_z$. This yields a
quadratic equation in $B_\varphi$ with the solution
\begin{equation}\label{eq:Bphi}
B_\varphi = \frac{-v_\varphi v_zB_z\mp\sqrt{v_\varphi^2B_z^2 +
H^2(1-v_z^2)}}{1-v_z^2}.
\end{equation}
Here we consider the negative branch because it guarantees that
$B_\varphi$ and $v_\varphi$ have opposite signs, as suggested by
acceleration models \citep[see e.g.,][]{Blandford_Payne82, Ferreira_Pelletier95,
Zanni_etal07}. We choose to characterize the magnetic field configuration by specifying the
absolute value of the pitch on the axis, $P_c$, and the ratio of the energy density of the
matter to the magnetic energy density, $M_a^2$,
\begin{equation}\label{eq:P_cnd_MA}
P_c \equiv \left|\frac{rB_z}{B_\varphi}\right|_{r=0} \,,\qquad M_a^2
\equiv \frac{\rho \gamma_c^2}{\av{\vec{B}^2}} \,,
\end{equation}
where $\av{\vec{B}^2}$ represents the radially averaged magnetic energy density across the beam
\begin{equation}\label{eq:Bav}
\av{\vec{B}^2} = \frac{\int_0^{r_j} (B_z^2 + B_\varphi^2)r\,dr}
{\int_0^{r_j} r \,dr},
\end{equation}
and $r_j=1$ in our units. $M_a$ is related to the standard magnetization
parameter $\sigma=B^2/(\rho h)$ ($h$ is the specific enthalpy)
used in other studies via $M_a^2=\gamma_c^2/\sigma$ and
to the relativistic form of the Alfv\'en speed, $v_a=B/\sqrt{\rho h+B^2}$,
via $M_a^2=\gamma_c^2(1-v_a^2)/v_a^2$ ($c=1$ and in the cold limit $h=1$).
The constants $B_{zc}$ and $H_c$ appearing in
the above equations can be found in terms of
$P_c$, $M_a$ and $\Omega_c$ by simultaneously solving equations
(\ref{eq:P_cnd_MA}) and (\ref{eq:Bav}) using expressions
(\ref{eq:Bz_prof}) and (\ref{eq:Bphi}) for the magnetic field
components. In particular, from the definition of the pitch
parameter, after some algebra, we find (in the $r\rightarrow 0$
limit)
\begin{equation}
a^4 B^2_{zc} = \frac{H_c^2 P_c^2}{1- \left( P_c\Omega_c + v_{zc}
\right)^2}.
\end{equation}
\begin{figure*}
\centering
\includegraphics[width=12cm]{radial_structure.png}
\caption{Radial profiles of the Lorentz factor, magnetic field components and axial current
density for the equilibrium at $\alpha=1$, $M_a^2=1$, $P_c=1$ and different $\gamma_c=1.01
~(blue), 2~(green), 5~(red)$.}\label{fig:radial}
\end{figure*}
Fig. \ref{fig:radial} shows the typical radial profile of this equilibrium solution for the
special/representative case of maximal rotation, $\alpha=1$, and different $\gamma_c$. However,
as discussed in Paper II for the non-relativistic case, not
all the combinations of $\Omega_c$, $\gamma_c$, $P_c$ and $M_a$ are
allowed, because, in order to have a physically meaningful solution,
we have to impose the additional constraints that $B_\varphi^2$ and
$B_z^2$ must be everywhere positive. We note that, since $B_z^2$
decreases with radius monotonically, the condition $\lim_{r
\rightarrow \infty}B_z^2 > 0$ ensures that $B_z^2$ is positive
everywhere.
\begin{figure*}
\centering
\includegraphics[width=5cm]{fig1a.png}
\includegraphics[width=5cm]{fig1b.png}
\includegraphics[width=5cm]{fig1c.png}
\includegraphics[width=5cm]{fig1d.png}
\includegraphics[width=5cm]{fig1e.png}
\includegraphics[width=5cm]{fig1f.png}
\includegraphics[width=5cm]{fig1g.png}
\includegraphics[width=5cm]{fig1h.png}
\includegraphics[width=5cm]{fig1i.png}
\caption{\small Regions of allowed equilibria in the ($\Omega_c,
\gamma_c P_c$)-plane shaded in light green. The different panels
refer to different values of $\gamma_c$ and $M_a$, the values
corresponding to each panel are reported in the legend. The red
curves mark the boundary where $\lim_{r \rightarrow \infty}B_z^2 =
0$ and the green curves mark the boundary where $B_z^2$ is constant
with radius ($\alpha = 1$). Insets show the maximum $\Omega_c$ of the
possible equilibria when the latter extend beyond the range of $\Omega_c$
represented in these plots.}\label{fig:equil}
\end{figure*}
Fig. \ref{fig:equil} shows the allowed region (shaded in light
green) in the ($\Omega_c, \gamma_c P_c$)-plane, with the red curve
marking the boundary where $\lim_{r \rightarrow \infty}B_z^2 = 0$
and the green curve marking the boundary where $B_z^2$ becomes
constant with radius (i.e., $\alpha = 1$). (As discussed in Paper
II, there are also possible equilibria where $B_z^2$ increases with
radius, outside the green curve, but they are not considered here.)
We used $\gamma_c P_c$ on the ordinate axis, since it represents the
pitch measured in the jet rest frame. The three panels in each row
refer to decreasing values of $M_a^2$ (from left to right, $M_a^2 =
100, 1, 0.1$) corresponding to increasing strength of the magnetic
field, while the three panels in each column refer to increasing
value of the Lorentz factor (from top to bottom, $\gamma_c = 1.01,
2, 5$). The top leftmost panel with the lowest magnetization and $\gamma_c=1.01$
should correspond to the Newtonian limit shown in Fig. 1 of Paper II,
comparing these two figures we can see that the shape of the permitted
light green regions are nearly the same, while the values of $\Omega_c$ are different because of the different normalization, in Paper II it was normalized by $v_a/r_j$, whereas here it is normalized by $c/r_j$. Even at this nearly Newtonian value of $\gamma_c=1.01$, the relativistic effects become noticeable starting from intermediate magnetization $M_a^2 = 1$ -- the corresponding permitted light green region with its green and red boundaries (top middle panel) differs from those in the Newtonian limit $M_a^2=100$ (top left panel), after taking into account the above normalization of the angular velocity. Generally, in Fig. \ref{fig:equil}, relativistic effects become increasingly stronger, on one hand, going from left to right, because the Alfv\'en speed approaches the speed of light, and on the other hand, going from top to bottom, because the jet velocity approaches the speed of light.
At high values of the pitch, there is a maximum allowed value of
$\Omega_c$, while decreasing the pitch we see that below a threshold
value $\gamma_c P_c = 0.8$, the jet must rotate in order to ensure a
possible equilibrium and the rotation rate has to increase as the
pitch decreases. For low values of $P_c$, the allowed range of $\Omega_c$ therefore tends to
become very narrow. Comparing the panels in the three columns, we see that
increasing the magnetic field, the maximum $\Omega_c$, found for
$P_c \rightarrow 0$, increases, scaling with $1/M_a$, and the
rotation velocities become relativistic. Increasing the value of
$\gamma_c$ (middle and bottom panels), the allowed values of
$\Omega_c$ in the laboratory frame (shown in the figure) decrease,
however, they increase when measured in the jet rest frame.
\begin{figure*}
\centering
\includegraphics[width=12cm]{fig2.png}
\caption{\small Plots of $\alpha_{min}$ as a function of $\gamma_c
P_c$. $\alpha_{min}$ represents the minimum value of $\alpha$ for
which equilibrium is possible. The three panels refer to three
different values of $\gamma_c$, the corresponding values are
reported in each panel. The different curves correspond to different
values of $M_a$ as indicated in the legend.} \label{fig:alphamin}
\end{figure*}
Since in the stability analysis we will often make use of the
parameter $\alpha$, defined in equation (\ref{eq:alfa}), for
characterizing the equilibrium solutions, in Fig. \ref{fig:alphamin}
we show the minimum value of $\alpha$ required for the existence of
the equilibrium as a function of $\gamma_c P_c$. We recall that
$\alpha=0$ corresponds to no rotation and $\alpha=1$ corresponds to
the case where $B_z$ is constant and the hoop stresses by $B_\phi$
are completely balanced by rotation. As discussed above, for
$\gamma_c P_c < 0.8$ some rotation is needed for maintaining the
equilibrium and this minimum rotation corresponds to $\alpha_{min}$
plotted in Fig. \ref{fig:alphamin}. The three panels refer to three
different values of $\gamma_c$ and in each panel the three curves
correspond to three different values of $M_a$. Decreasing $\gamma_c
P_c$ below the critical value, $\alpha_{min}$ increases tending to 1
as $\gamma_c P_c \rightarrow 0$. Comparing the curves for the same
value of $\gamma_c$ in each panel, we see that $\alpha_{min}$
decreases as $M_a$ is decreased. Comparing the different panels, the
corresponding curves for the same value of $M_a$ also show a
decrease of $\alpha_{min}$ with $\gamma_c$.
\section{Results}
\label{results}
In Paper II, we identified and described different linear modes of
instability existing in a non-relativistic rotating static (with $v_z=0$) column:
the CD mode as well as the toroidal and poloidal buoyancy modes
driven by the centrifugal force due to rotation. In the present
analysis, we have to consider additionally the instabilities driven
by the velocity shear between the jet and the ambient medium, that
is, KH modes. We will investigate these different perturbation modes
in the following subsections.
The small perturbations of velocity and magnetic field about the
above-described equilibrium are assumed to have the form $\propto \exp({\rm
i}\omega t-{\rm i}m\varphi-{\rm i}kz)$, where the azimuthal
(integer) $m$ and axial $k$ wavenumbers are real, while
the frequency $\omega$ is generally complex, so that there is instability if
its imaginary part is negative, ${\rm Im}(\omega)<0$, and the growth rate of the
instability is accordingly given by $-{\rm Im}(\omega)$. The related
eigenvalue problem for $\omega$ -- the linear differential equations
(with respect to the radial coordinate) for the perturbations
together with the appropriate boundary conditions in the vicinity of
the jet axis, $r\rightarrow 0$, and far from it, $r\rightarrow
\infty$ -- were formulated in Paper I in a general form for
magnetized relativistic rotating jets, but then only the
non-rotating case was considered. For reference, in Appendix A, we
give the final set of these main equations (\ref{eq:dxi/dr}) and
(\ref{eq:dPhii/dr}) with the boundary conditions (\ref{eq:bound_in}) and (\ref{eq:bound_out}), which are solved in the present rotating case
and the reader can consult Paper I for the details of the
derivation. In this study, we focus on $m=1$ modes for the following
reasons. For CDI, this kink mode is the most effective one, leading
to a helical displacement of the whole jet body, while CDI is absent
for $m=-1$ modes (Paper I). As for KHI, it is practically
insensitive to the sign of $m$ (Paper I), so we can choose only
positive $m$. Finally, as we have seen in Paper II, the centrifugal-buoyancy
modes also behave overall similarly at $m=-1$ and $m=1$ for large and
small $k$, which are the main areas of these modes’ activity.
Our equilibrium configuration depends on the four parameters $\gamma_c$, $\alpha$, $P_c$
and $M_a$, specifying, respectively, the jet bulk flow velocity along the axis, the strength of the centrifugal force, the magnetic pitch and the magnetization. As mentioned above, relativistic effects become important either at high values of $\gamma_c$, because the jet velocity approaches the speed of
light, or at low values of $M_a$, because the Alfv\'en speed approaches the speed of light, even
when $\gamma_c\sim 1$. For some of the parameters we are forced to make a choice of few
representative values since it would be impossible to have a full coverage of the four-dimensional
parameter space. For $\gamma_c$ we choose one value to be $1.01$ since at large $M_a$ we
make connection with the non-relativistic results (Paper II), while at low $M_a$ we can explore the
relativistic effects due to the high magnetization. As another value, we choose $\gamma_c = 10$,
which can be considered as representative of AGN jets \citep{Padovani_Urry92, Giovannini_etal01,
Marscher06, Homan12} (except for some cases in which lower values are used, since the growth
rates of the modes for $\gamma_c = 10$ becomes extremely low). For $\alpha$ we chose the two
limiting cases $\alpha = 0$ (no rotation) and $\alpha = 1$ (centrifugal force exactly balances
magnetic forces) and one intermediate value, $\alpha = 0.2$, for which the effects due to rotation
start to be substantial (notice that the relation between $\alpha$ and the rotation rate is not
linear). Finally, for $P_c$ we explore several different values, $P_c = 0.01, \, 0.1, \, 1, \, 10$,
depending on the allowed equilibrium configurations (see discussion above).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig3a.png}
\includegraphics[width=\columnwidth]{fig3b.png}
\includegraphics[width=\columnwidth]{fig3c.png}
\caption{\small Distribution of the growth rate, $-{\rm
Im}(\omega)$, as a function of the wavenumber $k$ and $M_a$ for
$\gamma_c = 1.01$ and $P_c = 10$. The three panels correspond to
three different values of $\alpha$ (Top panel: $\alpha=0$; Middle
panel: $\alpha=0.2$; Bottom panel: $\alpha=1$). In this and other analogous plots below,
the colour scale covers the range from 0 to the maximum value of the growth rate, while the
contours are equispaced in logarithmic scale from $10^{-5}$ up to this maximum growth rate.}
\label{fig:cdvar1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig4a.png}
\includegraphics[width=\columnwidth]{fig4b.png}
\includegraphics[width=\columnwidth]{fig4c.png}
\
\caption{\small Same as in Fig. \ref{fig:cdvar1}, but for $\gamma_c
= 1.01$ and $P_c = 1$.}\label{fig:cdvar2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig5a.png}
\includegraphics[width=\columnwidth]{fig5b.png}
\includegraphics[width=\columnwidth]{fig5c.png}
\
\caption{\small Same as in Fig. \ref{fig:cdvar1}, but for $\gamma_c
= 10$ and $P_c = 10$.}\label{fig:cdvar3}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig6a.png}
\includegraphics[width=\columnwidth]{fig6b.png}
\includegraphics[width=\columnwidth]{fig6c.png}
\caption{\small Same as in Fig. \ref{fig:cdvar1}, but for $\gamma_c
= 10$ and $P_c = 1$.} \label{fig:cdvar4}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig7a.png}
\includegraphics[width=\columnwidth]{fig7b.png}
\includegraphics[width=\columnwidth]{fig7c.png}
\caption{\small Same as in Fig. \ref{fig:cdvar1}, but for $\gamma_c
= 10$ and $P_c = 0.1$.} \label{fig:cdvar5}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig8.png}
\caption{\small Distribution of the growth rate, $-{\rm Im}(\omega)$, as a function of the
wavenumber $k$ and $M_a$ for $\gamma_c = 10$, $\alpha = 1$ and $P_c
= 0.01$.}\label{fig:cdvar6}
\end{figure}
\subsection{CD and KH modes}
CD and KH modes were already discussed in detail in Paper I, where
we found KH modes dominating at large values of $M_a$ and CD modes
dominating at small values of $M_a$. Here we are mainly interested in how they are affected by rotation.
In Fig. \ref{fig:cdvar1}, we show the behaviour of the growth rate (defined as $-{\rm Im}(\omega)$) as a function of the wavenumber $k$ and $M_a$ for $P_c=10$, $\gamma_c =1.01$ and three different values of
$\alpha$: $\alpha = 0$ in the top panel corresponds to no rotation, $\alpha = 1$ in the bottom
panel corresponds to the case where rotation exactly balances magnetic forces and as an
intermediate value, in the middle panel, we choose $\alpha = 0.2$ for which the influence of rotation is already appreciable. The case shown in the top panel is for zero rotation and has already been considered in Paper I, we
show it again here in order to highlight the effects of rotation by direct comparison. Increasing
rotation (middle and bottom panels), we see a stabilizing effect on CDI, which progressively
increases when $M_a$ decreases. This stabilizing effect of rotation has been already
discussed in Paper II \citep[see also][]{Carey09}. In this figure, the difference between the last two
values of $\alpha$ may still be small, however, as we will see below, the behaviour can be noticeably
different at $M_a < 1$ for other values of $P_c$ and
$\gamma_c$, especially, in the limit $M_a\rightarrow 0$. By
contrast, the KH modes, occurring at larger values of $M_a$, are
essentially unaffected by rotation and, for this value of $P_c$,
are the dominant modes. Fig. \ref{fig:cdvar2} shows the same
kind of plots, but for a lower value of the pitch, $P_c=1$. In the top
panel (no rotation, $\alpha = 0$) we see that, as discussed in Paper
I, the CD mode increases its growth rate and moves towards larger
values of the wavenumber. Rotation has, as before, a stabilizing
effect, that becomes stronger as we decrease $M_a$. At zero rotation,
CDI is the dominant mode, its growth rate is independent from $M_a$ (for $M_a < 1$),
and is about an order of magnitude larger than the growth rate of
KHI. As we increase rotation, the
growth rate of CDI decreases and the stability boundary moves
towards smaller and smaller $k$ as $M_a$ is decreased. This decrease
in the level of CDI in Fig. \ref{fig:cdvar2} is most dramatic when $\alpha$
goes from zero to $0.2$ (as it is visible by comparing the top and
middle panels), further increasing $\alpha$ to $1$ the decrease is then
much less pronounced. As a result, for $\alpha = 1$, the KHI is
again the mode with the highest growth rate, which, however, has
changed only slightly relative to its value in the non-rotating
case. The cases with the same high $\gamma_c=10$ and three different
values of the pitch, $P_c=10$, $P_c=1$ and $P_c=0.1$, are shown,
respectively, in Figs. \ref{fig:cdvar3}, \ref{fig:cdvar4} and
\ref{fig:cdvar5}. We note that the pitch measured in the jet rest
frame is given by $\gamma_c P_c$, so these three values would
correspond to $P_c=100$, $P_c=10$ and $P_c=1$ when measured in the
rest frame. (Additionally, we have to note that for $\gamma_c =
1.01$ there are no equilibrium solutions for $P_c=0.1$.) In the top
panel (no rotation) of Fig \ref{fig:cdvar3}, we therefore see that
the stability boundary of the CD modes moves further to the left,
i.e., towards smaller wavenumbers compared to the above case with
$\gamma_c=1.01$, because of the high value of the pitch measured in
the jet rest frame. Increasing rotation (middle and bottom panels),
below $M_a \sim 2$ the CD mode is stable (at least in the wavenumber
range considered). For lower values of the pitch in Figs.
\ref{fig:cdvar4} and \ref{fig:cdvar5}, in the absence of rotation,
the stability limit of the CD modes shifts again to larger $k$ with
decreasing pitch. The effect of rotation is thus similar also in
this highly relativistic case as it is for $\gamma_c=1.01$, being
most remarkable when $\alpha$ increases from zero to 0.2. Notice
that, as discussed in Paper I, we have a splitting of the CD mode as
a function of wavenumber for $P_c=1$, this splitting is,
however, only present at zero rotation (top panel of Fig.
\ref{fig:cdvar4}). As noted above, the KH mode is essentially
unaffected by rotation. Therefore, except for the case with $P_c =
0.1$ and no rotation, the mode with the highest growth rate remains
the KHI. Finally, in Fig. \ref{fig:cdvar6} we show the case with
$P_c=0.01$, $\gamma_c=10$ and $\alpha=1$. For this value of $P_c$,
equilibrium is possible only at high values of $\gamma_c$ and the
allowed values of $\alpha$ cannot be much smaller than 1. The
behavior is similar to those discussed above, on one hand the CDI
tends to move towards higher wavenumbers and increase its growth
rate due to the decreasing value of $P_c$, on the other hand,
rotation has the usual stabilizing effect and, as a result, creates
an inclined stability boundary that moves towards smaller
wavenumbers as $M_a$ is decreased.
These results show that the effect of rotation is generally
stabilizing for the CD mode, at $M_a \lesssim 1$ (high
magnetization). An interesting question is then what happens in the
limit $M_a \rightarrow 0$. This limit, corresponding to the
force-free regime in relativistic jets, was investigated by
\citet{Pariev94,Pariev96,Lyubarski99} and \citet{Tomimatsu01}.
\citet{Tomimatsu01} derived the following condition for instability:
\begin{equation}\label{eq:tomimatsu}
|B_\varphi| > r \Omega_F B_z
\end{equation}
where $\Omega_F$ is the angular velocity of field lines that can be
expressed as (see Paper I)
\begin{equation}\label{eq:OmegaF}
\Omega_F = \frac{v_\varphi}{r} - \frac{v_z}{P}.
\end{equation}
Using equation (\ref{eq:elec_field}) for the electric field $E_r$
and equation (\ref{eq:OmegaF}) for $\Omega_F$, the condition
(\ref{eq:tomimatsu}) can be written as
\begin{equation}\label{eq:tomimatsu2}
|B_\varphi| > |E_r|.
\end{equation}
The equilibrium condition (\ref{eq:radial_eq}) in the force-free
limit becomes
\begin{equation}
\frac{1}{2r}\frac{d(r^2H^2)}{dr} + \frac{r}{2}\frac{dB_z^2}{dr} = 0.
\end{equation}
If $B_z$ is constant, we have $H=0$ and $|B_\varphi| = |E_r|$.
According to Tomimatsu condition (\ref{eq:tomimatsu2}), in this
case, we are on the stability boundary and the system can be stable.
In fact, \citet{Pariev94, Pariev96} considered such a
situation and found stability. On the other hand, if $B_z$ decreases
radially outward, $H>0$, the Tomimatsu condition is satisfied and
there is instability. \citet{Lyubarski99} considered such
a case and indeed found instability, with a characteristic
wavenumber increasing inversely proportional to pitch. In our setup,
a constant $B_z$ corresponds to $\alpha = 1$, while a radially
decreasing vertical field corresponds to $\alpha<1$. From the above
figures we have seen that, in the presence of rotation, the
instability boundary moves towards smaller and smaller values of the
wavenumber as $M_a$ becomes low. However, it is hard to deduce from
this result how exactly the instability region changes along $k$
when approaching the force-free limit, $M_a\rightarrow 0$,
because of the limited interval of $M_a$ and $k$ values represented.
Nevertheless, we can see that, in general, at a given small $M_a\ll
1$, the stability boundary for $\alpha = 1$ tends to be at values of
$k$ smaller than those for $\alpha = 0.2$ (see e.g., Figs.
\ref{fig:cdvar2} and \ref{fig:cdvar5}), that is overall consistent
with the results of \citet{Pariev94, Pariev96,
Lyubarski99} and \citet{Tomimatsu01}.
\subsection{Centrifugal-buoyancy modes}
In Paper II, we demonstrated that in rotating non-relativistic jets,
apart from CD and KH modes, there exists yet another important class of
unstable modes that are similar to the Parker instability with the
driving role of external gravity replaced by the centrifugal force
\citep{Huang03} and analysed in detail their properties. At small
values of $k$, these modes operate by bending mostly toroidal field
lines, while at large $k$ they operate by bending poloidal field
lines. Accordingly, we labeled them the toroidal and poloidal
buoyancy modes \citep[see also][]{Kim00}. In this subsection, we
investigate how the growth of these modes is affected by
relativistic effects.
\subsubsection{Toroidal buoyancy mode}
The toroidal buoyancy mode operates at small values of $k$ and, in
fact, its instability is present only for wavenumbers $k<k_c$, where
the high wavenumber cutoff $k_c$ depends on the pitch parameter and
satisfies the condition $k_c P_c \sim 1$. Fig. \ref{fig:tor1}
presents the typical behaviour of the growth rate of the toroidal
mode as a function of $k$ and $M_a$ for $P_c=1$, $\gamma_c =1.01$
and $\alpha = 1$. It is seen that in the unstable region, the growth
rate is essentially independent from $k$ and reaches a maximum for
$M_a$ slightly larger than 1. At smaller and larger values of $M_a$
the mode is stable or has a very small growth rate, depending on the
parameters, as we will see below.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig9.png}
\caption{\small Distribution of the growth rate, $-{\rm Im}(\omega)$, of the toroidal
buoyancy mode as a function of the wavenumber $k$ and $M_a$ for
$\gamma_c = 1.01$, $\alpha = 1$ and $P_c = 1$.} \label{fig:tor1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig10.png}
\caption{\small Growth rate of the toroidal mode as a function of
$k$ for $\alpha=1$, $P_c=1$ and $\gamma_c=1.01,2,5$. The growth rate
is normalized by $\gamma_c \Omega_c^2$. The three different curves
refer to three different values of $\gamma_c$ as indicated in the
legend. $M_a$ is also different for the three curves and for each
curve it has the value at which the maximum growth rate is reached
at a fixed $k$: $M_a = 1.5,~4.5,~15$, respectively,
for $\gamma_c=1.01,~2,~5$.} \label{fig:tor2}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=7.5cm]{fig11a.png}
\includegraphics[width=7.5cm]{fig11b.png}
\includegraphics[width=7.5cm]{fig11c.png}
\includegraphics[width=7.5cm]{fig11d.png}
\caption{\small Growth rate of the toroidal mode as a function of
$M_a$. The four panels refer to different values of the pitch
parameter $P_c$ (Top left: $P_c = 10$; Top right: $P_c = 1$; Bottom
left: $P_c = 0.1$; Bottom right: $P_c = 0.01$). Solid curves are for
$\alpha = 1$, dashed curves are for $\alpha=0.2$, curves with
different colours refer to different values of $\gamma_c=1.01,2,5,10$
as indicated in the legend of each panel. The black lines represent
the growth rate calculated with non-relativistic ideal MHD equations
at zero thermal pressure and the same pitch and $\alpha$ in the respective panel. If the dashed curve is
absent in any panel, this means that the equilibrium with
$\alpha=0.2$ is not possible for that pair of $P_c$ and $\gamma_c$
associated with this panel. The wavenumber is $k=0.01$ in all the cases,
although the growth rate is essentially independent of this
value.}\label{fig:tor3}
\end{figure*}
In Paper II, we showed that the growth rate of the centrifugal-buoyancy
modes scales approximately as $\Omega_c^2$. In the present
case, we have to take into account the relativistic effects and, as
the mode tends to be concentrated inside the jet, we have to
consider quantities measured in the rest frame of the jet. The
growth rate in the rest frame should scale as in the
non-relativistic case, i.e., as the square of the rotation frequency
in this frame. Since the growth rate in the jet rest frame is ${\rm
Im}(\omega') = \gamma_c {\rm Im}(\omega)$, while the rotation
frequency is $\Omega'_c = \gamma_c \Omega_c$, we can write the
scaling law in the lab frame
\begin{equation}\label{eq:scaling}
-{\rm Im}(\omega) \sim \gamma_c \Omega_c^2,
\end{equation}
In Fig. \ref{fig:tor2}, we plot the growth rate normalized according
to this scaling, $-{\rm Im}(\omega)/(\gamma_c\Omega_c^2)$, as a
function of the wavenumber for $\alpha = 1$, $P_c = 1$ and three
different values of $\gamma_c=1.01,2,5$. For each $\gamma_c$, we
choose the value of $M_a$, which corresponds to the maximum growth
rate at a given $k$, these values are reported in the figure
caption. For each curve, the value of $\Omega_c$ is also different
and equal to $0.6$ for $\gamma_c=1.01$, $0.09$ for $\gamma_c = 2$
and $0.01$ for $\gamma_c = 5$. The growth rate in the unstable range
is independent from the wavenumber, as seen in Figs. \ref{fig:tor1}
and \ref{fig:tor2}, and the scaling law (\ref{eq:scaling})
reproduces quite well the behaviour of the growth rate at different
$\gamma_c$: the corresponding curves come close to each other
(collapse) when the normalized growth rate is plotted.
To study in more detail the dependence of the toroidal buoyancy
instability on the jet flow parameters, in Fig. \ref{fig:tor3}, we
plot the growth rate as a function of $M_a$ for a given $k$ and
various $P_c$ and $\gamma_c$. Although the value of $k$ is fixed in
these plots, we recall that, as we have seen above, there is no
dependence of the growth rate on $k$ when $k$ is sufficiently
smaller than the cut-off value. So, the curves in this figure would not
change for other choices of the unstable wavenumber. The four
different panels correspond to different values of the pitch
parameter (top left: $P_c=10$, top right: $P_c = 1$, bottom left:
$P_c = 0.1$, bottom right: $P_c = 0.01$). In each panel, different
colours refer to different values of $\gamma_c=1.01,2,5,10$, while
the solid curves are for $\alpha = 1$ and dashed ones for $\alpha =
0.2$. Not all curves are present in all panels because, as discussed
in subsection 2.1, there are combinations of parameters for which
equilibrium is not possible. At large values of $M_a \gtrsim 10$ (low
magnetization), the growth rate decreases as $1/M_a$ for all values of
$\gamma_c$, $P_c$ and $\alpha$ given in these panels. In particular, at $\gamma_c = 1.01$ the
behaviour coincides with the non-relativistic MHD case (black lines in Fig. \ref{fig:tor3} calculated with ideal non-relativistic MHD equations at zero thermal pressure, as in Paper II).
This behaviour can be explained as follows. At large
$M_a$, rotation is balanced by magnetic forces and hence both decrease with increasing $M_a$.
As a result, the growth rate of the centrifugal-buoyancy modes decreases too, because it scales
with the square of the rotation frequency (Eq. \ref{eq:scaling}).
As $M_a$ decreases, the growth rate first increases more
slowly, reaches a maximum and then decreases at small $M_a\ll 1$, as we
approach the force-free limit where the centrifugal instabilities should
eventually disappear. The behaviour of the growth rate as a function
of $\alpha$, $P_c$ and $\gamma_c$ can be understood from the same scaling
with the rotation frequency discussed above. Increasing $\alpha$,
the rotation frequency increases as well and, consequently, the growth
rate. A decrease in the pitch also leads to an increase of the
rotation frequency and, therefore, of the growth rate. Finally,
understanding the dependence on $\gamma_c$ is more complex since we
have to take into account the transformation of all the quantities
from the lab frame to the jet frame. The pitch in the jet frame is
given by $\gamma_c P_c$ and is, therefore, larger for larger
$\gamma_c$ and the rotation rate is consequently smaller. In
addition, the growth rate measured in the lab frame is smaller by a
factor of $\gamma_c$ than the growth rate measured in the jet frame,
as a result, the growth rate strongly decreases with $\gamma_c$.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig12.png}
\caption{\small Distribution of the growth rate, $-{\rm Im}(\omega)$, of the poloidal
buoyancy mode as a function of the wavenumber $k$ and $M_a$ for
$\gamma_c = 1.01$, $\alpha = 1$ and $P_c = 1$.}\label{fig:pol1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig13.png}
\caption{\small Growth rate of the poloidal buoyancy mode as a
function of $k$ for $\alpha=1$, $P=1$ and $\gamma_c=1.01,2,5$. The
growth rate is normalized by $\gamma_c\Omega_c^2$. The three
different curves refer to three different values of $\gamma_c$ as
indicated in the legend. $M_a$ is different for the three curves and
for each curve has the value at which the maximum growth rate is
found at fixed $k$: $M_a =1.8,~5.72,~14.2$,
respectively, for $\gamma_c=1.01,~2,~5$.}
\label{fig:pol2}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=7.5cm]{fig14a.png}
\includegraphics[width=7.5cm]{fig14b.png}
\includegraphics[width=7.5cm]{fig14c.png}
\includegraphics[width=7.5cm]{fig14d.png}
\caption{\small Growth rate of the poloidal buoyancy mode as a
function of $M_a$. The four panels refer to different values of the
pitch parameter $P_c$ (Top left: $P_c = 10$; Top right: $P_c = 1$;
Bottom left: $P_c = 0.1$; Bottom right: $P_c = 0.03$). Solid curves
are for $\alpha = 1$, dashed curves are for $\alpha=0.2$, curves
with different colours refer to different values of
$\gamma_c=1.01,2,5,10$ as indicated in the legend. As in Fig. \ref{fig:tor3},
the black lines represent the results obtained using ideal non-relativistic
MHD equations for the same pitch and $\alpha$ in each panel. The dashed curves are absent
in those panels with such a pair of $P_c$ and $\gamma_c$ that do not allow the equilibrium with
$\alpha=0.2$. For computation reasons, the wavenumber $k$ is different for each panel (Top left:
$k=5$; Top right: $k=10$; Bottom left: $k=14$; Bottom right: $k=40$), but corresponds to the
regime where the growth rate practically no longer depends on it.}\label{fig:pol3}
\end{figure*}
\subsubsection{Poloidal buoyancy mode}
Another type of the centrifugal-buoyancy mode existing in the
rotating jet is the poloidal buoyancy mode. As mentioned above, this
mode operates at large $k$ by bending mostly poloidal field lines.
In Fig. \ref{fig:pol1}, we present the behaviour of its growth rate
as a function of the wavenumber $k$ and $M_a$ for $P_c=1$, $\gamma_c
=1.01$ and $\alpha = 1$. This figure is almost specular with respect
to Fig. \ref{fig:tor1} and shows that the poloidal buoyancy
instability first starts from the same cutoff wavenumber $k_cP_c
\sim 1$ and extends instead to larger wavenumbers, $k>k_c$, having
the growth rate somewhat larger than that of the toroidal buoyancy
mode. At fixed $k$, it is concentrated in a certain range of $M_a$,
with the maximum growth rate being achieved around $M_a\sim 1$ at
every $k$ and decreasing at large and small $M_a$. At a given finite
$M_a$, in the unstable region, the growth rate initially increases
with $k$ and then tends to a constant value at $k\gg k_c$, as also
seen in Fig. \ref{fig:pol2}. Like the toroidal buoyancy mode, the
poloidal buoyancy mode obeys the same scaling law
(\ref{eq:scaling}), because it is also determined by the centrifugal
force. This is confirmed by Fig. \ref{fig:pol2}, which presents the
growth rate as a function $k$ for $\alpha = 1$, $P_c = 1$ and three
different values $\gamma_c=1.01,2,5$, while $M_a$ is chosen for each
curve such that to yield the maximum the growth rate at a given wavenumber.
Thus, modification (reduction) of the growth of both centrifugal-buoyancy
instabilities in the relativistic case compared to the
non-relativistic one is in fact mainly due to the time-dilation
effect -- in the jet rest frame their growth rate is determined by
$\Omega_c'^2$ in this frame, as it is in the non-relativistic case.
Fig. \ref{fig:pol3} shows the behaviour of the poloidal buoyancy mode
as a function of $M_a$ for fixed, sufficiently large values of $k$, when the
instability is practically independent of it (see Fig.
\ref{fig:pol2}). The values of $\alpha$, $\gamma_c$ and $P_c$ are the same as
used in Fig. \ref{fig:tor3} except that in the bottom right panel we used
$P_c = 0.03$ instead of 0.01 since for $P_c$ slightly below 0.03 the
poloidal mode becomes stable, when the cutoff wavenumber, $k_c$,
which is set by the pitch, becomes larger than the fixed wavenumber
($k=40$) used in this panel. Overall, the dependence of the growth
rate of the poloidal buoyancy mode on these parameters is quite
similar to that of the toroidal one described above. In particular, at large $M_a$ the growth rate
varies again as $1/M_a$ at all other parameters, coinciding at $\gamma_c=1.01$ with the behaviour
in the non-relativistic case (black lines). It then increases with decreasing $M_a$, reaches a
maximum and decreases at small $M_a$. The lower is $\gamma_c$ the
higher is this maximum and the smaller is the corresponding $M_a$.
On the other hand, with respect to pitch, the highest growth is
achieved at $P_c\sim 0.1$ at all values of $\gamma_c$ considered. Due
to the above scaling with the jet rotation frequency, the growth
rate also increases with $\alpha$. This behaviour of the poloidal
mode instability as a function of $\alpha$, $P_c$ and $\gamma_c$ can
be explained by invoking similar arguments as for the toroidal mode in the previous subsection.
\section{Summary}
\label{summary}
We have investigated the stability properties of a relativistic
magnetized rotating cylindrical flow, extending the results obtained
in Papers I and II. In Paper I, we neglected rotation, while in
Paper II we did not consider the presence of the longitudinal flow
and relativistic effects, here we considered the full case, still
remaining however in the limit of zero thermal pressure. In the
first two papers, we discussed several modes of instabilities that
in the present situation all exist. The longitudinal flow velocity
gives rise to the KHI, the toroidal component of the magnetic field
leads to CDI, while the combination of rotation and magnetic fields
gives rise to unstable toroidal and poloidal centrifugal-buoyancy modes. The instability behaviour depends, of course, on the chosen equilibrium configuration and our results can be considered representative of an equilibrium
configuration characterized by a distribution of current
concentrated in the jet, with the return current assumed to be
mainly found at very large distances. Not all combinations of
parameters are allowed: there are combinations for which equilibrium
solution does not exist. More precisely, for any given rotation
rate, there is a minimum value of the pitch, below which no
equilibrium is possible. Increasing the rotation rate, this minimum
value of the pitch decreases.
The behaviour of KHI and CDI is similar to that discussed in Paper I,
at high values of $M_a$ we find the KHI, while at low values we find
the CDI. The KHI is largely unaffected by rotation, which, on the
contrary, has a strong stabilizing effect on the CDI. Decreasing
$M_a$ and increasing rotation, the unstable region (stability
boundary) progressively moves towards smaller axial wavenumbers. A
decreasing value of the pitch, on the other hand, moves the
stability limits towards larger axial wavenumbers. For relativistic
flows, we have also to take into account that the pitch measured in
the jet rest frame, which determines the behaviour of the CDI, is
$\gamma_c$ times the value measured in the laboratory frame, therefore
relativistic flows, with the same pitch, are more stable.
In Paper I, we found a scaling law showing that the growth rate
of CDI strongly decreases with
$\gamma_c$, while an even stronger stabilization
effect is found here due to the combination of relativistic effects and rotation.
Extrapolating the behaviour found at low values of $M_a$ to the limit
$M_a \rightarrow 0$, the results are in agreement with Tomimatsu
condition \citep{Tomimatsu01}, applicable to the force-free limit.
Rotation drives centrifugal-buoyancy modes: the toroidal buoyancy
mode at low wavenumbers and the poloidal buoyancy mode at high axial
wavenumbers. Apart from the different range of these wavenumbers,
they have a similar behaviour and their growth rate scales with the
square of the rotation frequency, which, in turn, increases as the
pitch decreases; therefore they become important at low values of
the pitch. In the unstable range, the growth rate is independent
from the wavenumber. As the magnetic field increases they tend to be
more stable. The same happens increasing the Lorentz factor of the
flow.
In this paper, we have considered only the $m=1$ mode, as this mode
is thought to be the most dangerous one for jets. Higher order
modes $(m>1)$ can trigger instabilities internally in the jet,
instead of a global kink. These perturbations can cause inherent
breakup of current sheets, reconnection, etc. So, these modes are
also interesting to study in the future. If the global jet can remain stable
for long time scales and large distances, locally, higher order
modes can cause local instabilities that can or cannot disrupt the
jet.
In summary, rotation has a stabilizing effect on the CDI which
becomes more and more efficient as the magnetization is increased.
Rotation on the other hand drives centrifugal modes, which, however,
are also stabilized at high magnetizations. Finally, relativistic
jet flows tend to be more stable compared to their non-relativistic
counterparts.
\section*{Acknowledgments}
This project has received funding from the European Union's Horizon
2020 research and innovation programme under the Marie Sk{\l}odowska
-- Curie Grant Agreement No. 795158 and from the Shota Rustaveli
National Science Foundation of Georgia (SRNSFG, grant number
FR17-107). GB, PR, AM acknowledge support from PRIN MIUR 2015 (grant
number 2015L5EE2Y) and GM from the Alexander von Humboldt Foundation
(Germany).
|
{
"timestamp": "2019-03-01T02:03:26",
"yymm": "1902",
"arxiv_id": "1902.10781",
"language": "en",
"url": "https://arxiv.org/abs/1902.10781"
}
|
\section{Introduction}
Since the formulation of the periodic system in the 1860s, the quest for understanding its structure has intensively motivated research in different areas of chemistry and physics. However, almost 150 years after its announcement, the different approaches from quantum chemistry \cite{Schwarz2010a, Schwarz2013, Huang2016, Pyykko2011, Pyykko2017, Pyykko2016, Pyykko2012, Schwarz2010b, Wahiduzzaman2013, Geerlings2011, Schwarz2009}, group theory \cite{Thyssen2013, Dudek2002}, clustering \cite{Schwarz2009, Leal, RestrepoOUP2} and information theory \cite{Bonchev, Geerlings2011}, to name but a few \cite{mathPTbook,Imyanitov2011}, have not led to an unified picture \cite{Katriel2012}. Instead, they give insights on the possible chemical and physical causes of the patterns depicted by the system but have failed in providing a formal structure for it \cite{Katriel2012}.
As noted by Mendeleev:\footnote{According to Scerri \cite{Scerri2007}, Mendeleev was one of the six formulators of the periodic system, being the others B\'eguyer de Chancourtois, Newlands, Meyer, Odling and Hinrichs.} ``the reason for the absence of any explanation concerning the nature of the periodic law [Here, in general, periodic system] resides entirely in the fact that not a single rigorous, abstract expression of the law has been discovered (p. 221 of reference \cite{Jensen})." In this paper we report a formal structure for periodic system, based on a contemporary mathematical interpretation of 1869 Mendeleev publication and recent studies of the system.
\subsection{Periodic system, table and periodic law}
These are different terms that are usually treated as synonyms, but even if related, they make reference to different subjects \cite{RestrepoOUP2}. For the sake of clarity, here we discuss their differences.
A \emph{periodic system} of chemical elements is the structure resulting from considering order and similarity of chemical elements. A \emph{periodic table} is a mapping of the periodic system to another space, normally a bi-dimensional space. By \emph{periodic law} is understood the observed oscillation of some properties of chemical elements as a function of the atomic number $Z$.
There is not only one periodic system for the chemical elements, for they depend on the considered elements and on the setting up of similarity and order. Likewise, the intended generality of \emph{the} periodic law to \emph{all} properties of chemical elements does not hold, for there are properties that do not oscillate with $Z$.
In the current paper we explore the structure of a periodic system.
\subsection{The role of similarity and order}
In his 1869 publication, Mendeleev wrote: ``if one arranges the elements in vertical columns according to increasing atomic weight, such that the horizontal rows contain analogous elements, also arranged according to increasing atomic weight, one obtains the following table" (p. 16 of \cite{Jensen}). After considering that current tables interchange Mendeleev's columns and rows and that the ``arranging'' criteria has been replaced by the atomic number, two important relations are the salient structure keepers of the table, and in general of the periodic system: \emph{order} and \emph{similarity}.
Before going any further, let us analyse these two relations through examples. Let us take H, He and Li and their atomic numbers, which we order with the usual order on natural numbers, denoted by $\preceq$. An order relation holds that every element is related to itself, e.g. H $\preceq$ H. It holds that if He $\preceq x$ and if $x \preceq$ He, then $x$ is He. In addition, if H $\preceq$ He, and He $\preceq$ Li, then H $\preceq$ Li. In short, \emph{an order relation is reflexive, antisymmetric and transitive} (Appendix-Definition \ref{order}). If $E$ is the set of elements, its order by $\preceq$ is denoted as $(E,\preceq)$. In contrast to order, \emph{similarity}, represented as $\sim$, \emph{is only reflexive and symmetric}, that is, self similarity is allowed (Na $\sim$ Na) and if Na is similar to K (Na $\sim$ K), then K $\sim$ Na (Appendix-Definition \ref{similarity}). As similarity is used for classifying, it is worth mentioning that a customary outcome of a classification is a partition (Appendix-Definition \ref{partition}), i.e. a collection of subsets not sharing elements. The suitability of partitions for periodic systems is discussed later.
Despite the relevance of similarity and order for the periodic system \cite{Ashcroft2017},\footnote{Nicely accounted by the Oxford Dictionary when referring to the periodic table as: ``A table of the chemical elements arranged in order of atomic number, usually in rows, so that elements with similar atomic structure (and hence similar chemical properties) appear in vertical columns''\cite{Oxford}.} they are considered as separate aspects of it, with some emphasis on classification \cite{Scerri2009,Bengoetxea2014,Scerri2012a,Scerri2012b}\footnote{An example from outside the scientific literature is the definition of periodic table by the Cambridge Dictionary: ``An arrangement of the symbols of chemical elements in rows and columns, showing similarities in chemical behaviour, especially between elements in the same columns'' \cite{Cambridge}.} caused by the, taken for granted, ordering of the elements based upon atomic number. Whereas the possibilities for classifying are multiple given the huge number of properties chemical elements have, an exceptional example stressing ordering over similarity for the case of the table is the definition from Wikipedia: ``The periodic table is a tabular arrangement of the chemical elements, ordered by their atomic number, electron configurations, and recurring chemical properties'' \cite{Wikipedia}. In some other cases, as in \cite{Ashcroft2017,Scerri7tale}, it is said that similarity begets ordering, while, for example, the Encyclopaedia Britannica states the opposite: ``the organized array of all the chemical elements in order of increasing atomic number i.e., the total number of protons in the atomic nucleus. When the chemical elements are thus arranged, there is a recurring pattern called the `periodic law' in their properties, in which elements in the same column (group) have similar properties''\cite{Encyclopaediabritannica}.
Hence, there is confusion between order and similarity (classification), which are different binary relations, and the confusion has led to wrong statements that order leads to classifications and the other way round. The distinction between these two relations is central for the structure of the periodic system.
\section{The structure of the periodic system}
\subsection{Mendeleevian periodic system}
Back to Mendeleev's statement, the ``ingredients'' of a periodic system are: chemical elements ($E$), order by atomic number ($\preceq_Z$) and a classification ($C_P$) of the elements based on some properties $P$.\footnote{Although here we imply an unsupervised classification, it may be supervised too.}
Once elements are ordered, it is the bringing together of similar elements that ``twist'' the order giving place to ``periods''. Thus, the order by atomic number $Z$(Li) $<$ $Z$(Be) $< \ldots <$ $Z$(Ar) is twisted as $Z$(Li) $<$ $Z$(Be) $< \ldots <$ $Z$(Ne) and $Z$(Na) $<$ $Z$(Mg) $< \ldots <$ $Z$(Ar), for Li is brought together with Na, as they belong in a class. The twist is also caused by the other classes: Be-Mg, B-Al, etc. (Figure \ref{PT-chem-elts}a). An important consequence of the twists is the oscillating behaviour of some properties of chemical elements, which are nothing else than the product of considering similarity classes and order simultaneously.
A structure capturing the aspects of Mendeleev's periodic system is given by the following:
\begin{definition}
Let $E$ be the set of chemical elements, $Z$ the atomic number, $\preceq_Z$ the order relation by $Z$, $P$ some properties of the elements, $C_P$ a classification by $P$; then the \emph{Mendeleevian periodic system} is the ordered partition $(E, \preceq_Z, C_P)$.
\label{MendeleevianPS}
\end{definition}
A periodic table of the system highlighting the order relation $\preceq_Z$ among elements and among elements within classes is shown in Figure \ref{PT-chem-elts}b.
\begin{figure}[h]
\centering
\includegraphics[width=.9\textwidth,height=!, keepaspectratio]{fig1.pdf}
\caption{a) Order of elements, from H to Al, by $Z$ ($x\leftarrow y$ represents $Z(x)\preceq Z(y)$), where classes of similar elements are highlighted. The bringing together of these classes, preserving order, leads to b) A periodic table, where red arrows indicate order relationships for elements inside similarity classes, whereas green arrows between elements of different similarity classes.}
\label{PT-chem-elts}
\end{figure}
Similarity and order, however, can be treated in their broadest mathematical sense, giving place to richer structures, as we show in the next section.
\subsection{Generalised periodic system}
Just as similarity is customarily based on more than one property, order can also be based on several properties \cite{rainerbook,EST1}. Figure \ref{HD-elts} illustrates how eight chemical elements\footnote{They are selected to highlight the historical inversions of order resulting from atomic weight and atomic number.} can be ordered by atomic number $Z$ or by atomic weight $m_a$ (Figure \ref{HD-elts}a), independently (as usual), leading to Figures \ref{HD-elts}b and c, respectively. Note that, when using either $Z$ or $m_a$, it is always possible to compare any pair of elements $x$ and $y$ and assess whether $x \preceq y$ or $y \preceq x$; in both cases it is said that $x$ and $y$ are \emph{comparable}. A set endowed with an order satisfying this property is called a \emph{total order}. However, when both $Z$ and $m_a$ are simultaneously used, conflicts among properties may arise, e.g. $Z$(Ar) $ < Z$(K) and $m_a$(Ar) $>m_a$(K) (Appendix-Definition \ref{HDT-order}). Therefore, it is no longer possible to claim that $x \preceq y$ or $y \preceq x$, in this case we say that $x$ and $y$ are \emph{incomparable}. An ordering allowing comparabilities and incomparabilities is called a \emph{partially ordered set} (Appendix-Definition \ref{order}). Figures \ref{HD-elts}b to d are graphical representations of partially ordered sets, called \emph{Hasse diagrams}, where an arrow $x \leftarrow y$ between two elements $x$ and $y$ is depicted only if $x \preceq y$ and there is no $z$ such that $x \preceq z \preceq y$. This particular case of comparability is called a \emph{cover relation} and is denoted by $x \preceq: y$\footnote{Note that a cover relation is represented by adding a colon to the comparability.} (Appendix-Definition \ref{cover-preserving}). Hence, in a Hasse diagram any comparability $x \preceq y$ is represented as a sequence of cover relations; for example, as comparability $Z$(H) $\preceq Z$(K), in Figure \ref{HD-elts}b, can be inferred from the cover relations $Z$(H) $\preceq: Z$(Ar) along with $Z$(Ar) $\preceq: Z$(K), therefore $Z$(H) $\preceq Z$(K) is not graphically represented. Likewise H $\preceq$ Ni, in Figure \ref{HD-elts}d, is inferred from H $\preceq:$ Ar and Ar $\preceq:$ Ni, or H $\preceq:$ K and K $\preceq:$ Ni\footnote{Note that $x\leftarrow y$ can also be the sequence, which occurs if and only if $x\preceq: y$. Another depiction of Hasse diagrams takes the convention of replacing arrows by lines, where the direction of the order relation is inferred from the position of the elements on the drawing plane \cite{Trotter}.} (Appendix-Definition \ref{hasse-diagram}).
Thus, the ordering by more than one property may bring a structure of comparabilities and incomparabilities, which frees the periodic system from the historical total order imposed by $Z$.
\begin{figure}[h]
\centering
\includegraphics[width=.7\textwidth,height=!, keepaspectratio]{fig2.pdf}
\caption{a) Some elements and their atomic numbers ($Z$) and atomic masses ($m_a$). b) Elements ordered by $Z$ and by c) $m_a$. d) Resulting order by simultaneously considering $Z$ and $m_a$. In these Hasse diagrams (b to d) any sequence of arrows indicates that its extremes are comparable, e.g. in b, from the sequence H $\leftarrow$ Ar $\leftarrow$ K it follows that H $\preceq$ K; whereas the absence of such a sequence indicates that they are incomparable, e.g. Ar and K in d.}
\label{HD-elts}
\end{figure}
A further generalisation of periodic system can be obtained if classification is analysed. Here the question that arises is whether a classification leading to partitions is meaningful and general enough for the system of chemical elements. Is it always desirable to have disjoint classes? Could partitions be instances of a more general case for periodic systems? Chemistry helps to solve these questions. It has been found that a chemical element may belong to more than one class of similar elements,\footnote{Perhaps this is also the case of the discussion about which elements belong in group three of the periodic system of chemical elements \cite{group3}.} as is the case of Ti and Mn \cite{Rayner-Canham2018}. Other studies of hierarchies of similarity classes show that elements belong to multiple classes with different degrees of similarity \cite{RestrepoACS}, which contrast with the rigid structure of partitions of similarity classes, proper of the Mendeleevian system \cite{Rouvray}. Therefore, a more general structure for a periodic system is a collection of subsets of similar elements endowed with an order relation.
A mathematical object made of collections of subsets, called \emph{hyperedges}, is that of \emph{hypergraphs} (Appendix-Definition \ref{hypergraph}) \cite{Berge}. Elements belonging to a hyperedge (subset) are regarded as related, and the nature of their relation depends on the system to be modelled \cite{Klamt}, in our case, the relation is similarity. Figure \ref{Ti-Mn}a shows three similarity subsets with two overlaps caused by the dual similarities of Ti and Mn \cite{Rayner-Canham2018}. This system corresponds to a hypergraph.
\begin{figure}[h]
\centering
\includegraphics[width=.4\textwidth,height=!, keepaspectratio]{fig3.pdf}
\caption{a) Hypergraph with three hyperedges (similarity subsets) and b) ordered hypergraph, where the order relation is given by the arrows.}
\label{Ti-Mn}
\end{figure}
However, the periodic system is not complete without ordering. Therefore, the following definition provides a general structure for the periodic system:
\begin{definition}
Let $E$ be a non-empty set, $A$ a set of properties, $\preceq_A$ the order relation by $A$, $P$ some properties of the elements in $E$, $C_P$ a collection of subsets of similar elements regarding $P$ and $H=(E,C_P)$ a hypergraph; then the ordered hypergraph\footnote{Note that several authors \cite{Bollobas, Eslahchi} refer to ordered hypergraphs as $(H,<)$, with $H=(E,C_P)$; where $<$ is a linear order of the elements of $E$, i.e. the elements belonging in a subset (hyperedge) are totally ordered. In contrast, our definition of ordered hypergraph is more general by allowing general orders $\preceq$, which corresponds to the definition of partial-order hypergraph in \cite{He}.} $(H,\preceq_A)$ is a \emph{periodic system}.
\label{defps}
\end{definition}
The periodic system corresponding to Figure \ref{Ti-Mn}a is shown in Figure \ref{Ti-Mn}b, where the system of similarity subsets (Figure \ref{Ti-Mn}a) is endowed with an order (arrows), in this case given by atomic number. A depiction of a general periodic system is shown in Figure \ref{PT-abstract}, where the partially ordered structure and the generality of the collection of subsets is highlighted.
\begin{figure}[h]
\centering
\includegraphics[width=.4\textwidth,height=!, keepaspectratio]{fig4.pdf}
\caption{A periodic system (ordered hypergraph) where elements $E$ are nodes, which are ordered by $\preceq_A$ (green arrows) and whose order within similarity subsets (hyperedges represented as dotted lines) is shown as red arrows.}
\label{PT-abstract}
\end{figure}
Definition \ref{defps} shows that the Mendeleevian periodic system is one of the possible periodic systems. It comes up by ordering by atomic number, an order that has not incomparabilities (total order), and by taking subsets of similar elements, which depends on the properties used for the classification\footnote{In reference \cite{Jensen} (paper 2), Mendeleev discusses different classification criteria and also several for ordering.} and that leads to a partition, as a particular case of a collection of subsets. This chemical freedom in classification, which contrasts with the conservative ordering by $Z$, is the cause of the several periodic systems of chemical elements, which when combined with their possible representations gives place to the more than thousand periodic tables of the Mendeleevian periodic system.
Definition \ref{defps} frees the periodic system from the chemical domain, for it can now be used in other contexts as long as the elements be provided, as well as the criteria for ordering and classifying them. However, not to go far from chemistry, we show in the next section how Definition \ref{defps} can be used to devise a periodic system of bonds, in contrast to the traditional one of chemical elements.
\section{A periodic system of polarization of single covalent bonds}
Polarizability, i.e. the tendency of charge distribution to be distorted in response to an external electrical field, is an important property of materials at different levels, ranging from atomic and molecular to bulk scales \cite{Rupasinghe2015}. Its importance is given by its relationship with, e.g. stiffness of materials, compressibility and other properties \cite{Rupasinghe2015}. Not to mention its pedagogical chemical value \cite{JCE}. By addressing polarizability at a simple molecular level of atoms forming single covalent bonds, here we devise a periodic system tailored to such bonds.
Note that bond polarization is based on the definition of atomic charge, of which there are several, from different theoretical and experimental perspectives \cite{Schwarz1994}. Moreover, there are different properties to characterise bond polarizability, e.g. electronegativity, atomic radius, ionization potential, electron affinity, atomic volume and some others coming from natural bond orbital treatments, among others. In any case, the characterisation requires at least two properties related to the potential nucleus-electron attraction and the kinetic repulsion of electrons that make a single covalent bond a stable system. Two reasonable properties meeting this condition and readily available are electronegativity \cite{Pauling} and bond distance, as expressed by atomic radius of bonded atoms\footnote{Bond characterisation may also be attained through specific or averaged properties. Properties selected in this paper are averaged, but specific ones such as Allred-Rochow electronegativities and radii with reference to a particular parent group, e.g. methyl, could also be used. A systematic approach for the selection of properties, given a response variable, has been recently published in \cite{PhysRevMaterials.2.083802}.} \cite{Pyykko2009}.
We considered 94 single covalent bonds ($E$) of the form $x-y$, where $y$ is a chemical species as explained latter and $x$ is a chemical element.\footnote{The elements $x$ are: H, Li, Be, B, C, N, O, F, Na, Mg, Al, Si, P, S, Cl, K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Ge, As, Se, Br, Kr, Rb, Sr, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd, In, Sn, Sb, Te, I, Xe, Cs, Ba, La, Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Hf, Ta, W, Re, Os, Ir, Pt, Au, Hg, Tl, Pb, Bi, Po, At, Ra, Ac, Th, Pa, U, Np, Pu, Am, Cm, Bk, Cf, Es; which are the elements considered in \cite{Leal} and shown in Figure \ref{3plots}.} The properties for ordering are $A=$\{Pauling electronegativity, single-bond additive covalent radius\}. By additive covalent radius of bond $x-y$ is meant $r(x-y)=r(x)+r(y)$. These radii were obtained from either experimental or theoretical data of chemical species including bonds of the sort $x-x$, $x-\text{H}$, $x-\text{CH}_3$ and $x-y$, being $x$ and $y$ different chemical elements \cite{Pyykko2009}. The similarity subsets (hyperedges) $C_P$ for the bonds $x-y$ were based on the resemblance of the elements $x$ when forming binary compounds, which yielded 44 classes \cite{Leal} (Table \ref{44classes}). These classes are based on the chemical idea that elements are similar if they combine with the same substances to produce chemically similar compounds \cite{Schummer1998}. For instance, alkali metals are similar because they combine with water to produce alkalies, which when combined with hydracids of halogens produce simple salts, e.g. LiF, NaF, LiCl and NaCl, etc. Hence, alkali metals are regarded as similar, for a large amount of the compounds they form are common to the similarity class of alkali metals. As the notion of common compound is central to describe similarity of chemical elements, it was formalised in \cite{Leal} for binary compounds as follows: two elements $x$ and $y$ have a common binary compound if there exists a third element $z$, and binary compounds $x_az_b$ and $y_az_b$. Therefore, similarity between $x$ and $y$ increases with the number of common binary compounds they have. In \cite{Leal}, the notion of common compound includes similarity in proportions of combination to differentiate, for instance, between alkali and alkali earth metals. In such a setting, LiF and BeF$_2$ are not common binary compounds of Li and Be, but BeF$_2$ and MgF$_2$ are for Be and Mg. Having a set of binary compounds, similarity between two elements $x$ and $y$ is calculated as the number of common compounds between them, leading to a similarity matrix upon which classification algorithms may be applied, e.g. cluster analysis, yielding similarity classes \cite{Leal}.
The set of properties $P$ for the classification of single covalent bonds is made by the global neighbourhoods of each element. All calculations on order relationships here reported were performed with the free-ware Python package PyHasse \cite{Voigt2010,Rainer2013} developed by Rainer Bruggemann.\footnote{More information on PyHasse can be requested to its developer at \href{mailto:brg\_home@web.de}{brg\_home@web.de}.} This package has an on-line version \cite{PyHasse}, where some further order calculations are possible.
\begin{table}[!h]
\centering
\caption{44 similarity classes of chemical elements.}
\label{44classes}
\begin{tabular}{ c | c | c| c}
H & B & C & N \\
O & Si & P & S \\
Ti & Cr & Mn & Au \\
Bi & Po & La & Ce \\
Ac & Pu & Ir,Rh & Se,Te \\
Y,Sc & Ca,Mg & Mo,W & Ag,Cu \\
As,Sb & Am,Cf & Tb,Pr & Li,Na \\
Cm,Bk,Es & Sm,Eu,Yb & Be,Sr,Ba & K,Rb,Cs \\
V,Nb,Ta & Zn,Cd,Hg & Ru,Os,Pt & Ge,Sn,Pb \\
F,Cl,Br,I & Kr,Xe,At,Ra & Tm,Dy,Pm,Nd & Lu,Er,Ho,Gd \\
Tc,Re,Pa,Np & Fe,Co,Ni,Pd & Al,Ga,In,Tl & Zr,Hf,Th,U \\
\end{tabular}
\vspace*{-4pt}
\end{table}
To attain an ordering with chemical meaning where more polarized bonds involve electronegative elements and hold short radii, we reoriented the radius of element $x$, as $\bar r(x)=\text{max }r-r(x)$. In Figure \ref{3plots} it is seen that electronegativity and reoriented radius are highly correlated, as quantified by the 0.83 Spearman correlation.\footnote{Spearman coefficient quantifies whether two variables are monotonically related, not necessarily linearly, ranging from 0 (not correlated) to 1 (correlated).} However, these two properties lead to several incomparabilities in the periodic system, as we show later. A correlation of 1 would indicate that electronegativity and oriented radius hold the same total order of bonds, i.e. that when ordered by $\bar r(x)$ the order obtained is the same than when bonds are ordered by electronegativity of $x$. As the correlation is high, a large number of comparabilities is expected (few incomparabilities). On the other hand, since bonds involving Bk and Cf are equivalent, for they have the same electronegativity and $\bar r$, there are 93 different representative bonds accounting for $93\times 92/2=4,278$ order relationships, which are split in 3,548 comparabilities and 730 incomparabilities, i.e. 83\% of the order relationships are comparabilities and 17\% are incomparabilities. A depiction of this periodic system is shown in Figure \ref{PT-radii-electronegativity}.\footnote{As data uncertainty may affect the ordering, a fuzzy set theoretical approach has been devised to analyse uncertainty effects \cite{Rainer2011, Rainer2012}, where the order relation $\preceq$ is replaced by a fuzzy inclusion relation. In such a setting the ordering of any two elements is not any more given by the pairwise comparison of their properties (Appendix-Definition \ref{HDT-order}) but by the degree of subsethood of the two elements, now considered as sets of their property values.}
\begin{figure}[h!]
\centering
\includegraphics[width=.9\textwidth,height=!, keepaspectratio]{fig5.pdf}
\caption{Logarithmic distribution of a) reoriented single-bond covalent radius (pm) and b) Pauling electronegativity (dimensionless) and of c) bond dominance degree (dimensionless) over the elements ordered by atomic number. d) Conventional periodic table with elements classified by their bond dominance degree. Elements in the red hyperedge correspond to those forming the most polarized bonds, while those in the dark blue to the least.}
\label{3plots}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=.6\textwidth,height=!, keepaspectratio]{fig6.pdf}
\caption{A periodic system of polarized single covalent bonds, where 94 bonds $x-R=b(x)$, with $x$ a chemical element and $R$ another chemical species, are ordered by Pauling's electronegativity of element $x$ and its reoriented atomic radius in single bonds. For simplicity $b(x)$ is labelled as $x$. There are 93 nodes, for Bk and Cf are equivalent and are represented by Bk. At the top of the system highly polarized bonds appear and at the bottom the least ones. Hyperedges (subsets of similar elements as shown in Table \ref{44classes}) are depicted as bonds sharing filling and font colours.}
\label{PT-radii-electronegativity}
\end{figure}
The most polarized bonds correspond to those involving H, F and O (at the top of Figure \ref{PT-radii-electronegativity}), and the least is Cs (at the bottom). To know if a bond $b(x)=x-R$, with $x$ an atom of element $x$ and $R$ another chemical species \cite{Pyykko2009}, is more polarized than another $b(y)$, i.e. if $y-R \preceq x-R$, it must be found in Figure \ref{PT-radii-electronegativity} a sequence of arrows from $b(x)$ to $b(y)$\footnote{In mathematical terms it corresponds to finding a chain between $x$ and $y$ (Appendix-Definition \ref{chain}).} \cite{Trotter}; this is the case of, e.g. $b($I$) \preceq$ $b($F$)$.
A consequence of exploring order relationships is seen, e.g. in the case of bonds of Cl and N, or Pu and Am, which are incomparable, i.e. with no sequence of arrows connecting them.
Being located at the top of the periodic system does not necessarily mean that such bonds are more polarized than the others, e.g. $b($H$)$ is more polarized than 77 other bonds, while $b($O$)$ and $b($F$)$ more than 90 bonds; $b($H$)$, $b($O$)$ and $b($F$)$ are all at the top of the system. To quantify this degree, we devised the bond dominance degree:
\begin{definition}
Let $x$ and $y$ be single covalent bonds $b(x)$ and $b(y)$, with $x\neq y$. The \emph{bond dominance degree} of $x$ is given by:
\[Dom(x):=\frac{C_{y \prec x}}{n-1}\]
where $C_{y \prec x}=\vert\{y : y\prec x\}\vert$ and $n$ is the number of bonds. Note that $y \prec x$ indicates those $y$ such that $y \preceq x$ but that are not $y = x$.\footnote{Hence, $\prec$ is a relation that is not reflexive, it only holds antisymmetry and transitivity, while an order relation $\preceq$ holds the three of them.}
\label{eldomdeg}
\end{definition}
$Dom(x)=1$ indicates that all other bonds different from $x$ are dominated by $x$, i.e. that $b(x)$ is more polarized than all the other bonds. $Dom(x)=0$ shows that $x$ is less polarized than any other bond. A plot depicting the bond dominance degree is shown in Figure \ref{3plots}c, where, keeping the chemical tradition, bonds are ordered by the respective $Z$ of the bonded atom $x$. Figure \ref{3plots}b is actually a plot of the function $(Z,f(Z))$, where $f(Z)=Dom(Z)$, which is an oscillating function resulting from the oscillating nature of electronegativity and single-bond covalent radius (Figures \ref{3plots}a and b).
Figure \ref{3plots}d depicts the conventional periodic table with elements coloured by their bond dominance degree. Therein alkali metal, heavy alkaline earth and most of the lanthanoid and actinoid bonds are more polarized than only 20\% of the other bonds. Mg bond is more polarized than 40\% of the others, as some early transition metal bonds, e.g. Sc, Y, Zr and Hf. Be bond, whose Be is similar to Sr and Ba (Table \ref{44classes}), is more polarized than the bonds of Sr and Ba. In fact, $b($Be$)$ is more polarized than 60\% of the other bonds, as several transition metal and non metal bonds. Most of the platinum and coinage metals (except Ag), as well as B, P, Ge, As, I and Xe (dark yellow), form more polarized bonds than 80\% of the other bonds.
The bonds that are more polarized than the rest of the bonds are those of H, C, N, Kr, halogens and chalcogens, except Te. Specific details on the dominated, dominating and incomparable bonds for each bond\footnote{Bond $y$ is a dominating bond of $x$ if the relation $x \prec y$ holds. In mathematical terms the dominated bonds of $x$ correspond to the \emph{down set} or \emph{ideal} of $x$ without the bond $x$. Likewise, the dominating bonds of $x$ are the \emph{up set} or \emph{filter} of $x$ without the bond $x$ \cite{Trotter}.} are provided in Table S1.
So far, it has been discussed how order and classification shed light on bond polarization. However, the two relations can be further considered to explore order relations within and among subsets of bonds. These relationships correspond, respectively, to the red and green arrows in Figures \ref{PT-chem-elts} and \ref{PT-abstract}.
\subsection{Ordering bond polarizations within subsets of similar elements (hyperedges)}
The analysis of order relationships within hyperedges allows assessing whether a subset of bonds of similar elements also involves an ordered structure. This order relationship permits knowing whether, e.g. the well-known electronegative fluorine forms most polarized bonds than the other halogens. To analyse these within-hyperedge order relations, we quantified the degree of within-hyperedge comparability.
\begin{definition}
Let $C$ be a subset of single covalent bonds (hyperedge), the \emph{within-hyperedge dominance degree} $Dom(C)$ is given by
\[
Dom(C):=\frac{2T_{j\prec i}}{n(n-1)}
\]
with $n$ being the number of bonds in $C$ and $T_{j\prec i}=\vert\{(x_i,x_j): x_j\prec x_i, x_i,x_j \in C\}\vert$.
\label{withindegree}
\end{definition}
Hence, for a hyperedge with $n$ bonds, $n(n-1)/2$ relationships of the sort $x \prec y$ are expected, with $x,y \in C$. How many of them are actually $\prec$ (non self-comparabilities) is what within-hyperedge dominance degree quantifies. Note that this degree is only calculated for hyperedges of more than one bond, for the relation $\prec$ does not allow self comparisons. Hyperedges where all relationships are comparabilities, i.e. where there is a chain (Appendix-Definition \ref{chain}) containing the bonds of the hyperedge, are robust in terms of similarity and order. These hyperedges have 1 as degree of within-hyperedge comparability; likewise, if the bonds of the hyperedge are not comparable at all, the degree of comparability is 0.
There are 26 non-single hyperedges of similar chemical elements, out of the 44 discussed (Table \ref{44classes}), whose degrees of comparability are shown in (Table \ref{deg-comp-incomp-classes}). It is seen that almost half of the hyperedges have within-hyperedge comparability degrees greater than 0.5. This shows that these hyperedges not only gather bonds of similar elements, but that they have a rich order structure. This makes that similar elements, e.g. Ge, Sn and Pb, that form a hyperedge with $Dom(C)=1$, can be ordered by bond polarization, in this case being $b($Ge$)\succ b($Sn$)\succ b($Pb$)$. This kind of trend is well-known for halogens, which actually form a hyperedge with $Dom(C)=1$, being $b($F$)$ the most polarized single covalent bond. There are hyperedges with non-vertical similarities on the table having $Dom(C)=1$, e.g. $b($Tc$)\succ b($Re$)\succ b($Pa$)\succ b($Np$)$ and $b($Ru$)\succ b($Os$)\succ b($Pt$)$. Figure \ref{PT-dom1} shows the hyperedges with $Dom(C)=1$. There is an average of 0.73 for within-hyperedge dominance degree, which is expected given the high amount of comparabilities in the periodic system.
\begin{table}[!h]
\centering
\caption{Within-hyperedge dominance degree for the 26 non-single subsets of similar bonds. Hyperedges with degree 1 are shown in non-increasing order of polarizability, e.g. Ge $\succ$ Sn $\succ$ Pb. For simplicity $b(x)$ is labelled $x$.}
\label{deg-comp-incomp-classes}
\begin{tabular}{ l | c }
Subset (Hyperedge) & \makecell{Within-hyperedge \\dominance degree} \\ \hline
Ge, Sn, Pb & 1 \\
Zr, Hf, Th, U & 0.5 \\
Al, Ga, In, Tl & 0.66 \\
Am, Cf & 0 \\
As, Sb & 1 \\
Cu, Ag & 0 \\
Ru, Os, Pt & 1 \\
Mo, W & 1 \\
Fe, Co, Ni, Pd & 0.5 \\
Zn, Cd, Hg & 0.33 \\
V, Nb, Ta & 0.66 \\
Tc, Re, Pa, Np & 1 \\
Li, Na & 1 \\
K, Rb, Cs & 1 \\
Ca, Mg & 1 \\
Be, Sr, Ba & 1 \\
Lu, Er, Ho, Gd & 1 \\
Tm, Dy, Pm, Nd & 1 \\
Sm, Eu, Yb & 0.66 \\
Y, Sc & 1 \\
Cm, Bk, Es & 0.66 \\
Tb, Pr & 0 \\
Ra, Kr, Xe, At & 1 \\
Ir, Rh & 0 \\
Se, Te & 1 \\
F, Cl, Br, I & 1 \\
\end{tabular}
\vspace*{-4pt}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[width=.8\textwidth,height=!, keepaspectratio]{fig7.pdf}
\caption{Periodic table where hyperedges with within-dominance degree equal to 1 are coloured. Elements (bonds) sharing cell colour and font belong in the same hyperedge and hold dominance degree 1.}
\label{PT-dom1}
\end{figure}
\subsection{Ordering bond polarizations among subsets of similar elements (hyperedges)}
By ordering hyperedges we can address questions like: Are lanthanoid bonds more polarized than actinoid ones?, which have technological, as well as geochemical implications related to the materials they may form and the extraction from ores these elements undergo. Similar questions aiming at comparing sets of elements can be addressed. To do so, we applied the dominance degree for hyperedges.
\begin{definition}
Given $C_i$ and $C_j$ as hyperedges of bonds, the \emph{inter-hyperedge dominance degree} $Dom(C_i,C_j)$ of $C_i$ over $C_j$ is given by:
\[Dom(C_i,C_j):=\frac{T_{j\prec i}}{n_in_j}\]
with $T_{j\prec i}=\vert\{(x_i,x_j) : x_i\in C_i, x_j\in C_j, x_j\prec x_i\}\vert$.
\label{domdegclass}
\end{definition}
Hence, for a given couple of hyperedges of bonds $C_i$ and $C_j$, $Dom(C_i,C_j)$ quantifies how many bonds of $C_i$ dominate those of $C_j$, i.e. how many bonds of $C_i$ are more polarized than bonds in $C_j$. Figure \ref{in-out-deg-scatter}a shows a schematic representation of a \emph{dominance diagram} \cite{EST1}, where the most dominated hyperedges are at the bottom with a high number of incoming arrows (high in-degree) and most dominating hyperedges are located at the top, holding high number of outgoing arrows (out-degree).\footnote{Note that the arrows of the dominance diagram are not cover relations as in a Hasse diagram. In particular, dominance diagrams do not hold transitivity \cite{EST1, Restrepo2008}.} When the dominance diagram turns too complex (with many arrows), it is better to depict each hyperedge in a coordinate system given by its in- and out-degrees (Figure \ref{in-out-deg-scatter}b), which we call the \emph{dominance profile}.\footnote{Similar diagrams to represent complex partially ordered sets are devised in \cite{Rainer2013,Quintero2018}.} As the dominance diagram for hyperedges of bonds is too complex, even for $Dom(C_i,C_j)>0.95$; we show in Figure \ref{in-out-deg-scatter}c its respective dominance profile.
Figure \ref{in-out-deg-scatter}c shows that the least polarized bonds are those where the most electropositive alkali metals, La and Ac are involved. These bonds are dominated by almost all other hyperedges and they dominate no other hyperedge or just a couple of them. A cluster of a bit more polarized bonds is made by those involving some transition metals such as \{Y, Sc\}, electropositive alkaline earths \{Mg, Ca\} and most of the lanthanoids and actinoids. Bonds with intermediate dominances are \{V, Nb, Ta\} and Ti that are more polarized than about one third of the other hyperedges and less polarized than about half of the other hyperedges. A cluster of dominating bonds is made by several transition metals, with in-degrees about 8 and out-degrees close to 20. These hyperedges of bonds of metals are more polarized than about half of the other hyperedges and are only less polarized than about 8 others, where O, N, C, S and H bonds are included. It is found that actinoids dominate more hyperedges than lanthanoids, except for La and Ac, where La dominates more hyperedges than Ac.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth,height=!, keepaspectratio]{fig8.pdf}
\caption{a) Dominance diagram of 10 hyperedges (black nodes) and its respective b) dominance profile, where some hyperedges are highlighted. The diagonal limiting the profile corresponds to all possible sums of in and out-degrees being equal to the number of hyperedges. c) Dominance profile for the 44 hyperedges of single covalent bonds.}
\label{in-out-deg-scatter}
\end{figure}
\section{Conclusions and outlook}
Based on an analysis of the periodic system of chemical elements, we have formalised and generalised the periodic system as a set endowed with a system of similarity classes, whose elements hold an order relation. This structure corresponds to an ordered hypergraph, where similarity classes are hyperedges.
An advantage of having a mathematical structure for a periodic system is that it opens the possibility of exploring and formally characterising the relationships among periodic systems, i.e. given two periodic systems $P_1$ and $P_2$ , it can be determined whether one is a substructure of the other $P_1 \subseteq P_2$, if they are isomorphic $P_1 \simeq P_2$ , equivalent $P_1 \equiv P_2$ or equal $P_1 = P_2$ (Appendix-Definitions \ref{hypergraph} to \ref{hypergraph-isomorphism}). This brings up new questions. How many different periodic systems of the chemical elements have been devised? Which of them are isomorphic or equivalent? (Appendix-Definitions \ref{hypergraph-equivalence} and \ref{periodic-system-relations}). Which systems are the most populated by their projections into periodic tables? Which is the super-structure formed by all the devised periodic systems? Are there some sort of embedding relations between them?
The structure here reported is flexible enough to accommodate new chemical elements, all of them located in the region of superheavy elements (SHEs), right after oganesson (Z=118).\footnote{Current estimations indicate that $Z$=173 is the heaviest possible element \cite{Indelicato2011}.} Although the hypergraph structure was actually the framework in which Mendeleev predicted elements and several of their properties; he did it through interpolative methods. This is no longer possible because the expansion of the system is in the SHE region. Instead, predicting new elements requires relativistic quantum theoretical methods \cite{Pyykko2011}, which is how some SHEs properties are addressed, e.g. ionization potentials. Once such calculations are provided, elements can be classified using properties derived from relativistic methods, e.g. electronic configurations in the ground state of the neutral atom.\footnote{As noted by Haba \cite{Haba2019}, the electronic configurations of SHEs are difficult to estimate, for valence orbitals are energetically close to each other. This is especially difficult for elements with $Z >$121.} As SHEs have associated atomic numbers $Z$, these elements can therefore be ordered. Thus, new elements can be incorporated into the structure, for their similarity classes can be determined as well as their order relationships.
Although there is no room for interpolations, structures encoding chemical information about similarity and order can be used, as shown by Klein and coworkers \cite{Klein1995, Restrepo2011, Panda2013}.\footnote{A case in point is the ordering of substituted cubanes, which were ordered by the relation established between molecular structures when one can be obtained, by H-substitution, from the other \cite{Restrepo2011}. By using three different interpolative methods that take into account the order relationships of couples of cubanes, not experimentally measured densities of nitro-cubanes could be estimated. The results showed that the estimation of known densities were very close to experimental values.} They have estimated properties of unknown substances, which makes foreseeable using the ordered hypergraph structure to estimate unknown properties of known elements.
Another instance of the relevance of the structure of the system is its recent use in the prediction of enthalpies of formation of several compounds \cite{C8SC02648C}. There, Zhang and coworkers show how sensitive their neural networks predictions are to the input structure, which is a periodic table. As we have discussed, a periodic table is a mapping of the ordered hypergraph to a bi-dimensional space and there are many possibilities for the mapping. The striking result is that by randomising the input structure, the quality of the estimations drops down. Also relevant is that the input table is a traditional one containing the most well-known similarity classes of chemical elements, ordered by atomic number, and that such a table yields the best predictions. The destruction of such a structure by shuffling the elements reveals how important the ordered hypergraph is for the system.
Taking into account the structure of periodic system, we devised a periodic system for polarized single covalent bonds, which shows not only the similarity and order relationships for bonds, but allows exploring order relationships inside classes of similar bonds and among classes of bonds. This last order was of interest for Mendeleev, as noted when writing ``The objective of my memoir will be fully achieved if I can successfully direct the attention of investigators to those relationships involving the atomic weights of dissimilar elements, which, as far as I know, have so far been entirely ignored'' (p. 145 in \cite{Jensen}).
We found that most of the classes of similar bonds have an internal ordered structure, ranging from the typical example of halogens, where bonds of F are more polarized than those of Cl, Br and I; to cases involving transition metals and actinoids as \{Tc, Re, Pa, Np\}, where Tc bonds are more polarized than those of Re, Pa and Np.
The order relationships for classes of similar bonds show that there are few classes of poorly polarized bonds, which are less polarized than almost all other classes (hyperedges). They are the heavy alkali metals \{K, Rb, Cs\} and La and Ac. There are also hyperedges of strongly polarized bonds, as those of O and N that are more polarized than almost all the other classes of bonds. Halogens, with the electronegative F, are only more polarized than about half of the other classes of bonds, as the inclusion of not so electronegative elements as I makes that the polarization as a class decreases.
The periodic system of polarized bonds relies on the similarity of chemical elements calculated from their presence in binary compounds. This methodology is chemically general, for it can be extended not only to binary but to $n$-ary compounds. Thus, the method can be applied to any dataset of compounds to assess elemental similarity. The current electronic availability of this information in databases such as Reaxys\textsuperscript{TM} and SciFinder\textsuperscript{TM} make possible the automatisation of the process. Results on the periodic system of chemical elements based on compounds gathered in Reaxys from 1800 up to date are the subject of a forthcoming publication.
The structure of the periodic system here reported frees the concept from the chemical domain and makes it readily applicable to other fields of knowledge. In fact, ordered hypergraphs are also found in information systems and web mining, as recently reported in \cite{He}. Klamt, a decade ago, drew attention to the suitability of hypergraphs for the description of biological, chemical and computational processes \cite{Klamt}. Other examples of systems able to be endowed with similarity subsets are, e.g. ordered systems of countries rated by child development indicators \cite{rainerbook} or by scientific production \cite{Restrepo2014}. Similar examples are found in engineering, hydrology, environmental sciences, sociology, to name but a few areas. Therefore, we envision periodic systems not only of chemical interest but of applicability in other disciplines.
Our results contribute to the undergoing generalisation of network theory to hypergraphs, where the traditional network description as a graph is being abstracted to that of hypergraphs as a mean to model complex relations among multiple entities \cite{He,leal2018formanricci}. We show that hypergraphs can be ordered and that the resulting structure has been at the core of chemistry for more than 150 years.
\vskip6pt
\enlargethispage{20pt}
\section*{Ethics}
No human or animal subjects were involved in this work.
\section*{Data Accessibility}
All calculations on order relationships were performed with the free-ware Python package PyHasse available at \url{https://pyhasse.org/}
\section*{Authors\text{'} Contributions}
WL and GR conceived the study, developed the mathematical formalisation and drafted the manuscript; GR collected and processed the data for the tailored periodic system and wrote the document. Both authors gave final approval for publication
\section*{Competing Interests}
We have no competing interests.
\section*{Acknowledgements}
The authors thank Douglas Klein for motivating this research and Eugen Schwarz and Rainer Br\"uggemann for their valuable comments.
\section*{Appendix}
\begin{definition}
\label{order}
A binary relation $\preceq$ on $X$ is a \emph{partial order} (or order relation) if for all $x,y,z \in X$:
\begin{itemize}
\item $x \Rightarrow x\preceq x$ (reflexivity)
\item $x\preceq y$ and $y \preceq x \Rightarrow x=y$ (antisymmetry)
\item $x \preceq y$ and $y \preceq z \Rightarrow x \preceq z$ (transitivity)
\end{itemize}
The couple $(X,\preceq)$ is called a \emph{partially ordered set}.
\end{definition}
\begin{definition}
\label{similarity}
A binary relation $\sim$ on $X$ is a \emph{similarity relation} (or tolerance relation) if for all $x,y \in X$:
\begin{itemize}
\item $x \sim x$ (reflexivity)
\item $x \sim y \Rightarrow y \sim x$ (symmetry)
\end{itemize}
\end{definition}
\begin{definition}
\label{partition}
A family of sets $P$ is a \emph{partition} of $X$ if and only if:
\begin{itemize}
\item $\varnothing \notin P$
\item $\cup_{A\in P}A=X$
\item For all $A,B \in P$, if $A \neq B \Rightarrow A \cap B = \varnothing$
\end{itemize}
\end{definition}
\begin{definition}
\label{HDT-order}
Let $X$ be a non-empty set and $P$ a set of properties $p_i$ of $x\in X$. For any $x,y \in X$, we say that $x\preceq y$ if $p_i(x) \preceq p_i(y)$, for all $p_i\in P$.
\end{definition}
\begin{definition}
\label{cover-preserving}
Given a partially ordered set $(X,\preceq)$, we say that $x$ \emph{covers} $y$, denoted by $x\preceq : y$, if $x\preceq y$ and there is no $z$ such that $x\preceq z\preceq y$.
A \emph{cover-preserving map} from a partially ordered set $X$ to another $X'$ is a function $f$ such that, if $x,y\in X$ and $x\preceq : y$, then $f(x)\preceq : f(y)$.
\end{definition}
\begin{definition}
\label{hasse-diagram}
Given a partially ordered set $(X,\preceq)$, its corresponding \emph{Hasse diagram} is a directed graph $(X,E)$, such that $(x,y)\in E$ if $x\preceq : y$, i.e. arrows associated to $(x,y)$ in the directed graph are the cover relations of the partially ordered set $(X,\preceq)$.
\end{definition}
\begin{definition}
\label{hypergraph}
Given a set $X$ and a collection $\{X_i\}_{i \in I}$ of subsets of $X$, a \emph{hypergraph} on $X$ is the pair $H=(X,\{X_i\}_{i \in I})$.
\end{definition}
\begin{definition}
\label{chain}
Given a partially ordered set $(X,\preceq)$, $(X',\preceq)$ is called a \emph{chain} if $X'\subseteq X$ and if for every distinct pair of elements $x,y\in X'$, $x\preceq y$ or $y \preceq x$ holds.
\end{definition}
\begin{definition}
\label{substructure}
Let $H=(X,Y)$ and $H'=(X',Y')$ be hypergraphs. $H'$ is said to be a \emph{sub-hypergraph} of $H$ if $X' \subseteq X$ and $Y' \subseteq \{X_i\cap X' : X_i \in Y\}$, which is written as $H'\subseteq H$.
\end{definition}
\begin{definition}
\label{hypergraph-isomorphism}
Let $H=(X,\{X_i\}_{i \in I})$ and $H'=(X',\{X'_i\}_{i \in I})$ be hypergraphs on $X$ and $X'$, respectively, they are isomorphic if there exists a bijection $\psi:X \rightarrow X'$ and a permutation $\pi$ of $I$ such that $\psi(X_i)=X_{\pi(i)}'$ . $\psi$ is called an \emph{isomorphism} and $H$, $H'$ are called isomorphic, denoted by $H\simeq H'$.
\end{definition}
If the elements of $X$ and $X'$ are labelled, the notions of equivalence and equality between hypergraphs arise:
\begin{definition}
\label{hypergraph-equivalence}
Let $H=(X,Y)$ and $H'=(X',Y')$ be isomorphic hypergraphs under $\psi$, they are \emph{equivalent}, denoted by $H\equiv H'$, if $\psi(x)=x'$ and $\psi(X_i)=X_{\pi(i)}'$, where $X_i \in Y$ and $X'_i \in Y'$. Moreover, if $\pi$ is the identity map, the two hypergraphs are \emph{equal}, denoted by $H = H'$.
\end{definition}
According to Definition \ref{defps}, a periodic system is the couple $(H,\preceq)$ where $H$ is a hypergraph on a set $X$ and $\preceq$ is an order relation.
\begin{definition}
\label{periodic-system-relations}
Let $H=(X,Y)$ and $H'=(X',Y')$ be two hypergraphs such that either $H' \subseteq H$, or $H' \simeq H$ or $H' \equiv H$ or $H' = H$ under $\psi$. If $\psi$ is a cover-preserving map (Definition \ref{cover-preserving}) between $X$ and $X'$, then the periodic systems $PS=(H,\preceq)$ and $PS'=(H',\preceq)$ are $PS'\subseteq PS$, or $PS'\simeq PS$ or $PS'\equiv PS$ or $PS' = PS$, respectively.
\end{definition}
|
{
"timestamp": "2019-03-01T02:01:46",
"yymm": "1902",
"arxiv_id": "1902.10752",
"language": "en",
"url": "https://arxiv.org/abs/1902.10752"
}
|
\subsection{Black-box \& White-box Restrictions}
\label{BlackWhiteBoxRestrictions}
While this formulation of the GATN is sufficient for white-box attacks where we have access to the attacked model $f$ or the student model $s$, this assumption is unrealistic in the case of black-box attacks. For a black-box, we are not permitted access to either the internal model (a neural network or a classical model) or to the dataset that the model was trained on. Furthermore, for black-box attacks, we impose a restriction on the predicted labels, such that we utilize only the class label predicted, and not the probability distribution produced after softmax scaling (for neural networks), or the scaled probabilistic approximations of classical model predictions.
To further restrict ourselves to realistic attack vectors, we stratify the available dataset $D$, which will be used to train the GATN, into two halves, such that we train the GATN on one subset, $D_{eval}$, and are able to perform evaluations on both this train set and the wholly unseen test set, $D_{test}$. Note that this available dataset $D$ is not the dataset on which the attacked model $f$ was trained on. As such, we never utilize the train set available to the attacked classifier to either train or evaluate the GATN model. In order to satisfy these constraints on available data, we define our available dataset $D$ as the test set of the UCR Archive \cite{UCRArchive2018}. As the test set was never used to train any atacked model $f$, it is sufficient to utilize it as an unseen dataset. We then split the test dataset into two class-balanced halves, $D_{eval}$ and $D_{test}$. Another convenience is the availability of test set labels, which can be harnessed as a strict check when evaluating adversarial generators.
When we evaluate under the constraints of black-boxes, we further limit ourselves to “unlabeled” train sets, where we assume the available dataset is unlabeled, and thereby utilize only the predicted label from the attacked classifier $f$ to label the dataset prior to attacks. We state this as an important restriction, considering that it is far more difficult to freely obtain or create datasets for time series than for images which are easily understood and interpreted. For time series, significant expertise may be required to distinguish one sample amongst multiple classes, whereas natural images can be coarsely labeled with relative ease without sophisticated equipment or expertise.
\subsection{Time Series Classification Models}
\label{TSClassifiers}
\subsubsection{1-NN Dynamic Time Warping}
\par The equations and definitions below are obtained from Kate et al. \cite{kate2016using} and Xi et al. \cite{xi2006fast}. Dynamic Time Warping is a measures of similarity between 2 time series, $Q$ and $C$, which is detected by finding their best alignment. Time series $Q$ and $C$ are defined as:
\begin{align}
\label{eq:1}
Q &= q_1, q_2, q_3, ..., q_i, ..., q_n \\
\label{eq:2}
C &= c_1, c_2, c_3, ..., c_i, ..., c_n.
\end{align}
To align both the time series data, the distance between each timestep of $Q$ and $C$ is calculated, $(q_i - c_j)^2$, to generate a $n$-by-$m$ matrix. In other words, the $i^{\text{th}}$ and $j^{\text{th}}$ of the matrix is the $q_i$ and $c_j$. The optimal alignment between $Q$ and $C$ is considered the warping path, $W$, such that $W = w_1, w_2, w_3, ..., w_k, ..., w_K$. The warping path is computed such that,
\begin{enumerate}
\item $w_1 = (1,1)$,
\item $w_k=(n,m)_k$,
\item given $w_k = (a, b)$ then $w_{k - 1} = (a^{\prime}, b^{\prime})$ where $0 \leq a - a^{\prime} \leq 1$ and $0 \leq b - b^{\prime} \leq 1$.
\end{enumerate}
The optimal alignment is the warping path that minimizes the total distance between the aligning points,
\begin{equation}
\label{eq:3}
DTW(Q,C) = \operatorname*{argmin}_{W=w_1,w_2,...,w_K}\sqrt{\sum_{k=1,w_k=(i,j)}^{k} (q_i - c_j)^2}.
\end{equation}
\subsubsection{Fully Convolutional Network}
The Fully Convolutional Network (FCN) is one of the earliest deep learning time series classifier. \cite{wang2017time} It contains 3 convolutional layers, with convolution kernels of size 8, 5 and 3 respectively, and emitting 128, 256 and 128 filters respectively. Each convolution layer is followed by a batch normalization \cite{ioffe2015batch} layer that is applied with a ReLU activation layer. A global average pooling layer is employed after the last ReLU activation layer. Finally, softmax is applied to determine the class probability vector.
\subsection{Adversarial Transformation Network}
\label{AdversarialTransformNetwork}
Several methods have been proposed to generate adversarial samples that attack deep neural networks that are trained for computer vision tasks. Most of these methods use the gradient with respect to the image pixels of these neural networks. Baluja and Fischer \cite{baluja2017adversarial} propose Adversarial Transformation Networks (ATNs) to efficiently generate an adversarial sample that attacks various networks by training a feed-forward neural network in a self-supervised method. Given the original input sample, ATNs modify the classifier outputs slightly to match the adversarial target. ATN works similarly to the generator model in the Generative Adversarial Training framework.
According to Baluja and Fischer et al. \cite{baluja2017adversarial}, an ATN can be parametrize as a neural network ${g_f (x) : x \rightarrow \hat{x}}$, where $f$ is the target model (either a classical model or another neural network) which outputs either a probability distribution across class labels or a sparse class label, and $\hat{x} \sim x$, but argmax $f(x)$ $\neq$ argmax $f(\hat{x})$. To find $g_f$, we minimize the following loss function :
\begin{equation}
L = \beta * L_x (g_f (\textbf{x}_i), \textbf{x}_i) + L_y (f(g_f (\textbf{x}_i)), f(\textbf{x}_i))
\end{equation}
where $L_x$ is a loss function on the input space (e.g. $L_2$ loss function), $L_y$ is the specially constructed loss function on the output space of $f$ to avoid learning the identity function, $\textbf{x}_i$ is the i-th sample in the dataset and $\beta$ is the weighing term between the two loss functions.
It is necessary to carefully select the loss function $L_y$ on the output space to successfully avoid learning the identity function. Baluja and Fischer et al. \cite{baluja2017adversarial} define the loss function $L_y$ as $L_y (\textbf{y}', \textbf{y}) = L_2 (\textbf{y}', r(\textbf{y}, t))$, where $\textbf{y} = f(x)$, $\textbf{y}' = f(g_f (x))$ and $r(\cdot)$ is a reranking function that modifies \textbf{y} such that $y_k < y_t, \forall k \neq t$. This reranking function $r(\textbf{y}, t)$ can either be the simple one hot encoding function $onehot(t)$ or be formulated to take advantage of the already present $\textbf{y}$ to encourage better reconstruction. We therefore utilize the reranking function proposed by Baluja and Fischer et al. \cite{baluja2017adversarial}, which can be formulated as:
\begin{equation}
r_\alpha (\textbf{y}, t) = norm \left( \left.
\begin{cases}
\alpha * max \: y,& \text{if } k = t\\
y_k,& \text{otherwise}
\end{cases}
\right \}_{k \in y}
\right)
\end{equation}
where $\alpha > 1$ is an additional hyper parameter which defines how much larger $y_t$ should be than the current max classification and $norm$ is a normalizing function that rescales its input to be a valid probability distribution
\subsection{Transferability Property}
Papernot et al. \cite{papernot2016transferability} propose a black-box attack by training a local substitute network, $s$, to replicate or approximate the target DNN model, $f$. The local substitute model is trained using synthetically generated samples and the output of these samples are labels from $f$. Subsequently, $s$ is used to generate adversarial samples that it misclassifies. Generating adversarial samples for $s$ is much easier, as its full knowledge/parameters are available, making it susceptible to various attacks. The key criteria to successfully generate adversarial samples of $f$ is the transferability property, where adversarial samples that misclassify $s$ will also misclassify $f$.
\subsection{Knowledge Distilation}
Knowledge distillation, first proposed by Bucila et al. \cite{bucilua2006model}, is a model compression technique where a small model, $s$, is trained to mimic a pre-trained model, $f$. This process is also known as the model distillation training where the teacher is $f$ and the student is $s$. The knowledge that is distilled from the teacher model to the student model is done by minimizing a loss function, where the objective of the student model is to imitate the distribution of the class probabilities of the teacher model. Hinton et al. \cite{hinton2015distilling} note that there are several instances where the probability distribution is skewed such that the correct class probability would have a probability close to 1 and the remaining classes would have a probability closer to 0. Hence, Hinton et al. \cite{hinton2015distilling} recommend computing the probabilities $q_i$ from the pre-normalized logits $z_i$, such that:
\begin{equation}\label{eq:scaled-softmax}
q_i = \sigma(z; T) = \frac{exp \: (z_i / T)}{\sum_j exp \: (z_j / T)}
\end{equation}
where $T$ is a temperature factor normally set to 1. Higher values of $T$ produce softer probability distributions over classes. The loss that is minimized is the model distillation loss, further explained in Section \ref{sec:TrainingMethodology}.
\section{Introduction}
\input{introduction.tex}
\section{Background \& Related Works}
\label{Background Works}
\input{literature_review.tex}
\section{Methodology}
\label{Methodology}
\subsection{Gradient Adversarial Transformation Network}
\input{gatn.tex}
\subsection{Training Methodology}
\input{train_methodology.tex}
\section{Experiments}
\label{Experiments}
\input{experiments_intro.tex}
\subsection{Experiments}
\input{experiments}
\subsection{Results}
\input{results.tex}
\section{Conclusion \& Future Work}
\label{conclusion}
\input{conclusion.tex}
\bibliographystyle{IEEEtran}
\subsection{Evaluation Methodology}
Due to the different restrictions imposed between available information depending on whether the attack is a white-box or black-box attack, we train the GATN on one of two models. We assert that we train the GATN by attacking the target neural network $f$ directly only when we perform a white-box attack on a neural network. In all other cases, whether the attack is a white-box or black-box attack, and whether the attacked model is a neural network or a classical model, we select the student model $s$ as the model which is attacked to train the GATN, and then use the GATN's predictions ($\hat{x}$) to check if the teacher model $f$ is also attacked when provided the predicted adversarial input ($\hat{x}$) as a sample.
During evaluation of the trained GATN, we compute the number of adversaries of the attacked model $f$ that have been obtained on the training set $D_{eval}$. During the evaluation, we can measure any metric under two circumstances. Provided a labeled dataset which was split, we can perform a two-fold verification of whether an adversary was found or not. First, we check that the ground truth label matches the predicted label of the classifier when provided with an unmodified input ($y = y'$ when input $x$ if provided to $f$), and then check whether this predicted label is different from the predicted label when provided with the adversarial input ($y’ \neq \hat{y}'$ when input $\hat{x}$ is provided to $f$). This ensures that we do not count an incorrect prediction from a random classifier as an attack.
Another circumstance is that we do not have any labeled samples prior to splitting the dataset. This training set is an unseen set for the attacked model $f$, therefore we consider that the dataset is “unlabeled”, and assume that the label predicted by the base classifier is the ground truth ($y = y' $ by default, when sample $x$ is provided to $f$). This is done prior to any attack by the GATN and is computed just once. We then define an adversarial sample as a sample $\hat{x}$ whose predicted class label is different than the predicted ground truth label ($y’ \neq \hat{y}'$, when sample $\hat{x}$ is provided to $f$). A drawback of this approach is that it is overly optimistic and rewards sensitive classifiers that misclassify due to very minor alterations.
In order to adhere to an unbiased evaluation, we chose the first option, and utilize the provided labels that we know from the test set to properly evaluate the adversarial inputs. In doing so, we acknowledge the necessity of a labeled test set, but as shown above, it is not strictly necessary to follow this approach.
|
{
"timestamp": "2019-03-04T02:05:29",
"yymm": "1902",
"arxiv_id": "1902.10755",
"language": "en",
"url": "https://arxiv.org/abs/1902.10755"
}
|
\section*{\textbf{Introduction}}
Given an $n$-Iwanaga-Gorenstein ring $R$, we know that if $\mathcal{GP}(R)$ denotes the class of Gorenstein projective left $R$-modules, and $\mathcal{P}(R)$ the class of projective left $R$-modules, using Auslan-der-Buchweitz approximation theory (see \cite{ABtheory,BMPS}, for instance), we can assert that every left $R$-module $M$ can be covered by an epimorphism $\varphi \colon P \twoheadrightarrow M$ with $P \in \mathcal{GP}(R)$ and whose kernel has projective dimension at most $n-1$. Moreover, the orthogonality relation $\mathsf{Ext}^i_R(\mathcal{GP}(R),\mathcal{P}(R)) = 0$ is satisfied for every $i \geq 1$. We are interested in considering the latter condition in more general contexts and only for indexes $1 \leq i \leq n$.
In the present paper, we comprise the previous properties in the concept of \emph{left $n$-cotorsion pairs}. In the general setting provided by an abelian category $\mathcal{C}$, these will be defined by two classes $\mathcal{A}$ and $\mathcal{B}$ of objects of $\mathcal{C}$ such that: (1) $\mathcal{A}$ is closed under direct summands, (2) $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ for every $1 \leq i \leq n$, and (3) for every object $C \in \mathcal{C}$ there exists an exact sequence
\[
0 \to B_{n-1} \to B_{n-2} \to \cdots \to B_1 \to B_0 \to A \to C \to 0
\]
with $A \in \mathcal{A}$ and $B_k \in \mathcal{B}$ for every $0 \leq k \leq n-1$.
This concept and its dual, that we shall call \emph{right $n$-cotorsion pair}, will represent an approach to what, roughly speaking, we may call \emph{higher cotorsion}: that is, the study of the possible outcomes of considering two classes of objects of $\mathcal{C}$ which are complete with respect to the orthogonality relation defined by the vanishing of the bifunctor $\mathsf{Ext}^i_{\mathcal{C}}(-,-)$ for ``higher'' indexes $i > 1$. The case $i = 1$, on the other hand, is already covered by the theory of complete cotorsion pairs, widely considered in fields such as relative homological algebra or representation theory.
The present paper is organised as follows. In the first section we give some preliminaries on homological dimensions, orthogonality and approximations. The next section is devoted to present the concept of left and right $n$-cotorsion pairs and its relation with complete cotorsion pairs. In Proposition~\ref{prop:cotorsion_vs_ncotorsion} and Theorem~\ref{theo:left-n-cotorsion}, we give necessary and sufficient conditions for an $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$ to form a complete cotorsion pair $(\mathcal{A},\mathcal{B}^\wedge_{n-1})$, where $\mathcal{B}^\wedge_{n-1}$ will denote the class of objects of $\mathcal{C}$ with $\mathcal{B}$-resolution dimension $\leq n-1$.
In the third section we study how to construct covers and envelopes from $n$-cotorsion pairs $(\mathcal{A,B})$ in $\mathcal{C}$. We also define new type of approximations, that we call \emph{special $(\mathcal{A},k,\mathcal{B})$-precovers and preenvelopes}. We shall consider the class of objects in $\mathcal{C}$ having such approximations, and analyse some conditions under which this class is closed under extensions (see Corollary~\ref{Coro-k-prec}). Moreover, given an $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$, we give in Corollaries~\ref{Aperp y Bvee} and \ref{corUMP2} some necessary and sufficient conditions to obtain precovers and envelopes constructed from $\mathcal{A}$ and $\mathcal{B}$ that satisfy the unique mapping property. At this point, we shall make some comparisons with other approaches to higher cotorsion, like for instance the remarkable study \cite{CriveiTorrecillas} by S. Crivei and B. Torrecillas, where the authors establish several equivalent conditions for a class $\mathcal{A} \subseteq \mathcal{C}$ under which every object in $\mathcal{C}$ has an epic $\mathcal{A}$-envelope and a monic $\mathcal{A}$-cover. These conditions have to do with the concept of $(m,n)$-cotorsion pairs, $n$-special precovers and $m$-special preenvelopes.
Section 4 is devoted to explain what does it mean for a left or right $n$-cotorsion pair to be \emph{hereditary}. We shall see in Proposition~\ref{equiv hered y n-cot} that a (left and right) hereditary $n$-cotorsion pair coincides with the usual concept of hereditary complete cotorsion pair. Thus, we propose a notion for being hereditary that is not trivial for either left or right higher cotorsion pairs.
In Section 5 we present applications and examples of the theory of $n$-cotorsion pairs, developed in Sections \ref{sec:approximations} and \ref{sec:hereditary}, in the context of relative Gorenstein homological algebra and cluster-tilting subcategories. We shall see in Example~\ref{ex:GProj_ncot} and Proposition~\ref{prop:ncot_spli_silp} that the classes $\mathcal{GP}(R)$ and $\mathcal{P}(R)$ of Gorenstein projective and projective $R$-modules will form a left $n$-cotorsion pair provided that $R$ is an $n$-Iwanaga-Gorenstein ring or a Gorenstein ring (in the sense of \cite{BR}). Moreover, we give characterisations of Gorenstein rings in terms of the pair $(\mathcal{GP}(R),\mathcal{P}(R))$ and its dual $(\mathcal{I}(R),\mathcal{GI}(R))$, formed by the classes of injective and Gorenstein injective $R$-modules. As an application in this setting, we shall prove for example that every module over $2$-Iwanaga-Gorenstein ring has a Gorenstein injective cover with the unique mapping property, and that the existence of Gorenstein projective envelopes implies the existence of such envelopes with the unique mapping property (see Corollaries~\ref{coro:GI_unique_mapping} and \ref{coro:GP_unique_mapping}). An analogous study is done for the notions of Ding projective and Ding injective modules over a ring, but in addition we find some finiteness conditions for the global Ding projective and Ding injective dimensions of a ring. Later, we study some consequences of having the classes $\mathcal{F}(R)$ and $\mathcal{GF}(R)$ of flat and Gorenstein flat $R$-modules as halves of left and right $n$-cotorsion pairs. This will lead for instance to some characterisations of left perfect rings with null global Gorenstein flat dimension (Proposition~\ref{prop:perfect_global_flat}), and of left perfect rings that are also quasi-Frobenius (Proposition~\ref{prop:perfect_QF}). Another interesting fact about pairs of the form $(\mathcal{GF}(R),\mathcal{F}(R))$ is its relation with the pair $(\mathcal{I}(R),\mathcal{GI}(R))$, mentioned before, in terms of the Pontryagin duality functor $M \mapsto M^+ := \mathsf{Hom}_{\mathbb{Z}}(M,\mathbb{Q / Z})$ (see Theorems \ref{theo:GFGI_Pontryagin} and \ref{theo:GFGI_Pontryagin}). Besides its applications in Gorenstein homological algebra, we also study the interplay between the $n$-cotorsion pairs and cluster tilting subcategories in the sense of Iyama \cite{IyamaCluster}. For an abelian category $\mathcal{C}$ with enough projective and injective objects, we shall give a one-to-one correspondence between $n$-cotorsion pairs in $\mathcal{C}$ of the form $(\mathcal{D,D})$ and $(n+1)$-cluster tilting subcategories of $\mathcal{C}$ (see Proposition~\ref{nclustA=B y ff} and Theorem~\ref{ncot y ct}).
In the last section we show how to induce certain left and right $n$-cotorsion pairs of chain complexes from a given $n$-cotorsion pair $(\mathcal{A,B})$ in an abelian category $\mathcal{C}$. These induced pairs will involve the classes $\widetilde{\mathcal{A}}$ of $\mathcal{A}$-complexes, $\widetilde{\mathcal{B}}$ of $\mathcal{B}$-complexes, and ${\rm dg}\widetilde{\mathcal{A}}$ and ${\rm dg}\widetilde{\mathcal{B}}$ of differential graded complexes of objects in $\mathcal{A}$ and $\mathcal{B}$. The results presented in this section are motivated by the works of J. Gillespie \cite{GillespieFlat}, and X. Yang and N. Ding \cite{YangDingQuestion}, where they show that every complete and hereditary cotorsion pair $(\mathcal{A,B})$ gives rise to two complete cotorsion pairs of complexes of the form $(\widetilde{\mathcal{A}},{\rm dg}\widetilde{\mathcal{B}})$ and $({\rm dg}\widetilde{\mathcal{A}},\widetilde{\mathcal{B}})$. We also prove that if any of these pairs is a left or a right $n$-cotorsion pair of complexes, then so is $(\mathcal{A,B})$ in $\mathcal{C}$, provided that $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ for every $1 \leq i \leq n+1$, extending thus an important result in \cite{YangDingQuestion}.
\section{\textbf{Preliminaries}}\label{sec:preliminaries}
Let us recall some categorical and homological preliminaries that will be used in the sequel.
Throughout this paper, $\mathcal{C}$ will denote an abelian category (not necessarily with enough projective or injective objects). The main example of such category considered here will be the category $\mathsf{Mod}(R)$ of left $R$-modules and $R$-homomorphisms, where $R$ is an associative ring with identity. By a module $M$ we shall mean a left $R$-module unless otherwise specified. Right $R$-modules will be regarded as left modules over the opposite ring $R^{\rm op}$. We shall also consider the category $\mathsf{Ch}(R)$ of complexes of (left) $R$-modules, and the category $\mathsf{mod}(\Lambda)$ of finitely generated modules over an Artin algebra $\Lambda$.
Every subcategory of $\mathcal{C}$ is assumed to be full, and so any class $\mathcal{A} \subseteq \mathcal{C}$ of objects of $\mathcal{C}$ may be regarded as a (full) subcategory of $\mathcal{C}$. If two objects $C$ and $D$ in $\mathcal{C}$ are isomorphic, we write $C \simeq D$. The notation $F \cong G$ will be reserved to denote the existence of a natural isomorphism between two functors $F$ and $G$. Monomorphisms and epimorphisms in $\mathcal{C}$ may sometimes be denoted using arrows $\rightarrowtail$ and $\twoheadrightarrow$, respectively.
The results presented in this paper have their corresponding dual version, which sometimes will be omitted for simplicity.
\subsection*{Resolution and coresolution dimension}
Let $\mathcal{B} \subseteq \mathcal{C}$ be a class of objects of $\mathcal{C}$. Given an object $C \in \mathcal{C}$ and a nonnegative integer $m \geq 0$, a \emph{$\mathcal{B}$-resolution of $C$ of length $m$} is an exact sequence
\[
0 \to B_m \to B_{m-1} \to \cdots \to B_1 \to B_0 \to C \to 0
\]
in $\mathcal{C},$ where $B_k \in \mathcal{B}$ for every integer $0 \leq k \leq m$. The \emph{resolution dimension of $C$ with respect to $\mathcal{B}$} (or the \emph{$\mathcal{B}$-resolution dimension} of $C$), denoted ${\rm resdim}_{\mathcal{B}}(C)$, is defined as the smallest nonnegative integer $m \geq 0$ such that $C$ has a $\mathcal{B}$-resolution of length $m$. If such $m$ does not exist, we set ${\rm resdim}_{\mathcal{B}}(C) := \infty$. Dually, we have the concepts of \emph{$\mathcal{B}$-coresolutions of $C$ of length $m$} and of \emph{coresolution dimension of $C$ with respect to $\mathcal{B}$}, denoted by ${\rm coresdim}_{\mathcal{B}}(C)$.
With respect to these two homological dimensions, we shall frequently consider the following two classes of objects in $\mathcal{C}$:
\begin{align*}
\mathcal{B}^\wedge_m & := \{ C \in \mathcal{C} \mbox{ : } {\rm resdim}_{\mathcal{B}}(C) \leq m \}, \\
\mathcal{B}^\vee_m & := \{ C \in \mathcal{C} \mbox{ : } {\rm coresdim}_{\mathcal{B}}(C) \leq m \}.
\end{align*}
\subsection*{Orthogonality with respect to extension functors}
In any abelian category $\mathcal{C}$, we can define extension bifunctors $\mathsf{Ext}^i_{\mathcal{C}}(-,-)$ in the sense of Yoneda. See for instance \cite{Sieg} for a detailed treatise on this matter. Recall that $\mathsf{Ext}^1_{\mathcal{C}}(X,Y)$ is defined as the abelian group formed by classes of short exact sequences $0 \to Y \to Z \to X \to 0$ under certain equivalence relation. In case we work in the category $\mathsf{Ch}(\mathcal{C})$ of complexes in $\mathcal{C}$, we shall write $\mathsf{Ext}^i_{\mathsf{Ch}(\mathcal{C})}(-,-)$ as $\mathsf{Ext}^i_{\mathsf{Ch}}(-,-)$ for simplicity.
Given two classes of objects $\mathcal{A,B} \subseteq \mathcal{C}$ and an integer $i \geq 1$, the notation $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ will mean that $\mathsf{Ext}^i_{\mathcal{C}}(A,B) = 0$ for every $A \in \mathcal{A}$ and $B \in \mathcal{B}$. In the case where $\mathcal{A} = \{ M \}$ or $\mathcal{B} = \{ N \}$, we shall write $\mathsf{Ext}^i_{\mathcal{C}}(M,\mathcal{B}) = 0$ and $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A},N) = 0$, respectively.
Recall that the \emph{right $i$-th orthogonal complement} of $\mathcal{A}$ is defined by
\[
\mathcal{A}^{\perp_i} := \{ N \in \mathcal{C} \mbox{ : } \mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A},N) = 0 \},
\]
and the \emph{total right orthogonal complement} of $\mathcal{A}$ by
\[
\mathcal{A}^\perp := \bigcap_{i \geq 1} \mathcal{A}^{\perp_i}.
\]
Dually, we have the \emph{$i$-th and total left orthogonal complements} ${}^{\perp_i}\mathcal{B}$ and ${}^{\perp}\mathcal{B},$ respectively.
\subsection*{Approximations}
Let $\mathcal{A}$ be a class of objects of $\mathcal{C}$. A morphism $f \colon A \to C$ is said to be an \emph{$\mathcal{A}$-precover} (or a \emph{right $\mathcal{A}$-approximation}) \emph{of $C$} if $A \in \mathcal{A}$ and if for every morphism $f' \colon A' \to C$ with $A' \in \mathcal{A},$ there exists a morphism $h \colon A' \to A$ such that $f' = f \circ h$. If in addition, in the case $A' = A$ and $f' = f$, the previous equality can only be completed by automorphisms $h$ of $A$, then $f$ is called an \emph{$\mathcal{A}$-cover} (or a \emph{minimal right $\mathcal{A}$-approximation}). Furthermore, an $\mathcal{A}$-precover $f \colon A \to C$ of $C$ is \emph{special} if ${\rm CoKer}(f) = 0$ and ${\rm Ker}(f) \in \mathcal{A}^{\perp_1}$. The class $\mathcal{A}$ is said to be \emph{precovering} if every object of $\mathcal{C}$ has an $\mathcal{A}$-precover. Similarly, we have the concepts of \emph{covering} and \emph{special precovering} classes in $\mathcal{C}$.
Dually, we have the notions of \emph{$\mathcal{A}$-preenvelopes} (\emph{left $\mathcal{A}$-approximations}), \emph{$\mathcal{A}$-envelopes} (\emph{minimal left $\mathcal{A}$-approximations}) and \emph{special $\mathcal{A}$-preenvelopes} in $\mathcal{C}$, along with the corresponding notions of \emph{preenveloping}, \emph{enveloping} and \emph{special preenveloping} classes.
With these preliminaries in hand, we are ready to begin our approach to higher cotorsion in abelian categories.
\section{\textbf{{\textit n}-Cotorsion pairs}}\label{sec:ncotorsion}
The notion of cotorsion pair was first introduced by L. Salce in \cite{Salce}. It is the analog of a torsion pair where the bifunctor $\mathsf{Hom}_{\mathcal{C}}(-,-)$ is replaced by $\mathsf{Ext}^1_{\mathcal{C}}(-,-)$. Roughly speaking, two classes $\mathcal{A}$ and $\mathcal{B}$ of objects in an abelian category $\mathcal{C}$ form a cotorsion pair $(\mathcal{A,B})$ if they are complete with respect to the orthogonality relation defined by the vanishing of the functor $\mathsf{Ext}^1_{\mathcal{C}}(-,-)$. Specifically, and for the purpose of this paper, it comes handy to recall this concept as follows.
\begin{definition}\label{def:cotorsion_pair}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects in $\mathcal{C}$. We say that $\mathcal{A}$ and $\mathcal{B}$ form a \textbf{complete left cotorsion pair} $(\mathcal{A,B})$ in $\mathcal{C}$ if $\mathcal{A} = {}^{\perp_1}\mathcal{B}$ and if every object of $\mathcal{C}$ has an epic $\mathcal{A}$-precover with kernel in $\mathcal{B}$. Dually, we have the concept of \textbf{complete right cotorsion pair} in $\mathcal{C}$.
\end{definition}
Note that $(\mathcal{A,B})$ is a complete cotorsion pair in $\mathcal{C}$ if, and only if, it is both a complete left and right cotorsion pair in $\mathcal{C}$.
Motivated by the properties of Gorenstein projective and Gorenstein injective modules over Iwanaga-Gorenstein rings mentioned in the introduction, below we present a ``higher'' version of cotorsion pairs, which will cover complete left and right cotorsion pairs in the sense of Definition~\ref{def:cotorsion_pair}, as particular cases. By ``higher'' we mean that orthogonality with respect to $\mathsf{Ext}^i_{\mathcal{C}}(-,-)$ will be considered for indices $i \geq 1$. We shall also see how some well known properties of cotorsion pairs are transferred to the higher context resulting from Definition~\ref{def:ncotorsion} below.
Throughout, $n > 0$ will be a positive integer.
\begin{definition}\label{def:ncotorsion}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects in $\mathcal{C}$. We say that $(\mathcal{A,B})$ is a \textbf{left $\bm{n}$-cotorsion pair} in $\mathcal{C}$ if the following conditions are satisfied:
\begin{enumerate}
\item $\mathcal{A}$ is closed under direct summands.
\item $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ for every $1 \leq i \leq n$.
\item For every object $C \in \mathcal{C}$, there exists a short exact sequence
\[
0 \to K \to A \to C \to 0
\]
where $A \in \mathcal{A}$ and $K \in \mathcal{B}^{\wedge}_{n-1}$.
\end{enumerate}
Dually, we say that $(\mathcal{A,B})$ is a \textbf{right $\bm{n}$-cotorsion pair} in $\mathcal{C}$ if condition (2) above is satisfied, with $\mathcal{B}$ closed under direct summands, and if every object of $C$ can be embedded into an object of $\mathcal{B}$ with cokernel in $\mathcal{A}^\vee_{n-1}$. Finally, $\mathcal{A}$ and $\mathcal{B}$ form a \textbf{$\bm{n}$-cotorsion pair} $(\mathcal{A,B})$ in $\mathcal{C}$ if $(\mathcal{A,B})$ is both a left and right $n$-cotorsion pair in $\mathcal{C}$.
\end{definition}
\begin{example}\label{ex:trivial}
In what follows, let us denote by $\mathcal{P}(\mathcal{C})$ and $\mathcal{I}(\mathcal{C})$ the classes of projective and injective objects of $\mathcal{C}$, respectively. It is clear that $\mathcal{C}$ has enough projectives (resp., enough injectives) if, and only if, $(\mathcal{P}(\mathcal{C}),\mathcal{C})$ (resp., $(\mathcal{C},\mathcal{I}(\mathcal{C}))$) is an $n$-cotorsion pair in $\mathcal{C}$ for every $n \geq 1$.
In what follows, we shall call $(\mathcal{P}(\mathcal{C}),\mathcal{C})$ and $(\mathcal{C},\mathcal{I}(\mathcal{C}))$ the \textbf{trivial $\bm{n}$-cotorsion pairs} in the case where $\mathcal{C}$ has enough projectives and injectives. Some nontrivial examples will be presented later on in Section~\ref{sec:applications}.
\end{example}
\subsection*{\textbf{Relations between cotorsion and higher cotorsion}}
It is clear that left (resp., right) $1$-cotorsion pairs coincide with complete left (resp., right) cotorsion pairs in $\mathcal{C}$. However, we can say more on how (left and right) $n$-cotorsion pairs interact with the concept of complete cotorsion pairs. Specifically, we shall study under which conditions a complete left cotorsion pair induces a left $n$-cotorsion pair. Conversely, we shall prove that every left $n$-cotorsion pair induces a complete left cotorsion pair.
Let us begin establishing certain conditions under which two classes of objects $\mathcal{A}$ and $\mathcal{B}$ form a left $n$-cotorsion pair in $\mathcal{C}$. The following lemma can be deduced from a standard dimension shifting argument.
\begin{lemma} \label{lema1}
For any class $\mathcal{B}$ of objects of $\mathcal{C}$, the following containment holds:
\[
\bigcap^n_{i = 1} {}^{\perp_i}\mathcal{B} \subseteq {}^{\perp_1}\mathcal{B}^\wedge_{n-1}.
\]
\end{lemma}
\begin{proposition}\label{prop:cotorsion_vs_ncotorsion}
Let $\mathcal{C}$ be an abelian category with enough injectives, and let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects of $\mathcal{C}$ such that $\mathcal{I}(\mathcal{C}) \subseteq \mathcal{B}$. Then, $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{A}, \mathcal{B}^\wedge_{n-1}) = 0$ if, and only if, $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ for every $1 \leq i \leq n$. In particular, $(\mathcal{A}, \mathcal{B}^\wedge_{n-1})$ is a complete left cotorsion pair in $\mathcal{C}$ if, and only if, $(\mathcal{A,B})$ is a left $n$-cotorsion pair in $\mathcal{C}$.
\end{proposition}
\begin{proof}
The ``if'' part follows from Lemma~\ref{lema1}. In order to show the ``only if'' statement, note that since $\mathcal{C}$ has enough injectives and $\mathcal{I}(\mathcal{C}) \subseteq \mathcal{B},$ for every injective $(i-1)$-cosyzygy $K$ of $B \in \mathcal{B}$ we have that ${\rm resdim}_{\mathcal{B}}(K) \leq i - 1 \leq n - 1$ with $1 \leq i \leq n$. Then, $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{A}, K) = 0$ since $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{A}, \mathcal{B}^\wedge_{n-1}) = 0$. Therefore, we have that $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A},B) \cong \mathsf{Ext}^1_{\mathcal{C}}(\mathcal{A},K) = 0$ for every $B \in \mathcal{B}$ and $1 \leq i \leq n$.
\end{proof}
In the ``if'' part of the previous proposition, we actually do not need that $\mathcal{C}$ has enough injectives or $\mathcal{I}(\mathcal{C}) \subseteq \mathcal{B}$ either. As a matter of fact, we only require a complete left cotorsion pair of the form $(\mathcal{A},\mathcal{B}^\wedge_{n-1})$. Before showing this in Theorem~\ref{theo:left-n-cotorsion}, let us state and prove the following properties derived from the orthogonality relations $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ with $1\leq i\leq n.$
\begin{proposition}\label{prop8}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects of $\mathcal{C}$ satisfying $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ for every $1 \leq i \leq n$. If $Y \in \mathcal{B}^\wedge_k$ with $0 \leq k \leq n-1$, then $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A},Y) = 0$ for every $1 \leq i \leq n - k$. In particular, $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{A},\mathcal{B}^\wedge_{n-1}) = 0$.
\end{proposition}
\begin{proof}
Note that the case $n = 1$ is clear. Thus, we may assume that $n \geq 2$. We use induction on $k$. The case $k = 0$ is also clear, so we may take $1 \leq k \leq n-1$ for $n \geq 2$.
Let $A \in \mathcal{A}$ and $Y \in \mathcal{B}^\wedge_k$. First, for the case $k = 1$, we have that ${\rm resdim}_{\mathcal{B}}(Y) \leq 1$, and thus there is an exact sequence
\[
0 \to B_1 \to B_0 \to Y \to 0
\]
with $B_0, B_1 \in \mathcal{B}$. Then, we obtain an exact sequence
\[
\mathsf{Ext}^i_{\mathcal{C}}(A,B_0) \to \mathsf{Ext}^i_{\mathcal{C}}(A,Y) \to \mathsf{Ext}^{i+1}_{\mathcal{C}}(A,B_1)
\]
of abelian groups with $\mathsf{Ext}^i_{\mathcal{C}}(A,B_0) = 0$ and $\mathsf{Ext}^{i+1}_{\mathcal{C}}(A,B_1) = 0$ if $1 \leq i \leq n-1$. Hence, $\mathsf{Ext}^i_{\mathcal{C}}(A,Y) = 0$ for every $A \in \mathcal{A}$, $Y \in \mathcal{B}^\wedge_1$ and $1 \leq i \leq n-1$.
Now for the successor case, suppose that for every object $Y' \in \mathcal{B}^\wedge_k$ with $1 \leq k < n-1$ (the case $k = n - 1$ follows from Lemma~\ref{lema1}), we have that $\mathsf{Ext}^i_{\mathcal{C}}(A,Y') = 0$ for every $1 \leq i \leq n - k$. Now let $Y \in \mathcal{C}$ be an object with ${\rm resdim}_{\mathcal{B}}(Y) \leq k + 1$, so that there is an exact sequence
\[
0 \to Y' \to B \to Y \to 0
\]
with $B \in \mathcal{B}$ and ${\rm resdim}_{\mathcal{B}}(Y') \leq k$. Consider an integer $1 \leq i \leq n - (k + 1)$. Then, we have an exact sequence
\[
\mathsf{Ext}^i_{\mathcal{C}}(A,B) \to \mathsf{Ext}^i_{\mathcal{C}}(A,Y) \to \mathsf{Ext}^{i+1}_{\mathcal{C}}(A,Y')
\]
of abelian groups where $\mathsf{Ext}^i_{\mathcal{C}}(A, B) = 0.$ Since $1 \leq i \leq n - (k + 1)$ and $\mathsf{Ext}^{i+1}_{\mathcal{C}}(A, Y') = 0$ by the induction hypothesis, we get that $\mathsf{Ext}^i_{\mathcal{C}}(A,Y) = 0$ for every $1 \leq i \leq n - (k + 1)$.
\end{proof}
\begin{theorem}\label{theo:left-n-cotorsion}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects in $\mathcal{C}.$ Then, the following two conditions are equivalent:
\begin{itemize}
\item[(a)] $(\mathcal{A,B})$ is a left $n$-cotorsion pair in $\mathcal{C}$.
\item[(b)] $\mathcal{A} = \bigcap_{i=1}^n {}^{\perp_i}\mathcal{B}$ and for any $C \in \mathcal{C}$ there is a short exact sequence
\[
0 \to K \to A \to C \to 0,
\]
with $A \in \mathcal{A}$ and $K \in \mathcal{B}^\wedge_{n-1}$.
\end{itemize}
Moreover, if one of the above conditions holds true, then $(\mathcal{A},\mathcal{B}^\wedge_{n-1})$ is a complete left cotorsion pair in $\mathcal{C}$.
\end{theorem}
\begin{proof} Note that the implication (b) $\Rightarrow$ (a) is trivial. We prove that (a) implies (b). So let us assume that $(\mathcal{A,B})$ is a left $n$-cotorsion pair in $\mathcal{C}$. Then, by Lemma~\ref{lema1}, we get the containments
\[
\mathcal{A} \subseteq \bigcap_{i = 1}^n\,{}^{\perp_i}\mathcal{B} \subseteq {}^{\perp_1}(\mathcal{B}^\wedge_{n-1}).
\]
Thus, we only need to prove the remaining containment $\mathcal{A} \supseteq {}^{\perp_1}(\mathcal{B}^\wedge_{n-1})$. It suffices to note that for every $X \in {}^{\perp_1}(\mathcal{B}^\wedge_{n-1}),$ there exists a split epimorphism $A \twoheadrightarrow X$ with kernel in $\mathcal{B}^\wedge_{n-1}$.
\end{proof}
Normally, if we are given a cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$, a natural question is whether this pair is complete or hereditary, in order to construct special $\mathcal{A}$-precovers and special $\mathcal{B}$-preenvelopes. Now that we have explored the interplay between cotorsion and higher cotorsion, we shall study in the next section the relation between left and right $n$-cotorsion pairs, and left and right approximations by $\mathcal{A}$ and $\mathcal{B}$. In Section~\ref{sec:hereditary}, on the other hand, we shall deal with the hereditary aspect of $n$-cotorsion pairs $(\mathcal{A,B})$ for which $\mathcal{A}$ is resolving or $\mathcal{B}$ is coresolving.
\section{\textbf{Covers and envelopes from {\textit n}-cotorsion pairs}}\label{sec:approximations}
Approximations has been considered before in the study of higher cotorsion. For instance, in \cite{CriveiTorrecillas} Crivei and Torrecillas presented the concept of $n$-special $\mathcal{A}$-precovers and $m$-special $\mathcal{B}$-preenvelopes (epic $\mathcal{A}$-precovers and monic $\mathcal{B}$-preenvelopes with kernel in $\mathcal{A}^{\perp_n}$ and cokernel in ${}^{\perp_m}\mathcal{B}$, respectively), and established several conditions under which it is possible to obtain such approximations from an $(m,n)$-cotorsion pair $(\mathcal{A,B})$ (that is, $\mathcal{A} = {}^{\perp_m}\mathcal{B}$ and $\mathcal{B} = \mathcal{A}^{\perp_n}$). See \cite[Proposition 3.14 and Theorem 3.15]{CriveiTorrecillas}.
In this section, we study precovers and preenvelopes coming from an $n$-cotorsion pair $(\mathcal{A,B})$ in an abelian category $\mathcal{C}$. We also define a new family of approximations which we call $(\mathcal{A},k,\mathcal{B})$-precovers and $(\mathcal{A},k,\mathcal{B})$-preenvelopes, as analogs of the special precovers and special preenvelopes coming from a complete cotorsion pair. Among the properties of these new concepts, we prove that the class of objects having a $(\mathcal{A},k,\mathcal{B})$-precover is closed under extensions if we put certain orthogonality condition on $\mathcal{B}$. Later, we shall study some other conditions under which left and right $n$-cotorsion pairs are sources of precovers and preenvelopes with the unique mapping property.
\subsection*{\textbf{Special $(\mathcal{A},k,\mathcal{B})$-precovers and $(\mathcal{A},k,\mathcal{B})$-preenvelopes}}
One consequence of Theorem~\ref{theo:left-n-cotorsion} is that left and right $n$-cotorsion pairs are always sources of precovers and preenvelopes, as stated in the following result.
\begin{proposition}\label{A-precub,B-preenv}
If $(\mathcal{A,B})$ is a left $n$-cotorsion pair in $\mathcal{C}$, then $\mathcal{A}$ is a special precovering class.
\end{proposition}
\begin{proof}
Let $(\mathcal{A,B})$ be a left $n$-cotorsion pair in $\mathcal{C}.$ In particular, for any $C\in\mathcal{C},$ there is an exact sequence
\[
0 \to K \to A \to C \to 0,
\]
where $K \in \mathcal{B}^\wedge_{n-1}$ and $A \in \mathcal{A}$. Moreover, by Theorem~\ref{theo:left-n-cotorsion}, we get that $K \in \mathcal{B}^\wedge_{n-1} \subseteq \mathcal{A}^{\perp_1}$ and thus $A \to C$ is a special $\mathcal{A}$-precover of $C$.
\end{proof}
However, special precovers and preenvelopes are not the only type of approximations coming from left and right $n$-cotorsion pairs. Under certain conditions, we can find more information about the approximations resulting from Proposition~\ref{A-precub,B-preenv}. Following the spirit of \cite{CriveiTorrecillas} concerning the relation between $(m,n)$-cotorsion pairs, $n$-special precovers and $m$-special preenvelopes, we propose the following family of approximations and study their relation with left and right $n$-cotorsion pairs.
\begin{definition}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects of $\mathcal{C}$, and $k $ be a positive integer. Given an object $C \in \mathcal{C}$, we say that an $\mathcal{A}$-precover $f \colon A \to C$ of $C$ is a \textbf{special $\bm{(\mathcal{A},k,\mathcal{B})}$-precover} if $f$ is epic and ${\rm Ker}(f) \in \mathcal{B}_{k-1}^{\wedge}$. The notion of \textbf{special $\bm{(\mathcal{A},k,\mathcal{B})}$-preenvelopes} is defined dually.
\end{definition}
\begin{remark} \
\begin{enumerate}
\item Let $\mathcal{A}$ be a class of objects of $\mathcal{C}$ and $C \in \mathcal{C}$. Note that a morphism $f \colon A \to C$ with $A \in \mathcal{A}$ is a special $\mathcal{A}$-precover if, and only if, it is a special $(\mathcal{A},1,\mathcal{A}^{\perp_1})$-precover.
\item Any $C\in\mathcal{C}$ admits an $(\mathcal{A},n,\mathcal{B})$-precover if $(\mathcal{A,B})$ is a left $n$-cotorsion pair in $\mathcal{C}$.
\end{enumerate}
\end{remark}
The following is clear by \cite[Theorem 2.10]{Holm}.
\begin{example}\label{ex:special_AkB-precover}
Given a class of modules $\mathcal{Y} \subseteq \mathsf{Mod}(R)$, a chain complex $X = (X_m)_{m \in \mathbb{Z}}$ is called \emph{$\mathsf{Hom}_R(-,\mathcal{Y})$-acyclic} if $\mathsf{Hom}_R(X,Y) := (\mathsf{Hom}_R(X_m,Y))_{m \in \mathbb{Z}}$ is an exact complex of abelian groups for every $Y \in \mathcal{Y}$.
Recall that $\mathcal{GP}(R)$ denotes the class of \emph{Gorenstein projective} $R$-modules, that is, modules $M \in \mathsf{Mod}(R)$ such that $M \simeq Z_0(P)$ for some exact and $\mathsf{Hom}_R(-,\mathcal{P}(R))$-acyclic complex $P$ of projective modules. \emph{Gorenstein injective} modules are defined dually, that is, as cycles of exact and $\mathsf{Hom}_R(\mathcal{I}(R),-)$-acyclic complexes of injective modules. For the class of Gorenstein injective $R$-modules, we shall write $\mathcal{GI}(R)$.
Let us recall also that the \emph{Gorenstein projective dimension} of an $R$-module $M$, which we denote by ${\rm Gpd}(M)$, is defined as the $\mathcal{GP}(R)$-resolution dimension of $M$, that is,
\[
{\rm Gpd}(M) := {\rm resdim}_{\mathcal{GP}(R)}(M).
\]
The \emph{Gorenstein injective dimension of $M$}, denoted ${\rm Gid}(M)$, is defined similarly. It is known from \cite[Theorem 2.10]{Holm} that every $R$-module $M$ with finite Gorenstein projective dimension, say ${\rm Gpd}(M) = m < \infty$, has a Gorenstein projective special precover whose kernel has projective dimension at most $m-1$, that is, $M$ has a special $(\mathcal{GP}(R),m,\mathcal{P}(R))$-precover.
\end{example}
In \cite[Theorem 3.1]{Akinci}, Akinci and Alizade proved that for every hereditary cotorsion pair $(\mathcal{A,B})$ in $\mathsf{Mod}(R)$, the class of objects having a special $\mathcal{A}$-precover is closed under extensions. In what follows, we generalise this result for special $(\mathcal{A},k,\mathcal{B})$-precovers. Let us denote by ${\mathrm{Prec}}^k(\mathcal{A},\mathcal{B})$ the class of all $C\in\mathcal{C}$ admitting a special $(\mathcal{A},k,\mathcal{B})$-precover.
\begin{theorem}\label{Teo-k-prec}
Let $n$ be a positive integer and $1\leq k\leq \max(1,n-1)$, and let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects of $\mathcal{C}$ such that $\mathsf{Ext}^i_\mathcal{C}(\mathcal{A},\mathcal{B}) = 0$ for every $1 \leq i \leq n$, or $\mathsf{Ext}^2_{\mathcal{C}}(\mathcal{A,B}) = 0$ if $n = 1$. If $\mathcal{A}$ and $\mathcal{B}^\wedge_{k-1}$ are closed under extensions, then so is ${\mathrm{Prec}}^k(\mathcal{A},\mathcal{B})$.
\end{theorem}
\begin{proof}
We prove this result by adapting the arguments given in \cite[Theorem 3.1]{Akinci} to our approach of higher cotorsion. Let
\[
0 \to X \xrightarrow{f} Y \xrightarrow{g} Z \to 0
\]
be a short exact sequence in $\mathcal{C}$ such that $X, Z \in {\mathrm{Prec}}^k(\mathcal{A},\mathcal{B})$. First, consider a special $(\mathcal{A},k,\mathcal{B})$-precover of $Z$, say a short exact sequence
\[
0 \to K^Z \xrightarrow{\beta^Z} A^Z \xrightarrow{\alpha^Z} Z \to 0,
\]
with $K^Z \in \mathcal{B}^\wedge_{k-1}$ and $A^Z \in \mathcal{A}$. Taking the pullback of $Y \to Z \leftarrow A^Z$ yields the following commutative diagram with exact rows and columns:
\begin{equation}\label{fig3}
\parbox{1.75in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em,text height=1.25ex, text depth=0.25ex]
{
{} \& K^Z \& K^Z \\
X \& E \& A^Z \\
X \& Y \& Z \\
};
\path[>->]
(m-1-2) edge node[left] {\footnotesize$\tilde{\beta}^Z$} (m-2-2) (m-1-3) edge node[right] {\footnotesize$\beta^Z$} (m-2-3)
(m-2-1) edge node[above] {\footnotesize$\overline{f}$} (m-2-2) (m-3-1) edge node[below] {\footnotesize$f$} (m-3-2)
;
\path[->>]
(m-2-2) edge node[left] {\footnotesize$\tilde{\alpha}^Z$} (m-3-2) (m-2-3) edge node[right] {\footnotesize$\alpha^Z$} (m-3-3)
(m-2-2) edge node[above] {\footnotesize$\overline{g}$} (m-2-3) (m-3-2) edge node[below] {\footnotesize$g$} (m-3-3)
;
\path[->]
(m-2-2)-- node[pos=0.5] {\footnotesize$\mbox{\bf pb}$} (m-3-3)
;
\path[-,font=\scriptsize]
(m-1-2) edge [double, thick, double distance=2pt] (m-1-3)
(m-2-1) edge [double, thick, double distance=2pt] (m-3-1)
;
\end{tikzpicture}
}
\end{equation}
Now let us consider a special $(\mathcal{A},k,\mathcal{B})$-precover of $X$, say
\[
0 \to K^X \xrightarrow{\beta^X} A^X \xrightarrow{\alpha^X} X \to 0,
\]
with $A^X \in \mathcal{A}$ and $K^X \in \mathcal{B}^\wedge_{k-1}$. We obtain the following exact sequence of abelian groups:
\[
\mathsf{Ext}^1_{\mathcal{C}}(A^Z,A^X) \xrightarrow{\mathsf{Ext}^1_{\mathcal{C}}(A^Z,\alpha^X)} \mathsf{Ext}^1_{\mathcal{C}}(A^Z,X) \to \mathsf{Ext}^2_{\mathcal{C}}(A^Z,K^X).
\]
The morphism $\mathsf{Ext}^1_{\mathcal{C}}(A^Z,\alpha^X)$ is epic since $\mathsf{Ext}^2_{\mathcal{C}}(A^Z,K^X) = 0$. The latter can be shown as follows: for the case $n = 1$, we use that $\mathsf{Ext}^2_\mathcal{C}(\mathcal{A},\mathcal{B}) = 0$. If $n \geq 2$, on the other hand, then $k \leq n - 1$ and so $2 \leq n - k + 1$. Thus, by Proposition~\ref{prop8}, we get that $\mathsf{Ext}^i_\mathcal{C}(\mathcal{A},\mathcal{B}^\wedge_{k-1}) = 0$ for $i = 1, 2$.
Knowing that $\mathsf{Ext}^1_{\mathcal{C}}(A^Z,\alpha^X)$ is surjective, we can assert that for the central row
\[
\eta \colon 0 \to X \xrightarrow{\overline{f}} E \xrightarrow{\overline{g}} A^Z \to 0,
\]
there exists a short exact sequence
\[
\eta' \colon 0 \to A^X \xrightarrow{\hat{f}} A^Y \xrightarrow{\hat{g}} A^Z \to 0
\]
such that $\eta$ can be obtained as the pushout of $\eta'$ along $\alpha^X \colon A^X \to X$:
\begin{equation}\label{fig4}
\parbox{1.75in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.25ex, text depth=0.25ex]
{
K^X \& K^X \& {} \\
A^X \& A^Y \& A^Z \\
X \& E \& A^Z \\
};
\path[->]
(m-2-1)-- node[pos=0.5] {\footnotesize$\mbox{\bf po}$} (m-3-2)
;
\path[>->]
(m-1-1) edge node[left] {\footnotesize$\beta^X$} (m-2-1) (m-1-2) edge node[right] {\footnotesize$\overline{\beta}^X$} (m-2-2)
(m-2-1) edge node[above] {\footnotesize$\hat{f}$} (m-2-2)
(m-3-1) edge node[below] {\footnotesize$\overline{f}$} (m-3-2)
;
\path[->>]
(m-2-1) edge node[left] {\footnotesize$\alpha^X$} (m-3-1) (m-2-2) edge node[right] {\footnotesize$\overline{\alpha}^X$} (m-3-2)
(m-2-2) edge node[above] {\footnotesize$\hat{g}$} (m-2-3) (m-3-2) edge node[below] {\footnotesize$\overline{g}$} (m-3-3)
;
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(m-1-1) edge [double, thick, double distance=2pt] (m-1-2)
(m-2-3) edge [double, thick, double distance=2pt] (m-3-3)
;
\end{tikzpicture}
}
\end{equation}
Since $A^X, A^Z \in \mathcal{A}$ and $\mathcal{A}$ is closed under extensions, we have that $A^Y \in \mathcal{A}$. From central columns in diagrams \eqref{fig3} and \eqref{fig4}, we obtain the following commutative diagram with exact rows and columns after taking the pullback of $K^Z \to E \leftarrow A^Y$:
\begin{equation}\label{fig5}
\parbox{1.75in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.25ex, text depth=0.25ex]
{
K^X \& K^X \& {} \\
K^Y \& A^Y \& Y \\
K^Z \& E \& Y \\
};
\path[->]
(m-2-1)-- node[pos=0.5] {\footnotesize$\mbox{\bf pb}$} (m-3-2)
;
\path[>->]
(m-1-1) edge node[left] {\footnotesize$\tilde{f}$} (m-2-1) (m-1-2) edge node[right] {\footnotesize$\overline{\beta}^X$} (m-2-2)
(m-2-1) edge node[above] {\footnotesize$\beta^Y$} (m-2-2) (m-3-1) edge node[below] {\footnotesize$\tilde{\beta}^Z$} (m-3-2)
;
\path[->>]
(m-2-1) edge node[left] {\footnotesize$\tilde{g}$} (m-3-1) (m-2-2) edge node[right] {\footnotesize$\overline{\alpha}^X$} (m-3-2)
(m-2-2) edge node[above] {\footnotesize$\alpha^Y$} (m-2-3) (m-3-2) edge node[below] {\footnotesize$\tilde{\alpha}^Z$} (m-3-3)
;
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(m-1-1) edge [double, thick, double distance=2pt] (m-1-2)
(m-2-3) edge [double, thick, double distance=2pt] (m-3-3)
;
\end{tikzpicture}
}
\end{equation}
Note that $K^Y \in \mathcal{B}_{k-1}^{\wedge}$ since $K^X, K^Z \in \mathcal{B}_{k-1}^{\wedge}$ and $\mathcal{B}_{k-1}^{\wedge}$ is closed under extensions. Therefore, by using that $\mathsf{Ext}^1_\mathcal{C}(\mathcal{A},\mathcal{B}^\wedge_{k-1}) = 0$, it follows that the central row of \eqref{fig5} is a special $(\mathcal{A},k,\mathcal{B})$-precover of $Y$.
\end{proof}
\begin{remark}\label{rem:closure_Prec-n}
Note that the closure under extensions for the class ${\mathrm{Prec}}^n(\mathcal{A},\mathcal{B})$ is not covered in the previous result. This will be studied in Section \ref{sec:hereditary}.
\end{remark}
Note that in Theorem \ref{Teo-k-prec} we require the assumption that $\mathcal{B}^\wedge_{k-1}$ is closed under extensions. In the following result, we provide a sufficient condition that guarantees this closure property.
\begin{lemma}\label{Bk-1ext}
Let $k $ be a positive integer and $\mathcal{B}$ be a class of objects of $\mathcal{C}$ that is closed under extensions. If
$\mathsf{Ext}_{\mathcal{C}}^{1}(\mathcal{B},\mathcal{B}^\wedge_{k-1}) = 0$, then $\mathcal{B}_{k-1}^{\wedge}$ is closed under extensions.
\end{lemma}
\begin{proof}
Consider a short exact sequence in $\mathcal{C}$
\[
\eta \colon 0 \to X \to Y \to Z \to 0,
\]
with $X, Z \in \mathcal{B}_{k-1}^{\wedge}$. We show that ${\rm resdim}_{\mathcal{B}}(Y) \leq {\rm resdim}_{\mathcal{B}}(Z).$ In order to do that, we proceed by induction on $m := {\rm resdim}_{\mathcal{B}}(X).$
\begin{itemize}
\item For the initial case, suppose $m = 0$. If ${\rm resdim}_{\mathcal{B}}(Z) = 0$, it follows that $Y \in \mathcal{B}$ and so ${\rm resdim}_{\mathcal{B}}(Y) = 0 = {\rm resdim}_{\mathcal{B}}(Z)$, since $\mathcal{B}$ is closed under extensions. We can thus assume that ${\rm resdim}_{\mathcal{B}}(Z) \geq 1$. Then, there is an exact sequence
\[
\delta \colon 0 \to Z' \to B_0 \to Z \to 0,
\]
where ${\rm resdim}_{\mathcal{B}}(Z') +1= {\rm resdim}_{\mathcal{B}}(Z)$ and $B_0 \in \mathcal{B}$. Taking the pullback of $Y \to Z \leftarrow B_0$ produces the following commutative diagram with exact rows and columns:
\begin{equation}\label{fig1}
\parbox{1.5in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.25ex, text depth=0.25ex]
{
{} \& Z' \& Z' \\
X \& L \& B_0 \\
X \& Y \& Z \\
};
\path[>->]
(m-2-1) edge (m-2-2) (m-3-1) edge (m-3-2)
(m-1-2) edge (m-2-2) (m-1-3) edge (m-2-3)
;
\path[->>]
(m-2-2) edge (m-2-3) (m-3-2) edge (m-3-3)
(m-2-2) edge (m-3-2) (m-2-3) edge (m-3-3)
;
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(m-2-2)-- node[pos=0.5] {\footnotesize$\mbox{\bf pb}$} (m-3-3)
;
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(m-1-2) edge [double, thick, double distance=2pt] (m-1-3)
(m-2-1) edge [double, thick, double distance=2pt] (m-3-1)
;
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}
\end{equation}
Note that $L\in \mathcal{B},$ since $\mathcal{B}$ is closed under extensions. Hence, from the central column in \eqref{fig1}, we get the inequality ${\rm resdim}_{\mathcal{B}}(Y) \leq 1+{\rm resdim}_{\mathcal{B}}(Z') = {\rm resdim}_{\mathcal{B}}(Z).$ Therefore, we have $Y \in \mathcal{B}_{k-1}^{\wedge}$.
\item For the successor case, let $1 \leq m \leq k-1$ and suppose that ${\rm resdim}_{\mathcal{B}}(Y') \leq {\rm resdim}_{\mathcal{B}}(Z')$ in any short exact sequence
\[
0 \to X' \to Y' \to Z' \to 0,
\]
with $X', Z' \in \mathcal{B}^\wedge_{k-1}$ and ${\rm resdim}_{\mathcal{B}}(X') < m$.
Now for the object $Z$ appearing in the sequence $\eta$, consider a short exact sequence as $\delta$ above. Take the pullback of $Y \to Z \leftarrow B_0$ to construct a diagram as \eqref{fig1}, and consider the resulting central row and central column:
\begin{align*}
\varepsilon \colon & 0 \to X \to L \to B_0 \to 0, \\
\tau \colon & 0 \to Z' \to L \to Y \to 0.
\end{align*}
Since $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{B},\mathcal{B}^\wedge_{k-1}) = 0$, the sequence $\varepsilon$ splits, and so $L = X \oplus B_0$. On the other hand, consider a short exact sequence
\[
\varepsilon' \colon 0 \to X' \to B_1 \to X \to 0
\]
with $B_1 \in \mathcal{B}$ and ${\rm resdim}_{\mathcal{B}}(X') +1= {\rm resdim}_{\mathcal{B}}(X)$, and form the exact sequence
\[
\varepsilon'' \colon 0 \to X' \to B_1 \oplus B_0 \to X \oplus B_0 \to 0
\]
by adding to $\varepsilon'$ the identity on $B_0$. Now, take the pullback of $Z' \to X \oplus B_0 \leftarrow B_1 \oplus B_0$ in order to obtain the following commutative diagram with exact rows and columns:
\begin{equation}\label{fig2}
\parbox{1.75in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.25ex, text depth=0.25ex]
{
X' \& X' \& {} \\
Y' \& B_1 \oplus B_0 \& Y \\
Z' \& X \oplus B_0 \& Y \\
};
\path[>->]
(m-1-1) edge (m-2-1) (m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2) (m-3-1) edge (m-3-2)
;
\path[->>]
(m-2-2) edge (m-2-3) (m-3-2) edge (m-3-3)
(m-2-1) edge (m-3-1) (m-2-2) edge (m-3-2)
;
\path[->]
(m-2-1)-- node[pos=0.5] {\footnotesize$\mbox{\bf pb}$} (m-3-2)
;
\path[-,font=\scriptsize]
(m-1-1) edge [double, thick, double distance=2pt] (m-1-2)
(m-2-3) edge [double, thick, double distance=2pt] (m-3-3)
;
\end{tikzpicture}
}
\end{equation}
Note that $X', Z' \in \mathcal{B}^\wedge_{k-1}$ and ${\rm resdim}_{\mathcal{B}}(X') < m$ in the left column of \eqref{fig2}, so we can apply the induction hypothesis to conclude that ${\rm resdim}_{\mathcal{B}}(Y') \leq {\rm resdim}_{\mathcal{B}}(Z').$ On the other hand, $B_1 \oplus B_0 \in \mathcal{B}$ and so we have that
\[
{\rm resdim}_{\mathcal{B}}(Y) \leq {\rm resdim}_{\mathcal{B}}(Y') + 1 \leq {\rm resdim}_{\mathcal{B}}(Z') +1={\rm resdim}_{\mathcal{B}}(Z).
\]
Therefore, $Y \in \mathcal{B}_{k-1}^{\wedge}$.
\end{itemize}
\end{proof}
The condition $\mathsf{Ext}_{\mathcal{C}}^{1}(\mathcal{B},\mathcal{B}^\wedge_{k-1}) = 0$ in Lemma~\ref{Bk-1ext} seems to be more or less difficult to satisfy for a class $\mathcal{B} \subseteq \mathcal{C}$ closed under extensions. However, we can find classes satisfying this condition, such as the $m$-rigid subcategories. Following \cite[Definition 1.1]{IyamaCluster}, for an integer $m \geq 1$ we say that a subcategory $\mathcal{D} \subseteq \mathcal{C}$ is \emph{$m$-rigid} if $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{D,D}) = 0$ for any $0 < i < m$.
Let us show the following characterisation of $m$-rigid subcategories which involves the hypotheses of Lemma~\ref{Bk-1ext}.
\begin{proposition}\label{prop3.5}
If $\mathcal{D}$ is $m$-rigid subcategory of $\mathcal{C}$ for some $m \geq 2$, then the equality $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{D},\mathcal{D}^\wedge_{k-1}) = 0$ holds for every $1 \leq k \leq m-1$. Moreover, if $\mathcal{C}$ has enough injectives and $\mathcal{I}(\mathcal{C}) \subseteq \mathcal{D}$, then the converse statement also holds. That is, if there exists $m \geq 2$ such that $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{D},\mathcal{D}^\wedge_{k-1}) = 0$ for every $1 \leq k \leq m - 1$, then $\mathcal{D}$ is $m$-rigid.
\end{proposition}
\begin{proof}
The ``only if'' part follows by Proposition~\ref{prop8}. Now for the ``if'' part in the case where $\mathcal{C}$ has enough injectives and $\mathcal{I}(\mathcal{C}) \subseteq \mathcal{D}$, suppose that $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{D},\mathcal{D}^\wedge_{k-1}) = 0$ for every $0 \leq k-1 \leq m-2$. For $0 < i < m$, we have that $\mathsf{Ext}^i_{\mathcal{C}}(D,D') \cong \mathsf{Ext}^1_{\mathcal{C}}(D,K)$ where $D, D' \in \mathcal{D}$ and $K$ is an injective $(i-1)$-cosyzygy of $D'$. Note that $K \in \mathcal{D}^\wedge_{i-1}$, and so from the assumption it follows that $\mathsf{Ext}^1_{\mathcal{C}}(D,K) = 0$, that is, $\mathsf{Ext}^i_{\mathcal{C}}(D,D') = 0$. Therefore, $\mathcal{D}$ is $m$-rigid.
\end{proof}
\begin{corollary}\label{c3.6} Let $\mathcal{D}$ be a $m$-rigid class closed under finite coproducts and with $m \geq 2$. Then, the class $\mathcal{D}^\wedge_k$ is closed under extensions, for every $0 \leq k \leq m-2.$
\end{corollary}
\begin{proof}
Notice that $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{D,D}) = 0$ and $\mathcal{D}$ being closed under finite coproducts imply that $\mathcal{D}$ is closed under extensions. Then, the result follows from Lemma~\ref{Bk-1ext} and Proposition~\ref{prop3.5}.
\end{proof}
\begin{remark}\label{rem:special_n-1}
Under the hypothesis of Theorem \ref{Teo-k-prec}, given a short exact sequence
\[
0 \to X \xrightarrow{f} Y \xrightarrow{g} Z \to 0
\]
with $X$ and $Z$ having a special $(\mathcal{A},n,\mathcal{B})$-precover, say
\begin{align*}
\rho_X \colon & 0 \to B^X_{n-1} \xrightarrow{\beta^X_{n-1}} B^X_{n-2} \to \cdots \to B^X_1 \xrightarrow{\beta^X_1} B^X_0 \xrightarrow{\beta^X_0} A^X \xrightarrow{\alpha^X} X \to 0, \\
\rho_Z \colon & 0 \to B^Z_{n-1} \xrightarrow{\beta^Z_{n-1}} B^Z_{n-2} \to \cdots \to B^Z_1 \xrightarrow{\beta^Z_1} B^Z_0 \xrightarrow{\beta^Z_0} A^Z \xrightarrow{\alpha^Z} Z \to 0,
\end{align*}
it is possible in some cases to construct a special $(\mathcal{A},n,\mathcal{B})$-precover of $Y$ \emph{compatible} with $\rho_X$ and $\rho_Z$, that is, an exact sequence
\[
\rho_Y \colon 0 \to B^Y_{n-1} \xrightarrow{\beta^Y_{n-1}} B^Y_{n-2} \to \cdots \to B^Y_1 \xrightarrow{\beta^Y_1} B^Y_0 \xrightarrow{\beta^Y_0} A^Y \xrightarrow{\alpha^Y} Y \to 0
\]
along with a commutative diagram with exact rows and columns:
\begin{equation}\label{fig_compatible}
\parbox{4.75in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.25ex, text depth=0.25ex]
{
B^X_{n-1} \& B^X_{n-2} \& \cdots \& B^X_1 \& B^X_0 \& A^X \& X \\
B^Y_{n-1} \& B^Y_{n-2} \& \cdots \& B^Y_1 \& B^Y_0 \& A^Y \& Y \\
B^Z_{n-1} \& B^Z_{n-2} \& \cdots \& B^Z_1 \& B^Z_0 \& A^Z \& Z \\
};
\path[->]
(m-1-2) edge (m-1-3) (m-1-3) edge (m-1-4) (m-1-4) edge node[above] {\footnotesize$\beta^X_1$} (m-1-5) (m-1-5) edge node[above] {\footnotesize$\beta^X_0$} (m-1-6)
(m-2-2) edge (m-2-3) (m-2-3) edge (m-2-4) (m-2-4) edge node[above] {\footnotesize$\beta^Y_1$} (m-2-5) (m-2-5) edge node[above] {\footnotesize$\beta^Y_0$} (m-2-6)
(m-3-2) edge (m-3-3) (m-3-3) edge (m-3-4) (m-3-4) edge node[above] {\footnotesize$\beta^Z_1$} (m-3-5) (m-3-5) edge node[above] {\footnotesize$\beta^Z_0$} (m-3-6)
;
\path[>->]
(m-1-1) edge node[above] {\footnotesize$\beta^X_0$} (m-1-2)
(m-2-1) edge node[above] {\footnotesize$\beta^Y_0$} (m-2-2)
(m-3-1) edge node[above] {\footnotesize$\beta^Z_0$} (m-3-2)
(m-1-1) edge node[right] {\footnotesize$f_{n-1}$} (m-2-1) (m-1-2) edge node[right] {\footnotesize$f_{n-2}$} (m-2-2) (m-1-4) edge node[right] {\footnotesize$f_1$} (m-2-4) (m-1-5) edge node[right] {\footnotesize$f_0$} (m-2-5) (m-1-6) edge node[right] {\footnotesize$\hat{f}$} (m-2-6) (m-1-7) edge node[right] {\footnotesize$f$} (m-2-7)
;
\path[->>]
(m-1-6) edge node[above] {\footnotesize$\alpha^X$} (m-1-7)
(m-2-6) edge node[above] {\footnotesize$\alpha^Y$} (m-2-7)
(m-3-6) edge node[above] {\footnotesize$\alpha^Z$} (m-3-7)
(m-2-1) edge node[right] {\footnotesize$g_{n-1}$} (m-3-1) (m-2-2) edge node[right] {\footnotesize$g_{n-2}$} (m-3-2) (m-2-4) edge node[right] {\footnotesize$g_1$} (m-3-4) (m-2-5) edge node[right] {\footnotesize$g_0$} (m-3-5) (m-2-6) edge node[right] {\footnotesize$\hat{g}$} (m-3-6) (m-2-7) edge node[right] {\footnotesize$g$} (m-3-7)
;
\end{tikzpicture}
}
\end{equation}
To prove this assertion, we shall need the analog of hereditary cotorsion pairs for left $n$-cotorsion pairs, presented later in Section~\ref{sec:hereditary}.
For now, we can show the case $n = 1$. That is, we are given two classes of objects $\mathcal{A}$ and $\mathcal{B}$ in $\mathcal{C}$, closed under extensions, such that $\mathsf{Ext}^2_{\mathcal{C}}(\mathcal{A,B}) = 0$. Following the proof of Theorem~\ref{Teo-k-prec}, we have a short exact sequence
\[
0 \to X \xrightarrow{f} Y \xrightarrow{g} Z \to 0
\]
where $X$ and $Z$ have special $\mathcal{A}$-precovers with kernel in $\mathcal{B}$, say:
\begin{align*}
\rho_X \colon & 0 \to B^X \xrightarrow{\beta^X} A^X \xrightarrow{\alpha^X} X \to 0, \\
\rho_Z \colon & 0 \to B^Z \xrightarrow{\beta^Z} A^Z \xrightarrow{\alpha^Z} Z \to 0.
\end{align*}
From the diagrams \eqref{fig3}, \eqref{fig4} and \eqref{fig5}, we construct the following diagram with exact rows and columns:
\begin{equation}\label{fig_caso_n1}
\parbox{1.75in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.25ex, text depth=0.25ex]
{
B^X \& A^X \& X \\
B^Y \& A^Y \& Y \\
B^Z \& A^Z \& Z \\
};
\path[>->]
(m-1-1) edge node[above] {\footnotesize$\beta^X$} (m-1-2) (m-2-1) edge node[above] {\footnotesize$\beta^Y$} (m-2-2) (m-3-1) edge node[above] {\footnotesize$\beta^Z$} (m-3-2)
(m-1-1) edge node[left] {\footnotesize$\tilde{f}$} (m-2-1) (m-1-2) edge node[left] {\footnotesize$\hat{f}$} (m-2-2) (m-1-3) edge node[right] {\footnotesize$f$} (m-2-3)
;
\path[->>]
(m-1-2) edge node[above] {\footnotesize$\alpha^X$} (m-1-3) (m-2-2) edge node[above] {\footnotesize$\alpha^Y$} (m-2-3) (m-3-2) edge node[above] {\footnotesize$\alpha^Z$} (m-3-3)
(m-2-1) edge node[left] {\footnotesize$\tilde{g}$} (m-3-1) (m-2-2) edge node[left] {\footnotesize$\hat{g}$} (m-3-2) (m-2-3) edge node[right] {\footnotesize$g$} (m-3-3)
;
\end{tikzpicture}
}
\end{equation}
We check that \eqref{fig_caso_n1} commutes:
\begin{align*}
\beta^Y \circ \tilde{f} & = \overline{\beta}^X = \hat{f} \circ \beta^X & \mbox{(by \eqref{fig5} and \eqref{fig4})}, \\
\alpha^Y \circ \hat{f} & = \tilde{\alpha}^Z \circ \overline{\alpha}^X \circ \hat{f} = \tilde{\alpha}^Z \circ \overline{f} \circ \alpha^X = f \circ \alpha^X & \mbox{(by \eqref{fig5}, \eqref{fig4} and \eqref{fig3})}, \\
\beta^Z \circ \tilde{g} & = \overline{g} \circ \tilde{\beta}^Z \circ \tilde{g} = \overline{g} \circ \overline{\alpha}^X \circ \beta^Y = \hat{g} \circ \beta^Y & \mbox{(by \eqref{fig3}, \eqref{fig5} and \eqref{fig4})}, \\
\alpha^Z \circ \hat{g} & = \alpha^Z \circ \overline{g} \circ \overline{\alpha}^X = g \circ \tilde{\alpha}^Z \circ \overline{\alpha}^X = g \circ \alpha^Y & \mbox{(by \eqref{fig4}, \eqref{fig3} and \eqref{fig5})}.
\end{align*}
\end{remark}
\begin{corollary}\label{Coro-k-prec}
Let $(\mathcal{A,B})$ be a left $n$-cotorsion pair in $\mathcal{C}$ with $n \geq 2$, such that:
\begin{itemize}
\item[(i)] $\mathcal{B}$ is closed under finite coproducts in $\mathcal{C}$, and
\item[(ii)] $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{B},\mathcal{B}^\wedge_{k-1}) = 0$ for any $1 \leq k \leq \max(1,n-1)$.
\end{itemize}
Then, the class ${\mathrm{Prec}}^k(\mathcal{A},\mathcal{B})$ is closed under extensions for any $1\leq k\leq \max(1,n-1)$.
\end{corollary}
\begin{proof}
First, note that since $\mathcal{B}$ is closed under finite coproducts in $\mathcal{C}$ and $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{B},\mathcal{B}^\wedge_{0}) = 0$, it follows that $\mathcal{B}$ is closed under extensions. Furthermore, from Lemma~\ref{Bk-1ext} we obtain that $\mathcal{B}^\wedge_{k-1}$ is closed under extensions. On the other hand, Theorem~\ref{theo:left-n-cotorsion} allows us to conclude that $\mathcal{A}$ is also closed under extensions. Thus, Theorem~\ref{Teo-k-prec} gives us the result.
\end{proof}
\subsection*{\textbf{\textit{n}-Cotorsion and approximations having the unique mapping property}}
Now let us study the relation between higher cotorsion and approximations with the unique mapping property. This point has been tackled in other contexts of higher cotorsion. For instance, Crivei and Torrecillas considered $(m,n)$-cotorsion pairs $(\mathcal{A,B})$ in Grothendieck categories $\mathcal{G}$ with enough projectives, and studied some conditions under which it is possible to obtain special precovers with the unique mapping property. Namely, they proved in \cite[Theorem 3.15]{CriveiTorrecillas} that every object in $\mathcal{G}$ has an $\mathcal{A}^{\perp_{n+1}}$-preenvelope with the unique mapping property if, and only if, $\mathcal{G} = \mathcal{A}^{\perp_{n+1}}$.
Recall that an $\mathcal{A}$-precover $f \colon A \to C$ of $C \in \mathcal{C}$ is said to have the \emph{unique mapping property} if for every morphism $f' \colon A' \to C$ with $A' \in \mathcal{A}$ there exists a unique $h \colon A' \to A$ such that $f' = f \circ h$. The notion of an $\mathcal{A}$-preenvelope having the unique mapping property is defined dually.
The importance of the unique mapping property lies in its applications, which go from the description of certain categories of modules, to characterisations of rings that involve its global or weak dimension. One of these applications has to do with the existence of flat envelopes. Specifically, Asensio Mayor and Mart\'inez Hern\'andez proved in \cite[Proposition 2.1]{AsensioMartinez} that for any ring $R$, every module has a flat envelope with the unique mapping property if, and only if, $R$ is right coherent and with weak dimension ${\rm wd}(R) \leq 2$. The dual version of this result was proved by Mao and Ding in \cite[Corollary 2.4]{MaoDing_wgd}, that is, the latter condition is also equivalent to saying that every right $R$-module has an absolutely pure cover with the unique mapping property. One interesting question about the class $\mathcal{AP}(R)$ of absolutely pure $R$-modules (also known as $\text{FP}$-injective $R$-modules) is whether it is closed under direct limits. Mao and Ding also proved in \cite[Proposition 6.7]{MaoDing_finite} that if every module has an absolutely pure cover with the unique mapping property, then $\mathcal{AP}(R)$ is closed under direct limits. There are also other characterisations of the weak dimension of coherent rings involving the unique mapping property with respect to flat and projective envelopes (the reader can see \cite[Corollaries 3.4 and 3.9]{Ding96}, also by Ding).
In the following lines, we provide other conditions, within the context of $n$-cotorsion pairs, under which one can obtain approximations with the unique mapping property. The results obtained in this direction will be applied in Section~\ref{sec:hereditary} to comment more on Mao a Ding's \cite[Corollary 2.4]{MaoDing_wgd}, and in Section~\ref{sec:applications} in the field of Gorenstein homological algebra, where we extend some results concerning Gorenstein projective envelopes and Gorenstein injective covers with the unique mapping property.
Recall that the classes $\mathcal{B}^\wedge_{k}$ play an important role in the concept of left $n$-cotorsion pairs $(\mathcal{A,B})$ in abelian categories. In the study of approximations having the unique mapping property, we shall need to consider the classes $\mathcal{A}^\wedge_{k}$ instead. Let us begin showing the following two properties.
\begin{proposition}\label{Bvee sub Aperp}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects of $\mathcal{C}$ such that $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ for every $1 \leq i \leq n$. Then, the containment $\mathcal{A}^\wedge_{k} \subseteq {}^{\perp_{k+1}}\mathcal{B}$ holds for every $0 \leq k \leq n - 1$.
\end{proposition}
\begin{proof}
It follows by induction on $k$.
\end{proof}
\begin{proposition}\label{ncotder y cotder}
The following conditions are equivalent for any left $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$:
\begin{itemize}
\item[(a)] $\mathcal{A} = {}^{\perp_1}\mathcal{B}$.
\item[(b)] The equality ${}^{\perp_{k+1}}\mathcal{B} = \mathcal{A}^\wedge_k$ holds for every $0 \leq k \leq n - 1$.
\end{itemize}
\end{proposition}
\begin{proof}
The implication (b) $\Rightarrow$ (a) follows by setting $k = 0$, since $\mathcal{A}^\wedge_0 = \mathcal{A}$.
Now if we assume that condition (a) holds true, note that the case $k = 0$ is clear. Thus, we may suppose that $k \geq 1$. The containment $\mathcal{A}^\wedge_k \subseteq {}^{\perp_{k+1}}\mathcal{B}$ follows by Proposition~\ref{Bvee sub Aperp}. For the remaining containment $\mathcal{A}^\wedge_k \supseteq {}^{\perp_{k+1}}\mathcal{B}$, consider an object $M \in {}^{\perp_{k+1}}\mathcal{B}$. Since $\mathcal{A}$ is precovering, by Proposition~\ref{A-precub,B-preenv} we can construct an exact sequence of the form
\[
0 \to K \to A_{k-1} \to \cdots \to A_1 \to A_0 \to M \to 0,
\]
where $A_j \in \mathcal{A}$ for every $0 \leq j \leq k-1$. By using the relation $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ for $1 \leq i \leq n,$ along with dimension shifting, we have that $\mathsf{Ext}^1_\mathcal{C}(K,B)\simeq\mathsf{Ext}^{k+1}_\mathcal{C}(M,B)=0$ and thus
$K \in {}^{\perp_1}\mathcal{B} = \mathcal{A}$. Hence, $M \in \mathcal{A}^\wedge_k$.
\end{proof}
\begin{remark}\label{RkAortB}
Given a left $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$, one may ask for possible cases where condition (a) in the previous proposition holds.
The most obvious situation is when $n = 1$, as we get from Theorem~\ref{theo:left-n-cotorsion} that $\mathcal{A} = {}^{\perp_1}\mathcal{B}$, that is, we have that $(\mathcal{A,B})$ is a complete left cotorsion pair in $\mathcal{C}$.
The equality $\mathcal{A} = {}^{\perp_1}\mathcal{B}$ is not necessarily true if $n \geq 2$, like for instance in the left $n$-cotorsion pair $(\mathcal{GP}(R),\mathcal{P}(R))$ in $\mathsf{Mod}(R)$, with $R$ an $n$-Iwanaga-Gorenstein ring, considered at the beginning of Section~\ref{sec:applications}. However, in the case where $\mathcal{B}$ is closed under taking cokernels of monomorphisms between objects in $\mathcal{B}$, one can note that $\mathcal{A} = {}^{\perp_1}\mathcal{B}$.
\end{remark}
In \cite[dual of Proposition 3.3]{CriveiTorrecillas}, Crivei and Torrecillas proved that for every class $\mathcal{B}$ of objects of $\mathcal{C}$, an abelian category with enough projectives, if every object of $\mathcal{C}$ has a ${}^{\perp_{k+1}}\mathcal{B}$-precover with the unique mapping property, then $\mathcal{C} = {}^{\perp_{k+1}}\mathcal{B}$. This fact is extended in the following result using Proposition~\ref{ncotder y cotder}. Moreover, we shall note that the consequences of having $\mathcal{A} = {}^{\perp_1}\mathcal{B}$ for a left $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$ have an impact on how $\mathcal{C}$ can be described in terms of relative homological dimensions.
\begin{corollary}\label{Aperp y Bvee}
Let $\mathcal{C}$ be an abelian category with enough projectives, and $(\mathcal{A,B})$ be a left $n$-cotorsion pair in $\mathcal{C}$. If the equality $\mathcal{A} = {}^{\perp_1}\mathcal{B}$ holds, then the following conditions are equivalent for any integer $0 \leq k \leq n - 1$:
\begin{enumerate}
\item[(a)] Every object in $\mathcal{C}$ has a ${}^{\perp_{k+1}}\mathcal{B}$-precover with the unique mapping property.
\item[(b)] ${\rm resdim}_{\mathcal{A}}(\mathcal{C}) \leq k.$
\item[(c)] $\mathcal{C} = {}^{\perp_{k+1}}\mathcal{B}$.
\end{enumerate}
\end{corollary}
\begin{proof}
It follows from \cite[dual of Proposition 3.3]{CriveiTorrecillas} and Proposition~\ref{ncotder y cotder}.
\end{proof}
\begin{remark}
The equivalence (a) $\Leftrightarrow$ (c) of Corollary~\ref{Aperp y Bvee} is also proven in \cite[dual of Theorem 3.7]{CriveiTorrecillas} under different conditions. Namely, the authors work in the case where $\mathcal{C}$ is a Grothendieck category with enough projectives. Also, the class $\mathcal{B}$ need not be part of an $n$-cotorsion pair in this reference.
\end{remark}
Corollary~\ref{Aperp y Bvee} establishes some conditions under which it is possible to construct, from a left $n$-cotorsion pair of the form $({}^{\perp_1}\mathcal{B},\mathcal{B})$, right approximations with the unique mapping property. With respect to left approximations, we have the following.
\begin{corollary}\label{corUMP2}
Let $\mathcal{C}$ be an abelian category with enough projectives and $(\mathcal{A,B})$ be a left $n$-cotorsion pair with $n \geq 3$. If the equality $\mathcal{A} = {}^{\perp_1}\mathcal{B}$ holds, then the following conditions are equivalent:
\begin{itemize}
\item[(a)] Every object in $\mathcal{C}$ has an $\mathcal{A}$-envelope with the unique mapping property.
\item[(b)] Every object in $\mathcal{C}$ has an $\mathcal{A}$-envelope and ${\rm resdim}_{\mathcal{A}}(\mathcal{C}) \leq 2.$
\end{itemize}
\end{corollary}
\begin{proof}
First, we get from Proposition~\ref{A-precub,B-preenv} that $\mathcal{A}$ is special precovering. Moreover, by Theorem~\ref{theo:left-n-cotorsion}, $\mathcal{A} = {}^{\perp_1}\mathcal{B} \subseteq {}^{\perp_i}\mathcal{B}$ for $i = 2, 3$ since $n \geq 3.$ Thus, by Proposition~\ref{ncotder y cotder} we have ${}^{\perp_3}\mathcal{B} = \mathcal{A}_{2}^{\wedge}$. Hence, the result follows by \cite[dual of Theorem 3.16]{CriveiTorrecillas}.
\end{proof}
\section{\textbf{Higher cotorsion from hereditary cotorsion pairs}}\label{sec:hereditary}
In this section we analyse the situation where we are given a (left or right) $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$ where $\mathcal{A}$ is resolving or $\mathcal{B}$ is coresolving. This will be presented in three approaches. First, we study the relation between (left and right) $n$-cotorsion pairs and hereditary cotorsion pairs. We shall see that the only $n$-cotorsion pairs $(\mathcal{A,B})$ with $\mathcal{A}$ resolving are precisely the hereditary complete cotorsion pairs. Then, we shall comment on hereditary cotorsion pairs $(\mathcal{A,B})$ satisfying the property $\mathcal{A} \subseteq \mathcal{B}$, and provide several characterisations for them. These will allow us to note that the only $n$-cotorsion pair $(\mathcal{A,B})$ with $\mathcal{A}$ resolving and $\mathcal{A} \subseteq \mathcal{B}$ is the trivial $n$-cotorsion pair $(\mathcal{P}(\mathcal{C}),\mathcal{C})$ from Example~\ref{ex:trivial}. The previous suggest that being hereditary for higher cotorsion should be a one-sided notion. Thus, we shall propose in the last part of this section the concept of hereditary left $n$-cotorsion pair, and show that for any such pair the class ${\mathrm{Prec}}^{n}(\mathcal{A,B})$ is closed under extensions (see Remark \ref{rem:closure_Prec-n}). The latter is an important result that will help us to construct certain $n$-cotorsion pairs of chain complexes from an $n$-cotorsion pair in the ground category $\mathcal{C}$.
Recall that a cotorsion pair $(\mathcal{A,B})$ in an abelian category $\mathcal{C}$ is \emph{hereditary} if $\mathcal{A}$ is resolving and $\mathcal{B}$ is coresolving. The term \emph{resolving} means that $\mathcal{A}$ is closed under extensions and under taking kernels of epimorphisms between its objects, and that the class $\mathcal{P}(\mathcal{C})$ of projective objects of $\mathcal{C}$ is contained in $\mathcal{A}$. Coresolving classes are defined dually.
It is well known that in any abelian category $\mathcal{C}$ with enough injectives (like for instance any Grothendieck category), for any cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$ one has that $\mathcal{B}$ is coresolving if, and only if, $\mathsf{Ext}^2_{\mathcal{C}}(\mathcal{A,B}) = 0$. Dually, the latter is also equivalent to $\mathcal{A}$ being resolving provided that $\mathcal{C}$ has enough projectives. For $n$-cotorsion pairs, with $n \geq 2$, we can obtain a similar equivalence without having either enough projectives or injectives, as we show in the following result.
\begin{proposition}\label{equiv hered y n-cot}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects of $\mathcal{C}$, and $n \geq 2$ be a positive integer. Consider the following conditions.
\begin{enumerate}
\item[(a)] $(\mathcal{A,B})$ is an $n$-cotorsion pair in $\mathcal{C}$ and $\mathcal{B}$ is a coresolving class.
\item[(b)] $(\mathcal{A,B})$ is an $n$-cotorsion pair in $\mathcal{C}$ and $\mathcal{A}$ is a resolving class.
\item[(c)] $(\mathcal{A,B})$ is a hereditary complete cotorsion pair in $\mathcal{C}$.
\end{enumerate}
Then, conditions (a) and (b) are equivalent, and any of them implies (c). If in addition, $\mathcal{C}$ has enough projectives or injectives, then (c) also implies (a) and (b).
\end{proposition}
\begin{proof}
We first show that (a) and (b) are equivalent. Suppose that condition (a) holds, and so $(\mathcal{A,B})$ is an $n$-cotorsion pair with $\mathcal{B}$ coresolving. By Theorem~\ref{theo:left-n-cotorsion}, $(\mathcal{A},\mathcal{B}^\wedge_{n-1})$ is a complete left cotorsion pair, where $\mathcal{B}^\wedge_{n-1} = \mathcal{B}$ since $\mathcal{B}$ is coresolving. It follows that $\mathcal{A} = {}^{\perp_1}\mathcal{B}$, and thus it is clear that $\mathcal{P}(\mathcal{C}) \subseteq \mathcal{A}$ and that $\mathcal{A}$ is closed under extensions. To show that $\mathcal{A}$ is also closed under taking kernels of epimorphisms between objects of $\mathcal{A}$, suppose we are given a short exact sequence
\[
0 \to A_1 \to A_2 \to A_3 \to 0
\]
with $A_2, A_3 \in \mathcal{A}$. For every $B \in \mathcal{B}$, we have an exact sequence
\[
\mathsf{Ext}^1_{\mathcal{C}}(A_2,B) \to \mathsf{Ext}^1_{\mathcal{C}}(A_1,B) \to \mathsf{Ext}^2_{\mathcal{C}}(A_3,B),
\]
where $\mathsf{Ext}^1_{\mathcal{C}}(A_2,B) = 0$ and $\mathsf{Ext}^2_{\mathcal{C}}(A_3,B) = 0$ since $(\mathcal{A,B})$ is an $n$-cotorsion pair with $n \geq 2$. It follows that $A_1 \in {}^{\perp_1}\mathcal{B} = \mathcal{A}$. Hence, $\mathcal{A}$ is resolving. The implication (b) $\Rightarrow$ (a) follows similarly.
Note that if we assume (a) or (b), we obtain a complete left and a complete right cotorsion pair $(\mathcal{A,B})$, that is, a complete cotorsion pair in $\mathcal{C}$, which is also hereditary.
Now if we suppose that (c) holds and that $\mathcal{C}$ has enough projectives or injectives. One can see that this implies that $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ holds for every $i \geq 1$, and showing hence conditions (a) and (b).
\end{proof}
Note that the previous proposition basically says that there is no much hope in finding an $n$-cotorsion pair $(\mathcal{A,B})$ in a Grothendieck category with $\mathcal{A}$ resolving (and $\mathcal{B}$ coresolving) that is not actually a hereditary complete cotorsion pair. In Section~\ref{sec:applications}, we shall find some examples of $n$-cotorsion which do not come from such cotorsion pairs.
A first application of Proposition~\ref{equiv hered y n-cot} has to do with extending the result \cite[Proposition 6.7]{MaoDing_finite} about the existence of absolutely pure covers with the unique mapping property. For a precise statement, we need to recall some concepts.
Let $R$ be a ring and $M \in \mathsf{Mod}(R)$ be an $R$-module. Recall that $M$ is \emph{absolutely pure} (or \emph{FP-injective}) if $\mathsf{Ext}^1_R(F,M) = 0$ for every finitely presented module $F \in \mathsf{Mod}(R)$. In what follows, we shall denote by $\mathcal{AP}(R)$ the class of absolutely pure $R$-modules. The \emph{absolutely pure dimension} of $M$, denoted ${\rm apd}(M)$, is defined as the smallest nonnegative integer $m \geq 0$ such that $\mathsf{Ext}^{m+1}_R(F,M) = 0$ for every finitely presented module $F \in \mathsf{Mod}(R)$, or equivalently, as
\[
{\rm apd}(M) = {\rm coresdim}_{\mathcal{AP}(R)}(M).
\]
The (\emph{left}) \emph{global absolutely pure dimension} of $R$, denoted by ${\rm gl.APD}(R)$, is defined as the supremum
\[
{\rm gl.APD}(R) := \sup \{ {\rm apd}(M) \mbox{ : } M \in \mathsf{Mod}(R) \}.
\]
The dual concepts are the \emph{flat dimension} of $M$ and the \emph{weak global dimension} of $R$, denoted by ${\rm fd}(M)$ and ${\rm wd}(R)$, respectively.
\begin{corollary}
For any ring $R$ with ${\rm gl.APD}(R^{\rm op}) \leq 2$, the following statements are equivalent:
\begin{itemize}
\item[(a)] $R$ is a right coherent ring.
\item[(b)] Every right $R$-module has an absolutely pure cover with the unique mapping property.
\end{itemize}
Moreover, if any of above conditions holds true, then ${\rm wd}(R) \leq 2$ and the class $\mathcal{AP}(R{}^{\rm op})$ is closed under direct limits.
\end{corollary}
\begin{proof}
On the one hand, if $R$ is a right coherent ring, then from \cite[Proposition 3.6]{MaoDing_finite} and \cite[Corollary 2.7]{Pinzon}, we have that $({}^{\perp_{1}}\mathcal{AP}(R{}^{\rm op}),\mathcal{AP}(R{}^{\rm op}))$ is a complete hereditary cotorsion pair in $\mathsf{Mod}(R^{\rm op})$ and that every right $R$-module has an absolutely pure cover. Thus, by Proposition~\ref{equiv hered y n-cot} and the dual of Corollary~\ref{corUMP2} we get the implication (a) $\Rightarrow$ (b). The converse, on the other hand, holds true by \cite[Remark 6.8]{MaoDing_finite}. Finally, the second part follows by \cite[Theorem 6.6 and Proposition 6.7]{MaoDing_finite}.
\end{proof}
\begin{corollary}
For any ring $R,$ the following conditions are equivalent:
\begin{itemize}
\item[(a)] Every $R$-module has a flat envelope with the unique mapping property.
\item[(b)] Every $R$-module has a flat envelope and ${\rm wd}(R)\leq 2$.
\item[(c)] $R$ is a right coherent ring with ${\rm wd}(R)\leq 2$.
\end{itemize}
\end{corollary}
\begin{proof}
From \cite[Theorem 3.4]{MaoDing_finite}, Proposition~\ref{equiv hered y n-cot} and Corollary~\ref{corUMP2}, we have the equivalence (a) $\Leftrightarrow$ (b), while (a) $\Leftrightarrow$ (c) follows by \cite[Proposition 2.1]{AsensioMartinez}.
\end{proof}
Another application of Proposition~\ref{equiv hered y n-cot}, along with Proposition \ref{ncotder y cotder} and its dual, has to do with descriptions for the classes $\mathcal{A}^\wedge_n$ and $\mathcal{B}^\vee_m$ coming from a hereditary complete cotorsion pair $(\mathcal{A,B})$.
\begin{corollary}
Let $(\mathcal{A,B})$ be a hereditary complete cotorsion pair in an abelian category $\mathcal{C}$ with enough projectives and injectives. Then, for any pair of integers $m, n \geq 0$, the equalities $\mathcal{A}^{\perp_{m + 1}} = \mathcal{B}^\vee_m$ and ${}^{\perp_{n+1}}\mathcal{B} = \mathcal{A}^\wedge_n$ hold true.
\end{corollary}
Now let us focus on $n$-cotorsion pairs satisfying the condition $\mathcal{A} \subseteq \mathcal{B}$.
\begin{proposition}\label{Asubseteq B en ncot}
Let $(\mathcal{A,B})$ be an $n$-cotorsion pair in $\mathcal{C}$. Then, the following assertions are equivalent.
\begin{itemize}
\item[(a)] $\mathcal{A} \subseteq \mathcal{B}$.
\item[(b)] $\mathcal{C} = \mathcal{B}_n^{\wedge}$.
\item[(c)] $\mathsf{Ext}_{\mathcal{C}}^1(\mathcal{A}_{n-1}^{\vee},\mathcal{A}) = 0$.
\end{itemize}
\end{proposition}
\begin{proof}
We first show (a) $\Leftrightarrow$ (b). The implication (a) $\Rightarrow$ (b) follows by Definition~\ref{def:ncotorsion} (3). Now suppose that condition (b) $\mathcal{C} = \mathcal{B}_{n}^{\wedge}$ holds true, and let $A \in \mathcal{A}$. Then, $A$ is isomorphic to the image of an epimorphism from $\mathcal{B}$ with kernel in $\mathcal{B}_{n-1}^{\wedge}$. By Proposition~\ref{prop8}, this epimorphism splits, and therefore $A \in \mathcal{B}$ since $\mathcal{B}$ is closed under direct summands. Finally, the equivalence (a) $\Leftrightarrow$ (c) follows by the dual of Theorem~\ref{theo:left-n-cotorsion}.
\end{proof}
We can use the previous proposition to show the following generalisation of Akinci and Alizade's \cite[Remark 2.4]{Akinci}.
\begin{corollary}\label{coro:cot-hered-completo-trivial}
For every hereditary complete cotorsion pair $(\mathcal{A,B})$ in an abelian category $\mathcal{C}$, the following assertions are equivalent.
\begin{itemize}
\item[(a)] $\mathcal{A} \subseteq \mathcal{B}.$
\item[(b)] $\mathcal{C} = \mathcal{B}$.
\item[(c)] $\mathsf{Ext}_{\mathcal{C}}^{1}(\mathcal{A,A}) = 0$.
\item[(d)] $\mathcal{A} = \mathcal{P}(\mathcal{C})$.
\end{itemize}
\end{corollary}
\begin{proof}
Note that $(\mathcal{A},\mathcal{B})$ is $1$-cotorsion pair in $\mathcal{C}$ with $\mathcal{B}^\wedge_1 = \mathcal{B}$ and $\mathcal{A}^\vee_1 = \mathcal{A}$. Thus, by Proposition~\ref{Asubseteq B en ncot} we get the equivalences between (a), (b) and (c). The equivalence between (b) and (d), on the other hand, is straightforward.
\end{proof}
\begin{remark}\label{Rk-n-cot-trivial} \
\begin{enumerate}
\item Let $(\mathcal{A,B})$ be an $n$-cotorsion pair in an abelian category $\mathcal{C},$ with $n\geq 2$ and $\mathcal{A}$ resolving (or $\mathcal{B}$ coresolving). Then, by Proposition~\ref{equiv hered y n-cot} we know that $(\mathcal{A},\mathcal{B})$ is an hereditary complete cotorsion pair in $\mathcal{C}$. Thus, by Corollary~\ref{coro:cot-hered-completo-trivial}, we get the equivalences
\[
\mathcal{A} \subseteq \mathcal{B}\;\Leftrightarrow\;\mathcal{C} = \mathcal{B}\;\Leftrightarrow\;\mathsf{Ext}_{\mathcal{C}}^{1}(\mathcal{A,A}) = 0\;\Leftrightarrow\;\mathcal{A} = \mathcal{P}(\mathcal{C}).
\]
\item The previous implies that, in an abelian category $\mathcal{C}$ with enough projectives and injectives, the only $n$-cotorsion pair $(\mathcal{A,B})$ with $\mathcal{A}$ resolving and satisfying the continment $\mathcal{A} \subseteq \mathcal{B}$ is the trivial $n$-cotorsion pair $(\mathcal{P}(\mathcal{C}),\mathcal{C})$.
\end{enumerate}
\end{remark}
Now we are ready to propose the following definition of hereditary unilateral $n$-cotorsion.
\begin{definition}\label{def:hereditary_n-cotorsion}
Let $(\mathcal{A,B})$ be a left $n$-cotorsion pair in $\mathcal{C}$. We say that $(\mathcal{A,B})$ is \textbf{hereditary} if $\mathsf{Ext}^{n+1}_{\mathcal{C}}(\mathcal{A,B}) = 0$. Hereditary right $n$-cotorsion pairs are defined in the same way.
\end{definition}
One can note the following property of left $n$-cotorsion pairs.
\begin{proposition}\label{prop:hereditary_n-cotorsion}
If $(\mathcal{A,B})$ is a hereditary left $n$-cotorsion pair in $\mathcal{C}$, then the class $\mathcal{A}$ is resolving.
\end{proposition}
\begin{proof}
Notice by Theorem~\ref{theo:left-n-cotorsion} that $\mathcal{A} = \bigcap^n_{i = 1} {}^{\perp_i}\mathcal{B}$. Then, it is clear that $\mathcal{A}$ is closed under extensions and that $\mathcal{P}(\mathcal{C}) \subseteq \mathcal{A}$. Finally, if we are given a short exact sequence
\[
0 \to A_1 \to A_2 \to A_3 \to 0
\]
in $\mathcal{C}$ with $A_2, A_3 \in \mathcal{A}$, then we have an exact sequence
\[
\mathsf{Ext}^i_{\mathcal{C}}(A_2, B) \to \mathsf{Ext}^i_{\mathcal{C}}(A_1, B) \to \mathsf{Ext}^{i+1}_{\mathcal{C}}(A_3,B)
\]
of abelian groups where $\mathsf{Ext}^i_{\mathcal{C}}(A_2,B) = 0$ for every $1 \leq i \leq n$, and $\mathsf{Ext}^{i+1}_{\mathcal{C}}(A_3, B) = 0$ for every $1 \leq i \leq n$ since $(\mathcal{A,B})$ is hereditary. It follows that $\mathsf{Ext}^i_{\mathcal{C}}(A_1, B) = 0$ for every $1 \leq i \leq n$ and $B \in \mathcal{B}$, that is, $A_1 \in \cap^n_{i = 1} {}^{\perp_i}\mathcal{B} = \mathcal{A}$, and so $\mathcal{A}$ is closed under taking kernels of epimorphisms between its objects.
\end{proof}
\begin{remark}
Notice by the previous proposition and its dual, that is $(\mathcal{A,B})$ is a hereditary (left and right) $n$-cotorsion pair, then $\mathcal{A}$ is resolving and $\mathcal{B}$ is coresolving. Then, Theorem~\ref{theo:left-n-cotorsion} and its dual imply that $\mathcal{A} = {}^{\perp_1}(\mathcal{B}^\wedge_{n-1})$ and $\mathcal{B} = (\mathcal{A}^\vee_{n-1})^{\perp_1}$, where $\mathcal{B}^\wedge_{n-1} = \mathcal{B}$ and $\mathcal{A}^\vee_{n-1} = \mathcal{A}$. Hence, $(\mathcal{A,B})$ is a hereditary complete cotorsion pair in $\mathcal{C}$. The converse of this fact is clearly true in the case where $\mathcal{C}$ has enough projective and injectives.
\end{remark}
\begin{theorem}\label{theo:compatible}
Let $n \geq 2$ be a positive integer and $(\mathcal{A,B})$ be a hereditary left $n$-cotorsion pair in $\mathcal{C}$ with $\mathcal{B}$ closed under extensions and such that $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{B},\mathcal{B}^\wedge_{n-1}) = 0$. Let
\[
\eta \colon 0 \to X \xrightarrow{f} Y \xrightarrow{g} Z \to 0
\]
be a short exact sequence in $\mathcal{C}$ where $X$ and $Z$ have special $(\mathcal{A},n,\mathcal{B})$-precovers $\rho_X$ and $\rho_Z$ as in Remark~\ref{rem:special_n-1}. Then, $Y$ has a special $(\mathcal{A},n,\mathcal{B})$-precover $\rho_Y$ compatible with $\rho_X$ and $\rho_Z$ in the sense that it fits into a commutative diagram as \eqref{fig_compatible}. In particular, the class ${\rm Prec}^n(\mathcal{A,B})$ is closed under extensions.
\end{theorem}
\begin{proof}
For a better understanding of the arguments below, we only show the case $n = 2$. The more general cases $n > 2$ follow inductively.
We are given two short exact sequences
\begin{align*}
0 & \to K^X \xrightarrow{i^X} A^X \xrightarrow{\alpha^X} X \to 0, \\
0 & \to K^Z \xrightarrow{i^Z} A^Z \xrightarrow{\alpha^Z} Z \to 0,
\end{align*}
where $A^X, A^Z \in \mathcal{A}$ and $K^X, K^Z \in \mathcal{B}^\wedge_1$, that is, we also have two short exact sequences
\begin{align*}
0 & \to B^X_1 \xrightarrow{\beta^X_1} B^X_0 \xrightarrow{\gamma^X} K^X \to 0, \numberthis \label{eqn:BX1BX0KX} \\
0 & \to B^Z_1 \xrightarrow{\beta^Z_1} B^Z_0 \xrightarrow{\gamma^Z} K^Z \to 0,
\end{align*}
with $B^X_0, B^Z_0, B^X_1, B^Z_1 \in \mathcal{B}$. Following the same arguments as in the proof of Theorem \ref{Teo-k-prec}, we can find the following commutative diagrams with exact rows and columns:
\begin{equation}\label{fig7}
\parbox{2.5in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.25ex, text depth=0.25ex]
{
{} \& {} \& X \& X \\
B^Z_1 \& B^Z_0 \& Y' \& Y \\
B^Z_1 \& B^Z_0 \& A^Z \& Z \\
};
\path[->]
(m-2-3)-- node[pos=0.5] {\footnotesize$\mbox{\bf pb}$} (m-3-4)
(m-2-2) edge node[above] {\footnotesize$\hat{\beta}^Z_0$} (m-2-3)
(m-3-2) edge node[below] {\footnotesize$\beta^Z_0 $} (m-3-3)
;
\path[>->]
(m-1-3) edge node[left] {\footnotesize$f'$} (m-2-3) (m-1-4) edge node[left] {\footnotesize$f$} (m-2-4)
(m-2-1) edge node[above] {\footnotesize$\hat{\beta}^Z_1$} (m-2-2)
(m-3-1) edge node[below] {\footnotesize$\beta^Z_1$} (m-3-2)
;
\path[->>]
(m-2-3) edge node[left] {\footnotesize$g'$} (m-3-3) (m-2-4) edge node[left] {\footnotesize$g$} (m-3-4)
(m-2-3) edge node[above] {\footnotesize$\tilde{\alpha}^Z$} (m-2-4)
(m-3-3) edge node[below] {\footnotesize$\alpha^Z$} (m-3-4)
;
\path[-,font=\scriptsize]
(m-1-3) edge [double, thick, double distance=2pt] (m-1-4)
(m-2-1) edge [double, thick, double distance=2pt] (m-3-1)
(m-2-2) edge [double, thick, double distance=2pt] (m-3-2)
;
\end{tikzpicture}
}
\end{equation}
\hfill(where $\beta^Z_0 := i^Z \circ \gamma^Z$)
\begin{equation}\label{fig8}
\parbox{1.75in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.25ex, text depth=0.25ex]
{
K^X \& K^X \& {} \\
A^X \& A^Y \& A^Z \\
X \& Y' \& A^Z \\
};
\path[->]
(m-2-1)-- node[pos=0.5] {\footnotesize$\mbox{\bf po}$} (m-3-2)
;
\path[>->]
(m-1-1) edge node[left] {\footnotesize$i^X$} (m-2-1)
(m-1-2) edge node[right] {\footnotesize$\overline{i}^X$} (m-2-2)
(m-2-1) edge node[above] {\footnotesize$\hat{f}$} (m-2-2)
(m-3-1) edge node[below] {\footnotesize$f'$} (m-3-2)
;
\path[->>]
(m-2-1) edge node[left] {\footnotesize$\alpha^X$} (m-3-1)
(m-2-2) edge node[right] {\footnotesize$\overline{\alpha}^X$} (m-3-2)
(m-2-2) edge node[above] {\footnotesize$\hat{g}$} (m-2-3)
(m-3-2) edge node[below] {\footnotesize$g'$} (m-3-3)
;
\path[-,font=\scriptsize]
(m-1-1) edge [double, thick, double distance=2pt] (m-1-2)
(m-2-3) edge [double, thick, double distance=2pt] (m-3-3)
;
\end{tikzpicture}
}
\end{equation}
\hfill (where $A^Y \in \mathcal{A}$ since $\mathcal{A}$ is closed under extensions)
\begin{equation}\label{fig9}
\parbox{2.5in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.25ex, text depth=0.25ex]
{
{} \& K^X \& K^X \& {} \\
B^Z_1 \& Q \& A^Y \& Y \\
B^Z_1 \& B^Z_0 \& Y' \& Y \\
};
\path[->]
(m-2-2)-- node[pos=0.5] {\footnotesize$\mbox{\bf pb}$} (m-3-3)
(m-2-2) edge (m-2-3)
(m-3-2) edge node[below] {\footnotesize$\hat{\beta}^Z_0$} (m-3-3)
;
\path[>->]
(m-1-2) edge (m-2-2)
(m-1-3) edge node[right] {\footnotesize$\overline{i}^X$} (m-2-3)
(m-2-1) edge (m-2-2)
(m-3-1) edge node[below] {\footnotesize$\hat{\beta}^Z_1$} (m-3-2)
;
\path[->>]
(m-2-3) edge node[above] {\footnotesize$\alpha^Y$} (m-2-4)
(m-3-3) edge node[below] {\footnotesize$\tilde{\alpha}^Z$} (m-3-4)
(m-2-2) edge (m-3-2)
(m-2-3) edge node[right] {\footnotesize$\overline{\alpha}^X$} (m-3-3)
;
\path[-,font=\scriptsize]
(m-1-2) edge [double, thick, double distance=2pt] (m-1-3)
(m-2-1) edge [double, thick, double distance=2pt] (m-3-1)
(m-2-4) edge [double, thick, double distance=2pt] (m-3-4)
;
\end{tikzpicture}
}
\end{equation}
Since $B^Z_0 \in \mathcal{B}$, $K^X \in \mathcal{B}^\wedge_1$ and $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{B},\mathcal{B}^\wedge_1) = 0$, the column $K^X \rightarrowtail Q \twoheadrightarrow B^Z_0$ is split exact and so $Q \simeq B^Z_0 \oplus K^X$. Thus, the diagram \eqref{fig9} can be rewritten as:
\begin{equation}\label{fig9mod}
\parbox{3.5in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=3.5em, column sep=4em, text height=1.25ex, text depth=0.25ex]
{
{} \& K^X \& K^X \& {} \\
B^Z_1 \& B^Z_0 \oplus K^X \& A^Y \& Y \\
B^Z_1 \& B^Z_0 \& Y' \& Y \\
};
\path[->]
(m-2-2) edge node[above] {\scriptsize$\left( \begin{array}{cc} a & \overline{i}^X \end{array} \right)$} (m-2-3)
(m-3-2) edge node[below] {\footnotesize$\hat{\beta}^Z_0$} (m-3-3)
;
\path[>->]
(m-1-2) edge node[left] {\scriptsize$\left( \begin{array}{c} 0 \\ {\rm id}_{K^X} \end{array} \right)$} (m-2-2)
(m-1-3) edge node[right] {\footnotesize$\overline{i}^X$} (m-2-3)
(m-2-1) edge node[below] {\scriptsize$\left( \begin{array}{c} \beta^Z_1 \\ 0 \end{array} \right)$} (m-2-2)
(m-3-1) edge node[below] {\scriptsize$\hat{\beta}^Z_1$} (m-3-2)
;
\path[->>]
(m-2-3) edge node[above] {\footnotesize$\alpha^Y$} (m-2-4)
(m-3-3) edge node[below] {\footnotesize$\tilde{\alpha}^Z$} (m-3-4)
(m-2-2) edge node[right] {\scriptsize$\left( \begin{array}{cc} {\rm id}_{B^Z_0} & 0 \end{array} \right)$} (m-3-2)
(m-2-3) edge node[right] {\scriptsize$\overline{\alpha}^X$} (m-3-3)
;
\path[-,font=\scriptsize]
(m-1-2) edge [double, thick, double distance=2pt] (m-1-3)
(m-2-1) edge [double, thick, double distance=2pt] (m-3-1)
(m-2-4) edge [double, thick, double distance=2pt] (m-3-4)
;
\end{tikzpicture}
}
\end{equation}
The existence of the arrow $a \colon B^Z_0 \to A^Y$ is a consequence of the pullback construction, and satisfies the relations
\begin{align}\label{eqn:arrow_a}
\overline{\alpha}^X \circ a & = \hat{\beta}^Z_0.
\end{align}
Moreover, following the arguments that show the commutativity of the diagram \eqref{fig_caso_n1} in Remark \ref{rem:special_n-1}, we have the following commutative diagram with exact rows and columns:
\begin{equation}\label{fig_kernel}
\parbox{1.75in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.25ex, text depth=0.25ex]
{
K^X \& K^Y \& K^Z \\
A^X \& A^Y \& A^Z \\
X \& Y \& Z \\
};
\path[>->]
(m-1-1) edge node[left] {\footnotesize$i^X$} (m-2-1)
(m-1-2) edge node[left] {\footnotesize$i^Y$} (m-2-2)
(m-1-3) edge node[left] {\footnotesize$i^Z$} (m-2-3)
(m-1-1) edge node[above] {\footnotesize$\tilde{f}$} (m-1-2)
(m-2-1) edge node[above] {\footnotesize$\hat{f}$} (m-2-2)
(m-3-1) edge node[above] {\footnotesize$f$} (m-3-2)
;
\path[->>]
(m-2-1) edge node[left] {\footnotesize$\alpha^X$} (m-3-1)
(m-2-2) edge node[left] {\footnotesize$\alpha^Y$} (m-3-2)
(m-2-3) edge node[left] {\footnotesize$\alpha^Z$} (m-3-3)
(m-1-2) edge node[above] {\footnotesize$\tilde{g}$} (m-1-3)
(m-2-2) edge node[above] {\footnotesize$\hat{g}$} (m-2-3)
(m-3-2) edge node[above] {\footnotesize$g$} (m-3-3)
;
\end{tikzpicture}
}
\end{equation}
On the other hand, let us add the identity ${\rm id}_{B^Z_0}$ to the sequence \eqref{eqn:BX1BX0KX}, in order to obtain the exact sequence
\[
0 \to B^X_1 \xrightarrow{{\scriptsize\left( \begin{array}{c} 0 \\ \beta^X_1 \end{array} \right)}} B^Z_0 \oplus B^X_0 \xrightarrow{\scriptsize{\left( \begin{array}{cc} {\rm id}_{B^Z_0} & 0 \\ 0 & \gamma^X \end{array} \right)}} B^Z_0 \oplus K^X \to 0.
\]
Now take the pullback of $B^Z_1 \to B^Z_0 \oplus K^X \leftarrow B^Z_0 \oplus B^X_0$ in order to obtain the following commutative diagram with exact rows and columns:
\begin{equation}\label{fig10}
\parbox{2.5in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=4em, column sep=5.5em, text height=1.25ex, text depth=0.25ex]
{
B^X_1 \& B^X_1 \& {} \\
B^Y_1 \& B^Z_0 \oplus B^X_0 \& K^Y \\
B^Z_1 \& B^Z_0 \oplus K^X \& K^Y \\
};
\path[->]
(m-2-1)-- node[pos=0.5] {\footnotesize$\mbox{\bf pb}$} (m-3-2)
;
\path[>->]
(m-1-1) edge node[left] {\footnotesize$f_1$} (m-2-1)
(m-1-2) edge node[right] {\scriptsize$\left( \begin{array}{c} 0 \\ \beta^X_1 \end{array} \right)$} (m-2-2)
(m-2-1) edge node[above] {\scriptsize$\left( \begin{array}{c} \beta^Z_1 \circ g_1 \\ b \end{array} \right)$} (m-2-2)
(m-3-1) edge node[below] {\scriptsize$\left( \begin{array}{c} \beta^Z_1 \\ 0 \end{array} \right)$} (m-3-2)
;
\path[->>]
(m-2-1) edge node[left] {\footnotesize$g_1$} (m-3-1)
(m-2-2) edge node[right] {\scriptsize$\left( \begin{array}{cc} {\rm id}_{B^Z_0} & 0 \\ 0 & \gamma^X \end{array} \right)$} (m-3-2)
(m-2-2) edge node[above] {\scriptsize$\left( \begin{array}{cc} c & \tilde{f} \circ \gamma^X \end{array} \right)$} (m-2-3)
(m-3-2) edge node[below] {\scriptsize$\left( \begin{array}{cc} c & \tilde{f} \end{array} \right)$} (m-3-3)
;
\path[-,font=\scriptsize]
(m-1-1) edge [double, thick, double distance=2pt] (m-1-2)
(m-2-3) edge [double, thick, double distance=2pt] (m-3-3)
;
\end{tikzpicture}
}
\end{equation}
where $b \colon B^Y_1 \to B^X_0$ and $c \colon B^Z_0 \to K^Y$ are arrows given by te pullback construction that satisfy the following relations:
\begin{align}
b \circ f_1 & = \beta^X_1, \label{eqn:arrow_b} \\
c \circ \beta^Z_1 & = 0. \label{eqn:arrow_c}
\end{align}
Finally, we form the following diagram with exact rows and columns:
\begin{equation}\label{fig11}
\parbox{4.25in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=4em, column sep=5.5em, text height=1.25ex, text depth=0.25ex]
{
B^X_1 \& B^X_0 \& A^X \& X \\
B^Y_1 \& B^Z_0 \oplus B^X_0 \& A^Y \& Y \\
B^Z_1 \& B^Z_0 \& A^Z \& Z \\
};
\path[->]
(m-1-2) edge node[above] {\footnotesize$\beta^X_0$} (m-1-3)
(m-2-2) edge node[below] {\scriptsize$\left( \begin{array}{cc} a & \hat{f} \circ \beta^X_0 \end{array} \right)$} (m-2-3)
(m-3-2) edge node[below] {\footnotesize$\beta^Z_0$} (m-3-3)
;
\path[>->]
(m-1-1)-- node[pos=0.25] {\scriptsize$\circled{1}$} (m-2-2)
(m-1-2)-- node[pos=0.75] {\scriptsize$\circled{2}$} (m-2-3)
(m-1-3)-- node[pos=0.5] {\scriptsize$\circled{3}$} (m-2-4)
(m-2-1)-- node[pos=0.15] {\scriptsize$\circled{4}$} (m-3-2)
(m-2-2)-- node[pos=0.75] {\scriptsize$\circled{5}$} (m-3-3)
(m-2-3)-- node[pos=0.5] {\scriptsize$\circled{6}$} (m-3-4)
(m-1-1) edge node[above] {\footnotesize$\beta^X_1$} (m-1-2)
(m-2-1) edge node[above] {\scriptsize$\left( \begin{array}{c} \beta^Z_1 \circ g_1 \\ b \end{array} \right)$} (m-2-2)
(m-3-1) edge node[below] {\footnotesize$\beta^Z_1$} (m-3-2)
(m-1-1) edge node[left] {\footnotesize$f_1$} (m-2-1)
(m-1-2) edge node[right] {\scriptsize$\left( \begin{array}{c} 0 \\ {\rm id}_{B^X_0} \end{array} \right)$} (m-2-2)
(m-1-3) edge node[left] {\footnotesize$\hat{f}$} (m-2-3)
(m-1-4) edge node[left] {\footnotesize$f$} (m-2-4)
;
\path[->>]
(m-1-3) edge node[above] {\footnotesize$\alpha^X$} (m-1-4)
(m-2-3) edge node[above] {\footnotesize$\alpha^Y$} (m-2-4)
(m-3-3) edge node[below] {\footnotesize$\alpha^Z$} (m-3-4)
(m-2-1) edge node[left] {\footnotesize$g_1$} (m-3-1)
(m-2-2) edge node[left] {\footnotesize$\left( \begin{array}{cc} {\rm id}_{B^Z_0} & 0 \end{array} \right)$} (m-3-2)
(m-2-3) edge node[left] {\footnotesize$\hat{g}$} (m-3-3)
(m-2-4) edge node[left] {\footnotesize$g$} (m-3-4)
;
\end{tikzpicture}
}
\end{equation}
where $\beta^X_0 := i^X \circ \gamma^X$. The commutativity of squares $\scriptsize\circled{3}$ and $\scriptsize\circled{6}$ was already verified for the diagram \eqref{fig_kernel}, while for $\scriptsize\circled{2}$ and $\scriptsize\circled{4}$ is clear. We check that the remaining squares also commute:
\begin{itemize}
\item[\scriptsize$\circled{1}$] We have by the first equality in \eqref{eqn:arrow_b} that
\[
\left( \begin{array}{c} \beta^Z_1 \circ g_1 \\ b \end{array} \right) \circ f_1 = \left( \begin{array}{c} \beta^Z_1 \circ g_1 \circ f_1 \\ b \circ f_1 \end{array} \right) = \left( \begin{array}{c} 0 \\ \beta^X_1 \end{array} \right) = \left( \begin{array}{c} 0 \\ {\rm id}_{B^X_0} \end{array} \right) \circ \beta^X_1.
\]
\item[\scriptsize$\circled{5}$] Using the diagram \eqref{fig8}, we have that $\hat{g} = g' \circ \overline{\alpha}^X$, and so $\hat{g} \circ a = g' \circ \overline{\alpha}^X \circ a$, where $\overline{\alpha}^X \circ a = \hat{\beta}^Z_0$ by \eqref{eqn:arrow_a} and $g' \circ \hat{\beta}^Z_0 = \beta^Z_0$ by \eqref{fig7}. Thus,
\begin{align*}
\hat{g} \circ \left( \begin{array}{cc} a & \hat{f} \circ \beta^X_0 \end{array} \right) & = \left( \begin{array}{cc} \hat{g} \circ a & \hat{g} \circ \hat{f} \circ \beta^X_0 \end{array} \right) = \left( \begin{array}{cc} g' \circ \hat{\beta}^Z_0 & 0 \end{array} \right) = \left( \begin{array}{cc} \beta^Z_0 & 0 \end{array} \right) = \beta^Z_0 \circ \left( \begin{array}{cc} {\rm id}_{B^Z_0} & 0 \end{array} \right).
\end{align*}
\end{itemize}
Note that $B^Z_0 \oplus B^X_0, B^Y_1 \in \mathcal{B}$ since $\mathcal{B}$ is closed under extensions. Therefore, the central row of \eqref{fig11} defines a special $(\mathcal{A},2,\mathcal{B})$-precover compatible with $\rho_X$ and $\rho_Z$.
\end{proof}
\begin{remark}
One can include the case $n = 1$ in the previous theorem, for which the hypothesis $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{B,B}) = 0$ is not needed (see Remark~\ref{rem:special_n-1}).
\end{remark}
\section{\textbf{Applications and examples}}\label{sec:applications}
Below we present some examples of (left and right) $n$-cotorsion pairs along with some applications, which are related to the characterisation of certain rings as well as to finding covers and envelopes with the unique mapping property.
We need to mention a couple of considerations. Let us denote the projective and injective dimensions of an object $C$ in an abelian category $\mathcal{C}$ by ${\rm pd}(C)$ and ${\rm id}(C)$, respectively. Recall that ${\rm pd}(C)$ is defined as the smallest nonnegative integer $m \geq 0$ such that $\mathsf{Ext}^i_{\mathcal{C}}(C,\mathcal{C}) = 0$ for every $i > m$. If such $m$ does not exist, one sets ${\rm pd}(C) := \infty$. Note that if $\mathcal{C}$ has enough projectives, then ${\rm pd}(C)$ coincides with the $\mathcal{P}(\mathcal{C})$-resolution dimension of $C$. Similarly ${\rm id}(C)$, defined dually, coincides with the $\mathcal{I}(\mathcal{C})$-coresolution dimension of $C$ in the case where $\mathcal{C}$ has enough injectives. For simplicity, if $\mathcal{C} = \mathsf{Mod}(R)$, we write the classes $\mathcal{P}(\mathsf{Mod}(R))$ and $\mathcal{I}(\mathsf{Mod}(R))$ as $\mathcal{P}(R)$ and $\mathcal{I}(R)$, respectively.
\subsection*{\textbf{Gorenstein projective modules}}
Recall the classes $\mathcal{GP}(R)$ and $\mathcal{GI}(R)$ of Gorenstein projective and Gorenstein injective $R$-modules from Example~\ref{ex:special_AkB-precover}. Over an arbitrary ring $R$, it is well known by \cite[Theorem 2.5]{Holm} that $\mathcal{GP}(R)$ is closed under direct summands. On the other hand, \cite[Proposition 2.3]{Holm} asserts that $\mathsf{Ext}^i_R(C,P) = 0$ for every $C \in \mathcal{GP}(R)$, $P \in \mathcal{P}(R)$ and $i \geq 1$. So $(\mathcal{GP}(R),\mathcal{P}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$ if, and only if, every module has a Gorenstein projective special precover whose kernel has projective dimension at most $n-1$. The most obvious choice of a ring $R$ over which the latter condition holds, is when $R$ is an $n$-Iwanaga-Gorenstein ring, that is, $R$ is two-sided noetherian with ${\rm id}({}_R R) = {\rm id}(R_R) = n$. Over such rings $R$, it is known that every module has Gorenstein projective dimension at most $n$. Therefore, we have the following example of a left $n$-cotorsion pair, which is also a consequence of Proposition~\ref{equiv hered y n-cot}, Hovey's \cite[Theorem 8.3]{Hovey} and Holm's \cite[Theorem 2.5]{Holm}.
\begin{example}\label{ex:GProj_ncot}
Let $R$ be an $n$-Iwanaga-Gorenstein ring with $n\geq 1$. Then, $(\mathcal{GP}(R),\mathcal{P}(R))$ is a left $n$-cotorsion pair and $(\mathcal{I}(R),\mathcal{GI}(R))$ is a right $n$-cotorsion pair in $\mathsf{Mod}(R)$.
For the case $n = 0$, a $0$-Iwanaga-Gorenstein ring is just a \textbf{quasi-Frobenius ring} (or \textbf{QF ring}, for short) by Bland's \cite[Proposition 10.2.14]{Bland}. Moreover, $\mathcal{P}(R) = \mathcal{I}(R)$ and $\mathcal{P}(R^{\rm op}) = \mathcal{I}(R^{\rm op})$ by \cite[Proposition 10.2.15]{Bland}, if $R$ is a QF ring, and so one can note that every module in $\mathsf{Mod}(R)$ is Gorenstein projective. Thus, in the case $n = 0$, $(\mathcal{GP}(R),\mathcal{P}(R))$ and $(\mathcal{I}(R),\mathcal{GI}(R))$ coincide with the trivial cotorsion pairs $(\mathsf{Mod}(R),\mathcal{I}(R))$ and $(\mathcal{P}(R),\mathsf{Mod}(R))$, respectively.
\end{example}
The previous example is not necessarily an equivalence. Indeed, there are slightly more general conditions for $R$ under which $(\mathcal{GP}(R),\mathcal{P}(R))$ is still a left $n$-cotorsion pair in $\mathsf{Mod}(R)$. These conditions will involve the following two relative homological dimensions:
\begin{align}
{\rm pd}\,(\mathcal{I}(R)):= {\rm sup}\{ {\rm pd}(I) \mbox{ : } I \in \mathcal{I}(R) \}, \label{eqn:pdi} \\
{\rm id}\,(\mathcal{P}(R)):={\rm sup}\{ {\rm id}(P) \mbox{ : } P \in \mathcal{P}(R) \}. \label{eqn:idp}
\end{align}
Recall from Beligiannis and Reiten's \cite[Definitions 2.1 and 2.5]{BR07} that a ring $R$ is a \emph{left Gorenstein ring} if
$\mathsf{Mod}(R)$ is a Gorenstein category, that is, if ${\rm pd}\,(\mathcal{I}(R)) $ and $ {\rm id}\,(\mathcal{P}(R))$ are both finite. Every $n$-Iwanaga-Gorenstein ring is a Gorenstein ring, but the converse is not necessarily true.
Below we give a characterisation and properties of Gorenstein rings in terms of left and right $n$-cotorsion pairs involving the classes $\mathcal{GP}(R)$, $\mathcal{GI}(R)$, $\mathcal{P}(R)$ and $\mathcal{I}(R)$, the homological dimensions \eqref{eqn:pdi} and \eqref{eqn:idp}, and the global Gorenstein homological dimensions. Recall that the (\emph{left}) \emph{global Gorenstein projective dimension} of a ring $R$ is defined as the supremum
\[
{\rm gl.GPD}(R) := {\rm sup}\{ {\rm Gpd}(M) \mbox{ : } M \in \mathsf{Mod}(R) \}.
\]
Dually, we have the \emph{global Gorenstein injective dimension} ${\rm gl.GID}(R)$ of $R$.
\begin{proposition}\label{prop:ncot_spli_silp}
The following conditions hold true for any ring $R$:
\begin{enumerate}
\item If $(\mathcal{GP}(R),\mathcal{P}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$, then
\[
{\mathrm{gl.GPD}}(R) = {\rm id}\,(\mathcal{P}(R))\leq n.
\]
Dually, if $(\mathcal{I}(R),\mathcal{GI}(R))$ is a right $m$-cotorsion pair in $\mathsf{Mod}(R)$, then
\[
{\mathrm{gl.GID}}(R) = {\rm pd}\,(\mathcal{I}(R))\leq m.
\]
\item The following assertions are equivalent:
\begin{itemize}
\item[(a)] $R$ is a left Gorenstein ring which is not QF.
\item[(b)] There exist integers $n, m \geq 1$ such that $(\mathcal{GP}(R),\mathcal{P}(R))$ is a left $n$-cotorsion pair and $(\mathcal{I}(R),\mathcal{GI}(R))$ is a right $m$-cotorsion pair in $\mathsf{Mod}(R)$.
\end{itemize}
Moreover, if any of the previous holds true, we can choose
\[
n = m = {\rm id}\,(\mathcal{P}(R)) = {\rm pd}\,(\mathcal{I}(R)).
\]
\end{enumerate}
\end{proposition}
\begin{proof}
Let us first show part (1). Suppose $(\mathcal{GP}(R),\mathcal{P}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$. Then, every module has Gorenstein projective dimension at most $n$. It follows by \cite[Corollary 5.19]{BecerrilMendozaSantiago} that ${\mathrm{gl.GPD}}(R) = {\rm id}\,(\mathcal{P}(R))\leq n$.
Now for part (2), let us show first the implication (a) $\Rightarrow$ (b). If $R$ is a Gorenstein ring which is not QF, we have that both ${\rm pd}\,(\mathcal{I}(R)) $ and $ {\rm id}\,(\mathcal{P}(R))$ are finite. By \cite[Proposition VII.1.3 (vi)]{BR07}, we have ${\rm pd}\,(\mathcal{I}(R)) = {\rm id}\,(\mathcal{P}(R))$. Thus, let $n := {\rm id}\,(\mathcal{P}(R))$ and note that $n \geq 1$ since $R$ is not QF. The first two conditions of Definition~\ref{def:ncotorsion} are well known for $\mathcal{GP}(R)$ and $\mathcal{P}(R)$. By \cite[Theorem VII.2.2 ($\gamma$)]{BR07}, we have that every module has Gorenstein projective dimension at most $n$. Thus, the remaining condition (3) in Definition~\ref{def:ncotorsion} follows after setting $\mathcal{X} = \mathcal{GP}(R)$ and $\omega = \mathcal{P}(R)$ in \cite[Theorem 2.8]{BMPS}. In a similar way, we can show that $(\mathcal{I}(R),\mathcal{GI}(R))$ is a right $n$-cotorsion pair in $\mathsf{Mod}(R)$.
Finally, the converse implication (b) $\Rightarrow$ (a) in part (2) follows by part (1).
\end{proof}
For the following observations, recall that an $R$-module $M \in \mathsf{Mod}(R)$ is \emph{Gorenstein flat} if $M \simeq Z_0(F)$, where $F = (F_m)_{m \in \mathbb{Z}}$ is an exact complex of flat $R$-modules such that for every injective right $R$-module $E \in \mathcal{I}(R^{\rm op})$, the induced complex of abelian groups
\[
E \otimes_R F = \cdots \to E \otimes_R F_1 \to E \otimes_R F_0 \to E \otimes_R F_{-1} \to \cdots
\]
is exact. We shall denote the class of Gorenstein flat $R$-modules by $\mathcal{GF}(R)$.
\begin{remark}\label{rem:consequences_InjGInj}
Let us mention some other consequences of having a right $n$-cotorsion pair of $R$-modules $(\mathcal{I}(R),\mathcal{GI}(R))$ for some $n \geq 1$.
\begin{enumerate}
\item If $R$ is a right coherent ring, then the \textbf{Pontryagin dual} $M^+ := \mathsf{Hom}_{\mathbb{Z}}(M,\mathbb{Q / Z})$ of every Gorenstein injective left $R$-module $M \in \mathsf{Mod}(R)$ is a Gorenstein flat right $R$-module. For this, consider the value
\[
\mathrm{fd}\,(\mathcal{I}(R)):= {\rm sup}\{ {\rm fd}(I) \mbox{ : } I \in \mathcal{I}(R) \}\leq{\rm pd}\,(\mathcal{I}(R)).
\]
Under the assumption that $(\mathcal{I}(R),\mathcal{GI}(R))$ is a right $n$-cotorsion pair, we have by Proposition~\ref{prop:ncot_spli_silp} that $\mathrm{fd}\,(\mathcal{I}(R)) \leq n$. Thus, Iacob's \cite[Theorem 4]{IacobAuslandercondition} implies that $M^+ \in \mathcal{GF}(R{}^{\rm op})$ for every $M \in \mathcal{GI}(R)$.
\item If $R$ is a left Noetherian and right coherent ring, then both ${\rm pd}\,(\mathcal{I}(R)) $ and $ {\rm id}\,(\mathcal{P}(R))$ are finite. Indeed, we already know ${\rm pd}\,(\mathcal{I}(R)) \leq n$ by Proposition \ref{prop:ncot_spli_silp}. Now let $P$ be a projective $R$-module. Then by Fieldhouse's \cite[Theorem 2.2]{Fieldhouse}, we have that ${\rm id}(P) = {\rm fd}(P^+)$, where $P^+$ is an injective $R^{\rm op}$-module. By (1) above, it follows that ${\rm id}(P) = {\rm fd}(P^+) \leq n$, and hence ${\rm id}\,(\mathcal{P}(R)) \leq n$.
\item If $R$ is a two sided Noetherian ring, then $\mathcal{GI}(R)$ is covering and $\mathcal{GF}(R)$ is preenveloping. This follows by part (1) of Propositon~\ref{prop:ncot_spli_silp} and \cite[Theorem 4]{IacobAuslandercondition}.
\item If $M$ is an $R$-module with finite injective dimension, then $M$ has projective dimension at most $n.$ Indeed, let $M\in\mathcal{I}(R)^\vee.$ Then, by \cite[Lemma 2.6]{BecerrilMendozaSantiago}, we get ${\rm pd}\,(M)\leq{\rm pd}\,(\mathcal{I}(R)^\vee)={\rm pd}\,(\mathcal{I}(R))\leq n.$
\item By (4) and \cite[Proposition VII.1.3(iii)]{BR07}, it follows that the \emph{big finitistic injective dimension} of $R$ is finite. Specifically,
\[
{\rm FID}(R) := {\rm sup}\{ {\rm id}(M) \mbox{ : $M$ has finite injective dimension} \} \leq n.
\]
\end{enumerate}
\end{remark}
In what remains of this section, we mention some consequences of Section~\ref{sec:approximations}. Most of our comments below have to do with right and left approximations by $\mathcal{GP}(R)$ and $\mathcal{GI}(R)$ with the unique mapping property. We begin with the following application of Corollary~\ref{Aperp y Bvee} in the context of Gorenstein homological algebra.
\begin{corollary}\label{corUMP1}
Let $R$ be an $n$-Iwanaga-Gorenstein ring with $n \geq 1$. Then, the following equalities hold:
\begin{enumerate}
\item $\mathsf{Mod}(R) = \mathcal{GP}(R)_n^\wedge = {}^{\perp_{n}}(\mathcal{P}(R)^\wedge_{n-1})$.
\item $\mathsf{Mod}(R) = \mathcal{GI}(R)_{n}^{\vee} = (\mathcal{I}(R)^\vee_{n-1})^{\perp_{n}}$.
\end{enumerate}
\end{corollary}
\begin{proof}
We only focus on the Gorenstein projective case (1). Firstly, by Example~\ref{ex:GProj_ncot} we have that $(\mathcal{GP}(R),\mathcal{P}(R))$ is a left $n$-cotorsion pair. Then, by Proposition~\ref{prop:ncot_spli_silp} (1) the first equality $\mathsf{Mod}(R) = \mathcal{GP}(R)_n^\wedge$ holds true. The equality $\mathsf{Mod}(R) = {}^{\perp_{n}}(\mathcal{P}(R)^\wedge_{n-1})$, on the other hand, will follow by condition (c) in Corollary~\ref{Aperp y Bvee} after showing that $\mathcal{GP}(R) = {}^{\perp_1}(\mathcal{P}(R)^\wedge_{n-1})$ and that $(\mathcal{GP}(R),\mathcal{P}(R)^\wedge_{n-1})$ is a left $(n+1)$-cotorsion pair in $\mathsf{Mod}(R)$. The former follows by the already known fact that $(\mathcal{GP}(R),\mathcal{P}(R))$ is a left $n$-cotorsion pair and by Theorem~\ref{theo:left-n-cotorsion}, while the latter can be noticed from the inclusion $\mathcal{P}(R)^\wedge_{n-1}\subseteq (\mathcal{P}(R)^\wedge_{n-1})^\wedge_n.$
\end{proof}
It is known that every module over an $n$-Iwanaga-Gorenstein ring has a Gorenstein injective cover (see, for instance \cite[Theorem 11.1.3]{EJ}). We can deduce a stronger assertion for the case $n = 2$, due to the dual of Corollary~\ref{corUMP2}.
\begin{corollary}\label{coro:GI_unique_mapping}
Let $R$ be a $2$-Iwanaga-Gorenstein ring. Then, every module has a Gorenstein injective cover with the unique mapping property.
\end{corollary}
\begin{proof}
From the dual of the proof of Corollary~\ref{corUMP1}, for the case $n = 2$, we can note that the pair $(\mathcal{I}(R)^\vee_1,\mathcal{GI}(R))$ is a right $3$-cotorsion pair such that $\mathcal{GI}(R) = (\mathcal{I}(R)^\vee_1)^{\perp_1}$. Then, the result follows by dual of Corollary~\ref{corUMP2}, since over a $2$-Iwanaga-Gorenstein ring, every module has Gorenstein injective dimension at most $2$.
\end{proof}
The existence of Gorenstein projective envelopes with the unique mapping property, on the other hand, has been studied by Mao in \cite{MaoPiCoherent}. Mao establishes a series of equivalent conditions under which a finitely generated module over a ring $R$ has a Gorenstein projective envelope with the unique mapping property \cite[Theorem 3.7]{MaoPiCoherent}. For (not necessarily finitely generated) modules over a $2$-Iwanaga-Gorenstein ring, we can say that if a module has a Gorenstein projective envelope, then we can always find for this module a Gorenstein projective envelope with the unique mapping property.
\begin{corollary}\label{coro:GP_unique_mapping}
Let $R$ be a $2$-Iwanaga-Gorenstein ring. Then, the following conditions are equivalent.
\begin{itemize}
\item[(a)] Every module has a Gorenstein projective envelope.
\item[(b)] Every module has a Gorenstein projective envelope with the unique mapping property.
\end{itemize}
\end{corollary}
\begin{proof}
It follows by Corollary \ref{corUMP2} after noting that $(\mathcal{GP}(R),\mathcal{P}(R))$ is a left $3$-cotorsion pair in $\mathsf{Mod}(R)$ over any $2$-Iwanaga-Gorenstein ring $R$.
\end{proof}
\subsection*{\textbf{Ding projective modules}}
In what follows, let us denote by $\mathcal{F}(R)$ the class of flat $R$-modules. Recall from Gillespie's \cite[Definition~3.7]{GillespieDing} that an $R$-module $M$ is \emph{Ding projective} (also called \emph{strongly Gorenstein flat} in Ding, Li and Mao's \cite{DLM}) if $M = Z_0(P)$ for some exact and $\mathsf{Hom}_R(-,\mathcal{F}(R))$-acyclic complex $P$ of projective $R$-modules. Dually, \emph{Ding injective} $R$-modules are defined as cycles in an exact and $\mathsf{Hom}_R(\mathcal{AP}(R),-)$-acyclic complexes of injective $R$-modules. We denote the classes of Ding projective and Ding injective $R$-modules by $\mathcal{DP}(R)$ and $\mathcal{DI}(R)$, respectively.
After a careful revision of the results cited from \cite{Holm} in the previous example, we can assert that the same results carry over to the context of Ding projective modules. Specifically, one can show that, over an arbitrary ring $R$, the class $\mathcal{DP}(R)$ is closed under direct summands and that $\mathsf{Ext}^i_R(C,F) = 0$ for every $C \in \mathcal{DP}(R)$, $F \in \mathcal{F}(R)$ and $i \geq 1$. The dual statements hold for the classes $\mathcal{DI}(R)$ and $\mathcal{AP}(R)$. On the other hand, condition (3) in Definition~\ref{def:ncotorsion} and its dual are valid for certain rings introduced by J. Chen and N. Ding \cite{DingChen93,DingChen96}. These rings $R$ are known as \emph{$n$-FC rings} (or \emph{Ding-Chen rings}): $R$ is left and right coherent and ${\rm apd}({}_R R) = {\rm apd}(R_R) = n$.
Similar to Gorenstein projective and Gorenstein injective dimensions, the \emph{Ding projective} and \emph{Ding injective dimensions} of a module $M \in \mathsf{Mod}(R)$, denoted by ${\rm Dpd}(R)$ and ${\rm Did}(M)$, are defined as the $\mathcal{DP}(R)$-resolution and the $\mathcal{DI}(R)$-coresolution dimensions of $M$, respectively. For these two homological dimensions, it is not true in general that the equality $\mathsf{Mod}(R) = \mathcal{DP}(R)_n^\wedge$ holds for an $n$-FC ring $R$. An example of such ring $R$ for which $\mathsf{Mod}(R) \neq \mathcal{DP}(R)_n^\wedge$ is constructed by Wang in \cite[Example 3.3]{Wang}. It follows that we can not always have the Ding projective analog of Example~\ref{ex:GProj_ncot}. As a matter of fact, the condition $\mathsf{Mod}(R) = \mathcal{DP}(R)_n^\wedge$ is strong enough to guarantee the existence of $(\mathcal{DP}(R),\mathcal{F}(R))$ as a left $n$-cotorsion pair in $\mathsf{Mod}(R)$.
For the rest of this section, recall that the global Ding projective and Ding injective dimensions of a ring $R$ are defined by:
\begin{align*}
{\rm gl.DPD}(R) & = {\rm sup}\{ {\rm Dpd}(M) \mbox{ : } M \in \mathsf{Mod}(R) \}, \\
{\rm gl.DID}(R) & = {\rm sup}\{ {\rm Did}(M) \mbox{ : } M \in \mathsf{Mod}(R) \}.
\end{align*}
\begin{example}\label{ex:DP_n-cotorsion}
Let $n \geq 1$ be an integer and $R$ be any ring with ${\mathrm{gl.DPD}}(R) \leq n.$ Then, the pair $(\mathcal{DP}(R),\mathcal{F}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$. Indeed, by the previous comments it suffices to show that for every module $M \in \mathsf{Mod}(R)$ there is an epimorphism $P \twoheadrightarrow M$ with $P \in \mathcal{DP}(R)$ and kernel in $\mathcal{F}(R)^\wedge_{n-1}$. This follows by setting $\mathcal{X} = \mathcal{DP}(R)$ and $\omega := \mathcal{P}(R)\subseteq\mathcal{F}(R)$ in \cite[Theorem 2.8]{BMPS}, since ${\rm Dpd}(M) \leq n$.
\end{example}
We can obtain characterisations of Von Neumann regular rings by considering the situation in which $(\mathcal{DP}(R),\mathcal{F}(R))$ is a left and right $n$-cotorsion pair in $\mathsf{Mod}(R)$.
\begin{proposition}
For any ring $R$, the following conditions are equivalent.
\begin{itemize}
\item[(a)] $(\mathcal{DP}(R),\mathcal{F}(R))$ is an $n$-cotorsion pair in $\mathsf{Mod}(R)$ and $\mathcal{DP}(R) \subseteq \mathcal{F}(R)$.
\item[(b)] $R$ is a Von Neumann regular ring (that is, $\mathcal{F}(R) = \mathsf{Mod}(R)$).
\end{itemize}
\end{proposition}
\begin{proof}
Indeed, let us suppose that (a) holds true. Then, by using the fact that $\mathcal{DP}(R)$ is resolving, we get (b) from Remark~\ref{Rk-n-cot-trivial} (2).
Assume now that $\mathcal{F}(R) = \mathsf{Mod}(R)$. In order to prove (a), it is enough to show that $\mathcal{DP}(R) = \mathcal{P}(R)$. Note that in this case, we can choose any $n \geq 1$. Let $M \in \mathcal{DP}(R).$ Then, there is an exact sequence
\[
\eta \colon 0 \to M \to P \to M' \to 0,
\]
where $P \in \mathcal{P}(R)$ and $M' \in \mathcal{DP}(R)$. Since $\mathcal{F}(R) = \mathsf{Mod}(R)$, it follows that $\mathsf{Hom}_R(\eta,M)$ is exact and thus $\eta$ splits, proving that $M \in \mathcal{P}(R)$.
\end{proof}
Let us give in the next result some finiteness conditions for the global Ding injective dimension ${\mathrm{gl.DID}}(R)$ of any ring $R$.
\begin{lemma}\label{glDIDfinite}
For any ring $R$, the following statements are equivalent:
\begin{itemize}
\item[(a)] $(\mathcal{I}(R)^\vee,\mathcal{DI}(R))$ is a hereditary complete cotorsion pair in $\mathsf{Mod}(R)$, and ${\rm pd}(\mathcal{I}(R)) < \infty$.
\item[(b)] $\mathcal{DI}(R) = \mathcal{I}(R)^\perp$ and ${\rm pd}(\mathcal{I}(R)) < \infty$.
\item[(c)] ${\mathrm{gl.DID}}(R) < \infty$.
\end{itemize}
Moreover, if one of the above conditions holds true, then
\[
{\mathrm{gl.DID}}(R) = {\rm pd}(\mathcal{AP}(R)) = {\rm pd}(\mathcal{I}(R)).
\]
\end{lemma}
\begin{proof}
We use freely the notation and results from \cite{BecerrilMendozaSantiago}. Note that the pair $(\mathcal{AP}(R),\mathcal{I}(R))$ is GI-admissible and WGI-admissible in the sense of \cite[Definitions 3.6 and 4.5]{BecerrilMendozaSantiago}. Then, by the dual of \cite[Corollaries 5.12 (c2) and 5.17]{BecerrilMendozaSantiago}, the result follows.
\end{proof}
Motivated by Example~\ref{ex:DP_n-cotorsion}, we present the following family of rings.
\begin{definition}
We say that a ring $R$ is \textbf{left Ding-finite} if
\begin{align*}
{\mathrm{gl.DPD}}(R) & < \infty & & \mbox{and} & {\mathrm{gl.DID}}(R) & < \infty.
\end{align*}
\end{definition}
\begin{proposition}\label{Prop:nDPDI}
If a ring $R$ is left Ding-finite, then the following statements hold true:
\begin{enumerate}
\item The cotorsion pairs $(\mathcal{DP}(R),\mathcal{P}(R)^\wedge)$ and $(\mathcal{I}(R)^\vee,\mathcal{DI}(R))$ are hereditary and complete.
\item $\mathcal{DP}(R) = {}^\perp\mathcal{P}(R)$ and $\mathcal{DI}(R)) = \mathcal{I}(R)^\perp$.
\item Both ${\rm pd}(\mathcal{I}(R))$ and ${\rm id}(\mathcal{P}(R))$ are finite, and
\[
{\mathrm{gl.DID}}(R) = {\rm pd}(\mathcal{AP}(R)) = {\rm pd}(\mathcal{I}(R)) = {\rm id}(\mathcal{P}(R)) = {\mathrm{gl.DPD}}(R) = {\rm id}(\mathcal{F}(R)).
\]
\end{enumerate}
\end{proposition}
\begin{proof}
It follows by Lemma~\ref{glDIDfinite}, \cite[Corollary 5.18]{BecerrilMendozaSantiago} and \cite[Proposition VII.1.3 (vi)]{BR07}.
\end{proof}
\begin{remark} \
\begin{enumerate}
\item By Proposition~\ref{Prop:nDPDI}, note that every left Ding-finite ring is a left Gorenstein ring in the sense of \cite{BR07}.
\item In case (3) holds in Proposition~\ref{Prop:nDPDI}, we have that ${\mathrm{gl.DPD}}(R)$ also coincides with the (left) global Gorenstein dimension (see Bennis and Mahdou's \cite[Theorem 1.1]{BMglobal} and Mahdou and Tamekkante's \cite[Theorem 3.2]{MahdouTamekkante}).
\item Let $R$ be a Ding-Chen ring. Then, by \cite{Yang} ${\mathrm{gl.DPD}}(R) = {\mathrm{gl.DID}}(R)$. The latter may include the case where ${\mathrm{gl.DPD}}(R) = \infty$ and ${\mathrm{gl.DID}}(R) = \infty$. Then, not every Ding-Chen ring is left Ding-finite, as shown by Wang in \cite[Example 3.3]{Wang}.
\end{enumerate}
\end{remark}
The following result is the Ding-Chen analogous of Proposition~\ref{prop:ncot_spli_silp}, and follows similarly.
\begin{proposition}\label{prop:ncot_DPDI}
The following conditions hold true for any ring $R$:
\begin{enumerate}
\item If $(\mathcal{DP}(R),\mathcal{P}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$, then
\[
{\mathrm{gl.DPD}}(R) = {\rm id}(\mathcal{P}(R)) \leq n.
\]
Dually, if $(\mathcal{I}(R),\mathcal{DI}(R))$ is a right $m$-cotorsion pair in $\mathsf{Mod}(R)$, then
\[
{\mathrm{gl.DID}}(R) = {\rm pd}(\mathcal{I}(R)) \leq m.
\]
\item The following assertions are equivalent:
\begin{itemize}
\item[(a)] $R$ is left Ding-finite with ${\mathrm{gl.DPD}}(R) = {\mathrm{gl.DID}}(R) = n \geq 1$.
\item[(b)] There exist integers $n, m \geq 1$ such that $(\mathcal{DP}(R),\mathcal{P}(R))$ is a left $n$-cotorsion pair and $(\mathcal{I}(R),\mathcal{DI}(R))$ is a right $m$-cotorsion pair in $\mathsf{Mod}(R)$.
\end{itemize}
Moreover, if any of the previous two conditions holds true, we can choose
\[
n = m = {\rm id}(\mathcal{P}(R)) = {\rm pd}(\mathcal{I}(R)).
\]
\end{enumerate}
\end{proposition}
\subsection*{\textbf{Gorenstein flat modules}}
We have previously mentioned in Section~\ref{sec:approximations} characterisations of certain rings which consider their global dimensions. In this example, given a left perfect ring $R$, we shall find equivalent conditions for which $R$ is quasi-Frobenius or has null global Gorenstein flat dimension. These conditions involve left and right $n$-cotorsion pairs formed by the classes $\mathcal{F}(R)$ and $\mathcal{GF}(R)$ of flat and Gorenstein flat $R$-modules.
The \emph{Gorenstein flat dimension} of an $R$-module $M \in \mathsf{Mod}(R)$, which we denote by ${\rm Gfd}(M)$, is defined as the $\mathcal{GF}(R)$-resolution dimension of $M$, that is,
\[
{\rm Gfd}(M) := {\rm resdim}_{\mathcal{GF}(R)}(M).
\]
Let us define the (\emph{left}) \emph{global Gorenstein flat dimension} of $R$ as the value
\[
{\rm gl.Gfd}(R) := {\rm sup}\{ {\rm Gfd}(M) \mbox{ : } M \in \mathsf{Mod}(R) \}.
\]
In the next results, we explore the situation where $(\mathcal{F}(R),\mathcal{GF}(R))$ is a left or a right $n$-cotorsion pair in $\mathsf{Mod}(R)$.
\begin{proposition}\label{prop:FlatGFlatLeft}
The following conditions are equivalent for any ring $R$ and any integer $n \geq 1$:
\begin{itemize}
\item[(a)] $(\mathcal{F}(R),\mathcal{GF}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$.
\item[(b)] $\mathsf{Ext}^1_R(\mathcal{F}(R),\mathcal{GF}(R)) = 0$ and ${\rm gl.Gfd}(R) \leq n$.
\end{itemize}
\end{proposition}
\begin{proof}
The implication (a) $\Rightarrow$ (b) is straightforward. Now suppose that condition (b) holds. It is well known that the class $\mathcal{F}(R)$ is closed under direct summands. Moreover, using the fact that $\mathcal{F}(R)$ is resolving, the condition $\mathsf{Ext}^1_R(\mathcal{F}(R),\mathcal{GF}(R)) = 0$ implies that $\mathsf{Ext}^i_R(\mathcal{F}(R),\mathcal{GF}(R)) = 0$ for every $1 \leq i \leq n$.
It was recently proved by J. {\v{S}}aroch and J. {\v{S}}\v{t}ov\'{\i}\v{c}ek \cite[Corollary 3.12]{SarochStovicek} that the class $\mathcal{GF}(R)$ is closed under extensions for any ring $R$ (that is, any ring $R$ is GF-closed). In particular, from \cite[Proposition 6.17]{BMPS} it follows that $\omega := \mathcal{F}(R) \cap \mathcal{F}(R)^{\perp_1}$ is a relative cogenerator in $\mathcal{X} := \mathcal{GF}(R)$. Thus, by \cite[Theorem 2.8]{BMPS} for every $M \in \mathsf{Mod}(R)$ we can obtain a short exact sequence
\[
0 \to K \to G \to M \to 0,
\]
where $G \in \mathcal{GF}(R)$ and $K \in (\mathcal{F}(R) \cap \mathcal{F}(R)^{\perp_1})^\wedge_{n-1}$, since ${\rm Gfd}(M) \leq n$. On the other hand, by the definition of $\mathcal{GF}(R)$, we have another short exact sequence
\[
0 \to G' \to F \to G \to 0,
\]
where $F \in \mathcal{F}(R)$ and $G' \in \mathcal{GF}(R)$. Taking the pullback of $K \to G \leftarrow F$, we have the following commutative diagram with exact rows and columns:
\begin{equation}\label{fig6}
\parbox{1.75in}{
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [ampersand replacement=\&, matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.25ex, text depth=0.25ex]
{
G' \& G' \& {} \\
E \& F \& M \\
K \& G \& M \\
};
\path[->]
(m-2-1)-- node[pos=0.5] {\footnotesize$\mbox{\bf pb}$} (m-3-2)
;
\path[>->]
(m-1-1) edge (m-2-1) (m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2) (m-3-1) edge (m-3-2)
;
\path[->>]
(m-2-1) edge (m-3-1) (m-2-2) edge (m-3-2)
(m-2-2) edge (m-2-3) (m-3-2) edge (m-3-3)
;
\path[-,font=\scriptsize]
(m-1-1) edge [double, thick, double distance=2pt] (m-1-2)
(m-2-3) edge [double, thick, double distance=2pt] (m-3-3)
;
\end{tikzpicture}
}
\end{equation}
By Bennis' \cite[Theorem 2.11]{Bennis}, we can note that ${\rm Gfd}(E) \leq n-1$. Hence, the central row in \eqref{fig6} completes the proof that $(\mathcal{F}(R),\mathcal{GF}(R))$ is a left $n$-cotorsion pair.
\end{proof}
If we assume that $(\mathcal{F}(R),\mathcal{GF}(R))$ is a right $n$-cotorsion pair instead, we can show that $R$ is a left perfect and a left IF ring. Recall from Colby's \cite{Colby} that a ring $R$ is a \emph{left IF ring} if every injective left $R$-module is flat. Before proving the previous assertion concerning $(\mathcal{F}(R),\mathcal{GF}(R))$, let us show the following characterisation of IF rings in terms of its global Gorenstein flat dimension.
\begin{lemma}\label{Lema:IF-ring}
Let $R$ be a ring. If ${\rm gl.Gfd}(R) = 0$, then $R$ is a left IF ring. If in addition ${\rm gl.Gfd}(R^{op}) = 0$, then the converse also holds. Moreover, if $R$ is commutative, then $R$ is an IF ring if, and only if, ${\rm gl.Gfd}(R) = 0$.
\end{lemma}
\begin{proof}
Suppose first that ${\rm gl.Gfd}(R) = 0$, and let $E$ be an injective module. Then, $E$ is also Gorenstein flat, and so there exists a short exact sequence
\[
0 \to E \to F \to N \to 0
\]
with $F \in \mathcal{F}(R)$ and $N \in \mathcal{GF}(R)$, which splits since $E$ is injective. It follows that $E$ is flat, and hence $R$ is a left IF ring.
Now, suppose that $R$ is a left IF ring with ${\rm gl.Gfd}(R^{op}) = 0$, and so $R$ is also a right IF ring. For every $M \in \mathsf{Mod}(R)$, note that we can find a chain complex
\[
F_\bullet = \cdots \to P_1 \to P_0 \to E^0 \to E^1 \to \cdots
\]
where $M = {\rm Ker}(E^0 \to E^1)$, $P_i \in \mathcal{P}(R)$ and $E^j \in \mathcal{I}(R)$, for every $i, j \geq 0$. This is a complex of flat modules, since every injective is flat. Moreover, injective right $R$-modules are also flat, and then $E \otimes_R F_\bullet$ is exact, for every injective $E \in \mathcal{I}(R{}^{\rm op})$.
\end{proof}
\begin{proposition}\label{prop:perfect_global_flat}
The following conditions are equivalent for any ring $R$.
\begin{itemize}
\item[(a)] $(\mathcal{F}(R),\mathcal{GF}(R))$ is a right $n$-cotorsion pair in $\mathsf{Mod}(R)$ for some integer $n \geq 1$.
\item[(b)] $R$ is left perfect and ${\rm gl.Gfd}(R) = 0$.
\end{itemize}
Moreover, in the case $R$ is commutative, we have that $R$ is a left perfect and an IF ring if, and only if, there exists an integer $n \geq 1$ such that $(\mathcal{F}(R),\mathcal{GF}(R))$ is a right $n$-cotorsion pair in $\mathsf{Mod}(R)$.
\end{proposition}
\begin{proof}
First, note that the implication (b) $\Rightarrow$ (a) is clear. To show (a) $\Rightarrow$ (b), let us assume that there exists an integer $n \geq 1$ such that $(\mathcal{F}(R),\mathcal{GF}(R))$ is a right $n$-cotorsion pair in $\mathsf{Mod}(R)$. Then, for every $M \in \mathsf{Mod}(R)$ there exists an exact sequence
\[
0 \to M \to G \to F^0 \to F^1 \to \cdots \to F^{n-2} \to F^{n-1} \to 0
\]
with $G \in \mathcal{GF}(R)$ and $F^k \in \mathcal{F}(R)$, for every $0 \leq k \leq n-1$. Since the class $\mathcal{GF}(R)$ is resolving by \cite[Corollary 3.12]{SarochStovicek}, we have that $M$ is Gorenstein flat, and hence ${\rm gl.Gfd}(R) = 0$. Now, let $F$ be a flat $R$-module, and consider an exact sequence
\[
0 \to K \to P \to F \to 0,
\]
with $P$ projective. Note that this sequence splits since $\mathsf{Ext}^1_R(\mathcal{F}(R),\mathsf{Mod}(R)) = 0$, and so $F$ is projective. Therefore, $R$ is a left perfect ring with ${\rm gl.Gfd}(R) = 0$.
\end{proof}
Two more interesting results occur if we switch the roles for the classes $\mathcal{F}(R)$ and $\mathcal{GF}(R)$, that is, if we analyse the implications of assuming that $(\mathcal{GF}(R),\mathcal{F}(R))$ is a left or a right $n$-cotorsion pair in $\mathsf{Mod}(R)$.
\begin{proposition}\label{prop:perfect_QF}
For any ring $R$ the following conditions are equivalent:
\begin{itemize}
\item[(a)] $(\mathcal{GF}(R),\mathcal{F}(R))$ is a right $n$-cotorsion pair in $\mathsf{Mod}(R)$ for some integer $n \geq 1$.
\item[(b)] $R$ is left perfect and QF.
\end{itemize}
\end{proposition}
\begin{proof}
Let us show first the implication (a) $\Rightarrow$ (b). Our first step is to show that every module is Gorenstein flat. Indeed, for every $M \in \mathsf{Mod}(R)$ we have an exact sequence
\[
0 \to M \to F \to G^0 \to G^1 \to \cdots \to G^{n-2} \to G^{n-1} \to 0
\]
with $F \in \mathcal{F}(R)$ and $G^k \in \mathcal{GF}(R)$ for every $0 \leq k \leq n-1$, since $(\mathcal{GF}(R),\mathcal{F}(R))$ is a right $n$-cotorsion pair. Using the fact that $\mathcal{GF}(R)$ is resolving, we obtain that $M$ is Gorenstein flat. Now, from the dual of Theorem \ref{theo:left-n-cotorsion}, we get that $\mathcal{F}(R) = (\mathcal{GF}(R)^\vee_{n-1})^{\perp_1} = \mathsf{Mod}(R)^{\perp_1} = \mathcal{I}(R)$. On the other hand, for every flat module $F$ we have a short exact sequence
\[
0 \to F' \to P \to F \to 0
\]
with $P$ projective and $F'$ flat (and so injective). Thus, this sequence splits, and then $F$ is projective. Therefore, we finally obtain $\mathcal{P}(R) = \mathcal{F}(R) = \mathcal{I}(R)$. This implies that $R$ is a left perfect and a QF ring.
Now, we prove that (b) implies (a). Assume that $R$ is left perfect and QF. Then, we have that the following conditions hold: (i) $\mathcal{P}(R) = \mathcal{F}(R) = \mathcal{I}(R)$, and (ii) $\mathcal{I}(R^{\rm op}) = \mathcal{P}(R^{\rm op}) \subseteq\mathcal{F}(R^{\rm op})$. We assert that $\mathcal{GF}(R) = \mathsf{Mod}(R)$. Indeed, for any $M \in \mathsf{Mod}(R)$ we can construct an exact complex
\[
\eta \colon \cdots \to P_1 \to P_0 \to I^0 \to I^1 \to \cdots,
\]
where $P_i \in \mathcal{P}(R)$ and $I^j \in \mathcal{I}(R)$ for any $i, j \geq 0$, and $M = {\rm Ker}(I^0 \to I^1)$. By condition (i), we get that $\eta$ is an acyclic complex of flat modules, and applying (ii) it follows that the complex $E \otimes_R \eta$ is acyclic for any injective $E \in \mathcal{I}(R^{\rm op})$. Then, $M \in \mathcal{GF}(R)$. Once we have the equality $\mathcal{GF}(R) = \mathsf{Mod}(R)$, it can be shown easily that $(\mathcal{GF}(R),\mathcal{F}(R))$ is a right $1$-cotorsion pair in $\mathsf{Mod}(R)$, since $\mathcal{I}(R) = \mathcal{F}(R)$.
\end{proof}
Now let us consider the remaining scenario where $(\mathcal{GF}(R),\mathcal{F}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$. It will be important to recall that for an arbitrary ring $R$, the Pontryagin duality functor $(-)^+ \colon \mathsf{Mod}(R) \longrightarrow \mathsf{Mod}(R^{\rm op})$ maps every flat $R$-module into an injective $R^{\rm op}$-module (see Enochs and Jenda's \cite[Theorem 3.2.9]{EJ}), and every Gorenstein flat $R$-module into a Gorenstein injective $R^{\rm op}$-module (as proved for example in Holm's \cite[Theorem 3.6]{Holm} or in Meng and Pan's \cite[Proposition 4.4]{MengPan}).
Recall also that for every $N \in \mathsf{Mod}(R{}^{\rm op})$ there is a \textbf{pure exact sequence}
\begin{align}\label{eqn:canonical_pure}
\rho_N \colon & 0 \to N \to N^{++} \to N^{++} / N \to 0,
\end{align}
that is, $\rho_N \otimes_R M$ is exact for every $M \in \mathsf{Mod}(R)$ (see \cite[Proposition 5.3.9]{EJ}).
\begin{proposition}\label{prop:GFF_globlaGID}
Let $R$ be a ring over which $(\mathcal{GF}(R),\mathcal{F}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$ for some integer $n \geq 1$. Then, ${\rm Gid}(N^{++}) \leq n$ for every $N \in \mathsf{Mod}(R^{\rm op})$. If in addition $R$ is a commutative noetherian ring with dualizing complex, then ${\mathrm{gl.GID}}(R^{\rm op}) \leq n$.
\end{proposition}
\begin{proof}
Let $N \in \mathsf{Mod}(R^{\rm op})$ and consider its character module $N^+ \in \mathsf{Mod}(R)$. Since $(\mathcal{GF}(R),\mathcal{F}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$ for some $n \geq 1$, we can find a short exact sequence
\[
0 \to K \to G \to N^+ \to 0
\]
where $G$ is a Gorenstein flat $R$-module and ${\rm fd}(K) \leq n-1$. Then, we have the following exact sequence involving $N^{++}$:
\[
0 \to N^{++} \to G^+ \to K^+ \to 0.
\]
Here, $G^+$ if Gorenstein injective and ${\rm id}(K^+) \leq n-1$ by previous comments. Hence, ${\rm Gid}(N^{++}) \leq n$ for every $N \in \mathsf{Mod}(R^{\rm op})$.
Now let us assume that $R$ is a commutative noetherian ring with dualizing complex. Under these conditions, it is known that the class $\mathcal{GI}(R^{\rm op})^\vee_n$ of modules with Gorenstein injective dimension $\leq n$ is closed under pure submodules by \cite[Lemma 2.5 (b) and Theorem 3.1]{HJduality}. On the other hand, for every $N \in \mathsf{Mod}(R^{\rm op})$ there is a canonical pure exact sequence
\[
0 \to N \to N^{++} \to N^{++} / N \to 0,
\]
where ${\rm Gid}(N^{++}) \leq n$ by the previous part. It follows that ${\rm Gid}(N) \leq n$ for every $N \in \mathsf{Mod}(R^{\rm op})$.
\end{proof}
The dual statement of the previous result holds in case $R$ is a two-sided noetherian ring.
\begin{proposition}\label{prop:plusplus}
Let $R$ be any ring and $n \geq 1$ be an integer. If $R$ is two-sided Noetherian and $(\mathcal{I}(R{}^{\rm op}), \mathcal{GI}(R{}^{\rm op}))$ is a right $n$-cotorsion pair in $\mathsf{Mod}(R{}^{\rm op})$, then ${\rm Gfd}(M^{++}) \leq n$ for every $M \in \mathsf{Mod}(R)$.
\end{proposition}
\begin{proof}
Recall that over a two-sided Noetherian ring $R$, a right $R$-module is injective if, and only if, its Pontryagin dual is flat. Furthermore, by Remark~\ref{rem:consequences_InjGInj} (1) we have that $N^+$ is a Gorenstein flat $R$-module, for any Gorenstein injective $N \in \mathsf{Mod}(R{}^{\rm op})$. Using these two facts, the rest of the proof follows as in Proposition~\ref{prop:GFF_globlaGID}.
\end{proof}
Let us consider again the left global Gorenstein flat dimension of $R$, but this time in the case where $(\mathcal{GF}(R),\mathcal{F}(R))$ is a left $n$-cotorsion pair.
\begin{proposition}\label{(GF,F)}
The following conditions are equivalent for any ring $R$ and any integer $n \geq 1$:
\begin{itemize}
\item[(a)] $(\mathcal{GF}(R),\mathcal{F}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$.
\item[(b)] $\mathsf{Ext}^1_R(\mathcal{GF}(R),\mathcal{F}(R)) = 0$ and ${\rm gl.Gfd}(R) \leq n$.
\end{itemize}
\end{proposition}
\begin{proof}
The implication (a) $\Rightarrow$ (b) is clear. Now, let us assume that (b) holds true. The condition $\mathsf{Ext}^i_R(\mathcal{GF}(R),\mathcal{F}(R)) = 0$ is clear for every $1 \leq i \leq n$, since $\mathcal{GF}(R)$ is resolving. Moreover, $\mathcal{GF}(R)$ is closed under direct summands by \cite[Corollary 3.12]{SarochStovicek}. The rest of the implication follows by applying \cite[Theorem 2.8]{BMPS} again, as in the proof of Proposition~\ref{prop:FlatGFlatLeft}.
\end{proof}
Propositions~\ref{prop:GFF_globlaGID} and \ref{(GF,F)} are not the only consequences of having $\mathcal{GF}(R)$ and $\mathcal{F}(R)$ forming a left $n$-cotorsion pair $(\mathcal{GF}(R),\mathcal{F}(R))$ in $\mathsf{Mod}(R)$. For the rest of this section, we shall study other possible results from this assumption, regarding the relation between the classes $\mathcal{F}(R)$, $\mathcal{GF}(R)$, $\mathcal{I}(R^{\rm op})$ and $\mathcal{GI}(R{}^{\rm op})$ via the Pontryagin duality functor $(-)^+$. Namely, we shall focus on the following:
\begin{enumerate}
\item To look for conditions under which it is possible to find an integer $k \geq 1$ such that $(\mathcal{I}(R{}^{\rm op}),\mathcal{GI}(R{}^{\rm op}))$ is a right $k$-cotorsion pair in $\mathsf{Mod}(R{}^{\rm op})$, provided that $(\mathcal{GF}(R),\mathcal{F}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$.
\item To see if the converse procedure is possible, that is, if there exists $k \geq 1$ for which $(\mathcal{GF}(R),\mathcal{F}(R))$ is a left $k$-cotorsion pair in $\mathsf{Mod}(R)$, assuming that $(\mathcal{I}(R{}^{\rm op}),\mathcal{GI}(R{}^{\rm op}))$ is a right $n$-cotorsion pair in $\mathsf{Mod}(R{}^{\rm op})$.
\end{enumerate}
\begin{theorem}\label{theo:GFGI_Pontryagin}
Let $R$ be any ring such that $(\mathcal{GF}(R),\mathcal{F}(R))$ is a left $n$-cotorsion pair in $\mathsf{Mod}(R)$. Then, the following conditions are equivalent.
\begin{itemize}
\item[(a)] $m := \sup \{{\rm Gid}(N^{++} / N) \mbox{ {\rm :} } N \in \mathsf{Mod}(R{}^{\rm op}) \} < \infty$.
\item[(b)] $(\mathcal{I}(R{}^{\rm op}),\mathcal{GI}(R{}^{\rm op}))$ is a right $(k+1)$-cotorsion pair in $\mathsf{Mod}(R{}^{\rm op})$ for some integer $k \geq 1$.
\end{itemize}
Moreover, if any of the previous conditions holds true, one can take $k = \max\{n, m\}$.
\end{theorem}
\begin{proof}
For the first implication, let $k := \max\{n, m\}.$ It suffices to show that, for every $N \in \mathsf{Mod}(R{}^{\rm op}),$ one can find an embedding into $\mathcal{GI}(R{}^{\rm op})$ whose cokernel has injective dimension at most $k$. Consider the canonical pure exact sequence $\rho_N$ from \eqref{eqn:canonical_pure}. By Proposition~\ref{prop:GFF_globlaGID}, we get that ${\rm Gid}(N^{++}) \leq k$. The latter, along with condition (a) implies that ${\rm Gid}(N) \leq k + 1$ (see \cite[Proposition 2.15]{MengPan}). Hence, (b) follows by \cite[dual of Theorem 2.8]{BMPS}.
Conversely, if $(\mathcal{I}(R{}^{\rm op}),\mathcal{GI}(R{}^{\rm op}))$ is a right $(k+1)$-cotorsion pair for some $k \in \mathbb{N}$, then ${\rm Gid}(N) \leq k + 1$ and ${\rm Gid}(N^{++}) \leq n$ for every $N \in \mathsf{Mod}(R{}^{\rm op})$. Using \cite[Proposition 2.15]{MengPan} again, we obtain that ${\rm Gid}(N^{++} / N) \leq {\rm max}\{ k, n \}$.
\end{proof}
\begin{theorem}\label{theo:GFGI_Pontryagin}
Let $R$ be a two-sided noetherian ring such that $(\mathcal{I}(R{}^{\rm op}), \mathcal{GI}(R{}^{\rm op}))$ is a right $n$-cotorsion pair in $\mathsf{Mod}(R{}^{\rm op})$. Then, $\mathrm{gl.Gfd}(R)\leq n$. Moreover, the following conditions are equivalent:
\begin{itemize}
\item[(a)] $\mathsf{Ext}_R^1(\mathcal{GF}(R),\mathcal{F}(R)) = 0$.
\item[(b)] $(\mathcal{GF}(R),\mathcal{F}(R))$ is a left Frobenius pair in $\mathsf{Mod}(R)$ (in the sense of \cite[Definition 2.5]{BMPS}).
\item[(c)] $(\mathcal{GF}(R),\mathcal{F}(R))$ is a left $k$-cotorsion pair in $\mathsf{Mod}(R)$ for some integer $k \geq 1$.
\end{itemize}
If any of the previous conditions holds, one can take $k = n$. Furthermore,
\[
\mathrm{gl.Gfd}(R) = {\rm id}(\mathcal{F}(R)).
\]
\end{theorem}
\begin{proof}
For the first part, let $M\in\mathsf{Mod}(R)$ and consider the canonical pure exact sequence
\[
\rho_M \colon 0 \to M \to M^{++} \to M^{++} / M \to 0,
\]
where ${\rm Gfd}(M^{++}) \leq n$ by Proposition~\ref{prop:plusplus}. On the other hand, by \cite[Lemma 2.5 (a) and Theorem 3.1]{HJduality}, we get that the class $\mathcal{GF}(R)^\wedge_n$ is closed under pure submodules, and thus from $\rho_M$ we get that ${\rm Gfd}(M) \leq n$.
Now, let us show the equivalence between (a), (b) and (c).
\begin{itemize}
\item (a) $\Rightarrow$ (b): From \cite[Corollary 3.12]{SarochStovicek}, we know that $\mathcal{GF}(R)$ is the left part of an hereditary cotorsion pair in $\mathsf{Mod}(R).$ Thus, the condition $\mathsf{Ext}_R^{1}(\mathcal{GF}(R),\mathcal{F}(R)) = 0$ implies that $\mathsf{Ext}_R^{i}(\mathcal{GF}(R),\mathcal{F}(R)) = 0$ for any $i \geq 1$. Therefore, $\mathcal{F}(R)$ is an $\mathcal{GF}(R)$-injective relative cogenerator in $\mathcal{GF}(R)$. Finally, it is clear that $\mathcal{F}(R)$ is closed under direct summands, thus proving (b).
\item (b) $\Rightarrow$ (c): Assume that (b) holds true. Since $\mathcal{GF}(R)^\wedge_n = \mathsf{Mod}(R)$, we get by \cite[Theorem 2.10]{BMPS} that
\[
\mathrm{gl.Gfd}(R) = {\rm pd}_{\mathcal{F}(R)}(\mathsf{Mod}(R)) = {\rm id}\,(\mathcal{F}(R)).
\]
Moreover, for any $M\in\mathsf{Mod}(R)$ we get by \cite[Theorem 2.8]{BMPS} an exact sequence
\[
0 \to K \to G \to M \to 0,
\]
where $G\in\mathcal{GF}(R)$ and $K \in \mathcal{F}(R)^\wedge_{n-1}$.
\item (c) $\Rightarrow$ (a): Trivial.
\end{itemize}
\end{proof}
\subsection*{\textbf{Cluster tilting subcategories}}
Following Iyama's \cite[Definition 1.1]{IyamaCluster}, for an integer $m \geq 1$, a subcategory $\mathcal{D} \subseteq \mathcal{C}$ is said to be \emph{$m$-cluster tilting} if it is precovering and preenveloping, and the following equalities hold true
\[
\mathcal{D} = \bigcap\limits_{0 < i < m}^{}{}^{\perp_{i}}\mathcal{D} = \bigcap\limits_{0 < i < m}^{}\mathcal{D}^{\perp_{i}}.
\]
\begin{remark}\label{rem:cluster_precover}
Note that if $\mathcal{D}$ is an $m$-cluster tilting subcategory (with $m \geq 2$) of an abelian category $\mathcal{C}$ with enough projectives and injectives, then $\mathcal{D}$-precovers and $\mathcal{D}$-preenvelopes are special, since $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{D,D}) = 0$.
\end{remark}
In this example, we prove that a subcategory $\mathcal{D}$ of an abelian category $\mathcal{C}$ is an $(n+1)$-cluster tilting subcategory if, and only if, it forms an $n$-cotorsion pair of the form $(\mathcal{D,D})$. The following result is straightforward.
\begin{lemma}\label{nclustA=B}
Let $\mathcal{A}$ and $\mathcal{B}$ be classes of objects of $\mathcal{C}$ such that $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ for any integer $1 \leq i \leq n$. If the containment
\[
\bigcap\limits_{i = 1}^{m} {}^{\perp_{i}}(\mathcal{A} \cap \mathcal{B}) \subseteq \mathcal{A} \cap \mathcal{B},
\]
holds for some integer $1 \leq m \leq n$, then $\mathcal{A} \subseteq \mathcal{B}$.
\end{lemma}
\begin{proposition}\label{nclustA=B y ff}
Let $(\mathcal{A,B})$ be an $n$-cotorsion pair in $\mathcal{C}$. Then, the following conditions hold true:
\begin{enumerate}
\item If there exists an integer $1 \leq m \leq n$ such that the equalities
\[
\mathcal{A} \cap \mathcal{B} = \bigcap\limits_{i=1}^{m} {}^{\perp_{i}}(\mathcal{A} \cap \mathcal{B}) = \bigcap\limits_{i=1}^{m} (\mathcal{A} \cap \mathcal{B})^{\perp_{i}}
\]
hold true, then $\mathcal{A} = \mathcal{B}$ and the class $\mathcal{A} \cap \mathcal{B} = \mathcal{A}$ is special precovering and special preenveloping.
\item The class $\mathcal{A} \cap \mathcal{B}$ is an $(n+1)$-cluster tilting subcategory if, and only if, $\mathcal{A} = \mathcal{B}$.
\end{enumerate}
\end{proposition}
\begin{proof}
Part (1) follows by Lemma~\ref{nclustA=B} and Proposition~\ref{A-precub,B-preenv} and their duals.
The ``only if'' statement of part (2) is a consequence of part (1). Now for the ``if'' statement, suppose that $\mathcal{A} = \mathcal{B}$. Then, by Proposition~\ref{A-precub,B-preenv} and its dual, we get that $\mathcal{A}$ is a special precovering and a special preenveloping class. Thus, it suffices to prove that
\[
\mathcal{A} = \bigcap_{i = 1}^n {}^{\perp_i}\mathcal{A} = \bigcap_{i = 1}^n \mathcal{A}^{\perp_i},
\]
but this follows from Theorem~\ref{theo:left-n-cotorsion} (2) and its dual.
\end{proof}
We are now ready to show the following characterisation of $(n+1)$-cluster tilting subcategories.
\begin{theorem} \label{ncot y ct}
Let $\mathcal{C}$ be an abelian category with enough projectives and injectives. Then, for any subcategory $\mathcal{D} \subseteq \mathcal{C}$ and any integer $n \geq 1$, the following statements are equivalent:
\begin{itemize}
\item[(a)] $(\mathcal{D,D})$ is an $n$-cotorsion pair in $\mathcal{C}$.
\item[(b)] $\mathcal{D}$ is an $(n+1)$-cluster tilting subcategory of $\mathcal{C}$.
\end{itemize}
Moreover, in case any of the above conditions holds true, we have $\mathcal{C} = \mathcal{D}^\wedge_n = \mathcal{D}^\vee_n$.
\end{theorem}
\begin{proof}
The implication (a) $\Rightarrow$ (b) follows by Proposition~\ref{nclustA=B y ff}. Now suppose that $\mathcal{D}$ is an $(n+1)$-cluster tilting subcategory of $\mathcal{C}$. Then, we have that $\mathcal{D}$ is closed under direct summands and that $\mathsf{Ext}^{i}_{\mathcal{C}}(\mathcal{D,D}) = 0$ for any integer $1 \leq i \leq n$. The result will follow after showing the equalities $\mathcal{C} = \mathcal{D}^{\wedge}_{n} = \mathcal{D}^{\vee}_{n}$.
By Remark~\ref{rem:cluster_precover}, for any $M \in \mathcal{C}$ we can consider an exact sequence
\[
\eta \colon 0 \to K_0 \to D_0 \xrightarrow{f_0} M \to 0,
\]
where $f_0$ is a special $\mathcal{D}$-precover. After applying the functor $\mathsf{Hom}_{\mathcal{C}}(D,-)$ to $\eta$, with $D$ running over $\mathcal{D}$, we get:
\begin{align*}
\mathsf{Ext}^1_{\mathcal{C}}(D,K_0) & = 0 \quad \text{and}\quad\mathsf{Ext}^{i+1}_{\mathcal{C}}(D,K_0) \cong \mathsf{Ext}^{i}_{\mathcal{C}}(D,M)\;\text{ for any}\; 1 \leq i \leq n-1.
\end{align*}
Inductively, we can construct an exact sequence
\begin{align*}
0 & \to K_{n} \to D_{n-1} \xrightarrow{f_{n-1}} D_{n-2} \to \cdots \to D_1 \xrightarrow{f_1} D_0 \xrightarrow{f_0} M \to 0,
\end{align*}
where $D_i \in \mathcal{D}$ and $K_i := {\rm Im}\,(f_i)$ for any $0 \leq i \leq n-1$, and such that the following relations hold:
\begin{align*}
\mathsf{Ext}^1_{\mathcal{C}}(D,K_n) &= 0,\\
\mathsf{Ext}^2_{\mathcal{C}}(D,K_n) & \cong \mathsf{Ext}^1_{\mathcal{C}}(D,K_{n-1}) = 0,\\
\vdots & \qquad\qquad\qquad \vdots\qquad\qquad\qquad \vdots \\
\mathsf{Ext}^n_{\mathcal{C}}(D,K_n) & \cong \mathsf{Ext}^{n-1}_{\mathcal{C}}(D,K_{n-1}) \cong \cdots \cong \mathsf{Ext}^1_{\mathcal{C}}(D,K_1) = 0.
\end{align*}
Therefore, we get that $K_n \in \bigcap_{i = 1}^n\,\mathcal{D}^{\perp_{i}} = \mathcal{D}$ and thus $M \in \mathcal{D}^{\wedge}_n$. Dually, we get $M \in \mathcal{D}^{\vee}_n$.
\end{proof}
One interesting fact to note about $(n+1)$-cluster tilting subcategories $\mathcal{D}$ is that $n$ is the biggest integer for which the condition $\mathsf{Ext}^n_{\mathcal{C}}(\mathcal{D,D}) = 0$ is true, in the sense that letting $\mathsf{Ext}^{n+1}_{\mathcal{C}}(\mathcal{D,D}) = 0$ forces $\mathcal{C}$ to be a Frobenius category. We specify this in the following result.
\begin{proposition}
Let $n \geq 1$ and $\mathcal{D}$ be an $(n+1)$-cluster tilting subcategory of an abelian category $\mathcal{C}$ with enough projectives and injectives. Then, the following conditions are equivalent:
\begin{itemize}
\item[(a)] $\mathsf{Ext}^{n+1}_{\mathcal{C}}(\mathcal{D,D}) = 0$.
\item[(b)] $\mathcal{P}(\mathcal{C}) = \mathcal{D}$.
\item[(c)] $\mathcal{I}(\mathcal{C}) = \mathcal{D}$.
\item[(d)] $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{D,D}) = 0$ for every $i \geq 1$.
\end{itemize}
\end{proposition}
\begin{proof}
It suffices to show that (a) implies (b) and (c). So let us assume that $\mathsf{Ext}^{n+1}_{\mathcal{C}}(\mathcal{D,D}) = 0$. By Proposition \ref{prop8}, we have that $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{D},\mathcal{D}^\wedge_n) = 0$, and since $\mathcal{C} = \mathcal{D}^\wedge_n$ by the proof of Theorem \ref{ncot y ct}, we obtain the containment $\mathcal{D} \subseteq {}^{\perp_1}\mathcal{C} = \mathcal{P}(\mathcal{C})$. Dually, we can also show that $\mathcal{D} \subseteq \mathcal{C}^{\perp_1} = \mathcal{I}(\mathcal{C})$ holds. On the other hand, we know that $\mathcal{P}(\mathcal{C}) \cup \mathcal{I}(\mathcal{C}) \subseteq \mathcal{D}$, since $\bigcap_{1 \leq i \leq n} {}^{\perp_i}\mathcal{D} = \mathcal{D} = \bigcap_{1 \leq i \leq n} \mathcal{D}^{\perp_i}$. Therefore, $\mathcal{P}(\mathcal{C}) = \mathcal{D} = \mathcal{I}(\mathcal{C})$.
\end{proof}
\begin{remark}\label{rem:cluster_trivial}
Theorem~\ref{ncot y ct} may constitute a nontrivial example of a two-sided $n$-cotorsion pair. Namely, let $\mathcal{D}$ be an $(n+1)$-cluster tilting subcategory of an abelian category $\mathcal{C},$ with enough projectives and injectives. Then by Theorem~\ref{ncot y ct} and Remark~\ref{Rk-n-cot-trivial} (2), $(\mathcal{D,D})$ is the trivial $n$-cotorsion pair (that is, $\mathcal{D} = \mathcal{P}(\mathcal{C})$) if, and only if, $\mathcal{D}$ is resolving.
\end{remark}
Using the previous theorem and \cite[Theorem 1.6]{IyamaCluster}, we obtain the following example.
\begin{example}
Let $\Lambda$ be an Artin $R$-algebra. Note that the category $\mathsf{mod}(\Lambda),$ of finitely generated left $\Lambda$-modules, is an abelian category with enough projectives and injectives, as it is well known that every finitely generated $\Lambda$-module has a finitely generated projective cover and a finitely generated injective envelope.
\begin{enumerate}
\item If ${\rm gl.dim}(\Lambda) \leq n+1$ and $\mathsf{mod}(\Lambda)$ has an $(n+1)$-cluster tilting object $T$, then there exists a unique $n$-cotorsion pair in $\mathsf{mod}(\Lambda)$ of the form $(\mathcal{D,D})$, where $\mathcal{D} := {\rm add}(T)$ is the class of direct summands of finite direct sums of copies of $T$. In this case, note that $\mathcal{D}$ is resolving if, and only if, $\mathcal{D} = {\rm add}(\Lambda)$.
\item If $\Lambda$ is not self-injective having an $(n+1)$-cluster tilting object $T$, we necessarily have that $\mathsf{Ext}^{n+1}_{\Lambda}(T,T) \neq 0$ and $\mathcal{P}(\Lambda) \cup \mathcal{I}(\Lambda) \subsetneq {\rm add}(T)$.
\end{enumerate}
\end{example}
\section{Higher cotorsion for chain complexes}\label{sec:complexes}
The last part of the present paper is devoted to study $n$-cotorsion pairs in the setting provided by the category $\mathsf{Ch}(\mathcal{C})$ of chain complexes of objects in $\mathcal{C}$. In the first part of this section, we characterise certain families of $n$-cotorsion pairs of complexes in terms of $n$-cotorsion pairs in the ground category $\mathcal{C}$. In the second part, we shall study how to induce $n$-cotorsion pairs of complexes from an $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$. The complexes involved in these $n$-cotorsion pairs are the $\mathcal{A}$-complexes, $\mathcal{B}$-complexes, and differential graded complexes considered by Gillespie in \cite{GillespieFlat}.
Let us set some notation for the category $\mathsf{Ch}(\mathcal{C})$. Given a chain complex $X \in \mathsf{Ch}(\mathcal{C})$ with differentials $\partial^X_m \colon X_m \to X_{m-1}$, we denote its cycle and boundary objects in $\mathcal{C}$ by $Z_m(X) := {\rm Ker}(\partial^X_m)$ and $B_m(X) := {\rm Im}(\partial^X_{m+1})$, respectively.
Let us also borrow some notation from \cite[Section 3]{GillespieDegreewise}. Let $(\mathscr{A,B})$ be an $n$-cotorsion pair in $\mathsf{Ch}(\mathcal{C})$. The symbol $\mathscr{A}'$ will denote the class of all objects $M \in \mathcal{C}$ such that $M = A_m$ for some $A \in \mathscr{A}$ and some $m \in \mathbb{Z}$. The class $\mathscr{B}'$ is defined similarly.
Motivated by \cite[Definition 3.4]{GillespieDegreewise}, we propose the following.
\begin{definition}
An $n$-cotorsion pair $(\mathscr{A,B})$ in $\mathsf{Ch}(\mathcal{C})$ is \textbf{degreewise orthogonal} if for every pair if integers $i, j \in \mathbb{Z}$ we have the relations:
\begin{enumerate}
\item $\mathsf{Ext}_{\mathcal{C}}^1(A_i, Y_j) = 0$ whenever $A \in \mathscr{A}$ and $Y \in \mathscr{B}_{n-1}^{\wedge}$, and
\item $\mathsf{Ext}_{\mathcal{C}}^1(X_i, B_j) = 0$ whenever $X \in \mathscr{A}_{n-1}^{\vee}$ and $B \in \mathscr{B}$.
\end{enumerate}
\end{definition}
Given an object $M \in \mathcal{C}$ and an integer $m \in \mathbb{Z}$, the \emph{$m$-th disk complex centred at $M$} is the chain complex denoted by $D^m(M)$, such that $M$ appears at degrees $m$ and $m\mbox{-}1$, and $0$ elsewhere. The only nonzero differential is the identity on $M$. The \emph{$m$-th sphere complex centred at $M$}, on the other hand, is the chain complex $S^m(M) \in \mathsf{Ch}(\mathcal{C})$ with $M$ at the $m$-th component and $0$ elsewhere.
The first relation we note between $n$-cotorsion in $\mathsf{Ch}(\mathcal{C})$ and $n$-cotorsion in $\mathcal{C}$ is described in the following result, which is the $n$-cotorsion version of \cite[Lemma 3.5]{GillespieDegreewise}.
\begin{lemma}\label{JKtoJ'K'}
Let $\mathcal{C}$ be an abelian category with enough injectives. Then, the following statements are equivalent for any $n$-cotorsion pair $(\mathscr{A,B})$ in $\mathsf{Ch}(\mathcal{C})$:
\begin{itemize}
\item[(a)] $(\mathscr{A,B})$ is degreewise orthogonal.
\item[(b)] If $A \in \mathscr{A}$ and $B \in \mathscr{B}$, then $D^{m}(A_i) \in \mathscr{A}$ and $D^n(B_j) \in \mathscr{B}$ for every $m, n, i, j \in \mathbb{Z}$.
\item[(c)] $(\mathscr{A}',\mathscr{B}')$ is an $n$-cotorsion pair in $\mathcal{C}$.
\end{itemize}
\end{lemma}
\begin{proof}
We prove the implications (a) $\Rightarrow$ (b) $\Rightarrow$ (c) $\Rightarrow$ (a).
\begin{itemize}
\item (a) $\Rightarrow$ (b): Let $A \in \mathscr{A}$ and $Y \in \mathscr{B}_{n-1}^{\wedge}$. By \cite[Lemma 3.1]{GillespieFlat} we have that
\[
\mathsf{Ext}_{\mathsf{Ch}}^1(D^{m}(A_i), Y) \cong \mathsf{Ext}_{\mathcal{C}}^1(A_i, Y_m) = 0
\]
where $\mathsf{Ext}_{\mathcal{C}}^1(A_i, Y_m) = 0$ by condition (a). Thus, $D^m(A_i) \in {}^{\perp_1}(\mathscr{B}_{n-1}^{\wedge}) = \mathscr{A}$ by Theorem~\ref{theo:left-n-cotorsion}. In a similar way, we can prove that $D^m(B_j) \in \mathscr{B}$ for any $j, m \in \mathbb{Z}$ whenever $B \in \mathscr{B}$.
\item (b) $\Rightarrow$ (c): We prove that $(\mathscr{A}', \mathscr{B}')$ is a left $n$-cotorsion pair in $\mathcal{C}$ assuming (b).
We first show that $\mathscr{A}'$ is closed under direct summands. Let $N \in \mathscr{A}'$ and $M$ be a direct summand of $A$. Then, $N = A_m$ for some complex $A \in \mathscr{A}$ and some $m \in \mathbb{Z}$. Note that $D^0(A_m) \in \mathscr{A}$ by condition (b), and that $D^0(M)$ is a direct summand of $D^0(A_m)$. Since $\mathscr{A}$ is closed under direct summands by hypothesis, we have that $D^0(M) \in \mathscr{A}$, that is, $M \in \mathscr{A}'$. Hence, $\mathscr{A}'$ is closed under direct summands.
Now let us show that $\mathsf{Ext}^i_{\mathcal{C}}(\mathscr{A}',\mathscr{B}') = 0$ for every $1 \leq i \leq n$. By Proposition~\ref{prop:cotorsion_vs_ncotorsion}, since $\mathcal{C}$ has enough injectives, it is equivalent to show that $\mathsf{Ext}^1_{\mathcal{C}}(\mathscr{A}',(\mathscr{B}')^\wedge_{n-1}) = 0$. So let $M \in \mathscr{A}'$ and $N \in (\mathscr{B}')^\wedge_{n-1}$. By condition (b), we can note that $D^{0}(M) \in \mathscr{A}$ and $D^1(N) \in \mathscr{B}_{n-1}^{\wedge}$. From \cite[Lemma 3.1]{GillespieFlat}, we have that
\[
\mathsf{Ext}_{\mathcal{C}}^1(M,N) \cong \mathsf{Ext}_{\mathsf{Ch}}^1(D^0(M),D^1(N)) = 0,
\]
where the last equality follows by Proposition~\ref{prop:cotorsion_vs_ncotorsion}. Hence, $\mathsf{Ext}_{\mathcal{C}}^1(\mathscr{A}', (\mathscr{B}')_{n-1}^\wedge) = 0$.
Finally, we show that for every object $C \in \mathcal{C}$ there exists a short exact sequence
\[
0 \to N \to M \to C \to 0
\]
where $M \in \mathscr{A}'$ and $N \in (\mathscr{B}')^\wedge_{n-1}$. For, consider the sphere complex $S^0(C) \in \mathsf{Ch}(\mathcal{C})$. Since $(\mathscr{A},\mathscr{B})$ is an $n$-cotorsion pair in $\mathsf{Ch}(\mathcal{C})$, there exists a short exact sequence
\[
0 \to Y \to A \to S^0(C) \to 0
\]
where $A \in \mathscr{A}$ and $Y \in \mathscr{B}^\wedge_{n-1}$. Thus, at degree $0$ we have the exact sequence
\[
0 \to N \to M \to C \to 0
\]
where $M = A_0 \in \mathscr{A}'$ and $N = Y_0 \in (\mathscr{B}')^\wedge_{n-1}$.
The previous shows that $(\mathscr{A}',\mathscr{B}')$ is a left $n$-cotorsion pair in $\mathcal{C}$. In a similar way, one can show that $(\mathscr{A}',\mathscr{B}')$ is also a right $n$-cotorsion pair in $\mathcal{C}$. Therefore, (c) follows.
\item (c) $\Rightarrow$ (a): It is clear due to the equalities $\mathscr{A}' = {}^{\perp_1}((\mathscr{B}')_{n-1}^{\wedge})$ and $\mathscr{B}' = ((\mathscr{A}')_{n-1}^{\vee})^{\perp_1}$.
\end{itemize}
\end{proof}
In what follows, we need to consider the subgroup $\mathsf{Ext}^1_{\mathsf{dw}}(X,Y)$ of $\mathsf{Ext}^1_{\mathsf{Ch}}(X,Y)$ of classes of short exact sequences
\[
0 \to Y \to Z \to X \to 0
\]
which are degreewise split, that is,
\[
0 \to Y_m \to Z_m \to X_m \to 0
\]
is a split exact sequence in $\mathcal{C}$ for every $m \in \mathbb{Z}$.
Recall also that given a chain complex $X \in \mathsf{Ch}(\mathcal{C})$ and an integer $k \in \mathbb{Z}$, the \emph{$k$-th suspension of $X$} is the complex $X[k] \in \mathsf{Ch}(\mathcal{C})$ with components $(X[k])_m := X_{m-k}$ and differentials $\partial^{X[k]}_m := (-1)^k \partial^X_{m-k}$.
The following result corresponds to \cite[Proposition 3.7]{GillespieDegreewise} in the context of $n$-cotorsion pairs. We provide a characterisation for the class $\mathscr{A}$ in every degreewise orthogonal $n$-cotorsion pair $(\mathscr{A,B})$ in $\mathsf{Ch}(\mathcal{C})$.
\begin{proposition}
Let $(\mathscr{A,B})$ be a degreewise orthogonal $n$-cotorsion pair in $\mathsf{Ch}(\mathcal{C})$ (where $\mathcal{C}$ is an abelian category with enough injectives), and let $(\mathscr{A}',\mathscr{B}')$ be the corresponding $n$-cotorsion pair in $\mathcal{C}$ from Lemma~\ref{JKtoJ'K'}. If $\mathscr{B}$ is closed under suspensions, then $\mathscr{A}$ equals the class of all complexes $A \in \mathsf{Ch}(\mathcal{C})$ for which $A_m \in \mathscr{A}'$ for every $m \in \mathbb{Z}$, and such that every chain map $A \to Y$ is null homotopic whenever $Y \in \mathscr{B}^{\wedge}_{n-1}$.
\end{proposition}
\begin{proof}
Suppose that $(\mathscr{A,B})$ is an $n$-cotorsion pair in $\mathsf{Ch}(\mathcal{C})$ with $\mathscr{B}$ closed under suspensions. Note that the latter implies that $\mathscr{B}^{\wedge}_{n-1}$ is also closed under suspensions.
Let us denote by $\mathscr{X}$ the class of complexes $X \in \mathsf{Ch}(\mathcal{C})$ such that $X_m \in \mathscr{A}'$ and such that every chain map $X \to Y$ is null homotopic whenever $Y \in \mathscr{B}^\wedge_{n-1}$. We show $\mathscr{A} = \mathscr{X}$ using the equality $\mathscr{A} = {}^{\perp_1}\mathscr{B}^\wedge_{n-1}$ from Theorem~\ref{theo:left-n-cotorsion}.
\begin{itemize}
\item $\mathscr{A} \supseteq \mathscr{X}$: Let $X \in \mathscr{X}$ and $Y \in \mathscr{B}^\wedge_{n-1}$. Since $(\mathscr{A}',\mathscr{B}')$ is an $n$-cotorsion pair by Lemma~\ref{JKtoJ'K'}, we have that $\mathsf{Ext}^1_{\mathcal{C}}(X_m, Y_m) = 0$ for every $m \in \mathbb{Z}$, and so $\mathsf{Ext}_{\mathsf{Ch}}^1(X, Y) = \mathsf{Ext}_{\mathsf{dw}}^1(X, Y)$. On the other hand, $\mathsf{Ext}_{\mathsf{dw}}^1(X,Y) \cong \mathsf{Hom}_{\mathsf{Ch}}(X, Y[1]) / \sim$ by \cite[Lemma 2.1]{GillespieFlat}, where $\sim$ represents the chain homotopy relation. Since $X \in \mathscr{X}$ and $Y[1] \in \mathscr{B}^\wedge_{n-1}$ being $\mathscr{B}^\wedge_{n-1}$ closed under suspensions, we have that $\mathsf{Hom}_{\mathsf{Ch}}(X, Y[1]) / \sim \mbox{} = 0$, and hence $\mathsf{Ext}_{\mathsf{Ch}}^1(X, Y) = 0$. Then, we have that $X \in \mathscr{A}$.
\item $\mathscr{A} \subseteq \mathscr{X}$: Let $A \in \mathscr{A}$ and $Y \in \mathscr{B}^\wedge_{n-1}$. We have that
\[
0 = \mathsf{Ext}^1_{\mathsf{Ch}}(A,Y[-1]) \supseteq \mathsf{Ext}^1_{\mathsf{dw}}(A,Y[-1]) \cong \mathsf{Hom}_{\mathsf{Ch}}(A,Y) / \sim.
\]
since $Y[-1] \in \mathscr{B}^\wedge_{n-1}$. It follows that every chain map $A \to Y$ is null homotopic whenever $Y \in \mathscr{B}^\wedge_{n-1}$.
Now let $N \in (\mathscr{B}')^\wedge_{n-1}$ and note that $D^{m+1}(N)\in \mathscr{B}_{n-1}^{\wedge}$ by Lemma~\ref{JKtoJ'K'}. Then, we have
\[
\mathsf{Ext}_{\mathcal{C}}^1(A_m, N) \cong \mathsf{Ext}_{\mathsf{Ch}}^1(A, D^{m+1}(N)) = 0,
\]
that is, $A_m \in {}^{\perp_1}[(\mathscr{B}')^\wedge_{n-1}] = \mathscr{A}'$ (by Lemma~\ref{JKtoJ'K'} and Theorem~\ref{theo:left-n-cotorsion}). Therefore, we have that $A_m \in \mathscr{A}'$ for all $m \in \mathbb{Z}$.
\end{itemize}
\end{proof}
Now let us show how to induce $n$-cotorsion pairs involving certain families of complexes from an $n$-cotorsion pair in $\mathcal{C}$. These families are presented below in Definition \ref{def:special_complexes}, which follows the spirit of Gillespie's \cite[Definition 3.3]{GillespieFlat}.
For the rest of this section, it will be important to recall that $\mathsf{Ch}(\mathcal{C})$ is equipped with an \emph{internal hom} functor $\mathcal{H}{om}(-,-)$ defined as follows: for every $X, Y \in \mathsf{Ch}(\mathcal{C})$, $\mathcal{H}{om}(X,Y)$ is the chain complex of abelian groups defined by
\[
\mathcal{H}{om}(X,Y)_m := \prod_{k \in \mathbb{Z}} \mathsf{Hom}_{\mathcal{C}}(X_k, Y_{m+k})
\]
for every $m \in \mathbb{Z}$, and with differentials given by $f \mapsto \partial^Y \circ f - (-1)^m f \circ \partial^X$ (see Garc\'ia Rozas' \cite{JRGR}, for instance). It is known that every chain map $X \to Y$ is null homotopic if, and only if, the complex $\mathcal{H}{om}(X,Y)$ is exact.
\begin{definition}\label{def:special_complexes}
Let $\mathcal{X}$ be a class of objects of $\mathcal{C}$. A chain complex $X \in \mathsf{Ch}(\mathcal{C})$ is:
\begin{enumerate}
\item a \textbf{complex (with terms) in $\bm{\mathcal{X}}$} (or a \textbf{degreewise $\bm{\mathcal{X}}$-complex}) if $X_m \in \mathcal{X}$ for every $m \in \mathbb{Z}$;
\item an \textbf{$\bm{\mathcal{X}}$-complex} if $X$ is exact and $Z_m(X) \in \mathcal{X}$ for every $m \in \mathbb{Z}$.
\end{enumerate}
We shall denote by $\mathsf{Ch}(\mathcal{X})$ the class of complexes in $\mathcal{X}$, and by $\widetilde{\mathcal{X}}$ the class of $\mathcal{X}$-complexes.
Now let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects in $\mathcal{C}$ such that $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{A,B}) = 0$. We can also define two new families of complexes from $\mathsf{Ch}(\mathcal{A})$, $\widetilde{\mathcal{A}}$, $\mathsf{Ch}(\mathcal{B})$ and $\widetilde{\mathcal{B}}$.
\begin{enumerate}
\setcounter{enumi}{2}
\item We shall say that a complex $X \in \mathsf{Ch}(\mathcal{C})$ is \textbf{$\bm{\mathcal{H}{om}(-,\widetilde{\mathcal{B}})}$-acyclic in $\bm{\mathsf{Ch}(\mathcal{A})}$} if $X \in \mathsf{Ch}(\mathcal{A})$ and if $\mathcal{H}{om}(X,B)$ is an exact complex of abelian groups whenever $B \in \widetilde{\mathcal{B}}$.
\item \textbf{$\bm{\mathcal{H}{om}(\widetilde{\mathcal{A}},-)}$-acyclic complexes in $\bm{\mathsf{Ch}(\mathcal{B})}$} are defined dually, that is, as those complexes $Y \in \mathsf{Ch}(\mathcal{B})$ such that $\mathcal{H}{om}(A,Y)$ is exact for every $A \in \widetilde{\mathcal{A}}$.
\end{enumerate}
We shall denote by $\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$ the class of $\mathcal{H}{om}(-,\widetilde{\mathcal{B}})$-acyclic complexes in $\mathsf{Ch}(\mathcal{A})$. Dually, $\mathsf{Ch}_{\rm acy}(\widetilde{\mathcal{A}};\mathcal{B})$ will denote the class of $\mathcal{H}{om}(\widetilde{\mathcal{A}},-)$-acyclic complexes in $\mathsf{Ch}(\mathcal{B})$.
\end{definition}
\begin{remark} \label{sumB'} \
\begin{enumerate}
\item If a class $\mathcal{X}$ of objects in $\mathcal{C}$ is closed under extensions, then $\widetilde{\mathcal{X}} \subseteq \mathsf{Ch}(\mathcal{X})$.
\item If $X \in \widetilde{\mathcal{X}}$, then $X[k] \in \widetilde{\mathcal{X}}$ for every $k \in \mathbb{Z}$.
\item If $0 \in \mathcal{X}$, then $D^m(X) \in \widetilde{\mathcal{X}}$ for every $X \in \mathcal{X}$ and $m \in \mathbb{Z}$.
\item In the case where $(\mathcal{A,B})$ is a cotorsion pair in $\mathcal{C}$, $\mathcal{H}{om}(-,\widetilde{\mathcal{B}})$-acyclic complexes in $\mathsf{Ch}(\mathcal{A})$ and $\mathcal{H}{om}(\widetilde{\mathcal{A}},-)$-acyclic complexes in $\mathsf{Ch}(\mathcal{B})$ are called in \cite{GillespieFlat} \textit{differential graded $\mathcal{A}$-complexes} and \textit{differential graded $\mathcal{B}$-complexes}, respectively. Since there may be more than two pairs $(\mathcal{A,B})$ of classes objects in $\mathcal{C}$ satisfying the condition $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{A,B}) = 0$, we have preferred to use the terminology specified in Definition~\ref{def:special_complexes} above in order to avoid confusion.
In fact, we can find an example for $\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$ where $\mathcal{A}$ and $\mathcal{B}$ do not form a cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$. This is the case for the classes $\mathcal{A} := \mathcal{DP}(R)$ and $\mathcal{B} := \mathcal{F}(R)$ of Ding projective and flat modules. By \cite[Theorem 3.7]{DPcomplexes}, a chain complex over an arbitrary ring $R$ is Ding projective if, and only if, it is $\mathcal{H}{om}(-,\widetilde{\mathcal{F}(R)})$-acyclic in $\mathsf{Ch}(\mathcal{DP}(R))$. Keep in mind that $\widetilde{\mathcal{F}(R)}$ is precisely the class of flat complexes.
\item If $\mathsf{Ext}_{\mathcal{C}}^1(\mathcal{A,B}) = 0$, $\mathcal{B}$ is closed under extensions and $0 \in \mathcal{A}$, then $S^m(A)\in \mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$ for every $A \in \mathcal{A}$ and $m \in \mathbb{Z}$.
\end{enumerate}
\end{remark}
The following result follows as \cite[Lemma 3.9]{GillespieFlat}.
\begin{lemma}\label{lem:null_homotopic}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects in $\mathcal{C}$. If $\mathsf{Ext}_{\mathcal{C}}^1(\mathcal{A,B}) = 0$ and $\mathcal{B}$ is closed under extensions, then every chain map from a $\mathcal{A}$-complex to a $\mathcal{B}$-complex is null homotopic.
\end{lemma}
Before inducing higher cotorsion pairs from an $n$-cotorsion pair in $\mathcal{C}$, we prove the following orthogonality relations between the classes (1), (2), (3) and (4) in Definition~\ref{def:special_complexes}. Recall that $\mathcal{C}$ is said to have \textit{enough $\mathcal{X}$-objects}, for some class $\mathcal{X}$ of objects of $\mathcal{C}$, if every object of $\mathcal{C}$ is an epimorphic image of an object in $\mathcal{X}$.
\begin{lemma}\label{B subseteq dgA}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects in $\mathcal{C}$. Then, the following statements hold true:
\begin{enumerate}
\item If $\mathsf{Ext}_{\mathcal{C}}^1(\mathcal{A,B}) = 0$, then $\mathsf{Ext}_{\mathsf{Ch}}^1(\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}}), \widetilde{\mathcal{B}}) = 0$ and $\mathsf{Ext}_{\mathsf{Ch}}^1(\widetilde{\mathcal{A}},\mathsf{Ch}_{\rm acy}(\widetilde{\mathcal{A}};\mathcal{B})) = 0$.
\item If $0 \in \mathcal{A}$, then $Z_m(Y) \in \mathcal{A}^{\perp_1}$ for every $Y \in (\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{A}^{\perp_1}}))^{\perp_1}$ and for each $m \in \mathbb{Z}$. Moreover, if $\mathcal{C}$ has enough $\mathcal{A}$-objects, then $Y$ is a $\mathcal{A}^{\perp_1}$-complex.
\item ${}^{\perp_1}\widetilde{\mathcal{B}} \subseteq \mathsf{Ch}_{\rm acy}({}^{\perp_1}\mathcal{B};\widetilde{\mathcal{B}})$.
\end{enumerate}
\end{lemma}
\begin{proof} \
\begin{enumerate}
\item Suppose the relation $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{A,B}) = 0$ holds true, and let $A \in \mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$ and $B \in \widetilde{\mathcal{B}}$. We aim to show that $\mathsf{Ext}^1_{\mathsf{Ch}}(A,B) = 0$. Consider the subgroup $\mathsf{Ext}^1_{\mathsf{dw}}(A,B) \subseteq \mathsf{Ext}^1_{\mathsf{Ch}}(A,B)$. We know that $A_m \in \mathcal{A}$ for every $m \in \mathbb{Z}$. On the other hand, we have short exact sequences
\[
0 \to Z_m(B) \to B_m \to Z_{m-1}(B) \to 0
\]
with $Z_{m-1}(B), Z_m(B) \in \mathcal{B}$, and so $\mathsf{Ext}^1_{\mathcal{C}}(A_m,B_m) = 0$ for every $m \in \mathbb{Z}$. This implies that $\mathsf{Ext}^1_{\mathsf{dw}}(A,B) = \mathsf{Ext}^1_{\mathsf{Ch}}(A,B)$. Now in oder to show that $\mathsf{Ext}^1_{\mathsf{dw}}(A,B) = 0$, it suffices to use the isomorphism
\[
\mathsf{Ext}^1_{\mathsf{dw}}(A,B) \cong {\rm H}_0(\mathcal{H}{om}(A,B[1]))
\]
from \cite[Lemma 2.1]{GillespieFlat}. Since the complex $\mathcal{H}{om}(A,B[1])$ is exact, we have that its $0$-th homology is zero, that is, ${\rm H}_0(\mathcal{H}{om}(A,B[1])) = 0$. Hence, $\mathsf{Ext}^1_{\mathsf{Ch}}(A,B) = 0$. The equality $\mathsf{Ext}_{\mathsf{Ch}}^1(\widetilde{\mathcal{A}},\mathsf{Ch}_{\rm acy}(\widetilde{\mathcal{A}};\mathcal{B})) = 0$ follows in the same way.
\item Let $Y \in (\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{A}^{\perp_1}}))^{\perp_1}$ and consider $Z_m(Y)$ and $A \in \mathcal{A}$. We show $\mathsf{Ext}^1_{\mathcal{C}}(A,Z_m(Y)) = 0$. By Gillespie's \cite[Lemma 4.2]{GillespieDegreewise}, we know that there is a monomorphism
\[
0 \to \mathsf{Ext}^1_{\mathcal{C}}(A,Z_m(Y)) \to \mathsf{Ext}^1_{\mathsf{Ch}}(S^m(A),Y).
\]
So it suffices to show that $S^m(A) \in \mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{A}^{\perp_1}})$. It is clear that $S^m(A) \in \mathsf{Ch}(\mathcal{A})$ since $0, A \in \mathcal{A}$. Now let $B \in \widetilde{\mathcal{A}^{\perp_1}}$. For each $m \in \mathbb{Z}$, we have that
\begin{align*}
{\rm H}_i(\mathcal{H}{om}(S^m(A),B)) & \cong \mathsf{Ext}^1_{\mathsf{dw}}(S^m(A),B[-i-1]).
\end{align*}
Note that $\mathsf{Ext}^1_{\mathcal{C}}((S^m(A))_k,(B[-i-1])_k) = 0$ for every $k \neq m$. Now for $k = m$, we have that $\mathsf{Ext}^1_{\mathcal{C}}((S^m(A))_m,(B[-i-1])_m) = \mathsf{Ext}^1_{\mathcal{C}}(A,B_{m+i+1})$. Since $B$ is an exact complex with cycles in $\mathcal{A}^{\perp_1}$, we can note that $\mathsf{Ext}^1_{\mathcal{C}}(A,B_{m+i+1}) = 0$ as we did in part (1). Hence, it follows that
\[
\mathsf{Ext}^1_{\mathsf{dw}}(S^m(A),B[-i-1]) = \mathsf{Ext}^1_{\mathsf{Ch}}(S^m(A),B[-i-1]).
\]
Using again \cite[Lemma 4.2]{GillespieDegreewise} and the fact that $B$ is exact yields an isomorphism
\[
\mathsf{Ext}^1_{\mathsf{Ch}}(S^m(A),B[-i-1]) \cong \mathsf{Ext}^1_{\mathcal{C}}(A,Z_m(B[-i-1])),
\]
where $\mathsf{Ext}^1_{\mathcal{C}}(A,Z_m(B[-i-1])) = 0$. Then, we have that ${\rm H}_i(\mathcal{H}{om}(S^m(A),B)) = 0$ for every $i \in \mathbb{Z}$, that is, $\mathcal{H}{om}(S^m(A),B)$ is an exact complex. Thus, $S^m(A) \in \mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{A}^{\perp_1}})$, and hence $\mathsf{Ext}^1_{\mathcal{C}}(A,Z_m(Y)) = 0$.
Now suppose that in addition $\mathcal{C}$ has enough $\mathcal{A}$-objects. Since we already know that $Y$ has cycles in $\mathcal{A}^{\perp_1}$, it suffices to show that $Y$ is exact, that is, that the equality $Z_m(Y) = B_m(Y)$ holds for every $m \in \mathbb{Z}$. The containment $B_m(Y) \subseteq Z_m(Y)$ is clear. For the converse containment, we have an epimorphism $f_m \colon A \to Z_m(Y)$ with $A \in \mathcal{A}$ since $\mathcal{C}$ has enough $\mathcal{A}$-objects. This induces a chain map $\tilde{f} \colon S^m(A) \to Y$ given by $\tilde{f}_m := i_m \circ f_m$ and $0$ elsewhere, where $i_m$ is the inclusion $Z_m(Y) \hookrightarrow Y_m$. On the other hand,
\begin{align*}
\mathsf{Hom}_{\mathsf{Ch}}(S^m(A),Y) / \sim \mbox{} & \cong {\rm H}_0(\mathcal{H}{om}(S^m(A),Y)) \cong \mathsf{Ext}^1_{\mathsf{dw}}(S^m(A),Y[-1]) \\
& \subseteq \mathsf{Ext}^1_{\mathsf{Ch}}(S^m(A),Y[-1]),
\end{align*}
where $\mathsf{Ext}^1_{\mathsf{Ch}}(S^m(A),Y[-1]) = 0$ since $S^m(A) \in \mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{A}^{\perp_1}})$ (by Remark~\ref{sumB'} (5)) and $Y[-1] \in (\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{A}^{\perp_1}}))^{\perp_1}$. It follows that the map $\tilde{f}$ is null homotopic, and so there exists a morphism $D_{m+1} \colon A \to Y_{m+1}$ such that $\partial^Y_{m+1} \circ D_{m+1} = i_m \circ f_m$. The latter implies $Z_m(Y) \subseteq B_m(Y)$, since $f_m$ is epic.
\item Finally, we show the containment ${}^{\perp_1}(\widetilde{B}) \subseteq \mathsf{Ch}_{\rm acy}({}^{\perp_1}\mathcal{B};\widetilde{\mathcal{B}})$. Let $X \in {}^{\perp_1}\widetilde{\mathcal{B}}$. We first show that $X_m \in {}^{\perp_1}\mathcal{B}$ for every $m \in \mathbb{Z}$. Let $B \in \mathcal{B}$. Then, we know that $D^{m+1}(B) \in \widetilde{\mathcal{B}}$ by Remark \ref{sumB'} (3), and so
\[
\mathsf{Ext}^1_{\mathcal{C}}(X_m,B) \cong \mathsf{Ext}^1_{\mathsf{Ch}}(X,D^{m+1}(B)) = 0
\]
since $X \in {}^{\perp_1}\widetilde{\mathcal{B}}$.
Now we show that $\mathcal{H}{om}(X,B)$ is an exact complex of abelian groups for every $B \in \widetilde{\mathcal{B}}$. We have natural isomorphisms
\[
{\rm H}_m(\mathcal{H}{om}(X,B)) \cong \mathsf{Ext}^1_{\mathsf{dw}}(X,B[-m-1]) = \mathsf{Ext}^1_{\mathsf{Ch}}(X,B[-m-1]) = 0
\]
where $\mathsf{Ext}^1_{\mathsf{dw}}(X,B[-m-1]) = \mathsf{Ext}^1_{\mathsf{Ch}}(X,B[-m-1])$ follows as in (1), and $B[-m-1] \in \widetilde{\mathcal{B}}$ by Remark~\ref{sumB'} (2).
\end{enumerate}
\end{proof}
We are know ready to show how to induce $n$-cotorsion pairs in $\mathsf{Ch}(\mathcal{C})$ involving the classes $\widetilde{\mathcal{A}}$, $\widetilde{\mathcal{B}}$, $\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$ and $\mathsf{Ch}_{\rm acy}(\widetilde{\mathcal{A}};\mathcal{B})$ from an $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$.
\begin{theorem}\label{theo:induced1}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects in an abelian category $\mathcal{C}$ with enough injectives, such that $\mathsf{Ext}_{\mathcal{C}}^{1}(\mathcal{A,B}) = 0$ and $\mathcal{B}$ is closed under extensions and contains the injectives of $\mathcal{C}$. If $(\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}}),\widetilde{\mathcal{B}})$ or $(\widetilde{\mathcal{A}},\mathsf{Ch}_{\rm acy}(\widetilde{\mathcal{A}};\mathcal{B}))$ is a left $n$-cotorsion pair in $\mathsf{Ch}(\mathcal{C})$, then $(\mathcal{A,B})$ is a left $n$-cotorsion pair in $\mathcal{C}$.
\end{theorem}
\begin{proof} \
Suppose first that $(\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}}),\widetilde{\mathcal{B}})$ is a left $n$-cotorsion pair in $\mathsf{Ch}(\mathcal{C})$. By Proposition~\ref{prop:cotorsion_vs_ncotorsion}, it suffices to show that $(\mathcal{A},\mathcal{B}^\wedge_{n-1})$ is a complete left cotorsion pair in $\mathcal{C}$.
We can apply Proposition~\ref{prop:cotorsion_vs_ncotorsion} in the setting of $\mathsf{Ch}(\mathcal{C})$, noticing that $\mathsf{Ch}(\mathcal{C})$ has enough injectives and that $\widetilde{\mathcal{B}}$ contains the injective complexes. Thus, $(\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}}),\widetilde{\mathcal{B}}^\wedge_{n-1})$ is a complete left cotorsion pair in $\mathsf{Ch}(\mathcal{C})$. In particular, the class $\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$ is closed under extensions, and so is $\mathcal{A}$ by Remark~\ref{sumB'} (5).
\begin{itemize}
\item[(i)] We first show that for every $C \in \mathcal{C}$, we can construct a short exact sequence
\[
0 \to N \to A \to C \to 0
\]
with $A \in \mathcal{A}$ and $N \in \mathcal{B}^\wedge_{n-1}$. For the complex $S^0(C)$, we can find a short exact sequence
\[
0 \to Y \to X \to S^0(C) \to 0
\]
where $X \in \mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$ and $Y \in \widetilde{\mathcal{B}}^\wedge_{n-1}$, since $(\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}}),\widetilde{\mathcal{B}}^\wedge_{n-1})$ is a complete left cotorsion pair in $\mathsf{Ch}(\mathcal{C})$. Then, it suffices to take $A = X_0$ and $N = Y_0$. Note that $N \in \mathcal{B}^\wedge_{n-1}$ since $\mathcal{B}$ is closed under extensions.
\item[(ii)] Now we prove the equality $\mathcal{A} = {}^{\perp_1}\mathcal{B}^\wedge_{n-1}$. So let $A \in \mathcal{A}$ and $N \in \mathcal{B}^\wedge_{n-1}$. Consider the complexes $S^0(A)$ and $D^1(N)$. We have that $S^0(A) \in \mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$ and $D^1(N) \in \widetilde{\mathcal{B}}^\wedge_{n-1}$ by Remark~\ref{sumB'}. This, along with \cite[Lemma 4.2]{GillespieDegreewise}, yields that
\[
\mathsf{Ext}^1_{\mathcal{C}}(A,N) = \mathsf{Ext}^1_{\mathcal{C}}(A,Z_0(D^1(N))) \cong \mathsf{Ext}^1_{\mathsf{Ch}}(S^0(A),D^1(N)) = 0.
\]
Then, the containment $\mathcal{A} \subseteq {}^{\perp_1}\mathcal{B}^\wedge_{n-1}$ follows. On the other hand, for every $M \in {}^{\perp_1}\mathcal{B}^\wedge_{n-1}$ there is an exact sequence
\[
0 \to N \to A \to M \to 0
\]
with $A \in \mathcal{A}$ and $N \in \mathcal{B}^\wedge_{n-1}$. We thus have that $M$ is a direct summand of $A$, since $\mathsf{Ext}^1_{\mathcal{C}}(M,N) = 0$. It follows that $M \in \mathcal{A}$.
\end{itemize}
Hence, (i) and (ii) show that $(\mathcal{A},\mathcal{B}^\wedge_{n-1})$ is a complete left cotorsion pair in $\mathcal{C}$. The same conclusion can be reached if we assume that $(\widetilde{\mathcal{A}},\mathsf{Ch}_{\rm acy}(\widetilde{\mathcal{A}};\mathcal{B}))$ is a left $n$-cotorsion pair instead.
\end{proof}
The converse of the previous result holds with some extra assumptions on the pair $(\mathcal{A,B})$ and the ground category $\mathcal{C}$. We will also need the following lemma, which follows after a careful revision of Yang and Ding's \cite[Lemma 2.1]{YangDingQuestion}.
\begin{lemma}
Let $(\mathcal{A,B})$ be a pair of classes of objects in a bicomplete abelian category $\mathcal{C}$ with enough $\mathcal{A}$-objects such that $\mathcal{B}$ is closed under extensions and satisfying $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{A,B}) = 0$. Then, for every complex $X \in \mathsf{Ch}(\mathcal{C})$, there exists a short exact sequence
\[
0 \to X \to E \to A \to 0
\]
with $E$ exact and $A \in \mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$.
\end{lemma}
\begin{theorem}\label{theo:induced_n-cotorsion_Ch}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of objects in a bicomplete abelian category $\mathcal{C}$ such that $\mathsf{Ext}_{\mathcal{C}}^{1}(\mathcal{A,B}) = 0$ and $\mathcal{B}$ closed under extensions.
\begin{enumerate}
\item If $(\mathcal{A,B})$ is a hereditary left $1$-cotorsion pair in $\mathcal{C}$, then $(\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}}),\widetilde{\mathcal{B}})$ is a left $1$-cotorsion pair in $\mathsf{Ch}(\mathcal{C})$. For $n \geq 2$ in the case where $\mathcal{C}$ has enough injectives and $\mathcal{B}$ contains the injectives of $\mathcal{C}$, if $(\mathcal{A,B})$ is a hereditary left $n$-cotorsion pair in $\mathcal{C}$ and $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{B},\mathcal{B}^\wedge_{n-1}) = 0$, then $(\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}}),\widetilde{\mathcal{B}})$ is a left $n$-cotorsion pair in $\mathsf{Ch}(\mathcal{C})$.
\item Suppose $\mathcal{C}$ has enough injectives. If $(\mathcal{A,B})$ is a hereditary left $1$-cotorsion pair in $\mathcal{C}$, then $(\widetilde{\mathcal{A}}, \mathsf{Ch}_{\rm acy}(\widetilde{\mathcal{A}};\mathcal{B}))$ is a left $1$-cotorsion pair in $\mathsf{Ch}(\mathcal{C})$. For $n \geq 2$, if $\mathcal{B}$ contains the injectives of $\mathcal{C}$ and $(\mathcal{A,B})$ is a hereditary left $n$-cotorsion pair in $\mathcal{C}$ with $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{B},\mathcal{B}^\wedge_{n-1}) = 0$, then $(\widetilde{\mathcal{A}},\mathsf{Ch}(\mathcal{B}))$ is a left $n$-cotorsion pair in $\mathsf{Ch}(\mathcal{C})$.
\end{enumerate}
\end{theorem}
\begin{proof}
We focus in the case where $n \geq 2$. The remaining case $n = 1$ follows in the same way.
We first show part (1). Again, as in the proof of Theorem~\ref{theo:induced1}, the idea is to use Proposition~\ref{prop:cotorsion_vs_ncotorsion} and show that $(\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}}),\widetilde{\mathcal{B}}^\wedge_{n-1})$ is a complete left cotorsion pair in $\mathsf{Ch}(\mathcal{C})$.
\begin{itemize}
\item[(i)] First, for every complex $X \in \mathsf{Ch}(\mathcal{C})$ we construct a short exact sequence
\[
0 \to B \to A \to X \to 0
\]
with $A \in \mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$ and $B \in \widetilde{\mathcal{B}}^\wedge_{n-1}$. It suffices to prove the case where $X$ is an exact complex. Indeed, the general case follows by using the previous lemma and a standard pullback argument, along with the fact that $\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$ is resolving since $(\mathcal{A,B})$ is hereditary. Thus, for each $m \in \mathbb{Z}$ we have a short exact sequence
\[
0 \to Z_m(X) \to X_m \to Z_{m-1}(X) \to 0.
\]
Since $(\mathcal{A,B})$ is a left $n$-cotorsion pair, each $Z_m(X)$ has a $\mathcal{A}$-precover with kernel in $\mathcal{B}^\wedge_{n-1}$. By Theorem~\ref{theo:compatible}, from these $\mathcal{A}$-precovers we can construct an exact sequence
\[
0 \to B^{n-1}_m \to B^{n-2}_m \to \cdots \to B^1_m \to B^0_m \to A_m \to X_m \to 0
\]
compatible with them, such that $A_m \in \mathcal{A}$ and $B_k \in \mathcal{B}$ for every $0 \leq k \leq n-1$. Thus, for each $m \in \mathbb{Z}$ we have a commutative diagram as in \eqref{fig_compatible}. Connecting these digrams yields an exact sequence
\[
0 \to B^{n-1} \to B^{n-1} \to \cdots \to B^1 \to B^0 \to A \to X \to 0
\]
in $\mathsf{Ch}(\mathcal{C})$ with $A \in \widetilde{\mathcal{A}} \subseteq \mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$ (see Lemma~\ref{lem:null_homotopic}) and $B^k \in \widetilde{\mathcal{B}}$ for every $0 \leq k \leq n-1$.
\item[(ii)] We now show the equality $\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}}) = {}^{\perp_1}(\widetilde{\mathcal{B}}^\wedge_{n-1})$. Note that $\mathsf{Ext}^1_{\mathcal{C}}(\mathcal{A},\mathcal{B}^\wedge_{n-1}) = 0$, then the containment $\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}}) \subseteq {}^{\perp_1}(\widetilde{\mathcal{B}}^\wedge_{n-1})$ follows by Lemma \ref{B subseteq dgA}. The converse inclusion follows after using part (i) and noticing that if $\mathcal{A}$ is closed under direct summands, then so is $\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}})$.
\end{itemize}
Therefore, $(\mathsf{Ch}_{\rm acy}(\mathcal{A};\widetilde{\mathcal{B}}),\widetilde{\mathcal{B}}^\wedge_{n-1})$ is a complete left cotorsion pair in $\mathsf{Ch}(\mathcal{C})$. For the proof concerning the pair $(\widetilde{\mathcal{A}},\mathsf{Ch}(\mathcal{B}))$, we only show that every complex is the epimorphic image of an objet in $\widetilde{\mathcal{A}}$ with kernel in $\mathsf{Ch}(\mathcal{B})^\wedge_{n-1}$. So let $X \in \mathsf{Ch}(\mathcal{C})$ be a complex. Since $\mathcal{C}$ has enough injectives, there exists a short exact sequence
\[
0 \to I \to E \to X \to 0
\]
where $E$ is exact and $I$ is a differential graded injective complex (see \cite[Lemma 2.1]{YangDingQuestion}). On the other hand, since $E$ is exact, there is a short exact sequence
\[
0 \to K^0 \to A \to E \to 0
\]
where $A \in \widetilde{\mathcal{A}}$ and $K^0 \in \widetilde{\mathcal{B}}^\wedge_{n-1}$. Taking the pullback of $I \to E \leftarrow A$ gives rise to two short exact sequences
\begin{align*}
0 & \to K^0 \to K \to I \to 0, \\
0 & \to K \to A \to X \to 0.
\end{align*}
Note that $I \in \mathsf{Ch}_{\rm acy}(\widetilde{\mathcal{A}};\mathcal{B})$ and $K^0 \in \mathsf{Ch}_{\rm acy}(\widetilde{\mathcal{A}};\mathcal{B})^\wedge_{n-1}$. By Lemma~\ref{Bk-1ext}, we have that $\mathcal{B}^\wedge_{n-1}$ is closed under extensions, and so $K \in \mathsf{Ch}(\mathcal{B})^\wedge_{n-1}$. For the case $n = 1$, it is not hard to see that also $K \in \mathsf{Ch}_{\rm acy}(\widetilde{\mathcal{A}};\mathcal{B})$.
\end{proof}
\begin{remark} \
\begin{enumerate}
\item From the previous theorem, we have hereditary left $1$-cotorsion pairs
\[
(\mathsf{Ch}(\mathcal{P}(R);\widetilde{\mathsf{Mod}(R)}),\widetilde{\mathsf{Mod}(R)}) \mbox{ \ and \ } (\widetilde{\mathcal{P}(R)},\mathsf{Ch}(\widetilde{\mathcal{P}(R)};\mathsf{Mod}(R)))
\]
where $\widetilde{\mathsf{Mod}(R)}$ is the class of exact complexes, $\widetilde{\mathcal{P}(R)}$ coincides with the class of projective complexes (which we denote by $\mathscr{P}(R)$), and $\mathsf{Ch}(\mathcal{P}(R);\widetilde{\mathsf{Mod}(R)})$ is the class of differential graded projective chain complexes. Note also that $\mathscr{P}(R)$ is part of another left $1$-cotorsion pair $(\mathscr{P}(R),\mathsf{Ch}(R))$.
\item Let $R$ be a Gorenstein ring which is not a QF ring. The results mentioned in Section \ref{sec:applications} for Gorenstein modules also hold in the context of $\mathsf{Ch}(R)$. So, if $\mathscr{GP}(R)$ denotes the class of Gorenstein projective chain complexes, we have that $(\mathscr{GP}(R),\mathscr{P}(R))$ is a left $n$-cotorsion pair in $\mathsf{Ch}(R)$. On the other hand, we know from \cite[Theorem 2.2]{GPcomplexes} that $\mathscr{GP}(R)$ is the class of complexes of Gorenstein projective modules, that is, $\mathscr{GP}(R) = \mathsf{Ch}(\mathcal{GP}(R))$. It follows that we have a left $n$-cotorsion pair in $\mathsf{Ch}(R)$ of the form $(\mathsf{Ch}(\mathcal{GP}(R)),\widetilde{\mathcal{P}(R)})$. Thus, the condition that $\mathcal{B}$ contains the injective objects required in Theorem \ref{theo:induced_n-cotorsion_Ch} is sufficient but not necessary.
Similar observations are also true for the classes of Gorenstein injective and injective complexes. Moreover, by \cite[Theorem 3.11]{GFcomplexes} and \cite[Corollary 3.12]{SarochStovicek}, we have that a chain complex $X \in \mathsf{Ch}(R)$ is Gorenstein flat if, and only if, each $X_m$ is a Gorenstein flat module. Thus, the results in Section \ref{sec:applications} for the classes $\mathcal{GF}(R)$, $\mathcal{GI}(R)$ and $\mathcal{F}(R)$ carry over to the classes $\mathsf{Ch}(\mathcal{F}(R))$, $\mathsf{Ch}(\mathcal{I}(R))$ and $\widetilde{\mathcal{F}(R)}$ of Gorenstein flat, Gorenstein injective and flat complexes, respectively.
\end{enumerate}
\end{remark}
\section*{\textbf{Funding}}
The authors thank Project PAPIIT-Universidad Nacional Aut\'onoma de M\'exico IN103317. The third named author is supported by a Comisi\'on Acad\'emica de Posgrado de la Universidad de la Rep\'ublica (CAP - UdelaR) posdoctoral fellowship.
\bibliographystyle{alpha}
|
{
"timestamp": "2019-03-01T02:08:05",
"yymm": "1902",
"arxiv_id": "1902.10863",
"language": "en",
"url": "https://arxiv.org/abs/1902.10863"
}
|
\section*{\large{Supplementary Information}}
\vspace{10mm}
\beginsupplement
\section*{Wafer growth and characterization}
\begin{figure}[!ht]
\includegraphics[width=.8\textwidth]{FigS1_V4.pdf}
\caption{(a), Layer stack of the InSb/GaAs heterostructure, where the layer constituents and thicknesses are indicated. (b), Scanning transmission electron micrograph of the structure in (a), obtained in High Angle Annular Dark Field Mode along the [110] zone axis.}
\label{SF.1}
\end{figure}
InSb-based 2DEGs were grown on semi-insulating GaAs (100) substrates by molecular beam epitaxy in a Veeco Gen 930 using ultra-high purity techniques and methods as described in Ref.~\citenum{Gardner2016}. The layer stack of the heterostructure is shown in Fig.~\ref{SF.1}a. The growth has been initiated with a $100\,\text{nm}$ thick GaAs buffer followed by a $1\,\mu\text{m}$ thick AlSb nucleation layer. The metamorphic buffer is composed of a superlattice of $300\,\text{nm}$ thick In$_{0.91}$Al$_{0.09}$Sb and $200\,\text{nm}$ thick In$_{0.75}$Al$_{0.25}$Sb layers, repeated 3 times, and directly followed by a $2\,\mu\text{m}$ thick In$_{0.91}$Al$_{0.09}$Sb layer. The active region consists of a $30\,\text{nm}$ thick InSb quantum well and a $40\,\text{nm}$ thick In$_{0.91}$Al$_{0.09}$Sb top barrier. The Si $\delta$-doping layer has been introduced at $20\,\text{nm}$ from the quantum well and the surface. The In$_\text{x}$Al$_{1-\text{x}}$Sb buffer, the InSb quantum well and the In$_\text{x}$Al$_{1-\text{x}}$Sb setback were grown at a temperature of $440$~\textcelsius\ under a p(1x3) surface reconstruction. The growth temperature was lowered to $340$~\textcelsius, where the surface reconstruction changed to c(4x4), just before the $\delta$-doping layer, to facilitate Si incorporation \cite{Liu1998}.
The scanning transmission electron micrograph of Fig.~\ref{SF.1}b reveals the efficiency of the metamorphic buffer to filter the dislocations.\\
The wafer is characterized by measuring the (quantum) Hall effect in a Hall bar geometry at $T=300~\text{mK}$. From a linear fit to the transversal resistance in a magnetic field range up to 1~T, we extract an electron density $n=2.71\cdot 10^{11}~\text{cm}^{-2}$, and by using the longitudinal resistivity at zero field, we obtain a mobility $\mu=146,400~\text{cm}^2/\text{Vs}$ (see Tab. \ref{tab:1}). We calculate the corresponding mean free path to be $l_{\text{e}}=1.26~\mu\text{m}$. In Tab. \ref{tab:1}, we also include $n$, $\mu$ and $l_e$ for the low mobility wafer, obtained from a quantum Hall measurement on this wafer. Data from the low mobility wafer is shown in Fig. 1d in the main text.
\begin{table}[H]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
&High mobility wafer&Low mobility wafer\\
\hline\hline
$n~(\text{cm}^{-2})$&$2.71\cdot10^{11}$&$2.71\cdot10^{11}$\\
\hline
$\mu~(\text{cm}^{2}/\text{Vs})$&146,400&61,500\\
\hline
$l_{\text{e}}~(\mu\text{m})$&1.26&0.53\\
\hline
\end{tabular}
\end{center}
\caption{Electron density, mobility and mean free path for the high and low mobility wafer, obtained from quantum Hall measurements at $T=300~\text{mK}$.}
\label{tab:1}
\end{table}
\section*{Device fabrication}
The devices are fabricated using electron beam lithography. First, mesa structures of width $W$ and length $L$ are defined by etching the InSb 2DEG in selected areas. We use a wet etch solution consisting of 560~ml deionized water, 9.6~g citric acid powder, 5~ml $\text{H}_{2}\text{O}_{2}$ and 7~ml $\text{H}_3\text{PO}_4$, and etch for 5~min, which results in an etch depth around $150~\text{nm}$. This is followed by the deposition of superconducting contacts in an ATC 1800-V sputtering system. Before the deposition, we clean the InSb interfaces in an Ar plasma for 3 min (using a power of 100~W and a pressure of 5~mTorr). Subsequently, without breaking the vacuum, we sputter NbTi (30~s) and NbTiN (330~s) at a pressure of 2.5~mTorr, resulting in a layer thickness of approximately $200~\text{nm}$. Next, a $45~\text{nm}$ thick layer of $\text{AlO}_{\text{x}}$ dielectric is added by atomic layer deposition at 105~\textcelsius, followed by a top-gate consisting of 10~nm/170~nm of Ti/Au.
\section*{Multiple Andreev reflections and excess current}
\begin{figure*}[h]
\includegraphics[scale=1.0]{FigS2_V6.pdf}
\caption{(a), Differential resistance, $\text{d}V/\text{d}I$, as a function of bias voltage, $V$, showing multiple Andreev reflections. Three dips at $V=2\Delta,\, 2\Delta/2 \text{ and } 2\Delta/3$ are highlighted. (b), Voltage measured as a function of bias current. The excess current, $I_\text{exc}$, and $V=2\Delta$ are indicated.}
\label{SF.2}
\end{figure*}
To further characterize the superconductivity in our JJs, we study multiple Andreev reflections (MAR) in a representative JJ, by measuring its differential resistance, $\text{d}V/\text{d}I$, as a function of applied bias voltage, $V$.
In Fig.~\ref{SF.2}a, we observe three dips in $\text{d}V/\text{d}I$, the first, at $2\Delta$, corresponding to the coherence peaks of the superconducting density of states, and two MAR peaks at $2\Delta/2$ and $2\Delta/3$. From these peaks we extract an induced superconducting gap $\Delta=0.9~\text{meV}$.
In addition, we estimate the transparancy of the same JJ by measuring its excess current, $I_\text{exc}$, and normal state resistance, $R_\text{n}$.
This measurement is shown in Fig.~\ref{SF.2}b, where we perform a linear fit in the high bias region of the $I-V$ curve ($V>2\Delta$) and obtain $I_\text{exc}=9~\mu\text{A}$ and $R_\text{n}=50~\Omega$.
Using the OBTK model \cite{Flensberg1988}, we find a value of 0.62 for the transparency of the JJ.
\section*{Weak anti-localization and spin-orbit interaction energy}
\begin{figure*}[!h]
\includegraphics[width=.5\textwidth]{SOI_WAL_V3.pdf}
\caption{Measured longitudinal conductivity difference, $\Delta\sigma$, as a function of magnetic field, $B$, displaying a weak anti-localization peak around zero field. We fit (red) the data (cyan) using the ILP model and extract the SOI energy at the Fermi energy, $\Delta_\text{SO}$, from which we calculate the Rashba spin-orbit parameter $\alpha$. The inset shows a schematic of the Hall bar device, indicating its length and width, and the magnetic field direction.}
\label{SF.3}
\end{figure*}
To obtain an estimate of the typical energy scale associated with the spin-orbit interaction, we performed weak anti-localization (WAL) measurements.
We use a Hall bar device (inset Fig.~\ref{SF.3}) fabricated on the high mobility wafer (Fig.~\ref{SF.1}), and apply magnetic field perpendicular to the Hall bar.
The measurement in Fig.~\ref{SF.3} reveals the typical WAL peak around zero field.
This peak is caused by suppression of coherent backscattering at small magnetic fields due to the spin-orbit interaction.
As we expect the Dyakonov Perel scattering mechanism to be dominating in our high mobility wafer, we use the theory developed by Iordanskii, Lyanda-Geller and Pikus~\cite{Iordanskii1994} to fit the data:
\begin{equation*}
\begin{aligned}
\dfrac{\Delta\sigma(B)}{e^2/2\pi h}\,=&\,-\dfrac{1}{a}-\dfrac{2a_0+1+H_\text{s}}{a_1(a_0+H_\text{s}-2H_\text{s})} - 2\ln H_\text{tr} - \Psi\left( 1/2+H_{\phi} \right) - 3C \\
& + \sum_{n=1}^{\infty}\left[\dfrac{3}{n}-\dfrac{3a^2_\text{n}+2a_\text{n}H_\text{s}-1-2(2n+1)H_\text{s}}{(a_\text{n}+H_\text{s}) a_\text{n-1} a_\text{n+1} - 2 H_\text{s} [(2n+1) a_\text{n}-1]}\right] \text{,}
\end{aligned}
\end{equation*}
where $\Psi$ is the Digamma function, $C$ the Euler constant, and
\begin{equation*}
a_\text{n}=n+\dfrac{1}{2}+H_{\phi}+H_\text{s} \quad \quad H_{\text{tr,$\phi$,s}}=\dfrac{\hbar}{4 e D B \tau_{\text{tr,$\phi$,s}}} \quad \quad\Delta_\text{SO}=\sqrt{\dfrac{2\hbar^2}{\tau_\text{tr} \tau_\text{s}}},
\end{equation*}
with $D=v_\text{F} l_e /2$, and $\tau_{\text{tr,$\phi$,s}}$ the scattering times for elastic, inelastic and spin-orbit scattering, respectively.
We find a spin-orbit energy splitting at the Fermi level ($\Delta_\text{SO}$) of 0.93~meV.
The Rashba spin-orbit parameter of $\alpha=36~\text{meV}\text{\AA}$ is calculated following $\alpha=\Delta_{\text{SO}}/k_\text{F}$, where $k_F$ is deduced from a classical Hall measurement.
Finally, we compare $\Delta_\text{SO}$ to the Zeeman energy.
For a Land\'{e} g-factor of 25, $\Delta_{\text{SO}}\,>\, E_{\text{Z}}$ up to 640\,mT. We are therefore in the spin-orbit dominated regime for the $0\!-\!\pi$ transition.
\section*{Magnetic field alignment}
\begin{figure*}[h]
\includegraphics[width=\textwidth]{FigS3_V5.pdf}
\caption{(a), Differential resistance, $\text{d}V/\text{d}I$ maps as a function of current bias, $I$, and out-of-plane magnetic field, $B'_\text{z}$, with increasing in-plane magnetic field, $B'_{\text{y}}$, in steps of $20$~mT. We track the central lobe of the interference pattern, labeled by white dash lines, to obtain $B_{\text{z,max}}$. (b), The $B'_{\text{z,max}}$ vs. $B'_{\text{y}}$ dependence showing the small perpendicular component of $B'_{\text{y}}$.}
\label{SF.4}
\end{figure*}
To ensure we are sweeping the magnetic field in the plane of the JJs only, we characterize the misalignment of our vector magnet axes, $B'_\text{y}$ and $B'_\text{z}$, used to apply the magnetic field in-plane and out-of-plane of the JJ, $B_\text{y}$ and $B_\text{z}$.
In Fig.~\ref{SF.4}a we present a systematic measurement of the Fraunhofer interference pattern induced by $B'_\text{z}$ with increasing $B'_{\text{y}}$.
We track the magnetic field at which the central lobe reaches its maximum $I_\text{s}$, $B'_\text{z,max}$ and plot this for all $B'_\text{y}$ in Fig.~\ref{SF.4}b.
The linear dependence observed, represents a small misalignment angle of $\theta = 1.4^\circ$.
We take this angle into account when sweeping the in-plane field, $B_\text{y}=\cos(\theta) B'_\text{y}+\sin(\theta) B'_\text{z}$, and disregard it for the out-of-plane direction, $B_\text{z}=B'_\text{z}$.
\section*{In-plane interference considerations}
\begin{figure*}[!b]
\includegraphics[width=1\textwidth]{crosssection.png}
\caption{Cross-sectional illustration of the InSb quantum well for a JJ with $L=1.1~\mu\text{m}$ and $d=30~\text{nm}$. The image is drawn to scale and the in-plane magnetic field direction, $B_\text{y}$ is indicated.}
\label{SF.FH}
\end{figure*}
We observe a switching current, $I_\text{s}$ modulation in a JJ with $L=1.1$~$\mu$m, with minima at 470~mT and 1250~mT, which are attributed to Zeeman induced $0\!-\!\pi$ transitions.
One might be inclined to believe that this modulation is caused by an in-plane Fraunhofer interference effect, due to the finite thickness ($d=30\,\text{nm}$) of the InSb quantum well.
The $I_s$ minima of such a Fraunhofer pattern are expected to occur at $B_{\text{node}}=\text{N}\Phi_0/A$, where $\Phi_0$ is the magnetic flux quantum, $A=d\cdot L$ is the cross-sectional 2DEG area and $\text{N}=1,2,3,...$.
The second minimum should thus occur at twice the value of the first, which is not the case here.
Moreover, based on the estimated cross-sectional area of the JJ (see Fig.~\ref{SF.FH}), one would expect the first node to be at 60~mT, inconsistent with the observation.
In fact, it has been shown~\cite{Monaco2009,Chiodi2012} that a oscillatory interference pattern is not expected at all in such an SNS junction with $L\gg d$.
Finally, for an in-plane interference effect one expects the $B$ value at which the $I_s$ minima occur to increase for more negative gate voltages, since the wavefunction is then squeezed and $d$ effectively reduced. However, we observe the opposite behavior (i.e., the minima move to lower $B$), as expected for Zeeman-induced $0\!-\!\pi$ transitions. To conclude, we rule out an in-plane interference effect as a possible explanation for the supercurrent modulation.
\section*{Additional gate-driven $0\!-\!\pi$ transitions}
\begin{figure*}[h]
\includegraphics[width=.98\textwidth]{FigS5_1_V4.pdf}
\caption{Differential resistance, $\text{d}V/\text{d}I$, as a function of current bias, $I$, and gate voltage, $V_\text{g}$, for the in-plane magnetic field values, $B_\text{y}$, indicated.}
\label{SF.5-1}
\end{figure*}
Here, we present additional data of the gate-driven $0\!-\!\pi$ transitions in the JJ with $L=1.1~\mu\text{m}$. The gate voltages of the $0\!-\!\pi$ transitions presented in the phase diagram are extracted from the plots in Fig.~\ref{SF.5-1}.
\section*{Error analysis for gate-driven $0\!-\!\pi$ transitions}
\begin{figure*}[!t]
\includegraphics[scale=0.8]{FigS5_V5.pdf}
\caption{(a-b), Linetraces of the differential resistance, $\text{d}V/\text{d}I$, in the JJ with $L=1.1~\mu\text{m}$ as a function of gate voltage, $V_\text{g}$, for magnetic fields, $B_\text{y}$ of $310$~mT for (a), and as indicated for (b), respectively. The peaks observed are fitted with a Gaussian curve, to obtain the standard deviation, $\sigma$. In (b) the traces are shifted for clarity.}
\label{SF.5}
\end{figure*}
To systematically extract the value where gate-driven $0\!-\!\pi$ transition occurs and its error, we use a fit of the linetraces from Fig.~\ref{SF.5-1}, at zero $I$.
At the transition point, a peak in $\text{d}V/\text{d}I$ indicates the $0\!-\!\pi$ transition.
As an example, we show a single linetrace at $310$~mT in Fig.~\ref{SF.5}a, and extract the standard deviation, $\sigma$, based on a Gaussian fit of the peak. Subsequently, we used the gate to density mapping to convert $\sigma$ to the error bar shown in the phase diagram. This fitting procedure is used for all magnetic fields (Fig.~\ref{SF.5}b).
\section*{Effective mass measurement}
To extract the effective mass of the electrons in the InSb 2DEG, the temperature dependence of the Shubnikov-de Haas (SdH) oscillation amplitude is measured in a Hall bar geometry. Figure \ref{SF.6}a shows the magnetoresistance oscillations after the subtraction of a polynomial background, $\Delta\rho_{\text{xx}}$, as a function of filling factor, $\nu$, for temperatures ranging from $T=1.73~\text{K}$ to $T=10~\text{K}$. At a fixed filling factor, the effective mass, $m^*$, can be obtained from a fit to the damping of the SdH oscillation amplitude with increasing temperature, using the expressio
\begin{equation}
\frac{\Delta\rho_{\text{xx}}(T)}{\rho_{\text{xx,0}}(T)}\propto\frac{\alpha T}{\text{sinh}(\alpha T)}~, \label{Eq.Meff}
\end{equation}
where $\rho_{\text{xx,0}}(T)$ is the temperature-dependent low-field resistivity and $\alpha=\pi k_{\text{B}} m^{*}\nu/(\hbar^2 n)$. Figure \ref{SF.6}b shows such fits to the oscillation minima and maxima of $\nu=10$ and $\nu=12$, resulting in a mean effective mass of $m^*=(0.022 \pm 0.002)\cdot m_e$, with $m_e$ being the free electron mass.
\begin{figure*}[!b]
\includegraphics[scale=0.8]{FigS_meff_V5.pdf}
\caption{(a), Shubnikov-de Haas oscillation amplitude after polynomial background subtraction as a function of filling factor for temperatures $T=1.73-10~\text{K}$. The symbols denote points that are used to extract the effective mass. (b), Temperature dependence of the oscillation amplitude (symbols). The solid lines are fits to the data (using Eq. \ref{Eq.Meff}) in order to obtain the effective mass.}
\label{SF.6}
\end{figure*}
\newpage
|
{
"timestamp": "2019-03-01T02:01:23",
"yymm": "1902",
"arxiv_id": "1902.10742",
"language": "en",
"url": "https://arxiv.org/abs/1902.10742"
}
|
\section{Introduction}
In structured prediction, the goal is to make multiple highly-correlated predictions jointly in a coordinated way~\citep{nowozin2011structured,smith2011linguistic}.
As an example of a structured-prediction task consider optical character recognition (OCR) where the main goal is to recognize a word given the sequence of images of hand-written letters.
One possible approach to this task is to train a classifier (can be, e.g., an MNIST-size convolutional neural network, CNN \citep{lecun1995convolutional}) that separately predicts each letter given its image (we will refer to such classifiers as unary predictors).
However, a unary predictor does not take into account the global word structure, i.e., which combinations of letters are likely to occur together.
The information about the word structure can be either taken into account by enriching the features fed into the unary predictor with information about other letter images or by explicitly modeling dependencies between letters.
We associate the models doing the latter with structured prediction (possibly together with the former).
On the OCR dataset collected by~\citet{taskar2003max}, sole unary predictors usually give 8-12\% letter recognition error and structured-prediction models can decrease the error to 1-3\% \citep{perez07cgm,searnn2018leblond}.
One popular approach to build structured-prediction systems is the framework of undirected probabilistic graphical models~\citep{wainwright2008} or closely related energy-based models~\citep{lecun2006tutorial}.
As of today, it is common to use structured models with parts parameterized by neural networks.
Such models give competitive results in many applications and are often called deep structured-prediction models (DSPM).
To train DSPMs, one typically needs to do inference of some form at the training time and combine it with back-propagation and stochastic optimization.
Modern neural network libraries (e.g., PyTorch \citep{paszke2017automatic} and TensorFlow \citep{tensorflow2015-whitepaper}) offer powerful automatic differentiation tools that allow to compute gradients of many inference schemes, but training such models is often difficult.
A common simplifications consists in training different system components separately, which provides a good initialization for the joint fine-tuning.
We refer to this approach as \emph{stage training} to contrast it to the \emph{joint training}, which amounts to a single stage that learns all the parameters.
Stage training is time-consuming as the procedure requires multiple runs of stochastic optimization until convergence and separate hyperparameter tuning for each stage.
The process also lacks stability due to the stage switching.
In the literature, there is contradicting evidence whether the stage or joint training procedure works better (see~Section~\ref{sec:relatedworks}).
\begin{figure*}[t]
\begin{tabular}{@{}c@{\!\!\!\!\!}c@{\!\!\!\!\!}c@{}}
\includegraphics[width=0.35\textwidth, clip=true, trim=6mm 0mm 6mm 0mm]{res_a.pdf}
&
\includegraphics[width=0.35\textwidth, clip=true, trim=6mm 0mm 6mm 0mm]{res_b.pdf}
&
\includegraphics[width=0.35\textwidth, clip=true, trim=6mm 0mm 6mm 0mm]{res_c_3.pdf}
\\[-1mm]
(a) & (b) & (c)
\end{tabular}
\vspace{-5mm}\caption{\label{motiv} OCR case study, part 2. Plot~(a) shows the dependence of trained model performance on the constant~$\alpha$ (the scaling factor of the unary potentials). The magenta vertical line corresponds to the initial setting with no scaling applied, i.e., $\alpha=1$, the yellow line corresponds to the best value of~$\alpha$, which is around $0.25$.
Plot~(b) presents the dynamics of the ratio of the unary to pairwise potentials during the joint training for the models with the different values of~$\alpha$ and different activations at the top of the net defining the unary potentials.
Plot~(c) presents the analogous dynamics for the stage training (for the first stage, we show the ratio of the unary potentials to the initialization of the pairwise ones).
The vertical dashed lines indicate the moments of stage switching.
The horizontal dashed lines indicate the quantities we compare the curve to.
}
\end{figure*}
\textbf{Contributions.}
In this paper, we experimentally compare the stage and joint procedures and provide evidence (see Section~\ref{sec:motivation} for the OCR case study) that the main reason for joint procedure to fail in some cases\footnote{Please note that we do not consider the cases when the joint training procedures fails because of the lack of fully labeled data or some engineering reasons such as GPU memory limits.} is an improper relative scaling of different model components.
We propose the online and offline algorithms to choose scaling that fixes the joint training (see Section~\ref{sec:scaling}).
To validate these methods, we apply them to the two particular types of DSPM, namely linear chain conditional random field, LCCRF \citep{lafferty2001conditional}, with application to OCR and text chunking, and Gaussian conditional random field~\citep{GAUSS} with application to the task of image segmentation (see Section~\ref{sec:models} for a review of these models with the inference and training approaches).
We present the experimental results in Section~\ref{sec:experiments} and conclude in Section~\ref{sec:conclusion}.
\section{Motivation: OCR Case Study\label{sec:motivation}}
To showcase the issues related to joint vs.\ stage training choice, we study a simple system for the task of optical character recognition (OCR).
We use the linear chain CRF model where the unary potentials are provided by the LeNet5-like CNN~\citep{lecun1998gradient} and the pairwise potentials are parameterized by a $26 \times 26$ matrix.
As the training objective we try the standard cross-entropy loss defined on the unary marginal distributions, structured SVM loss and log-likelihood.
See Sections~\ref{sec:models} and~\ref{sec:experiments} for details.
The joint training procedure consists in minimizing the objective with a stochastic optimization algorithm where at each iteration the gradient is obtained by automatically differentiating the inference algorithm (the sum-product or max-sum message passing).
The stage training procedure consists in the following three stages: train the unary predictor to classify the individual letter images independently, learn the matrix defining the pairwise potentials by minimizing the objective w.r.t.\ it (without back-propagating the gradients into the unary predictor), fine-tune all the components by minimizing the objective w.r.t.\ all the parameters jointly.
Table~\ref{tablemotiveocr} reports the comparison of the three different objectives for both stage and joint training procedures, where we observe that the stage training procedure is better for all objectives.
If we tweak the model slightly and change the ReLU activation of the last linear layer of the unary network to sigmoid, then the stage and joint procedure deliver results that are identical up to noise.
The motivation for our study is to find out why the models that are so close exhibit different behaviour.
We have seen evidence of such differences in a few related works (see Section~\ref{sec:relatedworks}) and those could even be the reason for the joint training procedure to perform worse than the sole unary predictors.
{\renewcommand{\arraystretch}{1.24}
\begin{table}[t]
\begin{center}
\begin{tabular}{c|c|c}
\textbf{OBJECTIVE} & \textbf{STAGE}& \textbf{JOINT} \\
\hline
Cross-entropy & \textbf{97.2} $\pm$ 0.1 & 96.5 $\pm$ 0.2 \\
Structured SVM & \textbf{97.0} $\pm$ 0.3 & 96.4 $\pm$ 0.4 \\
Log-likelihood & \textbf{97.2} $\pm$ 0.1 & 96.5 $\pm$ 0.3 \\
\end{tabular}
\end{center}
\caption{OCR case study, part 1. Stage denotes the stage training procedure and Joint stands for the joint training.
\label{tablemotiveocr}}
\end{table}
}
We now illustrate our hypothesis that the main reason for the joint training to underperform is the relative scaling of the unary and pairwise potentials in the LCCRF model.
We change the parameterization of the unary potentials by simply multiplying them by a the positive constant~$\alpha$ and check how this change affects the performance of the model after training.
In principle, this constant~$\alpha$ multiplies the output of a linear layer of a neural network so the model parametrization might compensate for it.
However, in the actual training runs this constant significantly affects the performance due to the fact that the training procedure optimization algorithm is not scale invariant (even if we use algorithms that try to approximate scale invariance, e.g., Adam~\citep{Adam}).
Figure~\ref{motiv}a illustrates the dependence of the final model performance (trained with the cross-entropy loss) on the constant~$\alpha$.
As the two baselines we show the accuracy of the unary predictors only and of the model trained by the stage procedure.
Please note that when $\alpha$ is too large or small the training fails, and there exists a sweet spot $\alpha_{\text{best}} \neq 1$ at which the joint training matches the stage training.
Furthermore, Figure~\ref{motiv}b shows the dynamics of the ratio of the unary and pairwise potentials\footnote{By the ratio of the unary and pairwise potentials, we mean the ratio of the average absolute values of the matrices defining the potentials. See Section~\ref{lcrf} for the details.}
at the three training runs:
the model trained with $\alpha = 1$, where the joint training underperforms;
same model with $\alpha = \alpha_{\text{best}}$, where the joint training succeeds. As another point of comparison we show the ratio for the model with the last ReLU non-linearity substituted with sigmoid, for which the joint training succeeds with $\alpha = 1$.
We observe that when the joint training underperforms the ratio of the unary to pairwise potentials diverges, but stabilizes otherwise.
As additional baseline, we show the behaviour of the ratio at the stage training procedure (Figure~\ref{motiv}c).
We observe that at the first stage the ratio grows since only unary potentials are trained, at the second stage the ratio goes down to the ``good'' level and is stable at the third stage.
Please also note that the stage training procedure needs significantly more iterations than the joint one.
\section{Related Works\label{sec:relatedworks}}
\textbf{Deep Structured-Prediction Models.} First, we review the works that combine neural networks with energy-based structured models and pay attention to whether they use joint or stage training.
The idea of combining networks and energy-based models was well known in the 90s, for example,
\citet{bottou97,lecun1998gradient} introduced graph transformers (a combination of hidden Markov models and neural networks) that were trained end-to-end and deployed as systems of hand-written character recognition.
More recently, \citet{tompson2014joint,jaderberg2014text,vu2015context,chen2015learning} used the stage training approach (with the end-to-end fine-tuning at the final stage) for the computer vision tasks of human pose estimation, free text recognition, multiple object detection, image tagging, respectively.
\citet{yang2016multi,chen2017improving} used the joint training approach for the BiLSTM-CRF models that combine neural networks techniques (word embeddings and recurrent nets with LSTM units) with LCCRF model on top for a variety of NLP tasks.
Recently, the DenseCRF model \citep{krahenbuhl2011efficient} received quite a bit of attention because it outperformed the previous state-of-the-art grid CRFs~\citep{shotton09textonboost} by a large margin on dense image labeling tasks, e.g., semantic image segmentation.
Naturally, there were many works that combined DenseCRF with CNN-based feature extractors to boost the performance even further.
In particular, \citet{zheng2015conditional,schwing15} used the joint training procedure, but \citet{chen2015deeplab,chen2018deeplab} used different variants of the stage procedure.
Because of complexity of tuning the parameters of DenseCRF, \citet{chen17deeplabv3} in their recent DeepLab-v3 system abandoned using DenseCRF at all.
\citet{vemulapalli2016gaussian,GAUSS,chandra2017dense} substituted the discrete DenseCRF model with the continuous Gaussian CRF, because it allowed relatively fast exact inference by solving a system of linear equations.
However, again due to the instabilities the joint training process (\citet{chandra2017dense} mentioned this explicitly), they used the stage training procedure.
Note that in many of the aforementioned works the authors used terminology ``joint end-to-end training'' to actually refer to the end-to-end fine-tuning of a system, some parts of which had been trained on the same data at an earlier stage.
For the purposes of our study, we refer to such procedures as stage training.
Also note that most of the aforementioned systems used some part of the networks pre-trained on different data, i.e., CNN feature extractors trained on ImageNet or word embeddings pre-trained on the Wikipedia corpus.
We chose not to consider such pre-training as a separate stage because from a practical point of view such pre-training is usually done separately from building a system for the task at hand, i.e., we just initialize the system from some pretrained model.
\textbf{Ill-conditioning.} Improper relative scaling, which we claim to be the main reason for the joint training to underperform, is closely related to ill-conditioning, which is a general problem in optimization and computational linear algebra.
Optimization literature offers a number of methods, mostly in the contexts of solving systems of linear and differential equations or convex optimization.
The most popular approach is to use preconditioning \citep{axelsson1985survey} and its
special cases, e.g., variable rescaling and various regularization schemes.
These ideas have been successfully applied for training neural networks \citep{bottou2012stochastic} and laid the foundation for layer-wise normalization techniques, such as Batch-Normalization \citep{ioffe2015batch} and input normalization~\citep{krizhevsky2012imagenet}. However, in the context of structured prediction, the aforementioned techniques are less efficient since the potentials are objects of different types (according to our experiments layer-wise normalization cannot improve the relative scaling of potentials provided by different models).
\textbf{Signal aggregation.} The problem of aggregation signals of different kinds naturally arises in multimodal learning (see, e.g., \citep{baltruvsaitis2019multimodal}). For models of this type, the most common way of aggregation is a simple concatenation of signals (e.g., \citet{simonyan2014twoStream} constructed a state-of-the-art model for action recognition in video by concatenating the RGB and optical flow features computed by separate networks).
In such settings, it is natural to rescale different signals before aggregating them to reduce the discrepancy between their contributions.
However, in case of DSPMs, naive normalization of the potentials leads to the problems of joint training, which we are studying.
To our knowledge, there is no general approach how to choose scaling of potentials in structured prediction models.
In this paper, we propose two approaches that can address the issue and can be readily applied to most DSPMs.
\section{Models\label{sec:models}}
In this section, we briefly review the models that we use for our study together with the corresponding learning and prediction algorithms.
\subsection{Linear Chain Conditional Random Field} \label{lcrf}
The linear chain conditional random field (LCCRF) \citep{lafferty2001conditional} is one of the most standard models for sequence labeling tasks.
Given a pair of input and output variables $X,Y$ (the inputs and outputs consist of $L$~elements, i.e., $X=\{x_j\}_{j=1}^L$ $~{\text{and\ } Y = \{y_j\}_{j=1}^L}$, output variables $y_j$ take one of the $M$~values)
the LCCRF model is defined by its score function (a.k.a. negative energy function)
\begin{equation}
F(Y\mid X, \theta) = \sum\limits_{j=1}^L U_{\theta}(x_j, y_j) + \sum\limits_{j=1}^{L-1} W_{\theta}(y_j, y_{j+1}),
\end{equation}
where the unary potential $U_{\theta}(x_j, y_j)$ measures the confidence of assigning the label $y_j$ to the input $x_j$ and the pairwise potential $W_{\theta}(y_j, y_{j+1})$ scores the assignment of the pair of consecutive output variables $(y_{j}, y_{j+1})$. In this model, each output variable $y_j$ is directly influenced only by the corresponding input variable $x_j$ and the neighbouring output variables~$y_{j-1}$~and~$y_{j+1}$.
The LCCRF likelihood is defined by the normalized exponent of the score function
\begin{equation}\label{prob}
P(Y\mid X,\theta) = \frac{1}{Z(\theta)}\exp\{F(Y\mid X, \theta)\},
\end{equation}
where $Z(\theta)$ denotes the partition function which ensures proper normalization of the probability function. Consequently, we can define the marginal distribution (which we call simply marginals) on the hidden variable $y_j$ as the summation of $P(Y|X,\theta)$ over all possible states of all output variables except~$y_j$:
\begin{equation}
p_j(y_j\mid X,\theta) = \sum\limits_{Y\setminus y_j}\frac{1}{Z(\theta)}\exp\{F(Y\mid X, \theta)\}.
\end{equation}
Given the parameters $\theta$ and input variables $X$ we can make prediction in several different ways, e.g., find the output variable assignment with the maximal score value, which coincides with the Maximum A Posteriori (MAP) estimate, or take the argmax of each marginal distribution.
To learn the parameters $\theta$ of LCCRF, we can use the maximum likelihood training, i.e., minimize the objective
\begin{equation}
\mathcal{L}_{\text{MLE}}(X,Y \mid \theta) := -\log{P(Y\mid X,\theta)},
\end{equation}
which depends on the computation of the partition function~$Z(\theta)$.
An alternative choice for the training objective is the marginal likelihood~\cite{kakade2002} (equivalent to the standard cross-entropy loss defined on the marginals):
\begin{equation}\label{crosstrain}
\mathcal{L}_{\text{CE}}(X,Y\mid \theta) := -\frac{1}{L} \sum\limits_{j=1}^L \log p_j(y_j \mid X,\theta).
\end{equation}
Instead of computing the marginals exactly, we can also approximate them, e.g., with the mean-field scheme, which in the case of LCCRF consists in the following updates:
\begin{align}
\log q_j(y_j) = \mathbb{E}_{q_{j+1}} W(y_j, y_{j+1}) &+ \mathbb{E}_{q_{j-1}} W(y_{j-1}, y_{j}) \nonumber\\
& + U(x_j,y_j) + C_j,
\end{align}
where $C_j$ is the normalization constant which can be computed in the linear in the number of labels~$M$ time.
For LCCRF, this approximation is not necessary since the exact computation of marginals is tractable, but for some more complicated models, such as DenseCRF \citep{krahenbuhl2011efficient}, the mean-field approximation is the only tractable scheme.
As an alternative to probabilistic training we can use the Structured Support Vector Machine, SSVM \citep{taskar2003max,tsochantaridis2005large}.
The SSVM objective seeks the parameters $\theta$ that push the score $F(Y\mid X,\theta)$ on the correct configuration $Y$ to be larger than the score $F(Y'\mid X,\theta)$ on other labelings~$Y'$ with some margin denoted as $\Delta(Y,Y')$:
\begin{align}
\mathcal{L}_{\text{SSVM}}(X,Y\mid\theta) := &\max_{Y'} \left\{F(Y'\mid X,\theta) + \Delta(Y,Y') \right\}\nonumber\\
&- F(Y\mid X,\theta).
\end{align}
In this work, we consider the margin $\Delta(Y,Y')$ defined by the normalized Hamming distance between $Y$ and $Y'$
\begin{equation}
\Delta(Y,Y') = \frac{1}{L} \sum\nolimits_{i=1}^L \mathbb{I}[y_i \neq y'_i],
\end{equation}
which is natural for sequence labeling tasks.
In the case of the SSVM and maximum likelihood training, we use the MAP inference during for prediction. In the case of the cross-entropy training, we take an argmax of each unary marginal.
In general, the computation of the partition function, marginals and MAP estimate has complexity exponential in $L$ as it involves the summation or maximization over all possible configurations of $Y$.
However, the chain-like structure of the model allows to solve all these problems efficiently in linear in the number of entities $L$ time by dynamic programming algorithms also known as max-sum and sum-product message-passing \citep{Bishop:2006}.
\subsection{Gaussian Conditional Random Field}
In this section, we review a model for dense labeling of images, e.g., the image segmentation task, known as Gaussian Conditional Random Field, GCRF \citep{GAUSS,chandra2017dense}.
The GCRF model is defined by the score function
\begin{equation}\label{GCRFenergy}
F(s\mid X,\Theta) = -\frac{1}{2}s^T(W_{\theta}(X)+\lambda I)s + U_{\theta}(X)s,
\end{equation}
where $W_{\theta}(X)$ is the symmetric positive semi-definite matrix of size $(LM)\times(LM)$ that corresponds to the pairwise potentials between pixel-label combinations and $U_{\theta}(X) \in \mathbb{R}^{LM}$ is the vector of unary terms.
Here $L$ stands for the number of pixels in the input image and $M$ for the number of possible pixel labels.
The vector $s$ consists of the stacked scores for each instance and is continuous (unlike LCCRF):
\begin{equation}
s = \left[s_{1,1},\dots,s_{1,L}|\dots|s_{M,1},\dots,s_{M,L}\right],
\end{equation}
where $s_{k,j}$ is the score for pixel $j$ to be labelled with label $k$.
The unary potentials $U_{\theta}(X)$ are of the structure similar to the score vector~$s$, i.e.,
\begin{equation}
U_{\theta}(X) = \left[u_{\theta}^1(X)|\dots|u_{\theta}^M(X)\right],
\end{equation}
where the symbols $u_{\theta}^k(X)$ stand for the per-class unary potentials.
The $\lambda I$ term of equation~\eqref{GCRFenergy} with $\lambda > 0$ enforces the matrix of the quadratic term to be strictly positive-definite, which allows to compute the maximum point in a closed form
\begin{equation}
\label{eq:gcrf_solution}
s^{*}(X,\theta) := (W_{\theta}(X)+\lambda I)^{-1}U_{\theta}(X).
\end{equation}
At the evaluation stage, the prediction is done by taking the instance-wise argmax of the score vector $s^{*}$
\begin{equation}
\hat{Y} := [\argmax_k s^{*}_{k,1}, \dots, \argmax_k s^{*}_{k,L}].
\end{equation}
Hence, the inference task only involves solving a system of linear equations, which can be efficiently done for large matrices with the conjugate gradient iterative algorithm~\citep{shewchuk1994introduction}.
Given the maximum point~\eqref{eq:gcrf_solution} we follow \citet{GAUSS} and define the marginal distribution of pixel~$j$ taking different labels by feeding the scores~$s^{*}_{:,j}(X,\theta)$ into the softmax function:
\begin{equation}\label{eq:softmaxmarginals}
p_j(y_j\mid X,\theta) = \text{softmax}_{y_j}\left(s^{*}_{:,j}(X,\theta)\right).
\end{equation}
Please note that the marginals~\eqref{eq:softmaxmarginals} are distributions on the discrete domain~$\{1,\dots,M\}$ and do not have any direct relationship to the actual marginal distributions of the Gaussian that corresponds to~\eqref{GCRFenergy}.
Using~\eqref{eq:softmaxmarginals} we learn the parameters~$\theta$ by optimizing the cross-entropy objective~\eqref{crosstrain}.
To optimize over $W_{\theta}(X)$ and $U_{\theta}(X)$ terms parameterized by an arbitrary model, we need to obtain the gradients of the loss function $\mathcal{L}(X,Y\mid\theta)$ w.r.t.\ them. However, the direct propagation of gradients through the linear system solver is inefficient as it involves roll-out of the whole iterative procedure. Fortunately, in this case, the backward pass can be computed by solving another system of linear equations \citep{chandra2017dense}:
\begin{align}
\frac{\partial \mathcal{L}}{\partial U} = (W+\lambda I)^{-1}\frac{\partial \mathcal{L}}{\partial s^{*}} \ , \ \ \ \ \ \ \ \ \frac{\partial \mathcal{L}}{\partial W} = - \frac{\partial \mathcal{L}}{\partial U} \otimes s^{*}, \label{eq:w}
\end{align}
where we omit the dependence of the potentials on $X$, $Y$ and $\theta$ for brevity and use $\otimes$ to denote the Kronecker product.
On the backward pass, we are given $\frac{\partial \mathcal{L}}{\partial s^{*}}$, hence, in order to compute both $\frac{\partial \mathcal{L}}{\partial U}$ and $\frac{\partial \mathcal{L}}{\partial W}$ we reuse $s^{*}$ from the forward pass and solve the system of linear equations~\eqref{eq:w}.
We now define the potentials of the score~\eqref{GCRFenergy}.
The unary potentials are the outputs of a CNN applied to the input image in the fully-convolutional way.
We use the pairwise term~$W$ in the form of the class-agnostic Potts model \citep{chandra2017dense}, where the pairwise potential for the connected pixels to have different labels is computed as the dot-product of the pixel embeddings $\mathcal{A}_i$ (each pixel has the embedding computed as the output of a CNN similar to the unaries).
In this setting, the matrix of pairwise potentials has the following form:
\begin{equation}
W = \begin{bmatrix}
0 & \mathcal{A}\mathcal{A}^T\\
\mathcal{A}\mathcal{A}^T & 0
\end{bmatrix}.
\end{equation}
\section{Scaling Methods\label{sec:scaling}}
\subsection{Online Scaling}\label{online}
We now introduce our first method to automatically choose a good relative scaling between the unary and pairwise potentials. Consider a score function $F(Y|X,\theta)$ as the function of the potentials $U_{\theta}(X)$ and $W_{\theta}(X)$, which are conditioned on the input and parameters:
\begin{equation}
F(Y\mid X,\theta) = F(Y\mid U_{\theta}(X), W_{\theta}(X)).
\end{equation}
In addition, we assume that we are given a training objective, which also can be viewed as a function of $U_{\theta}(X)$ and $W_{\theta}(X)$ as it depends on the score function and ground-truth output variable $Y$. We denote the training objective on a training pair $X,Y$ by~$\mathcal{L}(Y,U_{\theta}(X), W_{\theta}(X))$.
Now we define the scaling factor $\alpha$ that essentially represents the trade-off between the unary and pairwise components of the score function:
\begin{equation}
\hat{F}_{\alpha}(Y\mid X,\theta) = F(Y\mid \alpha U_{\theta}(X), W_{\theta}(X)).
\end{equation}
Consequently, the training objective is modified as~$\mathcal{L}(Y,\alpha U_{\theta}(X), W_{\theta}(X))$.
We denote the modified loss averaged over a subset $\mathcal{D}$ of objects by $\mathcal{L}_{\alpha}(\mathcal{D})$:
\begin{equation}
\mathcal{L}_{\alpha}(\mathcal{D}) = \frac{1}{|\mathcal{D}|}\sum\limits_{(X,Y)\in\mathcal{D}} \mathcal{L}(Y,\alpha U_{\theta}(X), W_{\theta}(X))
\end{equation}
Now we introduce our online scaling algorithm. After each training epoch (i.e., a pass over the training set), we choose the scaling factor $\alpha$ via the grid search on the training set (subset for speed-up) $\mathcal{D}$ with respect to~$\mathcal{L}_{\alpha}(\mathcal{D})$. At the evaluation stage, we perform inference according to~$\hat{F}_{\alpha}(Y\mid X,\theta)$. Algorithm~\ref{onlinealgo} summarizes this approach.
\begin{algorithm}[htbp]
\begin{algorithmic}[1]
\STATE{\textbf{initialize} $\theta$}
\STATE{\text{epoch := 0}, $\alpha := 1$}
\FOR{\text{epoch} $<$ \text{n\_epoch}}
\FOR{$(X,Y)$ \text{in training set}}
\STATE{\text{compute} $(U_{\theta}(X), W_{\theta}(X))$}
\STATE{\text{compute} $\mathcal{L}(Y,\alpha U_{\theta}(X), W_{\theta}(X))$}
\STATE{\text{update} $\theta$ \text{via an SGD step}}
\ENDFOR
\STATE{\text{epoch += 1}}
\STATE{$\mathcal{D} :=$ \text{training set or its subset}}
\STATE{pick the best~$\alpha_{\text{best}}$ minimizing~$\mathcal{L}_{\alpha}(\mathcal{D})$ on an $\alpha$-grid}
\STATE{$\alpha := \alpha_{\text{best}}$}
\ENDFOR
\end{algorithmic}
\caption{Online Scaling\label{onlinealgo}}
\end{algorithm}
\textbf{Remark.} When the unary potentials are defined as the output of a linear layer tweaking the constant $\alpha$ is equivalent to the special choice of initialization and learning rate for the parameters of the last linear layer when the stochastic optimization is done by the regular SGD (see Appendix~\ref{proof}).
\subsection{Offline Scaling}
The online scaling technique has a computational bottleneck, which is the grid search w.r.t.\ the scaling factor $\alpha$ after each training epoch. Unfortunately, for many models described in section \ref{online} it is impossible to determine a constant scaling factor $\alpha$ that works well for the whole training procedure.
We will now parameterize the potentials to approximately disentangle the potential individual norm and relative scaling between unary and pairwise terms.
We parameterize the potentials as follows:
\begin{align}
&\hat{U}_{\theta}(X) := \alpha \frac{U_{\theta}(X)}{\|U_{\theta}(X)\|}, \ \ \
\hat{W}_{\theta}(X) := \frac{W_{\theta}(X)}{\|W_{\theta}(X)\|},
\end{align}
for some fixed scaling factor $\alpha$, which can be tuned via validation.
Here, for a matrix $M \in \mathbb{R}^k\times \mathbb{R}^l$, we denote the averaged absolute value of its elements
by $\|M\|$.
Consequently, the score function and the training objective are modified in the following way:
\begin{align*}
&\hat{F}(Y\mid X,\theta) := F(Y\mid \hat{U}_{\theta}(X), \hat{W}_{\theta}(X))\\
&\hat{\mathcal{L}}(Y,U_{\theta}(X), W_{\theta}(X)) := \mathcal{L}(Y,\hat{U}_{\theta}(X), \hat{W}_{\theta}(X))
\end{align*}
In addition to the direct modification of the score function, we can also enforce proper scaling implicitly via the regularization term added to the training objective:
\begin{align}
\mathcal{R}(U_{\theta}(X), W_{\theta}(X)) := \lambda \left(\frac{\|U_{\theta}(X)\|}{\|W_{\theta}(X)\|} - \alpha\right)^2
\end{align}
The hyperparameters $\lambda$ and $\alpha$ may be tuned on the validation set. The key difference of this approach from the previous one is that we do not modify the potentials directly.
\textbf{Remarks.} We found that the score function parameterization that rescales both the unary and pairwise potentials with a constant factor $\alpha$, i.e., changes the temperature, did not affect the model performance (see results in Appendix~\ref{temperature}). In addition, the underperformance of the joint training is not related to the particular choice of the optimize. We report results for the Adam and SGD optimizers in Section~\ref{expsexps} and Appendix~\ref{sgd_study}, respectively.
\section{Experiments\label{sec:experiments}}
In this section, we evaluate the stage and joint training procedures together with the methods proposed in Section~\ref{sec:scaling} on the tree different tasks.
Section~\ref{sec:tasks} explains the tasks, models and evaluation methodology.
Section~\ref{expsexps} contains the experiment results and the discussion.
\subsection{Tasks}\label{sec:tasks}
\vspace{-0.1cm}\textbf{Optical Character Recognition.}\label{ocrdesc}
In the optical character recognition (OCR) task posed by \citet{taskar2003max},\footnote{\url{http://ai.stanford.edu/~btaskar/ocr/}}
we need to recognize an English word given images of its letters. All the first letters of the words in the dataset are cut off due to possible capitalization.
The OCR task uses the averaged accuracy~$\frac{1}{N}\sum\nolimits_{i=1}^N\frac{1}{L_i}\sum\nolimits_{j=1}^{L_i} \mathbb{I}[\hat{y}_j^i = y_j^i]$ as the performance metric.
Here $\{\hat{y}^i_j\}_{j=1}^{L_i}$ is the prediction for word $i$ and $\{y^i_j\}_{j=1}^{L_i}$ is its ground-truth labeling.
The OCR dataset contains approximately 7000 words and is divided into 10 folds. We take one fold for validation purposes and use it only for selecting hyperparameters.
On the remaining 9 folds, we do the 9-fold cross-validation (train on 8 folds and test on 1, repeat 9 times).
For this task, we use the linear-chain CRF model. The unary potentials are obtained by applying the LeNet5-like \citep{lecun1998gradient} convolutional neural network to letter images (each image is zero padded to the size of $32 \times 32$). This unary CNN consists of the two convolutional blocks with 10 and 20 filters, respectively, and the kernel size of 5. Each convolution is followed by the ReLU activation \citep{dahl2013improving} and the $2\times 2$ max-pooling.
The features obtained by the convolutional blocks are then passed through the two fully-connected layers with output sizes of 140 and 26, respectively, and the ReLU activation after the first fully-connected layer.
The pairwise potentials are parameterized with a real-valued matrix $W$ of size $26 \times 26$, where each matrix element represents the corresponding pairwise potential of the assignment of the neighboring variables~$(y_i, y_{i+1})$.
\textbf{CoNLL Chunking.}
Text chunking consists in dividing a text into syntactically correlated chunks of consecutive words. The CoNLL-2000 chunking dataset\footnote{\url{https://www.clips.uantwerpen.be/conll2000/chunking/}} \citep{chunk} contains 211727 tokens for training and 47377 tokens for the testing. Train and test data for this task are derived from the Wall Street Journal corpus (WSJ) and include words and part-of-speech (POS) tags as the input and chunking tags as the output (23 different chunking tags). For validation, we keep 20\% of the training data.
The performance on this task is measured by the chunk $F_1$, which we compute with the official evaluation script.
We now review the system for which joint training works well. We closely follow the hierarchical recurrent network with CRF on top \citep{yang2016multi}, but substitute the BiGRU cells with BiLSTM. In more details, the unary predictor has the following hierarchical structure: each word is first encoded with a character-level BiLSTM and the obtained feature vector is concatenated with the SENNA embedding of the word \citep{collobert2011natural} and the POS tag embedding. After that, each word combined feature vector is passed through the word-level BiLSTM to obtain the unary potentials. And finally, the unary potentials are passed to the linear-chain CRF model with the pairwise potentials parameterized by a matrix of size $23 \times 23$. The character-level LSTM has 2 layers with the state of size 50, the word-level LSTM has 2 layers with the state of size 300. The size of the letter embedding (encodes each letter for the character-level LSTM) equals 10, likewise the POS-tag embedding. Each SENNA embedding has the size of 50.
We add a learnable embedding which corresponds to the words that did not appear in SENNA embeddings.
The table below reports results of the stage and joint training procedures (three standard training objectives).
This table shows that joint training works well as is.
\setlength\extrarowheight{1pt}
\begin{center}
\vspace{-0.1cm}
\begin{tabular}{c|c|c}
\textbf{OBJECTIVE}& \textbf{STAGE}& \textbf{JOINT} \\
\hline
Cross-entropy & 94.7 & 94.9 \\
Structured SVM & 94.6 & 94.8 \\
Log-likelihood & 94.7 & 94.8 \\
\end{tabular}
\end{center}
\vspace{-0.1cm}We believe that the word-level BiLSTM cells in the model normalize the potentials properly so the scaling problem is not present.
If we substitute the word-level BiLSTM with the two linear layers (the ReLU activation in-between) the scaling breaks and the joint training starts to underperform.
We use this model for experiments in Section \ref{expsexps}.
{\renewcommand{\arraystretch}{1.2}
\begin{table*}[htbp]
\definecolor{Gray}{gray}{0.9}
\begin{center}
\resizebox{\textwidth}{!}{
\begin{tabular}{@{}c|c|c|c|c|c|c@{}}
\multirow{2}{*}{\textbf{TASK}} & \multirow{2}{*}{\textbf{OBJECTIVE}}& \multirow{2}{*}{\textbf{STAGE}} & \multirow{2}{*}{\textbf{JOINT}} & \multirow{2}{*}{\textbf{ONLINE}} & \multirow{2}{*}{\textbf{OFFLINE}} & \multirow{2}{*}{\shortstack{\textbf{OFFLINE}\\\textbf{(Reg)}}} \\[-1mm]
& & & & & & \\
\hline
\hline
\multirow{4}{*}{\textbf{OCR}} & Cross-entropy & 97.2 $\pm$ 0.1 , 41& \cellcolor[HTML]{FFF3F5}96.5 $\pm$ 0.2 , 23 & 97.3 $\pm$ 0.1 , 27 & \cellcolor[HTML]{F0F8FF}97.2 $\pm$ 0.1 , 23 &\cellcolor[HTML]{F0F8FF} 97.1 $\pm$ 0.2 , 23 \\
& Structured SVM & 97.0 $\pm$ 0.3 , 43 & \cellcolor[HTML]{FFF3F5}96.4 $\pm$ 0.4 , 23 & 97.0 $\pm$ 0.2 , 27&\cellcolor[HTML]{F0F8FF} 97.0 $\pm$ 0.3 , 23 &\cellcolor[HTML]{F0F8FF} 96.9 $\pm$ 0.4 , 23 \\
& Log-likelihood & 97.2 $\pm$ 0.1 , 42 & \cellcolor[HTML]{FFF3F5}96.5 $\pm$ 0.3 , 23 & 97.2 $\pm$ 0.1 , 27 &\cellcolor[HTML]{F0F8FF} 97.1 $\pm$ 0.1 , 23 &\cellcolor[HTML]{F0F8FF} 97.1 $\pm$ 0.2 , 23 \\
& Cross-entropy (MF) & 97.1 $\pm$ 0.1 , 50 & \cellcolor[HTML]{FFF3F5}96.6 $\pm$ 0.2 , 26 & 97.1 $\pm$ 0.1 , 30 &\cellcolor[HTML]{F0F8FF} 97.1 $\pm$ 0.1 , 26 &\cellcolor[HTML]{F0F8FF} 97.0 $\pm$ 0.1 , 27 \\
\hline
\multirow{4}{*}{\textbf{Chunking}} & Cross-entropy & 89.5 $\pm$ 0.3 , 94 & \cellcolor[HTML]{FFF3F5}87.9 $\pm$ 0.4 , 51 & 89.6 $\pm$ 0.3 , 55 &\cellcolor[HTML]{F0F8FF} 89.5 $\pm$ 0.3 , 53 &\cellcolor[HTML]{F0F8FF} 89.5 $\pm$ 0.3 , 53\\
& Structured SVM & 89.2 $\pm$ 0.5 , 97 & \cellcolor[HTML]{FFF3F5}87.6 $\pm$ 0.6 , 54 & 89.3 $\pm$ 0.4 , 57 &\cellcolor[HTML]{F0F8FF} 89.2 $\pm$ 0.5 , 56 &\cellcolor[HTML]{F0F8FF} 89.1 $ \pm$ 0.6 , 56 \\
& Log-likelihood & 89.4 $\pm$ 0.3 , 95 & \cellcolor[HTML]{FFF3F5}87.9 $\pm$ 0.4 , 53 & 89.5 $\pm$ 0.3 , 56 &\cellcolor[HTML]{F0F8FF} 89.5 $\pm$ 0.3 , 54 &\cellcolor[HTML]{F0F8FF} 89.4 $\pm$ 0.4 , 54\\
& Cross-entropy (MF) & 89.3 $\pm$ 0.3 , 104 & \cellcolor[HTML]{FFF3F5}87.7 $\pm$ 0.3 , 61 & 89.3 $\pm$ 0.2 , 66 &\cellcolor[HTML]{F0F8FF} 89.3 $\pm$ 0.3 , 64 &\cellcolor[HTML]{F0F8FF} 89.2 $\pm$ 0.3 , 64 \\
\hline
\textbf{Bin. segm.} & Cross-entropy & 86.5 $\pm$ 0.2 , 80 & \cellcolor[HTML]{FFF3F5}85.6 $\pm$ 0.3 , 44 & 86.6 $\pm$ 0.2 , 45 &\cellcolor[HTML]{F0F8FF} 86.6 $\pm$ 0.2 , 44 &\cellcolor[HTML]{F0F8FF} 86.5 $\pm$ 0.3 , 44\\[-1mm]
\end{tabular}
}
\end{center}
\caption{Results of the main experiment.
We report the performance (metrics are task specific, see Section~\ref{sec:tasks}) and the mean training time (in minutes) of the stage and joint training procedures together with the three proposed ways of fixing the joint training.
On the three tasks (OCR, chunking and binary segmentation), we studied the systems trained with the suitable objectives.
For all the performance metrics, we report the mean and standard deviation w.r.t.\ 8 random seeds.
\label{summary}
}
\vspace{-3mm}
\end{table*}
}
\textbf{Binary Segmentation.}
We consider the Weizmann Horses dataset \citep{borenstein2002class} for the binary segmentation task. The task consists in classifying image pixels into foreground or background. The dataset consists of 328 side-view color images of horses that were manually segmented. We keep 50 samples from the training set for validation. We randomly divide the remaining dataset into 5 almost equal folds and do 5-fold cross validation.
For the performance measure, this task uses the intersection over union scores (Jaccard index) between pixel labelings.
For the binary segmentation task, we use the GCRF model. The unary network is a UNet-like \citep{ronneberger2015u} fully-convolutional neural network which is a typical encoder-decoder model. The encoder consists of convolutional blocks, which are basically a stack of convolutions, batch normalization and ReLU activations. After each convolutional block, we downsample with max-pooling. The decoder consist of similar convolutional blocks and upsampling is done by the bilinear interpolation. The shortcut connections between the convolutional blocks of the encoder and decoder of the identical resolution allow to use the features of different resolutions for more robust segmentation. The architecture of pairwise network for pixel embeddings is similar to the unary network, but without the regression layer at the end.
\subsection{Results and Discussion}\label{expsexps}
Table~\ref{summary} reports the results of different training schemes (stage and regular joint training with~$\alpha=1$; the three methods to improve the joint training described in Section~\ref{sec:scaling}) on the three tasks described in Section~\ref{sec:tasks}.
On the OCR and chunking tasks, we also compare the four training objectives: the cross-entropy on the exact marginals, structured SVM, log-likelihood and the cross-entropy on the marginals estimated with the mean-field (MF) approximation.
For binary segmentation, we use the only objective available, which is the cross-entropy on the marginals estimated by the softmax of~\eqref{eq:softmaxmarginals}.
We always report the mean and standard deviation computed over 8 random seeds and the training time.
The details on the hyperparameter choice and stopping criterion (consequently, the time measurements) are provided in Appendix~\ref{experiment_details}.
As another baseline, the table below shows the performance of the unary predictors only, which is significantly worse than the performance of the full models in all tasks.
\vspace{-0.5em}
\begin{center}
\begin{tabular}{@{}c|c|c|c@{}}
& \textbf{OCR} & \textbf{CHUNKING} & \textbf{BIN. SEGM.} \\
\hline
\textbf{Unary} & 91.8 $\pm$ 0.2 & 86.2 $\pm$ 0.4 & 84.1 $\pm$ 0.2\\
\end{tabular}
\end{center}
\textbf{Discussion.} Table~\ref{summary} shows that the proposed scaling techniques perform up to noise identically to the stage training procedure (but are two times faster, and have less hyperparameters to tune, see Appendix~\ref{experiment_details} for the details) and are always superior to the joint training.
The online scaling scheme~\ref{online} has the most stable training at the cost of additional grid search steps, which increases the training time.
The offline scaling approaches allow to reduce computational complexity at the cost of having more hyperparameters.
The regularization approach has slightly higher variance and has one extra hyperparameter, which increases the tuning time.
To strengthen our conclusions, we run additional experiments to investigate possible confounding factors of the effects reported in Table~\ref{summary}.
All the models of Table~\ref{summary} were trained with the Adam optimizer~\cite{Adam}, but we have tried alternatives and observed similar story, except that all the methods performed worse. We report the full results for SGD with momentum in Appendix~\ref{sgd_study}.
As another possible confounding factor we investigate the joint scaling of the potentials (a.k.a. the temperature), which implies multiplying both unary and pairwise potentials on the same constant~$\alpha$.
Our experiments with temperature scaling showed that it did not solve underperformance of the joint training at all (see Appendix~\ref{temperature} for the results).
We would also like to explicitly mention that the problems of the joint training are not related to over-fitting, which is confirmed by the fact that the same discrepancy between joint and stage procedures appears in the training error as well as in the test error, which we report in all our experiments.
\section{Conclusion\label{sec:conclusion}}
In this paper, we study the situations when the joint training of the deep structured-prediction energy-based models unexpectedly performs worse than the inconvenient multistage training approach.
We conjecture that the source of the problem lies in the improper relative scaling of the summands of the energy (that by construction are of different nature) and propose several ways (with different engineering properties) to fix the problem.
We apply the proposed techniques to the two well-established models (linear-chain CRF and Gaussian CRF) on three different tasks and demonstrate that they indeed improve performance of the joint training to the level of the stage training procedure.
\section*{Acknowledgements}
This work was partly supported by Samsung Research, Samsung Electronics.
{
|
{
"timestamp": "2019-03-01T02:19:09",
"yymm": "1902",
"arxiv_id": "1902.11088",
"language": "en",
"url": "https://arxiv.org/abs/1902.11088"
}
|
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\title{Asymptotics for the Taylor coefficients of certain infinite products}
\author[S. Chern]{Shane Chern}
\address{Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA}
\email{shanechern@psu.edu}
\date{}
\begin{document}
\maketitle
\begin{abstract}
Let $\mathbf{m}=(m_1,\ldots,m_J)$ and $\mathbf{r}=(r_1,\ldots,r_J)$ be two sequences of $J$ positive integers satisfying $1\le r_j< m_j$ for all $j=1,\ldots,J$. Let $\boldsymbol{\updelta}=(\delta_1,\ldots,\delta_J)$ be a sequence of $J$ nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product
$$\prod_{j=1}^J\Bigg(\prod_{k\ge 1}\big(1-q^{r_j+m_j(k-1)}\big)\big(1-q^{-r_j+m_jk}\big)\Bigg)^{\delta_j}.$$
\Keywords{Infinite product, Taylor coefficient, asymptotics, circle method.}
\MSC{11P55.}
\end{abstract}
\tableofcontents
\section{Introduction}
\subsection{Motivations}
For complex variables $\alpha$ and $q$ with $|q|<1$, we denote
$$(\alpha;q)_n:=\prod_{k= 0}^{n-1}(1-\alpha q^k)\quad\text{and}\quad(\alpha;q)_\infty:=\prod_{k\ge 0}(1-\alpha q^k).$$
We also use the notation
$$(\alpha,\beta,\ldots,\gamma;q)_\infty:=(\alpha;q)_\infty(\beta;q)_\infty\cdots(\gamma;q)_\infty.$$
\medskip
Let $p(n)$ be the number of partitions of $n$; that is, the number of representations of $n$ written as a sum of a non-increasing sequence of positive integers. It is well known that $p(n)$ has the generating function
$$\sum_{n\ge 0}p(n)q^n=\frac{1}{(q;q)_\infty}.$$
The study of the asymptotic behavior of $p(n)$ originates from Hardy and Ramanujan \cite{HR1917}. A couple of decades later, Rademacher \cite{Rad1937} further proved the following formula
\begin{align}
p(n)=\frac{1}{2\sqrt{2} \pi}\sum_{k\ge 1}A_k(n)\sqrt{k}\;\frac{d}{dn}\left(\frac{2}{\sqrt{n-\frac{1}{24}}}\sinh\left(\frac{\pi}{k}\sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}\right)\right),
\end{align}
where
$$A_k(n)=\sum_{\substack{0\le h< k\\ \gcd(h,k)=1}}e^{\pi i(s(h,k)-2nh/k)}$$
with $s(h,k)$ being the Dedekind sum defined in \eqref{eq:dedekind-sum}.
Apart from ordinary partitions, partitions under symmetric congruence conditions also attract broad research interest. The most famous examples arise from the Rogers--Ramanujan identities (Rogers \cite{Rog1894}, Ramanujan \cite{Ram1919}). Here the first Rogers--Ramanujan identity states that (cf.~Corollary 7.67 in \cite{And1976})
$$\frac{1}{(q,q^4;q^5)_\infty}=\sum_{n\ge 0}\frac{q^{n^2}}{(q;q)_n}.$$
Using the language in partition theory, the above identity can be restated as follows. The number of partitions of $n$ such that each part is congruent to $\pm 1$ modulo $5$ equals the number of partitions of $n$ such that the adjacent parts differ by at least two. Let $p_{5,\pm 1}(n)$ be the number of partitions of $n$ such that each part is congruent to $\pm 1$ modulo $5$. Its asymptotic formula was shown by Lehner \cite{Leh1941}:
\begin{align}\label{eq:51}
p_{5,\pm 1}(n)\sim \frac{\csc(\pi/5)}{4\cdot 3^{1/4}\cdot 5^{1/4}}n^{-3/4}\exp\Bigg(2\pi\sqrt{\frac{n}{15}}\Bigg).
\end{align}
The interested reader may also refer to Niven \cite{Niv1940}, Livingood \cite{Liv1945}, Petersson \cite{Pet1954,Pet1956}, Grosswald \cite{Gro1958}, Iseki \cite{Ise1959,Ise1960,Ise1961}, Hagis Jr.~\cite{Hag1962}, Subrahmanyasastri \cite{Sub1972} and so forth for the asymptotic behaviors of other partition functions under symmetric congruence conditions.
As we have seen, the generating function of $p_{5,\pm 1}(n)$ is indeed an infinite product under a symmetric congruence condition. Further, similar infinite products are also of number-theoretic interest. One example is the Rogers--Ramanujan continued fraction defined by
$$R(q) := \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}=q^{1/5}\frac{(q,q^4;q^5)_\infty}{(q^2,q^3;q^5)_\infty}.$$
Let us focus on the infinite product part in $R(q)$ and write
$$\sum_{n\ge 0}C(n)q^n=\frac{(q,q^4;q^5)_\infty}{(q^2,q^3;q^5)_\infty}.$$
It is known from Richmond and Szekeres \cite{RS1978} that
\begin{align}\label{eq:C(n)}
C(n)\sim \frac{2^{1/2}}{5^{3/4}}\cos\Bigg(\frac{4\pi}{5}\bigg(n+\frac{3}{20}\bigg)\Bigg)n^{-3/4}\exp\Bigg(\frac{4\pi}{5}\sqrt{\frac{n}{5}}\Bigg).
\end{align}
Hence for sufficiently large $n$, $C(5n+0,2)>0$ and $C(5n+1,3,4)<0$. We also remark that in \cite{RS1978}, Richmond and Szekeres indeed studied the asymptotic behavior of the Taylor coefficients of the general infinite product
$$\prod_{j=1}^{m-1}(q^j;q^m)^{-\zeta \chi(j)}$$
where $m$ is a positive fundamental discriminant, $\chi(j)=(m | j)$ is the Kronecker symbol and $\zeta$ is either $1$ or $-1$.
Recently, there are a number of papers \cite{AG1994,AB1979,Hir2018,McL2015,Tan2018} studying vanishing Taylor coefficients of certain infinite products. For instance, Tang \cite{Tan2018} showed that the Taylor coefficients of
$$\sum_{n\ge 0} B(n)q^n=(-q^2,-q^3;q^5)_\infty^2(q^2,q^8;q^{10})_\infty=\frac{(q^2,q^8;q^{10})_\infty(q^4;q^6;q^{10})_\infty^2}{(q^2,q^3;q^5)_\infty^2}$$
satisfy $B(5n+1)=0$ for all $n\ge 0$. At the end of Tang's paper, he also provided numerical evidence of the inequalities $B(5n+0,2,3)>0$ and $B(5n+4)<0$ for sufficiently large $n$. Similar numerical evidences are also provided for inequalities of Taylor coefficients of other infinite products.
\medskip
Motivated by these work, it is natural to investigate a broad family of infinite products. Let $\mathbf{m}=(m_1,\ldots,m_J)$ and $\mathbf{r}=(r_1,\ldots,r_J)$ be two sequences of $J$ positive integers satisfying $1\le r_j< m_j$ for all $j=1,\ldots,J$. Let $\boldsymbol{\updelta}=(\delta_1,\ldots,\delta_J)$ be a sequence of $J$ nonzero integers. In this paper, we shall study the asymptotics for the Taylor coefficients of the following infinite product
\begin{equation}
\sum_{n\ge 0}g(n)q^n=\prod_{j=1}^J (q^{r_j},q^{m_j-r_j};q^{m_j})_\infty^{\delta_j}.
\end{equation}
\subsection{Notation and main result}
Let $\mathfrak{C}$ be the set of complex numbers and $\mathfrak{H}$ be the upper half complex plane. Let $\gcd$ and $\mathrm{lcm}$ be the greatest common divisor function and least common multiple function, respectively. For a positive integer $n$, we accept the convention that $\gcd(0,n)=n$.
We define the big-$O$ notation as usual: $f(x)=O(g(x))$ means that $|f(x)|\le C g(x)$ where $C$ is an absolute constant. Furthermore, $f(x)\ll g(x)$ means that $f(x)=O(g(x))$. Throughout this paper, we always assume that the constant $C$ depends on $\mathbf{m}$, $\mathbf{r}$ and $\boldsymbol{\updelta}$ unless otherwise stated.
\medskip
Below we assume that $0\le h<k$ are integers such that $\gcd(h,k)=1$. Let us define auxiliary functions
$$\lambda_{m,r}(h,k):=\myceil{\frac{rh}{\gcd(m,k)}}$$
and
$$\lambda^*_{m,r}(h,k):=\lambda_{m,r}(h,k)-\frac{rh}{\gcd(m,k)}.$$
We also put $\hbar_m(h,k)$ an integer such that
$$\hbar_m(h,k) \frac{m h}{\gcd(m,k)}\equiv -1 \pmod{\frac{k}{\gcd(m,k)}}.$$
Notice that one may always find such an integer since $\gcd(h,k)=1$.
\medskip
Next, we define
$$\Omega:=\sum_{j=1}^J\delta_j \Bigg(2m_j-12r_j+\frac{12r_j^2}{m_j}\Bigg),$$
$$\Delta(h,k):=-\sum_{j=1}^J\delta_j \Bigg(\frac{2\gcd^2(m_j,k)}{m_j}+\frac{12\gcd^2(m_j,k)}{m_j}({\lambda^*}^2_{\!\!\!m_j,r_j}(h,k)-\lambda^*_{m_j,r_j}(h,k))\Bigg)$$
and
\begin{align}
\omega_{h,k}:=\exp\left(-\pi i\sum_{j=1}^J \delta_j\cdot s\left(\frac{m_j h}{\gcd(m_j,k)},\frac{k}{\gcd(m_j, k)}\right)\right),
\end{align}
where $s(d,c)$ is the Dedekind sum defined by
\begin{equation}\label{eq:dedekind-sum}
s(d,c):=\sum_{n \bmod{c}} \bigg(\!\!\bigg(\frac{dn}{c}\bigg)\!\!\bigg)\bigg(\!\!\bigg(\frac{n}{c}\bigg)\!\!\bigg)
\end{equation}
with
$$(\!(x)\!):=\begin{cases}
x-\lfloor x\rfloor -1/2 & \text{if $x\not\in \mathbb{Z}$},\\
0 & \text{if $x\in \mathbb{Z}$}.
\end{cases}$$
We also define
\begin{align*}
\textup{\foreignlanguage{russian}{D}}_{h,k}&:=\exp\Bigg(\pi i \sum_{j=1}^J \delta_j\bigg(\frac{r_j h}{k}-\frac{r_j \gcd(m_j,k)}{m_j k}+\frac{2r_j \gcd(m_j,k)\lambda^*_{m_j,r_j}(h,k)}{m_j k}\\
&\ \quad+\frac{\hbar_{m_j}(h,k)\gcd(m_j,k)}{k}(\lambda^2_{m_j,r_j}(h,k)-\lambda_{m_j,r_j}(h,k))\bigg)\Bigg).
\end{align*}
One readily verifies that the choice of $\hbar_m(h,k)$ does not affect the value of $\textup{\foreignlanguage{russian}{D}}_{h,k}$. At last, we define
\begin{align*}
\Pi_{h,k}:=\begin{cases}
\displaystyle\prod_{j:\lambda^*_{m_j,r_j}(h,k)=0} \Bigg(1-\exp\bigg(2\pi i\frac{r_{j}\gcd(m_{j},k)+r_{j}\hbar_{m_{j}}(h,k)m_{j}h}{m_{j}k}\bigg)\Bigg)^{\delta_j} \\\hfill\text{{if there exists $j$ such that $\lambda^*_{m_j,r_j}(h,k)=0$}},\\
\quad\quad\ \ 1 \\\hfill\text{otherwise}.
\end{cases}
\end{align*}
Remark \ref{rmk:Pi-value} tells us that the choice of $\hbar_m(h,k)$ also does not affect the value of $\Pi_{h,k}$. Also, Proposition \ref{lem:pre} indicates that for any $j$ with $\lambda^*_{m_j,r_j}(h,k)=0$, we have
$$1-\exp\bigg(2\pi i\frac{r_{j}\gcd(m_{j},k)+r_{j}\hbar_{m_{j}}(h,k)m_{j}h}{m_{j}k}\bigg) \ne 0.$$
Hence the value $\Pi_{h,k}$ is well-defined and $\Pi_{h,k}\ne 0$.
\medskip
Given a real $0\le x<1$, we define
$$\Upsilon(x):=\begin{cases}
1 & \text{if $x=0$},\\
x & \text{if $0<x\le 1/2$},\\
1-x & \text{if $1/2<x<1$}.
\end{cases}$$
\medskip
Let $L=\mathrm{lcm}(m_1,\ldots,m_R)$. We define two disjoint sets:
\begin{align*}
\mathcal{L}_{>0}&:=\{(\varkappa,\ell) \; :\; 1\le \ell\le L,\; 0\le \varkappa<\ell,\; \Delta(\varkappa,\ell)>0\},\\
\mathcal{L}_{\le 0}&:=\{(\varkappa,\ell) \; :\; 1\le \ell\le L,\; 0\le \varkappa<\ell,\; \Delta(\varkappa,\ell)\le 0\}.
\end{align*}
\medskip
Our main result states as follows.
\begin{theorem}
If the inequality
\begin{equation}\label{eq:assump}
\min_{1\le j\le J}\left(\Upsilon\big(\lambda^*_{m_j,r_j}(\varkappa,\ell)\big)\frac{\gcd^2(m_j,\ell)}{m_j}\right)\ge \frac{\Delta(\varkappa,\ell)}{24}
\end{equation}
holds for all $1\le \ell \le L$ and $0\le \varkappa<\ell$, then for positive integers $n>-\Omega/24$, we have
\begin{align}
g(n)&=2\pi i^{\sum_{j=1}^{J}\delta_j}\underset{(\varkappa,\ell)\in\mathcal{L}_{>0}}{\sum_{1\le \ell\le L}\sum_{0\le \varkappa< \ell}} \left(\frac{24n+\Omega}{\Delta(\varkappa,\ell)}\right)^{-\frac{1}{2}}\notag\\
&\quad\quad\quad\quad\quad\quad\times \sum_{\substack{k\ge 1\\k \equiv \ell \bmod{L}}} \frac{1}{k} I_{-1}\left(\frac{\pi }{6k}\sqrt{\Delta(\varkappa,\ell)(24n+\Omega)}\right)\notag\\
&\quad\quad\quad\quad\quad\quad\times \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} e^{-\frac{2\pi i nh}{k}}(-1)^{\sum_{j=1}^{J}\delta_j \lambda_{m_j,r_j}(h,k)}\omega_{h,k}^2\;\textup{\foreignlanguage{russian}{D}}_{h,k}\Pi_{h,k},\label{eq:main-result}
\end{align}
where $I_s(x)$ is the modified Bessel function of the first kind.
\end{theorem}
\begin{remark}
In the main term of \eqref{eq:main-result}, if we truncate the summation with respect to $k$ at $k= \myfloor{\sqrt{2\pi \left(n+\frac{\Omega}{24}\right)}}$, then we will obtain an error term bounded by $O_{\mathbf{m},\mathbf{r},\boldsymbol{\updelta}}(1)$.
\end{remark}
\begin{remark}
To better understand the asymptotic behavior of $g(n)$, one may apply the asymptotic expansion of $I_s(x)$ (cf.~\cite[p.~377, (9.7.1)]{AA1972}): for fixed $s$, when $|\arg x|<\frac{\pi}{2}$,
\begin{align}\label{Bessel-order}
I_{s}(x)\sim \frac{e^x}{\sqrt{2\pi x}}\left(1-\frac{4s^2-1}{8x}+\frac{(4s^2-1)(4s^2-9)}{2!(8x)^2}-\cdots \right).
\end{align}
\end{remark}
\section{Applications of the main result}
Before moving to the proof of the main result, we first give some applications. In the first two examples, we reproduce the asymptotic formulas \eqref{eq:51} and \eqref{eq:C(n)}, respectively. We then confirm Tang's inequalities in \cite{Tan2018}. In this section, we always expand the infinite product as $\sum_{n\ge 0}g(n)q^n$.
In general, to obtain an explicit asymptotic formula of $g(n)$, we first compute $\mathcal{L}_{>0}$. Next, we find the largest number among $\{\sqrt{\Delta(\varkappa,\ell)}/k\}$ with $(\varkappa,\ell)\in\mathcal{L}_{>0}$ and $k\equiv \ell \pmod{L}$. Now one needs to check if the corresponding $I$-Bessel function vanishes for this choice. If it is nonvanishing, then the asymptotic formula shall be obtained from the $I$-Bessel term. Otherwise, we move to find the second largest number among $\{\sqrt{\Delta(\varkappa,\ell)}/k\}$ and carry out the same program. Notice that if there are multiple choices of $\varkappa$, $\ell$ and $k$ giving the same value of $\sqrt{\Delta(\varkappa,\ell)}/k$, one should sum up all such $I$-Bessel terms and check if the summation vanishes or not.
\subsection{Partitions into parts congruent to $\pm 1$ modulo $5$}
Let
$$\sum_{n\ge 0}g(n)q^n=\frac{1}{(q,q^4;q^5)_\infty}.$$
Then $\mathbf{m}=\{5\}$, $\mathbf{r}=\{1\}$ and $\boldsymbol{\updelta}=\{-1\}$. Hence $L=5$ and $\Omega=-2/5$. We now compute that
\begin{align*}
\mathcal{L}_{>0}&=\{(0, 1), (0, 2), (1, 2), (0, 3), (1, 3), (2, 3),\\
&\quad\;\;\; (0, 4), (1, 4), (2, 4), (3, 4), (0, 5), (1, 5), (4, 5)\}.
\end{align*}
First, the assumption \eqref{eq:assump} is satisfied. We next find that the largest number among $\{\sqrt{\Delta(\varkappa,\ell)}/k\}$ with $(\varkappa,\ell)\in\mathcal{L}_{>0}$ and $k\equiv \ell \pmod{L}$ is $\sqrt{\frac{2}{5}}$. Here we have two choices:
$$(\varkappa,\ell,k)=(0,1,1),\ (0,5,5).$$
When $k=1$, the admissible $(h,k)$ is $(0,1)$. We compute that the $I$-Bessel term is
$$\frac{\pi \csc\big(\frac{\pi}{5}\big)}{2\sqrt{15}}\Bigg(n-\frac{1}{60}\Bigg)^{-1/2}I_{-1}\Bigg(\frac{2\pi}{\sqrt{15}}\sqrt{n-\frac{1}{60}}\Bigg).$$
When $k=5$, noticing that $\gcd(0,5)=5\ne 1$, there is no admissible $(h,k)$. Hence,
\begin{align*}
g(n)&\sim \frac{\pi \csc\big(\frac{\pi}{5}\big)}{2\sqrt{15}}\Bigg(n-\frac{1}{60}\Bigg)^{-1/2}I_{-1}\Bigg(\frac{2\pi}{\sqrt{15}}\sqrt{n-\frac{1}{60}}\Bigg)\\
&\sim \frac{\csc\big(\frac{\pi}{5}\big)}{4\cdot 3^{1/4}\cdot 5^{1/4}}\Bigg(n-\frac{1}{60}\Bigg)^{-3/4}\exp\Bigg(\frac{2\pi}{\sqrt{15}}\sqrt{n-\frac{1}{60}}\Bigg).
\end{align*}
\subsection{The Rogers--Ramanujan continued fraction}
Let
$$\sum_{n\ge 0}g(n)q^n=\frac{(q,q^4;q^5)_\infty}{(q^2,q^3;q^5)_\infty}.$$
Then $\mathbf{m}=\{5,5\}$, $\mathbf{r}=\{1,2\}$ and $\boldsymbol{\updelta}=\{1,-1\}$. Hence $L=5$ and $\Omega=24/5$. We compute that
\begin{align*}
\mathcal{L}_{>0}&=\{(2, 5),(3, 5)\}.
\end{align*}
First, the assumption \eqref{eq:assump} is satisfied. We next find that the largest number among $\{\sqrt{\Delta(\varkappa,\ell)}/k\}$ with $(\varkappa,\ell)\in\mathcal{L}_{>0}$ and $k\equiv \ell \pmod{L}$ is $\frac{2\sqrt{6}}{5\sqrt{5}}$. Here we have two choices:
$$(\varkappa,\ell,k)=(2, 5,5),\ (3, 5,5).$$
When $k=5$, the admissible $(h,k)$ are $(2,5)$ and $(3,5)$. We compute that, in total, the $I$-Bessel term is
$$\frac{4\pi}{5\sqrt{5}}\cos\Bigg(\frac{4\pi}{5}\bigg(n+\frac{3}{20}\bigg)\Bigg)\Bigg(n+\frac{1}{5}\Bigg)^{-1/2}I_{-1}\Bigg(\frac{4\pi}{5\sqrt{5}}\sqrt{n+\frac{1}{5}}\Bigg).$$
Notice that $\cos\big(\frac{4\pi}{5}(n+\frac{3}{20})\big)$ does not vanish for all $n$. Hence,
\begin{align*}
g(n)&\sim\frac{4\pi}{5\sqrt{5}}\cos\Bigg(\frac{4\pi}{5}\bigg(n+\frac{3}{20}\bigg)\Bigg)\Bigg(n+\frac{1}{5}\Bigg)^{-1/2}I_{-1}\Bigg(\frac{4\pi}{5\sqrt{5}}\sqrt{n+\frac{1}{5}}\Bigg)\\
&\sim \frac{2^{1/2}}{5^{3/4}}\cos\Bigg(\frac{4\pi}{5}\bigg(n+\frac{3}{20}\bigg)\Bigg)\Bigg(n+\frac{1}{5}\Bigg)^{-3/4}\exp\Bigg(\frac{4\pi}{5\sqrt{5}}\sqrt{n+\frac{1}{5}}\Bigg).
\end{align*}
\subsection{Tang's inequalities}
Let
$$\sum_{n\ge 0}g(n)q^n=\frac{(q^2,q^8;q^{10})_\infty(q^4;q^6;q^{10})_\infty^2}{(q^2,q^3;q^5)_\infty^2}.$$
Then $\mathbf{m}=\{5,10,10\}$, $\mathbf{r}=\{2,2,4\}$ and $\boldsymbol{\updelta}=\{-2,1,2\}$. Hence $L=10$ and $\Omega=-8$. We compute that
\begin{align*}
\mathcal{L}_{>0}&=\{(0, 1), (0, 3), (1, 3), (2, 3), (0, 5), (2, 5), (3, 5), (0, 7), (1, 7), (2, 7), (3, 7),\\
&\quad\;\;\; (4, 7), (5, 7), (6, 7), (0, 9), (1, 9), (2, 9), (3, 9), (4, 9), (5, 9), (6, 9), (7, 9),\\
&\quad\;\;\; (8, 9), (1, 10), (2, 10), (3, 10), (4, 10), (6, 10), (7, 10), (8, 10), (9, 10)\}.
\end{align*}
First, the assumption \eqref{eq:assump} is satisfied. We next find that the largest number among $\{\sqrt{\Delta(\varkappa,\ell)}/k\}$ with $(\varkappa,\ell)\in\mathcal{L}_{>0}$ and $k\equiv \ell \pmod{L}$ is $\frac{1}{\sqrt{5}}$. Here we have four choices:
$$(\varkappa,\ell,k)=(0,1,1),\ (0,5,5),\ (2, 5,5),\ (3, 5,5).$$
When $k=1$, the admissible $(h,k)$ is $(0,1)$. We compute that the $I$-Bessel term is
$$\frac{\sqrt{2}\pi}{\sqrt{15}}\sin\Bigg(\frac{\pi}{5}\Bigg)\Bigg(n-\frac{1}{3}\Bigg)^{-1/2}I_{-1}\Bigg(\frac{\sqrt{2}\pi}{\sqrt{15}}\sqrt{n-\frac{1}{3}}\Bigg).$$
When $k=5$, the admissible $(h,k)$ are $(2,5)$ and $(3,5)$. We compute that, in total, the $I$-Bessel term is
$$\frac{\sqrt{2}\pi}{\sqrt{15}}\sin\Bigg(\frac{2\pi}{5}\big(2n+1\big)\Bigg)\Bigg(n-\frac{1}{3}\Bigg)^{-1/2}I_{-1}\Bigg(\frac{\sqrt{2}\pi}{\sqrt{15}}\sqrt{n-\frac{1}{3}}\Bigg).$$
In total, we therefore have
$$\frac{\sqrt{2}\pi}{\sqrt{15}}\Bigg(\sin\bigg(\frac{\pi}{5}\bigg)+\sin\bigg(\frac{2\pi}{5}\big(2n+1\big)\bigg)\Bigg)\Bigg(n-\frac{1}{3}\Bigg)^{-1/2}I_{-1}\Bigg(\frac{\sqrt{2}\pi}{\sqrt{15}}\sqrt{n-\frac{1}{3}}\Bigg).$$
Notice that $\sin\big(\frac{\pi}{5}\big)+\sin\big(\frac{2\pi}{5}(2n+1)\big)$ vanishes only if $n\equiv 1 \pmod{5}$. Hence, for $n\not\equiv 1 \pmod{5}$,
\begin{align*}
g(n)&\sim\frac{\sqrt{2}\pi}{\sqrt{15}}\Bigg(\sin\bigg(\frac{\pi}{5}\bigg)+\sin\bigg(\frac{2\pi}{5}\big(2n+1\big)\bigg)\Bigg)\Bigg(n-\frac{1}{3}\Bigg)^{-1/2}I_{-1}\Bigg(\frac{\sqrt{2}\pi}{\sqrt{15}}\sqrt{n-\frac{1}{3}}\Bigg)\\
&\sim \frac{1}{30^{1/4}}\Bigg(\sin\bigg(\frac{\pi}{5}\bigg)+\sin\bigg(\frac{2\pi}{5}\big(2n+1\big)\bigg)\Bigg)\Bigg(n-\frac{1}{3}\Bigg)^{-3/4}\exp\Bigg(\frac{\sqrt{2}\pi}{\sqrt{15}}\sqrt{n-\frac{1}{3}}\Bigg).
\end{align*}
It follows that $g(5n+0,2,3)>0$ and $g(5n+4)<0$ for sufficiently large $n$. If we further compute a number of lower $I$-Bessel terms, we still encounter the same vanishment for $n\equiv 1 \pmod{5}$. This highly suggests that $g(5n+1)=0$, which is, indeed, proved by Tang using elementary techniques in \cite{Tan2018}.
\medskip
All other inequalities conjectured by Tang can be proved in the same manner. We omit the details here.
\section{Dedekind eta function and Jacobi theta function}
In this section, we introduce the Dedekind eta function and Jacobi theta function. All results here are standard, which can be found in, for example, \cite{Apo1990} or \cite{Zwe2002}.
Let $\tau\in\mathfrak{H}$ and $\varsigma\in\mathfrak{C}$. The Dedekind eta function is defined by
$$\eta(\tau):=q^{1/24}(q;q)_\infty$$
with $q:=e^{2\pi i \tau}$. Further, the Jacobi theta function reads
$$\vartheta(\varsigma;\tau):=\sum_{\nu\in\mathbb{Z}+\frac{1}{2}}e^{2\pi i \nu (\varsigma+\frac{1}{2})+\pi i\nu^2 \tau}.$$
Notice that if we put $\zeta:=e^{2\pi i \varsigma}$, then the Jacobi triple product identity indicates that
$$\vartheta(\varsigma;\tau)=-i q^{1/8}\zeta^{-1/2}(\zeta,\zeta^{-1}q,q;q)_\infty.$$
It follows immediately that
\begin{align}
\textup{\foreignlanguage{russian}{Zh}}(\varsigma;\tau):=&\; (\zeta,\zeta^{-1}q;q)_\infty\notag\\[0.5em]
=&\; ie^{-\frac{\pi i \tau}{6}}e^{\pi i \varsigma}\frac{\vartheta(\varsigma;\tau)}{\eta(\tau)}.\label{eq:Zh-basic}
\end{align}
The Dedekind eta function and Jacobi theta function are of broad interest due to their transformation properties. Let $\displaystyle\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb{Z})$ where we assume that $c>0$. Recall that the M\"obius transformation for $\tau\in\mathfrak{H}$ is defined by
$$\gamma(\tau):=\frac{a\tau +b}{c\tau + d}.$$
Further, for the $\gamma$ given above, we write for convenience
$$\gamma^*(\tau):=\frac{1}{c\tau + d}.$$
If
$$\chi(\gamma)=\exp\Bigg(\pi i \left(\frac{a+d}{12c}-s(d,c)-\frac{1}{4}\right)\Bigg),$$
where, again, $s(d,c)$ is the Dedekind sum, then
\begin{align}\label{eta-trans}
\eta(\gamma(\tau))=\chi(\gamma)(c\tau+d)^{1/2}\eta(\tau)
\end{align}
and
\begin{align}\label{theta-trans1}
\vartheta(\varsigma\gamma^*(\tau);\gamma(\tau))=\chi(\gamma)^3(c\tau+d)^{1/2}e^{\frac{\pi i c \varsigma^2}{c\tau+d}}\vartheta(\varsigma;\tau).
\end{align}
Further, let $\alpha$ and $\beta$ be integers. The Jacobi theta function also satisfies
\begin{align}\label{theta-trans2}
\vartheta(\varsigma+\alpha\tau+\beta;\tau)=(-1)^{\alpha+\beta}e^{-\pi i \alpha^2 \tau}e^{-2\pi i\alpha \varsigma}\vartheta(\varsigma;\tau).
\end{align}
\section{Farey arcs and a transformation formula}\label{sec:Farey}
To study the asymptotics for the Taylor coefficients of $G(q)$, we turn to the celebrated circle method due to Rademacher \cite{Rad1937,Rad1943} whose idea originates from Hardy and Ramanujan \cite{HR1917}. Recalling that $G(q)$ is holomorphic inside the unit disk, we may directly apply Cauchy's integral formula to deduce
\begin{equation*}
g(n)=\frac{1}{2 \pi i} \oint_{\mathcal{C}:|q|=r} \frac{G(q)}{q^{n+1}}\ dq,
\end{equation*}
where the contour integral is taken counter-clockwise. Now one puts $r=e^{-2 \pi \varrho}$ with $\varrho=1/N^2$ where $N$ is a sufficiently large positive integer.
Next, we dissect the circle $\mathcal{C}$ by Farey arcs. Let $h/k$ with $\gcd(h,k)=1$ be a Farey fraction of order $N$. If we denote by $\xi_{h,k}$ the interval $[-\theta'_{h,k},\theta''_{h,k}]$ with $\theta'_{h,k}$ and $\theta''_{h,k}$ being the positive distances from $h/k$ to its neighboring mediants, then
\begin{equation*}
g(n)=\sum_{1\le k\le N} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1}} e^{-\frac{2\pi i nh}{k}} \int_{\xi_{h,k}} G\big(e^{2\pi i (h/k+i \varrho +\phi)}\big) e^{-2\pi i n \phi} e^{2 \pi n \varrho}\ d\phi.
\end{equation*}
Making the changes of variables $z=k(\varrho -i \phi)$ and $\tau = (h+i z)/k$ yields
\begin{equation}\label{eq:cauchy-var}
g(n)=\sum_{1\le k\le N} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1}} e^{-\frac{2\pi i nh}{k}} \int_{\xi_{h,k}} G\big(e^{2\pi i \tau}\big)e^{-2\pi i n \phi} e^{2 \pi n \varrho}\ d\phi.
\end{equation}
\medskip
Let $r<m$ be positive integers. Our next task is to apply the transformation properties of the Dedekind eta function and Jacobi theta function so that $\textup{\foreignlanguage{russian}{Zh}}(r\tau;m\tau)$ can be nicely reformulated around the Farey arc with respect to $h/k$. To do so, we need to construct a suitable matrix in $SL_2(\mathbb{Z})$.
Let $d=\gcd(m,k)$. For convenience, we write $m=d m'$ and $k= d k'$. Recalling that $\hbar_m(h,k)$ satisfies $\hbar_m(h,k) m' h\equiv -1 \pmod{k'}$, we put $b_{m'}=(\hbar_m(h,k) m' h+1)/k'$. It is straightforward to verify that the following matrix is in $SL_2(Z)$:
\begin{equation}
\gamma_{(m,h,k)}=\begin{pmatrix}\hbar_m(h,k) & -b_{m'}\\k' & -m' h\end{pmatrix}.
\end{equation}
\medskip
Since $\tau = (h+i z)/k = (h+i z)/dk'$, one may compute
\begin{align*}
&\gamma_{(m,h,k)}(m\tau)\\
&\quad=\dfrac{\hbar_m(h,k)\cdot m\frac{h+iz}{d k'}-b_{m'}}{k' \cdot m\frac{h+iz}{d k'}-m' h}=\frac{\hbar_m(h,k) m' h+\tilde{h}_{m'}(im'z)-(\hbar_m(h,k)m' h+1)}{m' h k'+k'(im'z)-m' hk'}\\
&\quad=\frac{\hbar_m(h,k)}{k'}+\frac{1}{m'k'z}i.
\end{align*}
Namely,
\begin{equation}\label{eq:gamma}
\gamma_{(m,h,k)}(m\tau)=\frac{\hbar_m(h,k) \gcd(m,k)}{k}+\frac{\gcd^2(m,k)}{mkz}i.
\end{equation}
On the other hand, we have
\begin{align*}
\gamma^*_{(m,h,k)}(m\tau)=\dfrac{1}{k' \cdot m\frac{h+iz}{d k'}-m' h}=-\frac{\gcd(m,k)}{mz}i
\end{align*}
and hence
\begin{equation}\label{eq:gamma-r}
r\tau\gamma^*_{(m,h,k)}(m\tau)=\frac{r \gcd(m,k)}{mk}-\frac{rh\gcd(m,k)}{mkz}i.
\end{equation}
Further,
\begin{align}
&r\tau\gamma^*_{(m,h,k)}(m\tau)+\lambda_{m,r}(h,k)\gamma_{(m,h,k)}(m\tau)\notag\\
&\quad=\frac{r \gcd(m,k)}{mk}+\lambda_{m,r}(h,k)\frac{\hbar_m(h,k) \gcd(m,k)}{k}+\lambda^*_{m,r}(h,k)\frac{\gcd^2(m,k)}{mkz}i.\label{eq:gamma-mix}
\end{align}
\medskip
Recalling from \eqref{eq:Zh-basic} that
\begin{align*}
\textup{\foreignlanguage{russian}{Zh}}(r\tau;m\tau)=ie^{-\frac{\pi i m\tau}{6}}e^{\pi i r\tau}\frac{\vartheta(r\tau;m\tau)}{\eta(m\tau)},
\end{align*}
one has, from \eqref{eta-trans}, \eqref{theta-trans1}, \eqref{theta-trans2} and the fact $s(-m'h,k')=-s(m'h,k')$, that
\begin{align*}
\textup{\foreignlanguage{russian}{Zh}}(r\tau;m\tau)&=ie^{-\frac{\pi i m\tau}{6}}e^{\pi i r\tau} \chi(\gamma_{(m,h,k)})^{-2}e^{-\frac{\pi i k' r^2\tau^2}{k'm\tau-m' h}}\\
&\quad\times\frac{\vartheta(r\tau\gamma^*_{(m,h,k)}(m\tau);\gamma_{(m,h,k)}(m\tau))}{\eta(\gamma_{(m,h,k)}(m\tau))}\\
&=ie^{-\frac{\pi i m\tau}{6}}e^{\pi i r\tau} \chi(\gamma_{(m,h,k)})^{-2}e^{-\frac{\pi i k r^2\tau^2}{km\tau-m h}}(-1)^{\lambda_{m,r}(h,k)}\\
&\quad\times e^{\pi i \lambda^2_{m,r}(h,k)\gamma_{(m,h,k)}(m\tau)}e^{2\pi i \lambda_{m,r}(h,k)r\tau \gamma^*_{(m,h,k)}(m\tau)}\\
&\quad\times \frac{\vartheta(r\tau\gamma^*_{(m,h,k)}(m\tau)+\lambda_{m,r}(h,k)\gamma_{(m,h,k)}(m\tau);\gamma_{(m,h,k)}(m\tau))}{\eta(\gamma_{(m,h,k)}(m\tau))}\\
&=i (-1)^{\lambda_{m,r}(h,k)}e^{-2\pi i s(m'h,k')}\\
&\quad\times \exp\Bigg(\pi i\bigg(\frac{rh}{k}-\frac{rd}{mk}+\frac{2rd\lambda^*_{m,r}(h,k)}{mk}\\
&\qquad\qquad\quad+\frac{\hbar_m(h,k)d}{k}(\lambda^2_{m,r}(h,k)-\lambda_{m,r}(h,k))\bigg)\Bigg)\\
&\quad\times \exp\Bigg(\frac{\pi}{12k}\bigg(\Big(2m-12r+\frac{12r^2}{m}\Big)z\\
&\qquad\qquad\quad-\Big(\frac{2d^2}{m}+\frac{12d^2}{m}({\lambda^*}^2_{\!\!\!m,r}(h,k)-\lambda^*_{m,r}(h,k))\Big)\frac{1}{z}\bigg)\Bigg)\\
&\quad\times \textup{\foreignlanguage{russian}{Zh}}\big(r\tau\gamma^*_{(m,h,k)}(m\tau)+\lambda_{m,r}(h,k)\gamma_{(m,h,k)}(m\tau);\gamma_{(m,h,k)}(m\tau)\big).
\end{align*}
Consequently, we deduce the following transformation formula.
\begin{align}
&G(e^{2\pi i \tau})=\prod_{j=1}^{J}\textup{\foreignlanguage{russian}{Zh}}^{\delta_j}(r_j\tau;m_j\tau)\notag\\
&\quad=i^{\sum_{j=1}^{J}\delta_j}(-1)^{\sum_{j=1}^{J}\delta_j \lambda_{m_j,r_j}(h,k)}\omega_{h,k}^2\;\textup{\foreignlanguage{russian}{D}}_{h,k}\notag\\
&\quad\quad\times \exp\Bigg(\frac{\pi}{12k}(\Omega z+\Delta(h,k)z^{-1})\Bigg)\notag\\
&\quad\quad\times \prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(r_j\tau\gamma^*_{(m_j,h,k)}(m_j\tau)+\lambda_{m_j,r_j}(h,k)\gamma_{(m_j,h,k)}(m_j\tau);\gamma_{(m_j,h,k)}(m_j\tau)\big).\label{eq:trans-main}
\end{align}
\begin{remark}\label{rmk:Im-part}
It follows from \eqref{eq:gamma-mix} that for all $j=1,2,\ldots,J$,
\begin{align*}
0\le \Im\big(r_j\tau\gamma^*_{(m_j,h,k)}(m_j\tau)+\lambda_{m_j,r_j}(h,k)\gamma_{(m_j,h,k)}(m_j\tau)\big)<\Im\big(\gamma_{(m_j,h,k)}(m_j\tau)\big).
\end{align*}
\end{remark}
\section{Some auxiliary results}
\subsection{Necessary bounds}
Now we are going to present some useful bounds, which were obtained in the previous work; see, for example, \cite{Rad1937}.
\medskip
First, it is well known (cf.~Chapter 3 in \cite{HW1979}) that for a Farey fraction $h/k$ of order $N$, one has
\begin{equation}\label{eq:theta-bound}
\frac{1}{2kN}\le \theta'_{h,k},\theta''_{h,k}\le \frac{1}{kN}
\end{equation}
and hence
\begin{equation}\label{eq:xi-bound}
\frac{1}{kN}\le |\xi_{h,k}| \le \frac{2}{kN}.
\end{equation}
Next, since $z=k(\varrho -i \phi)$, it follows that
\begin{equation}\label{eq:Re-z-bound}
\Re(z)=k\varrho=\frac{k}{N^2}.
\end{equation}
Further, one has
\begin{align}\label{eq:Re-1-z-bound}
\Re\left(\frac{1}{z}\right)\ge \frac{k}{2}
\end{align}
since
\begin{align*}
\Re\left(\frac{1}{z}\right)=\frac{1}{k}\frac{\varrho}{\varrho^2+\phi^2}\ge \frac{1}{k}\frac{N^{-2}}{N^{-4} + k^{-2} N^{-2}}=\frac{k}{k^2N^{-2}+1}\ge \frac{k}{1+1}=\frac{k}{2},
\end{align*}
where we use the fact $k\le N$ in the last inequality.
\subsection{A partition-theoretic result}\label{sec:ptn-res}
Let $\eta$ be a positive integer. Let $p^*_{\eta}(s,t;n)$ denote the number of 2-colored (say, red and blue) partition $\eta$-tuples with $s$ parts in total colored by red and $t$ parts in total colored by blue. Here we allow $0$ as a part. Let $q$, $\zeta$ and $\xi$ be such that $|q|<1$, $|\zeta|<1$ and $|\xi|<1$. The following infinite triple summation
$$\sum_{n\ge 0}\sum_{s\ge 0}\sum_{t\ge 0}p^*_{\eta}(s,t;n)\zeta^s \xi^t q^n=\left(\frac{1}{(\zeta,\xi;q)_\infty}\right)^{\eta}$$
is absolutely convergent. Further, considering another absolutely convergent infinite triple summation
$$\sum_{n\ge 0}\sum_{s\ge 0}\sum_{t\ge 0}d^*_{\eta}(s,t;n)\zeta^s \xi^t q^n:=(\zeta,\xi;q)_\infty^{\eta},$$
an easy partition-theoretic argument indicates that $|d^*_{\eta}(s,t;n)|\le p^*_{\eta}(s,t;n)$ for all $s,t,n\ge 0$. Also, we have $d^*_{\eta}(0,0;0)=p^*_{\eta}(0,0;0)=1$.
In general, for a nonzero integer $\delta$, if we write
$$\sum_{n\ge 0}\sum_{s\ge 0}\sum_{t\ge 0}a_{\delta}(s,t;n)\zeta^s \xi^t q^n:=(\zeta,\xi;q)_\infty^{\delta},$$
then
$$a_{\delta}(s,t;n)=\begin{cases}
p^*_{|\delta|}(s,t;n) & \text{if $\delta<0$},\\
d^*_{|\delta|}(s,t;n) & \text{if $\delta>0$},
\end{cases}$$
and hence $|a_{\delta}(s,t;n)|\le p^*_{|\delta|}(s,t;n)$ for all $s,t,n\ge 0$. Trivially, we also have
\begin{align*}
\Big|(\zeta,\xi;q)_\infty^{\delta}\Big|&=\Bigg|\sum_{n\ge 0}\sum_{s\ge 0}\sum_{t\ge 0}a_{\delta}(s,t;n)\zeta^s \xi^t q^n\Bigg|\\
&\le \sum_{n\ge 0}\sum_{s\ge 0}\sum_{t\ge 0}p^*_{|\delta|}(s,t;n)|\zeta|^s |\xi|^t |q|^n.
\end{align*}
Further, for real $0\le \alpha,\beta,x<1$, we have
\begin{align}
\sum_{n\ge 0}\sum_{s\ge 0}\sum_{t\ge 0}p^*_{1}(s,t;n)\alpha^s \beta^t x^n&=\frac{1}{(\alpha,\beta;x)_\infty}\notag\\
&=\exp\Bigg(-\sum_{k\ge 0}\log(1-\alpha x^k)-\sum_{\ell\ge 0}\log(1-\beta x^\ell)\Bigg)\notag\\
&\le \exp\Bigg(\frac{\alpha}{1-\alpha}+\frac{\alpha x}{(1-x)^2}+\frac{\beta}{1-\beta}+\frac{\beta x}{(1-x)^2}\Bigg).\label{eq:some-bound}
\end{align}
\section{Outline of the proof}
We know from \eqref{eq:cauchy-var} and \eqref{eq:trans-main} that
\begin{align*}
g(n)&=\sum_{1\le k\le N} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1}} e^{-\frac{2\pi i nh}{k}} \int_{\xi_{h,k}} G\big(e^{2\pi i \tau}\big)e^{-2\pi i n \phi} e^{2 \pi n \varrho}\ d\phi\\
&=i^{\sum_{j=1}^{J}\delta_j}\sum_{1\le k\le N} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1}} e^{-\frac{2\pi i nh}{k}}(-1)^{\sum_{j=1}^{J}\delta_j \lambda_{m_j,r_j}(h,k)}\omega_{h,k}^2\;\textup{\foreignlanguage{russian}{D}}_{h,k}\\
&\times \int_{\xi_{h,k}} \exp\Bigg(\frac{\pi}{12k}(\Omega z+\Delta(h,k)z^{-1})\Bigg)\\
&\quad\times \prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(r_j\tau\gamma^*_{(m_j,h,k)}(m_j\tau)+\lambda_{m_j,r_j}(h,k)\gamma_{(m_j,h,k)}(m_j\tau);\gamma_{(m_j,h,k)}(m_j\tau)\big)\\
&\quad\times e^{-2\pi i n \phi} e^{2 \pi n \varrho}\ d\phi.
\end{align*}
Let us fix a Farey fraction $h/k$. We first find integers $1\le \ell\le L$ and $0\le \varkappa< \ell$ such that $k\equiv \ell \pmod{L}$ and $h\equiv \varkappa \pmod{\ell}$. For convenience, we write $\rho(h,k):=(\varkappa,\ell)$. It is not hard to observe that for all $j=1,2,\ldots,J$,
$$\gcd(m_j,k)=\gcd(m_j,\ell)\quad\text{and}\quad \lambda^*_{m_j,r_j}(h,k)=\lambda^*_{m_j,r_j}(\varkappa,\ell).$$
It turns out that $\Delta(h,k)=\Delta(\varkappa,\ell)$. We now split $g(n)$ as follows.
\begin{align*}
g(n)&=i^{\sum_{j=1}^{J}\delta_j}\sum_{1\le \ell\le L}\sum_{0\le \varkappa< \ell}\sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} e^{-\frac{2\pi i nh}{k}}\\
&\times(-1)^{\sum_{j=1}^{J}\delta_j \lambda_{m_j,r_j}(h,k)}\omega_{h,k}^2\;\textup{\foreignlanguage{russian}{D}}_{h,k}\\
&\times \int_{\xi_{h,k}} \exp\Bigg(\frac{\pi}{12k}(\Omega z+\Delta(\varkappa,\ell)z^{-1})\Bigg)\\
&\quad\times \prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(r_j\tau\gamma^*_{(m_j,h,k)}(m_j\tau)+\lambda_{m_j,r_j}(h,k)\gamma_{(m_j,h,k)}(m_j\tau);\gamma_{(m_j,h,k)}(m_j\tau)\big)\\
&\quad\times e^{-2\pi i n \phi} e^{2 \pi n \varrho}\ d\phi\\
&=:i^{\sum_{j=1}^{J}\delta_j}\sum_{1\le \ell\le L}\sum_{0\le \varkappa< \ell} S_{\varkappa,\ell}.
\end{align*}
The minor arcs are those with respect to $h/k$ with $\rho(h,k)\in\mathcal{L}_{\le 0}$. We have the following bound.
\begin{theorem}\label{th:minor}
Let $(\varkappa,\ell)\in \mathcal{L}_{\le 0}$. For positive integers $n>-\Omega/24$, we have
$$S_{\varkappa,\ell}\ll_{\mathbf{m},\mathbf{r},\boldsymbol{\updelta}} \exp\Bigg(\frac{2\pi}{N^2} \bigg(n+\frac{\Omega}{24}\bigg)\Bigg) \xrightarrow{N\to\infty} 0.$$
In particular, if we take $N=\myfloor{\sqrt{2\pi \left(n+\frac{\Omega}{24}\right)}}$, then $S_{\varkappa,\ell}\ll_{\mathbf{m},\mathbf{r},\boldsymbol{\updelta}} 1$.
\end{theorem}
The arcs with respect to $h/k$ with $\rho(h,k)\in\mathcal{L}_{> 0}$ give us the main contribution.
\begin{theorem}\label{th:major}
Let $(\varkappa,\ell)\in \mathcal{L}_{> 0}$. If the inequality
\begin{align}\label{eq:condition-1}
\min_{1\le j\le J}\left(\Upsilon\big(\lambda^*_{m_j,r_j}(\varkappa,\ell)\big)\frac{\gcd^2(m_j,\ell)}{m_j}\right)\ge \frac{\Delta(\varkappa,\ell)}{24}
\end{align}
holds, then for positive integers $n>-\Omega/24$, we have
\begin{align*}
S_{\varkappa,\ell}&=E_{\varkappa,\ell}+\sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} e^{-\frac{2\pi i nh}{k}}(-1)^{\sum_{j=1}^{J}\delta_j \lambda_{m_j,r_j}(h,k)}\omega_{h,k}^2\;\textup{\foreignlanguage{russian}{D}}_{h,k}\Pi_{h,k}\\
&\quad\quad\quad\quad\times \frac{2\pi}{k} \left(\frac{24n+\Omega}{\Delta(\varkappa,\ell)}\right)^{-\frac{1}{2}} I_{-1}\left(\frac{\pi }{6k}\sqrt{\Delta(\varkappa,\ell)(24n+\Omega)}\right),
\end{align*}
where
\begin{align*}
E_{\varkappa,\ell}\ll_{\mathbf{m},\mathbf{r},\boldsymbol{\updelta}} e^{\frac{2\pi}{N^2} (n+\frac{\Omega}{24})}+\frac{N^2 e^{\frac{2\pi}{N^2} \left(n+\frac{\Omega}{24}\right)}}{n+\frac{\Omega}{24}}\xrightarrow{N\to\infty} 0.
\end{align*}
In particular, if we take $N=\myfloor{\sqrt{2\pi \left(n+\frac{\Omega}{24}\right)}}$, then $E_{\varkappa,\ell}\ll_{\mathbf{m},\mathbf{r},\boldsymbol{\updelta}} 1$.
\end{theorem}
Theorems \ref{th:minor} and \ref{th:major} immediately imply the main result. Before presenting proofs of the two results respectively in Sections \ref{sec:minor} and \ref{sec:major}, we make the following preparations.
\medskip
For fixed $\varkappa$ and $\ell$ with $1\le \ell\le L$ and $0\le \varkappa< \ell$, one may split the indexes $\{1,2,\ldots,J\}$ into two disjoint parts:
$$\mathcal{J}^*_{\varkappa,\ell}=\{j_1^*,\ldots,j_\alpha^*\} \quad\text{and}\quad \mathcal{J}^{**}_{\varkappa,\ell}=\{j_1^{**},\ldots,j_\beta^{**}\},$$
so that for $j^*\in \mathcal{J}^*_{\varkappa,\ell}$ we have $\lambda^*_{m_{j^*},r_{j^*}}(\varkappa,\ell)=0$ and for $j^{**}\in \mathcal{J}^{**}_{\varkappa,\ell}$ we have $\lambda^*_{m_{j^{**}},r_{j^{**}}}(\varkappa,\ell)\ne 0$.
\begin{proposition}\label{lem:pre}
Let $j^* \in \mathcal{J}^*_{\varkappa,\ell}$. For any Farey fraction $h/k$ such that $k\equiv \ell \pmod{L}$ and $h\equiv \varkappa \pmod{\ell}$, we have that
\begin{align}
&r_{j^*}\tau\gamma^*_{(m_{j^*},h,k)}(m_{j^*}\tau)+\lambda_{m_{j^*},r_{j^*}}(h,k)\gamma_{(m_{j^*},h,k)}(m_{j^*}\tau)\notag\\
&\quad=\frac{r_{j^*}\gcd(m_{j^*},k)+r_{j^*}\hbar_{m_{j^*}}(h,k)m_{j^*}h}{m_{j^*}k}
\end{align}
is a real noninteger. Further,
\begin{align}\label{eq:ineq-important}
\Big|1-e^{\frac{2\pi i}{m_{j^*}}}\Big|\le \Bigg|1-e^{2\pi i \big(r_{j^*}\tau\gamma^*_{(m_{j^*},h,k)}(m_{j^*}\tau)+\lambda_{m_{j^*},r_{j^*}}(h,k)\gamma_{(m_{j^*},h,k)}(m_{j^*}\tau)\big)}\Bigg|\le 2.
\end{align}
\end{proposition}
\begin{proof}
In this proof, we write for short $m=m_{j^*}$ and $r=r_{j^*}$. We also write $d=\gcd(m,k)$, $m=dm'$ and $k=dk'$. Since $j^* \in \mathcal{J}^*_{\varkappa,\ell}$, we have $\lambda^*_{m,r}(h,k)=\lambda^*_{m,r}(\varkappa,\ell)=0$. Hence $d$ divides $rh$ and $\lambda_{m,r}(h,k)=rh/d$. We know from \eqref{eq:gamma-mix} that
\begin{align*}
&r\tau\gamma^*_{(m,h,k)}(m\tau)+\lambda_{m,r}(h,k)\gamma_{(m,h,k)}(m\tau)\\
&\quad=\frac{r d}{mk}+\lambda_{m,r}(h,k)\frac{\hbar_m(h,k) d}{k}+\lambda^*_{m,r}(h,k)\frac{d^2}{mkz}i\\
&\quad=\frac{r d}{mk}+\lambda_{m,r}(h,k)\frac{\hbar_m(h,k) d}{k}\\
&\quad=\frac{r d}{mk}+\frac{rh}{d}\frac{\hbar_m(h,k) d}{k}\\
&\quad=\frac{r(1+\hbar_m(h,k)m'h)}{m'k}\\
&\quad=\frac{b_{m'}r}{m}=\frac{b_{m'}}{m'}\frac{r}{d},
\end{align*}
where as in Section \ref{sec:Farey}, we have put $b_{m'}=(\hbar_m(h,k) m' h+1)/k'$. Hence it is a real number.
Notice that $d=\gcd(m,k)$. Since $\gcd(h,k)=1$, $d\mid rh$ implies that $d\mid r$. Further, $b_{m'}=(\hbar_m(h,k) m' h+1)/k'$ implies that $\gcd(m',b_{m'})=1$. Hence, if $\frac{b_{m'}}{m'}\frac{r}{d}$ is an integer, then $m'\mid \frac{r}{d}$ so that $m=dm'\mid r$. This violates the assumption that $1\le r\le m-1$. Hence $r\tau\gamma^*_{(m,h,k)}(m\tau)+\lambda_{m,r}(h,k)\gamma_{(m,h,k)}(m\tau)$ is not an integer and \eqref{eq:ineq-important} follows immediately.
\end{proof}
\begin{remark}\label{rmk:Pi-value}
Recall that $\hbar_m(h,k)$ is defined to be an integer such that
$$\hbar_m(h,k) \frac{m h}{\gcd(m,k)}\equiv -1 \pmod{\frac{k}{\gcd(m,k)}}.$$
Let $n$ be an integer. It turns out that
\begin{align*}
&\exp\Bigg(2\pi i\frac{r_{j^*}\gcd(m_{j^*},k)+r_{j^*}\Big(\hbar_{m_{j^*}}(h,k)+n\frac{k}{\gcd(m_{j^*},k)}\Big)m_{j^*}h}{m_{j^*}k}\Bigg)\\
&\quad=\exp\Bigg(2\pi i\frac{r_{j^*}\gcd(m_{j^*},k)+r_{j^*}\hbar_{m_{j^*}}(h,k)m_{j^*}h}{m_{j^*}k}+2n\pi i\frac{r_{j^*}h}{\gcd(m_{j^*},k)}\Bigg)\\
&\quad=\exp\Bigg(2\pi i\frac{r_{j^*}\gcd(m_{j^*},k)+r_{j^*}\hbar_{m_{j^*}}(h,k)m_{j^*}h}{m_{j^*}k}\Bigg),
\end{align*}
since from the above proof we have $\gcd(m_{j^*},k)\mid r_{j^*}$. Hence the choice of $\hbar_{m_{j^*}}(h,k)$ does not affect the value of
\begin{align*}
&\exp\Bigg(2\pi i\bigg(r_{j^*}\tau\gamma^*_{(m_{j^*},h,k)}(m_{j^*}\tau)+\lambda_{m_{j^*},r_{j^*}}(h,k)\gamma_{(m_{j^*},h,k)}(m_{j^*}\tau)\bigg)\Bigg)\\
&\quad=\exp\Bigg(2\pi i\frac{r_{j^*}\gcd(m_{j^*},k)+r_{j^*}\hbar_{m_{j^*}}(h,k)m_{j^*}h}{m_{j^*}k}\Bigg).
\end{align*}
\end{remark}
\section{Minor arcs}\label{sec:minor}
Let $(\varkappa,\ell)\in\mathcal{L}_{\le 0}$, namely, $\Delta(\varkappa,\ell)\le 0$. We write $\mathcal{J}^*=\mathcal{J}^*_{\varkappa,\ell}$ and $\mathcal{J}^{**}=\mathcal{J}^{**}_{\varkappa,\ell}$. Notice that
\begin{align*}
|S_{\varkappa,\ell}|&\le \sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} \int_{\xi_{h,k}} \exp\Bigg(\frac{\pi}{12k}(\Omega \Re(z)+\Delta(\varkappa,\ell)\Re(z^{-1}))\Bigg)\\
&\times \Bigg|\prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(r_j\tau\gamma^*_{(m_j,h,k)}(m_j\tau)+\lambda_{m_j,r_j}(h,k)\gamma_{(m_j,h,k)}(m_j\tau);\gamma_{(m_j,h,k)}(m_j\tau)\big)\Bigg|\\
&\times e^{2 \pi n \varrho}\ d\phi.
\end{align*}
We now consider the Farey arcs with respect to $h/k$ with $k\equiv \ell \pmod{L}$ and $h\equiv \varkappa \pmod{\ell}$. Since $\Delta(\varkappa,\ell)\le 0$, it follows from \eqref{eq:Re-z-bound} and \eqref{eq:Re-1-z-bound} that
\begin{align*}
\exp\Bigg(\frac{\pi}{12k}(\Omega \Re(z)+\Delta(\varkappa,\ell)\Re(z^{-1}))\Bigg)&\le \exp\Bigg(\frac{\pi}{12k}\Big(\Omega \frac{k}{N^2}+\Delta(\varkappa,\ell)\frac{k}{2}\Big)\Bigg)\\
&=\exp\Bigg(\frac{\pi \varrho\; \Omega}{12}\Bigg)\exp\Bigg(\frac{\pi \Delta(\varkappa,\ell)}{24}\Bigg).
\end{align*}
For convenience, now we write $\lambda_j=\lambda_{m_j,r_j}(h,k)$ and $\lambda^*_j=\lambda^*_{m_j,r_j}(h,k)$. We also write for short $\tilde{\varsigma}_j=r_j\tau\gamma^*_{(m_j,h,k)}(m_j\tau)+\lambda_{m_j,r_j}(h,k)\gamma_{(m_j,h,k)}(m_j\tau)$ and $\tilde{\tau}_j=\gamma_{(m_j,h,k)}(m_j\tau)$. We know from \eqref{eq:gamma} and \eqref{eq:gamma-mix} that
$$\Im(\tilde{\tau}_j)=\frac{\gcd^2(m_j,k)}{m_jk}\Re(z^{-1})=\frac{\gcd^2(m_j,\ell)}{m_jk}\Re(z^{-1})$$
and
$$\Im(\tilde{\varsigma}_j)=\lambda^*_j\frac{\gcd^2(m_j,k)}{m_jk}\Re(z^{-1})=\lambda^*_j\frac{\gcd^2(m_j,\ell)}{m_jk}\Re(z^{-1}).$$
Notice that
$$0\le \Im(\tilde{\varsigma}_j)<\Im(\tilde{\tau}_j).$$
We write
\begin{align*}
\prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}(\tilde{\varsigma}_j;\tilde{\tau}_j)&=\prod_{j^*\in\mathcal{J}^*}(1-e^{2\pi i \tilde{\varsigma}_{j^*}})^{\delta_{j^*}}\\
&\quad\times \prod_{j^*\in\mathcal{J}^*}(e^{2\pi i (\tilde{\tau}_{j^*}+\tilde{\varsigma}_{j^*})},e^{2\pi i (\tilde{\tau}_{j^*}-\tilde{\varsigma}_{j^*})};e^{2\pi i \tilde{\tau}_{j^*}})_\infty^{\delta_{j^*}}\\
&\quad\times \prod_{j^{**}\in\mathcal{J}^{**}}(e^{2\pi i \tilde{\varsigma}_{j^{**}}},e^{2\pi i (\tilde{\tau}_{j^{**}}-\tilde{\varsigma}_{j^{**}})};e^{2\pi i \tilde{\tau}_{j^{**}}})_\infty^{\delta_{j^{**}}}.
\end{align*}
First, it follows from Proposition \ref{lem:pre} that
$$\prod_{j^*\in\mathcal{J}^*}(1-e^{2\pi i \tilde{\varsigma}_{j^*}})^{\delta_{j^*}}\ll 1.$$
Further, as we have seen in Section \ref{sec:ptn-res}, for $j^*\in\mathcal{J}^*$ (hence $\lambda^*_{j^*}=0$),
\begin{align*}
&\Big|(e^{2\pi i (\tilde{\tau}_{j^*}+\tilde{\varsigma}_{j^*})},e^{2\pi i (\tilde{\tau}_{j^*}-\tilde{\varsigma}_{j^*})};e^{2\pi i \tilde{\tau}_{j^*}})_\infty^{\delta_{j^*}}\Big|\\
& \le \sum_{n\ge 0}\sum_{s\ge 0}\sum_{t\ge 0}p^*_{|\delta_{j^*}|}(s,t;n)|e^{2\pi i (\tilde{\tau}_{j^*}+\tilde{\varsigma}_{j^*})}|^s |e^{2\pi i (\tilde{\tau}_{j^*}-\tilde{\varsigma}_{j^*})}|^t |e^{2\pi i \tilde{\tau}_{j^*}}|^n\\
&=\sum_{n\ge 0}\sum_{s\ge 0}\sum_{t\ge 0}p^*_{|\delta_{j^*}|}(s,t;n) e^{-2\pi \Im(\tilde{\tau}_{j^*}+\tilde{\varsigma}_{j^*})s}e^{-2\pi \Im(\tilde{\tau}_{j^*}-\tilde{\varsigma}_{j^*})t}e^{-2\pi \Im(\tilde{\tau}_{j^*})n}\\
&=\sum_{n\ge 0}\sum_{s\ge 0}\sum_{t\ge 0}p^*_{|\delta_{j^*}|}(s,t;n) \exp\bigg(-2\pi \frac{\gcd^2(m_{j^*},\ell)}{m_{j^*}k}\Re(z^{-1})s\bigg)\\
&\quad\times\exp\bigg(-2\pi \frac{\gcd^2(m_{j^*},\ell)}{m_{j^*}k}\Re(z^{-1})t\bigg)\exp\bigg(-2\pi \frac{\gcd^2(m_{j^*},\ell)}{m_{j^*}k}\Re(z^{-1})n\bigg)\\
&\le \sum_{n\ge 0}\sum_{s\ge 0}\sum_{t\ge 0}p^*_{|\delta_{j^*}|}(s,t;n) \exp\bigg(-\pi \frac{\gcd^2(m_{j^*},\ell)}{m_{j^*}}s\bigg)\\
&\quad\times\exp\bigg(-\pi \frac{\gcd^2(m_{j^*},\ell)}{m_{j^*}}t\bigg)\exp\bigg(-\pi \frac{\gcd^2(m_{j^*},\ell)}{m_{j^*}}n\bigg),
\end{align*}
where we use $\Re(z^{-1})\ge k/2$. It follows from \eqref{eq:some-bound} that
$$(e^{2\pi i (\tilde{\tau}_{j^*}+\tilde{\varsigma}_{j^*})},e^{2\pi i (\tilde{\tau}_{j^*}-\tilde{\varsigma}_{j^*})};e^{2\pi i \tilde{\tau}_{j^*}})_\infty^{\delta_{j^*}}\ll 1.$$
Likewise, for $j^{**}\in\mathcal{J}^{**}$,
\begin{align*}
&\Big|(e^{2\pi i \tilde{\varsigma}_{j^{**}}},e^{2\pi i (\tilde{\tau}_{j^{**}}-\tilde{\varsigma}_{j^{**}})};e^{2\pi i \tilde{\tau}_{j^{**}}})_\infty^{\delta_{j^{**}}}\Big|\\
&\quad\le \sum_{n\ge 0}\sum_{s\ge 0}\sum_{t\ge 0}p^*_{|\delta_{j^{**}}|}(s,t;n) \exp\bigg(-\pi \lambda^*_{j^{**}}\frac{\gcd^2(m_{j^{**}},\ell)}{m_{j^{**}}}s\bigg)\\
&\quad\quad\times\exp\bigg(-\pi (1-\lambda^*_{j^{**}})\frac{\gcd^2(m_{j^{**}},\ell)}{m_{j^{**}}}t\bigg)\exp\bigg(-\pi \frac{\gcd^2(m_{j^{**}},\ell)}{m_{j^{**}}}n\bigg)\\
&\quad\ll 1.
\end{align*}
Hence,
\begin{align*}
S_{\varkappa,\ell}&\ll \sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} \int_{\xi_{h,k}} e^{\frac{\pi \varrho\; \Omega}{12}} e^{2 \pi n \varrho}\ d\phi\\
&\ll \sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}}e^{2\pi \varrho(n+\frac{\Omega}{24})} \frac{1}{kN}\\
&\ll e^{2\pi \varrho(n+\frac{\Omega}{24})}=e^{\frac{2\pi}{N^2} (n+\frac{\Omega}{24})} \xrightarrow{N\to\infty} 0.
\end{align*}
\section{Major arcs}\label{sec:major}
Let $(\varkappa,\ell)\in\mathcal{L}_{> 0}$, namely, $\Delta(\varkappa,\ell)> 0$. Again, we write $\mathcal{J}^*=\mathcal{J}^*_{\varkappa,\ell}$ and $\mathcal{J}^{**}=\mathcal{J}^{**}_{\varkappa,\ell}$. Let us consider the Farey arcs with respect to $h/k$ with $k\equiv \ell \pmod{L}$ and $h\equiv \varkappa \pmod{\ell}$. For convenience, we write $\tilde{\varsigma}_j(h,k)=r_j\tau\gamma^*_{(m_j,h,k)}(m_j\tau)+\lambda_{m_j,r_j}(h,k)\gamma_{(m_j,h,k)}(m_j\tau)$ and $\tilde{\tau}_j(h,k)=\gamma_{(m_j,h,k)}(m_j\tau)$.
\smallskip
Recall that
\begin{align*}
S_{\varkappa,\ell}&=\sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} e^{-\frac{2\pi i nh}{k}}(-1)^{\sum_{j=1}^{J}\delta_j \lambda_{m_j,r_j}(h,k)}\omega_{h,k}^2\;\textup{\foreignlanguage{russian}{D}}_{h,k}\\
&\times \int_{\xi_{h,k}} \exp\Bigg(\frac{\pi}{12k}(\Omega z+\Delta(\varkappa,\ell)z^{-1})\Bigg) \prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(\tilde{\varsigma}_j(h,k);\tilde{\tau}_j(h,k)\big)\\
&\times e^{-2\pi i n \phi} e^{2 \pi n \varrho}\ d\phi.
\end{align*}
We split $S_{\varkappa,\ell}$ into two parts $\Sigma_1$ and $\Sigma_2$ where
\begin{align*}
\Sigma_1&:=\sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} e^{-\frac{2\pi i nh}{k}}(-1)^{\sum_{j=1}^{J}\delta_j \lambda_{m_j,r_j}(h,k)}\omega_{h,k}^2\;\textup{\foreignlanguage{russian}{D}}_{h,k}\\
&\;\times \int_{\xi_{h,k}} \exp\Bigg(\frac{\pi}{12k}(\Omega z+\Delta(\varkappa,\ell)z^{-1})\Bigg) \Pi_{h,k} e^{-2\pi i n \phi} e^{2 \pi n \varrho}\ d\phi
\end{align*}
and
\begin{align*}
\Sigma_2&:=\sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} e^{-\frac{2\pi i nh}{k}}(-1)^{\sum_{j=1}^{J}\delta_j \lambda_{m_j,r_j}(h,k)}\omega_{h,k}^2\;\textup{\foreignlanguage{russian}{D}}_{h,k}\\
&\;\times \int_{\xi_{h,k}} \exp\Bigg(\frac{\pi}{12k}(\Omega z+\Delta(\varkappa,\ell)z^{-1})\Bigg) \Bigg(\prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(\tilde{\varsigma}_j(h,k);\tilde{\tau}_j(h,k)\big)-\Pi_{h,k}\Bigg)\\
&\;\times e^{-2\pi i n \phi} e^{2 \pi n \varrho}\ d\phi.
\end{align*}
We first show that $\Sigma_2$ is negligible. Notice that by \eqref{eq:Re-z-bound}
\begin{align*}
|\Sigma_2|&\le\sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} e^{2\pi \varrho(n+\frac{\Omega}{24})}|\Pi_{h,k}|\\
&\times \int_{\xi_{h,k}} \exp\Bigg(\frac{\pi\Delta(\varkappa,\ell)}{12k}\Re(z^{-1})\Bigg) \Bigg|\frac{1}{\Pi_{h,k}}\prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(\tilde{\varsigma}_j(h,k);\tilde{\tau}_j(h,k)\big)-1\Bigg|\ d\phi.
\end{align*}
Let us fix $h$ and $k$ and write $\tilde{\varsigma}_j=\tilde{\varsigma}_j(h,k)$ and $\tilde{\tau}_j=\tilde{\tau}_j(h,k)$. We also write $\lambda^*_j=\lambda^*_{m_j,r_j}(h,k)$. Recalling the definition of $\Pi_{h,k}$ and Proposition \ref{lem:pre}, we have
\begin{align*}
\frac{1}{\Pi_{h,k}}\prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(\tilde{\varsigma}_j;\tilde{\tau}_j\big)-1&=\prod_{j^*\in\mathcal{J}^*}(e^{2\pi i (\tilde{\tau}_{j^*}+\tilde{\varsigma}_{j^*})},e^{2\pi i (\tilde{\tau}_{j^*}-\tilde{\varsigma}_{j^*})};e^{2\pi i \tilde{\tau}_{j^*}})_\infty^{\delta_{j^*}}\\
&\quad\times \prod_{j^{**}\in\mathcal{J}^{**}}(e^{2\pi i \tilde{\varsigma}_{j^{**}}},e^{2\pi i (\tilde{\tau}_{j^{**}}-\tilde{\varsigma}_{j^{**}})};e^{2\pi i \tilde{\tau}_{j^{**}}})_\infty^{\delta_{j^{**}}}-1.
\end{align*}
Let us write for short
$$\tilde{\varsigma}_j^{\textrm{New}}=\begin{cases}
\tilde{\tau}_{j}+\tilde{\varsigma}_{j} & \text{if $j\in\mathcal{J}^*$},\\
\tilde{\varsigma}_{j} & \text{if $j\in\mathcal{J}^{**}$}.
\end{cases}$$
It follows again from \eqref{eq:gamma} and \eqref{eq:gamma-mix} that
$$\Im(\tilde{\tau}_j)=\frac{\gcd^2(m_j,\ell)}{m_jk}\Re(z^{-1}),$$
$$\Im(\tilde{\varsigma}_j)=\lambda^*_j\frac{\gcd^2(m_j,\ell)}{m_jk}\Re(z^{-1})$$
and
$$\Im(\tilde{\varsigma}_j^{\textrm{New}})=\Phi(\lambda^*_j)\frac{\gcd^2(m_j,\ell)}{m_jk}\Re(z^{-1}),$$
where for real $0\le x<1$,
$$\Phi(x):=\begin{cases}
1 & \text{if $x=0$},\\
x & \text{otherwise}.
\end{cases}$$
We have
\begin{align*}
&\Bigg|\frac{1}{\Pi_{h,k}}\prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(\tilde{\varsigma}_j;\tilde{\tau}_j\big)-1\Bigg|\\
&\quad=\Bigg|\prod_{j=1}^J(e^{2\pi i \tilde{\varsigma}_j^{\textrm{New}}},e^{2\pi i (\tilde{\tau}_{j}-\tilde{\varsigma}_{j})};e^{2\pi i \tilde{\tau}_{j}})_\infty^{\delta_{j}}-1\Bigg|\\
&\quad\le \sum_{\mathbf{n}:=(n_1,\ldots,n_J)\in\mathbb{Z}_{\ge 0}^J}\sum_{\mathbf{s}:=(s_1,\ldots,s_J)\in\mathbb{Z}_{\ge 0}^J}\sum_{\mathbf{t}:=(t_1,\ldots,t_J)\in\mathbb{Z}_{\ge 0}^J}\\
&\quad\quad\prod_{j=1}^J p^*_{|\delta_{j}|}(s_j,t_j;n_j)|e^{2\pi i \tilde{\varsigma}_j^{\textrm{New}}}|^{s_j} |e^{2\pi i (\tilde{\tau}_{j}-\tilde{\varsigma}_{j})}|^{t_j} |e^{2\pi i \tilde{\tau}_{j}}|^{n_j}-1\\
&\quad=\underset{\mathbf{n}\times\mathbf{s}\times\mathbf{t}\in (\mathbb{Z}_{\ge 0}^J)^3\backslash (0,\ldots,0)^3}{\sum\sum\sum}\prod_{j=1}^J p^*_{|\delta_{j}|}(s_j,t_j;n_j)|e^{2\pi i \tilde{\varsigma}_j^{\textrm{New}}}|^{s_j} |e^{2\pi i (\tilde{\tau}_{j}-\tilde{\varsigma}_{j})}|^{t_j} |e^{2\pi i \tilde{\tau}_{j}}|^{n_j}\\
&\quad=\underset{\mathbf{n}\times\mathbf{s}\times\mathbf{t}\in(\mathbb{Z}_{\ge 0}^J)^3\backslash (0,\ldots,0)^3}{\sum\sum\sum}\prod_{j=1}^J p^*_{|\delta_{j}|}(s_j,t_j;n_j) e^{-2\pi \Im(\tilde{\varsigma}_j^{\textrm{New}})s_j} e^{-2\pi \Im(\tilde{\tau}_{j}-\tilde{\varsigma}_{j})t_j} e^{-2\pi \Im(\tilde{\tau}_{j})n_j}\\
&\quad=\underset{\mathbf{n}\times\mathbf{s}\times\mathbf{t}\in(\mathbb{Z}_{\ge 0}^J)^3\backslash (0,\ldots,0)^3}{\sum\sum\sum}\Bigg(\prod_{j=1}^J p^*_{|\delta_{j}|}(s_j,t_j;n_j)\Bigg) \\
&\quad\quad\times \exp\Bigg(-2\pi\frac{\Re(z^{-1})}{k}\sum_{j=1}^J\frac{\gcd^2(m_j,\ell)}{m_j}\big(\Phi(\lambda^*_j)s_j+(1-\lambda^*_j)t_j+n_j\big)\Bigg).
\end{align*}
Hence,
\begin{align*}
&\exp\Bigg(\frac{\pi\Delta(\varkappa,\ell)}{12k}\Re(z^{-1})\Bigg)\Bigg|\frac{1}{\Pi_{h,k}}\prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(\tilde{\varsigma}_j;\tilde{\tau}_j\big)-1\Bigg|\\
&\le \underset{\mathbf{n}\times\mathbf{s}\times\mathbf{t}\in(\mathbb{Z}_{\ge 0}^J)^3\backslash (0,\ldots,0)^3}{\sum\sum\sum}\Bigg(\prod_{j=1}^J p^*_{|\delta_{j}|}(s_j,t_j;n_j)\Bigg) \\
&\times \exp\Bigg(\!\!-2\pi\frac{\Re(z^{-1})}{k}\bigg(\!\!-\frac{\Delta(\varkappa,\ell)}{24}+\sum_{j=1}^J\frac{\gcd^2(m_j,\ell)}{m_j}\big(\Phi(\lambda^*_j)s_j+(1-\lambda^*_j)t_j+n_j\big)\bigg)\!\Bigg).
\end{align*}
Since at least one coordinate of $\mathbf{n}\times\mathbf{s}\times\mathbf{t}$ is nonzero, under the condition \eqref{eq:condition-1}, we know that
\begin{align*}
&-\frac{\Delta(\varkappa,\ell)}{24}+\sum_{j=1}^J\frac{\gcd^2(m_j,\ell)}{m_j}\big(\Phi(\lambda^*_j)s_j+(1-\lambda^*_j)t_j+n_j\big)\\
&\qquad\ge -\frac{\Delta(\varkappa,\ell)}{24}+\min_{1\le j\le J}\left(\Upsilon(\lambda^*_j)\frac{\gcd^2(m_j,\ell)}{m_j}\right)\ge 0
\end{align*}
for all $\mathbf{n}\times\mathbf{s}\times\mathbf{t}\in(\mathbb{Z}_{\ge 0}^J)^3\backslash (0,\ldots,0)^3$. Recalling that $\Re(z^{-1})\ge k/2$, it follows that
$$\exp\Bigg(\frac{\pi\Delta(\varkappa,\ell)}{12k}\Re(z^{-1})\Bigg)\Bigg|\frac{1}{\Pi_{h,k}}\prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(\tilde{\varsigma}_j;\tilde{\tau}_j\big)-1\Bigg|$$
is maximized when $\Re(z^{-1})=k/2$. Namely,
\begin{align*}
&\exp\Bigg(\frac{\pi\Delta(\varkappa,\ell)}{12k}\Re(z^{-1})\Bigg)\Bigg|\frac{1}{\Pi_{h,k}}\prod_{j=1}^J \textup{\foreignlanguage{russian}{Zh}}^{\delta_j}\big(\tilde{\varsigma}_j;\tilde{\tau}_j\big)-1\Bigg|\\
&\quad\le \exp\Bigg(\frac{\pi\Delta(\varkappa,\ell)}{24}\Bigg)\underset{\mathbf{n}\times\mathbf{s}\times\mathbf{t}\in(\mathbb{Z}_{\ge 0}^J)^3\backslash (0,\ldots,0)^3}{\sum\sum\sum}\Bigg(\prod_{j=1}^J p^*_{|\delta_{j}|}(s_j,t_j;n_j)\Bigg) \\
&\quad\quad\times \exp\Bigg(-\pi\sum_{j=1}^J\frac{\gcd^2(m_j,\ell)}{m_j}\big(\Phi(\lambda^*_j)s_j+(1-\lambda^*_j)t_j+n_j\big)\Bigg)\\
&\quad\ll 1.
\end{align*}
Together with the fact $\prod_{h,k}\ll 1$ which follows from \eqref{eq:ineq-important}, we conclude that
\begin{align*}
\Sigma_2&\ll\sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} e^{2\pi \varrho(n+\frac{\Omega}{24})} \int_{\xi_{h,k}} 1\ d\phi\\
&\ll \sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}}e^{2\pi \varrho(n+\frac{\Omega}{24})} \frac{1}{kN}\\
&\ll e^{2\pi \varrho(n+\frac{\Omega}{24})}=e^{\frac{2\pi}{N^2} (n+\frac{\Omega}{24})} \xrightarrow{N\to\infty} 0.
\end{align*}
Finally, we estimate the main contribution $\Sigma_1$. To do so, we need the following evaluation of an integral, which is a special case of Lemma 2.4 in \cite{Che2019}. For the sake of completeness, we sketch a brief proof.
\begin{lemma}\label{le:key-int}
Let $a\in \mathbb{R}_{>0}$ and $b\in\mathbb{R}$. Let $\gcd(h,k)=1$. Define
\begin{equation}
I:=\int_{\xi_{h,k}}e^{\frac{\pi}{12k}\left(\frac{a}{z}+bz\right)}e^{-2\pi i n\phi}e^{2\pi n \varrho}\ d\phi.
\end{equation}
Then for positive integers $n$ with $n> -b/24$, we have
\begin{equation}
I=\frac{2\pi}{k} \left(\frac{24n+b}{a}\right)^{-\frac{1}{2}} I_{-1}\left(\frac{\pi }{6k}\sqrt{a(24n+b)}\right)+E(I),
\end{equation}
where
\begin{equation}\label{eq:integral-error}
E(I)\ll_a \frac{e^{2\pi\varrho \left(n+\frac{b}{24}\right)}}{n+\frac{b}{24}}.
\end{equation}
\end{lemma}
\begin{proof}
Putting $w=z/k=\varrho-i\phi$ and reversing the integral order, one has
$$I=\frac{1}{2\pi i}\int_{\varrho-i\theta''_{h,k}}^{\varrho+i\theta'_{h,k}} 2\pi e^{\frac{\pi a}{12k^2 w}} e^{2\pi w \left(n+\frac{b}{24}\right)}\ dw.$$
We now split the integral into three parts:
\begin{align*}
I&=\frac{1}{2\pi i}\left(\int_\Gamma-\int_{-\infty-i\theta''_{h,k}}^{\varrho-i\theta''_{h,k}}+\int_{-\infty+i\theta'_{h,k}}^{\varrho+i\theta'_{h,k}}\right) 2\pi e^{\frac{\pi a}{12k^2 w}} e^{2\pi w \left(n+\frac{b}{24}\right)}\ dw\\
&=:J_1-J_2+J_3,
\end{align*}
where
\begin{align*}
\Gamma:=(-\infty-i\theta''_{h,k}) \to (\varrho-i\theta''_{h,k}) \to (\varrho+i\theta'_{h,k}) \to (-\infty+i\theta'_{h,k})
\end{align*}
is a Hankel contour.
\smallskip
The dominant contribution to $I$ comes from $J_1$. We make the following change of variables $t=wk\sqrt{(24n+b)/a}$. Then
$$J_1=\frac{2\pi}{k}\left(\frac{24n+b}{a}\right)^{-\frac{1}{2}} \frac{1}{2\pi i}\int_{\tilde{\Gamma}} e^{\frac{\pi }{12k}\sqrt{a(24n+b)}\left(t+\frac{1}{t}\right)} dt,$$
in which the new contour $\tilde{\Gamma}$ is still a Hankel contour. Recalling the contour integral representation of $I_s(x)$:
$$I_s(x)=\frac{1}{2\pi i}\int_{\Gamma} t^{-s-1}e^{\frac{x}{2}\left(t+\frac{1}{t}\right)}\ dt\quad\text{($\Gamma$ is a Hankel contour)},$$
we conclude that
$$J_1=\frac{2\pi}{k} \left(\frac{24n+b}{a}\right)^{-\frac{1}{2}} I_{-1}\left(\frac{\pi }{6k}\sqrt{a(24n+b)}\right).$$
\smallskip
We next bound the error term $E(I)$, coming from $J_2$ and $J_3$. Let us put $w=x+i\theta$ with $-\infty\le x\le \varrho$ and $\theta\in\{\theta'_{h,k},-\theta''_{h,k}\}$. We know that
$$\left|e^{2\pi w \left(n+\frac{b}{24}\right)}\right|= e^{2\pi x \left(n+\frac{b}{24}\right)}$$
and
\begin{align*}
\left|e^{\frac{\pi a}{12k^2 w}}\right|&=e^{\frac{\pi a}{12k^2}\Re\left(\frac{1}{w}\right)}=e^{\frac{\pi a}{12k^2}\frac{x}{x^2+\theta^2}}\le e^{\frac{\pi a}{12k^2}\frac{x}{\theta^2}}\le e^{\frac{\pi a}{12k^2}\varrho (2kN)^2}=e^{\frac{\pi a}{3}}\ll_a 1,
\end{align*}
where we use the fact $\frac{1}{2kN}\le |\theta|\le \frac{1}{kN}$. Hence for $j=2$ and $3$, we have
\begin{align*}
|J_j|\ll_a \int_{-\infty}^{\varrho} e^{2\pi x \left(n+\frac{b}{24}\right)}\ dx\ll_a \frac{e^{2\pi\varrho \left(n+\frac{b}{24}\right)}}{n+\frac{b}{24}}.
\end{align*}
This implies that
$$|E(I)|=|-J_2+J_3|\le |J_2|+|J_3|\ll_a \frac{e^{2\pi\varrho \left(n+\frac{b}{24}\right)}}{n+\frac{b}{24}},$$
which gives \eqref{eq:integral-error}.
\end{proof}
Recall that
\begin{align*}
\Sigma_1&=\sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} e^{-\frac{2\pi i nh}{k}}(-1)^{\sum_{j=1}^{J}\delta_j \lambda_{m_j,r_j}(h,k)}\omega_{h,k}^2\;\textup{\foreignlanguage{russian}{D}}_{h,k}\Pi_{h,k}\\
&\quad\times \int_{\xi_{h,k}} \exp\Bigg(\frac{\pi}{12k}(\Omega z+\Delta(\varkappa,\ell)z^{-1})\Bigg) e^{-2\pi i n \phi} e^{2 \pi n \varrho}\ d\phi.
\end{align*}
The main contribution to $\Sigma_1$ is
\begin{align*}
&\sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} e^{-\frac{2\pi i nh}{k}}(-1)^{\sum_{j=1}^{J}\delta_j \lambda_{m_j,r_j}(h,k)}\omega_{h,k}^2\;\textup{\foreignlanguage{russian}{D}}_{h,k}\Pi_{h,k}\\
&\times \frac{2\pi}{k} \left(\frac{24n+\Omega}{\Delta(\varkappa,\ell)}\right)^{-\frac{1}{2}} I_{-1}\left(\frac{\pi }{6k}\sqrt{\Delta(\varkappa,\ell)(24n+\Omega)}\right).
\end{align*}
The error term in $\Sigma_1$ is bounded by
\begin{align*}
\sum_{\substack{1\le k\le N\\k \equiv \ell \bmod{L}}} \sum_{\substack{0\le h< k\\ \gcd(h,k)=1\\ h\equiv \varkappa \bmod{\ell}}} \frac{e^{2\pi\varrho \left(n+\frac{\Omega}{24}\right)}}{n+\frac{\Omega}{24}}\ll \frac{N^2 e^{\frac{2\pi}{N^2} \left(n+\frac{\Omega}{24}\right)}}{n+\frac{\Omega}{24}} \xrightarrow{N\to\infty} 0.
\end{align*}
\bibliographystyle{amsplain}
|
{
"timestamp": "2019-03-01T02:05:49",
"yymm": "1902",
"arxiv_id": "1902.10839",
"language": "en",
"url": "https://arxiv.org/abs/1902.10839"
}
|
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\begin{document}
\title[A central idempotent]
{A central idempotent in the endomorphism algebra of a finite lattice}
\author{Serge Bouc}
\author{Jacques Th\'evenaz}
\date\today
\subjclass[2010]{06A05, 06A07, 06A11, 06A12, 06B05, 16S99}
\keywords{Poset, lattice, total order, idempotent, correspondence}
\begin{abstract}
We give a direct construction of a specific idempotent in the endomorphism algebra of a finite lattice $T$.
This idempotent is associated with all possible sublattices of $T$ which are total orders.
\end{abstract}
\maketitle
\section{Introduction}
\noindent
Let $T$ be a finite lattice and let $k$ be a commutative ring.
The set of all $k$-linear combinations of join-morphisms from $T$ to~$T$ is a $k$-algebra $\End_{k\CL}(T)$ which plays an important role in our work on correspondence functors \cite{BT1, BT2, BT3}.
This algebra is also of independent interest from a purely combinatorial point of view because it reflects the structure of~$T$ in an algebraic fashion.
Here $\CL$ refers to the category of finite lattices, defined in Section~\ref{Section-lattices}, and $k\CL$ is its $k$-linearization.\par
We introduced in~\cite{BT2} an idempotent $e_T^{\tot} \in \End_{k\CL}(T)$ which is associated with all possible subsets of~$T$ which are totally ordered.
We proved that $e_T^{\tot}$ is central and that $e_T^{\tot}\End_{k\CL}(T)$ is isomorphic to a product of matrix algebras.
Unfortunately, the definition of $e_T^{\tot}$ relies on some rather cumbersome constructions.\par
In the present paper, we give a new point of view for this idempotent.
We express it by means of a much easier and explicit formula, which also has the advantage of allowing for computer calculations.
This formula does not depend on our previous work, but of course the proof that the result coincides with the idempotent $e_T^{\tot}$ relies on~\cite{BT2}.
\section{Finite lattices} \label{Section-lattices}
\noindent
In this section, we recall the basic facts we need about the category of finite lattices.
For the rest of this paper, $T$ denotes a finite lattice.
We write $\leq$ for its partial order (or $\leq_T$ when necessary), $\vee$ for its join, $\meet$ for its meet, $\hat0=\hat0_T$ for its least element, and $\hat1=\hat1_T$ for its greatest element.
Recall that an empty join is equal to~$\hat0$, while an empty meet is equal to~$\hat1$.
If $T'$ is another finite lattice, a {\em join-morphism} $\varphi: T \to T'$ is a map such that, for any subset $X\subseteq T$, we have
$$\varphi\big(\bigvee_{x\in X} x\big) = \bigvee_{x\in X} \varphi(x) \mpoint$$
The case $X=\emptyset$ yields the property $\varphi(\hat0)=\hat0$.
Recall that, because $T$ is finite, the meet is uniquely determined by the join thanks to the finite expression
$$x\meet y = \bigvee_{\substack{a\in T \\ a\leq x, a\leq y}} a \mpoint$$
However, a join-morphism need not respect the meet, and in particular need not map $\hat1$ to~$\hat1$.\par
We let $\CL$ be the category whose objects are the finite lattices and morphisms are the join-morphisms.
We let $k\CL$ be the $k$-linearization of~$\CL$.
Its objects are again the finite lattices and $\Hom_{k\CL}(T,T')$ is the free $k$-module with basis $\Hom_\CL(T,T')$.
Composition in~$k\CL$ is the $k$-bilinear extension of composition in~$\CL$.
In particular, $\End_{k\CL}(T)=\Hom_{k\CL}(T,T)$ is a $k$-algebra with respect to composition and its $k$-basis is the monoid $\End_\CL(T)$ of all join-endomorphisms of~$T$.\par
The opposite partial order on a finite lattice~$T$ yields the {\em opposite lattice} $T\op$, swapping the role of $\vee$ and $\meet$,
and with $\hat0_{T\op}=\hat1_T$ and $\hat1_{T\op}=\hat0_T$.
Associated with a join-morphism $\varphi: T \to T'$, there is its opposite
$$\varphi\op : T'{\op} \longrightarrow T\op \,, \qquad \varphi\op(t')= \bigvee_{\substack{t\in T \\ \varphi(t) \leq t'}} t \mpoint$$
\result{Lemma} \label{opposite-morphism}
Let $\varphi: T \to T'$ be a join-morphism between two finite lattices.
\begin{enumerate}
\item $\varphi\op:T'{\op} \to T\op$ is a join-morphism (that is, a meet-morphism $T'\to T$).
\item $(\varphi\op)\op=\varphi$.
\item If $\varphi$ is surjective, then
$$\varphi\op(t')= \bigvee_{\substack{t\in T \\ \varphi(t) = t'}} t \;=\;\sup\{t\in T \mid \varphi(t) = t' \} \mpoint$$
\end{enumerate}
\fresult
\pf See Lemma~8.1 in~\cite{BT2}.
\endpf
Recall that a {\em chain} in~$T$ is a totally ordered subset of~$T$.
If $n\in\N$, we write $\sou n=\{0,1,\ldots,n\}$, a totally ordered lattice with $\hat0_{\sou n}=0$ and $\hat1_{\sou n}=n$.
It is straightforward to see that a join-morphism $\varphi:\sou n\to T$ is simply an order-preserving map such that $\varphi(\hat0)=\hat0$.
Therefore, an injective join-morphism $\varphi:\sou n\to T$ corresponds to a chain
$A=\{a_0,a_1,a_2,\ldots,a_n\}$ in~$T$ such that $a_0=\hat0$, where $a_i=\varphi(i)$.
We let $\CA_{T,n}$ be the set of all chains of size $(n+1)$ in~$T$ whose least element is $a_0=\hat0$.
If $n=0$, there is just one element in $\CA_{T,0}$, namely the chain consisting of $\hat0=a_0$.
\par
Similarly, a surjective join-morphism $\pi: T\to \sou n$ corresponds to a chain
$B=\{b_0,b_1,\ldots,b_{n-1}, b_n\}$ in~$T$ such that $b_n=\hat1$, where
$$b_i=\pi\op(i)=\sup\{t\in T \mid \pi(t) = i\} \mpoint$$
We let $\CB_{T,n}$ be the set of all chains of size $(n+1)$ in~$T$ whose greatest element is $b_n=\hat1$.\par
The set
$$\CA_T:=\bigcup_{n\geq0} \CA_{T,n}$$
is partially ordered by inclusion.
It has no greatest element (unless $T$ is totally ordered) and we let $\infty$ be an additional element,
larger than any $A\in \CA_T$. This allows us to consider the M\"obius function $\mu(A,\infty)$,
or in other words the reduced Euler characteristic $\widetilde\chi \big(\, ]A,\infty[ \,\big)$ of the interval $]A,\infty[$ of all chains containing~$A$.
For later use, we now show that this M\"obius function can be expressed in terms of the M\"obius function of~$T$.
\result{Lemma} \label{Mobius}
Let $A=\{a_0,a_1,a_2,\ldots,a_n\}$ be an element of~$\CA_{T,n}$. Then
$$\mu(A,\infty) = (-1)^{n+1}\prod_{k=1}^n \mu(a_{k-1},a_k) \mvirg$$
where $\mu(a_{k-1},a_k)$ denotes the M\"obius function of the interval $]a_{k-1},a_k[$ in~$T$.
\fresult
\pf
For any poset $X$, let $s_i(X)$ be the number of chains of cardinality~$i$ in~$X$.
For $i=0$, there is the empty chain, so $s_0(X)=1$.
It is well-known that
$$\mu(A,\infty)=\widetilde\chi \big(\, ]A,\infty[ \,\big)=\sum_{i\geq0} (-1)^{i-1} s_i(\, ]A,\infty[ \,) \mpoint$$
The sign is $(-1)^{i-1}$ because a chain of cardinality~$i$ is an $(i{-}1)$-simplex.
Now if $A'$ is a chain with $A\subseteq A'$, then $A'$ is obtained from~$A$ by inserting a chain in each interval $]a_{k-1},a_k[$ independently. Therefore
$$s_i(\, ]A,\infty[ \,) = \displaystyle \sum_{\substack{i_1,\ldots,i_n\geq0 \\ i_1+\ldots+i_n=i}} \prod_{k=1}^n s_{i_k}(\, ]a_{k-1},a_k[ \,)$$
and it follows that
\begin{eqnarray*}
\mu(A,\infty)&=& \displaystyle \sum_{i_1,\ldots,i_n\geq0} (-1)^{i_1+\ldots +i_n-1} \, \prod_{k=1}^n s_{i_k}(\, ]a_{k-1},a_k[ \,) \\
&=& (-1) \prod_{k=1}^n \Big( \sum_{i_k\geq 0} (-1)^{i_k} s_{i_k}(\, ]a_{k-1},a_k[ \Big) \\
&=& (-1)^{n+1} \prod_{k=1}^n \Big( \sum_{i_k\geq 0} (-1)^{i_k-1} s_{i_k}(\, ]a_{k-1},a_k[ \Big) \\
&=& (-1)^{n+1}\prod_{k=1}^n \mu(a_{k-1},a_k) \mvirg
\end{eqnarray*}
as was to be shown.
\endpf
There is one case when the M\"obius function vanishes.
\result{Lemma} \label{contraction}
Let $A=\{a_0,a_1,a_2,\ldots,a_n\}$ be an element of~$\CA_{T,n}$.
If $a_n<\hat1$, then $\mu(A,\infty)=0$.
\fresult
\pf
For any chain $B=\{b_0,b_1,b_2,\ldots,b_m\}$, we let $\overline B= \{b_0,b_1,b_2,\ldots,b_m, \hat1\}$ if $b_m<\hat1$
and $\overline B=B$ if $b_m=\hat1$.
The poset $]A,\infty[$ is conically contractible in the sense of Quillen (see 1.5 in~\cite{Qu}), via the contraction
$$B \leq \overline B \geq \overline A$$
and it follows that $\mu(A,\infty)=0$.
\endpf
Because of this lemma, we shall only be interested in the subset $\CZ_T\subseteq \CA_T$ consisting of all chains $A$ whose greatest element is~$\hat1$ (and least element~$\hat0$), i.e. such that $\overline A=A$. Thus for any chain $A=\{a_0,a_1,a_2,\ldots,a_n\}$ in~$\CZ_T$, we have
$$\hat0=a_0<a_1<\ldots<a_n=\hat1 \mpoint$$
\section{The idempotent corresponding to total orders} \label{Section-idempotent}
\noindent
In this section, we consider a two sided-ideal $\End_{k\CL}^{\tot}(T)$ of the $k$-algebra $\End_{k\CL}(T)$, corresponding to total orders.
This ideal was considered in Section~10 of~\cite{BT2} and it has a central identity element $e_T^{\tot}\in \End_{k\CL}^{\tot}(T)$.
Our main purpose is to prove that $e_T^{\tot}$ can be expressed by a much simpler formula and to prove it by direct combinatorial arguments.\par
We define $\End_\CL^{\tot}(T)$ to be the subset of~$\End_\CL(T)$ consisting of all join-morphisms $\alpha:T\to T$ such that the image $\alpha(T)$ is a totally ordered subset of~$T$.
We let $\End_{k\CL}^{\tot}(T)$ be the $k$-linear span of~$\End_\CL^{\tot}(T)$ in~$\End_{k\CL}(T)$.
\result{Lemma} \label{ideal}
$\End_{k\CL}^{\tot}(T)$ is a two-sided ideal of $\End_{k\CL}(T)$.
\fresult
\pf
Let $\alpha\in \End_\CL^{\tot}(T)$ and $\varphi \in \End_\CL(T)$. It is clear that the image of $\alpha\varphi$ is totally ordered,
so $\alpha\varphi\in \End_\CL^{\tot}(T)$.
On the other hand, the totally ordered subset $\alpha(T)$ is mapped by~$\varphi$ to a totally ordered subset, so $\varphi\alpha\in \End_\CL^{\tot}(T)$.
The result follows by considering $k$-linear combinations.
\endpf
The following result is Theorem~10.8 of~\cite{BT2} and is the starting point of the present work.
\result{Theorem} \label{decomposition}
There is a subalgebra $\CD$ of $\End_{k\CL}(T)$ such that
$$\End_{k\CL}(T) = \End_{k\CL}^{\tot}(T) \times \CD \mvirg$$
(where $\End_{k\CL}^{\tot}(T)$ is identified with $\End_{k\CL}^{\tot}(T)\times\{0\}$ and $\CD$ with $\{0\}\times\CD$, as usual).
\fresult
We let $e_T^{\tot}$ be the identity element of the factor~$\End_{k\CL}^{\tot}(T)$.
This is a central idempotent of~$\End_{k\CL}(T)$. The identity element $\Id_T\in \End_{k\CL}(T)$ decomposes as
$$\Id_T= e_T^{\tot} + (\Id_T- e_T^{\tot}) \mvirg$$
and $\Id_T- e_T^{\tot}\in\CD$.
The formula for $e_T^{\tot}$ given in Theorem~10.8 of~\cite{BT2} comes from rather elaborate constructions, which we revisit in Section~\ref{Section-original-approach} below. We now give an alternative formula for~$e_T^{\tot}$.\par
For any $B\in \CZ_T$, we define
$$\alpha_B:T\to T \,, \qquad \alpha_B(t):=\min\{b\in B \mid b\geq t \} \mpoint$$
Note that the set $\{b\in B \mid b\geq t \}$ is nonempty because $\hat1\in B$ (using our assumption that $B\in\CZ_T$).
The image of~$\alpha_B$ is equal to~$B$, hence totally ordered.
It follows easily that $\alpha_B$ is a join-morphism.
Thus $\alpha_B\in \End_\CL^{\tot}(T)$ and it is moreover clear that $\alpha_B(t)\geq t$ for any $t\in T$ and that $\alpha^2=\alpha$, because $\alpha(b)=b$ for any $b\in B$.
\result{Theorem} \label{main-theorem}
$e_T^{\tot} = \displaystyle - \sum_{B\in\CZ_T} \mu(B,\infty) \,\alpha_B$.
\fresult
\result{Remarks}
{\rm
(a) We could as well define $\alpha_B$ for $B\in\CA_T$ and sum over all $B\in\CA_T$, but since $\mu(B,\infty)=0$ whenever $B\in \CA_T-\CZ_T$ by Lemma~\ref{contraction}, we see that we only need to consider a sum indexed by~$\CZ_T$.\par
(b) The sum could be restricted further to all $B\in\CZ_T$ such that the lattice $[b_{k-1},b_k]$ is complemented for each~$k=1,\ldots,n$ (where $B=\{b_0,b_1,\ldots,b_n\}$ and $b_0=\hat0$, $b_n=\hat1$), because
$$\mu(B,\infty) = (-1)^{n+1}\prod_{k=1}^n \mu(b_{k-1},b_k)$$
by Lemma~\ref{Mobius} and $\mu(b_{k-1},b_k)=0$ whenever the lattice $[b_{k-1},b_k]$ is not complemented, by Crapo's formula (see Exercice 92 of Chapter 3 in~\cite{St}).\par
(c) The sum could also be indexed by all endomorphisms $\alpha\in\End_\CL^{\tot}(T)$
satisfying $\alpha\geq\Id$ and $\alpha^2=\alpha$,
because the latter two conditions imply that $\alpha=\alpha_B$ where $B$ is the image of~$\alpha$ (which is totally ordered).
}
\fresult
\bigskip\noindent
{\bf Proof of Theorem~\ref{main-theorem}~: }
Let $e:=\displaystyle - \sum_{B\in\CZ_T} \mu(B,\infty) \, \alpha_B$.
We claim that it suffices to prove that
\begin{equation} \label{e-psi}
e \psi=\psi \,,\qquad \forall \,\psi\in \End_\CL^{\tot}(T) \mpoint
\end{equation}
If (\ref{e-psi}) holds, then
$$e \,e_T^{\tot}=e_T^{\tot}$$
because $e_T^{\tot}$ belongs to~$\End_{k\CL}^{\tot}(T)$ and is therefore a $k$-linear combination of morphisms $\psi\in \End_\CL^{\tot}(T)$.
On the other hand, $e\, e_T^{\tot}=e$ because $e$ belongs to~$\End_{k\CL}^{\tot}(T)$ and $e_T^{\tot}$ is its identity element.
Thus $e_T^{\tot}=e\,e_T^{\tot}=e$, as required.\par
In order to establish~(\ref{e-psi}), we prove more generally that $e \psi=\psi$
for any map $\psi: S\to T$ such that $\Im(\psi)$ belongs to~$\CA_T$ where $S$ is some finite set
(i.e. $\Im(\psi)$ is a totally ordered subset of~$T$ starting with~$\hat0$).
Letting $X=\Im(\psi)$, we decompose $\psi$ as the composite of a surjection $S\to X$ followed by the inclusion map $i_X:X\to T$.
It suffices to prove that $e \,i_X=i_X$ for any chain~$X$ in~$T$ starting with~$\hat0$.
Now we have
\begin{equation} \label{e-iX}
e \,i_X= - \sum_{B\in\CZ_T} \mu(B,\infty) \,\alpha_B i_X
= \sum_{\varphi:X\to T}\big( - \sum_{\substack{B\in\CZ_T \\ \alpha_B i_X=\varphi}} \mu(B,\infty) \big) \,\varphi \mpoint
\end{equation}
By the definition of~$\alpha_B$, the equation $\alpha_B i_X=\varphi$ means that, for any $x\in X$,
the element $\varphi(x)$ is the least element of~$B$ such that $\varphi(x)\geq x$.
In other words, the condition $\alpha_B i_X=\varphi$ is equivalent to
\begin{equation} \label{condition1}
[x,\varphi(x)] \cap B = \{\varphi(x)\} \,, \;\forall \, x\in X \mpoint
\end{equation}
Any function $\varphi:X\to T$ appearing in the sum~(\ref{e-iX}) must satisfy the following 3 conditions~:
\begin{enumerate}
\item $\varphi$ is order-preserving and $\varphi(\hat0)=\hat0$ (that is, $\varphi$ is a join-morphism).
\item $\varphi(x)\geq x$, for all $x\in X$.
\item If $x,y\in X$ satisfy $x\leq y\leq \varphi(x)$, then $\varphi(y)= \varphi(x)$.
\end{enumerate}
In order to prove this, we note that the coefficient of~$\varphi$ in~(\ref{e-iX}) is nonzero only if there exists at least one $B\in\CZ_T$ such that $\alpha_B i_X=\varphi$.
Condition (a) follows from the fact that both $i_X$ and~$\alpha_B$ are order-preserving and map $\hat0$ to~$\hat0$, hence $\varphi=\alpha_B i_X$ has the same properties.
Condition~(b) is clear because $\alpha_B\geq\Id$.
For condition~(c), note that the only element of~$[x,\varphi(x)]\cap B$ is $\varphi(x)$,
so the definition of~$\alpha_B$ yields $\alpha_B(y)=\varphi(x)$,
that is, $\varphi(y)= \varphi(x)$.\par
Now we prove that, if $\varphi$ satisfies (b) and~(c), then (\ref{condition1}) is equivalent to
\begin{equation} \label{condition2}
\big( \bigcup_{x\in X} [x,\varphi(x)] \big) \cap B =\varphi(X) \mpoint
\end{equation}
It is clear that (\ref{condition1}) implies (\ref{condition2}).
Assume now (\ref{condition2}) and let $x\in X$.
Notice that $[x,\varphi(x)] \cap B$ is nonempty because $\varphi(x)\in\varphi(X)$, hence $\varphi(x)\in B$ by~(\ref{condition2}),
and so $\varphi(x)\in[x,\varphi(x)] \cap B$.
For any $b\in [x,\varphi(x)] \cap B$, we have $b\in\varphi(X)$ by~(\ref{condition2}), that is, $b=\varphi(y)$ for some $y\in X$.
Since $X$ is totally ordered, we have either $x\leq y\leq \varphi(y)=b\leq \varphi(x)$, hence $\varphi(y)=\varphi(x)$ by~(c),
or $y\leq x\leq b=\varphi(y)$, hence $\varphi(x)=\varphi(y)$ by~(c) again.
Therefore $b=\varphi(x)$, showing that $[x,\varphi(x)] \cap B=\{\varphi(x)\}$. This proves that (\ref{condition2}) implies~(\ref{condition1}).\par
We now fix a map $\varphi:X\to T$ satisfying (a), (b), (c), and we set
$$C=\bigcup_{x\in X} [x,\varphi(x)] \big) \qquad \text{ and } \qquad D=\varphi(X) \mvirg$$
so that (\ref{condition2}) becomes $C\cap B= D$.
Since $\varphi$ is order-preserving and $\varphi(\hat0)=\hat0$, we have $D\in\CA_T$.
Let $\overline C=C\cup\{\hat1\}$ and $\overline D=D\cup\{\hat1\}$, so that $\overline D\in\CZ_T$.
Clearly, $\hat1\notin C$ if and only if $\hat1\notin D$,
and therefore the condition $C\cap B= D$ is equivalent to $\overline C\cap B= \overline D$.
It follows that the coefficient of~$\varphi$ in~(\ref{e-iX}) is equal to
$$-\sum_{\substack{B\in\CZ_T \\ \alpha_B i_X=\varphi}} \mu(B,\infty)
= - \sum_{\substack{B\in\CZ_T \\ C\cap B= D}} \mu(B,\infty)
= - \sum_{\substack{B\in\CZ_T \\ \overline C\cap B= \overline D}} \mu(B,\infty) \mvirg$$
because the condition $\alpha_B i_X=\varphi$ is equivalent to~(\ref{condition2}) by the discussion above.
\par
By the defining property of the M\"obius function, we obtain
$$-\sum_{\substack{B\in\CZ_T \\ \overline C\cap B= \overline D}} \mu(B,\infty)
=\sum_{\substack{B,A\in\CZ_T \\ B\subseteq A \\ \overline C\cap B= \overline D}} \mu(B,A)
=\sum_{\substack{A\in\CZ_T \\ \overline D\subseteq A }}
\sum_{\substack{B\in\CZ_T \\ B\subseteq A \\ \overline C\cap B= \overline D}} \mu(B,A) \mpoint$$
For a fixed $A\in\CZ_T$, the chain $B$ runs over the interval $[\overline D,A]$
with the additional condition $\overline C\cap B= \overline D$,
which can also be written $(\overline C\cap A)\cap B= \overline D$ because $B\subseteq A$.
By a well-known property of the M\"obius function (Corollary~3.9.3 in \cite{St}), the corresponding sum
$$\sum_{\substack{B\in [\overline D,A] \\ (\overline C\cap A)\cap B= \overline D}} \mu(B,A)$$
is zero, provided the fixed element $\overline C\cap A$ is not equal to the top element~$A$.
If otherwise $\overline C\cap A=A$, then $A\subseteq \overline C$ and $B=\overline C\cap B=\overline D$,
so that the sum over~$B$ has the single term $\mu(\overline D,A)$ for $B=\overline D$.\par
Going back to the coefficient of~$\varphi$ in~(\ref{e-iX}), we obtain
$$ - \sum_{\substack{B\in\CZ_T \\ \alpha_B i_X=\varphi}} \mu(B,\infty)
= \sum_{\substack{A\in\CZ_T \\ \overline D\subseteq A \subseteq \overline C}} \mu(\overline D,A)
= \sum_{\substack{A\in\CZ_{\overline C} \\ \overline D\subseteq A}} \mu(\overline D,A)
= - \mu(\overline D,\infty)$$
where the latter symbol $\infty$ denotes a top element added to the poset $\CZ_{\overline C}$ (consisting of all chains in~$\overline C$ having least element~$\hat0$ and greatest element~$\hat1$).\par
Recall that $\varphi\geq i_X$ by condition~(b).
We now assume that $\varphi> i_X$ and we want to prove that $\mu(\overline D,\infty)=0$.
Let $y\in X$ be minimal such that $\varphi(y)>y$.
We claim that, for any $A\in \CZ_{\overline C}$, the union $A\cup\{y\}$ is totally ordered.
We have to prove that any $a\in A$ is comparable with~$y$.
Since $a\in \overline C$, either $a=\hat1$ and then we are done because $y\leq \hat1$, or there exists $x\in X$ such that $a\in [x,\varphi(x)]$.
If $y\leq x$, then $y\leq a$ and we are done again.
We can assume now that $y\not\leq x$, hence $x<y$ because $X$ is totally ordered.
By minimality of~$y$, we must have $\varphi(x)=x$, hence $[x,\varphi(x)]=\{x\}$ and $a=x$.
It follows that $a<y$. This completes the proof that $A\cup\{y\}$ is totally ordered.\par
We claim now that $y$ does not belong to~$D=\varphi(X)$.
Otherwise $y=\varphi(z)$ for some $z\in X$.
If we had $z=y$, we would obtain $\varphi(y)=y$, contrary to the choice of~$y$.
It follows that the relation $z\leq\varphi(z)=y$ must be a strict inequality $z<\varphi(z)$.
This contradicts the minimality of~$y$ and proves the claim.
Moreover, $y\notin \overline D$ because $y<\varphi(y)$, hence $y\neq\hat1$.
Consequently, the poset $]\overline D,\infty[$ is conically contractible (see 1.5 in~\cite{Qu})
via the contraction
$$A \leq A\cup\{y\} \geq \overline D\cup\{y\}$$
and it follows that $\mu(\overline D,\infty)=0$.\par
This shows that the coefficient of~$\varphi$ in~(\ref{e-iX}) is zero whenever $\varphi> i_X$.
Therefore we are left with a single term for $\varphi=i_X$, namely
$$e \,i_X= -\mu(\overline D, \infty) \, i_X \mpoint$$
But for $\varphi=i_X$, we have
$$C=\bigcup_{x\in X} [x,\varphi(x)]=\bigcup_{x\in X} \{x\} = X = i_X(X) = D \mvirg$$
and consequently the only chain in~$\overline C$ containing~$\overline D$ is $\overline D$ itself.
In other words $]\overline D, \infty[=\emptyset$ and $\mu(\overline D, \infty)=-1$.
The required equality $e \,i_X= i_X$ follows and this completes the proof of Theorem~\ref{main-theorem}.
\endpf
\result{Remark} \label{remark}
{\rm
It is easy to prove directly that the expression
$$e=-\sum_{B\in\CZ_T} \mu(B,\infty)\, \alpha_B$$
is idempotent, because (\ref{e-psi}) implies that
$e\alpha_B=\alpha_B$ for any $B\in\CZ_T$, hence
$$e^2=e\big(-\sum_{B\in\CZ_T} \mu(B,\infty)\, \alpha_B\big) = -\sum_{B\in\CZ_T} \mu(B,\infty)\, \alpha_B =e \mpoint$$
However, the proof that this idempotent is central is more elaborate and appears in Theorem~10.8 of~\cite{BT2}.
}
\fresult
\section{The original approach to the idempotent} \label{Section-original-approach}
\noindent
The idempotent $e_T^{\tot}$ was defined in Section~10 of~\cite{BT2} by an explicit formula.
Using this formula, we want to prove that $e_T^{\tot}$ satisfies the equation of Theorem~\ref{main-theorem}.
In other words, we are going to provide a second proof of that theorem, based on the original approach of~\cite{BT2}.
We first need to define the notation.\par
For any $n\in\N$, we use the set $\CB_{T,n}$ of all chains $B=\{b_0,b_1,\ldots,b_n\}$ in~$T$ whose greatest element is $b_n=\hat1$.
We have seen in Section~\ref{Section-lattices} that the set $\CB_{T,n}$ parametrizes the set of surjective join-morphism $\pi: T\to \sou n$ via the rule
$$b_i=\pi\op(i)=\sup\{t\in T \mid \pi(t) = i\} \mpoint$$
Instead of~$\sou n$, it will be convenient to use a totally ordered lattice~$P$ of cardinality $n+1$, that is, a lattice isomorphic to~$\sou n$,
and to define $r(p)=\sup\{q\in P\mid q<p\}$, for any $p\in P-\{\hat0\}$.
With this notation, a surjective join-morphism $\pi: T\to P$ corresponds to a chain $B=\{b_p\mid p\in P\}$ defined by
$$b_p=\pi\op(p)=\sup\{t\in T \mid \pi(t) = p\} \mvirg$$
and satisfying $b_p<b_q$ whenever $p<q$.
We write $\pi^B:T\to P$ for the surjective join-morphism corresponding to the chain $B\in\CB_{T,n}$.
Then $\pi^B(t)=\hat0$ if $t\leq b_0$ and otherwise we recall the rule
$$\pi^B(t)=p \qquad \text{ if } \;t\leq b_p \;\text{ and } \; t\not\leq b_{r(p)} \mpoint$$
For any given $B\in\CB_{T,n}$, we choose an element $a_p\in [b_{r(p)},b_p]$ for each $p\in P-\{\hat0\}$.
This defines a family $A=(a_p)_{p\in P-\{\hat0\}}$ of elements of~$T$.
We let $\CF_B$ be the set of all families $A=(a_p)_{p\in P-\{\hat0\}}$ of elements of~$T$ such that
$a_p\in [b_{r(p)},b_p]$ for every $p\in P-\{\hat0\}$.
If $A\in\CF_B$, we also set $a_{\hat 0}=\hat 0$ and we define
$$j_A^B: P \longrightarrow T \,, \qquad j_A^B(p)= a_p \mpoint$$
Clearly $j_A^B$ is order-preserving (because if $p<q$ in~$P$,
then $p\leq r(q)$, hence $a_p\leq b_p\leq b_{r(q)}\leq a_q$),
and it also maps $\hat0$ to~$\hat0$. Therefore $j_A^B$ is a join-morphism.
Now let $B^-=\{b_{r(p)} \mid p\in P-\{\hat0\} \}$ and for any $A\in\CF_B$, write
$$\mu(B^-,A)=\prod_{p\in P-\{\hat0\}} \mu(b_{r(p)},a_p) \mvirg$$
where $\mu(b_{r(p)},a_p)$ denotes the M\"obius function for the lattice~$T$.
Now we allow the family~$A$ to vary (i.e. $a_p$ varies in $[b_{r(p)},b_p]$ for each~$p\neq\hat0$) and we define
$$j^B=(-1)^n\sum_{A\in\CF_B}\mu(B^-,A)\, j_A^B \in \Hom_{k\CL}(P,T) \mpoint$$
By Proposition~10.2 of~\cite{BT2}, $f_B=j^B\pi^B$ is an idempotent in~$\End_{k\CL}(T)$
and when $n\geq0$ varies and $B\in \CB_{T,n}$ varies, the idempotents $f_B$ are pairwise orthogonal (Corollary~10.5 of~\cite{BT2}).
This allows us to define the idempotent
$$e_T^{\tot} = \sum_{n=0}^N \sum_{B\in \CB_{T,n}} f_B \mpoint$$
By Theorem 10.8 of~\cite{BT2}, $e_T^{\tot}$ is a central idempotent
and is the identity element of the two-sided ideal $\End_{k\CL}^{\tot}(T)$.
Thus we recover the notation of Section~\ref{Section-idempotent}.\par
\bigskip\noindent
{\bf Second proof of Theorem~\ref{main-theorem}~: }
For each $B\in\CB_{T,n}$, the idempotent $f_B$ is a linear combination of join-morphisms $j_A^B\pi^B$.
We are going to prove that most of these join-morphisms cancel pairwise in the sum
\begin{equation} \label{sum}
e_T^{\tot} = \sum_{n=0}^N \sum_{B\in \CB_{T,n}} (-1)^n\sum_{A\in\CF_B}\mu(B^-,A) \, j_A^B\pi^B \mpoint
\end{equation}
More precisely, we consider all triples $\{(n,B,A)\mid n\in\N, \, B\in\CB_{T,n}, \,A\in\CF_B\}$ such that $a_x<b_x$ for some $x\in P$.
For such a triple, we let $p\in P$ be minimal with respect to the condition $a_p<b_p$.
Since $a_p\in[b_{r(p)},b_p]$ for $p\neq\hat0$, we can either have $b_{r(p)}=a_p< b_p$ or
$b_{r(p)}<a_p<b_p$. The case $p=\hat0$ is special because we always have $a_{\hat0}=\hat0$.
It follows that $p$ must satisfy one of the following 4 cases~:
\par
\begin{enumerate}
\item[{\bf A1.}] $p\neq\hat0$, $r(p)\neq\hat0$, $b_{r(p)}=a_p< b_p$, and $a_x=b_x$ for any $x<p$.
\item[{\bf A2.}] $p\neq\hat0$, $r(p)=\hat0$, $b_{r(p)}=a_p< b_p$, and $\hat0=a_{\hat0}=b_{\hat0}$.
\item[{\bf B1.}] $p\neq\hat0$, $b_{r(p)}<a_p<b_p$, and $a_x=b_x$ for any $x<p$.
\item[{\bf B2.}] $p=\hat0$ and $\hat0=a_{\hat0}<b_{\hat0}$.
\end{enumerate}
{\bf Case A1.} Suppose we are in Case A1. Define
$$\widetilde P=P-\{p\}\,, \qquad\widetilde b_q = b_q \;\;\forall \,q\in \widetilde P-\{r(p)\} \,,\qquad \widetilde b_{r(p)} = b_p \mpoint$$
This defines a chain $\widetilde B$ in~$T$ and a surjective join-morphism $\pi^{\widetilde B}:T\to \widetilde P$, satisfying in particular $\pi^{\widetilde B}(b_p)=r(p)$.
Let $\widetilde A \in \CF_{\widetilde B}$ be the family defined by
$$\widetilde a_q= a_q \;\;\forall \,q\in \widetilde P-\{r(p)\} \,,\qquad \widetilde a_{r(p)}=b_{r(p)} \mvirg$$
and let $j_{\widetilde A}^{\widetilde B}:\widetilde P\to T$ be the corresponding join-morphism.
Then we obtain
$$b_{r(r(p))} < b_{r(p)} < b_p \,, \qquad\text{ that is, } \qquad \widetilde b_{r(r(p))} <\widetilde a_{r(p)}<\widetilde b_{r(p)} \mvirg$$
so that $\widetilde P$ and its element $r(p)$ are in Case~B1 (because $r(p)\neq\hat0$ by assumption~A1).
Moreover, $j_A^B\pi^B= j_{\widetilde A}^{\widetilde B}\pi^{\widetilde B}$.
This is easy to check on most elements of~$T$, the only nontrivial case being
$$j_A^B\pi^B(b_p) = j_A^B(p)=a_p=b_{r(p)} = \widetilde a_{r(p)}
= j_{\widetilde A}^{\widetilde B}(r(p)) = j_{\widetilde A}^{\widetilde B}\pi^{\widetilde B}(b_p) \mpoint$$
Finally, since $\mu(b_{r(p)},a_p)=\mu(b_{r(p)},b_{r(p)})=1$, the coefficient of $j_A^B\pi^B$ is equal to
\begin{eqnarray*}
(-1)^n \mu(B^-,A)&=&(-1)^n\prod_{x\in P-\{\hat0\}} \mu(b_{r(x)},a_x) \\
&=&(-1)^n\prod_{x\in P-\{\hat0,p\}} \mu(b_{r(x)},a_x) \\
&=&(-1)^n\prod_{x\in \widetilde P-\{\hat0\}} \mu(\widetilde b_{r(x)},\widetilde a_x) \\
&=& -(-1)^{n-1} \mu(\widetilde B^-,\widetilde A) \mvirg
\end{eqnarray*}
using the fact that, for $x=r(p)$, we have $\widetilde a_{r(p)}=b_{r(p)}$ and also $a_{r(p)}=b_{r(p)}$ by minimality of the choice of~$p$.
This shows that
$$(-1)^n \mu(B^-,A) \, j_A^B\pi^B \qquad \text{ and } \qquad (-1)^{n-1} \mu(\widetilde B^-,\widetilde A) \, j_{\widetilde A}^{\widetilde B}\pi^{\widetilde B}$$
cancel in the sum~(\ref{sum}).
Thus any Case~A1 cancels with some Case~B1.\par
\medbreak
{\bf Case B1.} Suppose we are in Case B1. Define
$$\widehat P=P_{< p} \sqcup \{s\} \sqcup P_{\geq p} \mvirg$$
with the total order defined by $x<s$ for all $x\in P_{< p}$ and $s<x$ for all $x\in P_{\geq p}$, so that $r(p)=s$.
Moreover, define
$$\widehat b_q = b_q \;\;\forall \,q\in \widehat P-\{s\} \,,\qquad \widehat b_s = a_p \mpoint$$
This defines a chain $\widehat B$ in~$T$ and a surjective join-morphism
$\pi^{\widehat B}:T\to \widehat P$,
satisfying in particular $\pi^{\widehat B}(a_p)=s$ and $\pi^{\widehat B}(b_p)=p$.
Finally, let $\widehat A \in \CF_{\widehat B}$ be the family defined by
$$\widehat a_q= a_q \;\;\forall \,q\in \widehat P-\{s\} \,,\qquad \widehat a_s=a_p \mvirg$$
and let $j_{\widehat A}^{\widehat B}:\widehat P\to T$ be the corresponding join-morphism.
Then we obtain
$$a_p < b_p \,, \qquad\text{ that is, } \qquad \widehat b_{r(p)} = \widehat a_p < \widehat b_p \mvirg$$
so that $\widehat P$ and its element $p$ are in Case~A1 (because $r(p)=s\neq\hat0$).
Applying the procedure described in Case~A1, we note that $\widehat P-\{p\}$ is isomorphic to~$P$
and it follows easily that we recover the Case~B1 we started with.
Thus every Case~B1 has been canceled with a corresponding Case~A1.\par
\medbreak
{\bf Case A2.} Suppose we are in Case A2.
Since $r(p)=\hat0$, $p$ is the least element of~$P-\{\hat0\}$.
Define
$$\widetilde P=P-\{p\}\,, \qquad\widetilde b_q = b_q \;\;\forall \,q\in \widetilde P-\{\hat0\} \,,\qquad \widetilde b_{\hat0} = b_p \mpoint$$
This defines a chain $\widetilde B$ in~$T$ and a surjective join-morphism $\pi^{\widetilde B}:T\to \widetilde P$, satisfying in particular $\pi^{\widetilde B}(b_p)=\hat0$.
Let $\widetilde A \in \CF_{\widetilde B}$ be the family defined by
$$\widetilde a_q= a_q \;\;\forall \,q\in \widetilde P-\{\hat0\} \,,\qquad \widetilde a_{\hat0}=b_{\hat0} \mvirg$$
and let $j_{\widetilde A}^{\widetilde B}:\widetilde P\to T$ be the corresponding join-morphism.
We have $\hat0=b_{\hat0}$ by minimality of~$p$ and we obtain
$$\hat0=b_{\hat0} < b_p \,, \qquad\text{ that is, } \qquad \hat0 = \widetilde a_{\hat0}<\widetilde b_{\hat0} \mvirg$$
so that $\widetilde P$ and its element $\hat0$ are in Case~B2.
The argument for the M\"obius function holds in the same way as in Case~A1 and it follows that any Case~A2 cancels with some Case~B2 in the sum~(\ref{sum}).\par
\medbreak
{\bf Case B2.} Suppose we are in Case B2.
Define
$$\widehat P= \{s\} \sqcup P \mvirg$$
with the total order defined by $s<x$ for all $x\in P$, so that $r(p)=s=\hat0_{\widehat P}$.
Moreover, define
$$\widehat b_q = b_q \;\;\forall \,q\in P \,,\qquad \widehat b_s = \hat0 \mpoint$$
This defines a chain $\widehat B$ in~$T$ and a surjective join-morphism
$\pi^{\widehat B}:T\to \widehat P$,
satisfying in particular $\pi^{\widehat B}(b_p)=p$.
Finally, let $\widehat A \in \CF_{\widehat B}$ be the family defined by
$$\widehat a_q= a_q \;\;\forall \,q\in P \,,\qquad \widehat a_p=\hat0 \mvirg$$
and let $j_{\widehat A}^{\widehat B}:\widehat P\to T$ be the corresponding join-morphism.
Then we obtain
$$\hat0 < b_p \,, \qquad\text{ that is, } \qquad \hat0=\widehat a_{\hat0}=\widehat b_{\hat0} = \widehat a_p < \widehat b_p \mvirg$$
so that $\widehat P$ and its element $p$ are in Case~A2.
Applying the procedure described in Case~A2, we note that $\widehat P-\{p\}$ is isomorphic to~$P$
and it follows easily that we recover the Case~B2 we started with.
Thus every Case~B2 has been canceled with a corresponding Case~A2.\par
\medbreak
Applying the cancelations described above, we can now eliminate all the join-morphisms $j_A^B\pi^B$
corresponding to a triple $(n\in \N, B\in\CB_{T,n}, A\in \CF_B)$ satisfying $a_x<b_x$ for some $x\in P$.
We are left with the triples satisfying $a_x=b_x$ for all $x\in P$.
In such a case, we have $b_{\hat0}=\hat0$, that is, $\hat0\in B$, hence $B\in\CZ_{T,n}$.
Moreover, $A=B-\{\hat0\}=:B^+$ and
$j_{B^+}^B\pi^B(t)=b_p$ if $\pi^B(t)=p$, that is, if $t\leq b_p$ but $t\not\leq b_{r(p)}$.
In other words,
$$j_{B^+}^B\pi^B(t)=\min\{b\in B\mid t\leq b\}$$
and this is exactly the definition of the endomorphism $\alpha_B$ considered in Section~\ref{Section-idempotent}. Thus
$$j_{B^+}^B\pi^B=\alpha_B \qquad \forall \; B \in\CZ_{T,n} \mpoint$$
Moreover, the coefficient of $j_{B^+}^B\pi^B$ in the expression for~$e_T^{\tot}$ is the M\"obius function
$$(-1)^n \mu(B^-,B^+) = (-1)^n \prod_{p\in P-\{\hat0\}} \mu(b_{r(p)},b_p)=-\mu(B,\infty) \mvirg$$
by Lemma~\ref{Mobius}, where the latter M\"obius function is the M\"obius function of the poset $\CZ_T\sqcup\{\infty\}$.
It follows that the expression for~$e_T^{\tot}$ given in~(\ref{sum}) reduces to
$$e_T^{\tot} = \sum_{n=0}^N \sum_{B\in \CZ_{T,n}} (-1)\mu(B,\infty) \, \alpha_B \mpoint$$
This completes the second proof of Theorem~\ref{main-theorem}.
\result{Remark} \label{semi-simple}
{\rm
It is proved in Theorem~10.6 of~\cite{BT2} that the two-sided ideal $\End_{k\CL}^{\tot}(T)$ is isomorphic to a direct sum of matrix algebras
$$\End_{k\CL}^{\tot}(T) \cong \bigoplus_{n=0}^N M_{|\CZ_{T,n}|} (k) \mvirg$$
where $N$ is the maximal length of a chain in~$T$.
It should be noticed that the new approach to the idempotent $e_T^{\tot}$ explained in the present paper does not simplify in any way the proof of this result.
In particular, if $T$ is totally ordered, then
$$\End_{k\CL}(T)=\End_{k\CL}^{\tot}(T)\cong \bigoplus_{n=0}^N M_{|\CZ_{T,n}|} (k) \mvirg$$
and this is a semi-simple algebra whenever $k$ is a field.
As noticed in Remark~11.3 of~\cite{BT2}, this result is similar, but not equivalent, to a theorem proved in~\cite{FHH} about the planar rook algebra.
}
\fresult
|
{
"timestamp": "2019-03-01T02:04:38",
"yymm": "1902",
"arxiv_id": "1902.10818",
"language": "en",
"url": "https://arxiv.org/abs/1902.10818"
}
|
\section{Introduction}
``Diffusion of innovations'' studies the diffusion processes of new ideas, technologies, products, or services through a society over time \cite{Rogers83}.
It has attracted a lot of interest in various fields such as anthropology, sociology, political science, and economics \cite{Greenhalgh04,Kiesling12}.
It is not trivial to predict that an innovation will diffuse to a society,
because people decide to adopt the innovation after careful consideration of the benefit from the adoption.
Many studies have reported that people adopt the innovation when the exposure (the ratio of adopters of the innovation in the neighborhood) is larger than a certain level of value - called a threshold \cite{Rogers83,Katz85,Valent96,Centola10}.
A threshold is an exposure value assigned for each agent to take innovation if it did not take innovation yet.
In the simplest case where every agent has the same number of neighbors ($\Delta$) and a constant threshold ($t$) smaller than $1/\Delta$, the final state is trivial: the whole population will adopt the innovation eventually. In other words, the innovation is spread over the population. If an innovation is spread over the population, then we say that it becomes {\it epidemic}.
On the other hand, if a given threshold is high enough, then the innovation stops diffusing at some time. Morris\cite{Morris} showed that with a threshold bigger than $1/2$ an innovation is never spread over any graph structure and both strategies will coexist eventually.
Inspired by Morris, Immorlica et al.\cite{Immorlica07} developed a model introducing a bilingual option at a cost $r$, which is compatible with both strategies. In this bilingual model a possible threshold for an innovation to be epidemic over an infinite regular graph depends on $r$, say $t_r$. They studied this problem and determined the maximum value for $t_r$ completely for every $r > 0$ on infinite regular trees, the grid, and the infinite thick-lines. We call this maximum value the {\it contagion threshold}. (See also \cite{Wortman08, Easley10}. For similar problems under different conditions, see \cite{Goyal97, Oyama15}.)
In this paper, we revisit this problem and obtain more results by finding a structure attaining the (universal) minimum of contagion thresholds among the whole class of infinite regular graphs. In section 2, we present basic preliminaries for the problem and our main results. In the section, we show that the contagion threshold of any infinite $\Delta$-regular graph containing the infinite $\Delta$-regular tree structure becomes the universal lower bound for contagion thresholds of all infinite $\Delta$-regular graphs. Also, we construct an infinite family of infinite $\Delta$-regular graphs (including the thick $\Delta$-line) that are the most advantageous to be epidemic as known so far. In section 3, we discuss further directions.
\section{Main Results}
\subsection{Preliminaries}
We consider connected infinite regular graphs unless mentioned otherwise and follow the model and setting introduced in \cite{Immorlica07}. We denote by $V(G)$ the vertex set of $G$.
\smallskip
Our main question is: what kind of graph structure on a population is more advantageous for a new strategy to be spread over the old strategy in the population?
An {\it $A$-$B$ coordination game} on an infinite regular graph $G$ is a pair $(G, q)$, where every vertex gets the payoffs from the payoff matrix in Table 1 while it plays a game with its neighbors.
\smallskip
\begin{table}[ht]
\centering
\begin{tabular}{c||c|c}
& $A$ & $B$ \\ \hline
$A$ & $1-q, 1-q$ & 0, 0 \\
$B$ & 0, 0 & $q, q$ \\
\end{tabular}
\caption{The payoffs in an $A$-$B$ coordination game $(G, q)$ \label{Table1}}
\end{table}
\smallskip
An {\it initial strategy profile} is a function $f_0: V(G) \longrightarrow \{A, B\}$. We call $S_0 = f_0^{-1}(A)$ an {\it initial set} of a game.
Let $\alpha = \{v_i\}_{i = 1}^{\infty}$ be a sequence of $V(G)$, where every vertex in $V(G)$ appears at least once in $\alpha$. An assignment $f_{\alpha}:\alpha \longrightarrow \{A, B\}$ in $(G, q)$ is called a {\it profile along $\alpha$} if $f_{\alpha}$ is determined as follows: (1) start with an initial strategy profile $f_0$ and let $f_{\alpha} = f_0$ and (2) for each $i$ in order, update $f_{\alpha}(v_i)$ as the strategy that gives the higher payoff to $v_i$ based on the payoff matrix in Table~\ref{Table1} after playing with its neighbors using the current assignment.
In an $A$-$B$ coordination game $(G, q)$, strategy $A$ becomes {\it epidemic} if there is a sequence $\alpha$ of $V(G)$ and a finite initial set $S_0$ such that the rule for $f_{\alpha}$ forces that for every $v \in V(G)$ there is an index $i$ such that $f_{\alpha}(v_i) = A$. We call $Q$ the {\it contagion threshold} of $G$ if (1) $A$ becomes epidemic in $(G, q)$ for every $q < Q$ and (2) $A$ never becomes epidemic in $(G, q)$ for any $q > Q$.
\smallskip
It is easy to see that $Q \ge 1/\Delta$ for every infinite regular graph.
Morris \cite{Morris} proved that the contagion threshold $Q$ is always at most $1/2$ and the sharpness is achieved by {\it thick lines}.
For even $\Delta$, the {\it thick line} $L_{\Delta}$ is a graph with a vertex set $\mathbf{Z} \times \{ 1, 2, \cdots, \Delta/2 \}$ and an edge set $\{((k, i), (l, j)) \colon |k-l| = 1 \}$.
\medskip
Our main concern in this paper is the case that there are two incompatible strategies $A$ and $B$, and another option $AB$ that is compatible with $A$ and $B$. (We call $AB$ a bilingual option.)
Immorlica et al. set this bilingual model using a two-player game called a contagion game \cite{Immorlica07}.
To prevent a game from being trivial such that the bilingual option $AB$ becomes epidemic in every graph, there needs a cost to pay for choosing $AB$.
\medskip
To begin with, all the vertices have (old) strategy $B$. Now a new strategy $A$ is introduced to finitely many vertices in $V(G)$, and the game starts along a given sequence. If a vertex takes a particular strategy, then the payoff that the player earns is the total of the payoff the player earns from its neighbors. While playing the game, each vertex at its turn chooses a strategy giving the largest payoff among $A$, $B$, or $AB$.
Given a sequence of vertices in $V(G)$ and for $0 < q < 1$ and $r > 0$, a {\it contagion game} $(G, q, r)$ is a coordination game where every vertex plays a game with its neighbors using the payoff matrix presented in Table~\ref{Table2}. In this paper, as mentioned above we consider a contagion game, where there are three strategies for each vertex to choose: $A, B$, or $AB$. If a vertex chooses one strategy, then the total payoff from this choice is the sum of the payoffs from playing with the neighbors of the vertex. For example, if a vertex chooses $A$, then the payoff it gets by playing with a neighbor having strategy $A$, $B$, and $AB$ (resp.) is $1-q$, $0$, and $\max(q, 1-q) - r$ (resp.).
\medskip
\begin{table}[ht]
\centering
\begin{tabular}{c||c|c|c}
& $A$ & $B$ & $AB$ \\ \hline
$A$ & $1-q, 1-q$ & 0, 0 & $1-q, 1-q-r$ \\
$B$ & 0, 0 & $q, q$ & $q, q-r$ \\
$AB$ & $1-q-r, 1-q$ & $q-r, q$ & $\max(q, 1-q)-r, \max(q, 1-q)-r$
\end{tabular}
\caption{The payoffs in a contagion game $(G, q, r)$ \label{Table2}}
\end{table}
\medskip
Note that the remaining part after deleting the row and column of $AB$ in Table~\ref{Table2} is the same as the payoff matrix in Table~\ref{Table1}. We mimic the definition for the strategy $A$ to be epidemic.
\smallskip
An {\it initial strategy profile} is a function $f_0: V(G) \longrightarrow \{A, B\}$. We call $S_0 = f_0^{-1}(A)$ an {\it initial set} of a game.
Let $\alpha = \{v_i\}_{i = 1}^{\infty}$ be a sequence of $V(G)$, where every vertex in $V(G)$ appears at least once in $\alpha$. An assignment $f_{\alpha}:\alpha \longrightarrow \{A, B, AB\}$ in a contagion game $(G, q, r)$ is called a {\it profile along $\alpha$} if $f_{\alpha}$ is determined as follows: (1) start with an initial strategy profile $f_0$ and let $f_{\alpha} = f_0$ and (2) for each $i$ in order, update $f_{\alpha}(v_i)$ as the strategy that gives higher payoff to $v_i$ based on the payoff matrix in Table~\ref{Table2} after playing with its neighbors using the current assignment.
In a contagion game $(G, q, r)$, strategy $A$ becomes {\it epidemic} if there is a sequence $\alpha$ of $V(G)$ and a finite initial set $S_0$ such that the rule for $f_{\alpha}$ forces that for every $v \in V(G)$ there is an index $i$ such that $f_{\alpha}(v_i) = A$.
In other words, the strategy $A$ becomes {\it epidemic} in $(G, q, r)$ if there are a finite set $S$ and a sequence of vertices in $V(G) - S$, say $v_1, v_2, v_3, \cdots $, satisfying that for every $v$ in $V(G) - S$ there is an index $k$ such that (1) $v = v_k$ and (2) at the $k$-th turn along the sequence the best strategy for $v$ is $A$ when having started with $A$s for the vertices of $S$ while other vertices had initially $B$. \smallskip
Note that the definition of epidemic status depends on $q$ and $r$ as well as $G$ because it is possible that $A$ is not epidemic for different $q$ and $r$ even with the same graph ($G$), the same finite set ($S$), and the same sequence ($\alpha$). \smallskip
For a fixed $r>0$, we call $Q_r$ the {\it contagion threshold} of $G$ if (1) $A$ becomes epidemic in $(G, q, r)$ for every $q < Q_r$ and (2) $A$ never becomes epidemic in $(G, q, r)$ for any $q > Q_r$. For an infinite regular graph $G$, the {\it epidemic region} denoted $\Omega_G$ is $\{ (q, r) \colon A \textrm{ becomes epidemic in } (G, q, r) \}$. Therefore, the boundary curve of an epidemic region is the points consisting of $(Q_r, r)$ for every $r > 0$. \smallskip
Let $\Omega_{\Delta} = \bigcup_G\Omega_G$, where the union is taken over all infinite $\Delta$-regular graphs. In this paper we focus on determining $\Omega_{\Delta}$ for every $\Delta \ge 2$.
\medskip
It is known that no vertex can change either from $A$ to $B$, from $A$ to $AB$, or from $AB$ to $B$ when taking its best response in a contagion game $(G, q, r)$ \cite{Immorlica07}. Hence there are only two possible types for $(q, r)$ to be in the epidemic region: (1) $A$ is always the best response for every turn of vertices and (2) $AB$ is the best response for some vertex but finitely many turns later the best strategy for the vertex is eventually $A$. \\
It is useful to apply the concept of {\it blocking structure} and results on blocking structures introduced in \cite{Immorlica07}. We say $(X, Y)$ a {\it non-trivial pair} of disjoint sets if either $X \neq \emptyset$ or $Y \neq \emptyset$. For a contagion game $(G, q, r)$, a non-trivial pair $(S_{AB}, S_B)$ of disjoint subsets of $V(G)$ is called a {\it blocking structure} for $(G, q, r)$ if the pair satisfies the following properties:
\begin{enumerate}
\item for every $v \in S_{AB}$, $\deg_{S_B}(v) > \frac{r}{q}\Delta$,
\item for every $v \in S_B$
\begin{enumerate}
\item $(1-q)\deg_{S_B}(v)+\min(q, 1-q)\deg_{S_{AB}}(v) > (1-q-r)\Delta$ and
\item $\deg_{S_B}(v) + q \deg_{S_{AB}}(v) > (1-q)\Delta,$
\end{enumerate}
\end{enumerate}
where $\deg_{S_{AB}}(v)$ and $\deg_{S_{B}}(v)$ are the number of neighbors of $v$ in $S_{AB}$ and $S_{B}$, respectively. \smallskip
\begin{thm}\cite{Immorlica07}\label{blocking}
For every contagion game $(G, q, r)$, strategy $A$ cannot be epidemic in $(G, q, r)$ if and only if every co-finite set of vertices of $G$ contains a blocking structure.
\end{thm}
\smallskip
Also, in the same reference, the authors showed that $(q, r)$ with $q > 1/2$ cannot be in the epidemic region for any infinite regular graph. Therefore, $\max(q, 1-q) = 1-q$ in Table~\ref{Table1} for payoffs to be epidemic. \medskip
\subsection{Our Results}
We find a sufficient condition for each vertex (i.e. agent) to adopt $A$ as its best strategy in the given game $(G, q, r)$. The condition will be formulated in terms of $r$ and $q$. \smallskip
\begin{lem} Let $G$ be an infinite $\Delta$-regular graph.
Suppose that there is an order of finite subsets of vertices $V_0, V_1, \cdots$ such that
\begin{enumerate}
\item
\[ \textrm{The union of $V_i$'s is $V(G)$, that is, } \ \ \bigcup_{i = 0}^{\infty} V_i = V(G) \textrm{ and }\]
\item Given $\epsilon$ with $0 < \epsilon < 1$, for each $i$ and $v \in V_i$,
\[ d_{V_0 \cup V_1 \cup \cdots \cup V_{i-1}}(v) \ge \epsilon \Delta .\]
\end{enumerate}
Then $(q, r)$ satisfying $r > (1-\epsilon)q$ and $q < \epsilon$ is in the epidemic region for $G$.
\end{lem} \vskip 0.1in
\begin{pf}
We prove that for each $i$ every $v \in V_i$ has $A$ as the best strategy under the given condition. We use induction on $i$. Starting from an initial situation that every vertex uses strategy $B$, we locate strategy $A$ at every vertex in $V_0$. This automatically satisfies the basis. Now assume that the vertices in $V_0\cup V_1 \cdots \cup V_{i-1}$ adopted $A$ as their strategies. For each vertex $u \in V_i$ the payoff of strategy $A$ is $p_1$ which is at least $(1-q)\epsilon\Delta$. The payoff of strategy $B$ is $p_2$ which is at most $(1-\epsilon)\Delta q$. The payoff of strategy $AB$ is $p_1+p_2 - r\Delta$. Hence the vertex $u$ has $A$ as the best strategy on $(G, q, r)$ game whenever $q$ and $r$ satisfy $r\Delta > p_2$ and $p_1 > p_2$. In particular, if $r > (1-\epsilon)q$ and $q < \epsilon$, then $p_1 - p_2 \ge (1-q)\epsilon \Delta - (1-\epsilon)\Delta q = \Delta(\epsilon - q) > 0$. Also, $p_1 - (p_1+p_2-r\Delta) = r\Delta - p_2 > (1-\epsilon)q\Delta - p_2 > (1-\epsilon)q\Delta - (1-\epsilon)\Delta q = 0$. \vskip 0.1in
After locating strategy $A$ at each vertex in $V_0$ we let vertices in sets $V_1, V_2, \cdots$ play their best strategy. Eventually $A$ becomes epidemic along the order. \qed
\end{pf}
\vskip 0.15in
An infinite regular-tree is a connected, acyclic, and infinite graph each of whose degree is the same.
As an easy consequence, if we pick a vertex in the infinite $\Delta$-regular tree, then by considering $V_i$ be the set of the vertices at depth $i$ and by letting $\epsilon \le \frac{1}{\Delta}$ the set $\{ (q, r) \colon r \ge \frac{\Delta-1}{\Delta}q, q \le \frac{1}{\Delta}\}$ is contained in $\Omega_{T_{\Delta}}$. In fact, it is not hard to see that $\Omega_{T_{\Delta}} = \{ (q, r) \colon r \ge \frac{\Delta-1}{\Delta}q, q \le \frac{1}{\Delta} \} \cup \{(q, r) \colon 2q + \Delta r \le 1 \}$.
\vskip 0.25in
\begin{thm}\label{tree}
Let $\Omega_{T_{\Delta}}$ be the epidemic region for the infinite $\Delta$-regular tree $T_{\Delta}$. For every infinite $\Delta$-regular graph $G$, $\Omega_{T_{\Delta}} \subseteq \Omega_G$. In other words, $\Omega_{T_{\Delta}}$ is the minimum epidemic region that any infinite $\Delta$-regular graph possibly has.
\end{thm} \vskip 0.1in
\begin{pf}
By Theorem \ref{blocking} for any $(q, r)$ in the complement of the epidemic region $\Omega_G$, every co-finite set of $V(G)$ contains a blocking structure $(S_B, S_{AB})$ for $(G, q, r)$. Therefore, $\{(q, r) \colon q, r > 0 \} - \Omega_G$ is the union of $\Omega_1$ and $\Omega_2$, where $\Omega_1$ consists of $(q, r)$ allowing a blocking structure with $S_{AB} \neq \emptyset$ and $\Omega_2$ consists of $(q, r)$ allowing a blocking structure with $S_{AB} = \emptyset$. \vskip 0.15in
Let $(q, r)$ be in$\{(q, r) \colon q, r > 0 \} - \Omega_G = \Omega_1 \cup \Omega_2$, and we consider a blocking structure $(S_B, S_{AB})$ in a co-finite set of $V(G)$.
\vskip 0.15in
Case 1) $S_{AB} \neq \emptyset$: \\
We let $a_v = \deg_{S_B}(v)$ and $b_v = \deg_{S_{AB}}(v)$ for any $v \in S_B$. Hence, $a_v, b_v \ge 0$ and $a_v + b_v \le \Delta$.
Now $r$ and $q$ must satisfy that $a_v + qb_v > (1-q)\Delta $ and $(1-q)a_v + qb_v > (1-q-r)\Delta$. If we let $d = \min_{u \in S_{AB}} \deg_{S_B}(u)$, then $d > \frac{r}{q}\Delta$ by the first condition in the definition of a blocking structure. In other words, $r < \frac{d}{\Delta}q$. Since $a_v + b_v \le \Delta$, we let $\Delta = a_v + b_v + t_v$ with $t_v \ge 0$.
From those inequalities we obtain that $\left(2-\frac{t}{\Delta-a_v}\right)q+\frac{\Delta}{\Delta-a_v}r > 1$ and $q > \frac{\Delta - a_v}{2\Delta-a_v-t_v}$.
Note that there is $v_0 \in S_B$ with $a=\deg_{S_B}(v_0) < \Delta$. Therefore $\frac{\Delta-a_v}{\Delta} \ge \frac{1}{\Delta}$. The line $\left(2-\frac{t}{\Delta-a}\right)q+\frac{\Delta}{\Delta-a}r = 1$ has the $q$-intercept that is as large as $\frac{1}{2}$ and the $r$-intercept that is as large as $\frac{1}{\Delta}$. Also, $\frac{\Delta - a_v}{2\Delta - a_v - t_v} - \frac{1}{\Delta+1} = \frac{{\Delta}^2 - a_v\Delta - \Delta + t_v}{(2\Delta-a_v-t_v)(\Delta+1)} = \frac{{\Delta}(\Delta-a_v) - (\Delta - t_v)}{(2\Delta-a_v-t_v)(\Delta+1)} \ge \frac{\Delta - (\Delta-t_v)}{(2\Delta-a_v-t_v)(\Delta+1)} \ge 0$. Moreover, $d \ge 1$. Hence, if we let $ W_1 = \{(q, r) \colon 2q + \Delta r > 1, q > \frac{1}{\Delta+1}, \ r < \frac{\Delta - 1}{\Delta}q \}$, then $\Omega_1 \subseteq W_1$. \vskip 0.1in
Case 2) $S_{AB} = \emptyset$: \\
We let $a_v = \deg_{S_B}(v)$ for any $v \in S_B$. Then $a_v \le \Delta - 1$.
Now $r$ and $q$ must satisfy that (1) $a_v > (1-q)\Delta $ and (2) $(1-q)a_v > (1-q-r)\Delta$. If we let $W_2 = \{(q, r) \colon q > \frac{1}{\Delta}, \ q+ \Delta r > 1\}$, then $\Omega_2 \subseteq W_2$. \vskip 0.15in
Now we can see that $\{(q, r) \colon q, r > 0 \} - (W_1 \cup W_2)$ is the epidemic region for $T_{\Delta}$, and it is contained in $\Omega_G$. \qed
\end{pf}
\vskip 0.2in
The next result is that if an infinite regular graph contains an infinite regular tree, then the epidemic region for the graph is just the same as the epidemic region for an infinite regular tree. We consider a rooted tree $RT_{\Delta}$ defined as follows: the root $x_0$ has $\Delta-1$ children $x_1^1, x_1^2, \cdots, x_1^{\Delta}$, and each child has $\Delta$ neighbors, and so on. In other words, $T$ is an infinite rooted tree, where the root has degree ${\Delta-1}$ and all the rest of the vertices have degree $\Delta$.
\vskip 0.2in
\begin{thm} If an infinite $\Delta$-regular graph $G$ contains $RT_{\Delta}$ as a subgraph, then the epidemic region for $G$ is the same as the epidemic region for the tree $\Omega_{T_{\Delta}}$.
\end{thm}
\vskip 0.2in
\begin{pf}
Let $H$ be a subgraph that is isomorphic to $RT_{\Delta}$ in $G$. Let $x_0$ be the root of $H$. For any vertex $x$ of $H$, the (induced) subtree starting at $v$ is in fact isomorphic to $H$. Therefore, for any finite subset of $V(G)$, say $C$, the remaining graph $G-C$ contains a vertex $v$ of $H$, and it contains a subtree isomorphic to $H$ with $v$ as the root. Also, note that an infinite component of $T_{\Delta} - C'$ for any finite set $C' \in V(T_{\Delta})$ contains a subtree isomorphic to $RT_{\Delta}$ with a root $v'$ for some $v'$. Therefore, we choose each blocking structure for $(q, r)$ in $H - C$ exactly the same way as in $T_{\Delta} - C'$ for $(q, r)$. \smallskip
Now the epidemic region $\Omega_G^c$ contains $\Omega_{T_{\Delta}}^c$, and therefore $\Omega_G \subseteq \Omega_{T_{\Delta}}$. But by Theorem \ref{tree}, $\Omega_{T_{\Delta}} \subseteq \Omega_G$. Therefore, $\Omega_G = \Omega_{T_{\Delta}}$. \qed
\end{pf}
\vskip 0.25in
\begin{lem}\label{b}
Let $G$ be an infinite $\Delta$-regular graph with even $\Delta$. If for every finite subset $C$ of $V(G)$,
\begin{enumerate}
\item there are two disjoint non-empty subsets $S_B$ and $S_{AB}$ of $V(G) - C$ such that
\begin{enumerate}
\item $\deg_{S_B}(v) \ge \frac{\Delta}{2}$ for every $v \in S_{AB}$
\item $\deg_{S_B}(u) + \deg_{S_{AB}}(u) = \Delta$ for every $u \in S_{B}$
\end{enumerate}
and
\item $G - C$ has a $\frac{\Delta}{2}$-regular subgraph,
\end{enumerate}
then the epidemic region $\Omega_G$ is a subset of $\{(q, r) \colon q, r > 0 \} - (\Omega_i \cup \Omega')$ for some $1 \le i \le \Delta - 1$, where $$\Omega_i =
\left\{(q, r) \colon r < \frac{1}{2}q, \ \ 2q + \frac{\Delta}{\Delta - i}r > 1, \ \ q > \frac{\Delta-i}{2\Delta - i} \right\}$$
$$\Omega' = \left\{ (q, r) \colon q + 2r > 1, \ \ q > \frac{1}{2} \right\} .$$
\end{lem}
\vskip 0.15in
\begin{pf}
We show that for any finite subset $C$ of $V(G)$ and for any $(q, r)$ in $\Omega_1 \cup \Omega_2$, there is a blocking structure. \vskip 0.15in
We have two cases. \\
Case 1) Let $(q, r)$ be in $\Omega_i$, where $i = \min_{v \in S_B} \deg_{S_B}(v)$. We use $S_B$ and $S_{AB}$ guaranteed in condition 1 of the statement of Lemma \ref{b}. Now for any $u \in S_{AB}$, $\deg_{S_B}(u) \ge \frac{\Delta}{2} > \frac{r}{q}\Delta$ since $r < \frac{1}{2}q$. Now for any $v \in S_B$, let $a = \deg_{S_B}(v)$ and $b = \deg_{S_{AB}}(v)$, where $a + b = \Delta$. Since $S_B$ is non-empty, $a \ge 1$. Now we obtain the following for $1 \le a \le \Delta-1$: (1) $(1-q)a+qb - (1-q-r)\Delta = (\Delta - a)[2q+\frac{\Delta}{\Delta-a}r - 1] > 0$ and (2) $a+qb-(1-q)\Delta = q(\Delta+b)-(\Delta-a) > 0$. Note that if $a' > a$, then $q$ and $r$ satisfy (1) and (2) after we replace $a$ by $a'$. We let $i = \min_{v \in S_B} \deg_{S_B}(v)$. Since $S_B \in G - C$, $1 \le i < \Delta$. \vskip 0.1in
Case 2) Let $(q, r)$ be in $\Omega'$. We let $S_{AB} = \emptyset$ and $S_B = V(H)$, where $H$ is a $\frac{\Delta}{2}$-regular subgraph guaranteed in condition 2 of the statement of Lemma \ref{b}. Since $S_{AB}$ is empty the first condition in the definition of a blocking structure is automatically satisfied. Now using the same notation as Case 1 we have $a = \frac{\Delta}{2}$ and $b = 0$ for every vertex $v \in S_B$. We check the second condition in the definition of a blocking structure: $(1-q)\frac{\Delta}{2} - (1-q-r)\Delta = \frac{\Delta}{2}(q+2r-1) > 0$. The third condition for being a blocking structure is satisfied because $a - (1-q)\Delta = \Delta(q-\frac{1}{2}) > 0$.
\vskip 0.1in
By Theorem \ref{blocking} every $(q, r)$ in $\Omega_i \cup \Omega'$ is non-epidemic in $(G, q, r)$. \qed
\end{pf}
\vskip 0.15in
\begin{cor}\label{c} If for every finite set $C$ there are two disjoint sets $S_B$ and $S_{AB}$ as in Lemma \ref{b} such that $\deg_{S_B}(v) = \deg_{S_{AB}}(v) = \frac{\Delta}{2}$ for every $v \in S_B$, then the epidemic region of $G$ is contained in $\Omega_{L_{\Delta}}$.
\end{cor}
\vskip 0.15in
\begin{pf}
The given condition implies that $i = \frac{\Delta}{2}$ in the proof of Lemma \ref{b}. Then,
$$\Omega_{\frac{\Delta}{2}} \cup \Omega' = \left\{(q, r) \colon r < \frac{1}{2}q, \ \ 2q + \frac{\Delta}{\Delta - \frac{\Delta}{2}}r > 1, \ \ q > \frac{\Delta-\frac{\Delta}{2}}{2\Delta - \frac{\Delta}{2}} \right\} \cup \left\{ (q, r) \colon q + 2r > 1, \ \ q > \frac{1}{2} \right\}$$
$$= \left\{(q, r) \colon r < \frac{1}{2}q, \ \ 2q+2r > 1, \ \ q > \frac{1}{3} \right\} \cup \left\{ (q, r) \colon 1+2r > 1, \ \ q > \frac{1}{2} \right\}.$$
Therefore, the epidemic region of $G$ is
$$ \left\{(q, r) \colon q, r > 0 \right\} - \left(\Omega_{\frac{\Delta}{2}} \cup \Omega' \right) = \left\{(q, r) \colon q \le \frac{1}{2}, \ \ r \ge \frac{1}{2}q \right\} \cup \left\{ (q, r) \colon r \le \frac{1}{2}q, \ \ 2q + 2r \le 1 \right\}.$$
Note that the resulting set is the same as $\Omega_{L_{\Delta}}$ \cite{Immorlica07}.
\qed
\end{pf}
\vskip 0.15in
We denote $HL_{\Delta}$ a thick half line graph for even $\Delta$. In other words, the vertex set of $HL_{\Delta}$ can be described as $\mathbf{N} \times \{1, 2, \cdots, \Delta/2\}$. There is an edge between $(k, i)$ and $(l, j)$ if and only if $|k - l| = 1$. In $HL_{\Delta}$ every vertex $(k, i), \ k \ge 2$ has degree $\Delta$, and every vertex $(1, i)$ has degree $\Delta/2$. \vskip 0.15in
We will construct a family of infinite $\Delta$-regular graphs. The family contains the thick line graph as a special case. \smallskip
For $N \ge 2$ and an even $\Delta$ we consider $N$ copies of $HL_{\Delta}$: $HL(1), HL(2), \cdots, HL(N)$. Let $V(i)$ be the vertices of $(1, 1), (1, 2), \cdots, (1, \Delta/2)$ of each copy $HL(i)$. We identify $t_{ij}$ vertices in $V(i)$ with $t_{ji}$ vertices in $V(j)$ for $i \neq j$ for some $t_i \ge 0$ and for some $t_{ij} \le \frac{\Delta}{2}$, where $t_{ij} = t_{ji}$. We add finitely many vertices to the above graph and add appropriately many edges such that every vertex has degree $\Delta$. Note that for $N = 2$ if $v_{12} = v_{21} = 0$ and we add a perfect matching between $V(1)$ and $V(2)$, then the resulting graph is isomorphic to $L_{\Delta}$.
\begin{thm}
The epidemic region for any graph from the construction is the same as $\Omega_{L_{\Delta}}$.
\end{thm} \vskip 0.1in
\begin{pf}
We denote $G$ an infinite $\Delta$-regular graph from the construction and $\Omega_G$ the epidemic region for $G$. We let $C$ be a set of vertices of $G$ consisting of $V(i), \ i = 1, 2, \cdots, N$ and the added vertices from the construction. Hence, $C$ is finite. Starting from the initial state where every vertex takes the strategy $B$, the vertices in $C$ take new strategy $A$. Now we consider $(G, q, r)$ game with every vertex taking its best strategy. \vskip 0.1in
We name group $g_m^i$ each group of vertices $(i, 1), (i, 2), \cdots , (i, \Delta/2)$ in $HL(m)$. We determine an order of moves of the vertices with strategy $B$ as follows: $g_1^1, g_1^2, \cdots, g_1^N, g_2^1, \cdots , g_2^N, g_3^1, \cdots$. Note that in each group any order of the $\Delta/2$ vertices works. Then at each stage the payoffs of strategies $A, B, AB$ are $\frac{\Delta}{2}(1-q), \frac{q\Delta}{2}$, and $\frac{\Delta}{2}-r\Delta$, respectively. (Note that it has the same pattern as in the thick line graphs.) Hence, for $(q, r)$ with $q \le \frac{1}{2}$ and $r \ge \frac{1}{2}q$ the best strategy of a vertex at its turn is $A$. \vskip 0.1in
On the other hand, for $(q, r)$ with $r \le \frac{1}{2}q$ and $2q + 2r \le 1$ the best strategy of a vertex at its turn (in the same order as above) is $AB$. As it is described in \cite{Immorlica07}, the payoff of strategies $A$, $B$, and $AB$ are $\frac{1-q}{2}\Delta$, $q\Delta$, and $\left( \frac{q+1-q}{2}\Delta \right) - r\Delta$, respectively. Hence, if $AB$ has the best payoff, then $\frac{\Delta}{2} - r\Delta \ge \frac{1-q}{2}\Delta$ and $\frac{\Delta}{2} - r\Delta \ge q\Delta$. The inequalities are simplified to $r \le q/2$ and $2q+2r \le 1$. We conclude that $\Omega_G$ contains $\{(q, r) \colon r \le \frac{1}{2}q, 2q + 2r \le 1 \} \cup \{ (q, r) \colon q \le \frac{1}{2}, r \ge \frac{1}{2}q \}$. \vskip 0.15in
To determine the complement of $\Omega_G$ we consider blocking structures. For any deletion of a finite set of vertices $C$ the remaining graph still contains a thick half line. Then we find the same blocking structures used for determining the complement of $\Omega_{L_{\Delta}}$. \vskip 0.15in
In $L_{\Delta}$, two blocking structures are used: A $\frac{\Delta}{2}$-regular subgraph $K_{\frac{\Delta}{2}, \frac{\Delta}{2}}$ satisfies condition 2 in the statement of Lemma \ref{b}. Also, $G - C$ contains a thick half line as an induced subgraph by a set of vertices $S = \{(k, i) \colon k \ge m, 1 \le i \le \frac{\Delta}{2} \}$ for some $m \ge 1$ from a copy of $HL_{\Delta}$. We partition $S$ into $S_{B}$ and $S_{AB}$. $S_{B} = \{ (k, i) \colon k = m, m+2, m+4, m+6, \cdots \}$ and $S_{AB} = \{ (k, i) \colon k = m+1, m+3, m+5, \cdots \}$. Now $S_{B}$ and $S_{AB}$ satisfy condition 1 in the statement of Lemma \ref{b}. Moreover, since $\deg_{S_B}(v) = \deg_{S_{AB}}(v) = \frac{\Delta}{2}$ for every $v \in S_B$, by Corollary \ref{c} $\Omega_G$ is contained in $\Omega_{L_{\Delta}}$. \qed
\end{pf}
\begin{figure}[tb!]
\centerline{\includegraphics[width=7cm]{fig_G_8_2.eps}}
\caption{ $G_{8,2}$ (in Corollary \ref{cor})
}
\label{FIG_G_8_2}
\end{figure}
\begin{cor}\label{cor}
For every even $\Delta > 2$, there are infinitely many infinite $\Delta$-regular graphs whose epidemic region is the same as $\Omega_{L_{\Delta}}$.
\end{cor}\vskip 0.1in
\begin{pf}
We let $k \ge 2$. We construct $G_{\Delta, k}$ for even $\Delta$ as follows. \\
Let $S$ be a set of $\Delta/2$ elements. We label the elements of $S$ as $x_1, x_2, \cdots, x_{\frac{\Delta}{2}}$. We consider $2k$ copies of $S$. Let $S_m$ be the $m$-th copy of $S$. Let $x_1^{m}$ and $x_{\frac{\Delta}{2}}^{m}$ be the first and the last elements respectively in $S_m$. We identify $x_1^{m}$ and $x_{\frac{\Delta}{2}}^{m-1}$. In other words, $S_{m-1} \cap S_{m} = \{ x_1^{m} \}$. We make the infinite half-line $HL(m)$ with degree $\Delta$ starting at $S_m$, where the elements of $S_m$ have degree $\frac{\Delta}{2}$.
Now we consider the $2k\left(\frac{\Delta}{2}-2\right)$ vertices in $\bigcup_{m = 1}^{2k}S_{m} - \bigcup_{m = 1}^{2k} \left\{x_1^{m}, x_{\frac{\Delta}{2}}^{m} \right\}$. \\
It is well-known that there is a connected $\frac{\Delta}{2}$-regular graph $G'$ on $2k\left(\frac{\Delta}{2} - 2\right)$ vertices. We consider the following graph $G_{\Delta, k}$ obtained by adding the edges of $G'$ to the union of $S_m$s for $m = 1, 2, , ..., 2k$. In other words, $G_{\Delta, k}= \bigcup_{m = 1}^{2k}{S_m} \cup E(G')$. Now the epidemic region of $G_{\Delta, k}$ is the same as $\Omega_{L_{\Delta}}$. \qed
\end{pf} \vskip 0.25in
\section{Discussion}
The simplest non-trivial infinite regular graph is when $\Delta$ is 2. There is only one possible such graph: infinite line $L_2$, which attains the smallest and the largest possible epidemic region simultaneously. For even $\Delta > 2$ there are many non-isomorphic infinite $\Delta$-regular graphs including $T_{\Delta}$ and $L_{\Delta}$. We showed that the regular tree has the smallest epidemic region, and we showed that any infinite regular graph containing the infinite regular tree must have the minimum epidemic region. \\
Our prediction is that for any infinite $\Delta$-regular graph $G$, $\Omega_{T_{\Delta}} \subseteq \Omega_G \subseteq \Omega_{L_{\Delta}}$.
In fact for bigger epidemic regions, the maximum possible epidemic region known so far is $\Omega_{L_{\Delta}}$. However, we still do not know whether $\Omega_{L_{\Delta}}$ has the maximum epidemic region among all infinite $\Delta$-regular graphs. We conjecture that $\Omega_{L_{\Delta}}$ is the maximum region among all infinite $\Delta$-regular graphs. \\
Along the lines of the results, here are various directions for further research.
One way is that we can apply the setting in a contagion game for non-regular graphs, where the (total) payoffs for a vertex $v$ are $(\alpha+\beta)(1-q), (\deg(v)-\alpha)q$, and $(\alpha+\beta)(1-2q)+q \deg(v)-c$ for strategies $A, B$, and $AB$, respectively.
Here each of $\alpha$, $\deg(v)-\alpha-\beta$, and $\beta$ is the number of neighbors with $A, B$, and $AB$, respectively.
It is reasonable to assume that the cost $c$ is constant so that a vertex with a high degree tends to adopt a bilingual strategy because $r_v = {c}/{\deg(v)}$.
For non-regular graphs, an epidemic region can be defined in a similar way. But in this case it is in the $(q, c)$-plane instead of the $(q, r)$-plane. \medskip
Another possible model reflecting reality more is that we consider a finite graph with $N$ vertices, where $N$ is the population of a network.
It is not clear whether an epidemic behavior on infinite graphs has the same characterization as the limit of the behavior on finite graphs with $N$ vertices as $N \to \infty$.
A corresponding definition is that we say $A$ is epidemic on a graph with $N$ vertices if there is a finite set of $S$ with size $f(N)$ and a finite sequence of vertices in $V(G) - S$ for playing the game, $v_1, v_2, v_3, \cdots, v_M$, satisfying the following conditions:
\begin{enumerate}
\item $\lim_{N \to \infty} \frac{f(N)}{N} = 0 $, and
\item for every $v$ in $V(G) - S$ there is $k$ such that (1) $v = v_k$ and (2) at the $k$-th turn in the sequence its best strategy for the vertex is $A$ when having started with $A$ for the vertices of $S$ while other vertices have $B$.
\end{enumerate} \vskip 0.15in
Is there an infinite family of graphs $\{ G_n \colon |V(G_n)| = n, n = 1, 2, 3, \cdots \}$ such that for each corresponding finite set $S_n$ making $G_n$ epidemic $f(n)$ is constant? In other words, $f(n) = o(1)$? Also, we can ask how large $f(N)$ could be for an $N$-vertex graph to be epidemic. How is $f(N)$ related to structure of graphs?
\vskip 0.15in
\section*{Acknowledgments}
This work was supported by GIST Research Institute (GRI) grant funded by the GIST in 2018. \vskip 0.05in
\section*{References}
|
{
"timestamp": "2019-03-01T02:09:53",
"yymm": "1902",
"arxiv_id": "1902.10908",
"language": "en",
"url": "https://arxiv.org/abs/1902.10908"
}
|
\section{Introduction and Problem Setup}
\label{sec:intro}
Modern data processing tasks aim to extract information from datasets or signals on graphs -- examples include identification of trends or patterns, learning of dynamics and data structures, or methods for comprehensive awareness of the underlying networks or systems generating the data~\cite{Becker18SPMag,Slavakis14SPMag}. In this domain, the present paper focuses on data processing methods for streams of (possibly high-dimensional) data, with particular emphasis on a setting where underlying computational constraints require one to process data on-the-fly, and with limited access to stored data.
One prominent task is clustering, which is utilized to cluster data points based on well-defined metrics modeling similarities (or distances) among data points, or capturing underlying data structures. For example, spectral clustering groups data points based on minimizing cuts of a similarity graph \cite{Lux}. In this context, this paper focuses on so-called sparse subspace clustering (SSC), a task utilized to build a similarity graph where points in the same subspace are considered ``similar'' \cite{Vidal,Becker}. SSC
uses the data itself as a dictionary,
where data points are represented as linear combinations of other data points in the same subspace. In this paper, we will consider an \emph{online} SSC methodology, where underlying computational considerations prevent one from solving pertinent optimization problems associated with a given set of data~\cite{Vidal} before a new datum arrives.
To outline the problem concretely, consider $N$-dimensional data points $\{x_{t}, t \in \mathbb{N}\}$ sequentially arriving at times $ t\cdot h$, with $h > 0$ a given interval\footnote{\emph{Notation}: hereafter, $(\cdot)^T$ denotes transposition. For a given vector $x \in \mathbb{R}^N$ or matrix $X \in \mathbb{R}^{N\times M}$,
$\|x\|$ and $\|X\|$ refer to a generic norm,
and $\vertiii{X}$ the spectral norm.
If $X_{ij}$ is the $(i,j)$ entry of $X$, then $\|X\|_F^2=\sum_{i,j} X_{ij}^2$ and $\|X\|_1 = \sum_{i,j} |X_{ij}|$.
The composition of two operators is denoted by $\circ$. Write $f(n)=\mathcal{O}(g(n))$ to denote that for sufficiently large $n$, $\exists c>0$ such that $|f(n)| \le c |g(n)|$.
}.
Assume that the data points lie in (or in the neighbourhood of) $S$ \emph{subspaces} $\{\Sc_i\}_{i=1}^S \subset \mathbb R^N$, with $\text{dim}(\Sc_i)=d_i$ for each $i$.
This paper studies the problem of repeatedly applying SSC to all observed data up to a given time, $\{x_{j}, j \le t\}$ with $\bar{X}_t = [x_{1}, \ldots, x_t]$. SSC is a two-step approach \cite{Vidal}: first, step [S1], based on the self-expressiveness property, a sparse representation (SR) problem is solved to identify the (sparse) coefficients $\{\bar{c}_j\}_{j = 1}^t$ so that $x_j = \bar{X}_t \bar{c}_j$ for all $j = 1, \ldots, t$; that is, data points are represented as linear combinations of data points in the same subspace (and we force the $j^\text{th}$ component of $\bar{c}_j$ to be $0$ to exclude the trivial solution). Second, step [S2], apply spectral clustering based on a similarity matrix $W := |\bar{C}_t| + |\bar{C}_t^T|$, where $\bar{C}_t := [\bar{c}_1, \ldots \bar{c}_t]$ and $|\cdot|$ is taken entry-wise. However, in the streaming setting, this setting has two drawbacks:
\begin{enumerate}[wide, labelwidth=!, labelindent=6pt]
\setlength\itemsep{.3em}
\setlength\parskip{0pt}
\setlength\parsep{0pt}
\item[(d2)] The dimensions of $\bar{X}_t$ and $\bar{C}_t$ grow with time, thus increasing the complexity of the associated SR and spectral clustering tasks; and,
\item[(d2)] Due to underlying computational complexity considerations,
steps [S1] and [S2] might not be executed to completion within a time interval $h$ (i.e., before a new datum arrives).
\end{enumerate}
Given (d1)-(d2), we address the problem of developing online algorithmic solutions to carry out steps [S1]-[S2] at each time $t$, based on a \emph{given computational budget}. The first step towards this goal involves the processing of data points using a ``sliding window'' $X_t:=[x_{t-T+1}, \ldots, x_t]\in \mathbb R ^{N\times T}$ of length $T$ (with $T$ determined by the computational budget, as explained later in the paper). Step [S1] is ideally carried out at time $t$ by solving the following SR problem:
\begin{align} \label{eq:srt}
C_t^* \in \argmin_{C\in\mathbb R^{T \times T}}&
\hspace{.3cm} \|C\|_1+\frac{\lambda_e}{2}\|X_t-X_tC\|_{\text{F}}^2 \tag{SR$_t$}\\
\text{s.t.}& \hspace{.3cm} \text{diag}(C)=0 \notag
\end{align}
with $\lambda_e > 0$ a given tuning parameter.
If $\lambda_e$ is too small, in particular if $\lambda_e\le \|\textrm{vec}(X_t^T X_t)\|_{\infty}^{-1}$, then the optimal minimizer $C_t^*$ may have all-zero columns, which is not informative, hence we always choose $\lambda_e$ sufficiently large.
Solving (SR$_t$) \emph{to convergence} within a time interval $h$ might not be possible for a given computational budget, especially for streams of high-dimensional vectors over a large window $T$. Section~\ref{sec:sparse} will address the design of \emph{online} algorithms, for the case where only one algorithmic step can be performed before a new datum $x_t$ arrives (the case of multiple steps follows easily).
With the minimizer (or approximate solution) $C_t^*$, one can compute the matrix $W_t=|C_t^*|+|C_t^*|^T$. Interpreting $W$ as the ``similarity'' matrix of a graph, one can compute the graph Laplacian $L_t = D_t - W_t$ where the degree matrix $D_t$ is a diagonal matrix attained by summing the rows of $W_t$.
The graph Laplacian is then normalized in one of two possible ways: as the symmetric graph Laplacian, $L_{sym,t}=D_t^{-1/2}L_tD_t^{-1/2}$, or as the random walk graph Laplacian, $L_{rw,t} = D_t^{-1} L_t$. We then compute the $S$ trailing eigenvectors of the normalized graph Laplacian and, viewing these eigenvectors as columns, we cluster their rows in $\mathbb R ^S$ into $S$ cluster using the k-means algorithm; see, e.g., \cite{Lux} for details on spectral clustering. This clustering is then applied to the original data points.
Section~\ref{sec:spectral} will elaborate further on this step.
\section{Online Sparse Representation}
\label{sec:sparse}
The proximal gradient descent algorithm and its accelerated version~\cite{Beck}
have rigorous convergence guarantees and can be applied to equations of the form \eqref{eq:srt}, which we now detail.
Let $f_t (C) = \frac{\lambda_e}{2}\|X_t-X_tC\|^2_F$ for brevity, and notice that $\nabla f_t(C)=\lambda_eX_t^T(X_tC-X_t)$ so $\nabla f_t$ is Lipschitz continuous with constant $M_t = \lambda_e\vertiii{X_t^TX_t}$. Let $n$ be the iteration index of the algorithm, and let $\gamma<\frac{2}{M_t}$. Then the (batch) proximal gradient descent algorithm, used to solve~\eqref{eq:srt} at time $t$, involves the following iterations for $n = 1, 2, \dots$ until convergence:
\begin{equation}
\label{eq:prox_batch}
C_{n+1}=\text{prox}_{\gamma\|\cdot\|_1,\text{diag}(\cdot)=0}\circ(I-\gamma\nabla f_t)(C_n)
\end{equation}
where $\text{prox}_{f,\mathcal{X}}(z)=\text{argmin}_{x\in \mathcal{X}}f(x)+\frac{1}{2}\|x-z\|^2$ is the proximal operator defined over a closed convex set $\mathcal{X}$ and for a function $f$. Convergence of~\eqref{eq:prox_batch} to a minimizer $C_t^*$ is shown in, e.g.,~\cite{Combettes}. Furthermore, $f_t(C_n)-f_t(C_t^*)\rightarrow 0$ by the continuity
of $f_t$; see Theorem 10.21 in \cite{Beck} for the rate.
Consider now the case where only one iteration~\eqref{eq:prox_batch} can be performed per time interval $h$ (see Remark 3 for the extension to multiple steps).
Then, an \emph{online} implementation of the proximal gradient descent algorithm involves the sequential execution of the following step at each time $t$:
\begin{equation}
\label{eq:prox_online}
C_{t+1}=\underbrace{\text{prox}_{\gamma_t\|\cdot\|_1,\text{diag}(\cdot)=0}}_{P_t}\circ\underbrace{(I-\gamma_t\nabla f_t)}_{F_t}(C_t)
\vspace{-.9em}
\end{equation}
where the coefficient $\gamma_t$ is selected so that $\gamma_t<\frac{2}{M_t}$.
The difference between \eqref{eq:prox_batch} and \eqref{eq:prox_online} is that $f_t$ changes per iteration in the latter.
The goal is to demonstrate that the online algorithm~\eqref{eq:prox_online} can \emph{track} the sequence of optimizers $\{C_t^*\}$. In the following, the performance of the online algorithm~\eqref{eq:prox_online} is investigated in two cases:
\noindent i) The cost of~\eqref{eq:srt} is \emph{strongly convex} for each $t$. In this case, we will derive bounds for $\|C_{t+1}-C_{t+1}^*\|$, where $C_t^*$ is the \textit{unique} minimizer of~\eqref{eq:srt} for each $t$.
\noindent ii) The cost of~\eqref{eq:srt} is \emph{not} strongly convex. In this case, we will derive dynamic regret bounds. Note that the argmin of ~\eqref{eq:srt} is nonempty, so let $C_t^*$ be in the argmin for each $t$.
Before proceeding, to capture the variability of the clustering solutions, define $\sigma_t = \|C_{t+1}^*-C_t^*\|_F$~\cite{Simonetto}.
As expected, it will be shown that high variability leads to poor tracking performance. The following assumptions are then introduced.
\begin{assumption}
The matrix $X_t^TX_t$ is positive definite for each time $t$ (e.g., $N>T$ and $X_t$ is full rank).
Let $m_t>0$ be the smallest eigenvalue of $X_t^TX_t$.
\end{assumption}
By inspection, the $m_t$ of Assumption 1 is the strong convexity constant of $f_t$.
Based on the assumptions below, the first result is stated next, where $\|\cdot\|$ is taken to be the Frobenius norm.
\begin{theorem}
Under Assumption 1,
\begin{equation}
\forall t \ge 1,
\|C_t-C_t^*\|\leq \Tilde{L}_{t-1}\left(\|C_0-C_0^*\|+\sum_{\tau=0}^{t-1} \frac{\sigma_{\tau}}{\Tilde{L}_{\tau}}\right)
\end{equation}
where $L_t=\max\{|1-\gamma_tm_t|,|1-\gamma_tM_t|\}$, $\Tilde{L}_t=\prod_{\tau=0}^t L_{\tau}$.
\end{theorem}
\begin{proof}
Define $F_t$ and $P_t$ as in \eqref{eq:prox_online},
so $C_{t+1}=(P_t\circ F_t) C_t$. Observe, for any $C,C'\in\mathbb R^{T\times T}$,
\begin{align*}
\|F&_t(C)-F_t(C')\|^2\\
&= \|C-\gamma_t \nabla f_t(C)-C'+\gamma_t \nabla f_t(C')\|^2\\
&\leq (1-2\gamma_t m_t + \gamma_t^2 M_t^2)\|C-C'\|^2\\
&\leq L_t^2\|C-C'\|^2.
\end{align*}
Also, by optimality, $C_t^*$ is a fixed point, so $C_t^* = P_tF_tC_t^*$.
Therefore, one has that $\|C_{t+1}-C_t^*\| = \|P_tF_tC_t-P_tF_tC_t^*\| \leq \|F_tC_t-F_tC_t^*\| \leq L_t\|C_t-C_t^*\|$,
where the second inequality comes from the nonexpansiveness of the prox operator~\cite{Combettes}. Finally, we get
\begin{align*}
\|C_{t+1}-C_{t+1}^*\| &\leq \|C_{t+1}-C_t^*\|+\|C_{t+1}^*-C_t^*\|\\
&\leq L_t\|C_t-C_t^*\|+\sigma_t
\end{align*}
and we apply this inequality recursively.
\end{proof}
\begin{corollary}
Define $\hat{L}_t = \underset{\tau=0,...,t}{\max}L_{\tau}$
and $\hat{\sigma}_t = \underset{\tau=0,...,t}{\max}\sigma_{\tau}$. If Assumptions 1 and 2 hold, then, for each $t$:
\begin{align}
\|C_t-C_t^*\|\leq \left(\hat{L}_{t-1}\right)^t\|C_0-C_0^*\|+\frac{\hat{\sigma}_t}{1-\hat{L}_{t-1}}.
\end{align}
If $m_\tau \ge m$ and $M_\tau\le M$ and $\gamma_\tau$ is chosen $\gamma_\tau = 2/(m_\tau+M_\tau)$ for all $\tau = 0, \ldots, t$, then $\hat{L}_t\le(M-m)/(M+m) < 1$.
\end{corollary}
\begin{proof}
Follows from Thm. 1 and the geometric series.
\end{proof}
These results closely follow the analysis of \cite{BoydPrimer,Simonetto,DallAnese}, applying it to SSC.
\begin{remark} Note that, due to the Lipschitz continuity of $\nabla f_t$, $f_t(C_t)-f_t(C_t^*)\leq\frac{M_t}{2}\|C_t-C_t^*\|^2$.
\end{remark}
In the following, we consider the case where the cost of~\eqref{eq:srt} is \emph{not} strongly convex. It is clear that contractive arguments cannot be utilized in this case since~\eqref{eq:prox_online} is no longer a strongly monotone operator.
Again, we use $\|\cdot\|$ for the Frobenius norm, and $\|f\|_\infty=\sup_{x}\,|f(x)|$.
\begin{theorem}
Let $\hat{C}_t^*\in\argmin_C \underset{\tau=0,...,t-1}{\max}\|C-C_{\tau}^*\|$, $\rho_t(\tau)=\|\hat{C}_t^*-C_{\tau}^*\|$, and $\delta_t=\|f_{t+1}-f_t\|_\infty$. Define $M=\max_t M_t$ and set $\gamma_t=\frac{1}{M}$. Then
\begin{align*}
&\frac{1}{t}\sum_{i=0}^{t-1}\left(f_i(C_i)-f_i(C_i^*)\right)\leq \frac{M}{2t}\left(\|C_0-\hat{C}_t^*\|^2+\sum_{i=0}^{t-1}\rho_t(i)^2\right)\\
&\hspace{2cm}+\frac{1}{t}\left(f_0(C_0)-f_{t-1}(C_t)+\sum_{i=0}^{t-2}\delta_i\right)
\end{align*}
\end{theorem}
\begin{proof}
From the descent lemma \cite[Thm. 2.1.14]{Nesterov}, we have
\begin{align*}
\frac{1}{t}\sum_{i=0}^{t-1}\left(f_i(C_{i+1})-f_i(\hat{C}_t^*)\right)\leq \frac{M}{2t}\|C_0-\hat{C}_t^*\|^2
\end{align*}
By the Lipschitz continuity of $\nabla f_t$,
\begin{align*}
\frac{1}{t}\sum_{i=0}^{t-1}\left(f_t(\hat{C}_t^*)-f_t(C_i^*)\right)\leq \frac{M}{2t}\sum_{i=0}^{t-1}\rho_t(i)^2
\end{align*}
Also,
\begin{align*}
&\frac{1}{t}\sum_{i=0}^{t-1}\left(f_i(C_i)-f_i(C_{i+1})\right) = \frac{1}{t}(f_0(C_0)-f_{t-1}(C_t)\\
&\hspace{3cm}+\sum_{i=0}^{t-2}\left(f_{i+1}(C_{i+1})-f_i(C_{i+1})\right)\\
&\hspace{1cm}\leq \frac{1}{t}\left(f_0(C_0)-f_{t-1}(C_t)+\sum_{i=0}^{t-2}\delta_i\right)
\end{align*}
Putting these together we get the result.
\end{proof}
\begin{corollary}
Define $\hat{\rho}_t=\underset{\tau=0,...,t-1}{\max}\rho_t(\tau)$ and\\ $\hat{\delta}_t=\underset{\tau=0,...,t-2}{\max}\delta_{\tau}$. Then
\begin{align*}
&\frac{1}{t}\sum_{i=0}^{t-1}\left(f_i(C_i)-f_i(C_i^*)\right)\leq\\
&\frac{1}{t}\left(\frac{M}{2}\|C_0-\hat{C}_t^*\|^2+f_0(C_0)-f_{t-1}(C_t)\right)+\frac{M\hat{\rho}_t^2}{2}+\hat{\delta}_t.
\end{align*}
\end{corollary}
\noindent The term $\hat{C}_t^*$ serves as a center point of all the $C^*_\tau$ and is the most meaningful ``best overall'' point. The corollary says that if $\hat{\delta}_t$ and $\hat{\rho}_t$ are well-behaved (i.e., the function changes slowly) then, on average, $f_t(C_t)$ tracks within a constant term of $f_t(C_t^*)$.
\begin{remark}
If $(C_t^*)$ is bounded, then $\hat{\rho}_t$ converges and $\hat{C}_t^*$ has a convergent subsequence. In that case, we can replace them with their limits in the bound in Corollary 2. The bound is only meaningful, though, when the $\delta_t$'s are finite. One way to make them finite is to impose boundedness with respect to the infinity norm as another constraint in~\eqref{eq:srt}. This particular constraint can be incorporated into our closed-form proximal projection \cite[Prop.\ 24.47]{Combettes}.
\end{remark}
\begin{remark} If $f_t$ is not strongly convex (which it will not be if $T>N$), then we can add a Tikhonov term to make it strongly convex. For example, we could add $\frac{\lambda_r}{2}\|C\|^2$. While this will provide the stronger result of Theorem 1, it incurs an error in the minimizer. That is, $\|C_{r,t}-C_t^*\|\leq\|C_{r,t}-C_{r,t}^*\|+\|C_{r,t}^*-C_t^*\|$ where $C_{r,t}$ and $C_{r,t}^*$ are the regularized sequence and optimal regularized sequence respectively.
\end{remark}
\begin{remark} \label{rmk:severalIterations}
If we take more than one iteration per time step, then we can modify $(f_t)$ accordingly in order to use the results in this paper. For example, with 2 iterations per time step, define $\Tilde{f}_t=f_{\floor*{t/2}}$. Alternatively, we can modify the Theorems. For example, if we take $n_t$ iterations at time $t$, then we just have to redefine $\Tilde{L}_t=\prod_{\tau=0}^t L_{\tau}^{n_{\tau}}$ in Theorem 1.
\end{remark}
Note that we did not consider accelerating our algorithm. In the non-strongly convex case, accelerated methods rely on global structure not just local descent. Because of this, it is not obvious that adapting an accelerated method to the online setting would lead to better tracking. However, this is a further research direction that we are currently exploring.
\section{Spectral Clustering}
\label{sec:spectral}
The main factor in determining how many iterations to take in step [S1] is the ratio between the costs of steps [S1] and [S2]. There are papers that explore online spectral clustering \cite{onlinespec}, but the results require relatively small changes in the graph. There are no such guarantees here. For example, when the oldest data point is thrown out and a new one added, all connections to the previous data point are thrown out as well, new connections have to be made for all of the points that were previously connected to the old data point, and connections have to be made for the new data point. While the end result of the spectral clustering may barely change, the change in the graph is catastrophic. Thus,
we do a full batch spectral clustering operation,
and leave a more general framework for online spectral clustering as a future research direction.
Note that the proximal operator in (2) simplifies to $\text{proj}_{\text{diag}(\cdot)=0}\circ \text{prox}_{\gamma_t\|\cdot\|_1}$. The closed form expression for the latter proximal operator is soft-thresholding: $\text{prox}_{\gamma_t\|\cdot\|_1}C = \text{sign}(C)(|C|-\gamma)_+$ component-wise where $(a)_+=\text{max}(a,0)$. Thus, the cost of each iteration of step [S1] is dominated by the gradient descent sub-step, and so the total cost of each step is $\mathcal{O}(NT^2)$ operations. If, instead of applying the proximal gradient operator just once to $C_t$, we apply it $n_t$ times, then the step at time $t$ will cost $\mathcal{O}(n_tNT^2)$ operations.
The cost of step [S2] depends on what method we use to compute the specified eigenvectors.
An upper bound on the cost is $\mathcal{O}(T^3)$ which can be achieved by classical dense algorithms and gives \emph{all} $T$ eigenvalues $\lambda$. To find $S \ll T$ eigenvalues approximately, the power method or Lanczos iterative methods can be used \cite{NMMC}.
First consider that $L_{rw}v=\lambda v$ iff $v-D^{-1}Wv=\lambda v$ iff $D^{-1}Wv=(1-\lambda)v$. Also, by the Gershgorin disc theorem, the eigenvalues of $L_{rw}$ are in [0,2] and the eigenvalues of $D^{-1}W$ are in [-1,1]. Thus, for $D^{-1}W$, we want the eigenvectors corresponding to the eigenvalues near 1. The leading eigenpairs of $D^{-1}W$ should correspond to the trailing eigenpairs of $L_{rw}$ as long as the trailing eigenvalues of $L_{rw}$ are closer to 0 than the leading eigenvalues are to 1. If this is not the case, then we have to use the power method to compute \textit{more} than S eigenpairs, so that we can take the eigenvectors corresponding to the S largest \textit{positive} eigenvalues. The power method costs $\mathcal{O}(nnz\cdot niter\cdot S)$ in the ideal case where we do not have to compute extra eigenpairs. Here, $nnz$ denotes the number of nonzero elements and $niter$ denotes the number of iterations to reach convergence.
There are efficient methods for computing the trailing eigenpairs of $L_{rw}$ directly. In particular, \cite{3methods} found that the Jacobi-Davidson method was superior to the Lanczos method, in terms of computation time, for spectral clustering. Both methods cost $\mathcal{O}(nnz\cdot niter)$, but $niter \approx S$ can vary.
The number of nonzeros, in the ideal case, can be estimated by the subspace dimensions, $d_i$. For a given subspace, the minimum number of points needed to represent another point, as long as it isn't cohyperplanar with a strict subset of the points, is exactly the dimension of the subspace. Thus, we can say that $nnz(C^*)\geq \mathcal{O}(\sum_i d_i T_i^t)$ where $T_i^t$ is the number of data points in $\Sc_i$ at time $t$. The same can be said of $W$, $L$, and $L_{rw}$.
In the case where each subspace has the same dimension, $d$, and the same number of points in it, $T/S$, then $nnz(C^*) \ge \mathcal{O}(dT)$ (and recall $d\le N$).
This means in the best case, [S2] costs $\mathcal{O}(dTS)$, while a single step of [S1] costs $\mathcal{O}(NT^2)$ so this suggests choosing $T \approx dS/N$ to balance the costs of the two steps; if $T$ is smaller than this, multiple steps to solve [S1] can be taken.
\section{Numerical Results}
\label{sec:numerical}
We performed tests on both synthetic data and the Yale Face Database \cite{Yale}. The synthetic data was composed of $S=10$ 5-dimensional subspaces in $R^{50}$ ($d=5$), each with 50 points. Noise was added to make the simulation more realistic. The sliding window had a capacity of $T=400$ data points and we took 100 time steps.
The Yale Face Database has $S=38$ subspaces, and we
let the sliding window have a capacity of $T=500$ points and took 200 time steps. The points in the Yale Face Database are in $R^{50}$. For both datasets, we took 50 iterations of the optimization algorithm per time step (cf.\ Remark~\ref{rmk:severalIterations}).
For the synthetic data, $T>N$, so the cost function is not strongly convex. On the other hand, for the Yale Face Database, $T<N$, so the cost function \textit{is} strongly convex. The numerical results show that for both datasets, though, the objective value trajectory converges to a region above the minimum trajectory. This can be seen in Figure 1.
\begin{figure}[htb]
\begin{minipage}[b]{.48\linewidth}
\centering
\centerline{\includegraphics[width=4.0cm]{figs/SynthObj.pdf}}
\centerline{(a) Synthetic dataset}\medskip
\end{minipage}
\hfill
\begin{minipage}[b]{0.48\linewidth}
\centering
\centerline{\includegraphics[width=4.0cm]{figs/YaleObj.pdf}}
\centerline{(b) Yale Face dataset}\medskip
\end{minipage}
\caption{Objective value of tracking sequence and actual time-varying minimum.}
\label{fig:res}
\end{figure}
The objective error seems to be driving whether or not the clustering error converges. Figure 2 shows the clustering error.
\begin{figure}[htb]
\begin{minipage}[b]{.48\linewidth}
\centering
\centerline{\includegraphics[width=4.0cm]{figs/SynthClust.pdf}}
\centerline{(a) Synthetic dataset}\medskip
\end{minipage}
\hfill
\begin{minipage}[b]{0.48\linewidth}
\centering
\centerline{\includegraphics[width=4.0cm]{figs/YaleClust.pdf}}
\centerline{(b) Yale Face dataset}\medskip
\end{minipage}
\caption{Clustering error of both the tracking sequence and optimal sequence.}
\label{fig:res}
\end{figure}
For both datasets, the clustering error of the tracking algorithm decreases. However, decreasing the number of iterations per time step too much causes the clustering error to no longer decrease. This suggests that there is some maximum value of the objective error for the clustering error to decrease.
Finally, for the synthetic dataset, step [S2] took 50 times as long as step [S1]. For the Yale Face dataset, step [S2] took 3 times as long as step [S1]. While we took a sufficient number of iterations in order for the clustering error of our algorithm to be small, for an actual system, its dynamics would dictate the number of iterations. The spectral clustering time would be subtracted from the length of a time step, and this value would be divided by the time for step [S1].
\bibliographystyle{IEEEbib}
|
{
"timestamp": "2019-03-01T02:05:59",
"yymm": "1902",
"arxiv_id": "1902.10842",
"language": "en",
"url": "https://arxiv.org/abs/1902.10842"
}
|
\section{Introduction}
Autonomous agents, such as self-driving cars and drones, need to make
decisions in real time, which is particularly important but difficult
in critical situations for example to avoid collisions. Such
decisions often need to be made in a sequential manner to achieve the
eventual goal ({\it e.g.}, avoiding collisions and recovering to
safe conditions), under partially observable environment, and by
taking into account how other agents behave. Towards this
far-reaching goal of realizing such autonomous agents, we propose
practical techniques of sequential decision making in real time and
demonstrate their effectiveness in Pommerman, a multi-agent
environment that has been used in one of the competitions held at the
Thirty-second Conference on Neural Information Processing Systems
(NeurIPS 2018) on Dec.~8, 2018 \cite{pommerman}. The techniques that
we propose in this paper have been used in the Pommerman agents
(HakozakiJunctions and dypm-final) who have won the first and third
places in the competition.
In Pommerman,
a team of two agents competes against another team of two agents on a
board of $11\times 11$ grids (see Figure~\ref{fig:pommerman}~(a) for
an initial configuration of the board). Each agent can observe only a
limited area of the board, and the agents cannot communicate with each
other. The goal of a team is to knock down all of the opponents.
Towards this goal, the agents
place bombs to destroy wooden walls and collect power-up items that
might appear from those wooden walls, while avoiding flames and
attacking opponents. See Figure~\ref{fig:pommerman}~(b) for an example
of the board in the middle of the game. See \citet{pommerman} and the
GitHub repository\footnote{
https://github.com/MultiAgentLearning/playground } for details of
Pommerman.
\begin{figure}[t]
\begin{minipage}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{pommerman0.png}\\
(a) Initial board
\end{minipage}
\begin{minipage}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{pommerman100.png}\\
(b) Board after 100 steps
\end{minipage}
\caption{An initial board (a) and a board after 100 steps in
Pommerman. The four small windows on the right most column
respectively denote the areas that the four agents can observe.}
\label{fig:pommerman}
\end{figure}
Although Pommerman has been developed recently (initial GitHub commit
was Dec.~25, 2017), it has been gaining much attention as a benchmark
of multi-agent study in the field of planning, game theory, and
reinforcement learning \cite{MARLsurvey2018a,MARLsurvey2018b}. Prior
to the one at NeurIPS 2018, the first competition was held on Jun.~3,
2018. The winning agent sets an intermediate goal with heuristics and
performs depth-limited tree search to achieve that intermediate goal
\cite{Zhou18}. \citet{backplay} propose a technique of imitation
learning with curriculum and show that, by using the behavioral data
of the winning agent, it can train an agent as strong as the winning
agent.
While the community has made progress in developing strong agents for
Pommerman, the level of those agents is not yet comparable to what has
been achieved for backgammon \cite{backgammon}, Chess \cite{DeepBlue},
Atari video games \cite{DQN}, Poker \cite{Bowling15,BroSan17}, and Go
\cite{AlphaZero}. Pommerman has its own difficulty that prohibits
effective applications of existing approaches that have seen success
in other games.
What makes Pommerman difficult is the constraint on real-time decision
making ({\it i.e.}, an agent needs to choose an action in 100
milliseconds). This constraint significantly limits the applicability
of Monte Carlo Tree Search, which would otherwise be a reasonable
approach to Pommerman \cite{matiisen2018pommerman}. In Pommerman, the
branching factor at each step can be as large as
$6^4=1,296$, because four agents take actions simultaneously in each
step, and there are six possible actions for each agent. The agents
should plan ahead and choose actions by taking into account the
explosion of bombs, whose lifetime is 10 steps. Tree search with
insufficient depth (less than 10) would ignore the explosion of bombs,
which in turn would make the agents easily caught up in flames. Tree
search with sufficient depth (at least 10) is practically infeasible
with the large branching factor. Other difficulties of
Pommerman include the following. Reward is only given at the end of
an episode, which can be as long as 800 steps. The agents can observe
only a limited part of the board, and some of the key information
cannot be directly observed. The agent needs to coordinate with its
teammate without explicit communication.
Here, we propose a practical approach to real-time tree search that
allows us to take into account critical events far ahead in the
future. In our approach, tree search after a specified depth is
performed under the assumption of a deterministic and pessimistic
scenario ({\it i.e.}, sequence of states). Because the scenario is
deterministic, there is no branching after the specified depth, which
allows us to perform the tree search with sufficient depth to take
into account the critical events far ahead in the future. This
deterministic scenario is designed to be pessimistic by allowing
multiple unfavorable events can happen ({\it e.g.}, by letting opponents
take multiple actions) simultaneously in a nondeterministic manner.
Hence, our pessimistic scenarios are unrealistic in general. Our key
idea is that an unrealistic scenario can capture critical events in
the future better than a small number of realistic scenarios that can
be sampled and explored under the constraint of real-time tree search.
We adjust the level of pessimism via self-play to achieve the best
overall performance.
Our approach is proposed particularly for Pommerman but might be
generally applicable to other domains that require sequential decision
making in real time. We demonstrate the flexibility of the proposed
approach by instantiating it as two variants of Pommerman agents, who
need to deal with the complex environment that involves multiple
agents and partial observability. The effectiveness of the proposed
approach is shown with Pommerman. The new approach of real-time tree
search with deterministic and pessimistic scenarios and its
application to Pommerman constitute the contributions of this paper.
\section{Related Work}
There has been a significant amount work on the techniques for
real-time tree search particularly for real-time (strategy) games. As
we will discuss it in the following, however, the focus of the prior work is on
the techniques for reducing the search space or guiding the search
towards the most relevant subspace. The novelty in our approach is in
synthesizing the deterministic and pessimistic scenarios.
The prior work has investigated various techniques to make Monte Carlo
Tree Search (MCTS) applicable to real-time games such as Ms.\ Pac-Man
\cite{MCTS-pacman,IkeIto11}, StarCraft \cite{MCTS-StarCraft}, Wargus
\cite{UCT-RTS}, Physical Traveling Salesman Problem
\cite{MCTS-TSP}, Quantified Constraint Satisfaction Problem
\cite{MCTS-QCSP}, and $\mu$RTS \cite{Puppet2,Mar19}. An example of a
recent work in this line is \citet{Mar19}, who study a technique of
action abstraction and apply it to MCTS among others to reduce the
search space. \citet{Puppet2} study a technique of using
non-deterministic rules to reduce the branching in MCTS. In all of
such prior work, MCTS is performed only with realistic or legal moves.
\citet{MTCS-Atari} study the approach of using MCTS to generate
training data for learning a deep neural network that approximates a
policy or a value function, and the effectiveness of the proposed
approach is demonstrated in Atari games. This approach is motivated
by the observation that tree search (planning-based approaches) can
perform far better than model-free approaches if it were not for the
real-time constraint.
Real-time tree search has also been studied for deterministic
settings. Here, the search tree is expanded on the basis of heuristic
values as long as time permits. Similar to real-time MCTS, key
questions are where to expand and what actions to take given the
search tree investigated. For example, \citet{Mit19} propose a
risk-sensitive approach to these questions.
Our approach of synthesizing a pessimistic scenario is also related to
the null-move heuristic, which has been studied particularly for Chess
\cite{nullmove}, in that it considers a scenario with illegal moves.
The null-move heuristic assumes that a player skips a move, which is
illegal in chess, to estimate a lower bound of the value of the best
move. The lower bound is then used to prune the search space. In
contrast, our approach can assume that a player takes multiple moves in
the pessimistic scenario.
\section{Real-time tree search for Pommerman}
\label{sec:tree}
In Pommerman, the dynamics of the environment is known, and much of
the uncertainties resulting from partial observability can be resolved
with careful analysis of historical observations. MCTS would thus be
a competitive approach if it were not for the real-time constraint.
For example, consider a situation where an agent can survive only by
following a particular route. Tree search is particularly suitable
for finding such a route, while model-free approaches of learning
policies or value functions, if not impossible, would require large scale
functional approximators ({\it e.g.}, deep neural networks)
and a large amount of data for training to be able to follow that
route. The applicability of MCTS or tree search in general is however
significantly limited in Pommerman due to the real-time constraint and
the large branching factor.
One approach of tree search is to push the depth as far as possible,
and this is the approach taken by the gorogm\_eisenach ({\tt
eisenach}) agents, who won the second place in the NeurIPS 2018
Pommerman competition. The {\tt eisenach} agent was implemented in
C++ with various engineering tricks to achieve the average depth of 2
in the tree search \cite{PommermanNeurIPS2018}.
\subsection{Tree search with pessimistic scenarios}
In our approach, the tree search after a specified depth is performed
under a deterministic and pessimistic scenario, which will be simply
referred to as a deterministic scenario or a pessimistic scenario in
the following. Figure~\ref{fig:tree} shows an example. Here, the
tree search is performed in a standard manner until the depth of 2.
In this example, the branching factor is 2, and there are 4 nodes at
the depth of 2. From each of these 4 nodes, ``tree search'' is
continued until it reaches the depth of 5 by assuming a deterministic
scenario. Because the scenario is deterministic, there are no
branches after the depth of 2.
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{tree.pdf}
\caption{Tree search with deterministic and pessimistic scenarios.}
\label{fig:tree}
\end{figure}
One may also interpret our approach as evaluating the 4 leaves (at
the depth of 2) on the basis of the deterministic scenario from the
depth of 2 to the depth of 5, and our following discussion will be
based on this interpretation. Our approach keeps the size of the
search tree small, because there are branches only until a limited
depth. At the same time, our approach can take into account far ahead
in the future, because the leaves (at the limited depth) can be
evaluated with a deterministic scenario that can be much longer than
what would be possible with branches. What differs from the rollout
in MCTS is that we let the deterministic scenario be pessimistic, as we will discuss in Section~\ref{sec:tree:pessimistic}.
More specifically, in the Pommerman agents that implement the proposed
approach ({\it i.e.}, HakozakiJunctions ({\tt hakozaki}) and
dypm-final ({\tt dypm})), the depth of tree search is limited to one.
The {\tt hakozaki} agent considers all of the leaves at depth 1 but
might need to choose an action before exhaustively searching all of
the leaves due to timeout. On the other hand, the {\tt dypm} agent
considers six leaves at depth 1 by taking into account only its own
actions, and the effect of the the actions by the other agents are
taken into account in evaluation with the deterministic scenario. In
both of {\tt hakozaki} and {\tt dypm}, the leaves are evaluated with a
deterministic scenario with the length of at least 10 to take into
account the explosion of bombs, whose lifetime is 10. Recall that the
{\tt eisenach} agent can perform the tree search only at the average
depth of 2. Hence, if we performed the standard tree search for 2
steps, there would be no computational budget left for the evaluation
with deterministic scenarios.
\subsection{Generating pessimistic scenarios}
\label{sec:tree:pessimistic}
From each of the leaves in the search tree, we generate a
deterministic scenario. The deterministic scenario is then used to
estimate the value of each leaf, which in turn is used to determine
the next best action of the agent. A key idea in our approach is to
make this deterministic scenario be pessimistic, which we will discuss
in this section with specific instantiation as Pommerman agents.
A pessimistic scenario can be generated in a systematic manner as
follows. We assume that the state of the environment can be
represented by the positions of objects. In Pommerman, these objects
are agents, bombs, flames, powerup items, and walls. Some of those
objects change their positions randomly or by depending on the actions
of the agents, which forces tree search to have branches. If one can
tell the worst sequence of the positions of an object among all of the
possibilities, one can place and move that object accordingly in the
pessimistic scenario. It is often the case, however, that the worst
positions are unknown. Instead, {\em we generate a pessimistic
scenario by allowing the objects to be located at multiple
positions} even if that is unrealistic or illegal. In Pommerman,
this typically corresponds to assuming that an opponent takes multiple
actions simultaneously in a nondeterministic manner, which means that
the opponent is copied into multiple positions in the next step.
Notice that such a pessimistic scenario can be more adversarial than
assuming the worst possible scenario, because an object cannot
actually be at multiple positions.
What is essential is that a pessimistic scenario is deterministic to
avoid the computational complexity resulting from branching in the
search tree. Our guideline is to make this deterministic scenario
rather pessimistic, because good actions are often the ones that
perform well under pessimistic scenarios particularly in cases where safety is a
primary concern. In Pommerman, an agent dies if it cannot escape from
flames, and our team loses if both of our agents die. It is thus of
critical importance to ensure that our agents can survive,
while attacking opponents or collecting powerup
items.
In Pommerman, a deterministic scenario is represented by a sequence of
boards, where each board is given by the state of the game at a
certain point in time. Such a deterministic scenario can be generated
by a forward model of Pommerman after resolving uncertainties. There
are two sources of uncertainties: the future actions of agents and
partial observability. These uncertainties can be resolved in
arbitrary ways, but our guideline is to resolve them in rather
pessimistic manners and to optimize the level of pessimism by tuning
hyperparameters via self-play. A caveat is that the purpose of a
pessimistic scenario is not to find a proper lower bound on the value
of a leaf in the search tree but to find a good action as a result of
evaluating that leaf with the pessimistic scenario. Therefore, a
pessimistic scenario can be more adversarial or less adversarial than
the worst scenario. The self-play allows us to optimize the level of
pessimism.
More specifically, both of {\tt hakozaki} and {\tt dypm} agents
generate a sequence of boards by letting the other agents move to
multiple positions simultaneously. They then records the time when
each position is first occupied by an agent. There are differences
between {\tt hakozaki} and {\tt dypm} in exactly what information is
recorded in the sequence of boards. In {\tt hakozaki}, each position
in the $t$-th board has the information about when that position was
occupied, if the position has been occupied by the $t$-th step. In
{\tt dypm}, each position in the $t$-th board has the information
about whether that position has been occupied by the $t$-th step.
Also, {\tt dypm} assumes that the other agents take actions only for a
predetermined number of steps (a hyperparameter tuned via self-play),
while {\tt hakozaki} does not have this limit.
Note that the sequence of such boards is in general illegal or
unrealistic. There may be multiple copies of an agent in the board
({\tt dypm}), and an agent that might occupy a position may be
replaced by the integer value representing when that position can be
occupied ({\tt hakozaki}). Also, some of the uncertainties are
resolved in a way that is not necessarily pessimistic. For example,
we ignore the possibility that an agent might kick bombs in the
sequence of boards. However, the purpose of the sequence of boards is
not to compute a proper lower bound of the value but to quickly
estimate the relative values of the leaves in the search tree in a way
that it gives the overall best performance. Note that the part of
tree search (with branches) can take into account all of
the details, unlike the part of evaluation with a deterministic
scenario. For example, if the action of an agent within the tree
search is to kick a bomb (by moving to the bomb), the movement of the
kicked bomb is taken into account in the remaining tree search as well
as in the deterministic scenario. Our approach of tree search with a
pessimistic scenario allows us to take into account all of the details
in the near future (via tree search) as well as possibly critical
events in distant future (via a pessimistic scenario).
\subsection{Evaluation with pessimistic scenarios}
This sequence of boards is then used to estimate the value of the
initial board in the sequence ({\it i.e.}, one of the leaves of the
search tree). The value needs to be defined in a way that choosing
actions that give high value tends to eventually achieving the goal.
The goal of a Pommerman agent is to knock down all of the opponents,
while the agent or its teammate is surviving. The value should thus
reflect some notion of the survivability of the agent itself, its
teammate, and its opponents. Roughly speaking, high value should
imply high survivability of the agent itself and its teammate, low
survivability of the opponents, or both.
In Pommerman, the survivability of an agent can be captured by the
number of positions that the agent can stay safely in the sequence of
boards. More specifically, given a deterministic scenario, {\tt dypm}
counts the number of the time-position pairs from which an agent can
survive at least until the end of the scenario. This number is used
as the survivability of the agent. Namely, the survivability of the
agent is computed by first searching the reachable time-position pairs
in the sequence of boards and then pruning those pairs from which one
cannot survive until the end of the sequence. On the other hand, {\tt
hakozaki} finds the positions that an agent can reach at the end of
the sequence of boards, and compute the survivability on the basis of
the integer values that represent when the positions might be occupied
by other agents. Intuitively, an agent is considered to have high
survivability if there are many positions that the agent can reach
without contacting the other agents.
Note that an agent $i$ computes the survivability $S(j,s)$ for each
leaf (state) $s$ and for each agent $j$ who is visible from $i$,
including $i$ itself. The survivability of an agent $j\neq i$ is
computed on the basis of the sequence of boards that is pessimistic to
$j$ ({\it i.e.}, the agents except $j$ move to multiple positions
simultaneously)\footnote{In {\tt dypm}, to save computational cost, a
single sequence of boards with no move of agents is used to compute
the survivabilities of agents $j\neq i$, and those survivabilities
are normalized by dividing them by the corresponding survivabilities
when the agent $i$ does not exist.}.
Now, one can choose the best action on the basis of these
survivabilities. Roughly speaking, our agent chooses the action that
maximizes the product the survivabilities of the agent
itself\footnote{To be more aggressive, the {\tt dypm} agent clips its
own survivability $S$ at a threshold $S_{\rm th}$ when $S$ exceeds
$S_{\rm th}$, where $S_{\rm th}$ is tuned via self-play. } and its
teammate divided by the product of the survivabilities of the
opponents. Because the survivability is defined for each leaf, which
corresponds to a combination of the actions of all agents, one needs
to marginalize out the actions of the other agents to define the
survivability of an agent with a particular action. The survivability
of an agent when that agent takes a particular action can be defined
to be the minimum survivability of that agent given that the agent
takes that action ({\it i.e.}, worst case). The survivability of a
teammate can also be defined as the minimum survivability. On the
other hand, we find that the survivability of an opponent with that
action should be defined as the average survivability rather than
minimum or maximum. These are how the survivabilities with a
particular action are defined in {\tt hakozaki}. On the other hand,
each leaf in th search tree of a {\tt dypm} agent corresponds to each
action of that agent, and there is no need for maginalization. A
caveat is that the action by the {\tt dypm} agent might be blocked by
other agent, resulting in no move. The survivability with such an
action is thus averaged with the survivability of no move.
\makeatletter
In this section, we have discussed how our agents choose actions in
most critical situations of Pommerman, where the agents interacts with
other agents. In those situations, the goal of an agent is to reduce
the survivabilities of the opponents, while keeping the
survivabilities of the agent itself or its teammate sufficiently high.
In other situations, the agent can ignore the behavior of other agents
and seek to attain intermediate objectives. Examples of such
objectives include (\@roman{1}) breaking wooden walls to make passages
or to find powerup items, (\@roman{2}) collecting powerup items, and
(\@roman{3}) moving towards the areas that cannot been observed to
obtain new information. Although there is a question of what
objective to pursue at each step, it is relatively easy to choose
actions once an objective is given, because we do not need to take
into account the actions of the other agents. Our agent heuristically
sets an objective and chooses actions by the use of a standard search
technique until the agent meets and starts interacting with other
agents.
\makeatother
\section{Experiments}
We conduct two sets of experiments to investigate the overall
performance of the proposed approach and the effectiveness of our key
idea ({\it i.e.}, the use of pessimistic scenarios in tree search).
Although the agents that implement our proposed approach have won the
first and third places in the NeurIPS 2018 Pommerman competition, the
number of matches in the competition is too limited to draw
conclusions. In the first set of experiments, we intensively evaluate
the performance among the top five teams, from the competition, that
implement state-of-the-art approaches, including ours. In the second
set of experiments, we study the effectiveness of pessimism by
changing one of the key hyperparameters of {\tt dypm} that controls
the level of pessimism in the proposed approach.
\subsection{Performance against state-of-the-art agents}
The competition was run according to a double
elimination style with two brackets, where the team that won two games
before the other moves on to the next round. A tie was replayed for
the first time, but another tie was resolved by matches on another
version of Pommerman, where walls can collapse. Namely, tie was not
the same as ``no game'' in the competition, and the results in this
section needs to be interpreted accordingly. See
\citet{PommermanNeurIPS2018} for more details about the settings of
the competition.
The top five teams were {\tt hakozaki}, {\tt eisenach}, {\tt dypm},
navocado ({\tt navocado}), and
nn\_team\_skynet955\_skynet955 ({\tt skynet}).
The top three teams are based on tree search, as we have discussed in Section~\ref{sec:tree}.
The other two are based on reinforcement learning. More specifically, {\tt navocado}
is based on advantage-actor-critic \cite{Navocado},
and {\tt skynet} is based on proximal policy optimization.
Table~\ref{tbl:neurips} shows the results of the direct matches among
the top five teams in the competition. For example, {\tt eisenach}
defeated {\tt dypm} four times, {\tt dypm} defeated {\tt eisenach}
once, and there were no ties between these two teams. Although the
winners of the competition were determined according to the rule of the
competition, the number of matches was clearly
limited to statistically determine which teams are stronger than
others. In particular, there were pairs of teams that had no direct
matches in the competition.
\begin{table}[tbh]
\centering
{\tt \scriptsize
\begin{tabular}{@{}rccccc@{}}
\toprule
& \hspace{-0.5mm}hakozaki$^\star$ & \hspace{-0.5mm}eisenach & \hspace{-0.5mm}dypm$^\star$ & \hspace{-0.5mm}navocado & \hspace{-0.5mm}skynet \\
\midrule
hakozaki$^\star$ & - & 4/2/1 & - & 2/0/2 & 2/0/0 \\
eisenach & 2/4/1 & - & 4/1/0 & - & - \\
dypm$^\star$ & - & 1/4/0 & - & 2/1/0 & - \\
navocado & 0/2/2 & - & 1/2/0 & - & 2/1/5 \\
skynet & 0/2/0 & - & - & 1/2/5 & - \\
\bottomrule\\
\end{tabular}
}
\caption{A summary of the matches among the top five teams in the
competition. Each entry shows the number of
``wins / losses / ties'' for a row agent against a column agent.
The $\star$ marks indicate the teams that implement our
approach.}
\label{tbl:neurips}
\end{table}
\begin{table*}[tbh]
\centering
{\tt \small
\begin{tabular}{@{}rrrrrrrrrrrrrrrr@{}}
\toprule
& hakozaki$^\star$ & eisenach & dypm$^\star$ & navocado & skynet & TOTAL \\
\midrule
hakozaki$^\star$ & 29/ 29/142 & 89/ 26/85 & 59/ 42/ 99 & 129/~~0/ 71 & 178/~~0/ 22 & 484/ 97/419\\
eisenach & 26/ 89/ 85 & 68/ 68/64 & 82/ 97/ 21 & 176/ 13/ 11 & 189/~~6/~~5 & 541/273/186\\
dypm$^\star$ & 42/ 59/ 99 & 97/ 82/21 & 22/ 22/156 & 194/~~0/~~6 & 197/~~0/~~3 & 552/163/285\\
navocado & 0/129/ 71 & 13/176/11 & 0/194/~~6 & 23/ 23/154 & 33/ 10/157 & 69/532/399\\
skynet & 0/178/ 22 & 6/189/ 5 & 0/197/~~3 & 10/ 33/157 & 15/ 15/170 & 31/612/357\\
\bottomrule\\
\end{tabular}
}
\caption{ A summary of the 200 matches between each pair among the
top five teams. Each entry shows the number of ``wins / losses /
ties'' for a row agent against a column agent. The rightmost
column shows the total number of ``wins / losses / ties'' for row
agents. The $\star$ mark indicates the team that implements the
proposed approach.}
\label{tbl:exp}
\end{table*}
The purpose of the following experiments is to complement the
competition by running a greater number of matches
between the top five teams. We use the Docker images\footnote{The
Docker images are available as
multiagentlearning/\{hakozakijunctions, eisenach, dypm.1, dypm.2,
navocado, skynet955\}. Note that the {\tt dypm} team uses two
agents, dypm.1 and dypm.2, that differ only in the values of their
hyperparameters. In the other teams, the two agents are identical.}
of the agents that have been used in the
competition. We run our experiments on a Ubuntu 18.04 machine having
eight Intel Core i7-6700K CPUs running at 4.00~GHz and 64~GB random
access memory. Note that these computational resources are
different from what has been used at the competition (exact
computational resources used at the competition are not known).
Therefore, the results of our experiments should not be considered as
a refinement but rather as complementary to the results from the
competition.
Table~\ref{tbl:exp} summarizes the results of the 200 matches that we
have run between each pair of the teams. Because there are two
essentially different configurations for the initial positions of the
two teams, half of the matches are initialized with one configuration,
and the other with the other configuration. Each team is also matched
against itself for 100 matches, and the results for both sides are
counted in the table (if a match is tied, two ties are counted). In
total, we run 2,500 matches, which take approximately three days in
our environment.
In our experiments, the top three teams ({\tt hakozaki}, {\tt
eisenach}, {\tt dypm}) have completely dominated the other two ({\tt
navocado}, {\tt skynet}), recording 1,063 wins (88.6~\%), 19 losses
(1.6~\%), and 118 ties (9.8~\%). In particular, {\tt hakozaki} and
{\tt dypm}, who implement our proposed approach, have never lost
against {\tt navocado} or {\tt skynet}. While the top three teams are
based on tree search, the other two are based on model-free
reinforcement learning. This indicates the advantages of tree search
in Pommerman, where precise planning in the following several steps is
critically important to survive from the explosion of bombs. The
results in our experiments are not necessarily consistent with those
in the competition (Table~\ref{tbl:neurips}), however. In particular,
{\tt dypm} has lost once against {\tt navocado} in the competition.
This may be because the {\tt dypm} agents have occasionally
experienced timeouts ({\it i.e.}, the agent does not respond in 100
milliseconds, which is treated as a ``stop'' action) in the
competition, because {\tt dypm} does not implement any mechanisms for
measuring the elapsed time. Also, the computational resources in our
experiments might be less favorable to the learning agents ({\tt
navocado} and {\tt skynet}) than what has been used in the
competition.
Among the top three teams, {\tt hakozaki} has consistently dominated
the others, although the relative advantages between {\tt hakozaki} and {\tt dypm}
are relatively small. Also, {\tt dypm} has slightly outperformed {\tt
eisenach}. Overall, {\tt hakozaki} and {\tt dypm}, who implement
the proposed approach, have recorded 186 wins (46.5~\%), 108 losses
(27.0~\%), and 106 ties (26.5~\%) against {\tt eisenach}. Note that
these top three teams implement their agents and forward models in
different programming languages: {\tt hakozaki} in Java\texttrademark, {\tt
eisenach} in C++, and {\tt dypm} in Python. Also, the {\tt dypm}
agent runs on a single thread, while {\tt hakozaki} and {\tt eisenach}
use multiple threads. Overall, our experimental results suggest the
superiority of the proposed approach in real-time tree search for
sequential decision making with limited computational resources.
\subsection{Effectiveness of pessimism}
We next study the effectiveness of the pessimism in the deterministic
scenario used in our proposed approach. Recall that a {\tt dypm}
agent generates a deterministic scenario by assuming that the other
agents take multiple actions simultaneously in a nondeterministic
manner but only for a limited number of steps. Here, we refer to this
limited number of steps as the pessimism level and study how the
performance of {\tt dypm} depends on the pessimism level.
Specifically, for each pessimism level, we run 1,000 matches
against a baseline and record the number of wins, losses, and times.
The baseline is either a team of default agents ({\tt
SimpleAgent}) or a team of the {\tt dypm} agents whose
pessimism level is set 0.
Figure~\ref{fig:pessimism} shows the number of wins, losses, and ties
of {\tt dypm} against each baseline, where the pessimism level in {\tt
dypm} is varied as indicated along the horizontal axis. For example,
{\tt dypm} with the pessimism level 3 has had 997 wins, 1 loss, and 2
ties against {\tt SimpleAgent} (Figure~\ref{fig:pessimism}~(a)), and
this pessimism level is the one that has actually been set in {\tt
dypm} in the competition. The winning rate of {\tt dypm} against
{\tt SimpleAgent} increases from 92.6~\% to 99.7~\% by increasing the
pessimism level from 0 to 3. Further increasing the pessimism level
can reduce the number of losses but increases the number of ties, but
these changes are insignificant with the very high winning rate.
\begin{figure}[tbh]
\begin{minipage}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{vs_simple.pdf}\\
(a) vs. {\tt SimpleAgent}
\end{minipage}
\begin{minipage}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{vs_zero.pdf}\\
(b) vs. no pessimism
\end{minipage}
\caption{The performance of {\tt dypm} with varying pessimism
levels. The rate of wins, losses, and ties of {\tt dypm} against
a baseline is shown for each pessimism level (from 0 to 10) as
indicated along the horizontal axis. The baseline is {\tt
SimpleAgent} in (a) and the {\tt dypm} with no pessimism (level
0) in (b).}
\label{fig:pessimism}
\end{figure}
To clarify these changes, Figure~\ref{fig:pessimism}~(b) shows the
results against a stronger baseline, which is {\tt dypm} with
pessimism level 0. Now, the winning rate of {\tt dypm} against the
baseline increases from 36.9~\% to 77.8~\% by increasing the pessimism
level from 0 to 3. Further increasing the pessimism level from 3 to 4
can reduce the rate of losses from 5.1~\% to 4.2~\% but increases the
rate of ties from 17.1~\% to 22.7~\%.
Overall, we find that pessimism in deterministic scenarios can
significantly improve the overall performance of Pommerman agents. Also,
the performance is sensitive to the particular level of pessimism, and
an appropriate level of pessimism may be determined via self-play.
Note, however, that Pommerman is an extensive-form game with imperfect
information, and the optimal strategy in general is to
probabilistically mix multiple levels of pessimism. Also, {\tt dypm}
has other hyperparameters, and the optimal level of pessimism depends
on the values of the other hyperparameters.
\section{Conclusion}
We have proposed an approach of real-time tree search, where tree
search is performed only with a limited depth, and the leaves are
evaluated under a deterministic and pessimistic scenario. Because
there is no branching with a deterministic scenario, our evaluation
can take into account far ahead in the future. Also, evaluation with
pessimistic scenarios can lead to selecting good actions, which are
often the ones that perform well under the pessimistic scenarios
particularly in cases where safety is a primary concern. We have
assumed that the state can be represented by the positions of objects
and generated pessimistic scenarios by allowing the objects to be
located at multiple positions even if that may be unrealistic or
illegal. One could, however, apply the general idea of our real-time
tree search with pessimistic scenarios to a broader range of domains
by designing pessimistic scenarios suitable for particular domains.
For example, a pessimistic scenario may be generated by assuming
objects that increases the size over time for domains on continuous
space.
Our experiments with Pommerman suggest that the performance of the
proposed approach is sensitive to the particular level of pessimism,
but it can be optimized via self-play. With the optimized level of
pessimism, the proposed approach is shown to outperform other
state-of-the-art approaches to real-time sequential decision making.
An interesting direction of future work is to combine the proposed
approach with model-free reinforcement learning, where the propose
approach is used to choose specific actions in each step to attain the
intermediate objective that is selected by model-free reinforcement learning.
\section*{Acknowledgments}
Takayuki Osogami was supported by JST CREST Grant Number JPMJCR1304, Japan.
\bibliographystyle{named}
|
{
"timestamp": "2019-03-01T02:08:38",
"yymm": "1902",
"arxiv_id": "1902.10870",
"language": "en",
"url": "https://arxiv.org/abs/1902.10870"
}
|
\section{Introduction}
Recommender systems are increasingly used to provide users with personalized product and information offerings \citep{benschafer01e,lu15recommender,covington16deep}. These systems employ user's personal characteristics and past behaviors to generate a list of items that are individually tailored to the user’s preferences.
Whilst extremely successful commercially, there are growing concerns that such systems might lead to a self-reinforcing pattern of narrowing exposure and shift in user's interest, problems that are often referred to in the literature as ``echo chamber'' and ``filter bubble''. A significant amount of research has therefore been devoted to deriving ways to favor diversity in the set of items an individual may be exposed to (see \citet{kunaver17diversity} for a review). However, current understanding of the echo chamber and filter bubble effects is limited and experimental analysis reports conflicting results.
In this paper, we define as echo chamber the effect of a user's interest being positively or negatively reinforced by repeated exposure to a certain item or category of items, thereby generalizing the definition in \citet{sunstein09republic}, where the term is used to refer to over- and limited-exposure to similar political opinions reinforcing one's existing beliefs. We focus the definition of filter bubble introduced by \citet{pariser11filter} to describe just the fact that recommender systems select limited content to serve users online. We provide a theoretical treatment that allows us to consider the echo chamber and filter bubble effects separately. We view user's interest as a dynamical system and treat interest extremes as degeneracy points of the system. We consider different models of dynamics and identify sets of sufficient conditions that make them degenerate over time. We then use this analysis to understand the role played by the recommender system. Finally, we showcase the interplay between the user's dynamics and the recommender system actions in a simulation study using synthetic data and several classic bandit algorithms. The results reveal several pitfalls of recommender system design and point towards mitigation strategies.
\section{Related Work}
Through an analysis on the MovieLens dataset, \citet{nguyen14exploring} found that the diversity of items recommended, and those users engage with, gets narrower over time. The paper asks whether there is a ``natural'' tendency of degeneration in user interest. Our paper takes steps toward answering this question by providing theoretical conditions for user interest degeneracy.
In the social sciences literature, \citet{flaxman16filter} found that online services are associated with increased political polarization between users as well as increased exposure to the less preferred side of political opinions. Their seemingly counter-intuitive findings are not contradictory according to our results: systems with some level of random exploration can be degenerative. \citet{Barbera15tweeting} also presented evidence of echo chamber related to political issues on Twitter. On the other hand, \cite{Borgesius16should,beam18facebook,Nechushtai18what} found counter-evidence on online news consumption. Another work by \citet{bakshy15exposure} measured the effect of user choices separately from that of the recommendation algorithm, and found that individual choices play a larger role than the algorithm in creating echo chamber on Facebook. This supports our viewpoint that user interests degenerate or not depending on their internal dynamics, the recommender system can only slow down or accelerate the process of degeneration.
\section{Model} \label{sec:method}
We consider a recommender system that interacts with a user over time\footnote{For simplicity, in this paper we restrict ourselves to the case of a single user, and leave the case of multiple users possibly influencing each-other interests to future work.}.
At every time step $t$, the recommender system serves $l$ items (or categories of items, \emph{e.g.}~news articles, videos, or consumer products)
to a user from a finite or countably infinite item\footnote{Throughout the paper, ``items'' also mean categories of items.} set $\M$.
In general, the goal of the system is to present items to a user that she is interested in: we assume that, at time step $t$, the user's interest in an item $a \in \M$ is described by a function $\mu_t: \M \to \mathbb R$ such that $\mu_t(a)$ is large (positive) if the user is interested in the item, and small (negative) if she is not\footnote{Whilst we focus on $\mu_t(a) \in \mathbb R$, we show in Remark~\ref{thm:rescale}, Appendix~\ref{app:rescale} that our results can be extended to the case where $\mu_t$ belongs to a bounded open interval.}.
Given a recommendation $a_t = (a^1_t,\ldots,a^l_t) \in \M^l$, the user provides some feedback $c_t$ based on her current interests $\mu_t(a^1_t),\ldots,\mu_t(a^l_t)$. This interaction has multiple effects:
in the traditional literature for recommender systems, the feedback $c_t$ is used to update the internal model $\theta_t$ of the recommender system that has been used to obtain the recommendation $a_t$, and the new model $\theta_{t+1}$ may depend on $\theta_t$, $a_t$, and $c_t$. In practice $\theta_t$ usually predicts the distribution of user feedback to determine which items $a_t$ should be presented to the user.
In this paper we focus on another effect and consider explicitly that the user's interaction with the recommender system may change her interest in different items for the next interaction, thus the interest $\mu_{t+1}$ may depend on $\mu_t$, $a_t$, and $c_t$. The full model of interaction is depicted in \figref{fig:state_graph}.
\begin{figure}[t]
\begin{center}
\scalebox{0.7}{
\begin{tikzpicture}
\node (theta) [state] {$\theta_{t+1}$};
\path (theta.west)+(-2.5, 0) node (theta0) [state] {$\theta_t$};
\path (theta.east)+(1.5, 0) node (theta1) [txt] {...};
\path (theta.south)+(0, -1) node (at) [state] {$a_{t+1}$};
\path (at.south)+(0, -1.5) node (ct) [state] {$c_{t+1}$};
\path (theta0.west) + (-1.5, 0) node (theta_1) [txt] {...};
\path (theta0.south) + (0, -1) node (at0) [state] {$a_t$};
\path (at0.south)+(0, -1.5) node (ct0) [state] {$c_t$};
\path (ct.south)+(0, -1.0) node (mu) [hstate] {$\mu_{t+1}$};
\path (ct0.south)+(0, -1.0) node (mu0) [hstate] {$\mu_t$};
\path (mu.east)+(1.5, 0) node (mu1) [txt] {...};
\path (mu0.west)+(-1.5, 0) node (mu_1) [txt] {...};
\path (theta_1) + (-1.85, -1.9) node (rec_sys) [txt_color] {Recommender system};
\node[inner sep=0pt] (recsys_pic) at (-6.65, -0.9)
{\includegraphics[width=.13\textwidth]{figures/computer.png}};
\path (theta_1) + (-2, -4.25) node (u_interest) [txt_color] {User};
\node[inner sep=0pt] (user_pic) at (-6.7,-3.15)
{\includegraphics[width=.1\textwidth]{figures/user.png}};
\path (theta_1) + (0, -2.15) node (line_start) [txt] {};
\path (theta1) + (0, -2.15) node (line_end) [txt] {};
\path [draw, ->] (theta.south) -- node [above] {} (at.north);
\path [draw, ->] (at.south) -- node [above] {} (ct.north);
\path [draw, ->, dashed] (theta.east) -- node [above] {} (theta1.west);
\path [draw, ->, dashed] (theta0.east) -- node [above] {} (theta.west);
\path [draw, ->] (ct0.east) -- node [above] {} (theta.250);
\path [draw, ->, dashed] (theta_1.east) -- node [above] {} (theta0.west);
\path [draw, ->] (theta0.south) -- node [above] {} (at0.north);
\path [draw, ->] (at0.south) -- node [above] {} (ct0.north);
\path [draw, ->] (mu.north) -- node [above] {} (ct.south);
\path [draw, ->] (mu0.north) -- node [above] {} (ct0.south);
\path [draw, ->] (mu0.east) -- node [above] {} (mu.west);
\path [draw, ->] (mu.east) -- node [above] {} (mu1.west);
\path [draw, ->] (mu_1.east) -- node [above] {} (mu0.west);
\path [draw, ->] (ct0.east) -- node [above] {} (mu.150);
\path [draw, ->] (at0.east) -- node [above] {} (theta.220);
\path [draw, ->] (at0.east) -- node [above] {} (mu.130);
\path [draw, teal, dashed] (line_start.east) -- node [above] {} (line_end.west);
\end{tikzpicture}}
\end{center}
\caption[State graph]{Model of interaction between a recommender system and user over time. Continuous and dashed links indicate existing or possible dependencies, respectively.}
\label{fig:state_graph}
\end{figure}
We are interested in studying the evolution of the user's interest. An example of such an evolution is that the interest is reinforced by user interactions with the recommended items, that is, $\mu_{t+1}(a)>\mu_t(a)$ if the user clicks on an item $a$ at time step $t$, while $\mu_{t+1}(a)<\mu_t(a)$ if $a$ is shown but not clicked (here $c_t \in \{0,1\}^l$ can be defined as the indicator vector of clicks to the corresponding items).
To analyze the echo chamber or filter bubble effect, we are interested in understanding when the user’s interest changes extremely, which, in our model, translates to $\mu_t(a)$ taking values arbitrarily different from the initial interest $\mu_0(a)$: large positive values indicate that the user becomes extremely interested in item $a$, while large negative values indicate that the user dislikes $a$.
Formally, for a finite item set $\M$, we can ask if the $L^2$ norm
$\|\mu_t-\mu_0\|_2 = \left(\sum_{a\in \M} (\mu_t(a)-\mu_0(a))^2\right)^{1/2}$
can grow arbitrarily large: the user's interest sequence $\mu_t$ is called \emph{weakly degenerate} if
\begin{equation}
\label{eq:weak}
\limsup_{t\rightarrow \infty}\|\mu_t - \mu_0\|_2 = \infty \quad \text{ \emph{almost surely\footnotemark}}.
\end{equation}
\footnotetext{\emph{i.e.}~ with probably 1.}
A stronger notion of degeneracy, which also requires that once $\mu_t$ drifted away from $\mu_0$ it remains so, is \emph{strong degeneracy}: the sequence $\mu_t$ is \emph{strongly degenerate} if
\begin{equation}
\label{eq:strong}
\lim_{t\rightarrow \infty}\|\mu_t - \mu_0\|_2 = \infty \qquad \text{\emph{almost surely}}.
\end{equation}
In the next section we show that weak or strong degeneracy occurs under mild sufficient conditions on the evolutionary dynamics of $\mu_t$.
There are multiple ways to extend the above definitions to the case of an infinite item set $\M$. For simplicity, we only consider here replacing $\|\mu_t - \mu_0\|_2$ with $\sup_{a\in\M}|\mu_t(a)-\mu_0(a)|$ in Eqs. (\ref{eq:weak}) and~(\ref{eq:strong}), which is equivalent to the original definitions when $\M$ is finite\footnote{As such, we could have used $\sup_{a\in\M}|\mu_t(a)-\mu_0(a)|$ in our original definitions, but we prefer $\|\mu_t - \mu_0\|_2$ as it also provides some information about the ``average'' deviation of the user's interest over the different items at any finite time $t$.}.
\section{User Interest Dynamics -- Echo Chamber}\label{sec:user}
As items often represent diverse categories of things, we make the simplifying assumption that they are independent from each other. By setting $l=1$ and $a^1_{t}=a$ for all $t$ (\emph{i.e.}, $\M=\{a\}$), we can remove the influence of the recommender system and consider the dynamics of the user's interest separately. This allows us to analyze the echo chamber effect: what happens to the interest $\mu_t(a)$ if item $a$ is served infinitely often (\emph{i.o.}).
Since $a$ is fixed, to simplify the notation, we write $\mu_t$ instead of $\mu_t(a)$ in this section. Given $a_t$, according to \figref{fig:state_graph}, $\mu_{t+1}$ is a---possibly stochastic---function of $\mu_t$ (as $\mu_{t+1}$ depends on $c_t$ and $\mu_t$, and $c_t$ depends on $\mu_t$).
Below we discuss the general case when the drift $\mu_{t+1}-\mu_t$ is a nonlinear stochastic function; deterministic models for the drift are considered in Appendix~\ref{app:deterministic}.
\vspace*{-5mm}
\paragraph{Nonlinear Stochastic Model.}
We assume that $\mu_0 \in \mathbb R$ is fixed and that $\mu_{t+1} = \mu_t + f(\mu_t, \xi_t)$, where
$(\xi_t)_{t=1}^\infty$ is an infinite sequence of independent uniformly distributed random variables that introduce noise into the system (\emph{i.e.}~$\mu_{t+1}$ is a stochastic function of $\mu_t$). The function $f : \mathbb R \times [0,1]$ is assumed to be measurable, but otherwise arbitrary. Denoting the uniform distribution over $[0,1]$ by $U([0,1])$, let
\begin{align*}
\bar f(\mu) = \mathbb E_{\xi \sim U([0,1])}[f(\mu, \xi)]
\end{align*}
be the expected increment $\mu_{t+1} - \mu_t$
when $\mu_t = \mu$. We also define
\begin{align*}
F(\mu, x) = \mathbb P_{\xi \sim U([0,1])}(f(\mu, \xi) \leq x)
\end{align*}
to be the cumulative distribution of the increment.
The asymptotic behavior of $\mu_t$
depends on $f$, but under mild assumptions the system degenerates weakly (Theorem~\ref{thm:weak}) or strongly (Theorem~\ref{thm:strong})\footnote{The proofs of these theorems are given in Appendix~\ref{app:proofs}.}.
\begin{theorem}[weak degeneracy]\label{thm:weak}
Assume that $F$ is continuous at $(\mu, 0)$ for all $\mu \in \mathbb R$ and there exists a $\mu_{\circ} \in \mathbb R$ such that 1) $F(\mu, 0) < 1$ for all $\mu \geq \mu_{\circ}$, 2) $F(\mu, 0) > 0$ for all $\mu \leq \mu_{\circ}$.
Then the sequence $\mu_t$ is weakly degenerate, \emph{\emph{i.e.}}~$\limsup_{t\to\infty} |\mu_t| = \infty$ almost surely.
\end{theorem}
The assumptions guarantee that within any closed bounded interval there is a constant probability that the random walk escapes to the left/right when starting to the left/right of $\mu_{\circ}$ respectively. Under stronger conditions it is possible to guarantee the divergence of the random walk. We state a simple version of the theorem, but note that the result can be generalized in many ways.
\begin{theorem}[strong degeneracy]\label{thm:strong}
Assume that the conditions of Theorem~\ref{thm:weak} hold,
and additionally that there exists $c \in \mathbb R$ such that $|\mu_{t+1} - \mu_t| \leq c$ almost surely and there exists an $\epsilon > 0$ such that for all sufficiently large $\mu$ it holds that
$\bar f(\mu) > \epsilon$, and for all sufficiently small $\mu$ it holds that $\bar f(\mu) < -\epsilon$.
Then $\lim_{t\to\infty} \mu_t = \infty$ or $\lim_{t\to\infty} \mu_t = -\infty$ almost surely.
\end{theorem}
Intuitively, weak degeneracy occurs in a stochastic environment if the user's interest has some non-zero probability of drifting up when above some threshold, and of drifting down when below. Strong degeneracy holds if additionally $|\mu_{t+1}-\mu_t|$ is bounded and for $\mu_t$ sufficiently large/small the increment $\mu_{t+1}-\mu_t$ has positive/negative drift that is larger than a constant.
Theorems~\ref{thm:weak} and~\ref{thm:strong} show that the user's interest degenerates under very mild conditions, in particular, in the model we consider in our simulation studies. Thus, in such cases degeneracy can only be avoided if an item (or item category) is showed only finitely many times; otherwise one can only hope to control how fast $\mu_t$ degenerates (\emph{i.e.}~tends to $\infty$).
\section{System Design Role -- Filter Bubble}\label{sec:recsys}
In the previous section we discussed conditions for degeneracy for different user interest dynamics. In this section we examine the other side of the story, the influence of recommender system actions in creating filter bubbles.
We typically do not know the dynamics of the user's interest in the real world. However, we consider the relevant scenario to the echo chamber/filter bubble problem where user's interest in some items has degenerative dynamics, and examine how to design a recommender system that slows down the degeneracy process. We consider three dimensions, namely model accuracy, amount of exploration, and growing candidate pool.
\vspace*{-3.5mm}
\paragraph{Model Accuracy.}
One common goal of recommender systems designers is to increase the prediction accuracy of the internal model $\theta_t$.
How does model accuracy coupled with greedy optimal $a_t$ affect the speed of degeneration? We examine this question for the extreme case of exact predictions, \emph{i.e.}~$\theta_t = \mu_t$, we call such a prediction model the \emph{oracle model}. We argue that under the {\it surfacing assumption} explained below, the oracle model coupled with greedily optimal action selection results in the quickest degeneracy.
In order to analyze the problem concretely, we focus on the degenerate linear dynamics model for $\mu_t(a)$ for $a \in \M$, \emph{i.e.}~ $\mu_{t+1}(a) = (1+k) \mu_t(a) + b$. Then we can solve for $\mu_t(a)$, obtaining
$$\mu_t(a) = (\mu_0(a) + \frac{b(a)}{k(a)})(1+k(a))^t - \frac{b(a)}{k(a)}\,,$$ for $|1+k(a)| > 1$ (see Appendix~\ref{app:deterministic}).
\emph{Surfacing Assumption:}
Let $[m]=\{1,2,\ldots,m\}$ be the candidate set of size $m$. If a subset of items $\s\subset[m]$ leads to positive degenerate dynamics (\emph{i.e.}~$\mu_t(a)\rightarrow +\infty$ for all $a\in\s$), then we assume that there exists a time $\tau > 0$ such that, for all $t \geq \tau$, $\s$ takes up the top $|\s|$ items in terms of values of $\mu_t$, sorted by the base value of the exponential function, $|1+k(a)|$.
The surfacing assumption makes sure that the quickest degenerating items surface out to the top list given enough time of exposure. It can be generalized to nonlinear stochastic dynamics of $\mu_t$ provided that the items from $\s$ have an almost surely stable ordering of degeneracy speed $|\mu_t(a) - \mu_0(a)|/t$ over time.
Under the general surfacing assumption, after time $\tau$, the quickest way to degeneration is to serve the top $l$ items according to $\mu_t$, or $\theta_t$ of the oracle model. Even if the assumption is violated to some degree, the oracle model still leads to degeneracy very efficiently by picking the top $l$ items according to $\mu_t$ which are likely to receive positive feedback due to high $\mu_t$, and therefore increasing $\mu_{t+1}$ and reinforcing the past choices.
In practice the recommender system models are inaccurate. We can think of inaccurate models as the oracle model with different levels of noises added to $\theta_t$. We discuss inaccurate models in the next section.
\vspace*{-3.5mm}
\paragraph{Amount of Exploration.}
Consider a type of $\epsilon$-random exploration where $a_t$ always picks the top $l$ items out of a finite candidate pool $[m]$ with uniform $\epsilon$ noise on $\theta_t$, \emph{i.e.}~according to $\theta'_t = \theta_t + U([-\epsilon, \epsilon])$.
Given the same model sequence $\theta_t$, the bigger $\epsilon$ is, usually the slower the system degenerates. However, in practice $\theta_t$ is learned from observations, and the random exploration added to an oracle model may in fact accelerate degeneration: random exploration can help reveal the most positively degenerating items over time making the surfacing assumption more likely to be true (we show this phenomenon in the simulation experiments below, \figref{fig:noise_oracle}). In addition, if user interests have degenerative dynamics, even recommending items uniformly at random leads to degeneration, albeit quite slowly.
How do we then make sure that the recommender system does not make user interests degenerate? One way is to limit the number of times an item for which the user's interest dynamics is degenerative is served to the user. In practice it is hard to detect which items correspond to degenerative dynamics, however we can generally prevent degeneration if all items are served only a finite number of times, which suggests having an ever growing pool of candidate items.
\begin{figure}[t!]
\vspace{-3.5mm}
\centering
\subfloat[][{\it Optimal Oracle}, $\mu_t$\label{fig:OptimalOracle_mu}]{\includegraphics[height=1.1in,trim={1cm 0 0 0},clip]{figures/presentation_rate_OptimalOracle_angle_200_mu_final.pdf}}
\subfloat[][{\it Optimal Oracle}, serving rate\label{fig:OptimalOracle_a}]{\includegraphics[height=1.1in,trim={1cm 0 0 0},clip]{figures/presentation_rate_OptimalOracle_angle_200_action_final.pdf}}
\hfill
\subfloat[][{\it Oracle}, $\mu_t$\label{fig:Oracle_mu}]{\includegraphics[height=1.1in,trim={1cm 0 0 0},clip]{figures/presentation_rate_Oracle_angle_200_mu_final.pdf}}
\subfloat[][{\it Oracle}, serving rate\label{fig:Oracle_a}]{\includegraphics[height=1.1in,trim={1cm 0 0 0},clip]{figures/presentation_rate_Oracle_angle_200_action_final.pdf}}
\hfill
\subfloat[][{\it TS}, $\mu_t$\label{fig:ThompsonSampling_mu}]{\includegraphics[height=1.in,trim={1cm 0 0 0},clip]{figures/presentation_rate_ThompsonSampling_angle_200_mu_final.pdf}}
\subfloat[][{\it TS}, serving rate\label{fig:ThompsonSampling_a}]{\includegraphics[height=1.in,trim={1cm 0 0 0},clip]{figures/presentation_rate_ThompsonSampling_angle_200_action_final.pdf}}
\hfill
\subfloat[][{\it UCB}, $\mu_t$\label{fig:UCB_mu}]{\includegraphics[height=1.1in,trim={1cm 0 0 0},clip]{figures/presentation_rate_UCB_angle_200_mu_final.pdf}}
\subfloat[][{\it UCB}, serving rate\label{fig:UCB_a}]{\includegraphics[height=1.1in,trim={1cm 0 0 0},clip]{figures/presentation_rate_UCB_angle_200_action_final.pdf}}
\hfill
\subfloat[][{\it Random}, $\mu_t$\label{fig:random_mu}]{\includegraphics[height=1.1in,trim={1cm 0 0 0},clip]{figures/presentation_rate_Random_angle_200_mu_final.pdf}}
\subfloat[][{\it Random}, serving rate\label{fig:random_a}]{\includegraphics[height=1.1in,trim={1cm 0 0 0},clip]{figures/presentation_rate_Random_angle_200_action_final.pdf}}
\caption[]{Echo chamber and filter bubble effect for {\it Optimal Oracle, Oracle, TS, UCB} and {\it Random} models. Sorted user interest $\mu_t$ and serving rates are plotted every 500 steps. Under all models except for the {\it Random Model}, very quickly both the top items served and the top user interests narrow down to the ($l=$) 5 most positively reinforced items.}
\label{fig:sim_filter_bubble}
\end{figure}
\vspace*{-3.5mm}
\paragraph{Growing Candidate Pool $\M$.}
With a growing candidate pool, at every time step an additional set of new items becomes available to be served to the user. Hence the domain of the function $\mu_t$ expands as $t$ increases. Adding new items at least linearly often
is a necessary condition to avoid possible degeneration, since in a finite or any sublinearly growing candidate pool, by the pigeon hole principle, there must exist at least one item that is served \emph{i.o.}, which is degenerate in the worst case scenario (also under general conditions described \emph{e.g.}~in Theorem~\ref{thm:strong}). However, with an at least linearly growing candidate pool $\M$ the system can potentially impose the maximum number of times any item is served to a user and prevent degeneration.
\section{Simulation Experiments}\label{sec:simulation}
In this section, we consider a simple degenerative dynamics for $\mu_t$ and examine degeneration speed under five different recommender system models. We further demonstrate that adding new items to the candidate pool can be an effective solution against system degeneracy.
We create a simulation for the model of interaction between a recommender system and a user of \figref{fig:state_graph}. Consider a possibly growing candidate pool of items of initial size $m_0$ and of size $m_t$ at time step $t$. At each time step $t$, a recommender system picks the top $l$ out of the $m_t$ items $a_t = (a_t^1, \ldots, a_t^l)$ according to the internal model $\theta_t$ to serve to a user. The user considers each of the $l$ items independently and chooses to click on a (possibly empty) subset of them, thereby generating a binary vector $c_t$ of size $l$ where $c_t(a_t^i)$ gives the user feedback on item $a_t^i$, according to $c_t(a_t^i) \sim Bernoulli(\phi(\mu_t(a_t^i)))$, where $\phi$ is the sigmoid function $\phi(x)=1/(1+e^{-x})$. The system then updates the model $\theta_{t+1}$ based on the past actions, feedbacks and the current model parameter $\theta_t$. We assume that the user's interest increases/decreases by $\delta(a')$ if the item $a'$ receives/does not receive a click, \emph{i.e.}~
\begin{align}
&\mu_{t+1}(a_t^i) - \mu_t(a_t^i) =
\begin{cases}
\delta(a_t^i) & \text{ if } c_t(a_t^i)=1,\\
-\delta(a_t^i) & \text{ otherwise, }
\end{cases}
\end{align}
where the function $\delta$ maps from the candidate set to $\mathbb{R}$. From Theorem~\ref{thm:strong}, we know that $\mu_t\rightarrow\pm\infty$ for every item.
In the experiment, we set $l=5$ and
sample the drift $\delta$ from a uniform random distribution $U([-0.01, 0.01])$. The user's initial interest $\mu_0$ for all items is independently sampled from a uniform random distribution $U([-1, 1])$.
\begin{figure}[t]
\centering
\includegraphics[height=1.9in]{figures/degeneracy_rate_fixed_pool_m0_100_final.pdf}
\caption[]{System evolves for 5,000 time steps with report interval 500. The results are averaged over 30 runs with the shaded area indicating the standard deviation. In terms of the degeneracy speed, {\it Optimal Oracle} $>$ {\it Oracle} $>$ {\it TS} $>$ {\it UCB} $>$ {\it Random}.}
\label{fig:m_100}
\end{figure}
\begin{figure}[t]
\centering
\subfloat[][Degeneracy rate vs candidate pool size vs time \label{fig:changing_m_T}]{\includegraphics[height=1.5in,trim={0.5cm 1.5cm 0 3cm},clip]{figures/3d_degeneracy_rate_m_t_30_final.pdf}}
\hfill
\subfloat[][Degeneracy rate vs candidate pool size\label{fig:changing_m}]{\includegraphics[height=1.9in]{figures/degeneracy_rate_vs_fixed_pool_sizes_final.pdf}}
\caption[]{
\subref{fig:changing_m_T} Degeneracy surfaces for {\it Optimal Oracle} (grey), {\it UCB} (green) and {\it TS} (orange) up to time $T=5,000$ while varying candidate pool sizes $\log_{10} m = 1, 2, 3, 4$. A larger candidate pool requires a longer time for exploration for the bandit algorithms, but among the three models {\it UCB} slows down system degeneracy the most given a large candidate pool.
\subref{fig:changing_m} Degeneracy speeds at $T=20,000$ of {\it Optimal Oracle} and the {\it Oracle} are higher given a larger size of the candidate set, but those of the {\it and Random Model, UCB}, and {\it TS} are lower.}
\label{fig:fixed_pool}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[height=1.9in]{figures/degeneracy_rate_vs_noise_levels_viridis_final.pdf}
\caption[]{Degeneracy speed for the {\it Oracle} model with different noise levels $\epsilon\in[0, 10]$ up to $T=20,000$. Adding noise to {\it Oracle} ($\epsilon=0$) accelerates degeneration but as the noise level grows, degeneracy slows down.}
\label{fig:noise_oracle}
\end{figure}
The internal recommender system model is updated according to following five algorithms:
\begin{itemize}
\item {\bf Random Model:}
Instead of picking top items, the set of items $a_t = (a_t^1, \ldots, a_t^l)$ is sampled from a uniform random distribution over the candidate set $U([m_t])$.
\item {\bf Oracle:}
$\theta_{t+1}(a_t^i) = \mu_{t+1}(a_t^i)$, $\forall i$.
\item {\bf Optimal Oracle:}
$\theta_{t+1}(a_t^i) = \delta(a_t^i)$, $\forall i$.
This model does not pick the highest $l$ items according to $\mu_t$ but according to $\delta$. Thus, it always picks the fastest degenerating items, therefore maximizing both the long term user engagement $\sum_{t=0}^\infty \|c_t\|_1$ and the degeneracy speed. For a fixed candidate pool $\M$, this model is equivalent to an \emph{Oracle} that satisfies the Surfacing Assumption.
\item {\bf Upper Confidence Bound Multi-armed Bandit Algorithm ({\it UCB})} \citep{lai87adaptive,auer02finite,lattimore19bandit}:
We use the version of UCB algorithm in Chapter 8 of \citet{lattimore19bandit}, however most UCB algorithms perform similarly to the purpose of this experiment. The algorithm prioritizes serving any item from the candidate set that has never been served before. This treatment includes the initial $m_0$ items as well as later whenever new items are added to the candidate pool. At time step $t$, UCB serves $l' (0\leq l'\leq l)$ previously unserved items and the top $l-l'$ items according to values of $\theta_t$. Define $f(t) = 1 + t\log^2(t)$ and we use the following model update
$\theta_{t+1} (a) = \hat{c}_t(a) + \sqrt{2\log f(t)/T_a(t)}$, where $\hat{c}_t$ is the empirical average of feedbacks on item $a$, \emph{i.e.}~$\hat{c}_t(a) = \sum_{0 \le i \le t, a\in a_i}^t c_t(a)/T_a(t)$, and $T_a(t)$ is the number of times item $a$ has been served up to time $t$, i.e. $T_a(t) = \sum_{0 \le i \le t, a\in a_i} 1$.
\item {\bf Thompson Sampling Multi-armed Bandit Algorithm ({\it TS})} \citep{thompson33on}:
We initialize $\alpha_0(a)=1, \beta_0(a)=1$ for any new item $a$. If $a$ is served at time $t$, we perform the update $\alpha_{t+1}(a) = \alpha_t(a) + c_t(a), \beta_{t+1}(a) = \beta_t(a) + 1 - c_t(a)$. At any time $t$, the internal model $\theta_t$ is sampled from the corresponding beta distribution $Beta(\alpha_t, \beta_t)$.
\end{itemize}
\subsection{Echo Chamber \& Filter Bubble Effect}
We examine the echo chamber and filter bubble effects by running the simulation on a candidate pool of fixed size $m_t=m = 100$ with time horizon $T=5,000$.
In \figref{fig:sim_filter_bubble}, we show the degeneration of user interest $\mu_t$ (left column) and the serving rate (right column) of every item as each recommender model evolves in time. The serving rate of an item shows how often it is served within the report interval. In order to see the distribution clearly, we sort the items according to the z-values at the report time. Although all models cause user interest degeneration, the degeneration speeds are quite different ({\it Optimal Oracle} $>$ {\it Oracle, TS, UCB} $>$ {\it Random Model}). The {\it Oracle, TS} and {\it UCB} optimize based on $\mu_t$ and so we see a positive degenerative dynamics for $\mu_t$. The {\it Optimal Oracle} optimizes on the degeneration speed directly and not on $\mu_t$ so we see both a positive and negative degeneration in $\mu_t$. The {\it Random Model} also drifts $\mu_t$ in both directions, but at a much slower rate. However, overall except for the {\it Random Model}, very quickly both the top items served and the top user interests narrow down to the ($l=$) 5 most positively reinforced items.
\subsection{Speed of Degeneracy}
Next, we compare the degeneracy speed for the five recommender system models on both fixed and growing candidate sets. As the $L^2$ distance that measures system degeneracy is asymptotically linear for all five models (see Appendix~\ref{app:linear_speed}), we quantify degeneracy speeds by compare empirically $\|\mu_t-\mu_0\|_2 / t$ in finite candidate pools for different experiment setups.
Figure~\ref{fig:m_100} shows the degeneracy speed of five models averaged across 30 runs when we take $m=100$ and evolve the system for $T=5,000$ steps. We see that the {\it Optimal Oracle} results in the fastest degeneration by far, followed by the {\it Oracle, TS} and {\it UCB}. The {\it Random Model} offers the slowest degeneracy speed.
\vspace*{-3mm}
\paragraph{The Effect of Candidate Pool Size.} In \figref{fig:changing_m_T} we compare the {\it Optimal Oracle}, {\it UCB} and {\it TS}' degeneracy speed $\|\mu_t-\mu_0\|_2 / t$ up to 5,000 time steps and candidate pool sizes $m = 10, 10^2, 10^3, 10^4$. Apart from the {\it Random} model, we see that {\it UCB} slows down system degeneracy the most given a large candidate pool since it is forced to explore any unserved item first. A larger candidate pool requires a longer time for exploration for the bandit algorithms. As the candidate pool size grows to 10,000 {\it UCB}'s degeneracy speed never peaks up given the time horizon, but will eventually grow given a longer time. {\it TS} has higher degeneracy speed due to weaker exploration on new items. The {\it Optimal Oracle} accelerates degeneration given a larger pool, as it can pick potentially faster degenerative items than from a smaller pool.
Additionally, in \figref{fig:changing_m} we plot all five models degeneracy speed for $T=20,000$ against the same changing candidate pool sizes. The degeneracy speed of the {\it Optimal Oracle} and the {\it Oracle} increases with the size of the candidate set, but that of the {\it and Random Model, UCB}, and {\it TS} decreases. In practice, having a large candidate pool can be a temporary solution to slow down system degeneration.
\vspace*{-3mm}
\paragraph{The Effect of the Noise Level.}
Next we show the influence of internal model inaccuracy on degeneracy speed. We compare the {\it Oracle} model with different amounts of uniformly random noises, \emph{i.e.}~the system serves the top $l$ items according to the noisy internal model $\theta'_t = \theta_t + U([-\epsilon, \epsilon])$. The candidate pool has fixed size $m=100$. In \figref{fig:noise_oracle}, we vary $\epsilon$ from 0 to 10. Counter-intuitively adding noise to {\it Oracle} accelerates degeneration since faster degenerative items may be selected by chance than those fixed set of top $l$ items ranked by $\mu_0$, and more likely satisfies the Surfacing Assumption. Given $\epsilon>0$, as expected, we see a nice monotonically increasing damping effect on degeneracy speed as the noise level grows.
\begin{figure}[t]
\centering
\hspace*{-.4cm}
\includegraphics[height=1.6in]{figures/degeneracy_rate_fixed_growth_rates_final.pdf}
\caption[]{Comparison of the five models with growing candidate pools at different rates $\eta=0, 0.5, 1.0$, degeneracy up to $T=10,000$, averaged over 10 runs. Both the {\it Oracle} and the {\it Optimal Oracle} for all growth rates are degenerate. The {\it Random Model} and {\it UCB} stop degeneration at sublinear growth while {\it TS} model requires linear growth to stop degeneration.}
\label{fig:growing_pool}
\end{figure}
\vspace*{-3mm}
\paragraph{Growing Candidate Pool.}
We extend the definition of degeneracy speed to an infinite candidate pool by computing $\sup_{a\in\M}|\mu_t(a)-\mu_0(a)| / t$ (see Appendix~\ref{app:linear_speed} for an asymptotic analysis). Since the degeneracy speed may not be asymptotically linear for all five models, we examine directly the sup distance $\sup_{a\in\M}|\mu_t(a)-\mu_0(a)|$ over 10,000 time steps. To construct growing candidate pools at different growth speed, we define a growth function $m_t = \left\lfloor m_0 + l t^{\eta} \right\rfloor$ by varying the growth parameter\footnote{$\eta=0$ gives a fixed candidate pool, $0<\eta<1$ gives sub-linear growth, $\eta=1$ gives linear growth.} $\eta=0, 0.5, 1$, where $m_0=100$. In \figref{fig:growing_pool} we average the results over 10 independent runs. Both the {\it Oracle} and the {\it Optimal Oracle} for all growth rates are degenerate. The {\it Random Model} stops degeneration at sublinear growth, $\eta=0.5$, so does {\it UCB} thanks to forced exploration on previously unserved items, although its trajectory has a small upward tilt. The {\it TS} model degenerates at sublinear growth but stops degeneration at linear growth $\eta=1$. For all models, the higher the growth rate $\eta$, the slower they degenerate, if they do at all. Overall when applicable, an ideally linearly growing candidate set and continuous random exploration seem to be good remedies against an adversarial dynamics of $\mu_t$ to best prevent degeneracy.
\section{Conclusion}
We provided a theoretical analysis of the echo chamber and filter bubble effects for recommender systems. We used the dynamical system framework to model user's interest and treated interest extremes as degeneracy points of the system. We gave formal definitions of system degeneracy and provided sufficient conditions which make the system degenerate with both deterministic and stochastic dynamics. On the recommender system side, we discussed the influence on degeneracy speed of three independent factors in system design, \emph{i.e.}~model accuracy, amount of exploration, and the growth rate of the candidate pool. An oracle model often leads to quick degeneracy of the system, while continuous exploration and a large candidate pool size can help slow it down. The best remedies against system degeneracy we found are continuous random exploration and growing the candidate pool at least linearly.
Our work has two main limitations. First, since user interests are hidden variables that are not directly observed, a good measure or proxy for user interests is necessary in practice to study degeneration reliably. Second, we assumed that items and users are independent from each other -- we will extend the theoretical analysis to the case of possibly mutually dependent items and users in a future work.
\section{Acknowledgments}
We would like to thank William Isaac, Michael Mathieu, Krishnamurthy Dvijotham, Timothy Mann and Dilan Gorur, for helpful discussions and advice.
\section{Proofs of theorems}
\label{proofs}
\addtocounter{theorem}{-7}
\subsection{Proof of Theorem~\ref{thm:s1_suff}}
\begin{lemma}
Let $\s$ be a finite set of options. If at least $k$ distinct elements are served from $\s$ every time step, then at least $k$ elements from $\s$ are served infinitely often.
\label{lem:s11}
\end{lemma}
\begin{proof}
Suppose the contrary is true. There are $k'$ elements served infinitely often and $k'<k$. Without loss of generality, let this set be $\{1, 2, \ldots k'\}$ or $[k']$. Let $m$ be the maximum number of times any element is served from $\s\setminus[k']$. Since at least $k$ distinct elements are served from $\s$ every time step, then at least $k-k'$ elements served are from the set $\s\setminus[k']$. The total number of service times from this set is at most $m\cdot|\s| < \infty$. However $(k-k')t\rightarrow \infty$ as $t\rightarrow\infty$. This is a contradiction!
\end{proof}
\begin{lemma}
Let $\s$ be a finite set of options. Let $d$ be an integer such that $0 < d \leq |\s|$. Using definitions from Theorem~\ref{thm:s1_suff}, if $\exists a^1, a^2, \ldots, a^d\in\s$ such that as $t\rightarrow\infty, \mu_t(a^i) \rightarrow \infty$, for $i=1,2\ldots,d$, and any other $a$ has bounded values of $\mu_t(a)$, i.e. for any other $a\in\s$, $\exists B>0$ s.t. $\mu_t(a)<B$ for any $t$. Then given any $t_0>0$, there must exist a time $t'>t_0$ such that $\theta_{t'}(a^1), \ldots, \theta_{t'}(a^d)$ constitute the top $d$ values of $\theta_{t'}(a)$ for $a\in\s$.
\label{lem:s12}
\end{lemma}
\begin{proof}
We argue by contradiction. Define the set of options $\A=\{a^1, a^2, \ldots, a^d\}$. Suppose the contrary is true, i.e. $\exists t_0>0$ such that for $\forall t\geq t_0, \exists a_t\in\s\setminus\A$ such that $\theta_t(a_t)$ is in the top $d$ values. Since $|\s|$ is finite whereas the sequence $\{a_{t_0}, a_{t_0+1}, \ldots\}$ is infinite, there must exist an option $a^*$ where $\theta_t(a^*)$ is in the top $d$ values for infinitely many times after time $t_0$. Thus there exists an infinite sequence $\{t_1, t_2, \ldots\}$ where $\theta_{t_i}(a^*) > \theta_{t_i}(a^{j_t})$ for some $j_t\in[d]$. Hence $\theta_{t_i}(a^*) > \theta_{t_i}(a^{j_t}) > \min_{i\in[d]}(\theta_{t_i}(a^i))$. Since $\mu_t(a^i) \rightarrow \infty$ as $t \rightarrow \infty$ for any $i=1,2\dots,d$, we have $\theta_t(a^i) \rightarrow \infty$ as $t \rightarrow \infty$ due to Condition~\ref{eq:_bdd_s1}. Since $d$ is finite, $\min_{i\in[d]}(\theta_{t_i}(a^i))\rightarrow \infty$ as $t\rightarrow \infty$ as well. Thus so do $\theta_{t_i}(a^*)$ and $\mu_{t_i}(a^*)$ by Condition~\ref{eq:_bdd_s1}. This is a contradiction with the assumption that any $a\in\s\setminus\A$ must have bounded $\mu_t(a)$.
\end{proof}
\addtocounter{theorem}{-2}
\begin{theorem}
Let $\mu_t, \theta_t:[m]\rightarrow \mathbb{R}$. Let $D\in\mathbb{R}$. If the following conditions hold,
\begin{align}
&a^1_t, a^2_t, \ldots a^k_t = \text{ top k } (\theta_t(a)),\label{eq:_opt_s1}\\
&c_t(a) = (\mu_t(a)>D), \text{ for } \forall a\in [m],\\
&c_t(a) = 1 \Leftrightarrow
\begin{cases}
\theta_{t+1}(a) > \theta_t(a), \label{eq:_pos_reinforce1_s1}\\
\mu_{t+1}(a) > \mu_t(a).
\end{cases}\\
&c_t(a) = 0 \Leftrightarrow
\begin{cases}
\theta_{t+1}(a) < \theta_t(a), \label{eq:_pos_reinforce2_s1}\\
\mu_{t+1}(a) < \mu_t(a).
\end{cases}\\
&\forall t_0 > 0, \forall a\in[m] \text{ where } |\{t|a_t=a\}|=\infty,\notag\\
&\qquad \sum_{\substack{t=t_0, \\a_t=a}}^\infty |\mu_{t+1}(a) - \mu_t (a)| = \infty,\label{eq:_inf_s1}\\
&\exists B>0, \text{ s.t. for } \forall t>0, \forall a\in[m], \notag\\
&\qquad |\theta_t(a) - \mu_t(a)|<B, \label{eq:_bdd_s1}
\end{align}
then we have a degenerate system, that is, $\lim_{t\rightarrow \infty}\|\mu_t - \mu_0\|_p = \infty$. Specifically, as $t \rightarrow \infty$, the system results in either of the two following scenarios.
\begin{enumerate}
\item $\mu_t(a^i) \rightarrow \infty$ for $k$ elements $a^i\in[m]$,
\item Strictly less than $k$ elements have $\mu_t(a^i) \rightarrow \infty$. For any other $a\in [m]$, $\mu_t(a)\rightarrow -\infty$ as $i\rightarrow \infty$.
\end{enumerate}
\end{theorem}
\begin{proof}
In the first scenario, we argue by contradiction. Suppose $\exists a^1, a^2, \ldots, a^{k'}\in[m]$ such that $\mu_t(a^i)\rightarrow\infty$ as $t\rightarrow\infty$, that is, $\forall N>0, \exists t_0>0$ s.t. $\mu_t(a^i)>N$ for all $t \geq t_0, i=[1,\ldots,k']$ and $k'>k$. Pick $N>D$. Moreover any $a\in[m]\setminus[k']$ has bounded $\mu_t(a)$. Invoking Lemma~\ref{lem:s12}, $\exists t'>t_0$ s.t. $\theta_{t'}(a^i)$ are the top $k'$ values for $i=1,2\dots,k'$. Without loss of generality, let $a^1,\ldots, a^{k'}$ be a decreasingly sorted list according to values of $\theta_{t'}(a^i)$. Since the system can only select $k$ options at time $t'$, it selects $a^1, \ldots, a^k$. Since $N>D$, we have $c_{t'}(a^i)=1$ for $i=1,\ldots,k$. Hence all top $k$ values of $\theta_t(a^i)$ increase and they remain in the top $k$ for time $t'+1$. Iterate the same argument for $t=t'+2, t'+3, \ldots$ and we conclude that $a^{k+1}, \ldots, a^{k'}$ will not be selected after $t'$. This contradicts with $\mu_t(a^i)\rightarrow \infty$ as $t\rightarrow \infty$ for $i=k+1, \ldots, k'$!
In the second scenario, suppose there are strictly less than $k$ elements with $\mu_t(a)\rightarrow\infty$. Let these elements be $a^1, a^2, \ldots, a^{k'}\in[m]$ and $k'<k$. For all $a\in[m]\setminus[k']$, $\mu_t(a)$ is bounded, that is, $\exists N_0>0$ such that $\forall t_0>0$, we have $\mu_t(a)<N_0$ for $\forall t\geq t_0$. Since at least $k-k'$ distinct options are served from the set $[m]\setminus [k']$ at every time step, by Lemma~\ref{lem:s11} we conclude at least $k-k'$ distinct options from $[m]\setminus[k']$ are served infinitely many times. Call this set $\h$. For $\forall a\in\h$, we discuss the following three cases based on values of the feedback $c_t(a)$.
\begin{enumerate}
\item $c_t(a) = 0$ infinitely often, $c_t(a) = 1$ finitely often. Thus $\exists t_m$ such that $c_t(a)=0$ for all $t>t_m$. By Condition~\ref{eq:_inf_s1}, we have
\begin{align}
&\sum_{\substack{t=t_m, \\a_t=a, \\c_t(a)=0}}^\infty |\mu_{t+1}(a) - \mu_t (a)| \\
=& \sum_{\substack{t=t_m, \\a_t=a}}^\infty |\mu_{t+1}(a) - \mu_t (a)| = \infty.
\end{align}
Due to Condition~\ref{eq:_pos_reinforce2_s1}, in this case $\mu_t(a)\rightarrow -\infty$ as $t\rightarrow \infty$.
\item $c_t(a) = 1$ infinitely often, $c_t(a) = 0$ finitely often. This case is the opposite of Case 1. We follow the same argument above and conclude $\mu_t(a)\rightarrow\infty$ as $t\rightarrow\infty$, which contradicts with the assumption that $\forall a\in\h\subset[m]\setminus[k']$ has bounded $\mu_t(a)$.
\item $c_t(a) = 1$ and $c_t(a) = 0$ infinitely often. This requires $\mu_t(a)$ to move above and below the threshold $D$ infinitely often. Suppose at some time $\mu_{t_0}(a)<D$, then $c_{t_0}(a)=0$. Thus $\mu_{t_0+1}(a)<\mu_{t_0}(a)<D$. When $a$ is selected again at $t'$, we have $c_{t'}(a)=0$ and $\mu_{t'}(a)$ will decrease again. $\mu_{t'}(a)$ can never move above $D$ again. This argument applies for the opposite case $\mu_{t_0}(a)>D$ as well. Thus this is an impossible case.
\end{enumerate}
Therefore, only Case 1 is possible. For $\forall a^*\in\h$, $\mu_t(a^*)\rightarrow -\infty$ as $t\rightarrow \infty$. Since $|\theta_t(a^*) - \mu_t(a^*)|<B$ (Condition~\ref{eq:_bdd_s1}), $\theta_t(a^*)\rightarrow -\infty$ as $t\rightarrow \infty$. For $\forall a\in [m]\setminus([k']\cup\h)$, option $a$ has been selected at most finite number of times. Thus $\exists t_a$ such that option $a$ is never selected after $t_a$ and $\theta_t(a)\equiv\theta_{t_a}(a)$. This contradicts with $\theta_{t_i}(a) < \theta_{t_i} (a^*)\rightarrow -\infty$ as $i\rightarrow \infty$ where $\{t_1, t_2\ldots\}, t_1>t_a,$ is the infinite sequence of time steps when $a^*$ is selected. Therefore $[m]\setminus([k']\cup\h) = \emptyset$, or $[m] = [k']\cup\h$.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:s2_suff}}
{\todo Specify in every proof whenever a condition in the theorem is used.}
\begin{lemma}
Let $d$ be a positive integer. Using definitions from Theorem~\ref{thm:s3_suff}, if $\exists a^1, a^2, \ldots, a^d\in\M$ s.t. $\mu_t(a^i) \rightarrow \infty$ as $t \rightarrow \infty$ for $i=1,2\ldots,d$, and for any other $a\in\M$, $\exists B>0$ s.t. $\mu_t(a)<B$ for any $t$. Then for any $t_0>0$ there must exist a time $t'>t_0$ such that $\theta_{t'}(a^1), \ldots, \theta_{t'}(a^d)$ constitute the top $d$ values of $\theta_{t'}(a)$ for $a\in\s_{t'}$.
\label{lem:s3}
\end{lemma}
\begin{proof}
Since $\mu_t(a^i) \rightarrow \infty$ as $t \rightarrow \infty$ for any $i=1,2\ldots,d$ and $d<\infty$, then it means that for any $N>0$, $\exists t_0$ s.t. for $\forall t>t_0$, we have $\mu_t(a^i)>N$. Take $N>C$ for $C$ defined in Theorem~\ref{thm:s3_suff}. Then we have $\mu_t(a^i)>N>C$. This means that for $\forall t>t_0$, the top k options at time $t$ can only be selected from $\s_{t_0}$ due to Condition~\ref{eq:_bdd_init}, which is a finite set of options containing $a^1, a^2, \ldots, a^d$. Therefore by Lemma~\ref{lem:s2}, for any $t_0>0$ there must exist a time $t'>t_0$ such that $a^1, \ldots, a^d$ constitute the top $d$ values of $\theta_{t'}(a)$ for $a\in\s_{t_0}\subset\s_{t'}$.
\end{proof}
\addtocounter{theorem}{-1}
\begin{theorem}
Let $\mu_t, \theta_t:\s_t\rightarrow \mathbb{R}$. Let $\phi:\mathbb{R}\rightarrow[0,1]$ be any non-decreasing surjective function. If the following conditions hold,
\begin{align}
&a^1_t, \ldots, a^k_t = \text{ top k } (\theta_t(a)),\\
&\text{ where }a\in\s_t, \s_{t-1}\subset\s_t\subset \M, |\s_t|<\infty,\label{eq:_subset}\\
&\exists C>0, \text{ s.t. } \theta_t(a)<C, \text{ for }\forall a\in\s_t\setminus\s_{t-1},\label{eq:_bdd_init}\\
&\exists \text{ a non-empty } \h\subset\M \text{ such that}\\
&\qquad \forall a\in\h \text{ is served infinitely often},\\
&c_t(a) = Bernoulli(p=\phi(\mu_t(a))), \forall a\in \M,\\
&c_t(a) = 1:
\begin{cases}
\theta_{t+1}(a) > \theta_t(a), \\
\mu_{t+1}(a) > \mu_t(a).
\end{cases}\\
&c_t(a) = 0:
\begin{cases}
\theta_{t+1}(a) < \theta_t(a), \\
\mu_{t+1}(a) < \mu_t(a).
\end{cases}\\
&\forall t_0 > 0, \forall a\in\M, s.t. |\{t|a_t=a\}|=\infty,\notag\\
&\qquad \sum_{\substack{t=t_0, \\a_t=a}}^\infty |\mu_{t+1}(a) - \mu_t (a)| = \infty,\\
&\exists B>0, \text{ s.t. for } \forall t>0, \forall a\in\M, |\theta_t(a) - \mu_t(a)|<B,
\end{align}
then we have a degenerate system, specifically the system results in either of the two following scenarios.
\begin{enumerate}
\item $\mu_t(a^i) \rightarrow \infty$ with probability 1 for $1$ to $k$ elements $a^i\in\M$,
\item $\mu_t(a^i) \rightarrow -\infty$ for $\forall a \in\h$. For $\forall a\in\mathcal{M}\setminus\h$, there exists an infinite sequence $\{t_1, t_2, \ldots\}$ where $\mu_{t_i}(a)\rightarrow -\infty$ as $i\rightarrow \infty$.
\end{enumerate}
\end{theorem}
\begin{proof}
We argue by contradiction. Suppose $\exists a^1, a^2, \ldots, a^{k'}\in\M$ s.t. for $\forall N>0, \exists t_0>0$ s.t. $\mu_t(a^i)>N$ for all $t \geq t_0, i=[1,\ldots,k']$ and $k'>k$. Take $N'>C+B$ and let $t'_0$ be the corresponding $t_0$. All other $a\in\M$ have bounded $\mu_t(a)$ for any $t>0$. Invoking Lemma~\ref{lem:s3}, $\exists t'>t'_0$ s.t. $\theta_{t'}(a^i)$ are the top $k'$ values in $\s_{t'}$ for $i=1,2\dots,k'$. Without loss of generality, let $a^1,\ldots, a^{k'}$ be a decreasingly sorted list according to values of $\theta_{t'}(a^i)$. Since the system can only select $k$ options at time $t'$, it selects $a^1, \ldots, a^k$. Since $N'$ can be arbitrarily large, with probability 1 we have $c_{t'}(a^i)=1$ for $i=1,\ldots,k$. Hence the top $k$ values of $\theta_{t'}(a^i)$ all increase at $t'+1$.
At $t'+1$, we have
\[
\mu_{t'+1}(a^i) > N' > C+B > \theta_{t'+1}(a)+B > \mu_{t'+1}(a),
\]
for $i=1,\ldots,k$ and $a\in\s_{t'+1}\setminus\s_{t'}$. For all other $a\in\s_{t'}$, $\theta_{t'+1}(a) = \theta_{t'}(a)$, thus remaining the same. Notice that this is true for all $t>t'$. By inferring on every time step, they maintain in the top $k$ of $\s_{t}$ for $t>t'$. Therefore $a^{k+1}, \ldots, a^{k'}$ will be selected after time $t'$ with probability 0. This is a contradiction with $\mu_t(a^i)\rightarrow \infty$ as $t\rightarrow \infty$ for $i=k+1, \ldots, k'$!
Now if for $\forall a\in\M$, $\theta_t(a)$ is always bounded from above. Then for $\forall a\in\h$, by the same argument as in the proof of Theorem~\ref{thm:s1_suff}, we have $c_t(a)=0$ for infinitely many times $t$. Thus we conclude that $\mu_t(a)\rightarrow-\infty$ as $t\rightarrow\infty$. For $\forall a\in\mathcal{M}\setminus\h$, since it can only be selected finitely many times, by the same argument as in Theorem~\ref{thm:s1_suff} again, we conclude that for $\forall a\in\mathcal{M}\setminus\h$, there exists an infinite sequence $\{t_1, t_2, \ldots\}$ where $\mu_{t_i}(a)\rightarrow -\infty$ as $i\rightarrow \infty$.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:s3_suff}}
\addtocounter{theorem}{-1}
\begin{lemma}
If $d(\theta^i_t, \theta^j_t) < \epsilon$, then $|\theta^i_t(a) - \theta^j_t(a)| < \epsilon$ for all $a\in \s_t$.
\label{lem:s41}
\end{lemma}
\begin{proof}
For $\forall a\in \s_t$, $|\theta^i_t(a) - \theta^j_t(a)| = |\theta^i_t(a) - \theta^j_t(a)|^{p\cdot\frac{1}{p}}\leq \left(\sum_{a\in\s_t} |\theta^i_t(a) - \theta^j_t(a)|^p\right)^{\frac{1}{p}} < \epsilon.$
\end{proof}
\begin{theorem}
Let $n$ be a positive integer. Consider a user $i\in[n]$. We denote variables for user $i$ by superscript $i$. Let $\mu^i_t, \theta^i_t:\s_t\rightarrow \mathbb{R}$ for $i\in[n]$. Let $\phi:\mathbb{R}\rightarrow[0,1]$ be any non-decreasing surjective function. Let $\epsilon\in\mathbb{R}^+$. Define the influence set of users $\U_i$ to user $i$, $\U^i_t:=\{u\in[n] \mid d(\theta^i_t, \theta^u_t)<\epsilon\}$. If the following conditions hold,
\begin{align}
&a^{1, i}_t, \ldots, a^{k, i}_t = \text{ top k } (\theta^i_t(a)), \\
&\text{ where } a\in\s_t, \s_{t-1}\subset\s_t\subset \M, |\s_t|<\infty, \forall i\in[n],\\
&\exists C>0, \text{ s.t. } \theta^i_t(a)<C, \text{ for }\forall a\in\s_t\setminus\s_{t-1}, \forall i\in[n],\label{eq:_bdd_init_u}\\
&\forall i\in[n], \exists \text{ a non-empty } \h^i\subset\M \text{ such that}\\
&\qquad \forall a\in\h^i \text{ is served infinitely often},\\
&c^i_t(a) = Bernoulli(p=\phi(\mu^i_t(a))), \forall a\in \M, \forall i\in[n],\\
&c^i_t(a) = 1:
\begin{cases}
\theta^i_{t+1}(a) > \theta^i_t(a), \\
\theta^j_{t+1}(a) > \theta^j_t(a) \text{ for } \forall j\in\U^i_t,\\
\mu^i_{t+1}(a) > \mu^i_t(a).
\end{cases}\\
&c^i_t(a) = 0:
\begin{cases}
\theta^i_{t+1}(a) < \theta^i_t(a), \\
\theta^j_{t+1}(a) < \theta^j_t(a) \text{ for } \forall j\in\U^i_t,\\
\mu^i_{t+1}(a) < \mu^i_t(a).
\end{cases}\\
&|\theta^i_{t+1}(a^{l, i}_t) - \theta^i_t(a^{l, i}_t)| > \sum_{u\in\U^i_t} |\theta^i_{t+1}(a^{l', u}_t) - \theta^i_t(a^{l', u}_t)|, \\
&\text{ for } l,l'\in[k].\label{eq:_direct_influence_u}\\
&\forall t_0 > 0, \forall j\in[n], \forall a\in\M, s.t. |\{t|a_t=a\}|=\infty,\notag\\
&\qquad \sum_{\substack{t=t_0, \\a_t=a}}^\infty |\mu^j_{t+1}(a) - \mu^j_t (a)| = \infty,\\
&\exists B>0, \text{ s.t. for } \forall t>0, \forall a\in\M, \\
&\qquad|\theta^i_t(a) - \mu^i_t(a)|<B, \forall i\in[n],\label{eq:_bdd_est_u}
\end{align}
then we have a degenerate system, specifically the system results in either of the two following scenarios. For any user $j\in[n]$,
\begin{enumerate}
\item $\mu^j_t(a) \rightarrow \infty$ with probability 1 for $1$ to $k$ elements $a\in\M$,
\item $\mu^j_t(a) \rightarrow -\infty$ for $\forall a \in\h^j$. For $\forall a\in\mathcal{M}\setminus\h^j$, there exists an infinite sequence $\{t_1, t_2, \ldots\}$ where $\mu^j_{t_i}(a)\rightarrow -\infty$ as $i\rightarrow \infty$.
\end{enumerate}
\end{theorem}
\begin{proof}
We argue by contradiction. Let $j\in[n]$ be any user. Suppose the contrary to the conclusion, i.e. $\exists a^1, a^2, \ldots, a^{k'}\in\M$ s.t. for $\forall N>0, \exists t_0>0$ s.t. $\mu^j_t(a^i)>N$ for all $t \geq t_0, i=[1,\ldots,k']$ and $k'>k$. Take $N'>C+B$ and let $t'_0$ be the corresponding $t_0$. All other $a\in\M$ have bounded $\mu^j_t(a)<B_m$ for any $t>0$. Invoking Lemma~\ref{lem:s3}, $\exists t'>t'_0$ s.t. $\theta^j_{t'}(a^i)$ are the top $k'$ values in $\s_{t'}$ for $i=1,2\dots,k'$. Without loss of generality, let $a^1,\ldots, a^{k'}$ be a decreasingly sorted list according to values of $\theta^j_{t'}(a^i)$. Since the system can only select $k$ options at time $t'$, it selects $a^1, \ldots, a^k$. Since $N'$ can be arbitrarily large, with probability 1 we have $c^j_{t'}(a^i)=1$ for $i=1,\ldots,k$.
The distance measure $d$ is symmetric, i.e. $d(\mu^i_t, \mu^j_t) = d(\mu^j_t, \mu^i_t)$. Then $\U^j_t$ contains all the users that may change $\theta^j_t$. By Lemma~\ref{lem:s41}, for any $u\in\U^j_t$ we have $|\theta^u_t(a) - \theta^j_t(a)| < \epsilon$ for all $a\in \s_t$. Thus if we consider the model update equations for user $u$, and let $N^u = N'-\epsilon-2B$, then
\begin{align}
&\mu^u_t(a^i) > \theta^u_t(a^i)-B > (\theta^j_t(a^i)-\epsilon)-B \\
&> (\mu^j_t(a^i)-B)-\epsilon-B > N'-\epsilon-2B = N^u,
\end{align}
for all $t \geq t'_0, i=[1,\ldots,k']$. Thus $\mu^u_t(a)\rightarrow \infty$ as $t\rightarrow\infty$ for $u\in\U^j_t, i=[1,\ldots,k']$. For all other options $a\in\M$, $\mu^u_t(a)$ must be bounded by $B_m+\epsilon+2B$ for all $t>0$. Invoking Lemma~\ref{lem:s3} again, $\exists t'>t'_0$ s.t. $\theta^u_{t'}(a^i)$ are the top $k'$ values in $\s_{t'}$ for $i=1,2\dots,k'$. The system selects the top $k$ values for user $u$ at time $t'$. Let the top $k$ options be $a^{\pi_1}, \ldots, a^{\pi_k}$. Since $N^u$ can be arbitrarily large, $c^u_{t'}(a^{\pi_i})=1$ with probability 1. Thus any user activity from the influence set $\U_j$ does not decrease any model parameter $\theta^j_t$ for user $j$.
Hence the top $k$ values of $\theta^j_{t'}(a^i)$ all increase at $t'+1$. We look at whether the top $k$ options can change in the next step by examining the model parameters for all options in $\s_{t'+1}$. Notice that
\[
\s_{t'+1} = (\s_{t'+1}\setminus\s_{t'}) \cup (\s_{t'}\setminus[k']) \cup ([k']\setminus[k]) \cup [k].
\]
At $t'+1$, we have
\[
\mu^j_{t'+1}(a^i) > N' > C+B > \theta^j_{t'+1}(a)+B > \mu^j_{t'+1}(a),
\]
for $i=1,\ldots,k$ and $a\in\s_{t'+1}\setminus\s_{t'}$. Since for $\forall u\in\U_j$, only $a^1, \ldots, a^{k'}$ are possible to be selected. The model parameters $\theta^j_{t'+1}(a), a\not\in[k']$ won't be affected by other users. Hence for all other $a\in\s_{t'}\setminus[k']$, $\theta^j_{t'+1}(a) = \theta^j_{t'}(a)$, remaining the same. The only options that may be listed into the top $k$ at time $t'+1$ are $a^1, \ldots, a^{k'}$. However from Condition~\ref{eq:_direct_influence_u}, for any $a^l, l=k+1, \ldots, k'$ that is served to any users $u\in\U^j_t$, $\Delta \theta^j_{t'}(a^i) > \sum_u \Delta \theta^j_{t'}(a^l)$ for $i=1,\ldots, k$. Thus $\theta^j_{t'+1}(a^l), l=k+1, \ldots,k'$ are smaller than any of $\theta^j_{t'+1}(a^i), i=1, \ldots, k$. The top $k$ options remain $a^1, \ldots, a^k$. Notice that the analysis is true for all $t>t'$.
Therefore by inferring on every time step, $a^1, \ldots, a^k$ maintain in the top $k$ options of $\s_{t}$ for $t>t'$. Therefore $a^{k+1}, \ldots, a^{k'}$ will be selected after time $t'$ with probability = 0. This is a contradiction with $\mu^j_t(a^i)\rightarrow \infty$ as $t\rightarrow \infty$ for $i=k+1, \ldots, k'$!
In the second scenario, with Condition~\ref{eq:_direct_influence_u} we can apply the same argument as in Theorem~\ref{thm:s2_suff} for each user $j\in[n]$, and reach the stated conclusion. {\todo double check carefully}
\end{proof}
\section{Summary of essential sufficient conditions}
\label{essential_suff}
Notice that Theorem~\ref{thm:s1_suff}-\ref{thm:s3_suff} still hold when we restrict the codomain of the function $\mu_t$ to a finite range $\mathcal{D}$ by simply replacing $\mu_t$ with $\psi(\mu_t)$ where $\psi:\M\rightarrow \mathcal{D}$. Thus some of the conditions in the sufficiency theorems are useful for derivation of their proofs but may not provide intuition for the main causes of degeneracy in a control system. Below we recall and summarize the essential conditions in each theorem, and their interpretation in practice.
\begin{enumerate}
\item Essential conditions in Theorem~\ref{thm:s1_suff}:
\begin{flalign*}
&a_t = \argmax_a \theta_t(a),\\
&\text{(The system optimizes for user feedback.)}\\
&c_t = 1:
\begin{cases}
\theta_{t+1}(a_t) > \theta_t(a_t), \\
\mu_{t+1}(a_t) > \mu_t(a_t).
\end{cases}\\
&c_t = 0:
\begin{cases}
\theta_{t+1}(a_t) < \theta_t(a_t), \\
\mu_{t+1}(a_t) < \mu_t(a_t).
\end{cases}\\
&\text{(Model estimate and user interest for an engaged option }\\
&\text{increase, and vice versa.) }\\
&\exists B>0, \text{ s.t. for } \forall t>0, \forall a\in[m], |\theta_t(a) - \mu_t(a)|<B, \\
&\text{(Model estimates cannot be completely inaccurate.) }
\end{flalign*}
\item Theorem~\ref{thm:s2_suff} is very similar to Theorem~\ref{thm:s1_suff} but just for top $k$ items.
\item New conditions in Theorem~\ref{thm:s3_suff}:
\begin{flalign*}
&a^1_t, \ldots, a^k_t = \text{ top k } (\theta_t(a)), \\
&\text{ where }a\in\s_t, \s_{t-1}\subset\s_t\subset \M, |\s_t|<\infty,\\
&\text{(At each time step, the system's reservoir of options }\\
&\text{expands with new options.)}\\
&\exists C>0, \text{ s.t. } \theta_t(a)<C, \text{ for }\forall a\in\s_t\setminus\s_{t-1},\\
&\text{(Model estimates for unseen new options are bounded.)}
\end{flalign*}
\item New conditions in Theorem~\ref{thm:s4_suff}:
\begin{flalign*}
&c^i_t(a) = 1: \theta^j_{t+1}(a) > \theta^j_t(a) \text{ for } \forall j\in[n] \text{ s.t. } d(\mu^i_t, \mu^j_t) < \epsilon,\\
&c^i_t(a) = 0: \theta^j_{t+1}(a) < \theta^j_t(a) \text{ for } \forall j\in[n] \text{ s.t. } d(\mu^i_t, \mu^j_t) < \epsilon,\\
&\text{(An engaged option increases model estimate of similar users'}\\
&\text{corresponding model parameter, and vice versa.)}\\
&|\theta^i_{t+1}(a^{l, i}_t) - \theta^i_t(a^{l, i}_t)| > \sum_{u\in\U_i} |\theta^i_{t+1}(a^{l', u}_t) - \theta^i_t(a^{l', u}_t)| \\
&\text{ for } l,l'\in[k]. \\
&\text{(Direct feedbacks result in more significant model updates}\\
&\text{than drawing inferences from similar users.)}
\end{flalign*}
\end{enumerate}
\end{appendices}
|
{
"timestamp": "2019-03-28T01:15:48",
"yymm": "1902",
"arxiv_id": "1902.10730",
"language": "en",
"url": "https://arxiv.org/abs/1902.10730"
}
|
\section{Introduction}
\vspace{-0.5em}
Learning the feature embedding representation that preserves the notion of similarities among the data is of great practical importance in machine learning and vision and is at the basis of modern similarity-based search \cite{facenet, npairs}, verification \cite{deepface}, clustering \cite{seanbell}, retrieval \cite{liftedstruct,facility}, zero-shot learning \cite{zeroshot1,zeroshot2}, and other related tasks. In this regard, deep metric learning methods \cite{seanbell,facenet,npairs} have shown advances in various embedding tasks by training deep convolutional neural networks end-to-end encouraging similar pairs of data to be close to each other and dissimilar pairs to be farther apart in the embedding space.
Despite the progress in improving the embedding representation accuracy, improving the inference efficiency and scalability of the representation in an end-to-end optimization framework is relatively less studied. Practitioners deploying the method on large-scale applications often resort to employing post-processing techniques such as embedding thresholding \cite{agrawal2014,zhai2017} and vector quantization \cite{survey_learningtohash} at the cost of the loss in the representation accuracy. Recently, Jeong \& Song \cite{jeong2018} proposed an end-to-end learning algorithm for quantizable representations which jointly optimizes the quality of the convolutional neural network based embedding representation and the performance of the corresponding sparsity constrained compound binary hash code and showed significant retrieval speedup on ImageNet \cite{imagenet} without compromising the accuracy.
In this work, we seek to learn hierarchically quantizable representations and propose a novel end-to-end learning method significantly increasing the quantization granularity while keeping the time and space complexity manageable so the method can still be efficiently trained in a mini-batch stochastic gradient descent setting. Besides the efficiency issues, however, naively increasing the quantization granularity could cause a severe degradation in the search accuracy or lead to dead buckets hindering the search speedup.
To this end, our method jointly optimizes both the sparse compound hash code and the corresponding embedding representation respecting a hierarchical structure. We alternate between performing cascading optimization of the optimal sparse compound hash code per each level in the hierarchy and updating the neural network to adjust the corresponding embedding representations at the active bits of the compound hash code.
Our proposed learning method outperforms both the reported results in \cite{jeong2018} and the state of the art deep metric learning methods \cite{facenet, npairs} in retrieval and clustering tasks on Cifar-100 \cite{cifar100} and ImageNet \cite{imagenet} datasets while, to the best of our knowledge, providing the highest reported inference speedup on each dataset over exhaustive linear search.
\vspace{-0.8em}
\section{Related works}\vspace{-0.5em}
Embedding representation learning with neural networks has its roots in Siamese networks \cite{signatureVerification, contrastive} where it was trained end-to-end to pull similar examples close to each other and push dissimilar examples at least some margin away from each other in the embedding space. \cite{signatureVerification} demonstrated the idea could be used for signature verification tasks. The line of work since then has been explored in wide variety of practical applications such as face recognition \cite{deepface}, domain adaptation \cite{domaintransduction}, zero-shot learning \cite{zeroshot1,zeroshot2}, video representation learning \cite{triplet_video}, and similarity-based interior design \cite{seanbell}, etc.
Another line of research focuses on learning binary hamming ranking \cite{xia2014supervised, zhao2015deep, hammingmetric, li2017deep} representations via neural networks. Although comparing binary hamming codes is more efficient than comparing continuous embedding representations, this still requires the linear search over the entire dataset which is not likely to be as efficient for large scale problems. \cite{cao2016deep, liu2017learning} seek to vector quantize the dataset and back propagate the metric loss, however, it requires repeatedly running k-means clustering on the entire dataset during training with prohibitive computational complexity.
We seek to jointly learn the hierarchically quantizable embedding representation and the corresponding sparsity constrained binary hash code in an efficient mini-batch based end-to-end learning framework. Jeong \& Song \cite{jeong2018} motivated maintaining the hard constraint on the sparsity of hash code to provide guaranteed retrieval inference speedup by only considering $k_s$ out of $d$ buckets and thus avoiding linear search over the dataset. We also explicitly maintain this constraint, but at the same time, greatly increasing the number of representable buckets by imposing an efficient hierarchical structure on the hash code to unlock significant improvement in the speedup factor.
\vspace{-0.5em}
\section{Problem formulation}\vspace{-0.5em}
Consider the following hash function
\[r(\mathbf{x})=\mathop{\rm argmin}_{\mathbf{h} \in \{0,1\}^d} -f(\mathbf{x}; \boldsymbol\theta)^\intercal \mathbf{h}~\] under the constraint that $\|\mathbf{h}\|_1 = k_s$. The idea is to optimize the weights in the neural network $f(\cdot; \boldsymbol\theta): \mathcal{X} \rightarrow \mathbb R^d$, take $k_s$ highest activation dimensions, activate the corresponding dimensions in the binary compound hash code $\mathbf{h}$, and hash the data $\mathbf{x} \in \mathcal{X}$ into the corresponding active buckets of a hash table $\mathcal{H}$. During inference, a query $\mathbf{x}_q$ is given, and all the hashed items in the $k_s$ active bits set by the hash function $r(\mathbf{x}_q)$ are retrieved as the candidate nearest items. Often times \cite{survey_learningtohash}, these candidates are reranked based on the euclidean distance in the base embedding representation $f(\cdot; \boldsymbol\theta)$ space.
Given a query $\mathbf{h}_q$, the expected number of retrieved items is $\sum_{i\neq q} \Pr(\mathbf{h}_i^\intercal \mathbf{h}_q \neq 0)$. Then, the expected speedup factor \cite{jeong2018} (SUF) is the ratio between the total number of items and the expected number of retrieved items. Concretely, it becomes $(\Pr(\mathbf{h}_i^\intercal \mathbf{h}_q \neq 0))^{-1} = (1 - {d-k_s \choose k_s}/{d \choose k_s})^{-1}$. In case $d \gg k_s$, this ratio approaches $d/{k_s}^2$.
Now, suppose we design a hash function $r(\mathbf{x})$ so that the function has total $\mathrm{dim}(r(\mathbf{x}))=d^k$ (\ie~ exponential in some integer parameter $k>1$) indexable buckets. The expected speedup factor \cite{jeong2018} approaches $d^k/k_s^2$ which means the query time speedup increases linearly with the number of buckets. However, naively increasing the bucket size for higher speedup has several major downsides. First, the hashing network has to output and hold $d^k$ activations in the memory at the final layer which can be unpractical in terms of the space efficiency for large scale applications. Also, this could also lead to \emph{dead buckets} which are under-utilized and degrade the search speedup. On the other hand, hashing the items uniformly at random among the buckets could help to alleviate the dead buckets but this could lead to a severe drop in the search accuracy.
Our approach to this problem of maintaining a large number of representable buckets while preserving the accuracy and keeping the computational complexity manageable is to enforce a hierarchy among the optimal hash codes in an efficient tree structure. First, we use $\mathrm{dim}(f(\mathbf{x})) = d k$ number of activations instead of $d^k$ activations in the last layer of the hash network. Then, we define the unique mapping between the $dk$ activations to $d^k$ buckets by the following procedure.
Denote the hash code as $\widetilde\mathbf{h} = [\mathbf{h}^1, \ldots, \mathbf{h}^k] \in \{0,1\}^{d\times k}$ where $\| \mathbf{h}^v \|_1 = 1 ~~\forall v \neq k$ and $\| \mathbf{h}^k \|_1 = k_s$. The superscript denotes the level index in the hierarchy. Now, suppose we construct a tree $\mathcal{T}$ with branching factor $d$, depth $k$ where the root node has the level index of $0$. Let each $d^k$ leaf node in $\mathcal{T}$ represent a bucket indexed by the hash function $r(\mathbf{x})$. Then, we can interpret each $\mathbf{h}^v$ vector to indicate the branching from depth $v-1$ to depth $v$ in $\mathcal{T}$. Note, from the construction of $\widetilde\mathbf{h}$, the branching is unique until level $k-1$, but the last branching to the leaf nodes is multi-way because $k_s$ bits are set due to the sparsity constraint at level $k$. \Cref{fig:hash_tree} illustrates an example translation from the given hash activation to the tree bucket index for $k\!=\!2$ and $k_s\!=\!2$. Concretely, the hash function $r(\mathbf{x}): \mathbb{R}^{d\times k} \rightarrow \{0,1\}^{d^k}$ can be expressed compactly as \Cref{eqn:hash_function}. \vspace{-0.5em}
\small
\begin{align}
\label{eqn:hash_function}
r&(\mathbf{x}) = \bigotimes_{v=1}^k~ \mathop{\rm argmin}_{\substack{\mathbf{h}^v}} -\left(f(\mathbf{x}; \boldsymbol\theta)^v\right)^\intercal \mathbf{h}^v \\
&\text{subject to } \|\mathbf{h}^v\|_1=\begin{dcases}1 & \forall v\neq k\\ k_s &v = k \end{dcases} \text{ and } \mathbf{h}^v \in \{0,1\}^d \nonumber
\end{align}
\normalsize
where $\bigotimes$ denotes the tensor multiplication operator between two vectors. The following section discusses how to find the optimal hash code $\widetilde\mathbf{h}$ and the corresponding activation $f(\mathbf{x}; \boldsymbol\theta) = [f(\mathbf{x}; \boldsymbol\theta)^1, \ldots, f(\mathbf{x}; \boldsymbol\theta)^k] \in \mathbb R^{d \times k}$ respecting the hierarchical structure of the code.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{hash_tree3.pdf}
\caption{Example hierarchical structure for $k\!=\!2$ and $k_s\!=\!2$. (Left) The hash code for each embedding representation $[f(\mathbf{x}_i; \boldsymbol\theta)^1, f(\mathbf{x}_i; \boldsymbol\theta)^2] \in \mathbb R^{2d}$. (Right) Corresponding activated hash buckets out of total $d^2$ buckets.}
\label{fig:hash_tree}
\end{figure} \vspace{-0.3em}
\section{Methods}\vspace{-0.7em}
To compute the optimal set of embedding representations and the corresponding hash code,
the embedding representations are first required in order to infer which $k_s$ activations to set in the hash code, but to learn the embedding representations, it requires the hash code to determine which dimensions of the activations to adjust so that similar items would get hashed to the same buckets and vice versa. We take the alternating minimization approach iterating over computing the sparse hash codes respecting the hierarchical quantization structure and updating the network parameters indexed at the given hash codes per each mini-batch. \Cref{sec:method_hash} and \Cref{sec:method_embed} formalize the subproblems in detail. \vspace{-0.5em}
\subsection{Learning the hierarchical hash code} \vspace{-0.3em}
\label{sec:method_hash}
Given a set of continuous embedding representation $\{f(\mathbf{x}_i; \boldsymbol\theta)\}_{i=1}^n$, we wish to compute the optimal binary hash code $\{\mathbf{h}_1, \ldots, \mathbf{h}_n\}$ so as to hash similar items to the same buckets and dissimilar items to different buckets. Furthermore, we seek to constrain the hash code to simultaneously maintain the hierarchical structure and the hard sparsity conditions throughout the optimization process. Suppose items $\mathbf{x}_i$ and $\mathbf{x}_j$ are dissimilar items, in order to hash the two items to different buckets, at each level of $\mathcal{T}$, we seek to encourage the hash code for each item at level $v$, $\mathbf{h}_i^v$ and $\mathbf{h}_j^v$ to differ. To achieve this, we optimize the hash code for all items per each level sequentially in cascading fashion starting from the first level $\{\mathbf{h}_1^1, \ldots, \mathbf{h}_n^1\}$ to the leaf nodes $\{\mathbf{h}_1^k, \ldots, \mathbf{h}_n^k\}$ as shown in \Cref{eqn:energy_seq}.
\footnotesize
\begin{align}
\label{eqn:energy_seq}
&\operatorname*{minimize}_{\mathbf{h}^k_{1:n},\ldots, \mathbf{h}^1_{1:n}}\quad \underbrace{\sum_{v=1}^{k} \sum_{i=1}^{n} -({f(\mathbf{x}_i; \boldsymbol\theta)^{v}})^\intercal~ \mathbf{h}^v_i}_{\text{unary term}}\\
&+ \underbrace{\sum_{v=2}^k \sum_{(i,j) \in \mathcal{N}} {\mathbf{h}^v_i}^\intercal Q' {\mathbf{h}^v_j}\prod_{w=1}^{v-1}\mathds{1}(\mathbf{h}^w_i=\mathbf{h}^w_j) }_{\text{sibling penalty}} + \underbrace{\sum_{v=1}^{k}\sum_{(i,j) \in \mathcal{N}} {\mathbf{h}_i^v}^\intercal P' {\mathbf{h}^v_j}}_{\text{orthogonality}} \nonumber\\
&\text{\ \ subject to }~~ \| \mathbf{h}^v_i \| = \begin{dcases}1 & \forall v\neq k\\ k_s &v = k \end{dcases},~ \mathbf{h}^v_i\in \{0,1\}^d,~ \forall i, \nonumber
\end{align}
\normalsize
where $\mathcal{N}$ denotes the set of dissimilar pairs of data and $\mathds{1}(\cdot)$ denotes the indicator function. Concretely, given the hash codes from \emph{all the previous levels}, we seek to minimize the following discrete optimization problem in \Cref{eqn:energy_tree}, subject to the same constraints as in \Cref{eqn:energy_seq}, sequentially for all levels\footnote{In \Cref{eqn:energy_tree}, we omit the dependence of $v$ for all $\mathbf{h}_1,\ldots,\mathbf{h}_n$ to avoid the notation clutter.} $v \in \{1,\ldots,k\}$. The unary term in the objective encourages selecting as large elements of each embedding vector as possible while the second term loops over \emph{all pairs of dissimilar siblings} and penalizes for their orthogonality. The last term encourages selecting as orthogonal elements as possible for a pair of hash codes from different classes in the current level $v$. The last term also makes sure, in the event that the second term becomes zero, the hash code still respects orthogonality among dissimilar items. This can occur when the hash code for all the previous levels was computed perfectly splitting dissimilar pairs into different branches and the second term becomes zero.
\vspace{-0.5em}
\scriptsize
\begin{align}
\label{eqn:energy_tree}
&\operatorname*{minimize}_{\mathbf{h}_1, \ldots, \mathbf{h}_n}\quad \underbrace{\sum_{i=1}^{n} -({f(\mathbf{x}_i; \boldsymbol\theta)^{v}})^\intercal~ \mathbf{h}_i}_{\text{unary term}} + \underbrace{\sum_{(i,j) \in \mathcal{S}^v} \mathbf{h}_i^\intercal Q' \mathbf{h}_j}_{\text{sibling penalty}} + \underbrace{\sum_{(i,j) \in \mathcal{N}} \mathbf{h}_i^\intercal P' \mathbf{h}_j}_{\text{orthogonality}}
\end{align}
\normalsize
\noindent where $\mathcal{S}^v = \left\{(i,j) \in \mathcal{N} \mid \mathbf{h}^w_i = \mathbf{h}^w_j, ~\forall w = 1,\ldots,v-1 \right\}$ denotes the set of pairs of siblings at level $v$ in $\mathcal{T}$, and $Q', P'$ encodes the pairwise cost for the sibling and the orthogonality terms respectively. However, optimizing \Cref{eqn:energy_tree} is NP-hard in general even in the simpler case of $k_s=1, k=1, d>2$ \cite{boykov_fast,jeong2018}. Inspired by \cite{jeong2018}, we use the average embedding of each class within the minibatch $\mathbf{c}_p^v = \frac{1}{m} \sum_{i:y_i = p} f(\mathbf{x}_i; \boldsymbol\theta)^v \in \mathbb R^{d}$ as shown in \Cref{eqn:energy_tree_avg}.
\small
\begin{align}
&\operatorname*{minimize}_{\mathbf{z}_1, \ldots, \mathbf{z}_{n_c}}\quad \underbrace{\sum_{p=1}^{n_c} -({\mathbf{c}_p^v})^\intercal \mathbf{z}_p +\sum_{\substack{(p,q) \in \mathcal{S}_z^v \\ p \neq q}} {\mathbf{z}_p}^\intercal Q \mathbf{z}_q + \sum_{p\neq q} {\mathbf{z}_p}^\intercal P \mathbf{z}_q}_{:=\hat{g}(\mathbf{z}_1, \ldots, \mathbf{z}_{n_c})} \nonumber\\
&\text{\ \ subject to }~~ \| \mathbf{z}_p \| = \begin{dcases}1 & \forall v \neq k\\k_s & v=k\end{dcases},~ \mathbf{z}_p\in \{0,1\}^d,~ \forall p,
\label{eqn:energy_tree_avg}
\end{align}
\normalsize
\noindent where $\mathcal{S}_z^v=\left\{(p,q)\mid \mathbf{z}_p^w=\mathbf{z}_q^w, ~\forall w =1,\ldots,v-1 \right\}$, $n_c$ is the number of unique classes in the minibatch, and we assume each class has $m$ examples in the minibatch (\ie~ \textit{npairs} \cite{npairs} minibatch construction). Note, in accordance with the deep metric learning problem setting \cite{facenet,npairs,jeong2018}, we assume we are given access to the label adjacency information only within the minibatch. \\
\noindent The objective in \Cref{eqn:energy_tree_avg} upperbounds the objective in \Cref{eqn:energy_tree} (denote as $g(\cdot; \boldsymbol\theta)$) by a gap $M(\boldsymbol\theta)$ which depends only on $\boldsymbol\theta$. Concretely, rewriting the summation in the unary term in $g$, we get
\small
\begin{align}
\label{eqn:bound}
&g(\mathbf{h}_1, \ldots, \mathbf{h}_n; \boldsymbol\theta) = \sum_p^{n_c} \sum_{i:y_i=p} -({f(\mathbf{x}_i; \boldsymbol\theta)^v})^\intercal~ \mathbf{h}_i\\
&\hspace{8em} + \sum_{(i,j)\in\mathcal{S}^v} \mathbf{h}_i^\intercal Q' \mathbf{h}_j + \sum_{(i,j)\in\mathcal{N}} \mathbf{h}_i^\intercal P' \mathbf{h}_j \nonumber\\
&\leq\sum_p^{n_c} \sum_{i: y_i=p} -({\mathbf{c}_p^{v}})^\intercal~ \mathbf{h}_i + \sum_{(i,j)\in\mathcal{S}^v} \mathbf{h}_i^\intercal Q' \mathbf{h}_j + \sum_{(i,j)\in\mathcal{N}} \mathbf{h}_i^\intercal P' \mathbf{h}_j \nonumber\\
&\quad +\underbrace{\operatorname*{maximize}_{\hat\mathbf{h}_1, \ldots, \hat\mathbf{h}_n} \sum_p^{n_c} \sum_{i:y_i = p} (\mathbf{c}_p^{v} - f(\mathbf{x}_i; \boldsymbol\theta)^{v})^\intercal~ \hat\mathbf{h}_i}_{:=M(\boldsymbol\theta)}. \nonumber
\end{align}
\normalsize
\noindent Minimizing the upperbound in \Cref{eqn:bound} over $\mathbf{h}_1,\ldots,\mathbf{h}_n$ is identical to minimizing the objective $\hat{g}(\mathbf{z}_1, \ldots, \mathbf{z}_{n_c})$ in \Cref{eqn:energy_tree_avg} since each example $j$ in class $i$ shares the same class mean embedding vector $\mathbf{c}_i$. Absorbing the factor $m$ into the cost matrices \ie~ $Q=mQ'$ and $P = mP'$, we arrive at the upperbound minimization problem defined in \Cref{eqn:energy_tree_avg}. In the upperbound problem \Cref{eqn:energy_tree_avg}, we consider the case where the pairwise cost matrices are diagonal matrices of non-negative values. \Cref{thm:equivalence} in the following subsection proves that finding the optimal solution of the upperbound problem in \Cref{eqn:energy_tree_avg} is equivalent to finding the minimum cost flow solution of the flow network $G'$ illustrated in \Cref{fig:mcf}. Section B in the supplementary material shows the running time to compute the minimum cost flow (MCF) solution is approximately linear in $n_c$ and $d$. On average, it takes 24 \textit{ms} and 53 \textit{ms} to compute the MCF solution (discrete update) and to take a gradient descent step with npairs embedding \cite{npairs} (network update), respectively on a machine with 1 TITAN-XP GPU and Xeon E5-2650.
\subsection{Equivalence of the optimization problem to minimum cost flow} \vspace{-0.5em}
\begin{figure*}[ht]
\centering
\resizebox{2.0\columnwidth}{!}{
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\path (b0q) edge [draw=none,bend right=85,midway] node {$\vdots$} (t);
\end{tikzpicture}
}
\vspace{-1em}
\caption{Equivalent flow network diagram $G'$ corresponding to the discrete optimization \Cref{eqn:energy_tree_avg}. Edge labels show the capacity and the cost respectively.}
\label{fig:mcf} \vspace{-1em}
\end{figure*}
\begin{thm}\label{thm:equivalence}
The optimization problem in \Cref{eqn:energy_tree_avg} can be solved exactly in polynomial time by finding the minimum cost flow solution on the flow network G'.
\end{thm}
\vspace{-1em}
\begin{proof}
Suppose we construct a vertex set $A = \{a_1,\ldots,a_{n_c}\}$ and partition $A$ into $\{A_r\}_{r=0}^l$ with the partition of $\{1,\ldots,n_c\}$ from equivalence relation $\mathcal{S}_{z}^v$\footnote{Define $(p,q) \in \mathcal{S}_z^v \iff a_p,a_q \in A_r, \forall r \geq 1$}. Here, we will define $A_0$ as a union of subsets of size $1$ (\ie~ each element in $A_0$ is a singleton without a sibling), and $A_1,\ldots,A_l$ as the rest of the subsets (of size greater than or equal to$2$). Concretely, $\left| A \right| = n_c$ and $A=\bigcup_{r=0}^l A_r$.
Then, we construct $l+1$ set of complete bipartite graphs $ \{G_r = \left( A_r \cup B_r, E_r \right) \}_{r=0}^l$ where we define $g_r\!=\!\left| A_r \right| $ and $\left| B_r \right| \!=\! d~~ \forall r$. Now suppose we construct a directed graph $G'$ by directing all edges $E_r$ from $A_r$ to $B_r$, attaching source $s$ to all vertices in $A_r$, and attaching sink $t$ to all vertices in $B_0$. Formally, $G' = \left(\bigcup_{r=0}^l \left(A_r \cup B_r\right) \cup \{s,t\}, E'\right)$. The edges in $E'$ inherit all directed edges from source to vertices in $A_r$, edges from vertices in $B_0$ to sink, and $\{E_r\}_{r=0}^l$. We also attach $g_r$ number of edges for each vertex $b_{r,q} \in B_r$ to $b_{0,q} \in B_0$ and attach $n_c$ number of edges from each vertex $b_{0,q} \in B_0$ to $t$. Concretely, $E'$ is
\vspace{-0.5em}
\scriptsize
\begin{align}
&\{(s,a_p)|a_p\in A\} \cup \bigcup_{r=0}^l E_r\cup \bigcup_{r=1}^l \{(b_{r,q}, b_{0,q})_i\}_{i=0}^{g_r-1} \cup \{(b_{0,q}, t)_j\}_{j=0}^{n_c-1}. \nonumber
\end{align}
\normalsize
\vspace{-1em}
Edges incident to $s$ have capacity $u(s,a_p) = k_s$ and cost $v(s, a_p)= 0$ for all $a_p \in A$. The edges between $a_p \in A_r$ and $b_{r,q} \in B_r$ have capacity $u(a_p, b_{r,q}) = 1$ and cost $v(a_p, b_{r,q}) = -\mathbf{c}_p[q]$. Each edge $i \in \{0, \ldots, g_r - 1\}$ between $b_{r,q} \in B_r$ and $b_{0,q} \in B_0$ has capacity $u\left( \left( b_{r,q}, b_{0,q} \right)_i \right) = 1$ and cost $u\left( \left( b_{r,q}, b_{0,q} \right)_i \right) = 2 \alpha i$. Each edge $j \in \{0, \ldots, n_c -1\}$ between $b_{0,q} \in B_0$ and $t$ has capacity $u\left( \left( b_{0,q}, t \right)_j \right) = 1$ and cost $v\left( \left( b_{0,q}, t \right)_j \right) = 2 \beta j$. Figure \ref{fig:mcf} illustrates the flow network $G'$. The amount of flow from source to sink is $n_c k_s$. The figure omits the vertices in $A_0$ and the corresponding edges to $B_0$ to avoid the clutter.\\
Now we define the flow $\{f_z(e)\}_{e \in E'}$ for each edge indexed both by flow configuration $\mathbf{z}_p \in \mathbf{z}_{1:n_c}$ where $\mathbf{z}_p \in \{0,1\}^d, \|\mathbf{z}_p\|_1 = k_s ~\forall p$ and $e \in E'$ below in \Cref{eqn:flow_def}.
\small
\begin{align}
&(i)~ f_z(s, a_p) = k_s,~ (ii)~ f_z(a_p, b_{r,q}) = \mathbf{z}_p[q]\nonumber\\
&(iii)~ f_z\left( \left( b_{r,q}, b_{0,q}\right)_i \right) = \begin{cases} 1 & ~\forall i < \sum_{p:a_p \in A_r} \mathbf{z}_p[q]\\ 0 & \text{otherwise}\end{cases} \nonumber\\
&(iv)~ f_z\left( \left(b_{0,q}, t\right)_j \right) = \begin{cases} 1 & ~\forall j < \sum_{p=1}^{n_c} \mathbf{z}_p[q]\\0 & \text{otherwise}\end{cases}
\label{eqn:flow_def}
\end{align}
\normalsize
To prove the equivalence of computing the minimum cost flow solution and finding the minimum binary assignment in \Cref{eqn:energy_tree_avg}, we need to show (1) that the flow defined in \Cref{eqn:flow_def} is feasible in $G'$ and (2) that the minimum cost flow solution of the network $G'$ and translating the computed flows to $\{\mathbf{z}_p\}$ in \Cref{eqn:energy_tree_avg} indeed minimizes the discrete optimization problem. We first proceed with the flow feasibility proof.\vspace{0.5em}
It is easy to see the capacity constraints are satisfied by construction in \Cref{eqn:flow_def} so we prove that the flow conservation conditions are met at each vertices. First, the output flow from the source $\sum_{a_p \in A} f_z(s,a_p) = \sum_{p=1}^{n_c} k_s = n_ck_s$ is equal to the input flow. For each vertex $a_p \in A$, the amount of input flow is $k_s$ and the output flow is the same $\sum_{b_{r,q}\in B_r} f_z(a_p,b_{r,q}) = \sum_{q=1}^d \mathbf{z}_p[q]=\| \mathbf{z}\|_1=k_s$.\\
For $r > 0$, for each vertex $b_{r,q} \in B_r$, denote the input flow as $y_{r,q} = \sum_{a_p \in A_r} f_z(a_p, b_{r,q}) = \sum_{p:a_p\in A_r} \mathbf{z}_p[q]$. The output flow is $\sum_{i=0}^{g_r-1} f_z((b_{r,q},b_{0,q})_i) = \sum_{p : a_p \in A_r} \mathbf{z}_p[q]= y_{r,q}$. The second term vanishes because of \Cref{eqn:flow_def} (iii). \vspace{0.5em}
The last flow conservation condition is to check the connections from each vertex $b_{0,q} \in B_0$ to the sink. Denote the input flow at the vertex as $y_{0,q} = \sum_{p:a_p\in A_0} \mathbf{z}_p[q] + \sum_{r=1}^l y_{r,q}= \sum_{p=1}^{n_c} \mathbf{z}_p[q]$. The output flow is $\sum_{j=0}^{n_c-1} f_z((b_{0,q}, t)_j) = \sum_{p=1}^{n_c} \mathbf{z}_p[q] = y_{0,q}$ which is identical to the input flow. Therefore, the flow construction in \Cref{eqn:flow_def} is feasible in $G'$. \vspace{0.5em}
The second part of the proof is to check the optimality conditions and show the minimum cost flow finds the minimizer of \Cref{eqn:energy_tree_avg}. Denote, $\{f_o(e)\}_{e \in E'}$ as the minimum cost flow solution of the network $G'$ which minimizes the total cost $\sum_{e \in E'} v(e) f_o(e)$. Also denote the optimal flow from $a_p \in A_r$ to $b_{r,q} \in B_r, f_o(a_p, b_q)$ as $\mathbf{z}'_p[q]$. By optimality of the flow, $\{f_o(e)\}_{e\in E'}$, $\sum_{e\in E'} v(e)f_o(e) \leq \sum_{e\in E'} v(e)f_z(e)~~ \forall z$. By \Cref{lemma:fo}, the \textit{lhs} of the inequality is equal to $\sum_{p=1}^{n_c} -{\mathbf{c}_p}^T\mathbf{z'}_p + \sum_{r=1}^l \sum_{p_1\neq p_2\in \{p|a_p\in A_r\}} \alpha {\mathbf{z'}_{p_1}}^T\mathbf{z'}_{p_2} + \sum_{p_1\neq p_2} \beta {\mathbf{z'}_{p_1}}^T\mathbf{z'}_{p_2}$. Additionally, \Cref{lemma:fz} shows the \textit{rhs} of the inequality is equal to $\sum_{p=1}^{n_c} -{\mathbf{c}_p}^T\mathbf{z}_p + \sum_{r=1}^l \sum_{p_1\neq p_2\in \{p|a_p\in A_r\}} \alpha {\mathbf{z}_{p_1}}^T\mathbf{z}_{p_2} + \sum_{p_1\neq p_2} \beta {\mathbf{z}_{p_1}}^T\mathbf{z}_{p_2}$.\vspace{0.5em} Finally, $\forall \{\mathbf{z}\}$
\scriptsize
\begin{align}
&\sum_{p=1}^{n_c} -{\mathbf{c}_p}^T\mathbf{z'}_p + \sum_{r=1}^l \sum_{p_1\neq p_2\in \{p|a_p\in A_r\}} \alpha {\mathbf{z'}_{p_1}}^T\mathbf{z'}_{p_2} +\sum_{p_1\neq p_2} \beta {\mathbf{z'}_{p_1}}^T\mathbf{z'}_{p_2}\nonumber\\
&\leq \sum_{p=1}^{n_c} -{\mathbf{c}_p}^T\mathbf{z}_p + \sum_{r=1}^l \sum_{p_1\neq p_2\in \{p|a_p\in A_r\}} \alpha {\mathbf{z}_{p_1}}^T\mathbf{z}_{p_2} + \sum_{p_1\neq p_2} \beta {\mathbf{z}_{p_1}}^T\mathbf{z}_{p_2}\nonumber.
\end{align}
\normalsize
This shows computing the minimum cost flow solution on $G'$ and converting the flows to $\mathbf{z}$'s, we can find the minimizer of the objective in \Cref{eqn:energy_tree_avg}.
\end{proof} \vspace{-1em}
\begin{lemma}\label{lemma:fo}
Given the minimum cost flow $\{f_o(e)\}_{e \in E'}$ of the network $G'$, the total cost of the flow is $\sum_{e\in E'} v(e)f_o(e) = \sum_{p=1}^{n_c} -{\mathbf{c}_p}^T\mathbf{z'}_p + \sum_{r=1}^l \sum_{p_1\neq p_2\in \{p|a_p\in A_r\}} \alpha {\mathbf{z'}_{p_1}}^T\mathbf{z'}_{p_2} + \sum_{p_1\neq p_2} \beta {\mathbf{z'}_{p_1}}^T\mathbf{z'}_{p_2}$.
\end{lemma}
\vspace{-1em}
\begin{proof}
Proof in section A.2 of the supplementary material.
\end{proof}
\vspace{-1em}
\begin{lemma}\label{lemma:fz}
Given a feasible flow $\{f_z(e)\}_{e \in E'}$ of the network $G'$, the total cost of the flow is $\sum_{e\in E'} v(e)f_z(e) = \sum_{p=1}^{n_c} -{\mathbf{c}_p}^T\mathbf{z}_p + \sum_{r=1}^l \sum_{p_1\neq p_2\in \{p|a_p\in A_r\}} \alpha {\mathbf{z}_{p_1}}^T\mathbf{z}_{p_2} + \sum_{p_1\neq p_2} \beta {\mathbf{z}_{p_1}}^T\mathbf{z}_{p_2}$.
\end{lemma}
\vspace{-1em}
\begin{proof}
Proof in section A.2 of the supplementary material.
\end{proof}
\vspace{-2em}
\subsection{Learning the embedding representation given the hierarchical hash codes} \vspace{-0.5em}
\label{sec:method_embed}
Given a set of binary hash codes for the mean embeddings $\{\mathbf{z}_1^v, \ldots, \mathbf{z}_{n_c}^v\}, ~\forall v=1, \ldots, k$ computed from \Cref{eqn:energy_tree_avg}, we can derive the hash codes for all $n$ examples in the minibatch, $\mathbf{h}_i^v := \mathbf{z}_p^v ~~\forall i : y_i = p$ and update the network weights $\boldsymbol\theta$ given the hierarchical hash codes in turn. The task is to update the embedding representations, $\{f(\mathbf{x}_i; \boldsymbol\theta)^v\}_{i=1}^n, ~\forall v=1, \ldots, k$, so that similar pairs of data have similar embedding representations indexed at the activated hash code dimensions and vice versa. Note, In terms of the hash code optimization in \Cref{eqn:energy_tree_avg} and the bound in \Cref{eqn:bound}, this embedding update has the effect of tightening the bound gap $M(\boldsymbol\theta)$.
We employ the state of the art deep metric learning algorithms (denote as $\ell_{\text{metric}}(\cdot)$) such as \emph{triplet loss with semi-hard negative mining} \cite{facenet} and \emph{npairs loss} \cite{npairs} for this subproblem where the distance between two examples $\mathbf{x}_i$ and $\mathbf{x}_j$ at hierarchy level $v$ is defined as $d_{ij}^v = \| \left(\mathbf{h}_i^v \lor \mathbf{h}_j^v \right) \odot \left( f(\mathbf{x}_i; \boldsymbol\theta)^v - f(\mathbf{x}_j; \boldsymbol\theta)^v \right) \|_1$. Utilizing the logical \emph{OR} of the two binary masks, in contrast to independently indexing the representation with respective masks, to index the embedding representations helps prevent the pairwise distances frequently becoming zero due to the sparsity of the code. Note, this formulation in turn accommodates the backpropagation gradients to flow more easily. In our embedding representation learning subproblem, we need to learn the representations which respect the tree structural constraint on the corresponding hash code $\mathbf{h} = [\mathbf{h}^1, \ldots, \mathbf{h}^k] \in \{0,1\}^{d\times k}$ where $\| \mathbf{h}^v \|_1 = 1 ~~\forall v \neq k$ and $\| \mathbf{h}^k \|_1 = k_s$. To this end, we decompose the problem and compute the embedding loss per each hierarchy level $v$ separately.
Furthermore, naively using the similarity labels to define similar pairs versus dissimilar pairs during the embedding learning subproblem could create a discrepancy between the hash code discrete optimization subproblem and the embedding learning subproblem leading to contradicting updates. Suppose two examples $\mathbf{x}_i$ and $\mathbf{x}_j$ are dissimilar and both had the highest activation at the same dimension $o$ and the hash code for some level $v$ was identical \ie~ $\mathbf{h}_i^v[o] = \mathbf{h}_j^v[o] = 1$. Enforcing the metric learning loss with the class labels, in this case, would lead to increasing the highest activation for one example and decreasing the highest activation for the other example. This can be problematic for the example with decreased activation because it might get hashed to another occupied bucket after the gradient update and this can repeat causing instability in the optimization process.
However, if we relabel the two examples so that they are treated as the same class as long as they have the same hash code at the level, the update wouldn't decrease the activations for any example, and the sibling term (the second term) in \Cref{eqn:energy_tree_avg} would automatically take care of splitting the two examples in the next subsequent levels.
To this extent, we apply \emph{label remapping} as follows. $y_i^v = remap(\mathbf{h}_i^v)$, where $remap(\cdot)$ assigns arbitrary unique labels to each unique configuration of $\mathbf{h}_i^v$. Concretely, $remap(\mathbf{h}_i^v) = remap(\mathbf{h}_j^v) \iff y_i^v = y_j^v$. Finally, the embedding representation learning subproblem aims to solve \Cref{eqn:embedding} given the hash codes and the remapped labels. Section C in the supplementary material includes the ablation study of label remapping.
\vspace{-0.5em}
\small
\begin{align}
\operatorname*{minimize}_{\boldsymbol\theta} \sum_{v=1}^k \ell_{\text{metric}} \left( \{f(x_i; \boldsymbol\theta)^v\}_{i=1}^n; ~\{\mathbf{h}_i^v\}_{i=1}^n, \{y_i^v\}_{i=1}^n \right)
\label{eqn:embedding}
\end{align}
\normalsize
Following the protocol in \cite{jeong2018}, we use the Tensorflow implementation of deep metric learning algorithms in \href{https://www.tensorflow.org/versions/master/api_docs/python/tf/contrib/losses/metric_learning}{\texttt{tf.contrib.losses.metric\_learning}}.\vspace{-0.5em}
\section{Implementation details} \vspace{-1em}
\begin{algorithm}[H]
\caption{Learning algorithm}
\label{alg:procedure}
\begin{algorithmic}
\item[\algorithmicinput] $\boldsymbol\theta_b^{\text{emb}}$ (pretrained metric learning base model); $\boldsymbol\theta_d$, $k$
\REQUIRE $\boldsymbol\theta_f = [\boldsymbol\theta_b, \boldsymbol\theta_d]$
\FOR{ $t=1,\ldots,$ MAXITER}
\STATE Sample a minibatch $\{\mathbf{x}_i\}$ and initialize $\mathcal{S}^1_{z}=\emptyset$
\FOR{$v=1,\cdots, k$}
\STATE Update the flow network $G'$ by computing class cost vectors\\
$~~~~\mathbf{c}_p^v = \frac{1}{m} \sum_{i:y_i = p} f(\mathbf{x}_i; \boldsymbol\theta_f)^v$
\STATE Compute the hash codes $\{\mathbf{h}^{v}_i\}$ via minimum cost flow on $G'$
\STATE Update $\mathcal{S}^{v+1}_{z}$ given $\mathcal{S}^v_{z}$ and $\{h_i^v\}$
\STATE Remap the label to compute $y^v$
\ENDFOR
\STATE Update the network parameter given the hash codes
\vspace{-0.5em}
$$\boldsymbol\theta_f \leftarrow \boldsymbol\theta_f - \eta^{(t)} \partial_{\boldsymbol\theta_f} \sum_{v=1}^k \ell_\text{metric}(\boldsymbol\theta_f; ~\mathbf{h}_{1:n_c}^v, y_{1:n_c}^v) $$ \vspace{-1em}
\STATE Update stepsize $\eta^{(t)} \leftarrow$ ADAM rule \cite{adam}
\ENDFOR
\item[\algorithmicoutput] $\boldsymbol\theta_{f}$ (final estimate);
\end{algorithmic}
\end{algorithm}
\label{sec:implementation}
\textbf{Network architecture}~ For fair comparison, we follow the protocol in \cite{jeong2018} and use the NIN \cite{NIN} architecture (denote the parameters $\boldsymbol\theta_b$) with \emph{leaky relu} \cite{xu2015empirical} with $\tau=5.5$ as activation function and train Triplet embedding network with semi-hard negative mining \cite{facenet}, Npairs network \cite{npairs} from scratch as the base model, and snapshot the network weights ($\boldsymbol\theta_b^{\text{emb}}$) of the learned base model. Then we replace the last layer in ($\boldsymbol\theta_b^{\text{emb}}$) with a randomly initialized $dk$ dimensional fully connected projection layer ($\boldsymbol\theta_d$) and finetune the hash network (denote the parameters as $\boldsymbol\theta_f = [\boldsymbol\theta_b, \boldsymbol\theta_d]$). \Cref{alg:procedure} summarizes the learning procedure.\vspace{-0.2em}
\textbf{Hash table construction and query}~ We use the learned hash network $\boldsymbol\theta_f$ and apply \Cref{eqn:hash_function} to convert $\mathbf{x}_i$ into the hash code $\mathbf{h}(\mathbf{x}_i; \boldsymbol\theta_f)$ and use the base embedding network $\boldsymbol\theta_b^{\text{emb}}$ to convert the data into the embedding representation $f(\mathbf{x}_i; \boldsymbol\theta_b^{\text{emb}})$. Then, the embedding representation is hashed to buckets corresponding to the $k_s$ set bits in the hash code. During inference, we convert a query data $\mathbf{x}_q$ into the hash code $\mathbf{h}(\mathbf{x}_q; \boldsymbol\theta_f)$ and into the embedding representation $f(\mathbf{x}_q; \boldsymbol\theta_b^{\text{emb}})$. Once we retrieve the union of all bucket items indexed at the $k_s$ set bits in the hash code, we apply a reranking procedure \cite{survey_learningtohash} based on the euclidean distance in the embedding space.\vspace{-0.2em}
\textbf{Evaluation metrics}~ Following the evaluation protocol in \cite{jeong2018}, we report our accuracy results using precision@k (Pr@k) and normalized mutual information (NMI) \cite{manningbook} metrics. Precision@k is computed based on the reranked ordering (described above) of the retrieved items from the hash table. We evaluate NMI, when the code sparsity is set to $k_s=1$, treating each bucket as an individual cluster. We report the speedup results by comparing the number of retrieved items versus the total number of data (exhaustive linear search) and denote this metric as SUF. \vspace{-0.7em}
\section{Experiments}
\label{sec:exp}
\begin{table*}[ht]
\centering
\fontsize{6pt}{6.5pt}\selectfont
\begin{tabular}{cc cccccccc cccccccc}
\addlinespace[-\aboverulesep]
\cmidrule[1pt](r){1-10} \cmidrule[1pt](r){11-18}
& & \multicolumn{8}{c}{Triplet} &\multicolumn{8}{c}{Npairs} \\
& & \multicolumn{4}{c}{\emph{test}} & \multicolumn{4}{c}{\emph{train}} &\multicolumn{4}{c}{\emph{test}} & \multicolumn{4}{c}{\emph{train}} \\
\cmidrule(r){1-6} \cmidrule(r){7-10} \cmidrule(r){11-14} \cmidrule(r){15-18}
&Method& SUF & Pr@1 & Pr@4 & Pr@16 & SUF & Pr@1 & Pr@4 & Pr@16 &SUF & Pr@1 & Pr@4 & Pr@16 & SUF & Pr@1 & Pr@4 & Pr@16 \\
\cmidrule(r){1-6} \cmidrule(r){7-10} \cmidrule(r){11-14} \cmidrule(r){15-18}
$k_s$
&Metric& 1.00 & 56.78 & 55.99 & 53.95 & 1.00 & 62.64 & 61.91 & 61.22 &1.00 & 57.05 & 55.70 & 53.91 & 1.00 &61.78 & 60.63 & 59.73 \\
\cmidrule(r){1-6} \cmidrule(r){7-10} \cmidrule(r){11-14} \cmidrule(r){15-18}
$1$
&LSH & \textbf{138.83}&52.52 & 48.67&39.71&\textbf{135.64} & 60.45 &58.10&54.00 & 29.74&53.55&50.75&43.03&30.75&59.87&58.34&55.35\\
&DCH & 96.13 & 56.26 & 55.65 & 54.26 & 89.60 & 61.06 & 60.80 & 60.81 &41.59 & 57.23 & 56.25 & 54.45 & 40.49 & 61.59 & 60.77 & 60.12 \\
&Th & 41.21 & 54.82 & 52.88 & 48.03 & 43.19 & 61.56 & 60.24 & 58.23 &12.72 & 54.95 & 52.60 & 47.16 & 13.65 & 60.80 & 59.49 & 57.27 \\
&VQ & 22.78 & 56.74 & 55.94 & 53.77 & 40.35 & 62.54 & 61.78 & 60.98 &34.86 & 56.76 & 55.35 & 53.75 & 31.35 & 61.22 & 60.24 & 59.34 \\
&\cite{jeong2018} & 97.67& 57.63 & 57.16 & 55.76 & 97.77 & 63.85 & 63.40 & 63.39&54.85 & 58.19 & 57.22 & 55.87 & 54.90 & \textbf{63.11}& 62.29 & 61.94 \\
&Ours & 97.67& \textbf{58.42}& \textbf{57.88}& \textbf{56.58}& 97.28 & \textbf{64.73}& \textbf{64.63}& \textbf{64.69}&\textbf{101.1}& \textbf{58.28} & \textbf{57.79}& \textbf{56.92}& \textbf{97.47} & 63.06 & \textbf{62.62}& \textbf{62.44} \\
\cmidrule(r){1-6} \cmidrule(r){7-10} \cmidrule(r){11-14} \cmidrule(r){15-18}
$2$
&Th & 14.82 & 56.55 & 55.62 & 52.90 & 15.34 & 62.41 & 61.68 & 60.89&5.09 & 56.52 & 55.28 & 53.04 & 5.36 & 61.65 & 60.50 & 59.50 \\
&VQ & 5.63 & 56.78 & 56.00 & 53.99 & 6.94 & 62.66 & 61.92 & 61.26&6.08 & 57.13 & 55.74 & 53.90 & 5.44 & 61.82 & 60.56 & 59.70 \\
&\cite{jeong2018} & 76.12 & 57.30 & 56.70 & 55.19 & 78.28 & 63.60 & 63.19 & 63.09&16.20 & 57.27 & 55.98 & 54.42 & 16.51 & 61.98 & 60.93 & 60.15 \\
&Ours & \textbf{98.38}& \textbf{58.39}& \textbf{57.51}& \textbf{56.09}& \textbf{97.20} & \textbf{64.35}& \textbf{63.91}& \textbf{63.81}& \textbf{69.48} & \textbf{57.60}& \textbf{56.98}& \textbf{55.82}& \textbf{69.91} & \textbf{62.19}& \textbf{61.71}& \textbf{61.27} \\
\cmidrule(r){1-6} \cmidrule(r){7-10} \cmidrule(r){11-14} \cmidrule(r){15-18}
$3$
&Th & 7.84 & 56.78 & 55.91 & 53.64 & 8.04 & 62.66 & 61.88 & 61.16& 3.10 & 56.97 & 55.56 & 53.76 & 3.21 & 61.75 & 60.66 & 59.73 \\
&VQ & 2.83 & 56.78 & 55.99 & 53.95 & 2.96 & 62.62 & 61.92 & 61.22& 2.66 & 57.01 & 55.69 & 53.90 & 2.36 & 61.78 & 60.62 & 59.73 \\
&\cite{jeong2018} & 42.12 & 56.97 & 56.25 & 54.40 & 44.36 & 62.87 & 62.22 & 61.84& 7.25 & 57.15 & 55.81 & 54.10 & 7.32 & 61.90 & 60.80 & 59.96 \\
&Ours & \textbf{94.55}& \textbf{58.19} & \textbf{57.42}& \textbf{56.02}& \textbf{93.69}&\textbf{63.60}&\textbf{63.35}&\textbf{63.32}& \textbf{57.09}& \textbf{57.56}& \textbf{56.70}& \textbf{55.41}& \textbf{58.62} & \textbf{62.30}& \textbf{61.44}& \textbf{60.91} \\
\cmidrule(r){1-6} \cmidrule(r){7-10} \cmidrule(r){11-14} \cmidrule(r){15-18}
$4$
&Th & 4.90 & 56.84 & 56.01 & 53.86 & 5.00 & 62.66 & 61.94 & 61.24 & 2.25 & 57.02 & 55.64 & 53.88 & 2.30 & 61.78 & 60.66 & 59.75 \\
&VQ & 1.91 & 56.77 & 55.99 & 53.94 & 1.97 & 62.62 & 61.91 & 61.22 & 1.66 & 57.03 & 55.70 & 53.91 & 1.55 & 61.78 & 60.62 & 59.73 \\
&\cite{jeong2018} & 16.19 & 57.11 & 56.21 & 54.20 & 16.52 & 62.81 & 62.14 & 61.58 & 4.51 & 57.15 & 55.77 & 54.01 & 4.52 & 61.81 & 60.69 & 59.77 \\
&Ours & \textbf{92.18}& \textbf{58.52}& \textbf{57.79}& \textbf{56.22}& \textbf{91.27} & \textbf{64.20}& \textbf{63.95}& \textbf{63.63} & \textbf{49.43} & \textbf{57.75} & \textbf{56.79}& \textbf{55.50}& \textbf{50.80} & \textbf{62.43} & \textbf{61.65}& \textbf{61.01} \\
\addlinespace[-\belowrulesep]
\cmidrule[1pt](r){1-6} \cmidrule[1pt](r){7-10} \cmidrule[1pt](r){11-14} \cmidrule[1pt](r){15-18}
\end{tabular}
\caption{Results with Triplet network with hard negative mining and Npairs network. Querying test data against a hash table built on \emph{test} set and a hash table built on \emph{train} set on Cifar-100.}
\label{tab:cifar}
\end{table*}
\begin{table}[ht]
\setlength{\tabcolsep}{4pt}
\centering
\fontsize{6pt}{6.5pt}\selectfont
\begin{tabular}{cccccccccc}
\addlinespace[-\aboverulesep]
\cmidrule[1pt](r){1-6} \cmidrule[1pt](r){7-10}
& & \multicolumn{4}{c}{Triplet} &\multicolumn{4}{c}{Npairs} \\
\cmidrule(r){1-6} \cmidrule(r){7-10}
&Method & SUF & Pr@1 & Pr@4 & Pr@16 & SUF & Pr@1 & Pr@4 & Pr@16 \\
\cmidrule(r){1-6} \cmidrule(r){7-10}
$k_s$
&Metric& 1.00 & 10.90 & 9.39 & 7.45 & 1.00 & 15.73 & 13.75 & 11.08\\
\cmidrule(r){1-6} \cmidrule(r){7-10}
$1$
& LSH & 164.25& 8.86& 7.23& 5.04& 112.31 & 11.71& 8.98& 5.56\\
& DCH & 140.77& 9.82& 8.43& 6.44& 220.52& 13.87& 11.77& 8.99\\
& Th & 18.81& 10.20& 8.58& 6.50& 1.74 & 15.06& 12.92& 9.92\\
& VQ & 146.26& 10.37& 8.84& 6.90& 451.42 & 15.20& 13.27& 10.96\\
& \cite{jeong2018}& 221.49& \textbf{11.00}& \textbf{9.59}& 7.83& 478.46 &16.95& 15.27& 13.06\\
& Ours&\textbf{590.41}& 10.91& 9.58& \textbf{7.85} &\textbf{952.49}& \textbf{17.00}&\textbf{15.53}&\textbf{13.54}\\
\cmidrule(r){1-6} \cmidrule(r){7-10}
$2$
&Th & 6.33& 10.82& 9.30& 7.32& 1.18& 15.70& 13.69& 10.96\\
&VQ & 32.83& 10.88& 9.33& 7.39& 116.26& 15.62& 13.68& 11.15\\
&\cite{jeong2018}& 60.25& 11.10& 9.64& 7.73& 116.61& 16.40& 14.49& 12.00\\
&Ours&\textbf{533.86}&\textbf{11.14}&\textbf{9.72}&\textbf{7.96}&\textbf{1174.35}& \textbf{17.22}&\textbf{15.57}& \textbf{13.63}\\
\cmidrule(r){1-6} \cmidrule(r){7-10}
$3$
&Th& 3.64& 10.87& 9.38& 7.42& 1.07& 15.73& 13.74& 11.07\\
&VQ& 13.85& 10.90& 9.38& 7.44& 55.80& 15.74& 13.74& 11.12\\
&\cite{jeong2018}& 27.16& 11.20& 9.55& 7.60& 53.98& 16.24& 14.32& 11.73\\
&Ours& \textbf{477.86}& \textbf{11.21}&\textbf{9.72}&\textbf{7.94}&\textbf{1297.98}&\textbf{17.09}&\textbf{15.37}&\textbf{13.39}\\
\addlinespace[-\belowrulesep]
\cmidrule[1pt](r){1-6} \cmidrule[1pt](r){7-10}
\end{tabular}
\caption{Results with Triplet network with hard negative mining and Npairs \cite{npairs} Network. Querying ImageNet \emph{val} data against hash table built on \emph{val} set.}
\label{tab:imagenet}
\end{table}
We report our results on Cifar-100 \cite{cifar100} and ImageNet \cite{imagenet} datasets and compare against several baseline methods. First baseline methods are the state of the art deep metric learning models \cite{facenet, npairs} performing an exhaustive linear search over the whole dataset given a query data (denote as `Metric'). Next baseline is the Binarization transform \cite{agrawal2014,zhai2017} where the dimensions of the hash code corresponding to the top $k_s$ dimensions of the embedding representation are set (denote as `Th'). Then we perform vector quantization \cite{survey_learningtohash} on the learned embedding representation from the deep metric learning methods above on the entire dataset and compute the hash code based on the indices of the $k_s$ nearest centroids (denote as `VQ'). Another baseline is the quantizable representation in \cite{jeong2018}(denote as \cite{jeong2018}). In both Cfar-100 and ImageNet, we follow the data augmentation and preprocessing steps in \cite{jeong2018} and train the metric learning base model with the same settings in \cite{jeong2018} for fair comparison. In Cifar-100 experiment, we set $(d,k)=(32,2)$ and $(d,k)=(128,2)$ for the npairs network and the triplet network, respectively. In ImageNet experiment, we set $(d,k)=(512,2)$ and $(d,k)=(256,2)$ for the npairs network and the triplet network, respectively. In ImageNetSplit experiment, we set $(d,k)=(64,2)$. We also perform LSH hashing \cite{jain2008fast} baseline and Deep Cauchy Hashing \cite{dch} baseline which both generate $n$-bit binary hash codes with $2^n$ buckets and compare against other methods when $k_s\!=\!1$ (denote as `LSH' and `DCH', respectively). For the fair comparison, we set the number of buckets, $2^n\!=\!dk$.
\subsection{Cifar-100}
Cifar-100 \cite{cifar100} dataset has $100$ classes. Each class has $500$ images for \emph{train} and $100$ images for \emph{test}. Given a query image from \emph{test}, we experiment the search performance both when the hash table is constructed from \emph{train} and from \emph{test}. The batch size is set to $128$ in Cifar-100 experiment. We finetune the base model for $70$k iterations and decayed the learning rate to $0.3$ of previous learning rate after $20$k iterations when we optimize our methods. \Cref{tab:cifar} shows the results from the triplet network and the npairs network respectively. The results show that our method not only outperforms search accuracies of the state of the art deep metric learning base models but also provides the superior speedup over other baselines. \vspace{-0.7em}
\begin{table}[ht]
\centering
\fontsize{6pt}{6.5pt}\selectfont
\begin{tabular}{c ccc ccc}
\cmidrule[1pt](r){1-4} \cmidrule[1pt](r){5-7}
& \multicolumn{3}{c}{Triplet} &\multicolumn{3}{c}{Npairs} \\
\cmidrule(r){1-4} \cmidrule(r){5-7}
& \multicolumn{2}{c}{Cifar-100} & ImageNet &\multicolumn{2}{c}{Cifar-100} & ImageNet \\
& train & test & val &train & test & val \\
\cmidrule(r){1-4} \cmidrule(r){5-7}
LSH & 62.94 & 53.11 & 37.90 & 43.80 & 37.45 & 36.00 \\
DCH & 86.11 & 68.88 & 45.55 & 80.74 & 65.62 & 50.01 \\
Th & 68.20 & 54.95 & 31.62 & 51.46 & 44.32 & 15.20 \\
VQ & 76.85 & 62.68 & 45.47 & 80.25 & 66.69 & 53.74 \\
\cite{jeong2018} & 89.11 & 68.95 & 48.52 & 84.90& 68.56 & 55.09 \\
Ours & \textbf{89.95} & \textbf{69.64}& \textbf{61.21} & \textbf{86.80} & \textbf{71.30} & \textbf{65.49} \\
\cmidrule[1pt](r){1-4} \cmidrule[1pt](r){5-7}
\end{tabular}
\caption{Hash table NMI for Cifar-100 and Imagenet.}
\label{tab:NMI}
\end{table}
\begin{table}[ht]
\centering
\fontsize{6pt}{6.5pt}\selectfont
\begin{tabular}{cccccccccc}
\addlinespace[-\aboverulesep]
\toprule
&Method & SUF & Pr@1 & Pr@4 & Pr@16 \\
\midrule
$k_s$
&Metric & 1.00 & 21.55 & 19.11 & 16.06 \\
\midrule
$1$
&LSH & 33.75 & 18.49 & 15.50 & 11.14 \\
& Th & 10.98 & 20.25 & 17.22 & 13.66 \\
& VQ-train& 54.30 & 20.15 & 18.10 & 14.85 \\
& VQ-test& 57.44 & 20.59 & 18.31 & 15.32 \\
& \cite{jeong2018}& 56.35& 21.35 & 18.49 & 15.32 \\
& Ours & \textbf{78.23}& \textbf{21.46} & \textbf{18.88} & \textbf{15.67} \\
\midrule
$2$
& Th & 4.55 & 21.27 & 18.86 & 15.68 \\
& VQ-train & 15.29 & 21.51 & 19.03 & 15.88 \\
& VQ-test & 16.43 & 21.58 & 18.93 & 15.94 \\
& \cite{jeong2018} & 15.99 & \textbf{22.12} & \textbf{19.21} & \textbf{15.95} \\
& Ours & \textbf{71.14} & \textbf{22.12} & 18.63 & 15.34 \\
\midrule
$3$
& Th & 2.79 & 21.53 & 19.11 & 15.99 \\
& VQ-train & 7.80 & 21.56 & 19.11 & 16.03 \\
& VQ-test & 8.20 & 21.58 & 19.09 & 16.06 \\
& \cite{jeong2018} & 7.24 & \textbf{22.18} & \textbf{19.40} & \textbf{16.10} \\
& Ours & \textbf{84.04} &21.97 & 18.87 & 15.56 \\
\addlinespace[-\belowrulesep]
\bottomrule
\end{tabular}
\caption{Results with Triplet network with hard negative mining. Querying ImageNet \emph{val} set in $C_{\text{test}}$ against hash table built on \emph{val} set in $C_{\text{test}}$.}
\label{tab:imagenetsplit}
\end{table}
\begin{figure*}[ht]
\centering
\includegraphics[width=\linewidth]{cifarhashtree.pdf}
\caption{Visualization of the examples mapped by our trained three level hash codes $[\mathbf{h}^{(1)}, \mathbf{h}^{(2)}]$ on Cifar-100. Each parent node (denoted as depth 1) is color coded in red, yellow, blue, and green in \emph{cw} order. Each color coded box (denoted as depth 2) shows examples of the hashed items in each child node.}
\label{fig:cifarhashtree}
\end{figure*}
\subsection{ImageNet}
ImageNet ILSVRC-2012 \cite{imagenet} dataset has $1,000$ classes and comes with \emph{train} ($1,281,167$ images) and \emph{val} set ($50,000$ images). We use the first nine splits of \emph{train} set to train our model, the last split of \emph{train} set for validation, and use \emph{validation} dataset to test the query performance. We use the images downsampled to $32\times32$ from \cite{imgnet-down}. We finetune npairs base model and triplet base model as in \cite{jeong2018} and add a randomly initialized fully connected layer to learn hierarchical representation. Then, we train the parameters in the newly added layer with other parameters fixed. When we train with npairs loss, we set the batch size to $1024$ and train for $15$k iterations decaying the learning rate to $0.3$ of previous learning rate after each $6$k iterations. Also, when we train with triplet loss, we set the batch size to $512$ and train for $30$k iterations decaying the learning rate of $0.3$ of previous learning rate after each $10$k iterations. Our results in \Cref{tab:imagenet} show that our method outperforms the state of the art deep metric learning base models in search accuracy while providing up to $1298\times$ speedup over exhaustive linear search. \Cref{tab:NMI} compares the NMI metric and shows that the hash table constructed from our representation yields buckets with significantly better class purity on both datasets and on both the base metric learning methods.
\subsection{ImageNetSplit}
In order to test the generalization performance of our learned representation against previously unseen classes, we performed an experiment on ImageNet where the set of classes for training and testing are completely disjoint. Each class in ImageNet ILSVRC-2012 \cite{imagenet} dataset has super-class based on WordNet \cite{wordnet}. We select $119$ super-classes which have exactly two sub-classes in $1000$ classes of ImageNet ILSVRC-2012 dataset. Then, we split the two sub-classes of each $119$ super-class into $C_{\text{train}}$ and $C_{\text{test}}$, where $C_{\text{train}}\cap C_{\text{test}}=\emptyset$. Section D in the supplementary material shows the class names in $C_{\text{train}}$ and $C_{\text{test}}$. We use the images downsampled to $32\times32$ from \cite{imgnet-down}. We train the models with triplet embedding on $C_{\text{train}}$ and test the models on $C_{\text{test}}$. The batch size is set to $200$ in ImageNetSplit dataset. We finetune the base model for $50$k iterations and decayed the learning rate to $0.3$ of previous learning rate after $40$k iterations when we optimize our methods. We also perform vector quantization with the centroids obtained from $C_{\text{train}}$ (denote as `VQ-train') and $C_{\text{test}}$ (denote as `VQ-test'), respectively. \Cref{tab:imagenetsplit} shows our method preserves the accuracy without compromising the speedup factor.\\
\vspace{-1em}
Note, in all our experiments in \Cref{tab:cifar,tab:imagenet,tab:NMI,tab:imagenetsplit}, while all the baseline methods show severe degradation in the speedup over the code compound parameter $k_s$, the results show that the proposed method robustly withstands the speedup degradation over $k_s$. This is because our method 1) greatly increases the quantization granularity beyond other baseline methods and 2) hashes the items more uniformly over the buckets. In effect, indexing multiple buckets in our quantized representation does not as adversarially effect the search speedup as other baselines. \Cref{fig:cifarhashtree} shows a qualitative result with npairs network on Cifar-100, where $d=32, k=2, k_s=1$. As an interesting side effect, our qualitative result indicates that even though our method does not use any super/sub-class labels or the entire label information during training, optimizing for the objective in \Cref{eqn:energy_seq} naturally discovers and organizes the data exhibiting a meaningful hierarchy where similar subclasses share common parent nodes.
\section{Conclusion}
We have shown a novel end-to-end learning algorithm where the quantization granularity is significantly increased via hierarchically quantized representations while preserving the search accuracy and maintaining the computational complexity practical for the mini-batch stochastic gradient descent setting. This not only provides the state of the art accuracy results but also unlocks significant improvement in inference speedup providing the highest reported inference speedup on Cifar100 and ImageNet datasets respectively.
\section*{Acknowledgements}
This work was partially supported by Kakao, Kakao Brain and Basic Science Research Program through the National Research Foundation of Korea (NRF) (2017R1E1A1A01077431). Hyun Oh Song is the corresponding author.
\newpage
{\small
\bibliographystyle{ieee}
|
{
"timestamp": "2019-03-08T02:10:19",
"yymm": "1902",
"arxiv_id": "1902.10990",
"language": "en",
"url": "https://arxiv.org/abs/1902.10990"
}
|
\section{Introduction}
\subsection{Motivation.}
Ergodic theory was created in the beginning of the last century motivated by the needs of homogenization
(more specifically the quest to justify the kinetic equations of statistical mechanics). By now ergodic theory is
a flourishing subject. Namely, ergodic theorems are established under very general conditions
and ergodic properties of a large number of smooth systems are known (see e.g. \cite{KH95}).
Moreover, ergodicity turns out to be useful in the questions of averaging and homogenization
(see e.g. \cite{JKO94, LM88, PV81, SVM07}). However, many dynamical systems appearing in applications
preserve infinite invariant measure and ergodic theory of infinite-measure-preserving systems is much less
developed. In fact, most of the work in infinite ergodic theory (see e.g. \cite{A97})
deals with local ($L^1$) observables
while from physical point of view it is more natural to consider extensive observables (\cite{Kh49, Ru78})
which admit an infinite-volume average. One explanation for this is that while for local observables
ergodic theorems can be obtained with minimal regularity assumptions on the observable,
this is not the case for global observables as the present paper shows.
The study of ergodic properties of infinite measure transformations
with respect to extensive functions started relatively recently \cite{L10}.
In particular, mixing properties of several systems with respect to global observables
were obtained in \cite{BGL18, DN18, L10, L17}.
A natural question is thus to investigate the law of large numbers
for global observables. A first step in this
direction was recently taken in \cite{LM18}. In this paper we carry out a
detailed analysis in the simplest possible setting: random walks on
$\mathbb{Z}^d$. Our goal is to ascertain the correct spaces for the law of
large numbers in various cases.
\subsection{Results.}
\label{SSResults}
Let $X_1, X_2,...$ be an iid sequence of $\mathbb Z^d$ valued random variables.
Let $S_0 = 0$ and $S_N = \sum_{n=1}^{N} X_n$ be the corresponding random walk.
We assume that
\begin{enumerate}
\item (non-degeneracy) the smallest group supporting the range of $X_1$ is $\mathbb Z^d$
\item (aperiodicity) $g.c.d.\{ n>0: \mathbb P (S_n = 0) > 0 \} = 1$
\end{enumerate}
We will also assume that $S_n$ is in the normal domain of attraction of
a stable law
with some index $\alpha$. That is, there is a non-degenerate $d$ dimensional random variable $Y $
such that
$$
\frac{S_n}{n^{1/\alpha }} \Rightarrow Y \text{ if } \alpha \in (0,1) \text{ and }
\frac{S_n - n \E(X_1)}{n^{1/\alpha }} \Rightarrow Y \text{ if } \alpha \in (1,2]
$$
To avoid uninteresting minor technical difficulties, we will mostly assume that $\alpha \neq 1$.
We define several function spaces which proved to be useful in the previous studies of global observables
\cite{DN18}.
Without further notice, we always assume that all functions are bounded.
Given a
non-empty
subset $V \subset \mathbb Z^d$ and $F \in L^{\infty}(\mathbb Z^d, \mathbb R)$, we write
$$
\bar F_V = \frac{1}{|V|} \sum_{v \in V} F(v).
$$
Given $(a_1, b_1, \dots a_d, b_d)$
with $a_i \le b_i$, for $j=1, \dots d$,
let
$$V(a_1, b_1, \dots a_d, b_d)=\{x\in {\mathbb{Z}}^d:\; x_j\in [a_j, b_j] \text{ for } j=1, \dots d\}. $$
Let $\mathbf{G}_+$ be the space of bounded functions on $\mathbb{Z}$ such that the limit
$
\displaystyle \bar F_+ = \lim_{v \to \infty} \bar F_{[0,v]}$
exists
and
$\mathbf{G}_-$ be the space of bounded functions on $\mathbb{Z}$ such that the limit
$\displaystyle \bar F_- = \lim_{v \to \infty} \bar F_{[-v,0]}$ exists. Set $\mathbf{G}_\pm=\mathbf{G}_+\cap\mathbf{G}_-.$
Define
$$\mathbf{G}_0=\{F\in L^\infty(\mathbb{Z}^d, \mathbb{R}): \exists {\bar F} \;\; \forall a_1, b_1, \dots a_d, b_d\quad \lim_{L\to \infty}
{\bar F}_{V(a_1 L, b_1 L, \dots, a_d L, b_d L)}={\bar F}\}.$$
Note that in dimension 1, $\mathbf{G}_0=\{F\in \mathbf{G}_\pm: {\bar F}_+={\bar F}_-\}.$
Let $\mathbf{G}_U$ be the space of functions such that for each ${\varepsilon}$ there is $L$
such that for all cubes $V$ with side larger than $L$ we have
\begin{equation}
\label{CubeAve}
|{\bar F}_V-{\bar F}|\leq {\varepsilon}.
\end{equation}
Let $\mathbf{G}_\gamma$ be the set of functions where \eqref{CubeAve} only holds if the center of $V$ is within
distance $L^\gamma$
of
the origin. Thus $\mathbf{G}_U \subset \mathbf{G}_\gamma .$ Also,
$\mathbf{G}_\gamma\subset \mathbf{G}_0$ if $\gamma>1.$
Finally, let
\begin{gather*}
\mathbf{G}_{\gamma}^{\beta} = \{
F \in L^{\infty}(\mathbb{Z}^d, \mathbb{R}): \exists {\bar F}\;\;
\forall a_1,b_1, ..., a_d,b_d\;\;
\exists C: \forall L,
\forall z \in \mathbb{Z}^d, |z| < L^{\gamma}, \\
|{\bar F}_{z + V(a_1L, b_1L,...,a_dL,b_dL)} - {\bar F} |< C L^{d(\beta -1)}
\}.
\end{gather*}
Clearly, $\mathbf{G}_{\gamma}^{\beta} \subset \mathbf{G}_{\gamma} $ for any $\beta <1$.
Also, let $\displaystyle \mathbf{G}_\infty^\beta=\bigcap_{\gamma>0} \mathbf{G}_\gamma^\beta.$
Our goal is to study Birkhoff sums
$$ T_N=\sum_{n=1}^N F(S_n) . $$
In particular we would like to know if $\frac{T_N}{N}$ converges to ${\bar F}$ for $F$ in each of the spaces
$\mathbf{G}_*$ introduced above.
Our results could be summarized as follows.
\begin{theorem}
\label{ThWLLN}
Suppose that $\E(X)=0$ and that
$S_N$ is in the normal domain of attraction of
a
stable law of some index $\alpha>1.$
Then for all $F\in \mathbf{G}_0$, $\dfrac{T_N}{N}\Rightarrow {\bar F}$ in law as $N\to \infty.$
\end{theorem}
\begin{theorem}
\label{ThArc}
Suppose that $d=1,$ $\E(X)=0$ and $V(X)<\infty$. Then
for all $F\in \mathbf{G}_\pm$, $\dfrac{T_N}{N}$ converges in law as $N\to \infty.$
In particular, if ${\bar F}_-=0$ and ${\bar F}_+=1$ then the limiting law has arcsine
distribution: for $z\in [0,1]$
$$ \lim_{N\to \infty} {\mathbb{P}}\left(\frac{T_N}{N}\leq z\right)=\frac{2}{\pi} \arcsin\sqrt{z}. $$
\end{theorem}
Note that Theorem \ref{ThArc} is a simple homogenization result: it says that the limit
distribution of $\frac{T_N}{N}$ remains the same if the oscillatory function $F$ is replaced by
a more regular function
${\bar F}_- 1_{x<0}+{\bar F}_+ 1_{x\geq 0}$ (see \cite{L39}).
This confirms the usefulness of global observables in applications.
\begin{theorem}
\label{ThGGamma}
Suppose that $S_N$ is in the normal domain of attraction of the stable law of
some index $\alpha$. Suppose that either
(i) $1 < \alpha \leq 2,$ $\E(X_1) \ne 0$ and $\gamma>1$ or
(ii) $1 < \alpha \leq 2$, $\E(X_1)=0$ and $\gamma>1/\alpha$ or
(iii) $\alpha\leq 1$ and $\gamma>1/\alpha.$
Then, for all $F\in \mathbf{G}_\gamma,$
$\frac{T_N}{N}\to{\bar F}$ almost surely.
\end{theorem}
\begin{theorem}
\label{ThGGammaBeta}
Suppose $E(X_1) = 0$ and $E(|X_1|^k) < \infty$ for all $k \in \mathbb{N}$.
For $d\in \mathbb{N}$, let
$$
\rho_d(\beta):=
\begin{cases}
\frac12 & \text{if } \beta \leq \frac{d-1}{d} \\
\frac{d}{2} (\beta -1) + 1 & \text{if } \beta > \frac{d-1}{d},
\end{cases} \quad \quad
\gamma(d, \beta, {\varepsilon}):=
\begin{cases} \frac{2}{\beta} & \text{if } d=1 \\
\frac{1}{{\varepsilon}} & \text{if } d\geq 2. \end{cases}
$$
Then for every $d \in \mathbb{N}$, for every
$\beta \in [0,1)$ every
$\varepsilon >0$
and any $F \in \mathbf{G}_{\gamma(d, \beta, {\varepsilon})}^{\beta}$ with ${\bar F} = 0$, we have
$$\frac{T_N}{N^{\rho_d(\beta)+ {\varepsilon}} }\to 0 \text{
almost surely as }N\to \infty.$$
\end{theorem}
\begin{corollary}
\label{cor:gammainf}
If $d=1$ and $F\in \mathbf{G}_{2/\beta}^\beta$ or $d\geq 2$ and $F\in \mathbf{G}_\infty^\beta$ then
with probability 1,
for all ${\varepsilon}$
$$ \lim_{N\to\infty} \frac{T_N}{N^{\rho_d(\beta)+{\varepsilon}}}=0. $$
\end{corollary}
\begin{remark}
Let us discuss two special cases.
(A) (Random walk in random scenery)
If $F(x)$, $x \in \mathbb{Z}^{d}$ are bounded and iid with expectation $0$,
then by moderate deviation estimates, for every $\gamma < \infty$ and for
every ${\varepsilon} >0$, $F \in \mathbf{G}_{\gamma}^{\frac12 + {\varepsilon}}$ holds with ${\bar F} = 0$
almost surely.
Now assuming that the random walker has zero expectation and finite moments of every order,
Theorem
\ref{ThGGammaBeta}
implies
$\frac{T_N}{N^{\frac34 + {\varepsilon} }}\to 0$ almost surely
in dimension $d=1$. Note that in this case,
$\frac{T_N}{N^{\frac34 }}$ has a non-trivial weak limit by \cite{KS79}.
If $d\geq 2$, Theorem \ref{ThGGammaBeta} gives
$\frac{T_N}{N^{\frac12 + {\varepsilon} }}\to 0$ almost surely
while $\frac{T_N}{N^{\frac12 }}$
($\frac{T_N}{\sqrt{N\ln N}}$ if $d=2$) has a non-trivial weak limit
(\cite{KS79}). We note that Theorem \ref{ThGGammaBeta} is not new for
$F$ as above (see \cite{KL98, GP01}) however, we would like to emphasize that
our space $\mathbf{G}_\gamma^{\frac{1}{2}+{\varepsilon}}$ includes many more functions than just realizations
of iid process, so both the result and the proof of Theorem \ref{ThGGammaBeta}
are new even for $\mathbf{G}_\gamma^{\frac{1}{2}+{\varepsilon}}.$
(B) If $F(x)$ is periodic, ${\bar F} = 0$, then $F \in
\mathbf{G}_\infty^{\frac{d-1}{d}}$.
Thus, assuming that the random walker has zero expectation and finite moments of every order,
Corollary
\ref{cor:gammainf} implies
\begin{equation}
\label{AlmDif}
\frac{T_N}{N^{\frac12 + {\varepsilon} }}\to 0
\end{equation}
almost surely for all $d$.
Note that by the central limit theorem for finite Markov chains,
$\frac{T_N}{\sqrt{N}}$
has a Gaussian weak limit.
In fact, our results also give \eqref{AlmDif} for quasi-periodic observables. That is, given
$d \in \N$ and
a $C^\infty$ function $\mathfrak{F}:\mathbb T^d\to\R,$
let $\hat{\mathfrak F} :\mathbb{R}^d \to \mathbb{R}$ be the $[0,1]^d$-periodic extension of $\mathfrak F$. Furthermore,
given
$d$ vectors $\alpha_{(1)}, \dots \alpha_{(d)}\in \R^d$
and an
initial phase $\omega\in [0,1]^d,$ let
$$
F(x)= \hat{\mathfrak{F}}\left(\omega+\sum_{j=1}^d x_j \alpha_{(j)} \right)$$
where $(x_1, \dots x_d)$ are coordinates of vector $x\in \Z^d.$
We say that a vector $\alpha \in \mathbb{Z}^d$ is Diophantine, if
there are constants $K$ and $\sigma$ such that
for each $m\in \Z^d$,
$$ \left|e^{2\pi \langle m, \alpha \rangle }-1\right|\geq \frac{K}{|m|^\sigma}.$$
If $\alpha_{(j)}$ is Diophantine for all $j=1,...,d$,
then $F \in \mathbf{G}_{\infty}^0$
(see e.g. \cite[\S 2.9]{KH95}) so \eqref{AlmDif} holds.
Thus in both cases (A) and (B) our results give an optimal exponent for the growth rate of $T_N.$
\end{remark}
\begin{remark}
Periodic (and quasi-periodic) observables are special case of stationary ergodic observables.
More precisely,
let $\mathfrak{T}_1, \dots \mathfrak{T}_d$ be
commuting measurable maps of a space $\Omega$ preserving a probability measure
$\nu.$ Given a bounded measurable function $\mathfrak{F}$ on $\Omega$ and an initial condition
$\omega\in \Omega$, define
\begin{equation}
\label{StErg}
F_\omega(k)=\mathfrak{F}(\mathfrak{T}^k \omega),
\end{equation}
where for $k=(k_1,\dots k_d)\in \Z^d$ we let
$\mathfrak{T}^k=\mathfrak{T}_d^{k_d}\dots \mathfrak{T}_1^{k_1}.$ If the family $\mathfrak{T}^k$ is ergodic,
then the ergodic theorem tells us that for almost all $\omega$, $F_\omega\in \mathbf{G}_0$ and
$\bar{F}=\nu(\mathfrak{F}).$ For the observables given by \eqref{StErg} the strong law of large numbers
for almost every $\omega$ follows from ergodicity of the environment viewed by the particle process (\cite{BS}).
Theorem \ref{ThWLLN}
only gives a weak law of large numbers, (except in dimension 1 in the ballistic case,
see Theorem \ref{ThD=1} below). On the other hand our result gives valuable additional information
even for stationary ergodic environments. Namely, the set of full measure where the weak law of large
numbers holds contains all environments where ergodic averages of $\mathfrak{F}$ exist. We also
note that Theorem \ref{ThGGammaBeta} provides new and non-trivial information even in the stationary ergodic case.
\end{remark}
\begin{theorem}
\label{ThD=1}
Suppose that $d=1,$ $v=\E(X_1)>0$ and for all $t\geq 1$, $\E(|X_1|>t)\leq C/t^\beta$ for some
$C>0$ and $\beta>1.$
If $F\in \mathbf{G}_+$, then $\frac{T_N}{N}\to{\bar F}_+$ almost
surely.
\end{theorem}
The next theorem shows that
in general the
strong law of large numbers fails in $\mathbf{G}_0.$
\begin{theorem}
\label{ThOcean}
Suppose that
$S_N$ is in the normal domain of attraction of the stable law of some index $\alpha.$
Moreover assume that one of the following assumptions is satisfied
(a) $\alpha>1$ and $\E(X_1)=0;$ or
(b) $\alpha<1.$
Then there exists $F\in \mathbf{G}_0$ such that, with probability 1, $\frac{T_N}{N}$ does not converge
as $N\to\infty.$
\end{theorem}
\begin{remark}
The same conclusion holds in case (a) even if $\E(X_1)\neq 0.$ However, in this case $\mathbf{G}_0$ is not an appropriate space to look at since we even do not have a weak law of large numbers in $\mathbf{G}_0.$
\end{remark}
\section{Weak convergence.}
\label{ScWeak}
Here we prove Theorems \ref{ThWLLN} and \ref{ThArc}.
\subsection{Preliminaries}
First, we recall two useful results.
\begin{theorem}(\cite[Section 50]{GK54})
\label{thm:llt}
Under the assumptions of Theorem \ref{ThWLLN},
$S_n$ satisfies the {\it local limit theorem}, i.e. there is a continuous probability density $g$ such that
$$
\lim_{n \to \infty}
\sup_{l \in \mathbb Z^d} | n^{d/\alpha} \mathbb P (S_n = l) - g(l / n^{1/\alpha}) | = 0.
$$
\end{theorem}
\begin{theorem}[Local global mixing, \cite{DN18}]
\label{thm:lgm}
Under the assumptions of Theorem \ref{ThWLLN}, $S_n$ is {\it local global mixing}, i.e.
$$
\lim_{n \to \infty} {\mathbb{E}}(F(S_n)) = {\bar F}.
$$
\end{theorem}
\subsection{Proof of Theorem \ref{ThWLLN}}
Replacing $F$ by $F - {\bar F}$, we can assume that ${\bar F} = 0$.
By Theorem \ref{thm:lgm}, we have $\displaystyle \lim_{N \to \infty} \frac{\E(T_N)}{N} = 0$.
Thus in order to prove Theorem
\ref{ThWLLN}, it suffices to verify that $\displaystyle \lim_{N \to \infty} \frac{\E(T^2_N)}{N^2} = 0$.
Let us fix some ${\varepsilon} >0$ and prove that $\E(T^2_N) < {\varepsilon} N^2$ for all sufficiently large $N$.
We have
$$
\E(T^2_N) = 2
\sum_{0\leq n_1 < n_2\leq N} \E (F(S_{n_1}) F(S_{n_2}))
+\sum_{n=1}^N \E( F^2(S_n))
$$
Now writing ${\varepsilon}_1 = \frac{{\varepsilon}}{50 \| F\|^2_{\infty}}$, we have
$$
\E(T^2_N) \leq \frac{{\varepsilon}}{10} N^2 +
\left|2 \sum_{{\varepsilon}_1N < n_1 < n_1+{\varepsilon}_1N < n_2\leq N} \E (F(S_{n_1})F(S_{n_2}))
\right|$$
Choose a constant $K$ such that
${\partial } (|S_N| > K N^{1/\alpha} /2) < \frac{{\varepsilon}}{10\| F\|^2_{\infty}}$
for all sufficiently large $N.$ Thus we have
$$
\E(T^2_N) \leq \frac{2{\varepsilon}}{10} N^2 +
\left| 2\sum_{{\varepsilon}_1N < n_1 < n_1+{\varepsilon}_1N < n_2\leq N}
\E (1_{\{|S_{n_1}|, |S_{n_2}| <KN^{1/\alpha}\}}F(S_{n_1})F(S_{n_2}))\right|. $$
Observe that by the Markov property of the random walk, we have
$$
\E (F(S_{n_1})F(S_{n_2})) = \E(F(S_{n_1}) \E_{S_{n_1}}(F({\tilde S}_{n_2-n_1}))).
$$
where
${\tilde S}_n = {\tilde S}_0 + \sum_{k=1}^n {\tilde X}_k$, $\{{\tilde X}_k\}$ are iid and have the same distribution as $X_1$
and ${\partial }_x$ is the measure defined by ${\partial }_x({\tilde S}_0 = x) = 1$.
Let us write
\begin{gather*}
e_1 = \E (1_{\{|S_{n_1}|, |S_{n_2}| <KN^{1/\alpha}\}}F(S_{n_1})F(S_{n_2})),\\
e_2 = \E(1_{\{|S_{n_1}| <KN^{1/\alpha}\}}F(S_{n_1})
\E_{S_{n_1}}(1_{\{|{\tilde S}_{n_2 - n_1}| <2KN^{1/\alpha}\}}F({\tilde S}_{n_2-n_1}))),\\
\mathcal{A} = \{ x,y: |x|, |y| < KN^{1/\alpha}\}, \quad \mathcal{B} = \{ x,y: |x| < KN^{1/\alpha}, |y-x| < 2KN^{1/\alpha}\}.
\end{gather*}
Note that $\mathcal{A} \subset \mathcal{B}$ and set $\mathcal{C} = \mathcal{B} \setminus \mathcal{A}$. Then
$$
e_2 - e_1 = \sum_{(x,y) \in \mathcal{C}} {\partial }(S_{n_1} = x, S_{n_2} = y) F(x) F(y)
$$
and we find
$$
|e_2 - e_1| \leq \| F \|_{\infty}^2 P (S_{n_1} > KN^{1/\alpha}) \leq \frac{{\varepsilon}}{10}.
$$
Consequently,
$$
|e_1| \leq |e_2| + \frac{{\varepsilon}}{10}.
$$
Recall that $g$ is the density function of the limiting distribution of $S_n / n^{1/\alpha}$ as in Theorem
\ref{thm:llt}.
Now we choose $\delta$ so that the oscillation of $g$ on any cube
of side length $ \delta/{\varepsilon}_1^{1/\alpha}$ within distance $2K$ from the origin
is less than
$\eta := \frac{{\varepsilon} {\varepsilon}_1^{d/\alpha}}{20 \cdot (2K)^d \| F\|_{\infty}^2}$.
Also note that if $F \in \mathbf{G}_0$ and ${\bar F} = 0$, then
\begin{equation}
\label{eq:weak1}
\lim_{N \to \infty} \sup_{-2K < a < 2K } | {\bar F}_{[Na,N(a + \delta)]^d} | = 0.
\end{equation}
Now given $n_1,n_2$ with ${\varepsilon}_1N < n_1 < n_1+{\varepsilon}_1N < n_2 <N$
$x < KN^{1/\alpha}$, we write
\begin{equation}
\label{eq:weak2}
\E_{x}(1_{\{|{\tilde S}_{n_2 - n_1}| <2KN^{1/\alpha}\}}F({\tilde S}_{n_2-n_1}))) =
\end{equation}
$$\sum_k
\E_{x}( 1_{\{ {\tilde S}_{n_2 - n_1} \in B_k \}}F({\tilde S}_{n_2-n_1})),
$$
where $B_k$'s are cubes of side length $\delta N^{1/\alpha}$ partitioning $[-KN^{1/\alpha},KN^{1/\alpha}]^d$.
For brevity, we write $m = n_2 - n_1$.
By Theorem \ref{thm:llt} and the choice of $\delta$, we have
\begin{equation}
\label{eq:weak3}
\E_{x}( 1_{\{ {\tilde S}_{m} \in B_k \}}F({\tilde S}_{m})) = \sum_{y \in B_k} (p_k + e_{x,k,m,y})m^{-d/\alpha} F(y),
\end{equation}
where $p_{k}=g\left(\frac{z_k}{m^{1/\alpha}}\right)$,
$z_k$ is the center of $B_k$ and $e_{x,k,m,y}< 2 \eta$ for $m$ sufficiently large (uniformly in $x,k,y$ as above).
Consequently,
$$\sum_k \sum_y e_{x,k,m,y} m^{-d/\alpha}\leq
m^{-d/\alpha} (2KN^{1/\alpha} )^d 2 \eta \leq \frac{{\varepsilon}}{10\|F\|_{\infty}^2}$$
for sufficiently large $m$. Thus dropping $e_{x,k,m,y}$ from the right hand side of
\eqref{eq:weak3} gives a negligible error.
The remaining term is $ \displaystyle p_{k}m^{-d/\alpha} \sum_{y \in B_k} F(y)$, which
when summed over $k$, is small by \eqref{eq:weak1}. Thus the absolute value of
\eqref{eq:weak2} is smaller than ${\varepsilon} /5$ for $N$ sufficiently large, completing the proof
of Theorem \ref{ThWLLN}.
\subsection{Proof of Theorem \ref{ThArc}}
We prove the second statement. The first one is a trivial corollary. Indeed,
given $F \in \mathbf{G}_{\pm}$ with ${\bar F}_- = {\bar F}_+$,
the convergence follows from Theorem \ref{ThWLLN}. On the other hand if ${\bar F}_- \neq {\bar F}_+$, then we can
consider
$\displaystyle \tilde F(x) = \frac{F(x) - {\bar F}_-}{{\bar F}_+ - {\bar F}_-}$ and note that
$\bar{\tilde F}_-=0$, $\bar{\tilde F}_+=1$ and
$\displaystyle \tilde{T}_N=
\sum_{n=1}^N \tilde F(S_n)= \frac{T_N-N {\bar F}_-}{{\bar F}_+-{\bar F}_-},$
whereby one derives the limit distribution of $T_N$.
Thus we assume ${\bar F}_- =0 $, ${\bar F}_+ =1$. The proof is similar to that of Theorem \ref{ThWLLN}
(with $\alpha =2$, $d=1$). The difference
is that, as the limit distribution is now non-degenerate, we need to verify the convergence of all moments,
not just the first two. Since the arcsine distribution is compactly supported, this implies the weak convergence.
We start with the following lemma.
\begin{lemma}
\label{lemma:integral}
Fix $F \in \mathbf{G}_{\pm}$ with ${\bar F}_- =0 $, ${\bar F}_+ =1$. Then for any continuous and compactly supported
function $\phi$, we have
$$
\lim_{n \to \infty} \frac{1}{n} \sum_{x \in \mathbb{Z}} F(x) \phi\left( \frac{x}{n}\right) = \int_0^{\infty} \phi(x) dx.
$$
Furthermore, for any compact interval $I$, the convergence is uniform for $\phi$
lying in a compact subset of
$\mathcal C(I, \mathbb{R})$.
\end{lemma}
\begin{proof}
The proof is a simple generalization of the computation at the end of the proof of Theorem \ref{ThWLLN}.
Let $L=\max\{|x|: \phi(x)\neq 0\}.$
Given ${\varepsilon} >0$, we choose $\delta$ so that the oscillation of $\phi$ on intervals of length $\delta$
is less than ${\varepsilon} \|F \|_{\infty}(L+1).$ Let $K$ be the smallest positive integer so that $K > L/\delta.$
Then
\begin{gather*}
\sum_{x >0} F(x) \phi\left( \frac{x}{n}\right) = \sum_{k=0}^{K-1}
\sum_{x = \lfloor k n / \delta \rfloor +1 } ^{\lfloor (k+1) n / \delta \rfloor}
F(x) \phi\left( \frac{x}{n}\right)
\end{gather*}
Next, by the
choice
of $\delta$,
$$
\left| \sum_{k=0}^{K-1}
\sum_{x = \lfloor k n / \delta \rfloor +1 } ^{\lfloor (k+1) n / \delta \rfloor}
F(x)
\left[
\phi\left( \frac{x}{n}\right) -
\phi\left( \frac{k}{\delta}\right)
\right]
\right|
\leq {\varepsilon} n.
$$
Since ${\bar F} _+ = 1$, we have
$$
\sup_{0=1,...,K-1}\left|
\frac1n \phi\left( \frac{k}{\delta}\right) \sum_{x = \lfloor k n / \delta \rfloor +1 } ^{\lfloor (k+1) n / \delta \rfloor} F(x)
- \frac{1}{\delta} \phi\left( \frac{k}{\delta}\right) \right| < {\varepsilon}
$$
for $n$ sufficiently large. Thus replacing a Riemann sum by an integral (further reducing $\delta$ if necessary),
we find
$$
\left| \frac1n \sum_{x >0} F(x) \phi\left( \frac{x}{n}\right) - \int_0^{\infty} \phi(x) dx \right| < 3 {\varepsilon}.
$$
Note that all the above estimates are uniform over compact subsets of $\mathcal C(I, \mathbb{R})$.
The computation for $x<0$ is similar but easier as ${\bar F}_- = 0$.
\end{proof}
Now we prove the convergence of the moments by induction.
The inductive hypothesis is the following:
({$\bf H_k$}) for every $\varepsilon >0$ and $K<\infty$ there exists some $N_0$ so that for all $N >N_0$
and for all $x \in [- K \sqrt N, K \sqrt N]$,
$$
|N^{-k}\E_x(T_N^k) - E (\tau_{x/\sqrt N}^k)| < {\varepsilon}.
$$
Here, $\tau_a$ is the total time spent on the positive half-line by a Brownian motion
$(B_t)_{t \in [0,1]}$ with
$B_0 \equiv a$ and such that $V(B_1) = V(X_1)$.
Furthermore, as in Theorem \ref{ThWLLN},
$E_x(.)$
means that the random walk starts from $x \in \mathbb Z$ (more precisely,
in the definition of $T_N$, one replaces $S$ with $\tilde S$, where
$\tilde S_0 \equiv x$ and $\tilde S_n - \tilde S_0$ has the same
distribution as $S_n$).
Clearly, ({$\bf H_0$}) holds. Now assume that ({$\bf H_k$}) holds for some $k$. To prove ({$\bf H_{k+1}$}),
first observe that
$$
\frac{\E_x(T_N^{k+1})}{N^{k+1}}=
\frac{1}{N}\E_x\left(\sum_{n=1}^N F({\tilde S}_n)
\frac{1}{N^k} \sum_{n_1,\dots n_k=1}^N F({\tilde S}_{n_1})\dots F({\tilde S}_{n_k})\right)$$
$$\approx \frac{k+1}{N}\E_x\left(\sum_{n=1}^N F({\tilde S}_n)
\frac{1}{N^k} \sum_{n_1,\dots n_k=n}^N F({\tilde S}_{n_1})\dots F({\tilde S}_{n_k})\right)$$
$$
\approx \frac{k+1}{N}
\sum_{n=\delta N}^{(1-\delta)N}
\sum_{y = - \lfloor K_1 \sqrt n\rfloor }^{\lfloor K_1 \sqrt n \rfloor }
{\partial }_x( {\tilde S}_n = y) F(y) N^{-k} \E_{y}(T^k_{N-n})
$$
$$
=\frac{k+1}{N}
\sum_{n=\delta N}^{(1-\delta)N} \left(1 - \frac{n}{N} \right)^k
\sum_{y = - \lfloor K_1 \sqrt n\rfloor }^{\lfloor K_1 \sqrt n \rfloor }
{\partial }_x( {\tilde S}_n = y) F(y) (N-n)^{-k}
\E_{y}(T^k_{N-n}).
$$
Here, $a_{x,\delta, K_1}(N) \approx b_{x,\delta, K_1}( N)$
means that for every $\varepsilon >0$, $K<\infty$
there exists some $\delta, K_1,N_0$ so that for all $N >N_0$, all
$x\in [-K\sqrt{N}, K\sqrt{N}]$
we have
$|a_{x,\delta, K_1}(N)- b_{x,\delta, K_1}( N)| < {\varepsilon}$
and the factor $k+1$ appears in the second line since we
imposed an additional restriction that $n$ is the smallest among $n, n_1,\dots n_k.$
Now using the inductive hypothesis and the local limit theorem, we find that
$$
\frac{\E_x(T_N^{k+1})}{N^{k+1}}
\approx (k+1) \frac{1}{N}
\sum_{n=\delta N}^{(1-\delta)N}
\left(1 - \frac{n}{N} \right)^k
\sum_{y \in \mathbb{Z}}
\frac{1}{\sqrt n}F(y) \phi_{x,N,n} \left(\frac{y}{\sqrt n}\right),
$$
where
$$
\phi_{x,N,n} (\mathfrak{y}) = \mathfrak g\left(\mathfrak{y}-\frac{x}{\sqrt n}\right) E\left(
\tau_{\mathfrak{y} \sqrt{\frac{n}{N-n}}}^k\right) 1_{|\mathfrak{y}|< K_1}
$$
and $\mathfrak g$ is the centered Gaussian density with variance $V(X)$. Now Lemma \ref{lemma:integral} implies that
\begin{equation}
\label{eq:arcsinf}
\frac{\E_x(T_N^{k+1})}{N^{k+1}}
\approx (k+1) \frac{1}{N}
\sum_{n=\delta N}^{(1-\delta)N} \left(1 - \frac{n}{N} \right)^k
\int_{y=0}^{K_1} \phi_{x,N,n} (y) dy.
\end{equation}
It remains to check that
\begin{equation}
\label{eq:arcsinrw}
E (\tau_{x/\sqrt N}^k) \approx
(k+1) \frac{1}{N}
\sum_{n=\delta N}^{(1-\delta)N} \left(1 - \frac{n}{N} \right)^k
\int_{y=0}^{K_1} \phi_{x,N,n} (y) dy.
\end{equation}
To prove \eqref{eq:arcsinrw}, we observe that in particular \eqref{eq:arcsinf}
holds for the heaviside function $F(x) =H(x) = 1_{x >0}$.
By classical theory \cite[\S XII.8]{Fel},
$T_N/N$ converges weakly to the arcsine law for $F(x) =H(x)$. Since $T_N/N$ is bounded,
the moments also converge and thus the left hand sides of \eqref{eq:arcsinf} and \eqref{eq:arcsinrw}
are asymptotically equivalent for $F(x) =H(x)$. Since the right hand side does not depend on the specific
choice of $F$, \eqref{eq:arcsinrw} follows. This completes the proof of Theorem \ref{ThArc}.
\section{SLLN in $\mathbf{G}_\gamma.$}
Here we prove Theorem \ref{ThGGamma}.
Fix $F \in \mathbf{G}_\gamma$. As before, we can assume w.l.o.g.\ that
$\bar F = 0$.
Since $F \in \mathbf{G}_\gamma$, the proof of Theorem\ 2.3 of \cite{DN18} shows that,
for any $\eta \in (0,1)$,
\begin{equation} \label{GGamma10}
\lim_{k \to \infty} \, \sup_{|x| \le k^{\eta\gamma}} \left|
\frac{\E_x(T_k)} {k} \right| = 0,
\end{equation}
where $\E_x$ denotes the expectation in the case where $S_0=x$.
\ignore{We start by proving the theorem for case $0 < \alpha < 1$, or
$1 < \alpha < 2$ and $E(X_1)=0$. Fix $\eta$ as in (\ref{GGamma10})
such that $\eta \gamma > 1/\alpha$ and define
\begin{equation} \label{GGamma15}
\mathcal{A}_N := \left\{ \max_{0 \le k \le N} |S_k| \le
N^{\eta \gamma} \right\}.
\end{equation}}
\begin{lemma}
\label{LmMax}
Suppose that $1< \gamma_1 $ in case (i) or
$1/\alpha<\gamma_1$ in cases (ii) and (iii). Then with probability one
we have that, for large $N$,
$$ \max_{0\leq k \leq N} |S_k|\leq N^{\gamma_1}. $$
\end{lemma}
\begin{proof}
In case (i) the statement follows from the Law of Large Numbers, so we only need to consider
cases (ii) and (iii).
We have for any ${\varepsilon} >0$ that $|S_N| >N^{1/\alpha + {\varepsilon}}$ holds only finitely many times almost surely
by \cite{M39} in case (ii) and by \cite{L31} in case (iii). The lemma follows.
\end{proof}
Choose $\gamma_1<\gamma$ as in Lemma \ref{LmMax}
and
$\eta<1$ such that
$\gamma_1<\gamma\eta.$
For
$j = 0, 1, \ldots, \lf N^{1-\eta} \rf$, set $\tilde T_j := \tilde T_{N,j} :=
N^{-\eta} \, T_{\lf j N^\eta \rf}$ (with the convention $T_0 \equiv 0$)
and denote by $\tilde \mathcal{F}_j := \tilde \mathcal{F}_{N,j}$ the $\sigma$-algebra
generated by $\{ S_k \}_{k=0}^{\lf j N^\eta \rf}$.
Denote $\mathcal{A}_{j,N}=\{|S_{\lf j N^\eta \rf}|\leq N^{\gamma_1}\}.$
Fix ${\varepsilon}>0$.
We claim that there exists $N_0 = N_0({\varepsilon})$ such that, for all $
N \ge N_0$ and $j < \lf N^{1-\eta} \rf$,
\begin{equation} \label{GGamma20}
\left| \E \! \left( \left. 1_{\mathcal{A}_{j,N}} \! \left( \tilde T_{j+1} - \tilde T_j \right) \right|
\tilde \mathcal{F}_j \right) \right| \le {\varepsilon}.
\end{equation}
Indeed if $\mathcal{A}_{j,N}$ occurs then \eqref{GGamma20} holds due to \eqref{GGamma10},
otherwise it holds since the LHS is zero.
\ignore{
To see this we use (\ref{GGamma10}) with $k := N^\eta$. For all $N$ larger
than or equal to some $N_0({\varepsilon})$ and $|x| \le N^{\eta \gamma}$, we have
that $| N^{-\eta} E_x(T_{N^\eta}) | \le {\varepsilon}$. All realizations of the RW
within $\mathcal{A}_N$ are such that $| S_{\lf j N^\eta \rf} | < N^{\eta \gamma}$.
So the Markov property of the RW proves (\ref{GGamma20}). Moreover
$| \tilde T_{j+1} - \tilde T_j | \le \| F \|_\infty$.}
Setting
\begin{equation} \label{GGamma30}
Y_j := 1_{\mathcal{A}_{j,N}} \! \left( \tilde T_{j+1} - \tilde T_j \right) - D_j
\end{equation}
where
\begin{equation} \label{GGamma40}
D_j:=\E \! \left( 1_{\mathcal{A}_{j,N}} \! \left( \tilde T_{j+1} - \tilde T_j \right) \Big|
\tilde \mathcal{F}_j \right)
\end{equation}
defines a martingale difference, w.r.t.\ $\{ \tilde \mathcal{F}_j \}$, with $|Y_j| \le
\| F \|_\infty + {\varepsilon}$. Applying Azuma's inequality we get that, for all
$\delta>0$,
$$
{\mathbb{P}} \!\left( \left| \sum_{j=0}^{N^{1 - \eta} -1} Y_j \right| \ge \delta N^{1 - \eta}
\right) \le 2 \exp \!\left( -\frac{\delta^2 N^{1 - \eta}} {2 (\| F \|_\infty +
{\varepsilon})^2} \right).
$$
Therefore, by Borel-Cantelli,
$$
\limsup_{N \to \infty} \, \frac1 {N^{1 - \eta}} \!
\left| \sum_{j=0}^{N^{1 - \eta} -1} Y_j \right|
\le \delta \quad \mbox{a.s.}
$$
Since $\delta$ is arbitrary,
with probability one we have
\begin{equation}
\label{LLNYJ}
\lim_{N\to\infty} \frac1 {N^{1-\eta}} \!\! \sum_{j=0}^{N^{1 - \eta} -1} Y_j =0.
\end{equation}
On the other hand,
definitions (\ref{GGamma30})-(\ref{GGamma40}) and Lemma \ref{LmMax} show
that, with probability one, for all large $N$ depending on the realization of the walk,
\begin{equation}
\label{NoExcNoGaps}
T_N=N^\eta \left(\sum_{j=0}^{N^{1 - \eta} -1} Y_j +\sum_{j=0}^{N^{1 - \eta} -1} D_j \right).
\end{equation}
In view of \eqref{GGamma20}, \eqref{LLNYJ}, and \eqref{NoExcNoGaps}
we have:
$$ \limsup_{N \to \infty}\, \left|\frac{T_N}{N} \right|
= \limsup_{N \to \infty} \, \frac{1}{N^{1 - \eta}} \!
\left| \sum_{j=0}^{N^{1 - \eta} -1} \left(Y_j+D_j\right) \right|$$
$$\hskip30mm=\limsup_{N \to \infty} \, \frac{1}{N^{1 - \eta}} \!
\left| \sum_{j=0}^{N^{1 - \eta} -1} D_j \right|
\le {\varepsilon} \quad \mbox{a.s.}$$
Since ${\varepsilon}$ is arbitrary, $\displaystyle \lim_{N\to\infty} \frac{T_N}{N}=0$
almost surely.
\section{Speed of convergence in $\mathbf{G}^{\beta}_\gamma.$}
\label{ScSpeed}
Here we prove Theorem \ref{ThGGammaBeta}.
Note that $\mathbf{G}_{\gamma}^{\beta_1} \subset \mathbf{G}_{\gamma}^{\beta_2}$ whenever $\beta_1 < \beta_2$.
Since $\rho_d(\beta)$ is constant for $\beta \in [0,(d-1)/d]$ and is continuous at $(d-1)/d$, it is sufficient to prove the theorem
for
\begin{equation}
\label{betabound}
\beta > \frac{d-1}{d} .
\end{equation}
Let $\mathbb P_x(.) = \mathbb P(.|S_0 = x)$, $\mathbb E_x(.) = \mathbb E(.|S_0 = x)$.
We start with the following
\begin{proposition}
\label{lemma:weak}
Under the conditions of Theorem \ref{ThGGammaBeta},
for every $d \in \mathbb{N}$, every $\beta \in ((d-1)/d,1)$ and every ${\varepsilon} >0$ there exists some $\delta >0$ so that
\begin{equation}
\label{eq:lemmaweak}
\sup_{x_0: |x_0| \leq N^{1/2 + \delta}}\mathbb E_{x_0} (T_{N}^2) < C N^{2\rho_d(\beta) + 2{\varepsilon}}.
\end{equation}
\end{proposition}
Note that Proposition \ref{lemma:weak} combined with Chebyshev's inequality implies that
$$
\frac{T_N}{N^{\rho_d(\beta)+ {\varepsilon}} }\Rightarrow 0
\text{ in law as } N\to \infty .
$$
Section \ref{ScSpeed} is divided into three parts.
In \S \ref{SSRefl} we derive Theorem \ref{ThGGammaBeta} from Proposition \ref{lemma:weak}.
In \S \ref{SSVar-1} we prove Proposition \ref{lemma:weak}
for $d=1$. In \S \ref{SSVar-2}, we extend the proof of Proposition \ref{lemma:weak} to arbitrary dimension $d$.
\subsection{Proof of Theorem \ref{ThGGammaBeta}}
\label{SSRefl}
Here, we derive the theorem from Proposition \ref{lemma:weak}. For simplicity we write $\rho = \rho_d(\beta)$.
We will show that
\begin{equation}
\label{eq:bc}
{\partial } (\exists n \leq N: |T_n| > 2 N^{\rho + {\varepsilon} /2}) \leq C N^{- {\varepsilon} /2}
\end{equation}
If \eqref{eq:bc} holds, then writing $N_k = 2^k$, we find
$$
{\partial } (\exists n = N_{k-1},... N_k : |T_n| > 2 N_k^{\rho + {\varepsilon} /2}) \leq C N_k^{- {\varepsilon} /2}.
$$
and the theorem follows from Borel Cantelli lemma.
To prove \eqref{eq:bc}, let us write
$$\tau_N = \min \{ \min \{ n: |T_n| > 2 N^{\rho + {\varepsilon} /2} \}, N \}.$$
Then
\begin{gather*}
{\partial } (\exists n \leq N: |T_n| > 2 N^{\rho + {\varepsilon} /2}) \\
\leq
{\partial } (|T_N| > N^{\rho + {\varepsilon} /2}) + {\partial } (|T_{\tau_N}| > 2 N^{\rho + {\varepsilon} /2}, |T_N| \leq N^{\rho + {\varepsilon} /2}) = : p_1 + p_2
\end{gather*}
By Proposition \ref{lemma:weak} for $x_0 = 0$ and by Chebyshev's inequality,
we have $p_1 \leq C N^{- {\varepsilon} /2}$. To bound $p_2$, we distinguish two cases: $S_{\tau_N} > N^{1/2 + \delta}$ and
$S_{\tau_N} \leq N^{1/2 + \delta}$.
The first case has negligible probablity by moderate deviation bound for random walks (see
formula \eqref{MDE} below). In the second case we compute
\begin{gather*}
{\partial } (|T_{\tau_N}| > 2 N^{\rho + {\varepsilon} /2}, |T_N| \leq N^{\rho + {\varepsilon} /2}, |S_{\tau_N}| \leq N^{1/2 + \delta})\\
\leq \sup_{x_0 : |x_0| \leq N^{1/2 + \delta} } \max_{n = 1,..., N} {\partial }_{x_0}( |T_n| \geq N^{\rho + {\varepsilon} /2})
\end{gather*}
which is again bounded by $ C N^{- {\varepsilon} /2}$ by Proposition \ref{lemma:weak} and Chebyshev's inequality.
We have verified \eqref{eq:bc} and finished the proof of the theorem.
\subsection{Proof of Proposition \ref{lemma:weak} for $d=1$}
\label{SSVar-1}
We have $\beta \in (0,1)$ and $2 \rho_d(\beta) = \beta +1$.
We start by recalling
some results on expansions in the LLT in case all moments are finite (a.k.a. Edgeworth expansion).
\begin{theorem}(\cite[Section 51]{GK54})
\label{thm:llt2}
Under the assumptions of Theorem \ref{ThGGammaBeta}, there are polynomials $Q_1, Q_2, ...$
so that for any $M \in \N$
\begin{equation}
\label{LLTerror}
{\partial } (S_n = l) = \frac{1}{\sqrt n} \mathfrak g (l /\sqrt n)
(1 + \sum_{m=1}^M Q_m(l /\sqrt n) n^{-m/2}) + e_{n,l,M}
\end{equation}
where $\mathfrak g$ is a Gaussian density and
$$
\limsup_{n \to \infty} \sup_{l \in \mathbb{Z}} e_{n,l,M} n^{M/2 +1} < \infty.
$$
\end{theorem}
Note that \eqref{LLTerror} implies the following estimate:
for any $\eta>0$ there exists $C<\infty$ such that
\begin{equation}
\label{MDE}
{\partial } (|S_n| > n^{1/2 + \eta}) < C n^{- 1/\eta}.
\end{equation}
Given a function $h = h(n,x) : \mathbb{Z}_+ \times \mathbb{Z} \to \mathbb{R}$, we write
${ \nabla}h$ for the discrete
derivative in the second coordinate, i.e.
$$\nabla h(n,x) = h(n,x) - h(n, x-1). $$
Note that
\begin{equation}
\label{productrule}
\nabla (gh)(n,x) = (\nabla g)(n,x)h(n,x) + g(n, x-1)(\nabla h)(n,x).
\end{equation}
We will also write $\nabla^k h$ for the $k$th discrete derivative.
Denote
\begin{equation}
\label{defH}
H(n,x) = {\partial }(S_n = x).
\end{equation}
With this notation, \eqref{MDE} can be rewritten as
\begin{equation}
\label{MDE2}
\sum_{x: |x| \geq n^{1/2+ \eta}} H(n,x) < C_{\eta} n^{-1/\eta} \text{ for any } \eta >0.
\end{equation}
Also \eqref{LLTerror} implies that there is a constant $c$ so that,
for every $k=0,1,2$,
\begin{equation}
\label{disder}
\sup_{x \in \mathbb{Z}}|\nabla^k H (n,x)| \leq c n^{-\frac{k+1}{2}}.
\end{equation}
Observe that
\begin{equation}
\label{eq:moment}
\mathbb E_{x_0}({T_{N}^2}) = \sum_{0 \leq n_1\leq n_2 \leq N} c_{n_1,n_2} E_{n_1,n_2}(x_0)\
\end{equation}
where $c_{n_1,n_2} = 1$ if $n_1 = n_2$
and $c_{n_1,n_2} = 2$ otherwise and
\begin{gather*}
E_{n_1,n_2}(x_0) = \mathbb E_{x_0}(F(S_{n_1}) F(S_{n_2})) \\
= \sum_{x_1, x_2 \in \mathbb{Z}} {\partial }(S_{n_2 - n_1} = x_2 - x_1)
{\partial }_{x_0}(S_{n_1} = x_1)
F(x_1) F(x_2)
\end{gather*}
We will show the following: for any $0 \leq n_1 \leq n_2 \leq N$ such that
\begin{equation}
\label{gapsqrt} n_1>N^\alpha, \quad
n_2 - n_{1} > { N^{\alpha} \text{ where } \alpha = 1/ \gamma = \beta / 2}
\end{equation}
we have
\begin{equation}
\label{KSK3}
|E_{n_1,n_2}(x_0)| \leq
\end{equation}
$$
C n_{1}^{\frac{\beta - 1}{2} + {\varepsilon}}
(n_{2}-n_{1})^{\frac{\beta - 1}{2} + {\varepsilon}}
+
C n_{1}^{\frac{\beta }{2} + {\varepsilon}}
(n_{2}-n_{1})^{\frac{\beta - 2}{2} + {\varepsilon}}.
$$
Summing the estimate \eqref{KSK3} for $n_1,n_2$ satisfying \eqref{gapsqrt} we obtain $N^{\beta +1 + 2 {\varepsilon}}$ as needed.
To complete the proof of the proposition,
it remains to
(I) prove \eqref{KSK3}
(II) verify that the contribution of $(n_1,n_2)$'s that do not satisfy \eqref{gapsqrt} is also negligible;
We start with (I).
We will use the following lemma:
\begin{lemma}
\label{lem:KS}
There is a constant ${\hat C}$ such that for any positive integer $n$ and any constants
$A,B$, the following holds.
If $g(x): \mathbb{Z} \to \mathbb{R}$ satisfies
\begin{itemize}
\item[(H1)] $\displaystyle \sup_{x: |x| \leq n^{1/2+ \delta}} |g(x)| \leq A
\item[(H2)] $\displaystyle \sup_{x: |x| \leq n^{1/2+ \delta}}|\nabla g(x)| \leq B
\end{itemize}
for some sufficiently small $\delta$, then, for $i=0,1$,
\begin{equation}
\label{KSlemma}
\sup_{y: |y| \leq n^{\frac{1/2 + \delta}{\alpha}}}
\left| \sum_{z \in \mathbb{Z}} \nabla^i H(n,z-y) g(z) F(z) \right| \leq {\hat C} ( \|g\|_{\infty} n^{-10}+
A n^{\frac{\beta - i- 1}{2} + {\varepsilon}} + B n^{\frac{\beta -i}{2} + {\varepsilon}} ).
\end{equation}
\end{lemma}
\begin{proof}
For the rest of the section $C$ will denote a constant
(independent of $A$ and $B$) whose value may change from line to line.
By \eqref{MDE2} and since $F$ is bounded, the sum for $z$'s with $|z-y| > n^{1/2 + \delta}$
is bounded by $C \|g\|_{\infty} n^{-10}$.
Denote $\displaystyle I(x) = \sum_{w=y- 2 n^{1/2 + \delta}}^x F(w)$.
Using summation by parts and (H1),
we find
\begin{gather}
\sum_{z:\; |z-y| \leq n^{1/2 + \delta}} \nabla^i H(n,z-y)g(z) F(z)
=
O\left(An^{- 10 }\right) + \nonumber \\
-
\sum_{z :\; |z -y| \leq n^{1/2 + \delta}} I(z -1) \nabla_z \left(\nabla^i H(n,z-y)g(z)\right).
\label{KSlemma1.2}
\end{gather}
Using \eqref{productrule}, \eqref{disder}, (H1) and (H2) we find that
\begin{equation}
\label{eq:prod}
\nabla_z \left(\nabla^i H(n,z-y)g(z)\right)
\leq C( A n^{-\frac{i+2}{2}} + B n^{-\frac{i+1}{2}}).
\end{equation}
Next, using $F \in \mathbf{G}_{\gamma}^{\beta}$,
with ${\bar F} = 0$, we find
\begin{equation}
\label{gammaapp}
|I(z)| \leq C n^{\frac{\beta}{2} + {\varepsilon}}
\end{equation} (assuming that $\delta = \delta({\varepsilon})$ is small enough).
The last two estimates imply that the sum in \eqref{KSlemma1.2}
is bounded by $C (A n^{\frac{\beta - 1 -i }{2} + {\varepsilon}} + B n^{\frac{\beta -i }{2} + {\varepsilon}} )$.
\end{proof}
Now we are ready to estimate $E_{n_1,n_2}(x_0)$. First, let
\begin{equation}
\label{DefG1}
g_{1}(x_{1}) := g_{1,n_2 - n_{1}}( x_{1}) := \sum_{x_2 \in \mathbb{Z}}
H({n_2 - n_{1}}, x_2 - x_{1}) F(x_2).
\end{equation}
By definition, $\| g_1 \|_{\infty}\leq \|F\|_{\infty}$.
Applying Lemma \ref{lem:KS} with $i=0$, $g = 1$, $n ={n_{2}-n_{1}}$, $A=1$, $B=0$
and using $n_2 - n_{1} > N^{\alpha}$, we find
\begin{equation}
\label{eqa1}
\sup_{x_1: |x_1|\leq N^{1/2 + \delta}} |g_{1}(x_1)| \leq C(n_{2}-n_{1})^{\frac{\beta - 1}{2} + {\varepsilon}}.
\end{equation}
Using Lemma \ref{lem:KS} the same way but now with $i=1$, we find
\begin{equation}
\label{eqb1}
\sup_{x_1: |x_1|\leq N^{1/2 + \delta}} |\nabla g_{1}(x_1)| \leq C(n_{2}-n_{1})^{\frac{\beta - 2}{2} + {\varepsilon}}
\end{equation}
Next, set
\begin{gather}
\label{DefG2}
g_{2}(x_{0}) := g_{2, n_1 }( x_{0})
:=
\sum_{x_{1} \in \mathbb{Z}}
H({n_{1} - n_{0}}, x_{1} - x_{0})
g_{1}( x_{1}) F(x_{1})
\end{gather}
Now we use Lemma \ref{lem:KS} with $i=0$,
$n= n_{1} $, $g=g_1$,
$A = (n_2 - n_{1})^{\frac{\beta - 1}{2} + {\varepsilon}}$, $B=(n_2 - n_{1})^{\frac{\beta - 2}{2} + {\varepsilon}}$.
Since $n \leq N$,
\eqref{eqa1} and \eqref{eqb1} give
(H1) and (H2).
Also using that $\|g_1\|_{\infty} \leq \|F\|_{\infty}^2$ and $n_{1} - n_{2} > N^{\alpha}$, we get
\begin{equation}
\label{KSK4}
\sup_{x_0: |x_0|\leq N^{1/2 + \delta}} |g_{2}(x_0)|
\leq
\end{equation}
$$
C n_{1}^{\frac{\beta - 1}{2} + {\varepsilon}}
(n_{2}-n_{1})^{\frac{\beta - 1}{2} + {\varepsilon}}
+
C n_{1}^{\frac{\beta }{2} + {\varepsilon}}
(n_{2}-n_{1})^{\frac{\beta - 2}{2} + {\varepsilon}}
$$
which gives \eqref{KSK3}.
It remains to verify (II), that is that the contribution of pairs $(n_1,n_2)$'s that do not satisfy \eqref{gapsqrt} is negligible.
First, assume that $n_1 > N^{\alpha}$ and $n_2 - n_1 { \leq} N^\alpha$. Then we derive as in \eqref{KSK4} but using the
trivial bounds $A= 1 + \| F\|_{\infty}$, $B = 2(1 + \| F\|_{\infty})$
that
\begin{equation*}
\sup_{x_0: |x_0|\leq N^{1/2 + \delta}} |g_{2}(x_0)|
\leq
C n_{1}^{\frac{\beta - 1}{2} + {\varepsilon}}
+
C n_{1}^{\frac{\beta }{2} + {\varepsilon}}.
\end{equation*}
Summing this estimate for $n_1 = N^{\alpha}, ..., N$ and multiplying by
$N^{\alpha}$ for the number of choices of $n_2$, we obtain
\begin{equation}
\label{eq:smalln}
O(N^{\alpha} N^{\frac{\beta + 1}{2}}) = O( N^{\beta + \frac12})
=o( N^{\beta + 1}).
\end{equation}
Next, assume that $n_1 <N^\alpha,$ $n_2 - n_1 < N^\alpha$. Using the bound
$|E_{n_1,n_2}(x_0)| \leq \| F \|_{\infty}^2$ we obtain
$$
\sum_{n_1=0}^{N^\alpha}\sum_{n_2=n_1}^{n_1 + N^\alpha}|E_{n_1,n_2}(x_0)|
\leq C N^{2 \alpha} = CN^{\beta}
=o( N^{\beta +1} ).
$$
Finally, assume that $n_1 \leq N^{\alpha}$ and $n_2 - n_1 > N^\alpha.$
By \eqref{MDE2}, we can assume that $S_{n_1} - S_{n_0} \leq N^{1/2 + \delta}$.
Then \eqref{eqa1} still holds and we
conclude that
\begin{equation}
\label{eq:smalln1}
|E_{n_1, n_2}(x_0)| \leq C (n_2-n_1)^{\frac{\beta -1}{2} + {\varepsilon}}.
\end{equation}
Summing for $n_2 = n_1+N^{\alpha}, ..., N$ and multiplying by $N^\alpha$, we obtain
the same error term as in \eqref{eq:smalln}.
This completes the proof.
\subsection{Proof of Proposition \ref{lemma:weak} for $d\geq 2$}
\label{SSVar-2}
\subsubsection{Preliminary estimates}
In dimension $d$, we have
\begin{equation}
\label{disder2}
\sup_{x \in \mathbb{Z}^2}|\nabla_{i_1} ... \nabla_{i_k} H (n,x)| \leq c n^{-\frac{k+d}{2}}.
\end{equation}
for any $i_1,...i_k =1,...,d$, where $\nabla_i$ denotes the discrete derivative
with respect to $x_i$, the $i$-th component of $x$.
We apply a similar approach as in $d=1$. That is, we perform summations by parts to estimate
$g_1$ and $g_2$ defined by \eqref{DefG1} and \eqref{DefG2}.
Each time we need $d$ summations by parts. For example, if $d=2$, then
\begin{gather*}
g_{1,m}(0) = \sum_{|x|< m^{1/2+\delta}} \sum_{|y|<m^{1/2+\delta}} H(m, (x,y)) F(x,y)\\
\approx \sum_{|x|< m^{1/2+\delta}} \sum_{|y|<m^{1/2+\delta}}
\nabla_{2} H(m,(x,y)) I_1(x,y) \\
\approx \sum_{|x|< n^{1/2+\delta}} \sum_{|y|<m^{1/2+\delta}}
\nabla_{1} \nabla_2 H(m,(x,y)) I(x,y)
\end{gather*}
where $I_1(x,y) = \sum_{z=0}^y F(x,z)$ and $I(x,y) = \sum_{w=0}^x \sum_{z=0}^y F(w,z)$
and $a_m \approx b_m$ means that the $a_m - b_m$ is superpolynomially small in $m$.
Using
that $|I(x,y)| \leq C m^{\beta(\frac12 + \delta)}$, we find
$$
|g_{1,m}(0)| \leq Cm^{\beta - 1 + {\varepsilon}}
$$
(with $m = n_2 - n_{1}$).
To simplify formulas, we will use the notation
$$
a_N \lesssim b_N \text{ if } a_N \leq C b_N N^{{\varepsilon}}.
$$
\ignore{
Set
\begin{equation}
\label{defalpha1}
\alpha_1 = \frac{\beta}{2-\beta}
\end{equation}
Proceeding as in case $d=1$, we find that if $\gamma > (2 - \beta) / \beta$, then for all
$n_1, n_2$ satisfying $n_1>N^{\alpha_1}, $
$n_2 - n_{1} \geq N^{\alpha_1}$ we have
\begin{equation}
\label{KSK3d2}
\sup_{x_0 \in \mathbb{Z}^d :|x_0|\leq N^{1/2+\delta} } |E_{n_1, n_2}(x_0)| \lesssim
\sum_{j = 0}^d n_1^{\frac{d\beta -j}{2}}
(n_2 - n_{1})^{\frac{d\beta -2d + j}{2}} .
\end{equation}
Since the proof of \eqref{KSK3d2}
is similar to that of \eqref{KSK3}, we only mention the main
difference. That is, now $j$ can take values $0,1,...,d$ and in dimension $d=1$
it could only take values $0,1$. This follows
from the fact that when applying $d$ summtions by parts to the function
$H(n_1-n_{0}, x_1 - x_{0}) g_1(x_1)$, we obtain
$$
\nabla_1 ... \nabla_d (Hg) =
\sum_{\{i_1,...,i_I\} \subset \{ 1,...,d\}}
(\nabla_{i_1} ... \nabla_{i_I} H) (\nabla_{j_1}...\nabla_{j_J} g )
$$
where $\{j_1,...,j_J\} = \{ 1,...,d\} \setminus \{i_1,...,i_I\}$.
Later we will need the following slight extension of \eqref{KSK3d2}.}
\begin{lemma}
\label{lemma:biggamma}
For any $a \in (0,1]$, if
$F \in \mathbf{G}_{1/a}^\beta$,
then for all
$n_1, n_2$ satisfying $n_1 \geq N^{a}, $
$n_2 - n_{1} \geq N^{a}$ we have
\begin{equation}
\label{ManyDer}
\sup_{x_0 \in \mathbb{Z}^d :|x_0|\leq N^{1/2+\delta} } |E_{n_1, n_2}(x_0)| \lesssim
\sum_{j = 0}^d n_1^{\frac{d\beta -j}{2}}
(n_2 - n_{1})^{\frac{d\beta -2d + j}{2}} .
\end{equation}
\end{lemma}
\begin{proof}
Since the proof of the lemma
is similar to that of \eqref{KSK3}, we only mention the main
difference. That is, now $j$ can take values $0,1,...,d$ and in dimension $d=1$
it could only take values $0,1$. This follows
from the fact that when applying $d$ summations by parts to the function
$H(n_1-n_{0}, x_1 - x_{0}) g_1(x_1)$, we obtain
$$
\nabla_1 ... \nabla_d (Hg) =
\sum_{\{i_1,...,i_I\} \subset \{ 1,...,d\}}
(\nabla_{i_1} ... \nabla_{i_I} H) (\nabla_{j_1}...\nabla_{j_J} g )
$$
where $\{j_1,...,j_J\} = \{ 1,...,d\} \setminus \{i_1,...,i_I\}$.
In the proof of \eqref{KSK3}, we only
used the definition of $\mathbf{G}_\gamma^\beta$ for boxes with side length
$L \geq N^{1/2 + \delta}$
(specifically in deriving \eqref{gammaapp}).
In order to extend that proof to the present setting, we only need to replace
$F(.)$ by $F(. - x_0)$.
Thus assuming $\gamma > 1/a$ and using
the definition of $\mathbf{G}_\gamma^\beta$, we can
repeat the previous proof.
\end{proof}
Next we show that the extreme terms in the right hand side of \eqref{ManyDer} provide
the main contribution.
\ignore{We can rewrite the right hand side of \eqref{KSK3d2} as
\begin{equation}
\label{aprioriest}
\sum_{j=0}^d m_1^{\frac{d \beta - j}{2} }m_2^{\frac{d \beta - 2d + j}{2} }
=: F_{m_1,m_2},
\end{equation}
where $m_2 = n_2- n_{1}$, $m_1=n_1.$
We will also use the following }
\ignore{
\begin{lemma}
\label{lem:ext}
For any $a,b>0$, $0 \leq c \leq b$ and for any positive numbers $m_1,m_2$, we have
\begin{equation}
\label{MExtreme}
m_1^{a-c}m_2^{a-b+c} + m_1^{a-b +c}m_2^{a-c} \leq
m_1^{a}m_2^{a-b} + m_1^{a-b}m_2^{a}
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:ext}]
Dividing both sides of \eqref{MExtreme} by $m_1^{a-b} m_2^a,$ we see that \eqref{MExtreme} is equivalent to
\begin{equation}
\label{M-X}
x^{b-c}+x^c\leq x^b+1
\end{equation}
where $x=m_1/m_2.$ Transferring all terms to the RHS we get
$$ 0\leq x^b-x^c-x^{b-c}+1=(x^c-1)(x^{b-c}-1).$$
Since both $c$ and $b-c$ are non-negative,
either we have $x=1$ and $x^c = x^{b-c} = 1$, or
$x \in \mathbb{R}_+ \setminus \{1\}$ and
$0 < (x^c-1)(x^{b-c}-1)$
\end{proof}
First we prove that for any positive number $x$,
\begin{equation}
\label{ineqj}
x^{c-b} + x^{-c} \leq x^{-b} + 1
\end{equation}
Indeed, if $c=0$ or $x=1$, we get an equality.
Now assume $c>0$. If $x<1$, then
we rewrite \eqref{ineqj} as
\begin{equation}
\label{ineqj2}
x^{c-b} - x^{-b} \leq 1 - x^{-c}
\end{equation}
and then divide \eqref{ineqj2} by $1-x^{-c}$ to obtain $x^{c-b} \geq 1$, which is true
since $c < b$.
If $x>1$, then dividing \eqref{ineqj2} by $1-x^{-c}$, we obtain $x^{c-b} \leq 1$ which is also true.
Multiplying the inequalities \eqref{ineqj} for $x= m_1$ and $x=m_2$ and subtracting the inequality \eqref{ineqj}
for $x= m_1m_2$ we obtain the lemma.
\end{proof}}
Set $m_1=n_1,$ $m_2=n_2-n_1.$
By Lemma \ref{lemma:biggamma}, we have for
$m_1\geq N^a,$ $m_2 \ge N^a$, $|x_0|\leq N^{1/2+\delta}$ that
\begin{equation}
\label{ExpCases}
|E_{n_1, n_2}(x_0)| \lesssim
\begin{cases} m_1^{\frac{(\beta-1)d}{2}} m_2^{\frac{(\beta-1)d}{2}}
& \text{if }m_2\geq m_1 \\[4pt]
m_1^{\frac{\beta d}{2}} m_2^{\frac{d\beta-2d}{2}} & \text{if } m_2<m_1.
\end{cases}
\end{equation}
Note that the second bound is quite bad if $m_2\ll m_1.$ However we can improve it
by bootstrap. Namely we have
\begin{lemma}
If $N^a<m_2<m_1$
then
$$ |E_{n_1, n_2}(x_0)| \lesssim m_2^{(\beta-1) d}$$
\end{lemma}
\begin{proof}
If $m_1\leq 2 m_2$ then the result follows from \eqref{ExpCases}. If $m_1>2 m_2$,
let $k=m_1-m_2$
and note that $k > m_2$ and
$$ E_{n_1, n_2}(x_0)=\sum_{y\in \Z^d} H(k, y-x_0) E_{m_2, 2 m_2}(y). $$
The sum of the terms where $|y|>2 N^{1/2+\delta}$ decays faster than $N^{-r}$
for any $r$. The terms where $|y|\leq 2 N^{1/2+\delta}$ can be estimated
by \eqref{ExpCases} with $m_1=m_2$ giving the result.
\end{proof}
We now combine the foregoing results in different regimes in case where $a={\varepsilon}.$ Then if
$\gamma = 1/{\varepsilon}$,
$F\in \mathbf{G}_\gamma^\beta$ we gather that
\begin{equation}
\label{ExpCases2}
|E_{n_1, n_2}(x_0)| \lesssim
\begin{cases} m_1^{\frac{(\beta-1)d}{2}} m_2^{\frac{(\beta-1)d}{2}}
& \text{if }m_2\geq m_1\geq N^{\varepsilon},\\[4pt]
m_2^{(\beta-1)d} & \text{if } m_1>m_2\geq N^{\varepsilon}, \\[4pt]
1 & \text{if } \min(m_1, m_2)<N^{\varepsilon}.
\end{cases}
\end{equation}
Summing the bounds of \eqref{ExpCases2} for $m_1, m_2\in \{1\dots N\}$ we obtain
Proposition \ref{lemma:weak}.
\section{SLLN in dimension 1.}
\label{ScD=1}
\subsection{Reduction to occupation times sum.}
Here we prove Theorem \ref{ThD=1}.
Let $\ell_n(x)$ be time spent by the walker at site $x$ before time $n$. Set
$\displaystyle \ell_\infty(x):= \lim_{n\to \infty} \ell_n(x).$
Thus $\ell_\infty(x)$ is the total time spent by the walker at site $x.$
\begin{lemma} \label{lem1}
There exist $C,c>0,$ $\mathfrak{p}\in (0,1),$ ${\varepsilon}_1$ such that for all $x \in \mathbb{N}$ and $m \in \mathbb{N}$
\begin{equation} \label{LTGeom}
{\mathbb{P}}( \ell_\infty(x) > m) < Ce^{-cm}. \end{equation}
Furthermore
\begin{equation} \label{InEqRen}
\left|{\mathbb{E}} (\ell_\infty(x))-\frac{1}{v}\right|\leq \frac{C}{x^{{\varepsilon}_1}},
\quad \left|{\mathbb{P}}(\ell_\infty(x)=0)-\mathfrak{p}\right| \leq \frac{C}{x^{{\varepsilon}_1}}.
\end{equation}
\end{lemma}
\begin{proof}
\eqref{InEqRen} follows from quantitative renewal theorem \cite{Rog73}.
\eqref{LTGeom} holds since for $k\geq 1,$
$\displaystyle {\mathbb{P}}(\ell_\infty(x)=k)={\mathbb{P}}(\ell_\infty(x)\neq 0)\;\mathfrak{p}_0^{k-1} (1-\mathfrak{p}_0)$
where $\mathfrak{p}_0$ is the probability that $S_n$ returns to the origin at some positive moment of time.
\end{proof}
Let $\displaystyle {\tilde T}_N=\sum_{x=1}^N \ell_\infty(x) F(x).$ We will show that
with probability 1
\begin{equation}
\label{LLNOcc}
\frac{{\tilde T}_N}{N}\to \frac{{\bar F}}{v}.
\end{equation}
We first deduce Theorem \ref{ThD=1} from \eqref{LLNOcc} and then prove \eqref{LLNOcc}.
Denote $\displaystyle \mathbb{L}_N=\sum_{x=1}^N \ell_\infty(x).$ By the strong law of large numbers
for~$S_N$
\begin{equation}
\label{LLNLocTime}
\frac{\mathbb{L}_N}{N}\to \frac{1}{v}.
\end{equation}
On the other hand for each ${\varepsilon}$ and for almost every $\omega$,
there is some $N_0 = N_0({\varepsilon}, \omega)$ so that for all $N>N_0$,
$$ \left|T_N-{\tilde T}_{Nv(1-{\varepsilon})}\right|\leq ||F||_\infty\left(\mathbb{L}^-+[\mathbb{L}_{Nv(1+{\varepsilon})}-\mathbb{L}_{Nv(1-{\varepsilon})}]\right),$$
where $\displaystyle \mathbb{L}^-=\sum_{x=-\infty}^0 \ell_\infty(x)$ is the total time spent on the negative halfline.
In view of \eqref{LLNLocTime},
$ \frac{T_N-{\tilde T}_{Nv(1-{\varepsilon})}}{N} $ can be made as small as we wish by taking ${\varepsilon}$ small. Hence
Theorem \ref{ThD=1} follows from \eqref{LLNOcc}.
In order to prove \eqref{LLNOcc} we observe that by Lemma \ref{lem1}
$$ {\mathbb{E}}\left(\frac{{\tilde T}_N}{N}\right)=\frac{1}{N}\sum_{x=1}^N F(x) {\mathbb{E}}(\ell_\infty(x))=
\frac{1}{v} \left[\frac{1}{N} \sum_{x=1}^{N} F(x)\right]+O\left(N^{-{\varepsilon}_1}\right)=
\frac{{\bar F}}{v}+o(1), $$
as $N\to \infty$.
We need the following bound, which will be proved in \S \ref{SSCovOcc}.
\begin{lemma}
\label{LmLTCov}
There are constants $C$ and ${\varepsilon}_2$ such that
for each $n_1<n_2$
$$\left|{\rm Cov}(\ell_\infty(n_1), \ell_\infty(n_2))\right|\leq
C\left(\frac{1}{n_1^{{\varepsilon}_2}}+\frac{1}{(n_2-n_1)^{{\varepsilon}_2}}\right). $$
\end{lemma}
Lemma \ref{LmLTCov} implies that
$${\rm Var}\left(\frac{{\tilde T}_N}{N}\right)\leq \frac{C}{N^{{\varepsilon}_2}}$$
and so
$$ {\mathbb{P}}\left(\frac{|{\tilde T}_N-{\mathbb{E}}({\tilde T}_N)|}{N}\geq \delta\right)\leq \frac{C}{\delta^2 N^{{\varepsilon}_2}}. $$
Set $r=2/{\varepsilon}_2.$ By Borel-Cantelli Lemma
$$\frac{{\tilde T}_{n^r}}{n^r}\to \frac{{\bar F}}{v} \text{ as } n\to \infty $$
almost surely. On the other hand,
\eqref{LTGeom} and the Borel-Cantelli Lemma imply that,
with probability 1, for all sufficiently large $x,$ $\ell_\infty(x)\leq \ln^2 x.$
Given $N,$ take $n$ such that
$n^r \leq N<(n+1)^r.$ Then
$$ \left| {\tilde T}_N-{\tilde T}_{n^r} \right| \leq ||F||_\infty \left(\mathbb{L}_N-\mathbb{L}_{n^r}\right)
\leq C N^{(r-1)/r} \ln^2 N .$$
It follows that
$\displaystyle \frac{{\tilde T}_N}{N}=\frac{{\tilde T}_{n^r}}{n^r}+
o(1),
$
for $N\to\infty$,
proving \eqref{LLNOcc}.
\subsection{Covariance of occupation times.}
\label{SSCovOcc}
The proof of Lemma \ref{LmLTCov} relies on the following estimates.
\begin{lemma}
\label{LmLDMin}
There are constants
$C$ and $ {\varepsilon}_3$ such that
for all $m \ge 1$,
$${\mathbb{P}}(\min_n(S_n)\leq -m)\leq \frac{C}{m^{{\varepsilon}_3}}. $$
\end{lemma}
Lemma \ref{LmLDMin} (with ${\varepsilon}_3 = \beta -1$) follows from Theorem 2(B) of \cite{V77}.
\begin{lemma}
\label{LmLTMC}
For each $\delta>0$ there is a constant $C(\delta)$ such that the following holds.
Consider a Markov chain with states $\{1, 2, 3\}$ and transition matrix
$$ \left(\begin{array}{ccc} p_1 & q_1 & \eta_1\\
q_2 & p_2 & \eta_2\\
0 & 0 & 1\end{array}\right)$$
and initial distribution $(\pi_1, \pi_2, \pi_3).$ Assume that
\begin{equation}
\label{Ell}
q_1>\delta, \quad \text{ and } \quad \eta_2 > \delta.
\end{equation}
Let $\mathfrak{l}_1$ and $\mathfrak{l}_2$ denote the occupation times of sites 1 and 2. Then
$$ \left|{\rm Cov}(\mathfrak{l}_1, \mathfrak{l}_2) \right|\leq C(\delta)\left( \frac{q_1}{q_1+\eta_1}(1-\pi_1)-\pi_2+q_2\right).$$
\end{lemma}
In the special case where $\eta_1=0$, $\pi_1=1,$
Lemma \ref{LmLTMC} follows from \\
\cite[Lemma 3.9(a)]{DG12}.
In this case the statement simplifies significantly since the first term in the RHS vanishes.
The proof of Lemma \ref{LmLTMC} will be given
in \S \ref{SS3State}.
We apply Lemma \ref{LmLTMC} to the states $(n_1, n_2, \infty)$
with $n_1 < n_2$.
This means that we define a 3-state Markov chain as a function of the
random walk $( S_k )$, such that the chain starts in the state 1,
if the random walk visits $n_1$ for the first time before it visits $n_2$; or in the state
2, if the random walk visits $n_2$ for the first time before it visits $n_1$; or in
in the state 3 if the random walk never visits $n_1$ or $n_2$. After
that, the chain transitions to state 1, 2 or 3, respectively, if the
next return of the random walk to the set $\{ n_1, n_2 \}$ occurs at $n_1$,
$n_2$, or never does. Clearly 3 is an absorbing state for this chain. So
$$\pi_1={\mathbb{P}}(n_1\text{ is visited before } n_2), \quad
\pi_2={\mathbb{P}}(n_2\text{ is visited before } n_1), $$
$$\pi_3={\mathbb{P}}(n_1\text{ and } n_2\text{ are not visited}).$$
Let $V_n$ be the event that $n$ is visited by our random walk.
Note that Lemma \ref{LmLDMin} implies $q_2 =O\left((n_2-n_1)^{-{\varepsilon}_3}\right).$ Hence
the probability that both $n_1$ and $n_2$ are visited with $n_2$ being
the first
is also $O\left((n_2-n_1)^{-{\varepsilon}_3}\right).$
Therefore
$$ \pi_1\asymp \mathfrak{q}, \quad \frac{q_1}{q_1+\eta_1}={\mathbb{P}}_{n_1}(V_{n_2})\asymp \mathfrak{q},
\quad \pi_2\asymp {\mathbb{P}}(V_{n_2})-
{\mathbb{P}}(V_{n_1} \cap V_{n_2})\asymp \mathfrak{q}-\mathfrak{q}^2, $$
where
$\mathfrak{q} = 1- \mathfrak{p}$ (see \eqref{InEqRen}) and
$\asymp$ means the difference between the LHS and the RHS is
$$ O\left(n_1^{-{\varepsilon}_2}\right)+O\left((n_2-n_1)^{-{\varepsilon}_2}\right)\quad\text{where}\quad
{\varepsilon}_2=\min({\varepsilon}_1, {\varepsilon}_3). $$
These estimates, combined with
Lemma \ref{LmLTMC} imply Lemma \ref{LmLTCov}.
\subsection{Analysis of three state chains.}
\label{SS3State}
\begin{proof}[Proof of Lemma \ref{LmLTMC}]
Under the assumptions of the lemma,
$\mathfrak{l}_1, \mathfrak{l}_2$ and $\mathfrak{l}_1\mathfrak{l}_2$ are uniformly integrable
(the uniformity is over all chains satisfying \eqref{Ell}). Let $\bar\mathfrak{l}_1$
be the time spent at 1 before the first visit to another state.
Then,
by the uniform integrability,
$$ {\mathbb{E}}(\mathfrak{l}_1-\bar\mathfrak{l}_1)=O(q_2), \quad {\mathbb{E}}((\mathfrak{l}_1-\bar\mathfrak{l}_1)\mathfrak{l}_2)=O(q_2), \quad
{\mathbb{E}}_2(\mathfrak{l}_1)=O(q_2), \quad {\mathbb{E}}_3(\mathfrak{l}_j)=0 .$$
Hence
$$ {\mathbb{E}}(\mathfrak{l}_1 \mathfrak{l}_2)=\pi_1 {\mathbb{E}}_1 (\bar\mathfrak{l}_1) \frac{q_1}{q_1+\eta_1} {\mathbb{E}}_2(\mathfrak{l}_2)+O(q_2); $$
$$ {\mathbb{E}}(\mathfrak{l}_1)=\pi_1 {\mathbb{E}}_1 (\bar\mathfrak{l}_1)+O(q_2) \quad\text{and}\quad
{\mathbb{E}}(\mathfrak{l}_2)=(\pi_1\frac{q_1}{q_1+\eta_1}+\pi_2) {\mathbb{E}}_2(\mathfrak{l}_2)+O(q_2).
$$
Therefore
\begin{align*} {\rm Cov}(\mathfrak{l}_1, \mathfrak{l}_2)&=&\pi_1 {\mathbb{E}}_1 (\bar\mathfrak{l}_1) \frac{q_1}{q_1+\eta_1} {\mathbb{E}}_2(\mathfrak{l}_2)
-\pi_1 {\mathbb{E}}_1 (\bar\mathfrak{l}_1) (\pi_1\frac{q_1}{q_1+\eta_1}+\pi_2) {\mathbb{E}}_2(\mathfrak{l}_2)+O(q_2) \\
&=&\pi_1 {\mathbb{E}}_1 (\bar\mathfrak{l}_1) {\mathbb{E}}_2(\mathfrak{l}_2) \left[\frac{q_1}{q_1+\eta_1} (1-\pi_1)-\pi_2\right]
+O(q_2) \hskip3cm
\end{align*}
as claimed.
\end{proof}
\section{Counterexamples to the strong law.}
\label{sec:oceans}
Here we prove Theorem \ref{ThOcean}.
Consider first the case $d=1$.
Assume that we are given a sequence $a_n \in \mathbb{Z}$, $a_n \nearrow \infty$ with $a_n -a_{n-1} \in \{0,1\}$
(to be specified later, see Lemma \ref{lem:stableprocess}).
Now define
$b_1 \gg 1$, $b_{n+1} = b_n + \lfloor b_n / a_n \rfloor$. By induction, we see that
\begin{equation}
\label{eq:b_nbounds}
a_n\leq n < b_n < b_{n+1} < 2^{n+1}.
\end{equation}
Consider the function $F$ defined by $F(0) = 0$,
\begin{equation}
\label{defcounterex}
F(x) = \begin{cases}
1 & \text{ if } b_{2k} \leq x < b_{2k+1} \text{ for some } k\\
0 & \text{ if } b_{2k +1} \leq x < b_{2k+2} \text{ for some } k\\
\end{cases}
\end{equation}
for $x>0$ and $F(x) = F(-x)$ for $x <0$. Since $b_{n+1} / b_n \to 1$, we have $F \in \mathbf{G}_0$ with $\bar F = 1/2$.
Let us denote $c_n = (b_n + b_{n+1})/2$, $t_n = \lfloor c_n^{\alpha} \rfloor$ and
$$I_n = [c_n - \lfloor b_n / 4 a_n \rfloor, c_n + \lfloor b_n / 4 a_n \rfloor]$$
We will show that almost surely, infinitely many of the events
$$
A_{2n} = \{ \forall k \in[ t_{2n}, 3 t_{2n}]: S_k \in I_{2n}\}
$$
occur, and likewise, infinitely many of the events
$$
A_{2n+1} = \{ \forall k \in [t_{2n+1}, 3 t_{2n+1}]: S_k \in I_{2n+1}\}
$$
occur. This
proves the theorem as $A_{2n}$ implies $T_{3t_{2n}} \geq 2 t_{2n}$ and
$A_{2n+1}$ implies $T_{3t_{2n+1}} \leq t_{2n+1}$.
To complete the proof, let us fix a sequence $D_n \nearrow \infty$ such that
$$ \sum_n {\mathbb{P}}(|S_n|\geq D_n)<\infty. $$
Then by the Borel-Cantelli Lemma,
$$
\mathbb P (\exists N: \forall n > N: |S_n| < D_n) =1.
$$
Now we choose a subsequence $n_k \in \mathbb{Z}$ inductively so that
\begin{equation}
\label{n_kparity}
n_{k+1} \equiv n_k \pmod{2}
\end{equation}
and
$$n_{k+1} > \max \left\{\exp \left(D_{ \lceil 2^{\alpha (n_k +1)} \rceil }\right),
\exp \left( \frac{1}{\alpha} 2^{(n_k +1)\alpha}\right) \right\} $$
These bounds, combined with \eqref{eq:b_nbounds}, give
\begin{equation}
\label{eq:b_nbd2}
b_{n_{k+1}} > \exp (D_{t_{n_k}}) \text{ and } t_{n_{k+1}} > \exp (t_{n_k}).
\end{equation}
We want
to show that for every ${\varepsilon} >0$ and every $K$,
\begin{equation}
\label{eq:BC}
\mathbb {\mathbb{P}} \left(\bigcap_{k=K}^{\infty} A_{n_k}^c\right) < {\varepsilon}.
\end{equation}
Since ${\varepsilon} >0$ is arbitrary, it follows that infinitely many of the events $A_{n_k}$ happen.
Choosing $n_k$ to be even for all $k$ we see
that almost surely infinitely many of the events $A_{2n}$
and happen. Likewise, choosing $n_k$ to be odd for all $k$ we see
infinitely many of the events $A_{2n+1}$ happen.
Thus it remains to verify \eqref{eq:BC}.
Given ${\varepsilon}$ and $K$, choose $K'>K$ so that
$\mathbb P (\mathcal B) < {\varepsilon} $, where
$$\mathcal B = \{ \exists n > n^{\alpha}_{K'}: |S_n| > D_n\}.$$
Then we write
\begin{equation}
\label{eq:BCbound}
{\partial } \left(\bigcap_{k=K}^{\infty} A_{n_k}^c\right) \leq
{\partial } \left(\bigcap_{k=K'}^{\infty} A_{n_k}^c\right)
\leq {\varepsilon} + {\partial } \left(\bigcap_{k=K'}^{\infty} A_{n_k}^c \cap \mathcal B^c\right).
\end{equation}
By construction, we have
\begin{gather}
{\partial } \left( A_{n_{k+1}}^c \cap \mathcal B^c \Big| \bigcap_{j=K'}^{k} A_{n_j}^c \cap \mathcal B^c\right)
\leq 1 - {\partial } \left( A_{n_{k+1}} \Big| \bigcap_{j=K'}^{k} A_{n_j}^c \cap \mathcal B^c\right) \nonumber\\
\leq 1- \min_{x: |x| < D_{ 3t_{n_k}}} {\partial } ( A_{n_{k+1}} | S_{ 3t_{n_k}} = x). \label{Anest}
\end{gather}
Next, we claim
\begin{lemma}
\label{lem:stableprocess}
There is a constant $K_0$ and a sequence $a_n \nearrow \infty$ with $a_n
- a_{n-1} \in \{ 0,1\}$
such that for any $k \geq K_0$,
$$
\min_{x: |x| < D_{ 3t_{n_k}}} {\partial } ( A_{n_{k+1}} | S_{3t_{n_k}} = x)
\geq \frac{1}{k}
$$
\end{lemma}
Clearly, Lemma \ref{lem:stableprocess}
combined with \eqref{Anest} and \eqref{eq:BCbound} implies \eqref{eq:BC}.
Thus the proof of Theorem \ref{ThOcean}
for the case $d=1$ will be completed once we prove Lemma \ref{lem:stableprocess}.
\begin{proof}[Proof of Lemma \ref{lem:stableprocess}]
Recall that the invariance principle gives
\begin{equation}
\label{invprinciple}
\frac{S_{\lfloor Nt \rfloor }}{N^{1/\alpha}} \Rightarrow Y_t,
\end{equation}
where $Y_t$ is a stable L\'evy process. In particular,
$Y_1$ is a stable random variable with parameter $\alpha$ and "skewness" $\beta \in [-1,1]$
(see e.g. \cite{B96}, Chapter VIII).
Now we distinguish two cases.
{\bf Case 1} $\alpha >1$ or $|\beta|\neq 1.$
Set
\begin{equation}
\label{DefQ}
q = \inf_{y \in [-1/16, 1/16]}
\mathbb P \left(\sup_{t \leq 1} |Y_t| < \frac14, |Y_1| < \frac{1}{16} \Big| Y_0 = y \right).
\end{equation}
We claim that $q>0.$
Indeed,
as $\alpha >1$ or $|\beta|\neq 1$, the stable process $Y_t$ cannot be a subordinator. In particular, the
density of $Y_t$ is positive everywhere for every $t>0$ and $Y_t$ has the scaling property (see page 216
in \cite{B96}). Thus we have
$$
\liminf_{{\varepsilon} \searrow 0} \inf_{y \in [-1/16, 1/16]}
\mathbb P \left(|Y_{\varepsilon}| < \frac{1}{16} \Big| Y_0 = y \right) = p >0.
$$
By Exercise 2 of Chapter VIII in \cite{B96}, there is some ${\varepsilon} >0$ such that
$$
\mathbb P \left(\sup_{t \leq {\varepsilon}} |Y_t| < \frac{3}{16} \Big| Y_0 = 0 \right) >1-p/2.
$$
Combining the last two displayed equations, we derive
$$
\inf_{y \in [-1/16, 1/16]}
\mathbb P \left(\sup_{t \leq {\varepsilon}} |Y_t| < \frac14, |Y_{\varepsilon}| < \frac{1}{16} \Big| Y_0 = y \right) \geq p/2.
$$
Applying this inequality inductively, we obtain that
$q \geq (p/2)^{\lceil 1/{\varepsilon} \rceil}$.
Next, we claim that there exist constants ${\bar c}>0,$ ${ {\bar p}}\in (0,1)$ such that
\begin{equation}
\label{eq:case1exp}
\min_{x: |x| < D_{t_{n_k}}} {\mathbb{P}} ( A_{n_{k+1}} | S_{t_{n_k}} = x)
\geq \frac{{\bar c}}{a_{n_k}} \;{ {\bar p}}^{a_{n_k}^\alpha}.
\end{equation}
\eqref{eq:case1exp}
implies the lemma since we can choose any sequence $a_n \nearrow \infty$ with $a_n - a_{n-1} \in \{ 0,1\}$
such that $a_{n_k} \leq (- \log k / \log \tilde p )^{1/\alpha}$ for a fixed $\tilde p \in (0,{ {\bar p}})$.
To prove \eqref{eq:case1exp}, we first
use the local limit theorem and
\eqref{eq:b_nbd2} to derive
\begin{equation}
\label{lltarrival}
\min_{x: |x| < D_{ 3t_{n_k}}} {\mathbb{P}} \left( \left| S_{t_{n_{k+1}}} - c_{n_{k+1}} \right| < \frac{b_{n_{k+1}}}{16 a_{n_{k+1}}}
\Big| S_{ 3t_{n_k}} = x \right) > \frac{{\bar c}}{a_{n_k}}.
\end{equation}
Now using \eqref{invprinciple} with $N = (b_{n_{k+1}} / 4a_{n_{k+1}})^{\alpha}$ {and \eqref{DefQ}}
we obtain \eqref{eq:case1exp} with $p = q^{2\cdot4^{\alpha}}$.
{\bf Case 2} $\alpha <1$ and $|\beta|=1$.
Let us assume $\beta = 1$ (otherwise apply the forthcoming argument to $-X_i$).
\eqref{lltarrival} still holds in case 2, however a new approach is required to estimate
${\mathbb{P}} ( A_{n_{k+1}} | S_{t_{n_k}} = x) $
since now the process $Y_t$
(a.k.a. stable subordinator)
is non-decreasing and thus $q=0$.
Note however that in case 2, $\sup_{t \leq 1} |Y_t| = Y_1$ and thus it suffices to estimate one
random variable instead of a stochastic process.
Recall that the density of $Y_1$ is strictly positive on $\mathbb{R}^+$.
Thus applying \eqref{invprinciple}
to $|X_i|$ (which is also in the standard domain of attraction of the totally
skewed $\alpha$-stable distribution)
we obtain the following: for any ${\varepsilon} >0$ there exists $N_0({\varepsilon})$ and $\delta({\varepsilon})>0$
such that for any $N \geq N_0({\varepsilon})$,
\begin{equation}
\label{case2pos}
{\partial }\left( \sum_{n = 1}^{3 N} |X_n| \leq \frac{{\varepsilon}}{8} N^{1/\alpha} \right) > \delta({\varepsilon}).
\end{equation}
Without loss of generality, we assume that $N_0$ and $\delta$ are, respectively,
non-decreasing and non-increasing functions of ${\varepsilon}$.
Now we define the sequence $a_n$ inductively. First, let $a_1 = 1$. Now assume that $a_{n_{k}}$ is defined.
Let $a_m = a_{n_{k}}$ for $m = n_{k} + 1,..., n_{k+1} -1$. Next, we define $a_{n_{k+1}} = a_{n_{k}}+1$
if both of the following conditions are satisfied:
\begin{enumerate}
\item[(A)] $ N_0\left(\frac{1}{a_{n_{k}} +1}\right)<n_{k+1}^{\alpha} $ and
\item[(B)] $\frac{{\bar c}}{2a_{n_{k}} +1} \;\delta \!\left( \frac{1}{a_{n_{k}} +1}\right) > \frac{1}{k+1}$.
\end{enumerate}
Here, ${\bar c}$ is the constant from \eqref{lltarrival}.
If either (A) or (B) fails, we put $a_{n_{k+1}} = a_{n_{k}}$.
Observe that by our construction, for all $k$, we have
\begin{equation}\label{NZero}
N_0\left(\frac{1}{a_{n_{k}}}\right)<n_{k}^{\alpha} ;
\end{equation}
\begin{equation} \label{Delta}\frac{{\bar c}}{2 a_{n_{k}}} \;\delta \!\left( \frac{1}{a_{n_{k}}} \right) > \frac{1}{k}.
\end{equation}
Indeed,
if $a_{n_{k+1}}=a_{n_k}+1$ then \eqref{NZero} and \eqref{Delta} follow from conditions (A) and (B) above.
If $a_{n_{k+1}}=a_{n_k}$ then \eqref{NZero} and \eqref{Delta} follow by induction since the LHSs
of both \eqref{NZero} and \eqref{Delta} do not change when we replace $k$ by $k+1,$ while
the RHS of \eqref{NZero} increases and the RHS of \eqref{Delta} decreases.
By construction, $a_n \nearrow \infty$. Let $K_0$ be the smallest integer $k$ so that $a_{n_k} = 2$.
We prove that the lemma holds with this choice of $K_0$ and $a_n$.
Recall that
by \eqref{eq:b_nbounds},
$b_{n_k}>n_k$ and so by \eqref{NZero}, $N:= b_{n_k}^{\alpha} > N_0({\varepsilon})$ with ${\varepsilon} = 1/a_{n_k}$. Applying
\eqref{case2pos} with this $N$ and ${\varepsilon}$ and using \eqref{Delta}, we obtain
\begin{equation*}
\label{eq:stayintube}
{\partial } \bigg( \left| S_m - S_{t_{n_{k}}} \right| <
\frac{b_{n_{k}}}{8 a_{n_{k}}} \;\; \forall m \leq 3 b^{\alpha}_{n_{k}}
\bigg)
> \frac{a_{n_k}}{{\bar c}} \frac{2}{k}.
\end{equation*}
and since $t_{n_k} < b_{n_k}^{\alpha}$ and $k-1 > k/2$, we arrive at
\begin{equation}
\label{eq:stayintube}
{\partial } \bigg( \left| S_m - S_{t_{n_{k}}} \right| <
\frac{b_{n_{k}}}{8 a_{n_{k}}} \;\; \forall m \leq 3t_{n_{k}}
\bigg)
> \frac{a_{n_{k-1}}}{{\bar c}} \frac{1}{k-1}.
\end{equation}
Combining \eqref{eq:stayintube} with $k$ replaced by $k+1$ and \eqref{lltarrival}, we obtain the estimate of the lemma.
\end{proof}
The above proof, with a few minor adjustments, applies to arbitrary dimension $d$.
Specifically, we need to consider the function
$\mathcal F \in \mathbf{G}_0$
defined by
$$\mathcal F(x_1,...,x_d) =
\begin{cases}
F(x_1) & \text{ if } |x_i| \leq |x_1| \text{ for } i=2,...,d\\
\frac12 & \text{ otherwise,}
\end{cases}
$$
where $F$ is given by \eqref{defcounterex} and we need to replace $I_n$ by
$$ [c_n - \lfloor b_n / 4 a_n \rfloor, c_n + \lfloor b_n / 4 a_n \rfloor]
\times \left[- \lfloor b_n / 4 a_n \rfloor, \lfloor b_n / 4 a_n \rfloor\right]^{d-1}.
$$
\begin{remark}
It is easy to adjust the above proof to derive the following stronger version of Theorem \ref{ThOcean}:
There is a function $F \in \mathbf{G}_0$ so that $F$ only takes values $\{0,1\}$, ${\bar F} = 1/2$ and for almost every $\omega$ and for any
$a \in [0,1]$,
there is a subsequence $n_k = n_k(a, \omega)$ such that
$T_{n_k} / n_k \to a$.
\end{remark}
\ignore{
\section{Auxiliary statements.}
\label{sec:app}
\subsection{Maximum of random walk.}
\begin{proof}[Proof of Lemma \ref{LmMax}]
In case (i) the statement follows from the Law of Large Numbers, so we only need to consider
{cases (ii) and (iii).
We have for any ${\varepsilon} >0$ that $|S_N| >N^{1/\alpha + {\varepsilon}}$ holds only finitely many times almost surely
by \cite{M39} in case (ii) and by \cite{L31} in case (iii). The lemma follows.
}
\end{proof}
{
Pick $1/\alpha<\gamma_2<\gamma_1.$ Observe that by Borel-Cantelli Lemma
$|X_n|<n^{\gamma_2},$ so it suffices to prove that for each $\gamma_3>\gamma_2$
\begin{equation}
\label{MaxCutOff}
\sum_{n=1}^N X_n 1_{|X_n|<n^{\gamma_2}}< N^{\gamma_3}.
\end{equation}
The claim of the lemma then follows by taking $\gamma_3<\gamma_1.$
To prove \eqref{MaxCutOff} we split
$$ X_n 1_{|X_n|<n^{\gamma_2}}=\mathcal{D}_n+\mathcal{X}_n,$$
where $\displaystyle \mathcal{D}_n={\mathbb{E}}\left(X_n 1_{|X_n|<n^{\gamma_2}}\right).$
We begin with estimating the drift.
\cite{BGT} shows that for each $\delta>0$
there is a constant $C=C_\delta$ such that
\begin{equation}
\label{RV-EXP}
|\mathcal{D}_n|\leq C n^{\gamma_2(1-\alpha+\delta)}
\end{equation}
To establish \eqref{RV-EXP} in case $\alpha\leq 1$ one can use that
$$ D_n=\int_{|x|<n^{\gamma_2}} x dP(X>x) $$
while in case $\alpha>1$ we need to use the fact ${\mathbb{E}}(X_n)=0$
and hence
$$ D_n=\int_{|x|>n^{\gamma_2}} x dP(X>x) .$$
\eqref{RV-EXP} gives
$$ \sum_{n=1}^N \mathcal{D}_N \leq C N^{\gamma_2(1-\alpha+\delta)+1}. $$
Observe that if $\gamma_2=1/\alpha$and $\delta=0$ then
$$ \gamma_2(1-\alpha+\delta)+1=\frac{1}{\alpha}<\gamma_3. $$
Therefore taking $\delta$ sufficiently small and $\gamma_2$ sufficiently close to
$1/\alpha$ we can get
$$ \gamma_2(1-\alpha+\delta)+1< \gamma_3$$
so $\displaystyle \sum_{n=1}^N \mathcal{D}_N$ is negligible.
\cite{BGT} also shows that ${\mathbb{E}}(\mathcal{X}_n^2)\leq C n^{\gamma_2(2-\alpha+\delta)}$ whence
$$ V\left(\sum_{n=1}^N \mathcal{X}_n\right)\leq C N^{\gamma_2(2-\alpha+\delta)+1}. $$
By the maximal inequality for each $p$
$$ {\mathbb{P}}\left(\sum_{n=1}^N \mathcal{X}_n\geq N^{\gamma_3}\right)\leq C
\frac{N^{p(\gamma_2(2-\alpha+\delta)+1)}}{N^{2p\gamma_3}}.
$$
Taking $p$ sufficiently large, $\delta$ sufficiently small, and $\gamma_2$ sufficiently
close to $1/\alpha$ we can make the RHS smaller than
$1/N^2.$ Now \eqref{MaxCutOff} follows from Borel-Cantelli Lemma.}}
\ignore{
\subsection{Large deviations for minimum.}
\label{SSLDMin}
We need the following fact.
\begin{lemma}
\label{LmLDMax}
Let $\mathcal{X}_n$ be iid random variables
such that ${\mathbb{E}}(\mathcal{X}_n)=0,$ and for some $C, \beta >0$ and for all $t>1$,
${\mathbb{P}}(|\mathcal{X}|>t)\leq \frac{C}{t^\beta}.$ Then for each $c$ there exists ${\bar C}$
such that the random walk $\displaystyle \mathcal{S}_N=\sum_{n=1}^N \mathcal{X}_n$
satisfies
$$ {\mathbb{P}}\left(\max_{1\leq n\leq N} (|\mathcal{S}_n|)>c N\right)\leq {\bar C} N^{1-\beta}. $$
\end{lemma}
\begin{proof}
Let $\displaystyle \mathcal{X}_n^*=\mathcal{X}_n 1_{|\mathcal{X}_n|\leq N}$.
We have
$$ {\mathbb{P}}\left(\max_{1\leq n\leq N} (|\mathcal{S}_n|)>N\right)\leq {\mathbb{P}}(\exists n\leq N: \mathcal{X}_n^*\neq \mathcal{X}_n)+
{\mathbb{P}}\left(\max_{1\leq n\leq N} (|\mathcal{S}_n^*|)>cN/2\right)$$
where $\mathcal{S}_N^*=\sum_{n=1}^N \mathcal{X}_n^*.$ The first term is $O\left(N^{1-\beta}\right).$
The second term is bounded by
$$
{\mathbb{P}}\left(\max_{1\leq n\leq N} (\mathcal{S}_n^*)>cN/2\right) + {\mathbb{P}}\left(\max_{1\leq n\leq N} ( - \mathcal{S}_n^*)>cN/2\right)
$$
We estimate the first probability, the second being identical.
Since $ \displaystyle \lim_{N\to\infty} {\mathbb{E}}\left(\mathcal{X}_n^*\right)=0$, we can assume that
$ |{\mathbb{E}}\left(\mathcal{X}_n^*\right)|< c/4$. Now by Doob's inequality
$${\mathbb{P}}\left(\max_{1\leq n\leq N} (\mathcal{S}_n^*)>cN/2\right) \leq
{\mathbb{P}}\left(\max_{1\leq n\leq N} (\mathcal{S}_n^* - {\mathbb{E}}(\mathcal{S}_n^*))^2>c^2N^2/16\right)$$
\hskip5cm $\displaystyle \leq \frac{16 {\rm Var}(\mathcal{S}_N^*)}{c^2 N^2} = \frac{{\hat C}}{N^{1-\beta}}$
\end{proof}
\begin{proof}[Proof of Lemma \ref{LmLDMin}] Let $\mathcal{X}_n=X_n-v.$
As before we write $\displaystyle S_n = \sum_{k=0}^{n-1} X_k$ and
$\displaystyle \mathcal{S}_n = \sum_{k=0}^{n-1} \mathcal{X}_k$.
Then
$$ \{\min(S_n) \leq -m\}\subset\bigcup_{j=0}^\infty A_j$$
where $\displaystyle A_0=\{\max_{1\leq n\leq m}|\mathcal{S}_n|\geq m\}$
and
$\displaystyle A_j=\{\max_{2^{j-1} m\leq n\leq 2^j m}|\mathcal{S}_n|\geq 2^{j-1} v m\}$
for $j\geq 1.$ By Lemma \ref{LmLDMax}
$\displaystyle {\mathbb{P}}(A_j)\leq {\bar C} \left(2^j m\right)^{1-\beta}.$ Summing over $j$ we obtain the result.
\end{proof}}
\section{Conclusions.} The results proven in this paper show that
for random walks the weak law of large numbers holds in the largest possible space of global
observables, namely $\mathbf{G}_0.$ On the other hand, the strong law of large numbers fails, in general,
except for the walks with drift in dimension 1. In that case the path of the walk is almost deterministic
and so the ergodic theory for occupation times could be used. The good news
is that the weak law of large numbers seems to be a good setting for homogenization theory
(cf Theorem~\ref{ThArc}), so the space
$\mathbf{G}_0$ could be useful for that purpose. If we have some control on fluctuations over the mesoscopic
scale as provided, for example, by the space $\mathbf{G}_\gamma,$ then we can ensure the strong law.
If we have polynomial control on the mesoscopic scale, as provided by the space $\mathbf{G}_\gamma^\beta$
then we can estimate the rate of convergence. In particular, our results give optimal rate of convergence
for two important special cases: random walks in random scenery and quasi-periodic observables.
We note that the main ingredient in most proofs is local limit theorem and its extensions, such as the
Edgeworth expansion used in Section \ref{ScSpeed}. This makes it plausible that similar results hold for
other systems where the local limit theorem hold, including the systems described in
\cite{BCR, CD15, DG13, DG18, DN-LLT, DN-Inf, DN18}. Another natural research direction motivated by the
present work is limit theorems for global observables. It is likely that just assuming that $F$ belongs to an
appropriate $\mathbf{G}_*$
will not be enough to derive limit theorems.
For example, the computation of variance for $T_N$ done in Sections \ref{ScWeak}
and \ref{ScSpeed} involve the expression of the form ${\tilde F}_z(x):=F(x) F(x+z)$ for a fixed $z\in \Z^d.$
Therefore additional restrictions seem to be required to obtain limit theorems.
Extending our results to more general systems as well as limit theorems for global observables will
be the subject of future work.
|
{
"timestamp": "2019-03-01T02:18:24",
"yymm": "1902",
"arxiv_id": "1902.11071",
"language": "en",
"url": "https://arxiv.org/abs/1902.11071"
}
|
\section{Introduction}
Within few years, it has been expected that tens of exabytes of global data traffic be handled on daily basis, and on-demand video streaming will account for about 70\% of them \cite{cisco}.
In on-demand video streaming services, a relatively small number of popular contents is requested at ultra high rates and playback delay is one of the most important measurement criteria of goodness \cite{youtube,mm17koo}.
To deal with the characteristics, wireless caching technologies have been studied for video streaming services by storing popular videos in caching helpers located nearby users during off-peak time~\cite{femtocaching,CM2014Bastug,CM2014Wang}.
Therefore, it is obvious that storing and streaming of video files are of major research interests in wireless caching networks.
There have been major research results for caching popular files in stochastic wireless caching networks~\cite{caching:ICC2015Blaszczyszyn,CL2017Chen,caching:TWC2016Chae,caching:TC2016Malak,JSAC2016Gregori}.
The major goal of those research results was to design the optimal caching policies according to the popularity distribution of contents and wireless network topology.
The probabilistic caching policy was proposed in \cite{caching:ICC2015Blaszczyszyn} to adapt characteristics of the stochastic network.
Many probabilistic caching methods have been proposed depending on various optimization goals, e.g., maximization of cache hit probability \cite{caching:ICC2015Blaszczyszyn}, cache-aided throughput \cite{CL2017Chen}, average success probability of content delivery \cite{caching:TWC2016Chae}, density of successful reception \cite{caching:TC2016Malak}, and average video quality \cite{caching:ICC2019Choi}.
The authors of \cite{JSAC2016Gregori} considered a joint optimization of caching and delivery when user demands were known in advance.
In addition, the optimal caching policy which maximizes the cache hit probability in two-tier networks with opportunistic spectrum access was designed in \cite{TWC2018Emara}.
However, these works on the caching policy do not consider the identical content with different qualities.
Since video files can be encoded to multiple versions which differ in the quality levels, the video caching policies having different quality levels have been widely studied in~\cite{cachingDiffQual:infocom2014Poularakis,cachingDiffQual:CL2017Zhan,infocom2016Poularakis,cachingDiffQual:CL2016Wu}.
Many researchers have proposed the static content placement policies under the consideration of differentiated quality requests for the same content, given probabilistic quality requests \cite{cachingDiffQual:infocom2014Poularakis}, \cite{cachingDiffQual:CL2017Zhan} or minimum quality requirements \cite{infocom2016Poularakis}.
Further, the probabilistic caching policy for video files of various quality levels was presented in \cite{cachingDiffQual:CL2016Wu} by using stochastic geometry, given the user preference for quality level.
The above works are focused only on the content placement problem with different qualities, however, the delivery policy of contents with different qualities has not yet been studied much.
For video delivery/streaming, there are some necessary decisions to be made: 1) which caching node will deliver the video, 2) which quality of video will be provided, and 3) how many video chunks will be transmitted.
The first one is
called {\em node association problem}, and in most research contributions that do not consider different quality levels for the same file, the file-requesting user is allowed to receive the desired video from the caching node under the strongest channel condition \cite{caching:TWC2016Chae,TWC2016Yang}.
The node associations for video delivery in heterogeneous caching networks have been studied in \cite{NA:TC2014Poularakis,NA:TC2016Zhang,NA:TMC2017Jiang}.
On the other hand, when videos with different qualities are independently cached, more elaborate node association algorithm is necessary, because the node association is consistent with decision on the content quality.
In this case, the video delivery policy was proposed in \cite{JSAC2018Choi} to pursue time-average video quality maximization while avoiding playback delays.
Since dynamic video streaming allows each chunk to have a different quality depending on time-varying network conditions, some researchers addressed transmission schemes which serve the video by dynamically selecting the quality level \cite{VD_DiffQ:TM2013Wang}.
In \cite{VD_DiffQ:TC2015Bethanabhotla} and \cite{VD_DiffQ:TON2016Kim}, the scheduling policies which maximize the network utility function of time-averaged video quality in small-cell networks and device-to-device networks were proposed.
The authors of \cite{TC2018Yang} considered scalable video coding (SVC) and proposed dynamic resource allocation and quality selection under the pricing strategy for interference.
While the video delivery policies of \cite{VD_DiffQ:TM2013Wang,VD_DiffQ:TC2015Bethanabhotla,VD_DiffQ:TON2016Kim,TC2018Yang} are operated at the base station (BS) side, however the decision policy of video quality level at user sides was not considered.
This scenario is consistent with the practical real-world software implementation of dynamic adaptive streaming over HTTP (DASH) \cite{dash}, in which users dynamically choose the most appropriate video quality.
Even though the work of \cite{JSAC2018Choi} can choose the video quality at the user side, however it cannot dynamically change the video quality without updates of node association.
Further, control of the amount of receiving chunks depending on stochastic network states has been largely neglected in above existing researches about video delivery.
Even though the authors of \cite{JSAC2018Choi} and \cite{TC2018Yang} maximize the long-term time-average video quality under the various constraints, their metrics representing video quality is obtained by averaging the number of quality selections at each time slot.
This method would be not enough to evaluate the user's quality of service, especially when the transmission rate varies over the video streaming service time.
In practice, when channel experiences deep fading and only the low-quality video is available, it would not be the best choice to receive as many chunks as the channel condition can provide.
Rather than receiving many low-quality chunks, the user could prefer to wait channel conditions to be better and then to receive high-quality chunks, if it guarantees no playback delay.
Therefore, by considering decision process of combinations of video quality and chunk amounts, we can formulate the optimization problem which maximizes the average video quality per each received chunk.
This paper proposes a video delivery policy in the wireless caching network for dynamic streaming services.
The main contributions are as follows:
\begin{itemize}
\item This paper proposes dynamic video delivery policy depending on stochastic network states. The proposed policy makes three different but necessary decisions for the streaming user: 1) caching node for video delivery, 2) video quality and 3) the quantity of video chunks to receive.
To the best of the authors' knowledge, no research has yet considered all of those video delivery decisions.
\item Caching node association and decisions of video quality and the amount of receiving chunks are conducted in different timescales.
Since wireless link activation for video delivery is time-consuming, it is reasonable that caching node association is performed slower than decisions of video quality and the amount of receiving chunks.
\item The optimization framework of video delivery policy is constructed based on frame-based Lypunov optimization theory \cite{TAC2013Neely} and Markov decision process.
The optimal caching node is found by Lyapunov optimization while decisions of video quality and the amount of receiving chunks are made by using dynamic programming \cite{DynamicProgram}.
\item The proposed technique maximizes the average streaming quality while averting playback latency, and can control the tradeoff between video quality and playback delay.
Different from \cite{JSAC2018Choi} and \cite{TC2018Yang}, we adopt the long-term average video quality per each received chunk as a performance metric.
\item We perform simulations to verify the proposed video delivery policy and to show the advantages of using Lyapunov optimization theory and Markov decision process.
\end{itemize}
The rest of the paper is organized as follows.
The wireless video caching network model is described in Sec.~\ref{sec:network_model}.
The optimization problem for dynamic video delivery is formulated in Sec.~\ref{sec:prob_formulation}.
The rule of caching node association and control policies of quality level and receiving chunk amounts are proposed in Sec.~\ref{sec:caching_node_decision} and Sec.~\ref{sec:qual_chunk_decisions}.
Simulation results are presented in Sec.~\ref{sec:simulation} and Sec.~\ref{sec:conclusion} concludes this paper.
\section{Network Model}
\label{sec:network_model}
\subsection{Wireless caching network model}
This paper considers wireless caching network model where a user requests certain video file for one of caching nodes
around the user, as shown in Fig.~\ref{fig:network_model}.
The BS has already pushed popular video files during off-peak hours to caching nodes which have the finite storage size.
Since we focus on video delivery, the caching policy is out of scope for this paper and only the desired video is considered.
Suppose that the desired video has $L$ quality levels.
Therefore, there are $L$ types of caching nodes, and the type-$l$ caching nodes can deliver the video of any quality $q \in \mathcal{L}_l$, where $\mathcal{L}_l=\{1, \cdots, l \}$ is the set of qualities which the type-$l$ caching node can provide.
Thus, the type-$L$ caching nodes can provide all quality levels from the quality set $\mathcal{L}_L$.
Note that simple definition of $\mathcal{L}_l=\{1, \cdots, l \}$ is assumed, but the proposed technique can be coordinated with any arbitrary quality set as long as multiple versions of the same video having different qualities are stored in caching nodes.
The identical files of different qualities are stored in multiple caching nodes, and the type-$l$ caching nodes are distributed by the independent Poisson Point Processes (PPPs) with intensity $\lambda p_q$ \cite{caching:ICC2015Blaszczyszyn}, where $p_q$ indicates the caching probability of the requested video encoded to provide any quality $q \in \{1,\cdots,l\}$.
Suppose that the caching policy is already determined, i.e., all $p_q$ for all $q$ are given.
In addition, videos of different qualities have different file sizes and $N_q$ denotes the file size of quality $q$ in bits, satisfying $N_m < N_q$ for all $m, q \in \mathcal{L}_L$ and $m < q$.
User mobility is also captured in network model.
The user is moving towards certain direction and periodically searches for a caching node to receive the desired video file.
As shown in Fig.~\ref{fig:network_model}, geological distribution of caching nodes around the user varies at each time slot, so the caching node decision should be appropriately updated.
Further, this paper also considers how many chunks of which quality level to be requested from the user depending on the stochastic network environment.
When there are other users who exploit the wireless caching network with the same resource, the target streaming user is interfering with them.
We adopt the distance-based interference management to limit the interference power lower than certain threshold, and details are explained in Section \ref{subsec:interference}.
\begin{figure} [t!]
\centering
\includegraphics[width=0.75\textwidth]{Network_Model.png}
\caption{Network Model}
\label{fig:network_model}
\end{figure}
\subsection{User queue model and channel model}
A video file consists of many sequential chunks.
The user receives the video file from a caching node and processes data for the streaming service in units of chunks.
Each chunk of a file is responsible for some playback time of the entire stream.
As long as all chunks are in correct sequence, each chunk can have different quality in dynamic streaming.
Therefore, the user can dynamically choose video quality level in each chunk processing time.
By using the queueing model, it can be said that the playback delay occurs when the queue does not have the chunk to be played.
In this sense, receiver queue dynamics collectively reflects the various factors which cause the playback delay.
In general, the user model has its own arrival and departure processes.
The user queue dynamics in each discrete time slot $t \in \{0,1,\cdots \}$ can be represented as follows:
\begin{equation}
Q(t+1) = \max \{ Q(t) - c, 0 \} + M(t) \text{ and } Q(0)=0, \label{eq:q_dynamics1}
\end{equation}
where $Q(t)$ stands for the queue backlog at time $t$.
In addition, the departure $c$ is a constant because the streaming user does not change the video playback rate in general.
The arrival $M(t)$ denotes the number of received chunks at time $t$.
Let the caching node which the user chooses for video delivery be $\alpha$.
Then, $h(\alpha,t)=\sqrt{D(\alpha, t)}u(t)$ represents the Rayleigh fading channel between the user and the caching node $\alpha$ at time $t$, where $D(\alpha, t)=1/d(\alpha, t)^2$ controls path loss with $d(\alpha, t)$ being the user-caching node distance at time $t$ and $u$ represents the fast fading component having a complex Gaussian distribution, $u\sim \mathcal{C}\mathcal{N}(0,1)$.
The link rate can be simply given by
$R(\alpha,t) = \mathcal{W} \log_2 \Big( 1+ \frac{\Psi|h(\alpha,t)|^2}{\Upsilon + 1} \Big)$,
where $\mathcal{W}$, $\Psi$, and $\Upsilon$ are bandwidth, transmit SNR, and interference-noise-ratio (INR), respectively.
The number of received chunks necessarily depends on the caching node decision $\alpha$ and its link rate.
In this paper, each slot interval is determined to be channel coherence time $t_c$.
Then, the number of received chunks $M(t)$ is constrained by
\begin{equation}
M(t) N_{q(t)} \leq t_c R(\alpha,t).
\end{equation}
Since $M(t)$ and $N_{q(t)}$ are nonnegative integers,
\begin{equation}
M(t) N_{q(t)} \leq B(\alpha,t) = \lfloor t_c R(\alpha,t) \rfloor. \label{eq:chunk_const}
\end{equation}
Therefore, the decision of $M(t)$ depends on the decisions of $\alpha(t)$ and $q(t)$ and the random network event $R(\alpha, t)$.
\subsection{Distance-based interference management}
\label{subsec:interference}
Although many existing works have investigated
complex interference management schemes such as interference alignment and interference cancellation, most of researches on the wireless caching and delivery policy have still
used simple interference avoidance based interference management schemes, e.g., by spectrum sharing \cite{TWC2017Chen} or assuming the protocol model \cite{TWC2018Qin}.
For simplicity, this paper considers the distance-based interference control for node association (i.e., link activation) for video delivery.
The design ideas can be extended to other more sophisticated interference management schemes \cite{DySPAN2005Etkin, ICC2009Blomer}.
Activation of the new link for video delivery in the wireless caching network means that the network allows the new streaming user to interfere with existing users.
A new user causes two types of interference, 1) from the caching nodes already
serving existing users to the new user, and 2) from the caching node associated with the new user to existing users.
Therefore, we define $R_U$ and $R_N$ as the safety distances for streaming users and their associated caching nodes respectively to keep the interference levels below the predetermined threshold of $\rho$.
In other words, a new streaming user who wants to exploit the wireless caching network should be generated outside the radius $R_N$ of all caching nodes associated with the existing users.
In addition, the new user has to find the caching node to receive the desired content outside the radius $R_U$ of all existing users.
The safety distances of $R_U$ and $R_N$ should be carefully chosen, and then
a new pair of a caching node and a user can be generated only when their interference power is acceptable for every existing video delivery link, as shown in Fig. \ref{fig:safety_radii}.
\begin{figure} [t!]
\centering
\includegraphics[width=0.6\columnwidth]{safety_radii.pdf}
\caption{Safety radius and activation of new link for video delivery}
\label{fig:safety_radii}
\end{figure}
In this regard, a new pair of a caching node and a streaming user is allowed for video delivery through following two steps.
The first step is to confirm the INR, say $\Upsilon_0$, at the new streaming user to be lower than $\rho$.
Here, $\Upsilon_0$ is the ratio of the aggregated interference power from all the activated caching nodes to noise variance.
If $\Upsilon_0 > \rho$, the system does not allow the new user to exploit the wireless caching network, and the new user should directly request the desired content from the server which has a whole file library or wait for content delivery in future.
For example, suppose that interference power from the nearest interfering caching node to the new user dominates $\Upsilon_0$.
We further let $\Psi_0$ be the transmit SNR of the interfering node, and $d_n$ be the distance from the interfering node to the new user.
Then, INR becomes $\Upsilon_0 = \frac{\Psi_0}{d_n^2}$, and $R_N \geq \sqrt{\Psi_0/\rho}$ to guarantee $\Upsilon_0 \leq \rho$.
Although the interference power at the new user is safe to exploit the wireless caching network, i.e., $\Upsilon_0 \leq \rho$, the caching node associated with the new user will be able to degrade the signal-to-interference-plus-noise ratios (SINRs) of existing users.
Thus, the second step is required, in which the new user should find the caching node to receive the desired content with sufficiently large link rate as well as not to significantly interfere with other users.
Let one of existing users have a margin of INR to guarantee $\rho$ before activating the new link, denoted by $\delta = \rho - \Upsilon_0$.
Since the interference signal from the new caching node is independent on other interfering nodes, $R_U = \sqrt{\Psi_0/\delta}$ is obtained similar to the case of $R_N$.
Therefore, the new caching node should be chosen outside the radius of $R_U$ from every existing user whose margin of INR would be different from each other.
Even though the new user and its caching node are found while limiting all interference levels at users lower than $\rho$, the newly generated link between them could not be enough to provide reliable content transmissions due to bad channel conditions.
Therefore, we investigate the existence of the caching node around the new user which stores the requested content and can deliver the content reliably.
Let the minimum SINR for reliable video delivery denoted by $\gamma_{\text{min}}$.
Then, the probability that at least one caching node can successfully deliver the desired content to the new user is represented by
\begin{align}
\eta = \mathrm{Pr} \bigg\{ \frac{\Psi|h_{n,1}|^2}{\Upsilon + 1} \geq \gamma_{\text{min}} \bigg\},
\label{eq:eta}
\end{align}
where $h_{n,1}$ is the channel gain between the new user and the caching node whose channel condition is the strongest among the nodes storing the desired content of the user.
According to order statistics and \cite{caching:TWC2016Chae}, the cumulative distribution function of the smallest reciprocal of channel power is $F_{\xi_{n,1}}(\xi) = 1-e^{-\pi\Gamma(2) \lambda_n \xi }$, where $\xi_{n,1}=1/|h_{n,1}|^2$ and $\lambda_n$ is the intensity of PPP of nodes caching the desired content.
According to \eqref{eq:eta}, $\eta$ can be found by
\begin{equation}
\eta = 1 - \exp \bigg\{ -\pi \Gamma(2) \lambda_n \frac{\Psi}{\gamma_{\text{min}} (\Upsilon+1)} \bigg\}.
\end{equation}
Then, by introducing the minimum probability of finding at least one caching node for reliable video delivery denoted by $\eta_{\text{min}}$, a set of $\{\gamma_{\text{min}}, \eta_{\text{min}} \}$ can be considered as a criterion for new reliable link activation which satisfies $\eta\geq \eta_{\text{min}}$.
In this regard, we can verify how much interference power is acceptable to satisfy the criterion of $\{\gamma_{\text{min}}, \eta_{\text{min}} \}$, as follows:
\begin{align}
&1 - \exp \bigg\{ -\pi \Gamma(2) \lambda_n \frac{\Psi}{\gamma_{\text{min}} (\Upsilon+1)} \bigg\} \geq \eta_{\text{min}} \nonumber \\
&~\iff \frac{\pi \Gamma(2) \lambda_n \Psi}{\gamma_{\text{min}} \ln (\frac{1}{1-\eta_{\text{min}}})} - 1 \geq \Upsilon.
\end{align}
Thus, if all network parameters are given, the threshold of interference power can be determined by
\begin{equation}
\rho = \frac{\pi \Gamma(2) \lambda_n \Psi}{\gamma_{\text{min}} \ln (\frac{1}{1-\eta_{\text{min}}})} - 1.
\end{equation}
On the other hand, if the network requires the target criterion of interference management, i.e., $\rho$, $\gamma_{\text{min}}$, and $\eta_{\text{min}}$ are given, the system can determine how much transmit power is required and/or how many caching nodes store the desired video.
In this paper, the minimum SINR threshold is set so that the chunk of the smallest size (i.e., the lowest quality) is deliverable at least,
i.e., $t_0 \mathcal{W}\log_2(1+\gamma_{\text{min}}) = N_1$.
Then, we can say that caching nodes which store the desired content should be distributed with the intensity of $\lambda_{\text{min}}$ at least, as follows:
\begin{equation}
\lambda_n \geq \lambda_{\text{min}} = \frac{ (2^{\frac{N_1}{t_0\mathcal{W}}}-1) \ln (\frac{1}{1-\eta_{\text{min}}}) (1+\Upsilon) }{ \pi \Gamma(2) \Psi }.
\end{equation}
\section{Dynamic Video Delivery Policies}
\label{sec:prob_formulation}
\subsection{Video delivery decisions}
The goal of this paper is to find the appropriate three decisions at each slot $t$ in the network model illustrated in Section \ref{sec:network_model}: 1) caching node for video delivery $\alpha(t)$, 2) video quality level $q(t)$, and 3) the quantity of receiving chunks $M(t)$.
However, to update the caching node association, the time-consuming process is required in which the user sends the request signal for video delivery and the caching node approves it.
Therefore, new caching node association is hardly performed as frequent as receiving chunks, and we suppose that the decision on $\alpha(t)$ is made at larger timescale than decisions on $q(t)$ and $M(t)$.
\begin{figure} [h!]
\centering
\includegraphics[width=0.75\textwidth]{decision_timescale.png}
\caption{Different timescales for decisions on $\alpha(t)$, $q(t)$ and $M(t)$}
\label{fig:decision_timescale}
\end{figure}
In this sense, the user decides $q(t)$ and $M(t)$ at time slots $t\in \{0,1,2\cdots \}$, but caching node decisions are performed at time slots $t=0,T,2T,\cdots$, where $T$ is the time interval for caching node association.
The time slot for the $k$-th caching node decision is denoted by $t_k = (k-1)T$ for $k\in \{1,2,\cdots \}$.
Different timescales of decisions on $\alpha(t)$, $q(t)$ and $M(t)$ are described in Fig.~\ref{fig:decision_timescale}.
Let the $k$-th frame for caching node decision be $\mathcal{T}_k = \{t_k,t_k+1,\cdots,t_k+T-1 \}$.
As shown in Fig.~\ref{fig:decision_timescale}, after associating with the caching node $\alpha(t_k)$ at time $t_k$, decisions on quality level $q(t)$ and chunk amounts $M(t)$ are performed over $t\in \mathcal{T}_k$ to receive the desired video from $\alpha(t_k)$.
Therefore, $q(t) \in \mathcal{L}_{l(\alpha(t_k))}$
and $M(t) N_{q(t)} \leq B(\alpha(t_k),t)$ should be satisfied for $t\in \mathcal{T}_k$, where $l(\alpha(t_k))$ is the type of the caching node $\alpha(t_k)$.
The user can make the candidate set of caching nodes denoted by $\mathcal{A}(t_k)$, and $\alpha(t_k)\in \mathcal{A}(t_k)$.
All caching nodes in $\mathcal{A}(t_k)$ should be outside the radius $R_U$ of all existing users to limit the interference power lower than $\rho$.
To avoid the situation in which no caching node can deliver the desired video, i.e., $\mathcal{A}(t_k) > 0$, the caching nodes which provide SINRs larger than $\gamma_{\text{min}}$ are assumed to be outside the radius $R_U$ of all existing users.
$\mathcal{A}(t_k)$ consists of up to $L$ caching nodes, i.e., $|\mathcal{A}(t_k)| \leq L$,
in which each caching node in $\mathcal{A}(t_k)$ belongs to different types.
If there are several nodes of type-$l$, the user takes one of them whose channel condition is the strongest.
There is no reason to choose another type-$l$ caching node while leaving the node with the strongest channel if another streaming user does not request the video from that strongest node.
In addition, $|\mathcal{A}(t_k)| < L$ means that $L-|\mathcal{A}(t_k)|$ caching node types do not exist around the user.
Suppose that the new streaming user is already generated outside the radius $R_N$ of all existing caching nodes and the INR $\Upsilon$ is observed at the new user.
Also, another user's link activation is banned around the target user due to the interference issue.
Then, we just consider the node association problem of the new streaming user with respect to the candidate set $\mathcal{A}(t_k)$ while the INR $\Upsilon$ is observed.
\subsection{Problem formulation}
For determining the appropriate video delivery policy, two performance metrics are considered: playback delay and average streaming quality.
Based on these goals, we can formulate the optimization problem which minimizes the quality degradation constrained on averting queue emptiness as follows:
\begin{align}
&\{ \boldsymbol{\alpha},\boldsymbol{q},\boldsymbol{M} \} = \underset{\alpha \in \mathcal{A},~q \in \mathcal{L}_{l(\alpha)} }{\argmin}~ \lim_{K\rightarrow \infty} \mathbb{E}\left[ \frac{1}{KT} \sum_{t=0}^{KT-1}(\bar{\mathcal{P}} - \mathcal{P}(q(t))) \cdot M(t) \right] \label{eq:opt_metric} \\
&~~~~~~~~~~~~~~\text{s.t.}~\underset{t'\rightarrow \infty}{\lim} \frac{1}{t'} \sum_{t=0}^{t'-1}\nolimits \mathbb{E}[Z(t)] < \infty \label{eq:opt_const_queue} \\
&~~~~~~~~~~~~~~~~~~~M(t)N_{q(t)} \leq B(\alpha,t) \label{eq:opt_const_chunk}
\end{align}
where $\mathcal{P}(q(t))$ is quality measure of $q(t)$ and $\bar{\mathcal{P}}$ is the maximum quality measure, i.e., equation \eqref{eq:opt_metric} is the time averaged video quality degradation.
Decision vectors are represented as $\boldsymbol{\alpha}=[\alpha(t_1), \cdots, \alpha(t_K) ]$, $\boldsymbol{q}=[q(0),q(1),\cdots, q(KT-1)]$ and $\boldsymbol{M}=[M(0),M(1),\cdots,M(KT-1)]$.
Specifically, the expectation of \eqref{eq:opt_metric} is with respect to random channel realizations and stochastic distributions of caching nodes.
As mentioned earlier, playback delay occurs when the next chunk is not arrived in the queue, therefore the constraint \eqref{eq:opt_const_queue} has a role of avoiding queue emptiness, where $Z(t) = \tilde{Q} - Q(t)$.
Here, $Z(t)$ is introduced to make $Q(t)$ large enough to avert playback delay, and $\tilde{Q}$ is a sufficiently large parameter which affects the maximal queue backlog.
From \eqref{eq:q_dynamics1}, the queue dynamics of $Z(t)$ can be represented as follows:
\begin{equation}
Z(t+1) = \min \{ Z(t) + c, \tilde{Q} \} - M(t) \text{ and }
Z(0) = \tilde{Q}.
\end{equation}
Even though the update rules of $Q(t)$ and $Z(t)$ are different, both queue dynamics mean the same video chunk processing.
Therefore, playback delay due to emptiness of $Q(t)$ can be explained by queueing delay of $Z(t)$.
By Littles' Law \cite{LittlesThm}, the expected value of $Z(t)$ is proportional to the time-averaged queueing delay.
We aim to limit the queuing delay by addressing \eqref{eq:opt_const_queue}, and it is well known that Lyapunov optimization with \eqref{eq:opt_const_queue} can make $Z(t)$ bounded \cite{Lyapunov}.
From the optimization problem \eqref{eq:opt_metric}-\eqref{eq:opt_const_chunk}, we can intuitively see how decisions are made depending on $Q(t)$.
Suppose that the queue is almost empty.
In this case, the user prefers the caching node whose channel condition is strong, pursues low-quality file, and tries to receive as many chunks as possible to stack many chunks in the queue.
However, all of those decisions could degrade the average streaming quality.
When the caching node with the strongest channel condition belongs to type-$1$, it can be better to associate with the caching node of another type in terms of average quality.
In addition, when low quality is chosen, receiving too many chunks may not be a good choice.
The user would prefer to receive the small number of chunks in current time-step and wait the better channel condition.
If the channel condition is improved at the next time-step, the user can request many chunks of high-quality video.
Thus, those decisions are strongly dependent on the queue state $Q(t)$, the caching node distribution, and channel conditions of caching node candidates.
\section{Caching Node Decision Policy}
\label{sec:caching_node_decision}
For avoiding the queue emptiness, i.e., pursuing queue stability of $Z(t)$, the optimization problem of \eqref{eq:opt_metric}-\eqref{eq:opt_const_chunk} are solved based on the Lyapunov optimization theory.
However, since the timescale of decision on $\boldsymbol{\alpha}$ is larger than that of decisions on $\boldsymbol{q}$ and $\boldsymbol{M}$, the frame-based Lyapunov optimization theory \cite{TAC2013Neely} is used for caching node decision.
Lyapunov function $L(t)$ can be defined as
$L(t) = \frac{1}{2} Z(t)^2$.
Then, let $\Delta(.)$ be a frame-based conditional Lyapunov function that can be formulated as $\Delta(t_k) = \mathbb{E}[L(t_k+T)-L(t_k)|Z(t_k)]$, i.e., the drift over the time interval $T$.
The dynamic policy is designed to solve the given optimization problem of \eqref{eq:opt_metric}-\eqref{eq:opt_const_chunk} by observing the current queue state, $Z(t_k)$, and determining the caching node to minimize a upper bound on frame-based \textit{drift-plus-penalty} \cite{Lyapunov}:
\begin{equation}
\Delta(t_k) + V \mathbb{E} \bigg[ \sum_{t=t_k}^{t_k+T-1}\nolimits (\bar{\mathcal{P}} - \mathcal{P}(q(t))) \cdot M(t) \bigg| Z(t_k) \bigg], \label{eq:drift-plus-penalty}
\end{equation}
where $V$ is an importance weight for quality improvement.
First of all, the upper bound on the drift can be found in the Lyapunov function.
\begin{align}
L(t+1) - L(t) &= \frac{1}{2} \Big\{ Z(t+1)^2 - Z(t)^2 \Big\} \nonumber \\
&= \frac{1}{2} \Big\{ \min\{ Z(t)-M(t) + c, \tilde{Q}-M(t) \}^2 - Z(t)^2 \Big\} \nonumber \\
&\leq \frac{1}{2} \Big\{ (Z(t)-M(t)+c)^2 - Z(t)^2 \Big\} \label{eq:Lypunov_drift}
\end{align}
By summing \eqref{eq:Lypunov_drift} over $t=t_k,\cdots,t_k+T-1$, the upper bound in the frame-based Lyapunov function is obtained by
\begin{equation}
L(t_k+T) - L(t_k) \leq \sum_{t=t_k}^{t_k+T-1} \bigg\{ Z(t)(c-M(t)) + \frac{1}{2}(c-M(t))^2 \bigg\}.
\end{equation}
Thus, according to \eqref{eq:drift-plus-penalty}, minimizing a bound on frame-based \textit{drift-plus-penalty} is equivalent to minimizing
\begin{align}
&\mathcal{D}(\alpha(t_k),Q(t_k), {\boldsymbol q}_k, {\boldsymbol M}_k) = \nonumber \\
&~~~~\mathbb{E}\Bigg[ \sum_{t=t_k}^{t_k+T-1}\nolimits \bigg\{ Z(t)(c-M(t)) + \frac{1}{2}(c-M(t))^2 + V (\bar{\mathcal{P}} - \mathcal{P}(q(t))) \cdot M(t) \bigg\} \bigg| Z(t_k) \Bigg],
\label{eq:exp_drift-plus-penalty}
\end{align}
where $\boldsymbol{q}_k = [q(t_k), q(t_k+1), \cdots, q(t_k+T-1)]$, $\boldsymbol{M}_k = [M(t_k), M(t_k+1), \cdots, M(t_k+T-1)]$ and recall that $Z(t) = \tilde Q - Q(t)$.
The above minimum is conditioned on $M(t)N_{q(t)}\leq B(\alpha(t_k),t)$ for all $t\in \mathcal{T}_k$.
This frame-based algorithm is shown to satisfy the queue stability constraint of \eqref{eq:opt_const_queue} while minimizing the objective function of \eqref{eq:opt_metric} in \cite{TAC2013Neely}.
For any $\alpha(t_k) \in \mathcal{A}(t_k)$, the minimum bound on frame-based drift-plus-penalty can be obtained by
\begin{equation}
\mathcal{D}(\alpha(t_k),Q(t_k)) = \min_{{\boldsymbol q}_k, {\boldsymbol M}_k} \mathcal{D}(\alpha(t_k),Q(t_k), {\boldsymbol q}_k, {\boldsymbol M}_k).
\label{eq:minimum-D-over-q-M}
\end{equation}
In Section \ref{sec:qual_chunk_decisions}, we will provide
an efficient method to find the minimum achieving ${\boldsymbol q}_k$ and ${\boldsymbol M}_k$.
System parameter $V$ in \eqref{eq:exp_drift-plus-penalty} is a weight factor for the term representing the measure of video quality degradation.
The value of $V$ is important to control the queue backlogs and quality measures at every time.
The appropriate initial value of $V$ needs to be obtained by experiment because it depends on the distribution of caching nodes, the channel environments, the playback rate $c$, and $\tilde{Q}$.
Also, $V\geq 0$ should be satisfied.
If $V<0$, the optimization goal is converted into maximizing the measure of video quality degradation.
Moreover, in the case of $V=0$, the user only aims at stacking queue backlogs without consideration of video quality.
On the other hand, when $V\rightarrow \infty$, users do not consider the queue state, and thus they just pursue to minimize the video quality degradation.
$V$ can be regarded as the parameter to control the trade-off between quality and delay, which captures the fact that the user can stack many low-quality chunks or relatively the small number of high-quality chunks in the queue, under the given channel condition.
From \eqref{eq:exp_drift-plus-penalty}, we can anticipate how the algorithm works.
When the queue is almost empty, i.e. $Z(t) \simeq \tilde{Q}$, the large arrivals $M(t)$ are necessary for the user not to wait the next chunk.
In this case, the user prefers the caching node which gives many chunks.
On the other hand, when the queue backlogs are stacked enough to avoid playback delay, i.e. $Z(t)\simeq 0$, the user would request the high quality level of $\mathcal{P}(q(t))$ without worrying about playback latency.
With the initial condition of $Q(t_k)$, the user computes $\mathcal{D}(\alpha(t_k),Q(t_k))$ for all $\alpha(t_k) \in \mathcal{A}(t_k)$.
Then, the caching node which minimizes $\mathcal{D}(\alpha(t_k),Q(t_k))$ is chosen at the user,
\begin{equation}
\alpha^*(t_k) = \underset{\alpha(t_k)\in \mathcal{A}(t_k)}{\argmin} \mathcal{D}(\alpha(t_k),Q(t_k)).
\label{eq:caching_node_decision}
\end{equation}
However, the user should estimate the average function value of future queue states $Z(t)$ and decisions of $q(t)$ and $M(t)$ for $t\in \mathcal{T}_k$.
For finding \eqref{eq:minimum-D-over-q-M}, the frame-based algorithm is formulated based on Markov decision process \cite{TAC2013Neely}, and it can be solved by dynamic programming as following section.
\section{Decisions on Quality Level and Receiving Chunk Amounts}
\label{sec:qual_chunk_decisions}
The goal of this section is to compute $\mathcal{D}(\alpha(t_k),Q(t_k))$, given the associated caching node $\alpha(t_k)$ and initial queue backlogs $Q(t_k)$.
\subsection{Stochastic shortest path problem}
According to \eqref{eq:exp_drift-plus-penalty}, we can formulate the drift-plus-penalty algorithm of the $k$-th frame as follows:
\begin{align}
\{ \boldsymbol{q}_k,\boldsymbol{M}_k \} &= \underset{q,M}{\argmin}~ \mathcal{D}(\alpha(t_k),Q(t_k), {\boldsymbol q}, {\boldsymbol M}) \label{eq:opt2_metric} \\
&~~\text{s.t.}~M(t)N_{q(t)} \leq B_k(t) \label{eq:opt2_const1} \\
&~~~~~~~q(t) \in \mathcal{L}_{l(\alpha(t_k))}, \label{eq:opt2_const2}
\end{align}
where $B_k(t)\triangleq B(\alpha(t_k), t)$.
The problem of \eqref{eq:opt2_metric}-\eqref{eq:opt2_const2} is similar to the stochastic shortest path problem based on Markov decision process.
In the network model, $B_k(t)$ and $Z(t)$ (i.e., $Q(t)$) are given before making decisions of $q_k(t)$ and $M_k(t)$ at every time $t$.
The queue backlog $Z(t)$ represents the current state which satisfies the Markov property.
Define $\mathcal{Z}=\{0,1,\cdots,\tilde{Q} \}$ as the state space of the user queue.
It is reasonable to set $\tilde{Q}$ be the arbitrarily predefined maximum queue backlog
because the queue size is finite in practical system.
The action set is defined as $\Theta(t) = \{M(t),q(t) \}$.
Then, incurred cost at $t \in \mathcal{T}_k$ can be formulated by
\begin{align}
g_k(Z(t),\Theta(t)) &= Z(t) (c-M(t)) + \frac{1}{2} (c-M(t))^2 + V(\bar{\mathcal{P}} - \mathcal{P}(q(t)))M(t).
\end{align}
The transition probabilities from $Z(t)$ to $Z(t+1)$ can be defined for all states $i$ and $j$ as
\begin{equation}
P_{ij}(\Theta, b) = {P}\{ Z(t+1)=j | Z(t) = i, \Theta(t) = \Theta, B_k(t) = b \}.
\end{equation}
Since the next state $Z(t+1)$ is deterministic given $Z(t)$ and action $M(t)$, it can be seen that $P_{ij}(\Theta,b) \in \{0,1\}$.
\subsection{Probability mass function of $B_k(t)$}
The constraint \eqref{eq:opt2_const1} indicates that the maximum number of chunks which the user can receive depends on the random network event $B_k(t)$ and the decision of quality $q(t)$.
It notifies that decisions on $q(t)$ and $M(t)$ should jointly made as well as the probability distribution of the random network event $B_k(t)$ is required.
Define a random variable $Y=\log_2(1+aX)$, where $X$ is a chi-square random variable and $a$ is a constant.
Then, we can obtain $P\{Y \geq y\}$, as given by
\begin{equation}
P\{Y \geq y\} = \exp\Big\{ -\frac{1}{2a}(2^y-1) \Big\}.
\end{equation}
Since $|h(\alpha,t)|^2$ follows the chi-squared distribution and a random variable $B_k(t)$ is a nonnegative integer, the probability mass function of $B_k(t)$ is found as follows:
\begin{align}
P\{B_k(t)=b\} &=P\{t_c R(\alpha(t_k), t) \geq b \} - P\{t_c R(\alpha(t_k), t) \geq b+1\} \\
&=e^{1/2\Gamma} D(\alpha, t) \Bigg\{ \exp \bigg\{ -\frac{2^{b/t_c\mathcal{W}}}{2\Gamma D(\alpha, t)} \bigg\} - \exp \bigg\{ -\frac{2^{(b+1)/t_c\mathcal{W}}}{2\Gamma D(\alpha, t)} \bigg\} \Bigg\}.
\end{align}
\subsection{Dynamic programming}
\label{subsec:dynamic_program}
Given $\alpha(t_k)$ and $Z(t_k)$, the user observes the queue state $Z(\tau)$ and the random network event $B_k(\tau)$, and decides the action $\Theta(\tau)$ for each time slot $\tau \in \mathcal{T}_k$.
Then, the minimum incurred cost based on measurements of $B_k(\tau)$ and $Z(\tau)$ is
\begin{equation}
G_k(\tau,z_0,b_0) = \underset{\Theta}{\min}~\mathbb{E} \left[ \sum_{t=\tau}^{t_k+T-1} g_k(Z(t),\Theta(t)) \Big| Z(\tau)=z_0, B_k(\tau)=b_0 \right],
\end{equation}
conditioned on $M(\tau) N_{q(\tau)} \leq B_k(\tau)$.
Let $J_k(\tau,z_0)$ be the marginalized function of $G_k(\tau,z_0,b_0)$ over all possible $B_k(\tau)=b_0$, and it can be approximated into
\begin{equation}
J_k(\tau,z_0) = \sum_{b_0=0}^{B_{\text{max}}} P\{ B_k(\tau)=b_0 \} G_k(\tau,z_0,b_0),
\end{equation}
where $B_{\text{max}}$ is a nonnegative integer such that $P\{B \geq B_{\text{max}} \} \approx 0$.
The dynamic programming provides the action that minimizes the following cost as given by \cite{DynamicProgram}
\begin{align}
G_k(\tau,z_0,b_0) &= \underset{ \Theta }{\min}~\mathbb{E} \Big[ g_{k}(Z(\tau)=z_0,\Theta(\tau)) + \sum_{y\in \mathcal{Z}} P_{z_0,y}(\Theta(\tau),b_0) \cdot G_k(\tau+1, y,B_k(\tau+1)) \Big] \label{eq:DP1} \\
&= \underset{\Theta}{\min} \Big[ g_{k}(Z(\tau)=z_0,\Theta(\tau)) + J_k(\tau+1,Z(\tau+1)) \Big], \label{eq:DP2}
\end{align}
where the expectation of \eqref{eq:DP1} is with respect to $\{ B_k(t):~\tau+1 \leq t \leq T-1 \}$ and $Z(\tau+1) = \min \{z_0 + c,\tilde{Q} \} - M(\tau)$.
The minimum cost is obtained over all $\Theta(\tau)$ such that $M(\tau) N_{q(\tau)} \leq B_k(\tau)=b_0$.
Given $B_k(t)=b_0$, the user can find the minimum value of \eqref{eq:DP2} by greedily testing all joint combinations of decisions on $q(t)$ and $M(t)$.
For example, let there exists $L=2$ quality levels and $q\in \{1,2\}$
correspond to the file size of $N\in \{10,20\}$
If $B_k(t) \in [20$:$30)$ Kbits, where $[a$:$b) \triangleq \{a, a+1, \cdots, b-1\}$ for simplicity, then there are four possible decisions: 1) $M(t)=0$, 2) $q(t)=1,M(t)=1$, 3) $q(t)=1,M(t)=2$, and 4) $q(t)=2,M(t)=1$.
The user computes costs for all those possible decision cases and picks the minimum one as an optimal cost.
We set the end time slot of the $k$-th frame as $t_{k+1}=t_k+T$, which is the start time of the $k+1$-th frame.
To find the optimal costs $\mathbf{J}_k(t) = [J_k(t,0),\cdots,J_k(t,\tilde{Q})]$ for $t\in \mathcal{T}_k$ by using dynamic programming equation \eqref{eq:DP2}, the end costs of $\mathbf{J}_k(t_{k+1})$ are required.
Since the playback delay occurs at the end state when the accumulated chunk amounts are smaller than the departure quantity, i.e., $Q(t_{k+1})<c$ and $Z(t_{k+1}) > \tilde{Q}-c$.
Therefore, the end costs for those states, i.e., $J_k(t_{k+1},z)$ for $z\in\{ \tilde{Q}-c+1, \cdots, \tilde{Q} \}$ should be very large.
Even when $Q(t_{k+1}) \geq c$, the more chunks are accumulated, the more likely there will be no playback delays.
In this sense, $J_k(t_{k+1},i) \geq J_k(t_{k+1},j)$ for $i \geq j$ is preferred for all $i,j \in \{0,\cdots,\tilde{Q}-c \}$.
Especially, as a large number of chunks are received in the queue, the effect of additional chunks to avert queue emptiness would be significantly decreased.
Therefore, $J_k(t_{k+1},i)$ for $i=\{1,\cdots \tilde{Q} \}$ are arbitrarily modeled as the truncated form of exponential distribution.
Thus, we can set the end costs for all states as follows:
\begin{align}
&J_k(t_{k+1},z) = A,~\forall z \in \{\tilde{Q}-c+1, \cdots, \tilde{Q} \} \\
&J_k(t_{k+1},i) = 10^{-3} \cdot A \mu e^{-\mu \cdot (\tilde{Q}-i)},~\forall i \in \{0,\cdots,\tilde{Q}-c \},
\end{align}
where $A$ is a predefined large constant to give penalties for playback delay occurrences and $\mu$ is the exponential distribution coefficient.
Given the end costs $\mathbf{J}_k(t_{k+1})$, the optimal costs $\mathbf{J}_k(t)$ for all $t \in \mathcal{T}_k$ can be obtained by backtracking the shortest path based on the dynamic programming equation \eqref{eq:DP2}.
Then, when the queue backlog at time $t=t_k$ is $Z(t_k)$, $J_k(t_k,Z(t_k))$ becomes the averaged drift-plus-penalty term \eqref{eq:exp_drift-plus-penalty}, i.e., $\mathcal{D}_k$.
Then, after the user finds all the averaged drift-plus-penalty terms for all $\alpha \in \mathcal{A}(t)$ by using dynamic programming, the user determines the caching node to receive the desired video file by comparing all drift-plus-penalty terms, as described in \eqref{eq:caching_node_decision}.
\subsection{Decisions of quality and chunk amounts}
\label{subsec:quality_chunk_decisions}
After determining the caching node $\alpha(t_k)$, the user should choose the video quality and the number of chunks to receive for every time slot $t\in \mathcal{T}_k$, depending on time-varying channel conditions and its queue state.
For this goal, we can simply use the principle of optimality in the dynamic programming algorithm \cite{DynamicProgram}, which argues that if the optimal policy $\Theta^* = \{\Theta^*(t_k), \cdots, \Theta^*(t_k+T-1) \}$
is a solution of the stochastic shortest path problem, then the truncated policy $\{\Theta^*(t_k+j), \cdots, \Theta^*(t_k+T-1) \}$ is optimal for the subproblem over $t\in \{t_k+j,\cdots, t_k+T-1 \}$, where $0\leq j \leq T-1$.
Based on this principle of optimality, the user can make the optimal decisions of $q^*(t)$ and $M^*(t)$ for $t\in \mathcal
{T}_k$ by using the minimum costs obtained while performing dynamic programming for caching node decision $\alpha^*(t_k)$.
When deciding $q^*(t)$ and $M^*(t)$, the channel gain can be observed, e.g. $B(\alpha(t_k),t) = b_0$,
so the optimal action $\Theta^*(t)$ is deterministic given $Z(t)=z_0$ and $B_k(t) = b_0$ which provides the minimum cost $G_k(t,z_0,b_0)$, as given by
\begin{equation}
\Theta_k^*(t,z_0,b_0) = \underset{ \Theta }{\argmin}~ \Big[ g_{k}(Z(t)=z_0,\Theta) + J_k(t+1,Z(t+1)) \Big],
\end{equation}
conditioned on $M(t)N_{q(t)} \leq b_0$ for $t \in \mathcal{T}_k$.
Thus, the user should store the optimal actions $\Theta_k^*(t,z,b)$ for all $t\in \mathcal{T}_k$, $z\in \mathcal{Z}$ and $b \in \mathcal{B} = \{0,1,\cdots,B_{\text{max}} \}$ to deal with all possible random network events.
Simply, $T\cdot \tilde{Q} \cdot (B_{\text{max}}+1)$ actions are required, but some of channel realizations can give the same optimal action.
Again, consider the example of $L=2$ quality levels and $q\in \{1,2\}$ corresponding to the file size of $N\in \{10,20\}$ in Kbits.
Then, any $B=b \in [20$:$30)$ Kbits allows four combinations of decisions of $q(t)$ and $M(t)$, as explained in Section \ref{subsec:dynamic_program}, and the user is enough to store the only one optimal action for all $B=b \in [20$:$30)$ Kbits.
In this sense, define $\mathcal{N}_B$ subsets of $\mathcal{B}$ denoted by $\mathcal{B}_n$ for $n \in \{1,\cdots, \mathcal{N}_B \}$, as follows:
\begin{align}
&\bigcup_{n=1}^{\mathcal{N}_B} \mathcal{B}_n = \mathcal{B} \\
&\mathcal{B}_n \cap \mathcal{B}_m = \phi,~\forall n\neq m,~n,m\in \{ 1, \cdots, \mathcal{N}_B \} \\
&\Theta^*_k(t,z,b_1) = \Theta^*_k(t,z,b_2),~\forall b_1, b_2 \in \mathcal{B}_n.
\end{align}
Thus, the user needs to store $T \cdot \tilde{Q} \cdot \mathcal{N}_B$ actions.
The whole steps for video delivery decisions on caching node, video quality, and receiving chunk amounts are presented in Algorithm \ref{alg:video_delivery}.
\begin{algorithm}[t!]
\caption{Dynamic video delivery decisions on $\boldsymbol{\alpha}$, $\boldsymbol{q}$, and $\boldsymbol{M}$ in different timescales
\label{alg:video_delivery}}
\begin{algorithmic}[1]
\Require{\\
\begin{itemize}
\item $V$: parameter for streaming quality-delay trade-offs
\item $\tilde{Q}$: threshold for queue backlog size
\item $K$: the number of caching node decisions
\item $T$: the time interval of updating caching node decision
\end{itemize}
}
\State{$t=0$ // $KT-1$: number of discrete-time operations}
\While{$k \leq K$}{
\State{$t_k = (k-1)T$: time for the $k$-th caching node decision}
\State{Observe $Z(t_k)$ and find $\mathcal{A}(t_k)$}
\State{Compute $\mathcal{D}_k(\alpha(t_k),Q(t_k))$ by using dynamic programming equation \eqref{eq:DP2} and store $\Theta_k^*(t,z,b)$ for every $\alpha(t_k) \in \mathcal{A}(t_k)$, $z\in \mathcal{Z}$ and $b\in \mathcal{B}$}.
\State{Make a decision of $\alpha^*(t_k)$ by using \eqref{eq:caching_node_decision}}
\For{$t=t_k:t_k+T-1$}{
\State{Observe $Z(t)$ and $B_k(t)$}
\State{Make a decision of $\Theta_k^*(t,Z(t),B_k(t))$}
\EndFor}
}
\EndWhile
\end{algorithmic}
\end{algorithm}
\subsection{Computational complexity of dynamic programming}
To determine the optimal policy at each time slot, it seems that at least $\tilde{Q} B_{\text{max}}$ computations are required, but some of channel realizations can perform the same computation as seen in Section \ref{subsec:quality_chunk_decisions}.
Since all realizations $b\in \mathcal{B}_n$ not only give the same $\Theta_k^*(t,Z(t),b)$ but also make the same computations of $g_{k}(Z(t)=z_0,\Theta(t)) + J_k(t+1,Z(t+1))$ for all possible combinations of $q(t)$ and $M(t)$, $\tilde{Q} \mathcal{N}_B$ computations are required at least.
However, in most of random network events, more computations are required to take the minimum function in \eqref{eq:DP2}.
As shown in the example of $L=2$ quality levels and
$q\in \{1,2\}$ corresponding to the file size of
$N\in\{10,20\}$ in Kbits, there are four combinations of decisions of $q(t)$ and $M(t)$ when
$B=b \in [20$:$30)$ Kbits.
Let the average number of these decision combinations of $q(t)$ and $M(t)$ for all $B_k(t) \in \mathcal{B}$ be $\mathcal{N}_{\theta}$, then total $\tilde{Q} \mathcal{N}_B \mathcal{N}_{\theta}$ computations are required at each time slot in dynamic programming.
Here, $\mathcal{N}_B$ and $\mathcal{N}_{\theta}$ obviously depend on $B_{\text{max}}$, $L$ and $N_q$ for $q \in \mathcal{L}_L$.
There are not many versions of the identical video of different quality levels, i.e. $L$ is small in general, and $N_q$ is not controllable unless the video encoding scheme is changed.
On the other hand, $B_{\text{max}}$ increases as transmit SNR grows, therefore large SNR could result in huge computational complexity as well as large number of registers to store the optimal costs for decisions of quality and chunk amounts.
However, the streaming user can receive a large number of high-quality chunks enough to avoid queue emptiness in the sufficiently large transmit SNR region.
Considering that the proposed video delivery scheme targets the streaming user who is worrying about playback delays as well as video quality degradation, however, huge complexity burden for large transmit SNR is out of scope in targeting scenarios.
Thus, $\mathcal{N}_B$ and $\mathcal{N}_{\theta}$ are expected not much large in our targeting scenarios where adjustments of the tradeoff between playback delay and video quality are necessary, so computational complexity for dynamic programming can be somewhat limited.
\begin{table}[t!]
\small
\caption{System Parameters}
\label{table:parameters}
\begin{center}
\scalebox{1}{
\begin{tabular}{l|c}
\toprule
Description & Value \\
\midrule [1.0pt]
No. of quality levels ($L$) & 3 \\
Default PPP intensity ($\lambda$) & 0.4 \\
Time interval of caching node decisions ($T$) & 5 \\
User radius ($R$) & 50 m \\
Caching probabilities ($\mathbf{p}=[p_1,\cdots,p_L]$) & $[\frac{4}{7}, \frac{2}{7}, \frac{1}{7}]$ \\
Transmit SNR ($\Psi$) & 20 dB \\
INR ($\Upsilon$) & 5dB \\
Minimum probability of finding the caching node ($\eta_{\text{min}}$) & 0.99 \\
Queue departure ($c$) & 1 \\
Bandwidth ($\mathcal{W}$) & 1 MHz \\
Coherence time ($t_c$) & 5 ms \\
End cost coefficient ($A$) & $10^4$ \\
End cost coefficient ($\mu$) & 1 \\
$\tilde{Q}$ & 100 \\
$B_{\text{max}}$ & 52 kbits \\
$V$ & 0.015 \\
\bottomrule
\end{tabular}
}
\end{center}
\end{table}
\section{Simulation Results}
\label{sec:simulation}
In this section, we show that the proposed algorithm for dynamic video delivery policy works well with video files of different quality levels in wireless caching network.
Simulation parameters are listed in Table \ref{table:parameters}, and these are used unless otherwise noted.
The proposed technique can be applied to any distribution model for caching nodes, but we suppose that caching nodes which store the desired content are modeled as an independent PPP with an intensity of $\lambda$, which is generally assumed for researches of wireless caching networks \cite{caching:ICC2015Blaszczyszyn, CL2017Chen, caching:TWC2016Chae}.
Then, the PPP intensity of type-$l$ caching nodes becomes $\lambda p_l$, where $p_l$ denote the caching probability of the video which can be encoded into any quality in $\mathcal{L}_l$.
Therefore, larger $p_l$, more caching nodes of type-$l$ around the streaming user.
Based on the network model described in Fig. \ref{fig:network_model}, the user is slowly moving towards certain direction.
In practice, the channel condition between the user and the caching node delivering the desired video could be varying due to Doppler shift as the user is moving, but this effect is not captured in this paper.
Peak-signal-to-noise ratio (PSNR) is considered as a video quality measure, and quality measures and file sizes depending on quality levels are supposed as $\mathcal{P}(q) = [34,~36.64,~39.11]$ dB and $N(q) = [2621,~5073,~10658]$ Kbits which are obtained from real-world video traces~\cite{ICTC2015Kim}.
Since we assume that $\eta_{\text{min}} = 0.99$ and $t_0 \mathcal{W} \log_2(1+\gamma_{\text{min}}) = N(1)$, the minimum intensity of PPP distributions to satisfy the performance criterion of $\{\gamma_{\text{min}}, \eta_{\text{min}} \}$ should be $\lambda_{\text{min}} = 0.1113$.
To verify the advantages of the proposed algorithm, this paper compares the proposed one with three other schemes:
\begin{itemize}
\item `Strongest': The user receives the desired video file from the caching node whose channel condition is the strongest among $\mathcal{A}(t_k)$ at time slots of $t_k$, for $k\in\{1,2,\cdots \}$.
Decisions of $q(t)$ and $M(t)$ are made based on dynamic programming results.
\item `Highest-Quality': The user receives the desired video file from the caching node which can provide the highest-quality file among $\mathcal{A}(t_k)$ at time slots of $t_k$, for $k\in \{1,2,\cdots \}$.
Decisions of $q(t)$ and $M(t)$ are made based on dynamic programming results.
\item `One-Step': The user decides the caching node for video delivery based on the frame-based Lyapunov optimization theory.
However, decisions of $q(t)$ and $M(t)$ are made by minimizing the incurred cost only at each slot $t$ without using dynamic programming results.
\end{itemize}
In summary, performance comparisons with `Strongest' and `Highest-Quality' can show the effects of caching node decision based on Lyapunov optimization, and comparison with `One-Step' can specify the advantage of using Markov decision process and dynamic programming for decisions on video quality and the amounts of receiving chunks.
\begin{figure}[t]
\minipage{0.47\textwidth}
\includegraphics[width=\linewidth]{final_delay_lambda.pdf}
\caption{Delay occurrence rates over $\lambda$}
\label{fig:delay_lambda}
\endminipage\hfill
\minipage{0.47 \textwidth}
\includegraphics[width=\linewidth]{final_quality_lambda.pdf}
\caption{Video quality measures over $\lambda$}
\label{fig:quality_lambda}
\endminipage
\end{figure}
\subsection{Caching node distribution}
\label{subsec:simul-node}
At first, impacts of the PPP intensity, i.e. how many caching nodes are distributed around the streaming user, are shown in Figs. \ref{fig:delay_lambda} and \ref{fig:quality_lambda}, which give the plots of playback delay occurrence rates and average video quality measures per received chunk versus $\lambda$, respectively.
`Strongest' is likely to receive many chunks from the caching node whose channel condition is the strongest, so this scheme accumulates queue backlogs enough to avoid playback delays.
Therefore, `Strongest' shows the best delay performance but its gain over the proposed one is very small, as shown in enlarged plots in Fig. \ref{fig:delay_lambda}.
There are two reasons.
The first one is that even though the channel condition of certain caching node at $t_k$ is the strongest, after that it could not be the strongest due to time-varying channels and user mobility.
Second, the delay performance does not increase in proportional to the number of chunks accumulated in the queue.
If enough chunks are already in the queue to prevent playback delays, then the delay performance is not dramatically improved as additional chunks arrive.
Similar delay occurrence rates of the proposed technique and `Strongest' in Fig. \ref{fig:delay_lambda} show that the proposed scheme can accumulate chunks in the queue enough to avert playback delays.
On the other hand, since `Highest-Quality' pursues the video quality when choosing the caching node for video delivery, it gives better quality performance than `Strongest' with large $\lambda$.
However, when $\lambda$ is small, the caching node chosen by `Highest-Quality' is likely to be much distanced from the streaming user and its channel condition would be usually too bad to deliver the high-quality video.
Therefore, even though the caching node chosen by `Highest-Quality' could provide the high quality level, the user requests large number of low-quality chunks owing to less accumulated backlogs.
Since we assume that the user cannot achieve any quality-of-service when delay occurs, the quality performance of `Highest-Quality' is even worse than that of `Strongest' with small $\lambda$.
As the proposed technique determines to associate with the caching node by balancing the video quality and channel condition, the proposed one can provide better quality than both `Strongest' and `Highest-Quality', as shown in Fig. \ref{fig:quality_lambda}.
`One-Step' gives
the highest average quality measure per received chunk but it suffers from much more frequent delay occurrences compared to other schemes.
Considering that streaming users are much more sensitive to playback delays, `One-Step' is not appropriate for practical systems.
From the result of `One-Step', we can see that the merit of using dynamic programming which stochastically reflects future subsequent decisions is very large when determining the video quality and the number of receiving chunks.
In addition, even when $\lambda = 0.6$, PPP intensity of highest-quality videos ($q=3$) becomes $\lambda p_3 = 0.0857 < \lambda_{\text{min}}=0.1113$.
Therefore, the highest-quality level is rarely selected and the average quality measures of all schemes are much lower than the highest-quality measure ($\mathcal{P}(q=3)=39.11\text{dB}$).
\begin{figure}[t]
\minipage{0.47\textwidth}
\includegraphics[width=\linewidth]{final_delay_probList.pdf} \caption{Delay occurrence rates over caching policies}
\label{fig:delay_prob}
\endminipage\hfill
\minipage{0.47\textwidth}
\includegraphics[width=\linewidth]{final_quality_probList.pdf}
\caption{Video quality measures over caching policies}
\label{fig:quality_prob}
\endminipage
\end{figure}
\subsection{Uniform and nonuniform caching probabilities}
\label{subsec:simul-cachingProb}
We set three cases of caching probabilities for the video file with different quality levels, as follows:
\begin{itemize}
\item Case 1: $p_1=4/7,~p_2=2/7,~p_3=1/7$
\item Case 2: $p_1=1/3,~p_2=1/3,~p_3=1/3$
\item Case 3: $p_1=1/7,~p_2=2/7,~p_3=4/7$
\end{itemize}
Note that Case 2 corresponds to the uniform caching probability case and Case 1 and Case 3 are nonuniform.
In Case 3, the streaming user is more likely to receive high-quality video than other cases, on the other hand, Case 1 represents an environment where there are not many caching nodes which can provide the high-quality video around the user.
The performances of playback delay and quality measure depending on those cases of caching probabilities are shown in Figs. \ref{fig:delay_prob} and \ref{fig:quality_prob}, respectively.
In Fig. \ref{fig:delay_prob}, delay incidence of `Highest-Quality' definitely decreases as $p_1$ decreases and $p_3$ grows, because the caching nodes storing the high-quality video are likely to be near to the streaming user.
However, since the distribution density of all caching nodes does not change according to the probabilistic caching policy which satisfies $p_1+p_2+p_3=1$, the delay performance of `Strongest' is not influenced much by different caching policies.
For the `Strongest' scheme, any caching probability case can deliver as many low-quality chunks of small size as possible when there are too few chunks in queue so the playback delay is about to occur.
In this sense, the proposed technique shows almost the same delay performance as `Strongest', because the proposed one strongly limits the playback delay compared to quality improvement.
The average quality measures of all schemes increase as $p_1$ decreases and $p_3$ grows as shown in Fig. \ref{fig:quality_prob}.
Even though `Highest-Quality' pursues the video quality, its average quality measure per received chunk does not differ much from that of `Strongest' for any caching probability case owing to its poor delay performance.
As we have seen in Section \ref{subsec:simul-node}, queue backlogs do not accumulate much in the `Highest-Quality' scheme, therefore the user usually requests the small number of low-quality chunks.
Especially in Case 3, caching nodes storing the highest-quality video are distributed more than nodes of other types, therefore the caching node whose channel condition is the strongest among candidate nodes would be highly probable to be type 3.
Thus, the difference between quality performances of `Strongest' and `Highest-Quality' is not large.
The performance rankings in Figs. \ref{fig:delay_prob} and \ref{fig:quality_prob} among comparison techniques are consistent with the results of Figs. \ref{fig:delay_lambda} and \ref{fig:quality_lambda}.
Compared to those comparison schemes, the proposed technique provides quite high average video quality, while limiting delay occurrence rate as low as `Strongest'.
Thus, the proposed scheme can be said to smooth out the tradeoff between quality and playback delay and to achieve both goals.
As observed here, `One-Step' provides higher quality than the proposed one but its delay performance is too poor to achieve user satisfaction.
\begin{figure}[t]
\minipage{0.47\textwidth}
\includegraphics[width=\linewidth]{final_delay_V.pdf}
\caption{Delay occurrence rates over $V$}
\label{fig:delay_V}
\endminipage\hfill
\minipage{0.47\textwidth}
\includegraphics[width=\linewidth]{final_quality_V.pdf}
\caption{Video quality measures over $V$}
\label{fig:quality_V}
\endminipage
\end{figure}
\subsection{System parameter $V$}
\label{subsec:simul-V}
Since $V$ has a role to weigh quality maximization compared to averting playback delay in Lyapunov optimization problem, delay occurrence rates increase and the expected quality measures of all techniques become improved, as $V$ grows, as shown in Figs. \ref{fig:delay_V} and \ref{fig:quality_V}, respectively.
Therefore, we can control the tradeoff between video quality and playback latency by adjusting the system parameter $V$.
Among comparison techniques, the proposed scheme improves the quality performance sufficiently while minimizing the increase in delay incidence by taking large $V$.
Quality improvements of other comparison techniques due to large $V$ are comparable to that of the proposed one, but delay performances of `Highest-Quality' and `One-Step' are still much worse than that of the proposed one and `Strongest'.
As we've seen in Sections \ref{subsec:simul-node} and \ref{subsec:simul-cachingProb}, the proposed technique provides higher average video quality than `Strongest' and delay performance almost same as `Strongest'.
We can also see that `One-Step' does not respond sensitively to changes in $V$ compared to other techniques, because the role of $V$ is not completely captured in this scheme.
To reflect the effect of $V$ properly, minimization of the frame-based drift-plus-penalty term is necessary, but decisions of `One-Step' on quality and chunk amounts are not frame-based.
Those decisions are just conducted and dependent on only each time slot.
\begin{figure}[t]
\minipage{0.47\textwidth}
\includegraphics[width=\linewidth]{final_delay_INR.pdf}
\caption{Delay occurrence rates over $\Upsilon$}
\label{fig:delay_INR}
\endminipage\hfill
\minipage{0.47\textwidth}
\includegraphics[width=\linewidth]{final_quality_INR.pdf}
\caption{Video quality measures over $\Upsilon$}
\label{fig:quality_INR}
\endminipage
\end{figure}
\subsection{SINR level}
\label{subsec:simul-SINR}
The delay and quality performances over INR levels are shown in Figs. \ref{fig:delay_INR} and \ref{fig:quality_INR}, respectively.
It is easily expected that quality performances decrease and delay occurrence rates increase as INR grows for all comparison techniques.
Almost all of the performance rankings among comparison techniques remain as seen former subsections, but the performance of `Highest-Quality' is influenced by INR levels much more than the proposed one and `Strongest'.
We can expect that `Highest-Quality' becomes more difficult to accumulate video chunks in the queue as the INR grows, therefore the quality level chosen by the user becomes increasingly degraded.
Rather, `Strongest' is not significantly affected by INR changes compared to `Highest-Quality', because the channel condition of its caching node is much stronger than that of the node chosen by `Highest-Quality'.
The proposed scheme still achieves the improved video quality while guaranteeing very low delay occurrence rate.
\section{Conclusion}
\label{sec:conclusion}
This paper studies the dynamic delivery policy of video files of various quality levels in the wireless caching network.
When the caching node distribution around the streaming user is varying, e.g. the user is moving, the streaming user makes decisions on caching node to receive the desired file, video quality, and the number of receiving chunks.
The different timescales are considered for the caching node association and decisions on quality and the number of receiving chunks.
The optimization framework of those video delivery decisions conducted on different timescales is constructed based on Lyapunov optimization theory and Markov decision process.
By using dynamic programming and the frame-based drift-plus-penalty algorithm, the dynamic video delivery policy is proposed to maximize average streaming quality while limiting playback delay quite low.
Further, the proposed technique can adjust the tradeoff between performances of video quality and playback delay by controlling the system parameter of $V$.
\section*{Acknowledgment}
This work has supported by Institute for Information \& communications Technology Planning \& Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2018-0-00170, Virtual Presence in Moving Objects through 5G).
|
{
"timestamp": "2019-03-01T02:11:08",
"yymm": "1902",
"arxiv_id": "1902.10932",
"language": "en",
"url": "https://arxiv.org/abs/1902.10932"
}
|
\section{Introduction}
Deep neural networks (DNNs) have dominated the field of computer vision because of superior performance in all kinds of tasks.
It is a tendency that the network architecture is becoming deeper and more complex~\cite{googlenetv1, vggnet, he2016deep, xie2017aggregated, hu2018squeeze} to yield higher accuracy.
However,
the great computing expense of deeper networks contradicts the demands of many resource-constrained applications,
which prefer lightweight networks~\cite{howard2017mobilenets, sandler2018mobilenetv2, zhang2018shufflenet, ma2018shufflenet} to meet limited computation or storage requirement.
\begin{figure}
\begin{center}
\begin{subfigure}[b]{0.45\linewidth}
\centering
\includegraphics[height=2.0in]{figures/skip.pdf}
\caption{Layer-wise}
\label{fig:skipnet}
\end{subfigure}
\hspace{4pt}
\begin{subfigure}[b]{0.45\linewidth}
\centering
\includegraphics[height=2.0in]{figures/skip_us_4.pdf}
\caption{DMNN}
\label{fig:skipnet_us_4}
\end{subfigure}
\end{center}
\vspace{-5pt}
\caption{DMNN provides more flexible and diverse inference paths.}
\vspace{-5pt}
\label{fig:DMNN}
\end{figure}
An elegant solution is to make use of dynamic inference mechanism~\cite{wang2017skipnet, veit2018convolutional, wu2018blockdrop, huang2017multi, dong2017more, figurnov2017spatially, figurnov2016perforatedcnns, teerapittayanon2016branchynet},
reconfiguring the inference path according to the input sample adaptively to meet a better accuracy-efficiency trade-off.
Prevalent dynamic inference techniques are mostly layer-wise methods~\cite{wang2017skipnet, veit2018convolutional, wu2018blockdrop, figurnov2017spatially, teerapittayanon2016branchynet},
as shown in Fig.~\ref{fig:skipnet}.
These methods are usually adopted to determine the execution status of a whole layer at runtime based on a specified mechanism.
All these existing dynamic inference methods only alter the depth of the network. The drawbacks are obvious.
First,
it is impractical to drop the whole layer/block since some channels of a skipped layer may be useful.
Second,
the redundant information between different channels may still exist in the remaining layers.
A recent study~\cite{zeiler2014visualizing} visualizes the hidden features of CNN models and
shows the performance contribution from different channels and different layers.
There exists different emphasis on extracting feature among different channels and layers.
In this work,
we attempt to improve the conventional dynamic inference scheme in terms of both network width and depth
and find an effective forward mechanism for different inputs at runtime from a new perspective of block design.
We propose Dynamic Multi-path Neural Network (DMNN),
a novel dynamic inference method that provides various inference path selections.
Fig.~\ref{fig:skipnet_us_4} gives an overview of our approach.
Different from conventional methods,
it is expected that each channel has its gate to predict whether to execute or not.
The primary technical challenge of DMNN is how to design an efficient and effective controller.
\textbf{Challenge of efficiency.}
Since DMNN is aimed to conduct channel-wise dynamic evaluation,
it is ideal for controlling the execution of each channel of the
network at runtime.
However, this would lead to a significant increase in computational complexity.
Moreover,
as controllers are used at each layer/block of the network,
they are desirable for lightweight design and generate only a small amount of computational cost.
\textbf{Challenge of effectiveness.}
The gate control mechanism is similar to SENet~\cite{hu2018squeeze},
which adaptively recalibrates channel-wise feature responses by explicitly modeling interdependencies between channels.
However,
SENet makes use of soft-weighted sum,
while DMNN adopts the hard-max mechanism for faster inference while maintaining or boosting accuracy.
In order to obtain a more reasonable inference path,
it would be better if we take both previous state information and object category into consideration.
Besides,
the resource-constrained loss is also required to make the computational complexity controllable.
To tackle the challenges,
considering that different channels have different representation characteristics, we split the original block of the network
into several sub-blocks. Thus the proposed method provides more optional inference paths.
A gate controller is introduced to decide whether to execute or skip one sub-block for the current input,
which only generates minor additional computational cost during inference.
Each block has its controller to control the status of every sub-blocks.
We also carefully design the gate controller to take both previous state information and object category into consideration.
Moreover,
we introduce resource-constrained loss which integrates FLOPs constraint into the optimization process to make the computational complexity controllable.
The proposed DMNN is easy to implement and can be incorporated into most modern network architectures.
The contributions are summarized as follows:
\begin{itemize}
\vspace{-3pt}
\item We propose a novel dynamic inference method called \textit{Dynamic Multi-path Neural Network}, which can provide more path selection choices in terms of network width and depth during inference.
\vspace{-3pt}
\item We carefully design a gate module controller, which takes into account both previous state and object category information. The resource-constrained loss is also introduced to control the computational complexity of the target network.
\vspace{-3pt}
\item Experimental results demonstrate the superiority of our method on both efficiency and overall classification accuracy. To be specific, DMNN-101 significantly outperforms ResNet-101 with an encouraging 45.1\% FLOPs reduction, and DMNN-50 performs comparable results to ResNet-101 with 42.1\% fewer parameters.
\end{itemize}
\section{Related Work}
\textbf{Adaptive Computation.}
Adaptive computation aims to reduce overall inference time by changing network topology based on different samples while maintaining or even boosting accuracy. This idea has been adopted in early cascade detectors~\cite{felzenszwalb2010cascade, viola2004robust}, relying on extra prediction modules or handcrafted control strategies. Learning based layer-wise dynamic inference schemes are widely investigated in the field of computer vision. Early prediction models like BranchyNet~\cite{teerapittayanon2016branchynet}and Adaptive Computation Time~\cite{figurnov2017spatially} adopt branches or halt units to decide whether the model could stop early. Some works use gate mechanism to determine the execution of a specific block. Wang et al.~\cite{wang2017skipnet} propose SkipNet which uses a gating network to selectively skip convolutional blocks based on the activations of the previous layer. A hybrid learning algorithm that combines supervised learning and reinforcement learning is used to address the challenges of non-differentiable skipping decisions. Wu et al.~\cite{wu2018blockdrop} propose BlockDrop and also make use of a reinforcement learning setting for the reward of utilizing a minimal number of blocks while preserving recognition accuracy. ConvNet-AIG is proposed in~\cite{veit2018convolutional}, which utilizes the Gumbel-Max trick~\cite{gumbel1954statistical} to optimize the gate module. However, the block-wise method can only alter the depth of the network, which could be too rough as some channels of an abandoned block may be useful.
On the other hand, the channel-wise method can manually adjust the number of active channels of a specific model. However, as far as we know, only~\cite{yu2018slimmable} is similar to such a method. The proposed Slimmable Neural Networks can adjust its width on the fly according to the on-device benchmarks and resource constraints. Strictly speaking, it is not a dynamic process as the procedure of choosing the active channels is finished before inference. Moreover, the pre-defined width multipliers negatively affect the flexibility of the dynamic inference mechanism. Our work is close to~\cite{veit2018convolutional}. However, we attempt to combine the merits of both the above two methods and propose a novel dynamic inference method which can provide more path selection choices in terms of network width and depth.
\textbf{Model Compression.} The great computing expense of deeper networks contradicts the demands of many resource-constrained applications, such as mobile platforms, therefore, reducing storage and inference time also plays an important role in deploying top-performing deep
neural networks. Lots of techniques are proposed to attack this problem, such as pruning~\cite{he2017channel, molchanov2016pruning, wen2016learning}, distillation~\cite{hinton2015distilling, polino2018model}, quantization~\cite{han2015deep, wei2018quantization, jacob2018quantization}, low-rank factorization~\cite{ioannou2015training}, compression with structured matrices~\cite{cheng2015exploration} and network binarization~\cite{courbariaux2016binarized}. However, these works are usually applied after training the initial networks and generally used as post-processing, while DMNN could be trained end-to-end without well-designed training rules.
On the other hand, lightweight architectures play important roles in various real scenarios, such as MobileNet~\cite{howard2017mobilenets, sandler2018mobilenetv2} and ShuffleNet~\cite{zhang2018shufflenet, ma2018shufflenet}. In this paper, by applying our methods, we prove even compact model like MobileNetV2 could be further improved.
\section{Methodology}
In this section,
we introduce the proposed dynamic multi-path neural network (DMNN) in detail,
including the subdivision of the block,
the architecture of the controller and the optimization approach.
\subsection{Block Subdivision}
It is ideal for controlling the execution of each channel of the network at runtime.
However,
this would lead to a significant increase in computational complexity.
In this work,
we divide the origin block of the network into several sub-blocks,
and each sub-block has its switch to decide whether to execute or not,
resulting to a dynamic inference path for different samples.
We interpret optimizing the network structure as executing or skipping of each sub-block during the inference stage.
A key issue is how to split one block into $N$ sub-blocks.
The guiding principle is that the parameters of the new block must be consistent with or approximate to the original block for fair comparison.
Fig.~\ref{fig::subdivide} shows the subdivision of blocks of MobileNetV2 and ResNet.
\begin{figure}
\begin{center}
\centering
\begin{subfigure}[b]{0.8\linewidth}
\centering
\includegraphics[width=\linewidth]{./figures/divide_rule/mobile_split.pdf}
\vspace{-20pt}
\caption{MobileNetV2 block}
\label{fig:orig_mobile}
\end{subfigure}
\\
\begin{subfigure}[b]{0.8\linewidth}
\centering
\includegraphics[width=\linewidth]{./figures/divide_rule/res_split.pdf}
\vspace{-20pt}
\caption{ResNet block}
\label{fig:orig_res}
\end{subfigure}
\end{center}
\vspace{-20pt}
\caption{Subdivision strategy.}
\vspace{-10pt}
\label{fig::subdivide}
\end{figure}
For the block of MobileNetV2,
we divide the origin block into $N$ sub-blocks,
the expansion ratio of each sub-block is set to $E/N$.
Thus the sum of every sub-block's computation and parameters are the same with the original block since it only consists of pixel-wise convs and depth-wise convs,
more detail can be seen in Fig~\ref{fig:orig_mobile}.
While for ResNet,
it is not that straightforward.
As shown in Fig.~\ref{fig:orig_res},
suppose the shape of the input tensor is $H \times W \times C_{in}$,
the output channels of each conv operation are $C_1, C_2, C_{out}$.
The parameter of the original block is
\begin{equation}
C_{in}\cdot C_1 + 9\cdot C_1 \cdot C_2 + C_2 \cdot C_{out}.
\label{eq:old_param}
\end{equation}
The original block is then split into $N$ sub-blocks.
The output channels of each sub-block are $\hat{C_1}, \hat{C_2}, C_{out}$.
Then the parameter becomes
\begin{equation}
N \cdot \left(C_{in}\cdot \hat{C_1} + 9 \cdot \hat{C_1} \cdot \hat{C_2}+ \hat{C_2} \cdot C_{out} \right).
\label{eq:new_param}
\end{equation}
If we simply set the number of channels of each sub-block to $1/N$ of the origin blocks, i.e.
$\hat{C_1} = C_1 / N, \hat{C_2} = C_2 / N$, Eqn.~\ref{eq:new_param} can be rewritten as follow:
\begin{equation}
C_{in}\cdot C_1 + 9\cdot C_1 \cdot C_2 / N + C_2 \cdot C_{out},
\end{equation}
which is not equal to Eqn.~\ref{eq:old_param}.
Thus,
to make subsequent extensive studies fair,
we make minor modifications to ResNet
and design the corresponding DMNN version to make Eqn.~\ref{eq:old_param} $\approx$ Eqn.~\ref{eq:new_param}.
For saving space,
the detailed architecture of DMNN-50 can be referred to supplementary materials.
\subsection{The Architecture of Controller}
\label{section:CAM}
The controller is elaborately designed to predict the status of each sub-block (on/off) with an minimal cost.
It is the inference paths optimizer of DMNN.
An overview of the dynamic path selection framework is shown in Fig.~\ref{fig:controller_architecture}.
Given an input image,
its forward path is determined by the gate controllers
and Fig.~\ref{fig:controller_overview} shows the gate mechanism of DMNN.
Suppose we split $l$-th block into $N$ sub-blocks,
the output of $l$-th block is the combination of the outputs of an identity connection and $N$ sub-blocks.
Formally,
\begin{equation}
\boldsymbol{X_l}=\boldsymbol{X_{l-1}} + \sum_{n}^{N}s_{l,n}\mathcal{F}_{l,n}(\boldsymbol{X_{l-1}}),
\end{equation}
where $\boldsymbol{X_l}$ is output of $l$-th block, $s_{l,n}\in\{0,1\}$ refers to the off/on status
which is predicted by the controller.
$\mathcal{F}_{l,n}(\boldsymbol{X_{l-1}})$ refers to the output of $n$-th sub-block of $l$-th block.
\begin{figure}[t]
\begin{center}
\begin{subfigure}[b]{0.8\linewidth}
\includegraphics[width=\linewidth]{figures/gate_block.pdf}
\vspace{-18pt}
\caption{The overview of our gate mechanism.}
\label{fig:controller_overview}
\end{subfigure}
\begin{subfigure}[b]{0.8\linewidth}
\includegraphics[width=\linewidth]{figures/structure.pdf}
\vspace{-18pt}
\caption{The architecture of controller.}
\label{fig:controller_module}
\end{subfigure}
\end{center}
\vspace{-20pt}
\caption{The framework of dynamic paths selection.}
\vspace{-10pt}
\label{fig:controller_architecture}
\end{figure}
\textbf{Spatial and previous state information embedding}.
On the one hand,
the control modules make decisions based on the global spatial information,
and we achieve this process by applying global average pooling to compress the high dimension features to one dimension along channels.
We further use a fully connected layer followed by an activation layer to map the pooling features to low-dimensional space.
Specifically,
$\boldsymbol{X_{l-1}} \in \mathbb{R}^{H \times W \times C}$ represents the input features of $l$-th block,
we calculate the $c$-th channel statistic by
\begin{equation}
z_c = \frac{1}{H \cdot W} \sum_{i=1}^{H} \sum_{j=1}^{W}x_{i,j,c}^{l-1},
\end{equation}
The final embedding feature $\boldsymbol{V_{l-1}} \in \mathbb{R}^d$ is
\begin{equation}
\boldsymbol{V_{l-1}} = \mathcal{F}(\boldsymbol{z}, \boldsymbol{W_1}) = \sigma(\boldsymbol{W_1}\boldsymbol{z}),
\end{equation}
where $\boldsymbol{z}=\left[z_1, z_2,\cdots, z_c \right]$, $\boldsymbol{W_1} \in \mathbb{R}^{d \times c}$,
$\sigma$ is the ReLU~\cite{glorot2011deep} function,
$d$ is the dimension of the hidden layer.
On the other hand,
there are some connections between the current controller and the previous controllers.
Thus the integration of previous state information is also crucial.
We first employ a fully connected layer followed by ReLU function to map the previous state hidden features into the same subspace with $\boldsymbol{V_{l-1}}$.
Then we perform an addition operation on the hidden feature and $\boldsymbol{V_{l-1}}$ to get the result of the current state.
Formally,
\begin{equation}
\begin{split}
\boldsymbol{h_{l-1}^{'}} &= \mathcal{F}(\boldsymbol{h_{l-1}},\boldsymbol{W_2}) = \sigma(\boldsymbol{W_2}*\boldsymbol{h_{l-1}}), \\
\boldsymbol{h_l} &= \boldsymbol{V_{l-1}} + \boldsymbol{h_{l-1}^{'}},
\end{split}
\end{equation}
where $\boldsymbol{W_2} \in \mathbb{R}^{d \times d}$,
$\sigma$ represents the ReLU function.
Bias terms are omitted for simplicity.
The status predictions of each sub-block at $l$-th block are made through $\boldsymbol{h_l}$ by using a softmax trick which we will introduce in section \ref{section:optimization}.
\textbf{Softmax Trick with Gumbel Noise.}
To decide whether to execute or omit a sub-block is inherently discrete and therefore non-differentiable.
In this work,
we use softmax trick with gumbel noise to solve this problem,
which has been proved to be successful in~\cite{veit2018convolutional}.
Formally,
let $N$ be the number of sub-blocks and $\boldsymbol{g_l}=\boldsymbol{W_3h_l}+\boldsymbol{b_3}$, $\boldsymbol{W_3} \in \mathbb{R}^{2N \times d}$, $\boldsymbol{b_3}$ is the bias term.
$\boldsymbol{g_l}$ is then reshaped to $N\times 2$ for the final predictions.
The activation can be written as follows
\begin{equation}
\boldsymbol{s_l} = \arg \max \left( \text{softmax} \left(\boldsymbol{g_l} + \Delta \right) \right),
\label{eq:decision}
\end{equation}
where $\boldsymbol{s_l}=\left[s_{l,1}, s_{l,2}, \cdots, s_{l,N} \right]$ refers to the status of each sub-block of $l$-th block,
and $\Delta \sim \text{ Gumbel}(0, 1)$ is a random noise following the Gumbel distribution,
which can increase the stability of the training process of our network.
\textbf{Supervised learning of controller}.
Deep CNNs compute feature hierarchies in each layer
and produce feature maps with different depths and resolutions.
This can also be considered as a feature extraction process from coarse to fine.
The proposed DMNN has a diversity of inference paths,
and we hope that different classes would select different paths.
However,
if the path selection mechanism is trained only by optimizing the classification loss at the last layer,
it will be difficult for the controller to learn the category information.
To solve this problem,
we introduce category loss to each controller to enable all of them to become category-aware.
Considering that predicting each class as a different category by the controller is computationally expensive,
we cluster samples into fewer categories than original classes.
For the ImageNet dataset~\cite{imagenet_cvpr09},
we cluster all the 1000 classes samples into 58 big categories with the help of the hierarchical structure of ImageNet provided in ~\cite{imagenet_cvpr09}.
For the CIFAR-100 dataset~\cite{krizhevsky2009learning},
it groups the 100 classes into 20 superclasses.
We use the 20 superclasses as the big categories directly.
Then cross entropy loss is employed to supervise all controllers as shown in Fig.~\ref{fig:controller_module}.
Formally,
the category loss of $l$-th controller can be written as follow
\begin{equation}
\mathcal{L}_{l}= \sum_{j}^{K}k_j \log(p_j),
\end{equation}
where $p_j$ represents the probability of $j$-th class.
$k_j=1$ if $j$ is the ground-truth class and 0 otherwise,
$K$ indicates the number of categories.
It is worth noting that the loss weights of each block's controller are not always equal since the features of different layers have different semantic information.
Deep layers have a stronger semantic information than shallow layers.
In DMNN-50,
there are four stages composed of 3, 4, 6, 3 stacked blocks respectively,
resulting in 16 controllers.
The loss weight of the first stage is set to 0.0001,
and it will increase by a factor of 10 in the next stages.
DMNN-101 follows the same principle.
The loss of supervised controller can be represented as follows
\begin{equation}
\mathcal{L}_{ctg} = \sum_l^L \alpha_l \mathcal{L}_{l},
\end{equation}
where $\alpha_l$ denotes the loss weight of $l$-th controller and $L$ denotes the number of blocks.
The category information will be removed after training,
so it will not generate any extra computational burden during testing.
The controller is desirable for its lightweight characteristic during the optimization of network structure.
The dimension of the hidden layer $d$ is set to 32 in all experiments.
This setting generates only little computational cost and can be omitted compared to the whole computation of the network.
If we take DMNN-50 as an example,
the total 16 controllers only generate about 0.02\% FLOPs of the original ResNet-50.
\subsection{Optimization}
\label{section:optimization}
\textbf{Resource-constrained Loss.} The resource constraint comes from two aspects: the block execution rate and the total FLOPs. The execution rate of each block in a mini-batch is used to constrain the average block activation rate to the target rate $e$. Let $z_l$ denotes the execution rate of $l$-th block within a mini-batch, we define the execution rate $z_l$ as
\begin{equation}
z_l = \frac{\sum_i^N b_i} {B \cdot N},
\end{equation}
where $B$ is the mini-batch size, $b_i$ is the executed number of $i$-th sub-block within a mini-batch. The total execution rate loss can be written as follow
\begin{equation}
\mathcal L_{exec} = \sum_l^L \left(e-z_l\right)^2.
\end{equation}
The other constraint is the total FLOPs. To meet the desired FLOPs, we explicitly introduce the target FLOPs rate to the loss function. In each mini-batch, we compute the actual FLOPs via
\begin{equation}
f=\sum_{l}^{L}\sum_{i}^{N} \frac{b_i}{B} \cdot f_{l,i},
\end{equation}
where $f_{l,i}$ indicates the FLOPs of $i$-th sub-block at $l$-th block of the network. The FLOPs loss can be formulated as
\begin{equation}
\mathcal{L}_{flops} = \left (\frac {f} {f_{total}} - r \right )^2,
\end{equation}
where $f_{total}$ and $f$ represent the full FLOPs and the actual execution FLOPs of the network respectively, and $r$ denotes the target FLOPs rate. We set $e=r$ in all experiments since they have strong positive correlation and similar values. Thus, the resource-constrained loss is defined as
\begin{equation}
\mathcal{L}_{res} = \mathcal{L}_{exec} + \mathcal{L}_{flops}.
\end{equation}
The total training loss is
\begin{equation}
\mathcal{L}_{total} = \alpha_1 \mathcal{L}_{ctg}
+ \alpha_2 \mathcal{L}_{res}
+ \alpha_3 {\mathcal{L}_{cls}},
\end{equation}
where $\mathcal{L}_{cls}$ is the classification loss. In our experiments $\alpha_1=\alpha_2=\alpha_3=1$. The joint loss would be optimized by mini-batch stochastic gradient descent.
\begin{table*}[t]
\begin{center}
\begin{threeparttable}
\begin{tabular}{lcccc}
\toprule[1pt]
Model & Top-1 Err. (\%)
& Params ($10^6$) & FLOPs ($10^9$) & FLOPs Ratio (\%)\\
\hline
ResNet-50~\cite{he2016deep}
& 24.7
& 25.56 & 3.8 & - \\
ResNet-50 (PyTorch Official)~\cite{pytorch_models}
& 23.85
& 25.56 & 3.96 & 100.0 \\
ResNet-50\tnote{\dag} (ours)
& 23.51
& 25.56 & 3.96 & 100.0 \\
ResNet-50 + Pruning~\cite{molchanov2016pruning}
& 23.91
& 20.45 & 2.66 & 70.0 \\
ResNeXt-50 [$2\times40d$] ~\cite{xie2017aggregated}
& 23.0
& 25.4 & 4.16 & 105.1 \\
ResNeXt-50 [$4\times24d$] ~\cite{xie2017aggregated}
& 22.6
& 25.3 & 4.20 & 106.1 \\
ConvNet-AIG-50 [$t=0.7$] ~\cite{veit2018convolutional}
& 23.82
& 26.56 & 3.06 & 77.3 \\
S-ResNet-50-0.75~\cite{yu2018slimmable}
& 25.1
& 19.2 & 2.3 & 58.1 \\
DMNN-50, $N=2$ [$r=0.4$] & 24.06
& 24.67 & 2.07 & 52.3 \\
DMNN-50, $N=2$ [$r=0.5$] & 23.50
& 24.67 & 2.28 & 57.6 \\
DMNN-50, $N=2$ [$r=0.6$] & 23.22
& 24.67 & 2.52 & 63.6 \\
DMNN-50, $N=2$ [$r=0.7$] & 22.57
& 24.67 & 3.12 & 78.8 \\
DMNN-50, $N=3$ [$r=0.7$] & 22.54
& 25.81 & 3.16 & 79.8 \\
DMNN-50, $N=4$ [$r=0.7$] & \textbf{22.32}
& 25.81 & 3.17 & 80.1 \\
\hline
ResNet-101~\cite{he2016deep}
& 23.6
& 44.54 & 7.6 & - \\
ResNet-101 (PyTorch Official)~\cite{pytorch_models}
& 23.63
& 44.55 & 7.67 & 100.0 \\
ResNet-101\tnote{\dag} (ours)
& 22.02
& 44.55 & 7.67 & 100.0 \\
ResNeXt-101 [$2\times40d$] ~\cite{xie2017aggregated}
& 21.7
& 44.46 & 7.9 & 103.0 \\
ConvNet-AIG-101[$t=0.5$]~\cite{veit2018convolutional}
& 22.63
& 46.23 & 5.11 & 66.6 \\
DMNN-101, $N=2$ [$r=0.3$] & 22.82
& 43.12 & 2.48 & 32.3 \\
DMNN-101, $N=2$ [$r=0.5$] & 21.95
& 43.12 & 4.21 & 54.9 \\
DMNN-101, $N=2$ [$r=0.7$] & \textbf{21.43}
& 43.12 & 5.57 & 72.6 \\
\bottomrule[1pt]
\end{tabular}
\begin{tablenotes}
\footnotesize
\item[\dag] Our implementations of ResNet-50, ResNet-101, DMNN-50, DMNN-101 use $stride=2$ in conv$3\times3$ layers just as the PyTorch community does~\cite{pytorch_models} which is slightly different from the original paper.
\end{tablenotes}
\end{threeparttable}
\end{center}
\vspace{-10pt}
\caption{Comparison on heavyweight networks on ImageNet. We compare our DMNNs with the heavyweight networks ResNet-50, ResNet-101, and other dynamic networks ConNet-AIGs and slimmable network. Results show that our models outperform the other models in both accuracy and computational complexity.}
\vspace{-5pt}
\label{tab:resnet result}
\end{table*}
\section{Experiments}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\linewidth]{figures/flops.pdf}
\end{center}
\vspace{-10pt}
\caption{
{Top-1 error vs. FLOPs on ImageNet.} The proposed DMNN models outperform other methods by a large margin in both computational cost and accuracy.
}
\vspace{-10pt}
\label{fig:flops}
\end{figure}
In this section, we evaluate the performance of the proposed DMNN on benchmark datasets including ImageNet and CIFAR-100.
\subsection{Training Setup}
\textbf{ImageNet.} The ImageNet dataset~\cite{imagenet_cvpr09} consists of 1.2 million training images and 50K validation images of 1000 classes. We train networks on the training set and report the top-1 errors on the validation set. We apply standard practice and perform data augmentation with random horizontal flipping and random-size cropping to 224$\times$224 pixels. We follow the standard Nesterov SGD optimizer with momentum 0.9 and a mini-batch of 256. The cosine learning rate scheduler is employed for better convergence and the initial learning rate is set to 0.1. For different scale models, We use different weight decays, 0.0001 for ResNet and 0.00004 for MobileNet. All models are trained for 120 epochs from scratch.
\textbf{CIFAR-100.} The CIFAR-100 datasets~\cite{krizhevsky2009learning} consist of 60,000 color images of 10, 000 classes. They are split into the training set and testing set by the ratio of 5:1. Considering the small size of images ($32 \times 32$) in CIFAR, we follow the same setting as \cite{he2016deep} to construct our DMNNs for a fair comparison. We augment the input image by padding 4 pixels on each side with the value of 0, followed by random cropping with a size of $32 \times 32$ and random horizontal flipping. We train the network using SGD with the momentum of 0.9 and weight decay of 0.0001. The mini-batch size is set to 256, and the initial learning rate is set to 0.1. We train the networks for 200 epochs and divide the learning rate by 10 twice, at the 100th epoch and 150th epoch respectively.
\subsection{Performance Analysis}
We compare our method with ResNet~\cite{he2016deep}, ResNeXt~\cite{xie2017aggregated},
MobileNetV2~\cite{sandler2018mobilenetv2}, pruning method~\cite{molchanov2016pruning} and other dynamic
inference methods~\cite{yu2018slimmable, veit2018convolutional}.
We denote $N$ as the number of sub-blocks of each block, $r$ as the FLOPs target rate.
\begin{table*}[t]
\begin{center}
\begin{threeparttable}
\begin{tabular}{lcccc}
\toprule[1pt]
Model & Top-1 Err. (\%)
& Params ($10^6$) & FLOPs ($10^9$) & FLOPs Ratio (\%)\\
\hline
MobileNet V2~\cite{sandler2018mobilenetv2}
& 28.0
& 3.47 & - & - \\
MobileNet V2 (ours)
& 28.09
& 3.50\tnote{\dag} & 0.30 & 100.0 \\
S-MobileNet V2-0.75~\cite{yu2018slimmable}
& 31.1
& 2.7 & 0.23 & 76.7 \\
DMNN-MobileNetV2, $N=2$ [$r=0.7$] & 28.30
& 3.63 & 0.22 & 73.3 \\
DMNN-MobileNetV2, $N=2$ [$r=0.8$] & 28.15
& 3.63 & 0.24 & 80.0 \\
DMNN-MobileNetV2, $N=2$ [$r=0.9$] & \textbf{27.74}
& 3.63 & 0.27 & 90.0 \\
\hline
MobileNetV2 (1.4)~\cite{sandler2018mobilenetv2}
& 25.3
& 6.06 & - & - \\
MobileNetV2 (1.4) (ours)
& 25.30
& 6.09\tnote{\dag} & 0.57 & 100.0 \\
DMNN-MobileNetV2 (1.4), $N=2$ [$r=0.7$] & 26.03
& 6.29 & 0.42 & 73.7 \\
DMNN-MobileNetV2 (1.4), $N=2$ [$r=0.8$] & 25.53
& 6.29 & 0.47 & 82.5 \\
DMNN-MobileNetV2 (1.4), $N=2$ [$r=0.9$] & \textbf{25.26}
& 6.29 & 0.52 & 91.2 \\
\bottomrule[1pt]
\end{tabular}
\begin{tablenotes}
\footnotesize
\item[\dag]
Our implementation of MobileNet V2 is based on PyTorch and its parameter quantities are counted by PyTorch Summary~\cite{pytorch_summary}.
\end{tablenotes}
\end{threeparttable}
\end{center}
\vspace{-10pt}
\vspace{-2pt}
\caption{Comparison on lightweight networks on ImageNet. Our DMNNs based on MobileNetV2 can achieve remarkable results comparing to other lightweight models.}
\vspace{-10pt}
\label{tab:mobilenet result}
\end{table*}
\textbf{Performance on heavy networks.}
Tab.~\ref{tab:resnet result} shows that our DMNN achieves remarkable results compared to other heavy models on ImageNet.
First of all, we compare DMNN with ResNet.
When $N=2$, $r=0.5$, our DMNN-50 achieves similar performance with ResNet-50 but saves more than \textbf{42.4}\% FLOPs.
When we set $N=4$, $r=0.7$, our DMNN-50 further reduces 1.19\% Top-1 error while still saving \textbf{19.9}\% FLOPs.
Our DMNN-101 outperforms ResNet-101 and save \textbf{45.1}\% FLOPs in the same time when we set $N=2$, $r=0.5$.
The above comparison demonstrates that DMNN can greatly reduce FLOPs and improve the accuracy when compared to the models with similar parameters.
On the other hand,
DMNN-50 achieves even better performance than origin ResNet-101 (closely to ResNet-101 by our implementation), with \textbf{42.0}\%
parameter size reduction,
which indicates that DMNN can greatly save the parameters and is feasible for pratical model deployment.
Then,
we make comparason with DMNN and stronger baseline models ResNeXt.
As we set $r=0.7$,
our method is superior to ResNeXt-50 ($2\times40d$) in both FLOPs and accuracy.
When we set $N=4$, $r=0.7$,
our DMNN-50 reduces 0.28\% Top-1 error while still saves \textbf{24.5}\% FLOPs.
Our DMNN-101 outperforms ResNet-101 and save \textbf{45.1}\% FLOPs in the same time when we set $N=2$, $r=0.5$.
Similar result can be found while comparing to ResNeXt-101 ($2\times40d$) if we set $r=0.7$.
DMNN shows great superiority over ResNeXt mainly because of better control for different convolution groups.
Our method outperforms ConvNet-AIG~\cite{veit2018convolutional} in both accuracy and computational complexity,
demonstrating that multi-path design is more elaborate and superior than roughly skipping the whole block.
Especially, our DMNN-50 with $N=2$ and $r=0.4$ achieves comparable performance with ConvNet-AIG yet greatly
reduces the FLOPs by approximately \textbf{33.3}\%.
Fig~\ref{fig:flops} shows the trade-off between computational cost and accuracy of our DMNN while
comparing to other dynamic inference methods including slimmable neural network S-ResNet~\cite{yu2018slimmable}.
Meanwhile,
as an end-to-end method, DMNN shows great advantages over post process method such as pruning methods.
We also conduct experiments on CIFAR-100 dataset,
as shown in Tab.~\ref{tab:cifar}. It can be seen that DMNN-50
with $N=4$ and $t=0.7$ can even outperform ResNet-50 by 1.4\% on
CIFAR-100 with only \textbf{78.7}\% FLOPs.
\textbf{Performance on lightweight networks.}
We apply DMNN to lightweight network MobileNetV2, as shown in Tab.~\ref{tab:mobilenet result}.
Although similar conclusions can be obtained,
it is normal that the improvements is not as large as heavy models because of the compact structures.
Specially, our DMNN with $N=2$ and $r=0.9$ can save 10.0\% FLOPs and achieves better top1 error than MobileNetV2. The proposed method is also better than other dynamic inference methods.
In summary, our method performs superbly in accuracy and computational complexity for both heavy and lightweight networks, which demonstrates its great applicability to different networks and robustness on different datasets.
\begin{table}[t]
\addtolength{\tabcolsep}{-4.2pt}
\begin{center}
\begin{tabular}{lcc}
\toprule[1pt]
Model & FLOPs ($10^9$) & Top-1 Err. (\%) \\
\hline
ResNet-50~\cite{he2016deep} & 0.33 & 27.55 \\
\hline
DMNN-50, $N=2$ [$t=0.5$] & 0.18 & 28.24 \\
DMNN-50, $N=2$ [$t=0.7$] & 0.22 & 27.34 \\
DMNN-50, $N=4$ [$t=0.7$] & 0.26 & \textbf{26.15} \\
\bottomrule[1pt]
\end{tabular}
\end{center}
\vspace{-10pt}
\vspace{-2pt}
\caption{Test error on CIFAR-100. The DMNNs reduce 1.4\% Top-1 error while saving about 21.2\% FLOPs.}
\vspace{-10pt}
\label{tab:cifar}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{l|cc|cc}
\toprule[1pt]
Method & PREV & CAT & Top-1 Err. (\%) \\
\hline
ResNet-50~\cite{he2016deep} & & & 23.51 \\
\hline
DMNN-50 & & & 23.25 \\
DMNN-50 & $\surd$ & & 23.09 \\
DMNN-50 & & $\surd$ & 23.20 \\
DMNN-50 & $\surd$ & $\surd$ & \textbf{22.57}\\
\bottomrule[1pt]
\end{tabular}
\end{center}
\vspace{-10pt}
\vspace{-2pt}
\caption{The effectiveness of well-designed controller with supervised learning on ImageNet.
The FLOPs target is set to 0.7, and the number of sub-block $N$ is set to 2.
``PREV'' represents employing previous state features
and ``CAT'' represents employing supervised learning in this table.}
\vspace{-10pt}
\label{tab:cacs}
\end{table}
\subsection{Ablation Study}
\textbf{Effectiveness of the gate controller.}
In order to show the effectiveness of the controllers,
we conduct four groups of experiments on ImageNet dataset with different configurations.
Tab.~\ref{tab:cacs} shows the comparison of different models.
If we employ previous features and
supervised learning separately, additional promotions are obtained. After aggregating
these two improvements, we can boost the performance by 0.68\%, demonstrating the benefits of previous state information embedding
and supervised learning of controllers.
It is worth noting that it only introduces a fully connected layer
with 32 hidden neurons while applying previous controller's features, the additional computation
cost can be omitted. The supervised learning of the controllers may generate minor additional
computational cost during training, yet it will be removed at the testing stage.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\linewidth]{figures/dist.pdf}
\end{center}
\vspace{-10pt}
\vspace{-3pt}
\caption{
{Distribution of FLOPs on the ImageNet validation set using DMNN-50 with $N=2$, $r=0.7$.}
}
\vspace{-10pt}
\label{fig:dist}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{./figures/rate/min-max-flops.pdf}
\begin{subfigure}[b]{0.120\linewidth}
\caption{tench}
\end{subfigure}
\begin{subfigure}[b]{0.120\linewidth}
\caption{hermit crab}
\end{subfigure}
\begin{subfigure}[b]{0.120\linewidth}
\caption{malamute}
\end{subfigure}
\begin{subfigure}[b]{0.120\linewidth}
\caption{colobus}
\end{subfigure}
\begin{subfigure}[b]{0.120\linewidth}
\caption{dwelling}
\end{subfigure}
\begin{subfigure}[b]{0.120\linewidth}
\caption{lifeboat}
\end{subfigure}
\begin{subfigure}[b]{0.120\linewidth}
\caption{quilt}
\end{subfigure}
\begin{subfigure}[b]{0.120\linewidth}
\caption{trombone}
\end{subfigure}
\vspace{-10pt}
\caption{
Examples of ``easy'' and ``hard'' samples, each column belongs to the same category.
Top row: samples with less computation. Bottom row: samples with more computation.}
\label{fig:easy_hard_samples}
\end{figure*}
\textbf{The impact of $N$ and $r$.}
We adopt different values of $N$ and $r$ to explore
their impacts on the performance. As shown in Tab.~\ref{tab:resnet result}, we set $N=2,3,4$,
while keep $r=0.7$ on DMNN-50. The model with $N=4$ obtains the lowest test error rate,
indicating that bigger $N$ can lead to more path selection choices and consequently better performance.
We further keep $N=2$ and change $r$ to 0.4, 0.5 and 0.6 respectively. Larger $r$ leads to more computational
cost that verifies the effectiveness of our resource-constrained mechanism.
The model with larger FLOPs rate gains higher performance since more computation units are involved.
The DMNN can achieve a better accuracy-efficiency trade-off in terms of the computational budgets.
We have not conducted more experiments on larger $N$ due to resource limitation. But we will explore the characteristic
of DMNN with larger $N$ in the future work.
\subsection{Visualization}
\textbf{Visualization of dynamic inference paths.}
The inference paths vary across images, which leads to different computation cost. Fig.~\ref{fig:dist}
shows the distribution of FLOPs on the ImageNet validation set using our DMNN-50 model with $N=2$, $r=0.7$.
The proportion of images with FLOPs in the middle is the highest, and images do occupy different computing
resources guided by computational constraint. We further visualize the execution rates of each
sub-block within the categories of animals, artifacts, natural objects, geological formations as shown
in Fig.~\ref{fig:layer_rate}. We can see that some sub-blocks, especially at the first two blocks of the
network, are executed all the time and the execution rates of other sub-blocks vary from categories.
One reason could be that different categories share the same shallow layers' features which are important
for classification. As the layer goes deeper, the semantic information of the features becomes stronger,
which depends on categories.
\textbf{Visualization of ``easy'' and ``hard'' samples.}
We find that even samples of the same category would have different inference paths.
A reasonable explanation is that hard samples need more computation than easy ones.
Fig~\ref{fig:easy_hard_samples} shows examples of easy and hard samples with different actual FLOPs.
Although for some classes such as malamute and lifeboat, the ``hard'' samples are difficult than ``easy'' ones, for most classes,
the quality gap is not indeed noticeable.
We infer that it is because the definition of easy and hard samples mainly depends on the representation property of the
neural networks,
rather than on the intuition of human beings.
\begin{figure}
\centering
\includegraphics[height=3.0in]{figures/rate/rate_1.pdf}
\hspace{10pt}
\includegraphics[height=3.0in]{figures/rate/rate_2.pdf}
\hspace{10pt}
\includegraphics[height=3.0in]{figures/rate/rate_3.pdf}
\hspace{10pt}
\includegraphics[height=3.0in]{figures/rate/rate_4.pdf}
\hspace{10pt}
\includegraphics[height=3.0in]{figures/rate/rate_5.pdf}
\vspace{-5pt}
\caption{Execution rates of sub-blocks for different categories on DMNN-50 with $N=2$, $r=0.7$.}
\label{fig:layer_rate}
\vspace{-10pt}
\end{figure}
\section{Conclusion}
In this paper, we present a novel dynamic inference method called Dynamic Multi-path Neural Network (DMNN).
The proposed method splits the original block into multiple sub-blocks, making the network become more flexibility to
handle different samples adaptively.
We also carefully design the structure of the gate controller to get reasonable inference path, and introduce
resource-constrained lose to make full use of the representation capacity of sub-blocks.
Experimental results demonstrate the superiority of our method.
{\small
\bibliographystyle{ieee}
|
{
"timestamp": "2019-04-09T02:16:09",
"yymm": "1902",
"arxiv_id": "1902.10949",
"language": "en",
"url": "https://arxiv.org/abs/1902.10949"
}
|
\section{Introduction} \label{sec:intro}
High redshift ($z>1$) star-forming galaxies show a more irregular and clumpy structure than local spiral galaxies \citep{Elmegreen2004ApJ...604L..21E, Elmegreen2006ApJ...650..644E}. The luminous clumps measure up to $1\rm~kpc$ in radius \citep{Swinbank2012ApJ...760..130S, Fisher2017MNRAS.464..491F}. They are sites of extreme star formation that collectively reach star formation rates (SFRs) up to $100\ M_{\odot} \rm{yr^{-1}}$ \citep{Genzel2006Natur.442..786G, Stark2008Natur.455..775S}. In contrast, local main sequence galaxies, such as the Milky Way, only support SFRs of a few $M_{\odot} \rm{yr^{-1}}$ \citep{Licquia2015ApJ...806...96L} and this star formation is hosted in much smaller `Giant' Molecular Clouds (GMCs), measuring less than $100\rm pc$ in radius \citep{Bolatto2008ApJ...686..948B}. This difference between high-redshift and local star-forming systems parallels the strong evolution of the global comoving SFR density, which peaked at $z\sim2-3$ \citep{Hopkins2006ApJ...651..142H, Yuksel2008ApJ...683L...5Y} and has since declined by more than an order of magnitude. Hence, understanding the origin and physics of massive star-forming clumps is an important jigsaw piece in modelling the evolution of galaxies.
The most common scenarios for the formation of star-forming clumps can be grouped into in-situ and ex-situ processes. Ex-situ clump formation relates to environmental interactions such as star-bursting major mergers and minor mergers, where the merging satellite becomes a clump of its new host galaxy.
In-situ clump formation normally invokes the theory of Violent Disk Instabilities (VDIs), in which a turbulent, rotating disk fragments into gravitationally bound sub-structures \citep{Bournaud2009}. Due to high velocity dispersion, the Jeans' scales, the length at which thermal expansion and contraction due to gravity are in equilibrium, can reach up to $1~\rm kpc$, only a few times shorter than the characteristic scale of the entire disk. Such large Jeans' lengths are a necessary, but insufficient condition for large clumps to form. It is also required that instabilities of this size are \textit{not} stabilised by shear forces -- a non-trivial requirement in rotating disks \citep{Burkert2010}. A metric to quantify these instabilities is the Toomre parameter $Q$ \citep{Toomre1964ApJ...139.1217T} which measures the ratio between the outward pressure (thermal$+$dynamical) and gravitational force within a gas cloud. In the approximation of an axially symmetric disk, the situation of marginal stability can be expressed as $Q\approx1$, where $Q$ is a two-component (gas+stars) extension \citep[e.g.][]{Romeo2011} of the Toomre stability parameter. Using this ansatz several studies found that the marginal stability of clumpy disks can be attributed to high gas fractions \citep{Dekel2009ApJ...703..785D,Genzel2011ApJ...733..101G,Fisher2014,Wisnioski2015ApJ...799..209W, White2017ApJ...846...35W} and/or low angular momentum, with the latter being likely the dominant cause \citep{Obreschkow2015}. However, this $Q$-based ansatz remains debated and may require the inclusion of additional non-linear processes \citep{Inoue2016}.
Distinguishing between different clump formation scenarios is not trivial from an observational viewpoint (e.g.~\citealp{Glazebrook2013PASA...30...56G}). A purely morphological analysis of the CANDELS data \citep{Guo2015ApJ...800...39G} suggests that the incidence of clumps in massive ($M_{\rm *}>10^{10}M_{\odot}$) star-forming galaxies at $z\approx0.5-3$ is consistent with the VDI model, whereas minor mergers might be important for lower mass galaxies and at lower redshifts. Using HST images of galaxies with spectroscopic redshifts from the VIMOS Ultra Deep Survey (VUDS), \citet{Ribeiro2017A&A...608A..16R} analyse the number and luminosity statistics of clumps in individual galaxies at $2 \lesssim z \lesssim 6$ and again conclude that VDIs are probably the dominant cause of clump formation, rather than mergers. Additional circumstantial support for VDIs as the dominant origin of massive clumps comes from resolved kinematic studies (e.g.~\citealp{Tacconi2013ApJ...768...74T}) revealing that the majority of high-redshift star-forming galaxies are rotationally supported disks. However, \citet{Forster2009ApJ...706.1364F} and \citet{Law2009ApJ...697.2057L} find that it is possible for galaxies undergoing strong mergers to display a rotation profile that closely resembles that of a rotating disk.
Of course, resolved imaging and spectroscopy of individual clumps would enable much more stringent tests, however, this is normally hampered by instrumental limitations -- even at HST resolution sub-kpc scales at $z=2$ are barely resolved. To beat this limitation \citet{Dessauges-Zavadsky2018MNRAS.479L.118D} analyzed a sample of strongly lensed galaxies. They found that the mass function of clumps follows a power law of slope~$-2$ which is consistent with clumps forming in-situ by turbulent fragmentation. However, since the magnification of strong lensing is model dependent and acts only in a single direction, the interpretation of such data remains difficult.
An alternative approach to studying high redshift galaxies consists of using their lower redshift `analogs'. This is the leading idea of the DYNAMO sample, detailed in Section~\ref{subsec:dynamo}. Relying on a $Q$-based approximation, \citet{Fisher2017ApJ...839L...5F} (hereafter F17) predicted and observationally confirmed that clumps formed in-situ obey a scaling relation between the clump radius and the velocity dispersion (and, by extension, the gas fraction) of their parent disk (see also \citealp{Wisnioski2012,Livermore2012MNRAS.427..688L}). Using this relation, F17 explicitly showed that expectations from a minor-merger scenario are not likely to form most clumps in DYNAMO galaxies. Hence, this relation is a promising way to distinguish between clump formation scenarios, as well as to probe the inner physics of these heavily star-forming objects.
The use of scaling relations to test ideas for the origin of clumps raises important challenges:
\begin{itemize}
\item \textit{Clump size measurement:} The definition and measurement of the characteristic clump size should be robust against (1) variations in observational resolution/noise, (2) the complex hierarchical substructure of clumps \citep{Elmegreen2011EAS....51...31E}, (3) the potential random overlap of clumps, and (4) the presence of other, similarly-sized galactic substructures, such as bars and spiral arms.
\item \textit{Physics of scaling relation:} The model of F17, relating characteristic clump size to the disk's velocity dispersion, relies on a simplification of the Toomre stability criterion \citep{Toomre1964ApJ...139.1217T}, which makes a number of approximations and bypasses the possible strong feedback-regulation \citep{Genel2012} within star-forming clumps.
\end{itemize}
Here, we address these challenges using an advanced statistical method applied to both observations and simulations of clumpy galaxies. We use the well-tested 2-point statistics-based method of \citet{Ali2017ApJ...845...37A} (summarized in Section \ref{sec:background}) to measure the `characteristic' clump size in resolved images of star-formation rate traces. In Section \ref{sec:observations}, this method is applied the full set of 10 nearby clumpy galaxies from the DYNAMO (DYnamics of Newly-Assembled Massive Objects) survey \citep{Green2014MNRAS.437.1070G} that have been followed up by the Hubble Space Telescope (HST) in H$\alpha$ and continuum emission by \citet{Fisher2017MNRAS.464..491F}. We show that the VDI scaling relation of F17 holds for the clumpy galaxies (except for `new' mergers) when analyzed in this way. In Section \ref{sec:simulations}, we use four realizations of a simulated control galaxy with four different stellar feedback modes in order to (1) verify the scaling relation of F17 in a more realistic model and (2) check if this scaling relation applies irrespective of the feedback model. Section \ref{sec:discussion} gives a synthesis of the results and brief conclusion.
\section{Background: 2-point clump scale} \label{sec:background}
This section summarizes the statistical estimator of the characteristic clump size introduced by \cite{Ali2017ApJ...845...37A} (hereafter A17). The interested reader is referred to that paper for details beyond the brief summary presented here.
In A17, we found that the characteristic scale $r_{\rm clump}$ of the clumps in a star-formation density map is related to the maximum point $r_{\rm peak}$ of the weighted two-point correlation function (w2PF)
\begin{equation} \label{eq:w2pf}
r^{\gamma}\xi_{2}(r)\ \forall\ \gamma>0,
\end{equation}
where $r$, $\xi_{2}(r)$ and $\gamma$ are the length scale, the two-point correlation function of the map and a positive exponent, respectively. For randomly positioned clumps with circular 2D-Gaussian density profiles of standard deviation $r_{\rm clump}$, the exact analytical expectation is
\begin{equation} \label{eq:clump=peak}
r_{\rm clump} = \frac{r_{\rm peak}}{\sqrt{2\gamma}}.
\end{equation}
\begin{figure*}
\includegraphics[width=1\textwidth]{figure_dynamo.pdf}
\caption{Example of four galaxies in the observed sample: (a) the `normal' control galaxy in the DYNAMO-HST sample with regular GMCs that cannot be resolved, (b) one of 7 clumpy DYNAMO-HST galaxies with rotation supported disks, (c) one of two merging starbursts in DYNAMO-HST, (d) NGC 5194, a local ($z\approx0$) spiral galaxy with GMCs. The H$\alpha$ and continuum maps are displayed in cyan and red, respectively. The bottom row shows their w2PF (for $\gamma=1/2$), in arbitrary units on the y-axis, 1$\sigma$ uncertainty (shaded region) and their $r_{\rm clump}=r_{\rm peak}$ value (dashed line), determined by fitting a parabola (blue) at around the maxima. A distance of $8\ \rm Mpc$ \citep{Karachentsev2004AJ....127.2031K} is used in our values for NGC 5194, but note that $r_{\rm clump}/r_{\rm disk}$ is distance-independent. }
\label{fig:dynamo_galaxies}
\end{figure*}
In particular, if $\gamma=1/2$, then $r_{\rm clump}=r_{\rm peak}$. Real clumps are neither Gaussians of identical size, nor are they randomly positioned across the galaxy. However, the w2PF method is robust against these deviations, as shown in A17 for the following reasons.
Firstly, star-forming clumps normally exhibit a size distribution (often following power law between number and size, see \citealp{Oey1998AJ....115.1543O}) and they exhibit a complex, hierarchical substructure down to the scale of individual star-forming `cores'. Using mock images of clumps with an overall Gaussian density, but fractal substructure, drawn from realistic size-distributions, we showed numerically that the w2PF method recovers the mass-weighted clump size of the input model within 20\%, irrespective of the precise size-distribution and fractal substructure. Furthermore, the w2PF is robust against changing resolution, as long as the Point Spread Function (PSF) is smaller than the mass-weighted clump size, and also robust to different types of noise (with white, blue, red spectra) up to an RMS pixel-noise as high as the integrated flux of the brightest clumps (Fig.~3 in A17).
Secondly, the clump positions in real galaxies are not random, but they follow the global density structure of the disk, such as a roughly exponentially decreasing surface density with spiral arms and rings. We found that these galactic structures impact the clump size measurement via the w2PF, but can be removed in the same way that window functions and selection functions can be removed when measuring the 2PCF of cosmic large-scale structure. That is, in expression (\ref{eq:w2pf}), we must define $\xi_{2}(r)$ as the 2PCF in excess of galactic substructure other than clumps. This can be done, for instance, using the classic Landy-Szalay estimator \citep{LS1993}
\begin{equation} \label{eq:LS}
\hat{\xi}_{\rm LS}(r)= \frac{DD(r)-2DR(r)+RR(r)}{RR(r)},
\end{equation}
where the fields $D$ and $R$ are 2D fields. The $D$ contains the clumpy structure which we wish to measure and extra spurious galactic structure while the $R$ field consists of only the galactic structure wish we wish to remove. Hence in measuring clump sizes via 2PCF estimator it is vital we select an $R$-field which masks the excess correlation. The functions $DD$, $DR$ and $RR$ are defined as
\begin{equation} \label{eq:normalise}
\begin{split}
XY(r)\equiv\frac{1}{\sum X\sum Y} \sum_{\rm |\pmb{r_1-r_2}|\in(r\pm\Delta r/2)} X(\pmb{r_1})Y(\pmb{r_2}).
\end{split}
\end{equation}
The parameter $\Delta r$ is the bin width of the regularly distributed scale lengths $r$. Since Eq.~(\ref{eq:LS}) effectively removes correlations present in the $R$-field from the $D$-field we take a map of the older stellar population as $R$-field while that of the newly formed stars as $D$-field. For the DYNAMO-HST sample we use H$\alpha$ map as the $D$-field and the continuum map by using it as $R$-field. In the case of the simulations we use stars formed within $10\ \rm Myr$, which corresponds to the lifetime of O-stars, as $D$-field while taking the whole stellar population as the $R$-field. This removes spurious correlation added to the 2PCF by the galaxy disk structure.
Finally, we must choose a value of $\gamma$ when computing the w2PF (Eq.~\ref{eq:w2pf}). For a hypothetical infinitely extended field of Gaussian clumps, any positive value will result in an accurate estimation of the clump scale $r_{\rm clump}$ (via Eq.~\ref{eq:clump=peak}). However, in realistic circumstances, larger values can help suppress spurious small-scale structure, not already removed via the R-field in Eq.~(\ref{eq:LS}), whereas smaller values can suppress spurious large-scale structure. In A17 we adopted the fiducial $\gamma=1/2$, which leads to good results for mock images of galaxies with realistic noise. We here apply this value to all observed galaxies. In the case of our simulated disks, we find that a slightly larger value (we pick $\gamma=1$) allows us to avoid contamination by spurious small-scale structures associated with two-body relaxation present in SPH-based simulations (e.g.~\citealp{Power2016MNRAS.462..474P}).
\section{Clump-scalings in observed galaxies} \label{sec:observations}
In this section we first describe the observational data used in this study. We then apply the w2PF and compare the robustly estimated clump sizes to those measured by F17. Finally, we gauge the degree to which the scaling relationship of F17 holds for the DYNAMO-HST sample.
\subsection{Sample of high-$z$-analogs}\label{subsec:dynamo}
The DYNAMO (DYnamics of Newly-Assembled Massive Objects) galaxies \citet{Green2014MNRAS.437.1070G} were selected from the Sloan Digital Sky Survey (SDSS) \citep{York2000AJ....120.1579Y} as the objects with the most extreme H$\alpha$ luminosities ($L_{\rm H\alpha} > 10^{42}\ \rm erg\ s^{-1}$). Follow-up integral field spectroscopy observations were used to identify a subsample of high-dispersion systems, which excludes AGN. A sub-sample of 9 such galaxies, as well as one control galaxy (A04-3) with normal H$\alpha$ luminosity, were then observed with the Hubble Space Telescope (HST) Advanced Camera for Surveys Wide-field Camera by \citet{Fisher2017MNRAS.464..491F}. All galaxies in this DYNAMO-HST sample, except for the control object, show massive clumps, reminiscent of those seen at higher redshift, when degraded to $z=1$ resolution. Two galaxies in the DYNAMO-HST clearly look like systems about to undergo a major merger, while the other 8 show regular morphologies and rotation-supported disks in H$\alpha$ kinematics.
The HST data used in this paper are H$\alpha$ maps showing newly formed stars and a continuum image showing the older stellar population. The H$\alpha$ emission was observed using the ramp filters FR716N and FR782N within a 2\% bandwidth and integrated for 45 minutes. The continuum maps used the FR647M filter with an integration time of 15 minutes. The final H$\alpha$ image is generated by subtracting the continuum map from the H$\alpha$ map. The complete reduction process is given in \citet{Fisher2017MNRAS.464..491F}.
The sample consists of galaxies which are consistent with clump formation scenarios resembling self-gravity instabilities as well as major mergers. For certain galaxies the resolution is not sufficient to measure the clump size $r_{\rm clump}$. Figure \ref{fig:dynamo_galaxies} shows the variety of galaxies analysed in this study along with their w2PF: (a) the control galaxy of the DYNAMO-HST sample without significant clumps, (b) one of 7 clumpy disk with no signs of mergers in the DYNAMO-HST sample, (c) one of two merging starbursts, and (d) the local spiral galaxy NGC 5194. Since the primary goal of our analysis is to measure the clump scale we mask the central region and foreground stars as shown in Figure \ref{fig:dynamo_galaxies} (black). We then compute the w2PF and fit a parabola around the maxima to infer $r_{\rm clump}$ at sub-pixel resolution. In the control object (a), the size of the star-forming regions lies below the resolution and hence the w2PF only provides an upper bound.
\begin{figure}
\includegraphics[width=1\columnwidth]{figure_david_compare.pdf}
\caption{Comparison of clump sizes estimated by the w2PF method, $2r_{\rm clump}$, and a previous study by F17 $r_{\rm clump, F17}$. The two estimates are in good agreement as they fall on the one-to-one (dotted) line within their 1-$\sigma$ uncertainties.}
\label{fig:david_compare}
\end{figure}
\subsection{Results}
Comparing \emph{twice} the clump size $r_{\rm clump}$ estimated using the w2PF of A17 with those measured by F17 ($r_{\rm clump, F17}$) we find a good agreement, within 1-$\sigma$ uncertainty, as shown in Figure \ref{fig:david_compare}. Each point in this figure corresponds to a galaxy-average. The method of A17 naturally returns the luminosity-weighted average clump size, which is a converged quantity, even in the presence of hierarchical substructure, given the steep power-law distribution of the substructure (see A17 for details). F17 measure the size of each clump individually and then take the average. This method would result in smaller clump sizes, if individual clumps were resolved into sub-clumps. However, given the current resolution limit, no such substructure is detected. Explicitly, F17 identify the brightest peaks (relative to a smoothing mask) as clumps. They then fit these clumps with a 2D-Gaussian profile and compute an effective radius as the geometric mean of the major and minor half-axes. Finally, they take twice the average of all clumps within the galaxy as $r_{\rm clump, F17}$ due to which we compare their values to $2r_{\rm clump}$.
The errors on $r_{\rm clump}$ are computed by adding in quadrature the uncertainty due to \emph{Sample variance}, \emph{deblurring} and \emph{image noise}. \emph{Sample variance} originates from the fact that we observe only one instance of the galaxy. \emph{Deblurring} is the process of removing the contribution of the PSF from $r_{\rm clump}$ and hence the uncertainty scales in proportion to width of PSF compared to $r_{\rm clump}$. Finally, \emph{image noise} is due to the noise present in H$\alpha$ map. In A17, we explored these three sources of uncertainty in detail and here use the tabulated values to estimate the uncertainty in clump size of each galaxy. The clump sizes have been corrected for the PSF.
Figure \ref{fig:all_points} is a reproduction of the F17 Figure 2 (left) and compares the clump sizes measured using the w2PF to the theoretical model of F17. This VDI model assumes a marginally stable disk, in the sense of an average Toomre \citep{Toomre1964ApJ...139.1217T} parameter $Q\approx1$. A simple calculation then results in the prediction that the clump-to-disk scale ratio is proportional to the gas dispersion-to-rotational velocity ratio,
\begin{equation}
\frac{r_{\rm clump}}{r_{\rm disk}} = a\frac{\sigma}{V},
\end{equation}
where we adopt the definition of A17 that $r_{\rm clump}$ is the Gaussian clump size of eq.~(\ref{eq:clump=peak}) and $r_{\rm disk}$ is the effective radius. (Note that F17 define both values a factor 2 higher, leaving their ratio unchanged.) The proportionality factor $a$ depends on the shape of the rotation curve and is expected to vary between $1/3$ (Keplerian potential) and $\sqrt{2}/3$ (isothermal potential). The allowed range between these two proportionality factors is shown as grey shading in Figure \ref{fig:all_points}. The measurements are consistent with this model, except in the case of the two merging systems (open circles). This confirms the findings of F17 that the clump size scaling relation can help distinguish between major mergers and other scenarios of clump formation.
Figure \ref{fig:all_points} also shows the local main sequence galaxy NGC 5194 (whirlpool galaxy, M51a) in H$\alpha$ and $I$-band maps obtained from the Advanced Camera for
Surveys on board the HST \citep{Mutchler2005AAS...206.1307M}. Using the measurements of $r_{\rm disk}$, $\sigma_{\rm gas}$ and $V$ from \citet{Leroy2008AJ....136.2782L} we notice that NGC 5194 lies below the F17 VDI scaling relation. This is in agreement with \citet{Leroy2008AJ....136.2782L} who find a median value of $Q\approx2-3$ indicating a stable disk (except for in the dense regions of the spiral arms). The new clump size measurements validate the scaling relations presented by F17 as a way of differentiating between clump formation scenarios.
Open squares in Figure \ref{fig:all_points} denote upper limits for three systems where the clumps are too small for a reliable size determination, in the sense that the peak position of the w2PCF is consistent with the standard deviation of the PSF. The lowest of these points is the control galaxy A04-3 shown in Figure~\ref{fig:dynamo_galaxies}~(a), which by choice does not exhibit large clumps. Two of upper limits seem to lie somewhat below the relation. This might be explained by the fact that these systems are, in fact, rather stable disks ($Q>1$), similarly to NGC 5194.
We also compare the measured clump sizes to the disk thickness estimated in previous studies for three galaxies within our study. The power spectrum, Fourier equivalent of the 2PCF, has been widely used as an indicator of the disk thickness (\citealp{Elmegreen2001ApJ...548..749E}, \citealp{Combes2012A&A...539A..67C}). The fractal nature of the 2D galaxy structure gives a different power law as compared to a 3D structure. As the exponent changes at the transition between 2D and 3D behaviour the correlation function is likely to give a turning point at the associated scale height. We have attempted to mitigate this effect by taking a non-flat $R$-field unlike prior studies, however, the estimated disk thickness changes depending on the wavelength band used to observe the galaxy\citep{Elmegreen2013ApJ...774...86E}. To ensure we are not simply measuring the thickness parameter we compare our $2r_{\rm clump}$ measurements to scale height values determined by previous studies for galaxies G04-1, G20-2 and NGC5194. The clump sizes of $\{590,614,58\}$ (in parsecs) only match one scale height measurement $\{131,562,200\}$ \citep{Bassett2014MNRAS.442.3206B, Pety2013ApJ...779...43P} for the galaxy G20-2 but differ significantly for other two galaxies including the most resolved local galaxy. Hence, we find it unlikely that the w2PF turns over at the scale height of the galaxy.
\begin{figure}
\includegraphics[width=1\columnwidth]{figure_david.pdf}
\caption{Relationship between the clump-to-disk size and velocity dispersion-to-rotation velocity ratios in the DYNAMO-HST sample (red), NGC 5194 (blue) and the simulated galaxy (black). In the case of disk galaxies the average clump properties fall within the maximum and minimum allowed F17 scaling relation (shaded region) while the merging systems deviate significantly from this region due to the large Toomre parameter. This agreement is also seen in simulations regardless of the feedback model.}
\label{fig:all_points}
\end{figure}
\begin{figure*}
\includegraphics[width=1\textwidth]{figure_sim.pdf}
\caption{Simulated Milky Way-like galaxy at $z=2$ with four different feedback models, described in Section~\ref{subsec:fb}. Upper row: false-color face-on galaxy images, showing the stellar surface density (red), star formation rate surface density (green) and cold gas (blue). Bottom row: weighted 2-point correlation functions as a function of scale $r$. Peak values $r_{\rm peak}$ values are shown as dashed lines.}
\label{fig:sim_galaxies}
\end{figure*}
\section{Clump-scalings in simulated galaxies}\label{sec:simulations}
The theoretical scaling relation for marginally stable ($Q\approx1$) disks shown as grey region in Figure \ref{fig:all_points} relies on a simple calculation that neglects local asymmetries, complex accretion dynamics and stellar feedback. To test whether the relation still applies in the presence of more complex processes, we now consider a zoom-simulation of a galaxy at $z=2$, near its peak star-formation, in a cosmological context.
The simulated galaxy is a main-sequence object that ends up in a $10^{12}\rm M_{\odot}$ halo at $z=0$. This is likely less massive than the descendants of clumpy systems typically studied at higher redshifts, however we do not expect this difference to affect the physics analysed in this study. The galaxy is simulated four times using four different feedback models, including a no-feedback model. All four runs are re-simulations of the galaxy g8.26e11 from the NIHAO simulation project using identical initial conditions and the same cosmological environment (see \citealp{Wang2015MNRAS.454...83W}). The new runs use the updated hydrodynamics code \texttt{GASOLINE2} \citep{Wadsley2017MNRAS.471.2357W}, which improves on the original NIHAO galaxy that used \texttt{GASOLINE} \citep{Wadsley2004NewA....9..137W}. The runs use a flat $\Lambda$CDM cosmology with \citet{Planck2014A&A...571A..16P} parameters: Hubble parameter $H_{0}=67.1\ {\rm km\ s^{-1}}$, matter density $\Omega_{\rm M}=0.3175$, dark energy density $\Omega_{\rm \Lambda}=0.6824$, radiation density $\Omega_{\rm r}=0.00008$, baryon density $\Omega_{\rm b}=0.0490$, power spectrum normalization $\sigma_{8} = 0.8344$ and power spectrum slope $n_{\rm s} = 0.9624$.
Simulated with the standard NIHAO feedback, this galaxy has global properties similar to those of the Milky Way, with a final dynamical mass at $z=0$ of about $10^{12} M_{\odot}$. At $z=2$, this galaxy has a dynamical mass of $5\cdot10^{11} M_{\odot}$ and a stellar+cold gas mass of $5\cdot10^{10} M_{\odot}$ to $10^{11} M_{\odot}$, depending on the feedback model. The galaxy is simulated with $10^{6}$ dark matter+baryonic particles. The different feedback models and their results are discussed in the following section.
\subsection{Feedback models}\label{subsec:fb}
The first feedback model we consider is purely thermal resulting from the blastwave of stars within the mass range of $8 M_{\odot}< M_{\rm star} < 40 M_{\odot} $ undergoing a core-collapse supernova (SN) explosion. This \emph{Sedov Feedback} is implemented using the formalism of \citet{Stinson2006MNRAS.373.1074S}, which ensures that the energy and metals are ejected by the wave with cooling turned off for the particles within the blast radius. However, the high-density gas in the vicinity of the blast radius is allowed to cool and generally radiates the energy away efficiently. The resulting galaxy, shown in Figure \ref{fig:sim_galaxies} (a), therefore exhibits a lack of star formation in the outer regions.
The fiducial model used in NIHAO simulations employs the Early Stellar Feedback (ESF) model, originally explored by \citet{Stinson2013MNRAS.428..129S}, in addition to the \emph{Sedov Feedback} model. The ESF model incorporates the feedback mechanism due to the radiation of pre-SN massive stellar population, which adds a pathway for young massive stars to provide an ionizing source to the surrounding media and hence release energy into the ISM. Typically, an O-type star releases about $\sim 2\times10^{50}$ erg of energy per $M_{\odot}$ during the few $\rm Myrs$ between formation and SN explosion. This is comparable to the energy released by the SN itself. The fraction of the flux emitted in the ionizing UV was taken as $10\%$ by \citet{Stinson2013MNRAS.428..129S}. However, in our study we increase this stellar feedback efficiency to $\epsilon_{ESF} = 13\%$ to ensure better agreement with the mean stellar-to-halo mass relation derived from abundance matching \citep{Behroozi2013ApJ...770...57B}. Radiative cooling is allowed in this process. Figure \ref{fig:sim_galaxies} (b) shows the result of the NIHAO feedback model with increased star formation in the outskirts leading to a more realistic galaxy.
The third feedback scheme used in our analysis treats the evolution of clustered young stellar population as \emph{Superbubbles} wherein the associated structure is multi-phased with the feedback energy in hot phase being thermal while the cold expanding shell contains kinetic energy. Numerical simulations of the early stages of the superbubble are resolution-dependent and therefore computed using the analytical formalism of \citet{Keller2014MNRAS.442.3013K}, which employs thermal conduction to smoothly transition between the phases and hence provide a resolution insensitive result. We modify the \texttt{GASOLINE} code as per their suggestions to implement this technique in the zoom-in isolated galaxy simulation. Figure \ref{fig:sim_galaxies} (c) shows that this feedback model enhances star formation in the outer regions of the galaxy.
Finally, we also evolve the galaxy in the absence of a feedback mechanism (Figure \ref{fig:sim_galaxies} (d)) to quantify the extent to which turning on a feedback mechanisms affects the clump physics.
\subsection{Results}
The galaxy simulations evolved using the four feedback models exhibit vastly different morphologies as shown in Figure \ref{fig:sim_galaxies}. In particular, in the absence of feedback, we run into the classical `angular momentum catastrophe' where the stellar disk (red) is too small and bulgy. Interestingly, the no-feedback model still produces an extended cold gas disk (blue), but its mass is negligible compared to the stellar mass of this galaxy ($9\%$), as well as compared to the cold gas of the other galaxies ($\sim20\%$).
To measure the clump-sizes of these galaxies, we treat them in a similar way to the observations: Each galaxy is projected (face-on) onto a two-dimensional grid with $700\times700$ cells. As detailed in Section~2, stellar particles younger than $10~\rm Myr$ are taken to represent the star-formation rate surface density (green channel in Figure~\ref{fig:sim_galaxies}), while all stellar particles are used for the normalising global stellar surface density (red channel). For reference, the cold gas ($T<10^4\rm~K$) is shown in the blue channel; hence regions where all three components are abundant appear white. No radiative transfer is accounted for in producing the images. The clump size is then determined using the w2PF of Section~2. The clump size does not depend on the number of grid cells (resolution) as long as the cells are smaller than the clumps. The only source of uncertainty applied to $r_{\rm clump}$ is due to \emph{Sample variance}.
To estimate velocity dispersion $\sigma$ we use the standard deviation of the line-of-sight (vertical) velocity of gas particles as would be observed in a real observation. We find this dispersion to deviate no more than 30\% from the radial velocity dispersion in all the runs. The maximum rotation velocity $V$ and the half mass radius are straightforward to compute and have negligible errors.
Figure \ref{fig:all_points} shows that the scaling relation of F17 holds for all feedback models. However, the galaxy can move along this relation depending on the feedback. The largest jump occurs in the case of no feedback, where $r_{\rm clump}/r_{\rm disk}$ increases by a factor of $\sim 2$. This is mostly because of the disk being too small, however the other parameters compensate this change, such the galaxy falls onto the F17 scaling relation, i.e.\ back onto the Jeans' length prediction. As the intensity of feedback increases the galaxy resembles the observed turbulent disks in morphology and lies close to the DYNAMO-HST sample. This indicates that the simple scaling model of F17 is a useful tool to diagnose \emph{in-situ} clump formation via VDIs.
\section{Discussion and Conclusion}\label{sec:discussion}
In this paper we used the two-point statistics of A17 to estimate the characteristic size of star-forming clumps in the DYNAMO-HST galaxies and isolated galaxy simulations with four feedback models. We found that the estimated clump sizes are in good agreement with the previous study of F17, which uses a more subjective clump-by-clump analysis to infer an average clump size (which would result in very different sizes if higher-resolution images were available). It follows that the updated clump sizes (measured using the two-point statistics) remain on the scaling relation displayed in Figure~\ref{fig:all_points}. This scaling relation is therefore robust under a more objective clump size determination, which would remain constant under increasing spatial resolution revealing increasing levels of fragmented substructure.
Secondly, using a zoom-in simulation of a single Milky Way-like galaxy with four different feedback models, we verified that the clump size scaling relation of F17 remains valid in the presence of realistic galaxy formation physics. Interestingly, the relation holds regardless of the feedback model. This finding aligns with the results of \citet{Hopkins2012MNRAS.427..968H}, who show that at low redshift, in the absence of mergers, the global Toomre parameter of isolated Milky Way-type galaxies is self-regulated and independent of the underlying microphysics.
An important ramification of the simulations presented in Figure~\ref{fig:sim_galaxies} is that, while all feedback models satisfy the basic Toomre model visualized in Figure~\ref{fig:all_points}, the supra-clump structure (e.g.~total number of clumps, their physical sizes and spatial extent) of these galaxies depends enormously on the feedback model. A more in-depth analysis of how these properties depend explicitly on the radiation pressure \citep{Mandelker2017MNRAS.464..635M} drew comparable conclusions. Similarly, a direct comparison of the global clump patterns produced by blastwave (Sedov) versus Superbubble feedback \citep{Mayer2016ApJ...830L..13M} predicts easily observable differences between these models in the macroscopic distribution of clumps.
Returning to the interesting finding that the $r_{\rm clump}/r_{\rm disk}$--$\sigma/V$ relation is almost universal without a strong dependence on the feedback model, we caution that this result is only based on simulations of a single halo. It would be interesting to vastly expand these simulations to cover a wide parameter space, especially a larger range of halo masses, merger scenarios and redshifts.
Overall, this work emphasizes the usefulness of the w2PF (A17) to measure clump sizes in observed and simulated datasets and it demonstrates the power of the clump size scaling relation of F17 to diagnose \emph{in-situ} clump formation via VDIs. This parallels recent developments on spatial correlations of star-forming disks on scales larger than clumps \citep{Combes2012A&A...539A..67C,Hopkins2012MNRAS.423.2016H,Grasha2017}, as well as within individual clumps \citep{Guszejnov2016MNRAS.459....9G}. Spatial correlations can therefore be regarded as an essential modern tool for studying the physics of star-forming disks.
\section*{Acknowledgements}
LW thanks Ben Keller for his help in setting up the Superbubble feedback model. The simulations were supported by high performance computing project pawsey0201 and was carried out on Magnus cluster at the Pawsey computing centre at Perth. DBF, DO and KG acknowledge support from Australian Research Council grants DP130101460 and DP160102235. DBF acknowledges support from an Australian Research Council Future Fellowship
(FT170100376) funded by the Australian Government. The Hubble Space Telescope data in this program are drawn from the HST program PID 12977 (PI Damjanov). DO thanks Chris Power for insightful discussions.
|
{
"timestamp": "2019-03-01T02:16:57",
"yymm": "1902",
"arxiv_id": "1902.11034",
"language": "en",
"url": "https://arxiv.org/abs/1902.11034"
}
|
\section{The Gross-Neveu model and its phase diagram in the limit of infinitely many fermion flavors}
Exploring the phase diagram of QCD using lattice computations is currently restricted to small chemical potential, because of the QCD sign problem (see e.g.\ \cite{Philipsen:2010gj,Aarts:2015tyj} and references therein). There are, however, several QCD-inspired models, e.g.\ the Gross-Neveu (GN) model \cite{Gross:1974jv}, which are technically simpler to treat, and which share certain symmetries with QCD. Studies of such models might, thus, provide insights concerning the phase diagram of strongly interacting matter. A notable feature of the GN model in 1+1 dimensions in the limit of infinitely many fermion flavors is the existence of a so-called inhomogeneous phase, where the chiral order parameter is not a constant, but spatially oscillating \cite{Thies:2003kk,Schnetz:2004vr} (for a review on inhomogeneous condensates and phases see \cite{Buballa:2014tba}). In this work we perform a lattice field theory study of the 1+1 dimensional GN model at \emph{finite number} of fermion flavors $N_{\text{f}}$, to explore, whether inhomogeneous phases also exist at finite $N_{\text{f}}$.
The Euclidean action of the GN model is
\begin{align}
\label{eq:action}
S = \int \mathrm{d}^2x \, \bigg( \sum_{n=1}^{N_{\text{f}}} \bar{\psi}_n \Big(\gamma^0(\partial_0 + \mu) + \gamma^1 \partial_1 \Big) \psi_n - \frac{\lambda}{2 N_{\text{f}}}\bigg(\sum_{n=1}^{N_{\text{f}}} \bar{\psi}_n \psi_n\bigg)^2 \bigg) ,
\end{align}
where $\psi$ denotes a fermionic field with $N_{\text{f}}$ flavors and $\mu$ is the chemical potential. One can get rid of the four-fermion interaction by introducing a scalar field $\sigma$, which leads to the following partition function:
\begin{align}
\label{EQN001}
& Z = \int D\sigma \, \exp{\underbrace{\bigg(-N_{\text{f}} \bigg(\frac{1}{2 \lambda} \int \mathrm{d}^2x \, \sigma^2 - \ln\big(\det\big((\partial_0 + \mu) \gamma_0 + \partial_1 \gamma_1 + \sigma\big)\big)\bigg)\bigg)}_{S_\textrm{eff}}}.
\end{align}
The effective action has a discrete chiral symmetry, $S_\textrm{eff}[+\sigma] = S_\textrm{eff}[-\sigma]$, where $\langle \sigma \rangle \propto \langle \sum_n \bar{\psi}_n \psi_n \rangle$ represents the chiral condensate and indicates, whether the symmetry is spontaneously broken.
The phase diagram of the 1+1 dimensional GN model in the limit \mbox{$N_{\text{f}} \rightarrow \infty$} has been calculated in \cite{Thies:2003kk,Schnetz:2004vr}. There are three phases (see Fig.\ \ref{Fig:anaphase}):
\begin{itemize}
\item a \textit{chirally symmetric phase}, where $\langle \sigma \rangle = 0$;
\item a \textit{homogeneously broken phase}, where $\langle \sigma \rangle = \text{const} \neq 0$;
\item an \textit{inhomogeneous phase}, where $\langle \sigma \rangle$ is a spatially oscillating function.
\end{itemize}
In the inhomogeneous phase $\langle \sigma \rangle$ exhibits a periodic kink-antikink structure close to the phase boundary to the homogeneously broken phase and gradually changes into a $\sin$-like structure for increasing $\mu$.
\begin{figure}[htb]
\centering
\includegraphics[width=7cm]{figures/ana_phase.pdf}
\caption{\label{Fig:anaphase}Phase diagram of the GN model in the large-$N_{\text{f}}$ limit (see \cite{Thies:2003kk,Schnetz:2004vr}).}
\end{figure}
\section{The phase diagram at finite number of fermion flavors}
We perform lattice Monte Carlo simulations of the 1+1 dimensional GN model defined in eq.\ (\ref{EQN001}) at finite $N_{\text{f}} \in \{ 8 , 16 , 24 , 32 , 48 \}$. We use two different discretizations of the fermionic determinant, naive fermions and SLAC fermions (see e.g.\ \cite{Wozar:2011gu}), which we consider to be an important cross check of our numerical results: the results obtained with the two discretizations agree within statiscal errors. We set the scale via the absolute value of the chiral condensate at chemical potential $\mu = 0$ and temperature $T = 0$, i.e.\ $\sigma_0 = \langle |\bar{\sigma} | \rangle_{\mu=0,T=0}$, where
\begin{align}
\bar{\sigma} = \frac{1}{V} \sum_{x,t} \sigma(x,t),
\end{align}
$V$ is the number of lattice sites and $\langle \ldots \rangle_{\mu,T}$ denotes the path integral expectation value at chemical potential $\mu$ and at temperature $T$, i.e.\ the average over the generated set of Monte Carlo field configurations. In other words, we express dimensionful quantities in units of $\sigma_0$, e.g.\ $\mu / \sigma_0$, $T / \sigma_0$.
$\langle |\bar{\sigma} | \rangle_{\mu,T}$ is also a suitable approximate order parameter to distinguish between a homogeneously broken phase on the one hand ($\langle |\bar{\sigma} | \rangle_{\mu,T} \neq 0$) and a restored or inhomogeneous phase on the other hand ($\langle |\bar{\sigma} | \rangle_{\mu,T} \approx 0$). Numerical results for $N_{\text{f}} = 8$ are shown in Fig.\ \ref{Fig:phase1}, left plot. A homogeneously broken phase is indicated by the yellow dots at small $\mu$ and small $T$, somewhat smaller, but in a similar region as for infinite $N_{\text{f}}$. Results from analogous computations for $N_{\text{f}} \in \{ 16, 24, 32, 48 \}$ restricted to $\mu = 0$ are shown in Fig.\ \ref{Fig:phase1}, right plot. When increasing $N_{\text{f}}$, the results approach the numerical result at infinite $N_{\text{f}}$ (the latter has been obtained using techniques developed and explained in \cite{deForcrand:2006zz,Wagner:2007he,Heinz:2015lua}).
\begin{figure}[htb]
\centering
\includegraphics[width=6cm]{figures/phasediagram_0.4080_sigma_single.pdf}
\includegraphics[width=6cm]{figures/szero_inv.pdf}
\caption{\label{Fig:phase1}\textbf{(left)}~$\langle |\bar{\sigma} | \rangle_{\mu,T} / \sigma_0$ as a function of $\mu/\sigma_0$ and $T/\sigma_0$ for $N_{\text{f}} = 8$. \textbf{(right)}~$\langle |\bar{\sigma} | \rangle_{\mu=0,T} / \sigma_0$ as a function of $T/\sigma_0$ and $\mu/\sigma_0=0$ for various $N_{\text{f}}$. }
\end{figure}
To check for the existence of an inhomogeneous phase at finite $N_{\text{f}}$, we compute the spatial correlation function of the chiral condensate $\langle C(x) \rangle_{\mu,T}$ and its Fourier transform $\langle \tilde{C}(k) \rangle_{\mu,T}$, where
\begin{align}
C(x) = \frac{1}{V} \sum_{y,t} \sum_x \sigma(y,t) \sigma(y+x,t) .
\end{align}
Both $\langle C(x) \rangle_{\mu,T}$ and $\langle \tilde{C}(k) \rangle_{\mu,T}$ are suited to distinguish the three phases we are expecting as illustrated by Fig.\ \ref{Fig:C}:
\begin{itemize}
\item \textit{Chirally symmetric phase:} $\langle C(x) \rangle_{\mu,T}$ quickly approaches $0.0$. The Fourier transform is a smooth function close to $0.0$ indicating a vanishing chiral condensate.
\item \textit{Homogeneously broken phase:} $\langle C(x) \rangle_{\mu,T}$ quickly approaches $\sigma_0^2$. The Fourier transform exhibits a pronounced peak at $k = 0$ representing the non-vanishing constant chiral condensate.
\item \textit{Inhomogeneous phase:} $\langle C(x) \rangle_{\mu,T}$ is an oscillating function. The Fourier transform exhibits a pronounced peak at $k \neq 0$ proportional to the inverse wave length of the chiral condensate.
\end{itemize}
Of particular interest are the plots at the bottom of Fig.\ \ref{Fig:C}, because they provide clear evidence for the existence of an inhomogeneous phase at finite $N_{\text{f}}$.
\begin{figure}[htb]
\centering
\includegraphics[width=6cm]{figures/C_hom.pdf}
\includegraphics[width=6cm]{figures/Ctilde_hom.pdf}
\includegraphics[width=6cm]{figures/C_inhom.pdf}
\includegraphics[width=6cm]{figures/Ctilde_inhom.pdf}
\caption{\label{Fig:C}$C(x)$ and $\tilde{C}(k)$ for $N_{\text{f}} = 8$. \textbf{(top)}~$\mu/\sigma_0 = 0$ and $T/\sigma_0 = 0.988$ (chirally symmetric phase) as well as $T/\sigma_0 = 0.082$ (homogeneously broken phase). \textbf{(bottom)}~$\mu/\sigma_0 \in \{ 0.5 , 0.7 , 1.0 \}$ and $T/\sigma_0 = 0.082$ (inhomogeneous phase).}
\end{figure}
To identify the boundary between the homogeneously broken phase and the inhomogeneous phase, we plot in Fig.\ \ref{Fig:kmax}
\begin{align}
k_\text{max} = \Big|\arg \max \Big(\langle \tilde{C}(k) \rangle_{\mu,T}\Big)\Big|
\end{align}
as a function of $\mu$ and $T$. The phase boundary is clearly visible at $\approx \mu/\sigma_0 \approx 0.45$ separating the blue points ($k_\text{max} \approx 0$, homogeneously broken phase) from the red points ($k_\text{max} \neq 0$, inhomogeneous phase).
\begin{figure}[htb]
\centering
\includegraphics[width=12.5cm]{figures/phasediagram_0.4080_kmax.pdf}
\caption{\label{Fig:kmax}$k_\text{max}/\sigma_0$ as a function of $\mu/\sigma_0$ and $T/\sigma_0$ for $N_{\text{f}} = 8$.}
\end{figure}
To exhibit the oscillations of the chiral condensate in the inhomogeneous phase in an even more direct way, we compute $\langle \sigma(x + \ensuremath{x_{\mathrm{shift}}},t) \rangle_{\mu,T}$. Here $\ensuremath{x_{\mathrm{shift}}}$ is the phase shift of the spatially oscillating chiral condensate $\sigma(x,t)$ determined individually for each Monte Carlo field configuration by a standard Fourier transform. In this way destructive interference is excluded, when averaging over the Monte Carlo field configurations. In Fig.\ \ref{Fig:shifted} we show $\langle \sigma(x + \ensuremath{x_{\mathrm{shift}}},t) \rangle_{\mu,T}$ at three different $(\mu,T)$. In the left plot (homogeneously broken phase) $\langle \sigma(x + \ensuremath{x_{\mathrm{shift}}},t) \rangle_{\mu,T}$ is almost constant, close to $\sigma_0$, while in the center plot and the right plot (inhomogeneous phase) spatial oscillations are clearly visible.
\begin{figure}[htb]
\centering
\includegraphics[width=4.0cm]{figures/sHom.pdf}
\includegraphics[width=4.0cm]{figures/sInhom2.pdf}
\includegraphics[width=4.0cm]{figures/sInhom3.pdf}
\caption{\label{Fig:shifted}$\langle \sigma(x + \ensuremath{x_{\mathrm{shift}}},t) \rangle_{\mu,T}$ as a function of $x/\sigma_0$ and $t/\sigma_0$ for $(\mu/\sigma_0,T/\sigma_0) = (0.0,0.038)$ (homogeneously broken phase, left plot) and $(\mu/\sigma_0,T/\sigma_0) \in \{ (0.5,0.038) , (0.7,0.038) \}$ (inhomogeneous phase, center and right plot). }
\end{figure}
\section*{Acknowledgements}
We thank M.\ P.\ Lombardo for very helpful discussions and suggestions.
We thank M.\ Ammon and F.\ Karbstein for interesting discussions
on variations of the Goldstone theorem.
We thank A.\ Königsstein and M.\ Steil for critical comments on this manuscript.
L.P.\ thanks the organizers of the ``Excited QCD 2019'' conference for the possibility to give this talk.
M.W.\ acknowledges support by the Heisenberg Programme of the DFG (German Research Foundation), grant WA 3000/3-1.
L.P.\ and M.W.\ acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR 211 ``Strong-interaction matter under extreme conditions'' -- project number 315477589 -- TRR 211.
This work was supported in part by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse.
Calculations on the FUCHS-CSC high-performance computer of the Frankfurt University and the local HPC cluster of the FS-University Jena were conducted for this research. We would like to thank HPC-Hessen, funded by the State Ministry of Higher Education, Research and the Arts, for programming advice.
|
{
"timestamp": "2019-03-01T02:17:49",
"yymm": "1902",
"arxiv_id": "1902.11066",
"language": "en",
"url": "https://arxiv.org/abs/1902.11066"
}
|
\section{Correlation Clustering on Complete Bipartite Graphs}
Let $(V = L \cup R, E)$ be a complete bipartite graph with $L$ and $R$ being the bipartition of the vertex set. In this section, we provide and analyze an algorithm for correlation clustering on complete graphs with an approximation guarantee of $5$ for minimizing the mistakes on one side of the bipartition (which without loss of generality will be $L$). The algorithm and analysis for complete bipartite graphs is very similar to the algorithm and analysis for complete graphs. At each step $t$ of our algorithm, we select a cluster center $w_t \in L$ and a cluster $C_t \subseteq (L \cup R)$ and remove it from the graph. This clustering step is repeated until all vertices in $L$ are part of some cluster. If there are any remaining vertices in $R$ which are unclustered, we put them in a single cluster.
Similar to the definition of $\Ball(w, \rho)$ in Section~\ref{sec:cor-clust-complete}, we define $\Ball_S(w, \rho) = \{u \in S : x_{uw} \leq \rho\}$. We select the cluster centers $w_t$ in step $t$ as follows. Let $V_t \subseteq V$ be the set of unclustered vertices at the start of step $t$. We redefine $L^S_t(w) = \sum_{u \in Ball_{V_t \cap S}(w, r)} r - x_{uw}$. We select $w_t$ as the vertex $w \in L$ that maximizes $L_t(w)$. We then select $Ball_{L \cup R}(w, 2r)$ as our cluster and repeat. A pseudocode for the above algorithm is provided in Figure~\ref{fig:Alg3}.
\begin{figure}
\notarxiv{\begin{center}}
\begin{algorithm
\openLP
\smallskip
\noindent \textbf{Input: } Optimal solution $x$ to the linear program (P).\notarxiv{\\}
\par\noindent \textbf{Output: } Clustering $\calC$.
\medskip
\begin{enumerate}
\item Let $V_0 = V$, $r = 1/5$, $t = 0$.
\item \textbf{while} ($V_t \cap L \neq \varnothing$)
\begin{itemize}
\item Find $w_t = \argmax\limits_{w \in L} L^R_t(w)$.
\item Create a cluster $C_t = \Ball_{L \cup R}(w_t,2r)\cap V_t$.
\item Set $V_{t+1} = V_t \setminus C_t$ and $t = t+1$.
\end{itemize}
\item Let $\calC_L = (C_0,\dots, C_{t-1})$.
\item \textbf{if} ($R \cap V_t \neq \emptyset$)
\begin{itemize}
\item Let $C_R = R \cap V_t$.
\end{itemize}
\item Return $\calC = \calC_L \cup \{C_R\}$.
\end{enumerate}
\notarxiv{\caption{Correlation Clustering on complete bipartite graphs}\label{alg:corelation-bipartite-complete}}
\end{algorithm}
\notarxiv{\end{center}}
\closeLP
\caption{Algorithm for Correlation Clustering on complete bipartite graphs.}\label{fig:Alg3}\label{alg:corelation-bipartite-complete}
\end{figure}
\subsection{Analysis}
In this section, we present an analysis of our algorithm.
\begin{theorem}
Algorithm 3 gives a $5$-approximation for Correlation Clustering on complete biparite graphs where disagreements are measured on only one side of the bipartition.
\end{theorem}
The proof of this theorem is almost identical to the proof of Theorem~\ref{thm:5-apx-main}. We define
$\lp{u,v}$, $\alg{u,v}$, $\prft{u,v}{t}$ for every edge $(u,v)$ and $\pft{u}$, $\prft{u}{t}$ for every vertex $u$
as in Section~\ref{sec:cor-clust-complete}. We then
show that for each vertex $u \in L$, we have $\pft{u} \geq 0$ and, therefore, the number of
disagreeing edges incident to $u$ is upper bounded by $5y(u)$:
$$ALG(u) = \smashoperator[r]{\sum_{v:(u,v) \in E}} \alg{u,v} \leq \frac{1}{r} \smashoperator[r]{\sum_{v:(u,v) \in E}} \lp{u,v} = 5y(u).$$
Thus, $\|ALG\|_q \leq 5 \|y\|_q$ for any $q\geq 1$. Consequently, the approximation ratio of the algorithm is at most $5$ for any norm $\ell_q$.
\begin{lemma}
For every $u\in L$, we have $\pft{u} \geq 0$.
\end{lemma}
As in Lemma~\ref{lem:pft}, we need to show that $\prft{u}{t}\geq 0$ for all $t$. Note that we only need
to consider $u\in V_t\cap L$ as $\prft{u}{t} = 0$ for $u\notin V_t$.
Consider a step $t$ of the algorithm and vertex $u\in V_t \cap L$. Let $w = w_t$ be the center of the cluster chosen at this step.
First, we show that since the diameter of the cluster $C_t$ is $4r$, for all negative edges $(u,v) \in E^-$ with $u,v \in C_t$, we can charge the cost of
disagreement to the edge itself, that is, $\prft{u,v}{t}$ is nonnegative for $(u,v)\in E^-$ (see Lemma~\ref{cl:neg-edge-profit-nenneg}). We then consider two cases:
$x_{uw}\in [0, r]\cup [3r,1]$ and $x_{uw}\in (r,3r]$.
The former case is fairly simple since disagreeing positive edges $(u,v)\in E^+$ (with $x_{uw}\in [0, r]\cup [3r,1]$) have a ``large'' LP cost. In Lemma~\ref{lem:0r} and Lemma~\ref{lem:r1},
we prove that the cost of disagreement can be charged to the edge itself and hence $\prft{u}{t} \geq 0$.
We then consider the latter case. Similarly to Lemma~\ref{lem:pft}, we split the profit at step $t$ for vertices $u$ with $x_{uw} \in (r, 3r]$
into the profit they get from edges $(u,v)$ with $v$ in $\Ball_R(w,r)\cap V_t$ and from edges with $v$ in $V_t \setminus \Ball_R(w,r)$. That is,
\begin{multline*}
\prft{u}{t} =\\= \underbrace{\sum_{v\in \Ball_R(w,r) \cap V_t}\prft{u,v}{t}}_{P_{high}(u)} + \underbrace{\sum_{v\in V_t\setminus \Ball_R(w,r)}\prft{u,v}{t}}_{P_{low}(u)}.
\end{multline*}
Denote the first term by $P_{high}(u)$ and the second term by $P_{low}(u)$. We show that $P_{low}(u)\geq -L^R_t(u)$ (see Lemma~\ref{lem:PLow-LtRu} )
and $P_{high}\geq L^R_t(w) = \sum_{v \in \Ball_R{w, r} \cap V_t} r - x_{vw}$ (see Lemma~\ref{lem:PHigh-LtRu} ) and
conclude that $\prft{u}{t} = P_{high}(u) + P_{low}(u)\geq L^R_t(w)-L^R_t(u)\geq 0$ since $L^R_t(w) = \max_{w'\in V_t} L^R_t(w') \geq L^R_t(u)$.
Consider $u$ such that $x_{uw} \in (r, 3r]$. First, we show that the profit we obtain from every edge $(u,v)$ with $v \in \Ball_R(w,r)$ is
at least $r - x_{vw}$, regardless of whether the edge is positive or negative.
\begin{claim}
If $x_{uw} \in (r,3r]$ and $v \in \Ball_R(w,r)\cap V_t$, then $\prft{u,v}{t}\geq r-x_{vw}$.
\end{claim}
\begin{proof}
First consider $u$ such that $x_{uw} \in (r, 2r]$. Note that $x_{uv} \geq x_{uw} - x_{vw} \geq r - x_{vw}$. Moreover, $x_{uv} \leq x_{uw} + x_{vw} \leq 2r + x_{vw}$. Thus, if $(u,v) \in E^+$, then $\prft{u,v}{t} \geq r - x_{vw}$. Otherwise, $\prft{u,v}{t} \geq (1 - 2r - x_{vw}) - r \geq 2r - x_{vw}$.
For $u \in (2r, 3r]$, note that $x_{uv} \geq x_{uw} - x_{vw} \geq 2r - x_{vw}$. Moreover, $x_{uv} \leq x_{uw} + x_{vw} \leq 3r + x_{vw}$.
Thus, if $(u,v) \in E^+$, then $\prft{u,v}{t} \geq (2r - x_{vw}) - r \geq r - x_{vw}$. Otherwise, $\prft{u,v}{t} \geq (1 - 3r - x_{vw}) \geq 2r - x_{vw}$.
\end{proof}
Using the above claim, we can sum up the profits from all vertices $v$ in $\Ball_R(w, r)$ and lower bound $P_{high}(u)$ as follows.
\begin{lemma}\label{lem:PHigh-LtRu}
If $x_{uw}\in (r,3r]$, then $P_{high}(u) \geq L^R_t(w)$.
\end{lemma}
\begin{proof}
By Claim , we have $\prft{u,v}{t}\geq r-x_{vw}$ for all $v\in R \cap V_t$. Thus,
\begin{align*}
P_{high}(u) &= \sum_{v\in \Ball_R(w,r)\cap V_t}\prft{u,v}{t}\\ &\geq \sum_{v\in \Ball_R(w,r)\cap V_t}r-x_{vw} = L^R_t(w).
\end{align*}
\end{proof}
We now lower bound $P_{low}(u)$. To this end. we estimate each term $\prft{u,v}{t}$ in the definition of $P_{low}$.
\begin{claim}\label{claim:lb-bp}
If $x_{uw} \in (r,3r]$ and $v \in V_t \setminus \Ball_R(w,r)$, then $\prft{u,v}{t}\geq \min(x_{uv} - r, 0)$.
\end{claim}
\begin{proof}
By Claim~\ref{cl:neg-edge-profit-nenneg}, if $(u,v)$ is a negative edge, then $\prft{u,v}{t} \geq 0$. The profit is $0$ if $x_{uv}\notin \Delta E_t$ (i.e., neither $u$ nor $v$ belong
to the new cluster). So let us assume that $(u,v)$ is a positive edge in $\Delta E_t$. Then, the profit obtained from $(u,v)$ is $x_{uv}$ if $(u,v)$ is in agreement
and $x_{uv} - r$ if $(u,v)$ is in disagreement. In any case, $\prft{u,v}{t} \geq x_{uv} - r \geq \min(x_{uv} - r, 0)$.
\end{proof}
Lemma~\ref{lem:PLow-LtRu} is an immediate corollary of Claim~\ref{claim:lb-bp}.
\begin{lemma}\label{lem:PLow-LtRu}
If $x_{uw}\in (r,3r]$, then $P_{low}(u) \geq -L^R_t(u)$.
\end{lemma}
\begin{proof}
By Claim~\ref{claim:prof-uv-lower-bound}, we have $\prft{u,v}{t}\geq \min(x_{uv} - r,0)$ for all $v\in V_t$. Thus,
\begin{align*}
P_{low}(u) &= \sum_{v\in V_t\setminus \Ball_R(w,r)}\prft{u,v}{t}\\
&\geq \sum_{v\in V_t\setminus \Ball_R(w,r)} \min(x_{uv} - r,0)\\
&\overset{a}{\geq} \;\;\;\;\;\sum_{v\in V_t} \min(x_{uv} - r,0) \\
&\overset{b}{=} \sum_{v\in \Ball_R(u,r) \cap V_t} x_{uv} - r \\
&= - L_t^R(u).
\end{align*}
Here we used that (a) all terms $\min(x_{uv} - r,0)$ are nonpositive, and (b) $\min(x_{uv} - r, 0) = 0$ if $v\notin \Ball(u,r)$.
\end{proof}
\section{Correlation Clustering on Complete Graphs}\label{sec:cor-clust-complete}
In this section, we present our algorithm for Correlation Clustering on complete graphs and its analysis. Our algorithm
achieves an approximation ratio of $5$ and is an improvement over the approximation ratio of $7$ by \citet*{CGS17}.
\subsection{The Algorithm}
Our algorithm is based on rounding an optimal solution to the convex relaxation~(P). We start with considering the entire vertex set of
unclustered vertices. At each step $t$ of the algorithm, we select a subset of vertices as a cluster $C_t$ and remove it
from unclustered vertices. Thus, each vertex is assigned to a cluster exactly once and is never removed from a cluster once it is assigned.
For each vertex $w \in V$, let $\Ball(w,\rho) = \{u \in V : x_{uw} \leq \rho\}$ be the set of vertices within a distance of $\rho$ from $w$.
For $r = 1/5$ the quantity $r - x_{uw}$ where $u \in Ball(w,r)$ represents the distance from $u$ to the boundary of the ball of
radius $1/5$ around $w$. Let $V_t \subseteq V$ be the set of unclustered vertices at step $t$, and define
$$L_t(w) = \sum_{u \in \Ball(w,r) \cap V_t} r - x_{uw}.$$
At each step $t$, we select the vertex $w_t$ that maximizes the quantity $L_t(w)$ over all unclustered vertices $w\in V_t$ and select the set $Ball(w_t,2r)$ as a cluster.
We repeat this step until all the nodes have been clustered. A pseudo-code for our algorithm is given in Figure~\ref{fig:Alg2}.
\begin{figure}
\notarxiv{\begin{center}}
\begin{algorithm
\openLP
\smallskip
\noindent \textbf{Input: } Optimal solution $x$ to the linear program (P).\notarxiv{\\}
\noindent \textbf{Output: } Clustering $\calC$.
\medskip
\begin{enumerate}
\item Let $V_0 = V$, $r = 1/5$, $t = 0$.
\item \textbf{while} ($V_t \neq \varnothing$)
\begin{itemize}
\item Find $w_t = \argmax\limits_{w \in V_t} L_t(w)$.
\item Create a cluster $C_t = \Ball(w_t,2r)\cap V_t$.
\item Set $V_{t+1} = V_t \setminus C_t$ and $t = t+1$.
\end{itemize}
\item Return $\calC = (C_0,\dots, C_{t-1})$.
\end{enumerate}
\end{algorithm}
\notarxiv{\end{center}}
\closeLP
\caption{Algorithm for Correlation Clustering on complete graphs.}\label{fig:Alg2}\label{alg:corelation-complete}
\end{figure}
\subsection{Analysis}
In this section, we present an analysis of our algorithm.
\begin{theorem}\label{thm:5-apx-main}
Algorithm 2 gives a $5$-approximation for Correlation Clustering on complete graphs.
\end{theorem}
For an edge $(u,v) \in E$, let $LP(u,v)$ be the LP cost of the edge $(u,v)$:
$\lp{u,v} = x_{uv} $ if $(u,v) \in E^+$ and $\lp{u,v} = 1 - x_{uv}$ if $(u,v) \in E^-$. Let $\alg{u,v} = \ONE( (u,v) \text{ is in disagreement )}$.
Define
$$\pft{u} = \sum_{(u,v) \in E} \lp{u,v} - r \sum_{(u,v) \in E} \alg{u,v},$$
where $r=1/5$. We show that for each vertex $u \in V$, we have $\pft{u} \geq 0$ (see Lemma~\ref{lem:pft} below) and,
therefore, the number of disagreeing edges incident to $u$ is upper bounded by $5y(u)$:
$$ALG(u) = \smashoperator[r]{\sum_{v:(u,v) \in E}} \alg{u,v} \leq \frac{1}{r} \smashoperator[r]{\sum_{v:(u,v) \in E}} \lp{u,v} = 5y(u).$$
Thus, $\|ALG\|_q \leq 5 \|y\|_q$ for any $q\geq 1$. Consequently, the approximation ratio of the algorithm is at most $5$ for any norm $\ell_q$.
\begin{lemma}\label{lem:pft}
For every $u\in V$, we have $\pft{u} \geq 0$.
\end{lemma}
At each step $t$ of the algorithm, we create a new cluster $C_t$ and remove it from the graph. We also remove all
edges with at least one endpoint in $C_t$. Denote this set of edges by
$$\Delta E_t=\{(u,v): u\in C_t \text{ or } v \in C_t\}.$$
Now let
\begin{multline*}
\prft{u,v}{t} = \begin{cases}\lp{u,v} - r \alg{u,v},& \text{if } (u,v)\in \Delta E\\0,&\text{otherwise}\end{cases}.
\end{multline*}
\begin{align}\label{eq:for-profit-u}
\prft{u}{t} &= \sum_{v\in V_t}\prft{u,v}{t}\\
\notag &= \smashoperator[r]{\sum_{(u,v) \in \Delta E_t}} \lp{u,v} - r \smashoperator[r]{\sum_{(u,v) \in \Delta E_t}}\alg{u,v}.
\end{align}
As all sets $\Delta E_t$ are disjoint, $\pft{u} = \sum_t \prft{u}{t}$. Thus, to prove Lemma~\ref{lem:pft},
it is sufficient to show that $\prft{u}{t}\geq 0$ for all $t$. Note that we only need to consider $u\in V_t$ as $\prft{u}{t} = 0$ for
$u\notin V_t$.
\iffalse
\fullversionOnly{
\begin{figure}
\centering
\input{image-5apx}
\caption{Illustration for Algorithm 2.}
\end{figure}}
\fi
Consider a step $t$ of the algorithm and vertex $u\in V_t$. Let $w = w_t$ be the center of the cluster chosen at this step.
First, we show that since the diameter of the cluster $C_t$ is $4r$, for all negative edges $(u,v) \in E^-$ with $u,v \in C_t$, we can charge the cost of
disagreement to the edge itself, that is, $\prft{u,v}{t}$ is nonnegative for $(u,v)\in E^-$ (see Lemma~\ref{cl:neg-edge-profit-nenneg}). We then consider two cases:
$x_{uw}\in [0, r]\cup [3r,1]$ and $x_{uw}\in (r,3r]$.
The former case is fairly simple since disagreeing positive edges $(u,v)\in E^+$ (with $x_{uw}\in [0, r]\cup [3r,1]$) have a ``large'' LP cost. In Lemma~\ref{lem:0r} and Lemma~\ref{lem:r1},
we prove that the cost of disagreement can be charged to the edge itself and hence $\prft{u}{t} \geq 0$.
We then consider the latter case. For vertices $u$ with $x_{uw} \in (r, 3r]$,
$\prft{u,v}{t}$ for some disagreeing positive edges $(u,v)$ might be negative. Thus, we split the profit at step $t$ for such vertices $u$ into the profit
they get from edges $(u,v)$ with $v$ in $\Ball(w,r)\cap V_t$ and from edges with $v$ in $V_t\setminus \Ball(w,r)$. That is,
\begin{multline*}
\prft{u}{t} =\\= \underbrace{\sum_{v\in \Ball(w,r)}\prft{u,v}{t}}_{P_{high}(u)} + \underbrace{\sum_{v\in V_t\setminus \Ball(w,r)}\prft{u,v}{t}}_{P_{low}(u)}.
\end{multline*}
Denote the first term by $P_{high}(u)$ and the second term by $P_{low}(u)$. We show that $P_{low}(u)\geq -L_t(u)$ (see Lemma~\ref{lem:PLow-Ltu})
and $P_{high}\geq L_t(w)$ (see Lemma~\ref{lem:PHigh-Ltw}) and
conclude that $\prft{u}{t} = P_{high}(u) + P_{low}(u)\geq L_t(w)-L_t(u)\geq 0$ since $L_t(w) = \max_{w'\in V_t} L_t(w') \geq L_t(u)$.
In the following claim, we show that we can charge the cost of disagreement of a negative edge to the edge itself.
\begin{claim}\label{cl:neg-edge-profit-nenneg}
For a negative edge $(u,v)\in E^-$, $\prft{u,v}{t}$ is always nonnegative.
\end{claim}
\begin{proof}
The only case when $(u,v)$ is in disagreement is when both $u$ and $v$ belong to the new cluster. In this
case, they lie in the ball of radius $2r$ around $w$ (and thus $x_{uw}, x_{vw} \leq 2r$). Thus the distance $x_{uv}$ between them is at most $4r$ (because $x_{uv} \leq x_{uw} + x_{vw} \leq 4r$). The LP cost of the edge $(u,v)$ is at least $LP(u,v) = 1 - x_{uv} \geq 1- 4r = r$. Thus, $ \prft{u,v}{t} = LP(u,v)-r ALG(u,v) = LP(u,v)- r \geq 0$.
\end{proof}
In Lemma~\ref{lem:0r} and Lemma~\ref{lem:r1}, we consider the case when $x_{uw} \in [0,r] \cup (3r, 1]$.
\begin{lemma}\label{lem:0r}
If $x_{uw}\leq r$, then $\prft{u,v}{t}\geq 0$ for all $v\in V_t$.
\end{lemma}
\begin{proof}
If $x_{uw}\in E^-$, then $\prft{u,v}{t}\geq 0$ by Claim~\ref{cl:neg-edge-profit-nenneg}. Assume that $x_{uw}\in E^+$. Since
$x_{uw}\leq r$, $u$ belongs to the cluster $C_t$. Thus, $(u,v)$ disagrees only if $v$ does not belong to that cluster.
In this case, $x_{wv}\geq 2r$ and by the triangle inequality $x_{uv}\geq x_{vw} - x_{uw}\geq r$. Therefore,
$\prft{u,v}{t} = x_{u,v}-r \geq 0$.
\end{proof}
\begin{lemma}\label{lem:r1}
If $x_{uw}\geq 3r$, then $\prft{u,v}{t}\geq 0$ for all $v \in V_t$.
\end{lemma}
\begin{proof}
As in the previous lemma, we can assume that $x_{uw}\in E^+$. If $x_{uw}\geq 3r$,
then $u$ does not belong to the new cluster $C_t$. Thus, $(u,v)$ disagrees only if $v$ belongs to $C_t$.
In this case, $x_{wv}\leq 2r$ and by the triangle inequality $x_{uv}\geq x_{uw} - x_{vw}\geq r$. Therefore,
$\prft{u,v}{t} = x_{u,v}-r \geq 0$.
\end{proof}
We next consider $u$ such that $x_{uw} \in (r, 3r]$. First, we show that the profit we obtain from every edge $(u,v)$ with $v \in \Ball(w,r)$ is
at least $r - x_{vw}$, regardless of whether the edge is positive or negative.
\begin{claim}\label{claim:prof-from-core-v}
If $x_{uw} \in (r,3r]$ and $v \in \Ball(w,r)\cap V_t$, then $\prft{u,v}{t}\geq r-x_{vw}$.
\end{claim}
\begin{proof}
First consider $u$ such that $x_{uw} \in (r, 2r]$. Note that $x_{uv} \geq x_{uw} - x_{vw} \geq r - x_{vw}$. Moreover, $x_{uv} \leq x_{uw} + x_{vw} \leq 2r + x_{vw}$. Thus, if $(u,v) \in E^+$, then $\prft{u,v}{t} \geq r - x_{vw}$. Otherwise, $\prft{u,v}{t} \geq (1 - 2r - x_{vw}) - r \geq 2r - x_{vw}$.
For $u \in (2r, 3r]$, note that $x_{uv} \geq x_{uw} - x_{vw} \geq 2r - x_{vw}$. Moreover, $x_{uv} \leq x_{uw} + x_{vw} \leq 3r + x_{vw}$.
Thus, if $(u,v) \in E^+$, then $\prft{u,v}{t} \geq (2r - x_{vw}) - r \geq r - x_{vw}$. Otherwise, $\prft{u,v}{t} \geq (1 - 3r - x_{vw}) \geq 2r - x_{vw}$.
\end{proof}
Using the above claim, we can sum up the profits from all vertices $v$ in $\Ball(w, r)$ and lower bound $P_{high}(u)$ as follows.
\begin{lemma}\label{lem:PHigh-Ltw}
If $x_{uw}\in (r,3r]$, then $P_{high}(u) \geq L_t(w)$.
\end{lemma}
\begin{proof}
By Claim~\ref{claim:prof-from-core-v}, we have $\prft{u,v}{t}\geq r-x_{vw}$ for all $v\in V_t$. Thus,
\begin{align*}
P_{high}(u) &= \sum_{v\in \Ball(w,r)\cap V_t}\prft{u,v}{t}\\ &\geq \sum_{v\in \Ball(w,r)\cap V_t}r-x_{vw} = L_t(w).
\end{align*}
\end{proof}
We now lower bound $P_{low}(u)$. To this end. we estimate each term $\prft{u,v}{t}$ in the definition of $P_{low}$.
\begin{claim}\label{claim:prof-uv-lower-bound}
If $x_{uw} \in (r,3r]$ and $v \in V_t \setminus \Ball(w,r)$, then $\prft{u,v}{t}\geq \min(x_{uv} - r, 0)$.
\end{claim}
\begin{proof}
By Claim~\ref{cl:neg-edge-profit-nenneg}, if $(u,v)$ is a negative edge, then $\prft{u,v}{t} \geq 0$. The profit is $0$ if $x_{uv}\notin \Delta E_t$ (i.e., neither $u$ nor $v$ belong
to the new cluster). So let us assume that $(u,v)$ is a positive edge in $\Delta E_t$. Then, the profit obtained from $(u,v)$ is $x_{uv}$ if $(u,v)$ is in agreement
and $x_{uv} - r$ if $(u,v)$ is in disagreement. In any case, $\prft{u,v}{t} \geq x_{uv} - r \geq \min(x_{uv} - r, 0)$.
\end{proof}
Lemma~\ref{lem:PLow-Ltu} is an immediate corollary of Claim~\ref{claim:prof-uv-lower-bound}.
\begin{lemma}\label{lem:PLow-Ltu}
If $x_{uw}\in (r,3r]$, then $P_{low}(u) \geq -L_t(u)$.
\end{lemma}
\begin{proof}
By Claim~\ref{claim:prof-uv-lower-bound}, we have $\prft{u,v}{t}\geq \min(x_{uv} - r,0)$ for all $v\in V_t$. Thus,
\begin{align*}
P_{low}(u) &= \sum_{v\in V_t\setminus \Ball(w,r)}\prft{u,v}{t}\\
&\geq \sum_{v\in V_t\setminus \Ball(w,r)} \min(x_{uv} - r,0)\\
&\overset{a}{\geq} \;\;\;\;\;\sum_{v\in V_t} \min(x_{uv} - r,0) \\
&\overset{b}{=} \sum_{v\in \Ball(u,r) \cap V_t} x_{uv} - r \\
&= - L(u).
\end{align*}
Here we used that (a) all terms $\min(x_{uv} - r,0)$ are nonpositive, and (b) $\min(x_{uv} - r, 0) = 0$ if $v\notin \Ball(u,r)$.
\end{proof}
\section{Correlation Clustering on Arbitrary Graphs}
In this section, we describe our algorithm for Correlation Clustering on arbitrary graphs. Our algorithm
relies on a procedure for partitioning arbitrary metric spaces into pieces of small diameter
which we describe first.
\input{padded-decomp}
\subsection{Approximation Algorithm}
We now show how to use our metric space partitioning scheme to obtain an approximation algorithm for Correlation Clustering.
\begin{theorem}
There exists a randomized polynomial-time $O(n^{\frac{q-1}{2q}}\log^{\frac{q+1}{2q}} n)$ approximation algorithm for Correlation Clustering with the $\ell_q$ objective ($q\geq 1$).
\end{theorem}
We remark that the same algorithm gives $O(\sqrt{n\log n})$ approximation for the $\ell_{\infty}$ norm. We omit the details in the conference version of the paper.
\begin{proof}
Our algorithm first finds the optimal solution $x,y,z$ to the convex relaxation (P) presented in Section~\ref{sec:lp}. Then, it defines a metric $d(u,v)= x_{uv}$ on
the vertices of the graph. Finally, it runs the metric space partitioning algorithm with $\Delta = 1/2$ from the previous section
(see Theorem~\ref{thm:part-metric-spaces}) and output the obtained partitioning $\calP$.
Let us analyze the performance of this algorithm. Denote the cost of the optimal solution $x,y,z$ by $LP$. We know that the cost of the optimal solution
$OPT$ is lower bounded by $LP$ (see Section~\ref{sec:lp} for details). By Theorem~\ref{thm:part-metric-spaces}, applied to the graph $G=(V,E^+)$
(note: we ignore negative edges for now),
\begin{multline}\label{eq:thm:part-metric-spaces:approx-alg}
\bE\Big[\|\cut(\cP, E^+)\|_q\Big] \leq \frac{C}{\Delta} n^{\frac{q-1}{2q}}\log^{\frac{q+1}{2q}} n \cdot \Big(\big(\sum_{u\in V} y_u^q\big)^{\frac{1}{q}} +
\\+ \big(\sum_{u\in V} z_u\big)^{\frac{1}{q}}\Big)\leq 4C n^{\frac{q-1}{2q}}\log^{\frac{q+1}{2q}} n \cdot LP.
\end{multline}
Recall that a positive edge is not in agreement if and only if it is cut. Hence, $\disagree(\cP,E^+,\varnothing) = \cut(\cP, E^+)$, and the bound above holds
for $\bE \|\disagree(\cP, E^+,\varnothing)\|_q $. By the triangle inequality,
$\bE\|\disagree(\cP, E^+,E^-)\|_q \leq \bE\|\disagree(\cP, E^+,\varnothing)\|_q + \bE\|\disagree(\cP, \varnothing, E^-)\|_q$.
Hence, to finish the proof, it remains to upper bound $\bE\|\disagree(\cP, \varnothing, E^-)\|_q$.
Observe that the diameter of every cluster returned by the algorithm is at most $\Delta = 1/2$. For all disagreeing
negative edges $(u,v)\in E^-$, we have $x_{uv}\leq 1/2$ and $1-x_{uv}\geq 1/2$. Thus, $\disagree_u(\cP, \varnothing, E^-)\leq 2y_u$ for
every $u$, and $\bE\|\disagree(\cP, \varnothing, E^-)\|_q\leq 2\|y\|_q\leq 2LP$.
This completes the proof.
\end{proof}
\section{Hardness of approximation}
In this section, we prove the following hardness result.
\begin{theorem}
It is NP-hard to approximate the min $\ell_\infty$ s-t cut problem within a factor of $2 - \varepsilon$ for every positive $\varepsilon$.
\end{theorem}
\begin{proof}
The proof follows a reduction from $3$SAT. We will describe a procedure that reduces every instance of a $3$CNF formula $\phi$ to a graph $G_\phi$ such that the minimum $\ell_\infty$ \textit{s-t} cut for $G_\phi$ has a certain value if and only if the formula $\phi$ is satisfiable.
\medskip
\noindent \textbf{Reduction from 3SAT:} Given a $3$CNF instance $\phi$ with $n$ literals and $m$ clauses, we describe a graph $G_\phi$ with $(2 + 4n + 5m)$ vertices
and $(6n + 8m)$ edges. We refer to the vertex and edge set of $G_\phi$ as $V(G_\phi)$ and $E(G_\phi)$. For every literal $x_i, i \in \{1,\dots, n\}$, we have four nodes,
$x^T_i$, $x^F_i$, $x^{\dagger}_i$ and $\br{x}_i^\dagger$. Additionally, we have a ``False'' and a ``True'' node. For every $i \in \{1,\dots,n\}$, we connect
``False'' with $x^F_i$ and ``True'' with $x^T_i$ using an infinite weight edge. Both $x^F_i$ and $x^T_i$ are connected to $x^{\dagger}_i$ and $\br{x}_i^\dagger$ using edges
of weight $1$.
For every clause $C$ in $\phi$, we will create a gadget in $G_\phi$ consisting of five nodes. We will refer to the subgraph induced by these nodes as $G_\phi[C]$. Let the
clause $C = (y_1 \lor y_2 \lor y_3)$. We have a node in the gadget for each $y_i, i \in \{1,2,3\}$, and two additional nodes $C_a$ and $C_b$. We connect $y_2$ and $y_3$ to $C_b$,
and $y_1$ and $C_b$ to $C_a$, all using unit weight edges.
We connect the gadget $G_\phi[C]$ for clause $C = (y_1 \lor y_2 \lor y_3)$ to the main graph as follows. For each $i \in \{1,2,3\}$, connect the vertex for the
literal $y_i$ to the vertex $y^\dagger_i$ with a unit weight edge. Finally connect the node $C_a$ to the ``True'' vertex using an infinite weight edge. An example of
a 3CNF formula $\phi$ and the corresponding $G_\phi$ is given in Figure~\ref{figure:hardness}.
\textbf{Fact 1.} Consider the gadget $G_\phi[C]$ for the clause $C = (y_1 \lor y_2 \lor y_3)$. If all three nodes $y_1, y_2$, and $y_3$ need to be disconnected
from $C_a$, then either $|\cut_{C_a}| = 2$ or $\cut_{C_b} = 2$. If at most two of the three nodes $y_1, y_2$ and $y_3$ need to be disconnected from $C_a$, then there
is a cut that separates those nodes from $C_a$ such that both $\cut_{C_a}$ and $\cut_{C_b}$ are at most $1$.
\begin{lemma}
Given a 3CNF formula $\phi$, consider the graph $G_\phi$ constructed according to the reduction described above. The formula $\phi$ is satisfiable
if and only if the minimum $\ell_\infty$ True-False cut $\cP$ for the graph $G_\phi$ has value $1$, that is, $||\cut_\cP||_\infty = 1$.
\end{lemma}
\begin{proof}
\textbf{3SAT $\Rightarrow$ minimum $\ell_\infty$ \textit{True-False cut}}: If the 3CNF formula $\phi$ is satisfiable, then the graph $G_\phi$ has a minimum $\ell_\infty$ \textit{s-t} cut of value exactly $1$. This can be seen as follows. Given a satisfying assignment $x^*$, we will construct a cut $E_\cP$ (and corresponding partition $\cP$) such that for every vertex $u \in V(G_\phi)$, $\cut_\cP(u) \leq 1$. For every $i \in \{1,\dots, n\}$, if $x^*_i$ is True, then include $(x^{\dagger}_i, x_i^F)$ and $(\br{x}_i^\dagger, x_i^T)$ as part of the cut $E_\cP$, else include $(x^{\dagger}_i, x_i^T)$ and $(\br{x}_i^\dagger, x_i^F)$ as part of the cut $E_\cP$. Note that this cuts exactly one edge incident to each vertex $x^{\dagger}_i, x_i^F, \br{x}_i^\dagger$ and $x_i^T$ for $i \in \{1,\dots, n\}$. Since $\phi$ has a satisfiable assignment, each clause $C$ has at least one literal which is True, and hence the node corresponding to this literal is not connected to the vertex False in $G_\phi - E_\cP$. Thus, each clause $C$ has at most two literals that are False, and thus there are at most two False-True paths that go through this gadget. From Fact 1, we can know that we can include edges from $E(G_\phi[C])$ in $E_\cP$ such that both $\cut_\cP(C_a)$ and $\cut_\cP(C_b)$ are at most $1$ and the False-True paths through this gadget are disconnected. Thus, cut $E_\cP$ disconnects True from False such that $||\cut_\cP(G_\phi)||_\infty = 1$.
\medskip
\textbf{minimum $\ell_\infty$ \textit{True-False cut} $\Rightarrow$ 3SAT}: Let $G_\phi$ be the graph constructed for the 3CNF formula $\phi$ such that there is a cut $E_\cP \subseteq E(G_\phi)$ (and corresponding partition $\cP$) such that $\cP$ separates True from False and $||\cut_\cP(G_\phi)||_\infty = 1$. We will construct a satisfying assignment $x^*$ from the formula $\phi$. Since $\cut_\cP(u) \leq 1$ for every $u \in V(G_\phi)$, none of the $(True, x^T_i)$, $(x^F_i, False)$ edges are part of the cut $\cP$ for $i \in \{1,\dots, n\}$. In order for True to be separated from False, either the edges $(x^{\dagger}_i, x_i^F)$ and $(\br{x}_i^\dagger, x_i^T)$ are part of the cut $E_\cP$, or the edges $(x^{\dagger}_i, x_i^T)$ and $(\br{x}_i^\dagger, x_i^F)$ are part of the cut $E_\cP$. This gives us our assignment; for each $i \in \{1,\dots, n\}$, if $(x^T_i, x^{\dagger}_i) \in E \setminus E_\cP$, then assign $x^*_i$ as True and $\br{x}^*$ as False. Otherwise $(x^F_i, x^{\dagger}_i) \in E \setminus E_\cP$, so assign $x^*_i$ as False and $\br{x}^*$ as True. Now, we show that $x^*$ is a satisfiable assignment for $\phi$. To see this, note that for each clause $C$, there exists at least one literal $y_i$ such that the node corresponding to $y_i$ is still connected to $C_a$. As the cut $E_\cP$ separates True and False, $(y^\dagger_i, y^T_i) \in E \setminus E(G_\phi)$ and hence $y^*_i = $ True. Thus, the assignment $x^*$ is satisfiable for $\phi$.
\end{proof}
Thus, we can conclude Theorem 5.1 from the reduction procedure provided and Lemma 5.2.
\end{proof}
\begin{figure*}
\centering
\input{cartoon}
\caption{$G_\phi$ for the 3CNF formula $\phi = (x_1 \lor \br{x}_2 \lor x_3) \land (x_2 \lor \br{x}_4 \lor \br{x}_5) \land (\br{x}_1 \lor x_4\lor x_5)$.}\label{figure:hardness}
\end{figure*}
\section{Integrality gap}
In this section, we present an integrality gap example for the convex program (P). We describe an instance of the $\ell_q$ $s-t$ cut problem on $\Theta(n)$ vertices that has an integrality gap of $\Omega(n^{\frac{1}{2} - \frac{1}{2q}})$. In our integrality gap example, we describe a layered graph with $\Theta(n^\frac{1}{2})$ layers, with each layer consisting of a complete bipartite graph on $\Theta(n^\frac{1}{2})$ vertices. Between each layer $i$ and $i+1$, there is a terminal $s_i$ which connects these two layers. Finally, the terminals $s$ and $t$ are located at opposite ends of this layered graph. We will observe that for any integral cut separating $s$ and $t$, there will be at least one vertex such that a large fraction of the edges incident to it are cut. We will show that there is a corresponding fractional solution that is cheaper compared to any integral cut as the fractional solution can ``spread'' the cut equally across the layers, thus not penalizing any individual layer too harshly. In doing so, we will prove the following theorem,
\begin{theorem}
The integrality gap for the convex relaxation (P) is $\Omega(n^{\frac{1}{2}-\frac{1}{2q}})$.
\end{theorem}
\begin{proof}
We now give a more formal description of the layered graph discussed above. The construction has two parameters $a$ and $b$, so we will call such a graph $G_{a,b}$. The graph consists of $b$ layers with each layer consisting of the complete bipartite graph $K_{a,a}$. We refer to layer $i$ of the graph as $G^i_{a,b}$ and refer to the left and right hand of the bipartition as $L(G^i_{a,b})$ and $R(G^i_{a,b})$ respectively. In addition to these layers, the graph consists of $b+1$ terminals $\{s, t, s_1, \ldots, s_{b-1}\}$ (we will refer to $s$ as $s_0$ and $t$ as $s_b$ interchangeably). For each $i \in \{1, \ldots, b-1\}$, the vertex $s_i$ is connected to all the vertices in $R(G^i_{a,b})$ and $L(G^{i+1}_{a,b})$. Finally, $s$ is connected to all the vertices in $L(G^1_{a,b})$ and $t$ is connected to all the vertices in $R(G^b_{a,b})$.
Consider any integral cut separating $s$ and $t$ in the graph $G_{a,b}$. Any such cut must disconnect at least one pair of consecutive terminals (if all pairs of consecutive terminals are connected, then $s$ is still connected to $t$). Thus let $j \in \{0, 1, \ldots, b\}$ be such that $s_{j-1}$ is disconnected from $s_{j}$ and consider the subgraph induced on $\{s_{j-1} \cup s_{j} \cup G^j_{a,b}\}$. We will show that this induced subgraph contains a vertex such that $\Omega(a^\frac{1}{2})$ of its incident edges are cut. Intuitively, since $s_{j-1}$ is separated from $s_j$, if the majority of the edges incident to $s_{j-1}$ and $s_j$ are not cut, then $s_{j-1}$ and $s_{j}$ have many neighbors in $L(G^{j}_{a,b})$ and $R(G^{j}_{a,b})$ respectively. As $G^{j}_{a,b}$ is highly connected, in order for $s_{j-1}$ to be separated from $s_j$, there must be a vertex in $G^j_{a,b}$ with many incident edges which are cut. If $cut(s_{j-1})$ or $cut(s_{j})$ is at least $a/2$, then we are done. Otherwise, $s_j$ is connected to at least $a/2$ vertices in $R(G^{j}_{a,b})$, so every $u$ adjacent to $s_{j-1}$ must have at least $a/2$ incident edges which are cut. Therefore, $OPT^q \geq \Omega(a^q)$.
We now present a fractional cut separating $s$ and $t$. If an edge $e$ connects $s_i$ to a vertex in $R(G^i_{a,b})$ for some $i \in \{1, \ldots, b\}$, set the length of the edge to be $1/b$; otherwise set the edge length to be $0$. We let $x_{uv}$ be the shortest path metric in this graph. It is easy to see that such a solution is feasible. We now analyze the quality of this solution.
For each $i \in \{1, \ldots, b\}$, we have $y_{s_i} = a/b$ and for each $u \in R(G^i_{a,b})$, we have $y_u = 1/b$. Thus
$$LP^q = ab\Big(\frac{1}{b}\Big)^q + b\Big(\frac{a}{b}\Big)^q.$$
If $b>a$, then
$$LP^q \leq ab\Big(\frac{1}{b}\Big) + b\Big(\frac{a}{b}\Big) = 2a$$
and if $b > a$, then
$$LP^q \leq ab\Big(\frac{1}{b}\Big) + b\Big(\frac{a}{b}\Big)^q \leq a^q\Big(a^{-(q-1)} + b^{-(q-1)}\Big).$$
Setting $a = b = \Omega({n^\frac{1}{2}})$ gives
$$\frac{OPT^q}{LP^q} = \Omega\Big(n^{\frac{q}{2}-\frac{1}{2}}\Big),$$
so the integrality gap is $\frac{OPT}{LP} = \Omega(n^{\frac{1}{2}-\frac{1}{2q}})$.
\end{proof}
\begin{figure*}
\centering
\input{image-intGap.tex}
\caption{Integrality gap example.}\label{fig:int-gap}
\end{figure*}
\section{Introduction}
A basic task in machine learning is that of clustering items based on similarity between them. This task can be elegantly captured by Correlation Clustering, a clustering framework first introduced by \citet*{BBC04}. In this model, we are given access to items and the \textit{similarity/dissimilarity} between them in the form of a graph $G$ on $n$ vertices. The edges of $G$ represent whether the items are \textit{similar} or \textit{dissimilar} and are labelled as (``$+$'') and (``$-$'') respectively. The goal is to produce a clustering that agrees with the labeling of the edges as much as possible, i.e., to group positive edges in the same cluster and place negative edges across different clusters (a positive edge that is present across clusters or a negative edge that is present within the same cluster is said to be in disagreement). The Correlation Clustering problem can be viewed as an agnostic learning problem, where we are given noisy examples and the task is to fit a hypothesis as best as possible to these examples. Co-reference resolution (see e.g., \citet*{CR01, CR02}), spam detection (see e.g., \citet{RFV07,BGL14}) and image segmentation (see e.g., \citet*{Wirth17}) are some of the application to which Correlation Clustering has been applied to in practice.
This task is made trivial if the labeling given is consistent (transitive): if $(u,v)$ and $(v,w)$ are similar, then $(u,w)$ is similar for
all vertices $u,v,w$ in $G$ (the connected components on similar edges would give an optimal clustering). Instead, it is assumed that the given labeling is inconsistent, i.e., it is possible that $(u,w)$ are dissimilar even though $(u,v)$ and $(v,w)$ are similar. For such a triplet $u,v,w$, every possible clustering incurs a disagreement on at least one edge and thus, no perfect clustering exists. The optimal clustering is the one which minimizes the disagreements. Moreover, as the number of clusters is not predefined, the optimal clustering can use anywhere from $1$ to $n$ clusters.
The classical metric that has been used in Correlation Clustering is that of minimizing the total number of edges in disagreement. Let the disagreements vector be an $n$ dimensional vector indexed by the vertices where the $v$-th coordinate equals the number of disagreements at $v$. Thus, minimizing the total number of disagreements is equivalent to minimizing the $\ell_1$ norm of the disagreements vector. \citet*{PM16} initiated the study of more local objectives in the Correlation Clustering framework. They focus on complete graphs and study the minimization of $\ell_q$ norms $(q \geq 1)$ of the disagreements vector. \citet*{CGS17} studied the problem of minimizing the $\ell_\infty$ norm of the disagreements vector
(also known as Min Max Correlation Clustering) for general graphs.
For higher values of $q$ (particularly $q=\infty$), a clustering optimized for minimizing the $\ell_q$ norm prioritizes reducing the
disagreements at vertices that are worst off. Thus, such metrics are very unforgiving in most cases as it is possible that in
the optimal clustering there is only one vertex with high disagreements while every other vertex has low disagreements. Thus, one is forced to infer the most pessimistic picture about the overall clustering. The $\ell_2$ norm is a solution to this tension between the $\ell_1$ and $\ell_\infty$ objectives. The $\ell_2$ norm of the disagreements vector takes into account the disagreements at each vertex while also penalizing the vertices with high disagreements more heavily. Thus, a clustering optimized for the minimum $\ell_2$ norm gives a more balanced clustering as it takes into consideration both the global and local picture.
\textbf{Our contributions. } In this paper, we provide positive and negative results for Correlation Clustering with the $\ell_q$ objective. We first study the problem of minimizing disagreements on arbitrary graphs. We present the first approximation algorithm minimizing any $\ell_q$ norm $(q \geq 1)$ of the disagreements vector.
\begin{theorem}\label{Main.Thm.}
There exists a polynomial time $O(n^{\frac{1}{2} - \frac{1}{2q}} \cdot \log^{\frac{1}{2} + \frac{1}{2q}} n)$ approximation algorithm for the minimum $\ell_q$ disagreements problem on general weighted graphs.
\end{theorem}
For the $\ell_2$ objective, the above algorithm leads to an approximation ratio of $\tilde{O}(n^{\nicefrac{1}{4}})$, thus providing the first known approximation ratio for optimizing the clustering for this version of the objective. Note that the above algorithm matches the best approximation guarantee of $O(\log n)$ for the classical objective of minimizing the $\ell_1$ norm of the disagreements vector. For the $\ell_\infty$ norm, our algorithm matches the guarantee of the algorithm by \citet*{CGS17} up to $\log$ factors. Fundamental combinatorial optimization problems like \textit{Multicut, Multiway Cut} and \textit{s-t Cut} can be framed as special cases of Correlation Clustering. Thus, Theorem \ref{Main.Thm.} leads
to the first known algorithms for \textit{Multicut, Multiway Cut} and \textit{s-t Cut} with the $\ell_q$ objective when
$q\neq 1$ and $q \neq \infty$. We can also
use the algorithm from Theorem~\ref{Main.Thm.} to obtain $O(n^{\frac{1}{2} - \frac{1}{2q}} \cdot \log^{\frac{1}{2} + \frac{1}{2q}} n)$
bi-criteria approximation for Min $k$-Balanced Partitioning with the $\ell_q$ objective (we omit details here).
Next, we study the case of complete graphs. For this case, we present an improved $5$ approximation algorithm for minimizing any $\ell_q$ norm $(q \geq 1)$ of the disagreements vector.
\begin{theorem}
There exists a polynomial time $5$ approximation algorithm for the minimum $\ell_q$ disagreements problem on complete graphs.
\end{theorem}
We also study the case of complete bipartite graphs where disagreements need to be bounded for only one side of the bipartition, and not the whole vertex set. We give an improved $5$ approximation algorithm for minimizing any $\ell_q$ norm $(q \geq 1)$ of the disagreements vector.
\begin{theorem}
There exists a polynomial time $5$ approximation algorithm for the minimum $\ell_q$ disagreements problem on complete bipartite graphs where disagreements are measured for only one side of the bipartition.
\end{theorem}
\confversionOnly{
Finally, in the full version of this paper, we present an integrality gap of $\Omega(n^{\frac{1}{2} - \frac{1}{2q}})$ and
prove a hardness of approximation of 2 for minimum $\ell_\infty$ $s-t$ cut.
}
\fullversionOnly{
Our algorithm for the minimum $\ell_q$ disagreements problem is based on rounding the natural convex programming relaxation for this problem. We show that our result is best possible according to this relaxation by providing an almost matching integrality gap. The integrality gap example we provide is for the minimum $\ell_q$ $s-t$ cut problem (a special case of correlation clustering) and show the following result.
\begin{theorem}
The natural convex programming relaxation for the minimum $\ell_q$ disagreements problem has an integrality gap of $\Omega(n^{\frac{1}{2} - \frac{1}{2q}})$ on arbitrary graphs.
\end{theorem}
Finally, we present a hardness of approximation result for minimum $\ell_\infty$ $s-t$ cut.
\begin{theorem}
There is no $\alpha$-approximation algorithm for the min $\ell_\infty$ \textit{s-t cut problem} for $\alpha<2$ unless P = NP.
\end{theorem}
}
\textbf{Previous work.} \citet*{BBC04} showed that it is NP-hard to find a clustering that minimizes the total disagreements, even on complete graphs. They give a constant-factor approximation algorithm to minimize disagreements and a PTAS to maximize agreements on complete graphs. For complete graphs, \citet*{ACN08} presented an elegant randomized algorithm with an approximation guarantee of $3$ to minimize total disagreements. They also gave a $2.5$ approximation algorithm based on LP rounding. This factor was improved to slightly less than $2.06$ by \citet*{CMSY15}. Since, the natural LP is known to have an integrality gap of $2$, the problem of optimizing the classical objective is almost settled with respect to the natural LP. For arbitrary graphs, the best known approximation ratio is $O(\log n)$ (see \citet*{CGW03, DEFI06}). Assuming the Unique Games Conjecture, there is no constant-factor approximation algorithm for Correlation Clustering is possible (see~\citet{CKKRS06}). \citet*{PM16} first studied Correlation Clustering with more local objectives. For minimizing $\ell_q$ $(q \geq 1)$ norms of the disagreements vector on complete graphs, their algorithm achieves an approximation guarantee of $48$. This was improved to $7$ by \citet*{CGS17}. \citet{CGS17} also studied the problem of minimizing the $\ell_\infty$ norm of the disagreements vector on general graphs. They showed that the natural LP/SDP has an integrality gap of $\nicefrac{n}{2}$ for this problem and provided a $O(\sqrt{n})$ approximation algorithm for minimum $\ell_\infty$ disagreements. \citet*{PM16} also initiated the study of minimizing the $\ell_q$ norm of the disagreements vector (for one side of the bipartition) on complete bipartite graphs. The presented a $10$ approximation algorithm for this problem, which was improved to $7$ by \citet*{CGS17}.
\section{Convex Relaxation}\label{sec:lp}
\begin{figure*}
\centering
\begin{equation*}
\begin{array}{ll@{}llr}
\text{minimize} & \displaystyle \max\Big(\|y\|_q, \big(\sum\limits_{u \in V} z_u\big)^{\frac{1}{q}}\Big) \tag{P}\\
\text{subject to}
& y_u=\displaystyle\sum_{v:(u,v)\in E^+} w_{uv} x_{uv} + \sum_{v:(u,v)\in E^-} w_{uv} (1 - x_{uv}) \quad &\text{for all } u \in V & \text{(P1)}\\
& z_u=\displaystyle\sum_{v:(u,v)\in E^+} w^q_{uv} x_{uv} + \sum_{v:(u,v)\in E^-} w^q_{uv} (1 - x_{uv}) \quad &\text{for all } u \in V& \text{(P2)}\\ \\
&x_{v_1v_2}+x_{v_2v_3} \geq x_{v_1v_3} \quad &\text{for all } v_1,v_2,v_3 \in V&\text{(P3)}\\
&x_{uv} = x_{vu}\quad &\text{for all } u,v \in V& \text{(P4)}\\
& x_{uv} \in [0,1] &\text{for all } u,v \in V & \text{(P5)}\\
\end{array}
\end{equation*}
\caption{Convex relaxation for Correlation Clustering with min $\ell_q$ objective for $q < \infty$.}\label{figure:LPRelaxation}
\end{figure*}
In both algorithms for arbitrary and complete graphs, we use a convex relaxation given in Figure~\ref{figure:LPRelaxation}. Our convex relaxation for
Correlation Clustering is fairly standard. It is similar to relaxations used in the papers by~\citet*{GVY96, DEFI06, CGW03}. For every pair
of vertices $u$ and $v$, we have a variable $x_{uv}$ that is equal to the distance between $u$ and $v$ in the ``multicut metric''.
Variables $x_{uv}$ satisfy the triangle inequality constraints~(P3). They are also symmetric~(P4) and $x_{uv}\in [0,1]$~(P5). Thus, the set of
vertices $V$ equipped with the distance function $d(u,v)= x_{uv}$ is a metric space.
Additionally, for every vertex $u\in V$, we have variables $y_u$ and $z_u$ (see constraints~(P1) and (P2)) that lower bound the number of disagreeing edges
incident to $u$. The objective of our convex program is to minimize $\max(\|y\|_q, (\sum_{u} z_u)^{\frac{1}{q}})$. Note that all constraints in
the program (P) are linear; however, the objective function of (P) is not convex as is. So in order to find the optimal solution, we raise
the objective function to the power of $q$ and find feasible $x,y,z$ that minimizes the objective $\max(\|y\|^q_q, \sum_{u} z_u)$.
Let us verify that program (P) is a relaxation for Correlation Clustering. Consider an arbitrary partitioning $\calP$ of $V$.
In the integral solution corresponding to $\calP$, we set $x_{uv} = 0$ if $u$ and $v$ are in the same cluster in $\calP$; and $x_{uv} = 1$ if $u$ and $v$ are
in different clusters in $\calP$. In this solution, distances $x_{uv}$ satisfy triangle inequality constraints~(P3) and
$x_{uv} = x_{vu}$ (P4). Observe that a positive edge $(u,v)\in E^+$ is in disagreement with $\calP$ if $x_{uv} = 1$; and a negative edge
$(u,v)\in E^-$ is in disagreement if $x_{uv} = 0$. Thus, in this integral solution, $y_u = \disagree_u(\calP, E^+,E^-)$ and
moreover, $z_u \leq y^q_u$. Therefore, in the integral solution corresponding to $\calP$, the objective function of (P) equals
$\|\disagree_u(\calP, E^+,E^-)\|_q$. Of course, the cost of the optimal fractional solution to the problem may be
less than the cost of the optimal integral solution. Thus, (P) is a relaxation for our problem. Below, we
denote the cost of the optimal fraction solution to (P) by $LP$.
We remark that we can get a simpler relaxation by removing variables $z$ and changing the objective function to $\|y\|_q$.
This relaxation also works for $\ell_{\infty}$ norm. We use it in our 5-approximation algorithm.
\subsection{Algorithm for Partitioning Metric Spaces}
We prove the following theorem.
\begin{theorem}\label{thm:part-metric-spaces}
There exists a polynomial-time randomized algorithm that given a metric space $(X,d)$ on $n$ points and parameter $\Delta$
returns a random partition $\calP$ of $X$ such that the diameter of every set $P$ in $\cP$ is at most $\Delta$ and for every $q\geq 1$ ($q\neq \infty$) and
every weighted graph $G=(X,E,w)$, we have
\begin{multline}\label{eq:thm:part-metric-spaces}
\bE\Big[\|\cut(\cP, E)\|_q\Big] \leq \\ \leq C n^{\frac{q-1}{2q}}\log^{\frac{q+1}{2q}} n \cdot \Big[
\Big(\sum_{u\in X}\smashoperator[r]{\sum_{v:(u,v) \in E}} w^q_{uv} \frac{d(u,v)}{\Delta}\Big)^{1/q} + \\ +
\Big(\sum_{u\in X}\Big(\sum_{v:(u,v) \in E} w_{uv} \frac{d(u,v)}{\Delta} \Big)^q\Big)^{1/q} \Big],
\end{multline}
for some absolute constant $C$.
\end{theorem}
We remark that our algorithm also works for $q=\infty$.
\confversionOnly{
We provide the details in the full version of the paper (see supplemental materials for details).}
\fullversionOnly{
Indeed, the behaviour of the algorithm does not depend on $q$ (in fact,
$q$ is not even a part of the algorithm's input). Hence, inequality~(\ref{eq:thm:part-metric-spaces}) holds for any $q<\infty$. In the limit
as $q$ tends to infinity, we get the following result.
\begin{corollary}
The following inequality holds for a random partition $\calP$ from Theorem~\ref{thm:part-metric-spaces}:
\begin{multline}
\bE\Big[\|\cut(\cP, E)\|_{\infty}\Big] \leq \\ \leq C n^{\frac{1}{2}}\log^{\frac{1}{2}} n \cdot \Big[
\max_{(u,v)\in E} w \cdot \ONE (d(u,v)\neq 0)+ \\ +
\max_{u\in V}\Big(\sum_{v:(u,v) \in E} w_{uv} \frac{d(u,v)}{\Delta} \Big)\Big].
\end{multline}
\end{corollary}
}
We will need the following definition.
\begin{definition
Let $(X,d)$ be a metric space. The $\varepsilon$-neighborhood of a set $S\subset X$ is the set of points at distance at most $\varepsilon$ from $S$:
$$N_\varepsilon(S) = \{u \in X: \exists v\in S \text{ such that } d(u,v) \leq \varepsilon\}.$$
The $\varepsilon$-neighborhood of the boundary of a partition $\calP$ is the set of points
\begin{multline*}
N_\varepsilon(\partial \cP) = \bigcup_{P\in\cP} (N_{\varepsilon}(P)\setminus P) = \\ = \{ u \in X: \exists v\in X \text{ s.t. }
d(u,v) \leq \varepsilon \text{ and } \cP(u) \neq \cP(v) \}.
\end{multline*}
\end{definition}
We first describe an algorithm which succeeds with probability at least $1/2$ and fails with probability at most $1/2$. If the algorithm succeeds it outputs
a random partition $\calP$ of $X$ such that the diameter of every set $P$ in $\cP$ is at most $\Delta$ and for every $q$ and every weighted
graph $G=(X,E,w)$, we have
\begin{multline}\label{eq:cond-bound-on-partition}
\bE\Big[\|\cut(\cP, E)\|_q\given \text{algorithm succeeds}\Big] \leq \\ \leq
C' n^{\frac{q-1}{2q}}\log^{\frac{q+1}{2q}} n \cdot \Big(\sum_{u\in X}
\smashoperator[r]{\sum_{v:(u,v) \in E}} w^q_{uv} \frac{d(u,v)}{\Delta}\Big)^{1/q} +\\+
\Big(\sum_{u\in X}\Big(\sum_{v:(u,v) \in E} w_{uv} \frac{d(u,v)}{\Delta} \Big)^q\Big)^{1/q}.
\end{multline}
To obtain a valid partition with probability 1, we repeat our algorithm for at most $\roundup{\log_2 n}$ iterations till it succeeds and
output the obtained solution. If the algorithm does not succeed after $\roundup{\log_2 n}$ iterations (which happens with probability at
most $1/n$), we partition the graph using a simple deterministic procedure which we describe in the end of this section.
Our algorithm is based on the procedure for generating bounded padded stochastic decompositions (see Section~\ref{sec:prelim}). First, the algorithm picks a random padded decomposition $\calP$ of the metric space $X$. Then, it finds the $\varepsilon$-neighborhood
$N_\varepsilon(\partial \cP)$ of the boundary of $\calP$.
Finally, it outputs $\calP$ if $|N_\varepsilon(\partial \cP)| \leq 2 D \varepsilon/\Delta$ and fails otherwise. We present a pseudo-code for our algorithm in
Figure~\ref{fig:Alg1}.
\begin{figure}
\notarxiv{\begin{center}}
\openLP
\smallskip
\begin{algorithm
\smallskip
\noindent\textbf{Input: } metric space $(X,d)$ and parameter $\Delta > 0$.\notarxiv{\\}
\noindent\textbf{Output: } a random partition $\calP$ of $X$.
\begin{enumerate}
\item Let $D=O(\log n)$ be the parameter from Theorem~\ref{prelim:thm:padded-decomposition},
$\varepsilon = 1/\sqrt{2Dn}$ and $M=2D\varepsilon n/\Delta$.
\item Draw a random padded decomposition $\cP$ of the metric space $(X, d)$ with parameter $\Delta$ using Theorem~\ref{prelim:thm:padded-decomposition}.
\item Find the neighborhood $N_\varepsilon(\partial \cP)$ of the partition boundary.
\item If $|N_\varepsilon(\cP)| \leq M$ then output $\cP$; else fail.
\end{enumerate}
\end{algorithm}
\closeLP
\notarxiv{\end{center}}
\caption{Metric decomposition algorithm.}\label{fig:Alg1}\label{alg:metric-decomposition}
\end{figure}
\subsection{Analysis}
Our algorithm is scale invariant i.e., its output does not change if we multiply all distances in the metric space $(X,d)$ and the parameter $\Delta$ by some
positive number $\lambda$. Thus, for the sake of analysis, we assume that $\Delta = 1$. Algorithm~\ref{alg:metric-decomposition} succeeds when $N_\varepsilon(\cP)$ has size at most $M$.
Denote this event by $\cE$. We first show that $\pr(\cE) \geq 1/2$.
\begin{lemma}
Algorithm~\ref{alg:metric-decomposition} succeeds with probability at least~$\nicefrac{1}{2}$.
\end{lemma}
\begin{proof}
Let $\bar\cE$ be the complement of the event $\cE$. We need to show that $\pr(\bar\cE)\leq 1/2$. To this end, we bound
the expected size of the set $N_\varepsilon(\cP)$ using the second property of padded decompositions:
\begin{align*}
\bE[|N_\varepsilon(\partial \cP)|]
&= \sum_{u \in X} \pr(u \in N_\varepsilon(\partial\cP))\\
&= \sum_{u \in X} \pr(\Ball(u, \varepsilon) \not\subset \cP(u))\\
&\leq \sum_{u \in X} D\varepsilon = D\varepsilon n.
\end{align*}
Here, we used that $u \in N_\varepsilon(\partial\cP)$ if and only if $\Ball(u, \varepsilon) \not\subset \cP(u)$.
Now, by Markov's inequality,
$$ \pr(\bar\cE) = \pr(|N_\varepsilon(\partial\cP)| > \underbrace{2D\varepsilon n}_M) \leq \frac{D\varepsilon n}{2D\varepsilon n} = \frac{1}{2}.$$
\end{proof}
Let $X_{uv}$ be the indicator of the event $\{\calP(u)\neq \calP(v)\}$ i.e., the event that points $u$ and $v$ are separated by the partition $\calP$.
By the second property of padded stochastic decompositions, we have $\bE(X_{uv})=\pr(\calP(u)\neq \calP(v))\leq D\cdot d(u,v)$.
Since $\pr(\calE)\geq 1/2$, for each $(u,v)\in E$, we have
$$\bE[X_{uv}\given \calE]\leq \frac{\bE[X_{uv}]}{\pr(\calE)}\leq 2\bE[X_{uv}] \leq 2D\cdot d(u,v).$$
Consequently,
\begin{align}
\bE[w_{uv}X_{uv}\given \calE]&\leq 2D\cdot w_{uv} d(u,v) \;\;\;\text{ and } \label{eq:ineq-for-EX-uv-A}\\
\bE[w^q_{uv}X^q_{uv}\given \calE] &\leq 2D\cdot w^q_{uv} d(u,v). \label{eq:ineq-for-EX-uv-B}
\end{align}
We split all edges $E$ into two groups: short edges, which we denote by $\Eshort$, and long edges,
which we denote by $\Elong$. Short edges are edges of length at most $\varepsilon$; long edges are edges
of length greater than $\varepsilon$. Note that
$\cut(\cP, E) = \cut(\cP, \Eshort) + \cut(\cP, \Elong)$.
For every subset $E'\subset E$ (in particular, for $E'= \Eshort$ and $E'=\Elong$), we have
\begin{equation}
\bE\Big[\|\cut(\cP, E')\|^q_q | \calE\Big] =
\sum_{u\in X} \bE\Big[\big(\smashoperator{\sum_{v:(u,v) \in E'}} w_{uv}X_{uv}\big)^q | \calE\Big].
\label{eq:formula-for-Lq}
\end{equation}\
We separately upper bound $\bE[\|\cut(\cP, \Eshort)\|^q_q\given \calE]$ and $\bE[\|\cut(\cP, \Elong)\|^q_q\given \calE]$ using
the formula above and inequalities (\ref{eq:ineq-for-EX-uv-A}), (\ref{eq:ineq-for-EX-uv-B}) and then
use the triangle inequality for $\ell_q$ norms to bound $\bE[\|\cut(\cP, E)\|_q\given \calE]$.
\medskip
\noindent\textbf{Long edges.} Fix a vertex $u$ and consider long edges incident to $u$. Their total weight is upper bounded by
\begin{align*}
\sum_{v:(u,v) \in \Elong} w_{uv} &\leq \sum_{v:(u,v) \in \Elong} w_{uv}\;\underbrace{\frac{d(u,v)}{\varepsilon}}_{\geq 1}.
\end{align*}
Thus,
\begin{align*}
\Big(\smashoperator[r]{\sum_{v:(u,v) \in \Elong}} & w_{uv}X_{uv}\Big)^q \leq
\\& \leq \Big(\smashoperator[r]{\sum_{v:(u,v) \in \Elong}} w_{uv}\Big)^{q-1}\Big(\smashoperator[r]{\sum_{v:(u,v) \in \Elong}} w_{uv}X_{uv}\Big)\\
&\leq \Big(\smashoperator[r]{\sum_{v:(u,v) \in \Elong}} \frac{w_{uv} d(u,v)}{\varepsilon} \Big)^{q-1} \Big(\smashoperator[r]{\sum_{v:(u,v) \in \Elong}} w_{uv} X_{uv}\Big).
\end{align*}
Plugging this expression into formula~(\ref{eq:formula-for-Lq}) with $E'=\Elong$ and using inequality~(\ref{eq:ineq-for-EX-uv-A}), we get the following upper bound on
$\bE\Big[\|\cut(\cP, \Elong)\|^q_q \Given \calE\Big]$:
\begin{multline*}
\sum_{u\in X}
\Big(\smashoperator[r]{\sum_{v:(u,v) \in \Elong}} \frac{w_{uv} d(u,v)}{\varepsilon} \Big)^{q-1} \bE\Big[\smashoperator{\sum_{v:(u,v) \in \Elong}} w_{uv}X_{uv}\given \calE\Big]
\leq \\ \leq\frac{2D}{\varepsilon^{q-1}} \sum_{u\in X}\Big(\sum_{v:(u,v) \in \Elong} w_{uv} d(u,v) \Big)^q.
\end{multline*}
Finally, by Jensen's inequality, we have
\begin{align}
\notag\bE\big[\|&\cut(\cP,\Elong)\|_q\given \calE] = \bE\big[(\|\cut(\cP, \Elong)\|^q_q)^{\frac{1}{q}}\given \calE\big]\\
&\;\;\notag\leq \Big(\bE\big[\|\cut(\cP, \Elong)\|^q_q\given \calE\big]\Big)^{\frac{1}{q}}\\
&\;\;\leq \Big(\frac{2D}{\varepsilon^{q-1}} \sum_{u\in X}\Big(\sum_{v:(u,v) \in \Elong} w_{uv} d(u,v) \Big)^q\Big)^{\frac{1}{q}}.\label{eq:ineq-long}
\end{align}
\medskip
\noindent\textbf{Short edges.} To bound $||\cut(\cP, E_{short})||_q $, we will make use of the following lemma.
\begin{lemma}\label{lem:jensen-bound}
Consider non-negative (dependent) random variables $X_1,\dots, X_n$. Suppose that at most $M$ of them are non-zero with probability 1. Then, for every $q\geq 1$,
the following bound holds:
$$\bE\big[(X_1+\cdots+X_n)^q\big]\leq M^{q-1}\sum_{i=1}^n \bE\big[X_i^q\big].$$
\end{lemma}
\begin{proof}
Let $x_{i_1}, \ldots, x_{i_m}$ be the non-zero random variables in a certain sampling of $X_1,\dots, X_n$ for some $m\leq M$. Suppose that $m\neq 0$.
Using Jensen's inequality, we have
$$\bigg( \frac{x_{i_1} + \ldots + x_{i_m}}{m} \bigg)^q \leq \frac{1}{m} \sum_{j = 1}^m x_{i_j}^q,$$
and, therefore,
$$\bigg( {x_{i_1} + \ldots + x_{i_m}} \bigg)^q \leq m^{q-1} \sum_{j = 1}^m x_{i_j}^q\leq M^{q-1} \sum_{j = 1}^m x_{i_j}^q.$$
The inequality above also holds when $m=0$. Thus, the expectation of the left hand side is upper bounded by the expectation of the right hand side.
This concludes the proof.
\end{proof}
Fix a vertex $u$. Observe that if $(u,v)$ is a short edge which is cut by $\calP$ then $v$ must belong to $N_\varepsilon(\partial\cP)$. Thus,
the number of non-zero random variables $X_{uv}$ for a given $u$ and $(u,v)\in\Eshort$ is upper bounded by $|N_\varepsilon(\partial\cP)|$.
If the algorithm succeeds, then $|N_\varepsilon(\partial\cP)| \leq M$. Thus, by Lemma~\ref{lem:jensen-bound},
$$\bE\big[\big(\smashoperator{\sum_{v:(u,v) \in \Eshort}} w_{uv}X_{uv}\big)^q\given \calE \big]
\leq M^{q-1}\smashoperator{\sum_{v:(u,v) \in \Eshort}} \bE\big[w_{uv}^qX_{uv}^q\given \calE \big].$$
Plugging this bound into formula~(\ref{eq:formula-for-Lq}) with $E'=\Eshort$ and using inequality~(\ref{eq:ineq-for-EX-uv-B}), we get
the following upper bound on $\bE\Big[\|\cut(\cP, \Eshort)\|^q_q \Given \calE\Big]$:
\begin{multline*}
\sum_{u\in X}\Big(M^{q-1}\smashoperator{\sum_{v:(u,v) \in \Eshort}} \bE\big[w_{uv}^qX_{uv}^q\given \calE \big] \Big)\leq \\ \leq
2D\,M^{q-1} \sum_{u\in X}\sum_{v:(u,v) \in \Eshort} w^q_{uv} d(u,v).
\end{multline*}
Finally, by Jensen's inequality, we have
\begin{align}\label{eq:ineq-short}
\bE[&\|\cut(\cP, \Eshort)\|_q\given \calE] \leq \\
\notag& \leq\Big(2D\,M^{q-1}\Big)^{1/q} \Big(\sum_{u\in X}\sum_{v:(u,v) \in \Eshort} w^q_{uv} d(u,v)\Big)^{1/q}.
\end{align}
\medskip
To obtain the desired bound~(\ref{eq:cond-bound-on-partition}), we substitute $D=O(\log n)$, $\varepsilon = 1/\sqrt{2Dn}$, and $M=2D\varepsilon n/\Delta$ in bounds~(\ref{eq:ineq-long}) and~(\ref{eq:ineq-short}) and then apply the triangle inequality for the $\ell_q$ norm.
\confversionOnly{
To finish the proof of Theorem~\ref{thm:part-metric-spaces}, we need to describe what we do in the unlikely event that Algorithm~\ref{alg:metric-decomposition} fails
$\roundup{\log_2 n}$ times. In this case, we create a new graph on $X$ with edges between pairs of vertices at distance at most $1/n$ from each other and partition it
into connected components. We analyze this algorithm in the full version of the paper (see supplemental materials for details).}
\fullversionOnly{
To finish the proof of Theorem~\ref{thm:part-metric-spaces}, we describe what we do in the unlikely event that Algorithm~\ref{alg:metric-decomposition} fails
$\roundup{\log_2 n}$ times.
\begin{lemma}\label{lem:simple-n-approx-alg}
There exists a polynomial-time deterministic algorithm that given a metric space $(X,d)$ on $n$ points and parameter $\Delta$
returns a partition $\calP$ of $X$ such that the diameter of every set $P$ in $\cP$ is at most $\Delta$ and
for every $q$ and every weighted graph $G=(X,E,w)$, we have
$$\|\cut(\cP, E)\|_q \leq n \Big(\sum_{u\in X}\Big(\sum_{v:(u,v) \in E} w_{uv} \frac{d(u,v)}{\Delta} \Big)^q\Big)^{1/q}.$$
\end{lemma}
\begin{proof}
Consider a graph $\tilde G = (X,\tilde E)$ on $X$ with edges $\tilde E = \{(u,v)\in X\times X: d(u,v) \leq \Delta/n\}$. The algorithm partitions
$\tilde G$ into connected components and outputs the result. Note that the diameter of each connected component $P\in \calP$
is less than $\Delta$, since the length of every edge in $\tilde G$ is less than $\Delta/n$. Let $E_{cut}$ be the set of cut edges in graph $G$.
If two vertices $(u,v)$ are separated by $\calP$, then $d(u,v) \geq \Delta/n$. Hence, for every cut edge $(u,v)\in E_{cut}$, we have $n d(u,v)/\Delta \geq 1$.
Thus,
\begin{align*}
\|\cut(\cP, E)\|_q &= \Big(\sum_{u\in X}\Big(\sum_{v:(u,v) \in E} w_{uv}\Big)^q\Big)^{1/q}\\
&\leq \Big(\sum_{u\in X}\Big(\sum_{v:(u,v) \in E} w_{uv} \frac{n\cdot d(u,v)}{\Delta} \Big)^q\Big)^{1/q}\\
&= n \Big(\sum_{u\in X}\Big(\sum_{v:(u,v) \in E} w_{uv} \frac{d(u,v)}{\Delta} \Big)^q\Big)^{1/q}.
\end{align*}
\end{proof}
}
\section{Preliminaries}\label{sec:prelim}
We now formally define the Correlation Clustering with $\ell_q$ objective problem. We will need the following
definition. Consider a set of points $V$ and two disjoint sets of edges on $V$: positive edges $E^+$ and negative edges
$E^-$. We assume that every edge has a weight $w_{uv}$. For every partition $\calP$ of $V$, we say that a positive
edge is in disagreement with $\calP$ if the endpoints $u$ and $v$ belongs to different parts of $\calP$; and
a negative edge is in disagreement with $\calP$ if the endpoints $u$ and $v$ belongs to the same part of $\calP$.
The vector of disagreements, denoted by $\disagree(\calP, E^+, E^-)$, is a $|V|$ dimensional vector
indexed by elements of $V$. Its coordinate $v$ equals
\begin{multline*}
\disagree_u(\calP, E^+, E^-) =\\= \smashoperator{\sum_{v:(u,v)\in E^+\cup E^-}} w_{uv} \ONE((u,v) \text{ is in disagreement with }\calP).
\end{multline*}
That is, $\disagree_u(\calP, E^+, E^-)$ is the weight of disagreeing edges incident to $u$. We similarly define a cut vector
for a set of edges $E$:
$$
\cut_u(\calP, E) = \smashoperator{\sum_{v:(u,v)\in E}} w_{uv} \ONE(u \text{ and } v \text{ are separated by }\calP).
$$
We use the standard definition for the $\ell_q$ norm of a vector $x$: $\|x\|_q= (\sum_u x_u^q)^{\frac{1}{q}}$ and
$\|x\|_{\infty}= \max_u x_u$. For a partition $\calP$, we denote by $\calP(u)$ the piece that contains vertex $u$.
\begin{definition}
In the Correlation Clustering problem with $\ell_q$ objective, we are given a graph $G$ on a set $V$ with
two disjoint set of edges $E^+$ and $E^-$ and a set of weights $w_{uv}$. The goal is find a partition $\calP$ that minimizes the $\ell_q$
norm of the disagreements vector, $\|\disagree(\calP, E^+, E^-)\|_q$.
\end{definition}
In our algorithm for Correlation Clustering on arbitrary graphs, we will use a powerful technique of padded
metric space decompositions~(see e.g., \citet*{Bartal96, Rao99, FT03, GKL03}).
\begin{definition}[Padded Decomposition]
Let $(X, d)$ be a metric space on $n$ points, and let $\Delta > 0$. A probabilistic distribution of partitions $\cP$ of $X$ is called a padded decomposition if it satisfies the following properties:
\begin{itemize}
\item Each cluster $C \in \cP$ has diameter at most $\Delta$.
\item For every $u \in X$ and $\varepsilon > 0$,
$$\pr(\Ball(u, \delta) \not\subset \cP(u)) \leq D\cdot \frac{\delta}{\Delta}$$
where $\Ball(u, \delta) = \{v \in X : d(u,v) \leq \delta\}$
\end{itemize}
\end{definition}
\begin{theorem}[\citet*{FRT03}]\label{prelim:thm:padded-decomposition}
Every metric space $(X,d)$ on $n$ points admits a $D=O(\log n)$ separating padded decomposition. Moreover, there is a polynomial-time algorithm that samples
a partition from this distribution.
\end{theorem}
|
{
"timestamp": "2019-03-01T02:05:05",
"yymm": "1902",
"arxiv_id": "1902.10829",
"language": "en",
"url": "https://arxiv.org/abs/1902.10829"
}
|
\section{Introduction}
The random utility model of multinomial choice (McFadden, 1973) has gained
immense popularity among applied economists. However, there has been limited
research on the micro-theoretic underpinning of such models, and in
particular, on the question of which choice probability functions are
logically consistent with a random utility model.\footnote
There has been comparatively more work on rationalizability in empirical
demand models with \textit{continuous} goods, c.f. Lewbel, 2001 and Hausman
and Newey, 2016.} Daly and Zachary 1978 provided a set of closed-form,
global conditions under which choice-probability \textit{functions} can be
justified as having arisen from preference maximization by a heterogeneous
population. These conditions were re-stated in Anderson et al, 1992, Theorem
3.1, and independently derived in Armstrong and Vickers, 2015, who improved
upon the Daly-Zachary results by including an outside option in the choice
set. In all of these results, a key condition for rationalizability is
Slutsky symmetry, analogous to the classic textbook case for demand systems
with continuous goods.
In this paper, we first show that in the multinomial setting, Daly-Zachary's
Slutsky symmetry is equivalent to the absence of income effects, i.e. that
conditional choice probabilities do not depend on the decision-makers'
income. The \textquotedblleft necessity\textquotedblright\ part is easy to
show. Showing \textquotedblleft sufficiency\textquotedblright , i.e. that
Slutsky symmetry implies absence of income effects is non-trivial, and
represents the first result of the paper. Next, in multinomial settings that
allow for income effects, we provide a set of alternative shape restrictions
on conditional choice probability functions, including a counterpart of
Slutsky symmetry, which together are shown to be sufficient for
rationalizability. The proof of this result is constructive, and the
rationalizing utility functions are obtained by inverting solutions of
certain partial differential equations (PDEs). The way in which PDEs arise
here is unrelated to Roy's Identity (c.f. Mas-Colell et al, 1995,
Proposition 3.G.4); in particular, the partial derivatives appearing in the
PDE are of the average \textit{demand} function, not the indirect utility
function.
Finally, we show that the rationalizability results can also be used to
nonparametrically identify the underlying preference distributions from
empirical choice-probabilities. A key restriction delivering this
identification result -- viz. invertibility of sub-utilities in the
numeraire due to non-satiation -- is based on \textit{economic} theory, as
opposed to statistical assumptions. Furthermore, achieving nonparametric
identification by solving PDEs appears to be novel in the discrete choice
literature.\smallskip
The plan for the rest of the paper is as follows. Section 2 discusses
Daly-Zachary's Slutsky symmetry condition, and its connection with lack of
income effects. Section 3 discusses rationalizability for multinomial choice
in presence of income effects, and presents Theorem 1, the key result of
this paper. Section 4 discusses some further points, including the
implication of the rationalizability result for nonparametric identification
of preference distributions. A short appendix at the end presents two
mathematical results on partial and ordinary differential equations that are
intensively used in the paper.
\section{The Daly-Zachary Result}
Consider a setting of multinomial choice, where the discrete alternatives
are indexed by $j=0,1,...,J$, individual income is $y$, price of alternative
$j$ is $p_{j}$; if alternative $0$ refers to the outside option, i.e. not
buying any of the alternatives, then $p_{0}\equiv 0$. Let the utility from
consuming the $j$th alternative and a quantity $z$ of the numeraire be given
by $U\left( j,z\right) $. The consumer's problem is $\max_{j\in \left\{
0,1,...,J\right\} ,z}\left[ U\left( j,z\right) +\varepsilon _{j}\right] $,
subject to the budget constraint $z\leq y-p_{j}$, where $y$ is the
consumer's income, $p_{j}$ is the price of alternative $j$ faced by the
consumer, and $\varepsilon _{j}$ is unobserved heterogeneity in the
consumer's preferences. If $U\left( j,\cdot \right) $ is strictly increasing
(i.e. non-satiation in the numeraire), then we can replace $z=y-p_{j}$, and
rewrite the consumer problem as $\max_{j\in \left\{ 0,1,...,J\right\} }\left[
U\left( j,a_{j}\right) +\varepsilon _{j}\right] $, where $a_{j}\equiv
y-p_{j} $. Denote the (structural) probability of choosing alternative $j\in
\left\{ 0,...,J\right\} $ at $\mathbf{a\equiv }\left( a_{0},..,a_{J}\right) $
by $q_{j}\left( \mathbf{a}\right) $. In words, if we randomly sample
individuals from the population, and offer the vector $\mathbf{a}$ to each
sampled individual, then a fraction $q_{j}\left( \mathbf{a}\right) $ will
choose alternative $j$, in expectation. It is easy to incorporate other
attributes of the alternatives and characteristics of consumers in our
analysis, and we outline how to that below, after Theorem 1. For now, we
suppress other covariates for clarity of exposition.\smallskip
\textbf{Slutsky-Symmetry}: In this set-up, Daly-Zachary's Slutsky symmetry
conditions are that for any two alternatives $k,l\in \left\{
0,1,...,J\right\} $, $k\neq l$
\begin{equation}
\frac{\partial }{\partial a_{l}}q_{k}\left( \mathbf{a}\right) =\frac
\partial }{\partial a_{k}}q_{l}\left( \mathbf{a}\right) \text{.\footnotemark
} \label{S}
\end{equation
\footnotetext
Daly-Zachary defines choice probabilities as functions of price and income,
\tilde{q}_{j}\left( p_{0},p_{1},...,p_{J},y\right) $. This is equivalent to
our notation of $q_{j}\left( a_{0},a_{1},...a_{J}\right) $ with $a_{0}=y$,
a_{1}=y-p_{1}$,...,$a_{J}=y-p_{J}$, in that one can move back and forth
between the two notations, sinc
\begin{eqnarray*}
q_{j}\left( a_{0},a_{1},...,a_{J}\right) &=&\tilde{q}_{j}\left(
a_{0}-a_{1},a_{0}-a_{2},...,a_{0}-a_{J}\right) \text{, and} \\
\tilde{q}_{j}\left( p_{1},p_{2},...,p_{J},y\right) &=&q_{j}\left(
y,y-p_{1},y-p_{2},...,y-p_{J}\right) \text{.}
\end{eqnarray*
\textquotedblleft Slutsky symmetry\textquotedblright\ in Daly-Zachary's
notation is that $\partial \tilde{q}_{k}/\partial p_{j}=\partial \tilde{q
_{j}/\partial p_{k}$ for all $j\neq k$ (if alternative $0$ is the ouside
option, then the corresponding condition is $\partial \tilde{q}_{0}/\partial
p_{j}=\partial \tilde{q}_{j}/\partial y$). which is identical to (\ref{S})
in our notation.}We first show that the classic random utility model with no
income effects implies (\ref{S}). We then show the first result of our
paper, viz. that Slutsky symmetry (\ref{S}) implies absence of income
effects.\smallskip
\textbf{Necessity}: The canonical random utility model of multinomial choice
assumes that utility from consuming the $j$th alternative at income $y$ and
price $p_{j}$ is given b
\begin{equation}
U\left( j,a_{j}\right) =a_{j}\text{,} \label{N}
\end{equation
where $a_{j}=y-p_{j}$ as above. Income effects are zero since demand depends
on the $a$'s via the differences $a_{j}-a_{k}=\left( y-p_{j}\right) -\left(
y-p_{k}\right) =p_{k}-p_{j}$. Suppose $\left( \varepsilon _{0},\varepsilon
_{1},...\varepsilon _{J}\right) $ are continuously distributed with joint
density $g\left( \cdot \right) $. Then, the choice probability for the $0$th
alternative is given b
\begin{eqnarray*}
&&q_{0}\left( \mathbf{a}\right) \\
&=&\Pr \left( \cap _{j\neq 0}\left\{ a_{0}+\varepsilon
_{0}>a_{j}+\varepsilon _{j}\right\} \right) \\
&=&\Pr \left( \cap _{j\neq 0}\left\{ a_{0}-a_{j}>\varepsilon
_{j}-\varepsilon _{0}\right\} \right) \\
&=&\int_{-\infty }^{\infty }\int_{-\infty }^{a_{0}-a_{1}+\varepsilon
_{0}}...\int_{-\infty }^{a_{0}-a_{J}+\varepsilon _{0}}g\left( \varepsilon
\right) d\varepsilon _{J}...d\varepsilon _{1}d\varepsilon _{0}\text{.}
\end{eqnarray*
Therefore, by the first fundamental theorem of calculus
\begin{eqnarray}
&&\frac{\partial }{\partial a_{1}}q_{0}\left( \mathbf{a}\right) \notag \\
&=&-\int_{-\infty }^{\infty }\int_{-\infty }^{\varepsilon
_{0}+a_{0}-a_{2}}...\int_{-\infty }^{^{\varepsilon _{0}+a_{0}-a_{J}}}g\left(
\varepsilon _{0},a_{0}-a_{1}+\varepsilon _{0},\varepsilon
_{2},...\varepsilon _{J}\right) d\varepsilon _{J}...d\varepsilon
_{2}d\varepsilon _{0}\text{.} \label{D}
\end{eqnarray
Similarly
\begin{equation*}
q_{1}\left( \mathbf{a}\right) =\int_{-\infty }^{\infty }\int_{-\infty
}^{\varepsilon _{1}+a_{1}-a_{0}}...\int_{-\infty }^{^{\varepsilon
_{1}+a_{1}-a_{J}}}g\left( \varepsilon \right) d\varepsilon
_{J}...d\varepsilon _{2}d\varepsilon _{0}d\varepsilon _{1}\text{,}
\end{equation*
implyin
\begin{eqnarray}
&&\frac{\partial }{\partial a_{0}}q_{1}\left( \mathbf{a}\right) \notag \\
&=&-\int_{-\infty }^{\infty }\int_{-\infty }^{a_{1}-a_{2}+\varepsilon
_{1}}...\int_{-\infty }^{a_{1}-a_{J}+\varepsilon _{1}}g\left( \left(
a_{1}-a_{0}+\varepsilon _{1},\varepsilon _{1},\varepsilon
_{2},...\varepsilon _{J}\right\vert ,\gamma \right) d\varepsilon
_{J}...d\varepsilon _{2}d\varepsilon _{1} \label{A} \\
&=&-\int_{-\infty }^{\infty }\int_{-\infty
}^{s_{0}+a_{0}-a_{2}}...\int_{-\infty }^{s_{0}+a_{0}-a_{J}}g\left(
s_{0},s_{0}-a_{1}+a_{0},\varepsilon _{2},...\varepsilon _{J}\right)
d\varepsilon _{J}...d\varepsilon _{2}ds_{0} \notag \\
&=&\frac{\partial }{\partial a_{1}}q_{0}\left( \mathbf{a}\right) \text{,
using (\ref{D}),} \notag
\end{eqnarray
where the second equality follows by substituting $s_{0}=a_{1}-a_{0}
\varepsilon _{1}$ in (\ref{A}).
The same argument can be repeated for any other pair of alternatives $l\neq
k $, to obtai
\begin{equation}
\frac{\frac{\partial }{\partial a_{k}}q_{l}\left( \mathbf{a}\right) }{\frac
\partial }{\partial a_{l}}q_{k}\left( \mathbf{a}\right) }=1\text{,}
\label{I}
\end{equation
for all $\mathbf{a}$. This shows that in the canonical random utility model
with no income effects, Daly-Zachary's Slutsky symmetry condition
holds.\smallskip
\textbf{Sufficiency}: We now show that Slutsky symmetry implies absence of
income effects. To see this, first note that because $\sum_{k=0}^{J}q_{k
\left( \mathbf{a}\right) =1$, differentiating both sides w.r.t. $a_{l}$ give
\begin{equation}
\frac{\partial }{\partial a_{l}}q_{l}\left( \mathbf{a}\right)
+\sum_{k=0,k\neq l}^{J}\frac{\partial }{\partial a_{l}}q_{k}\left( \mathbf{a
\right) =0\text{.} \label{12}
\end{equation
Substituting (\ref{S}) in (\ref{12}), we get:
\begin{equation}
\frac{\partial }{\partial a_{l}}q_{l}\left( \mathbf{a}\right)
+\sum_{k=0,k\neq l}^{J}\frac{\partial }{\partial a_{k}}q_{l}\left( \mathbf{a
\right) =0\text{.} \label{T}
\end{equation
This is a linear, homogeneous partial differential equation in $q_{l}\left(
\cdot \right) $, and can be solved via the method of characteristics (c.f.
Courant, 1962, Chapter I.5 and II.2). The characteristic curve, i.e. the $J
-dimensional subspace on which $q_{l}\left( \mathbf{a}\right) $ remains
constant, can be obtained as follows. Parametrize $a_{j}=a_{j}\left(
r\right) $, $j=0,1,...J$ and conside
\begin{equation*}
0=\frac{dq_{l}}{dr}=\frac{\partial q_{l}\left( \mathbf{a}\right) }{\partial
a_{l}}\frac{da_{l}\left( r\right) }{dr}+\sum_{k=0,k\neq l}^{J}\frac{\partial
q_{l}\left( \mathbf{a}\right) }{\partial a_{k}}\times \frac{da_{k}\left(
r\right) }{dr}\text{.}
\end{equation*
Comparing with (\ref{T}), we ge
\begin{equation*}
\frac{da_{k}\left( r\right) }{dr}=1\text{, }k=0,1,...J\text{,}
\end{equation*
implying the so-called \textquotedblleft characteristic\textquotedblright\
Ordinary Differential Equations
\begin{equation}
\frac{da_{k}}{da_{l}}=1\text{, }k=0,...l-1,l+1,...,J\text{,} \label{C}
\end{equation
with generic solutions $a_{k}-a_{l}=c_{k}$, $k=0,...l-1,l+1,...,J$. This
means that general solutions to (\ref{T}) are of the for
\begin{equation}
q_{l}\left( \mathbf{a}\right) =H^{l}\left(
a_{0}-a_{l},a_{1}-a_{l},...,a_{l-1}-a_{l},a_{l+1}-a_{l},....a_{J}-a_{l
\right) \text{,} \label{W}
\end{equation
where $H^{l}\left( \cdot \right) $ is any arbitrary continuously
differentiable function. Thus $q_{l}\left( \mathbf{a}\right) $ depends on
the $\left( J+1\right) $-dimensional argument $\left(
a_{0},a_{1},a_{2},...a_{J}\right) $ through a $J$-dimensional vecto
\begin{equation*}
\left(
a_{1}-a_{l},a_{2}-a_{l},...,a_{l-1}-a_{l},a_{l+1}-a_{l},....a_{J}-a_{l
\right) \text{.}
\end{equation*
That (\ref{W}) is a solution to (\ref{T}) can also be verified directly by
partially differentiating the RHS of (\ref{W}), and verifying that it
satisfies (\ref{T}). Finally, note tha
\begin{eqnarray*}
&&\left(
a_{0}-a_{l},a_{1}-a_{l},...,a_{l-1}-a_{l},a_{l+1}-a_{l},....a_{J}-a_{l
\right) \\
&=&\left(
p_{l},p_{l}-p_{1},...,p_{l}-p_{l-1},p_{l}-p_{l+1},....p_{l}-p_{J}\right)
\text{,}
\end{eqnarray*
and so (\ref{W}) implies that $q_{l}\left( \mathbf{a}\right) $ does not
depend on income. Since $l$ is arbitrary, we have shown that Slutsky
symmetry implies that income effects are absent.
\section{Rationalizability under Income-Effects}
The previous section raises the question of whether utility maximization in
a setting of multinomial choice that allows for (individually heterogeneous)
income effects impose any restriction on choice-probabilities. In other
words, is there a counterpart of Slutsky symmetry under income effects? In
this section, we state that counterpart, and show that this analog, plus a
set of shape-restrictions on choice-probabilities are together sufficient
for rationalizability.
\textbf{Counterpart of Slutsky Symmetry}: Let there be $J+1$ exclusive and
indivisible alternatives, indexed by $j=0,1,....,J$. A consumer can choose
one among these $J+1$ alternatives, plus a quantity $z$ of a continuous
numeraire that they can buy after paying for the indivisible good, subject
to the budget constraint $z\leq y-p_{j}$, where $y$ is the consumer's
income, and $p_{j}$ is the price of alternative $j$ faced by the consumer.
We assume preferences are non-satiated in the numeraire, and denote the
amount of numeraire consumed upon having bought alternative $j$ by
a_{j}=y-p_{j}$, with $a_{0}=y$ corresponding to choosing the outside option
0$. Denote the (structural) probability of choosing alternative $j\in
\left\{ 0,...,J\right\} $ at $\mathbf{a\equiv }\left( a_{0},..,a_{J}\right) $
by $q_{j}\left( \mathbf{a}\right) $. In words, if we randomly sample
individuals from the population, and offer the vector $\mathbf{a}$ to each
sampled individual, then a fraction $q_{j}\left( \mathbf{a}\right) $ will
choose alternative $j$, in expectation. Then our counterpart of Slutsky
symmetry is:\smallskip
\begin{quote}
(A): For any $\mathbf{a}$, and any pair of alternatives $k\neq l$, the ratio
$\frac{\partial }{\partial a_{k}}q_{l}\left( \mathbf{a}\right) /\frac
\partial }{\partial a_{l}}q_{k}\left( \mathbf{a}\right) $ depends only on
a_{k}$ and $a_{l}$.\smallskip
\end{quote}
\textbf{Motivation}: To see where this restriction comes from, consider the
above setting of multinomial choice, and let the utility from consuming the
j$th alternative and a quantity $z$ of the numeraire be given by $U\left(
j,z\right) +\varepsilon _{j}$. The $\left\{ \varepsilon _{j}\right\} $,
which represent unobserved heterogeneity in preferences, are allowed to have
any arbitrary and unspecified joint distribution in the population (subject
to the resulting choice probability functions being smooth). If $U\left(
j,\cdot \right) $ is strictly increasing (i.e. non-satiation in the
numeraire), then we can replace $z=y-p_{j}\equiv a_{j}$, and rewrite the
consumer problem a
\begin{equation}
\max_{j\in \left\{ 0,1,...,J\right\} }\left[ U\left( j,a_{j}\right)
+\varepsilon _{j}\right] . \label{n}
\end{equation
To allow for income effects, we let $U\left( j,a_{j}\right) \equiv
h_{j}\left( a_{j}\right) $, where $h_{j}\left( \cdot \right) $ are smooth,
possibly nonlinear, strictly increasing, \textit{unspecified} functions of
the $a_{j}$'s. When $h_{j}\left( \cdot \right) $ are nonlinear, the
conditional choice-probabilities will depend on income, i.e., there are
non-zero income effects. This structure is also observationally equivalent
to a utility structure where unobserved heterogeneity is not additively
separable from the $a_{j}$'s (see below) in the utility function.
Now, for the above set-up, the choice probability for the $0$th alternative
is given b
\begin{eqnarray}
&&q_{0}\left( \mathbf{a}\right) \notag \\
&=&\Pr \left( \cap _{j\neq 0}\left\{ h_{0}\left( a_{0}\right) +\varepsilon
_{0}>h_{j}\left( a_{j}\right) +\varepsilon _{j}\right\} \right) \notag \\
&=&\Pr \left[ \cap _{j\neq 0}\left\{ h_{0}\left( a_{0}\right) -h_{j}\left(
a_{j}\right) >\varepsilon _{j}-\varepsilon _{0}\right\} \right] \notag \\
&=&\int_{-\infty }^{\infty }\int_{-\infty }^{\left( h_{0}\left( a_{0}\right)
-h_{1}\left( a_{1}\right) \right) +\varepsilon _{0}}...\int_{-\infty
}^{\left( h_{0}\left( a_{0}\right) -h_{J}\left( a_{J}\right) \right)
+\varepsilon _{0}}g\left( \varepsilon \right) d\varepsilon
_{J}...d\varepsilon _{1}d\varepsilon _{0}\text{.} \label{d}
\end{eqnarray
Therefore, by the first fundamental theorem of calculus
\begin{eqnarray}
&&\frac{\partial }{\partial a_{1}}q_{0}\left( \mathbf{a}\right) \notag \\
&=&-h_{1}^{\prime }\left( a_{1}\right) \left[
\begin{array}{c}
\int_{-\infty }^{\infty }\int_{-\infty }^{\substack{ \varepsilon _{0} \\
+h_{0}\left( a_{0}\right) \\ -h_{2}\left( a_{2}\right) }}...\int_{-\infty }
^{\substack{ \varepsilon _{0} \\ +h_{0}\left( a_{0}\right) \\ -h_{J}\left(
a_{J}\right) }}g\left(
\begin{array}{c}
\varepsilon _{0}, \\
\left( h_{0}\left( a_{0}\right) -h_{1}\left( a_{1}\right) \right)
+\varepsilon _{0}, \\
\varepsilon _{2},...\varepsilon _{J
\end{array
\right) \\
d\varepsilon _{J}...d\varepsilon _{2}d\varepsilon _{0
\end{array
\right] \text{.} \label{15}
\end{eqnarray
Similarly
\begin{equation*}
q_{1}\left( \mathbf{a}\right) =\int_{-\infty }^{\infty }\int_{-\infty }
^{\substack{ \varepsilon _{1} \\ +h_{1}\left( a_{1}\right) \\ -h_{0}\left(
a_{0}\right) }}...\int_{-\infty }^{\substack{ \varepsilon _{1} \\
+h_{1}\left( a_{1}\right) \\ -h_{J}\left( a_{J}\right) }}g\left(
\varepsilon \right) d\varepsilon _{J}...d\varepsilon _{2}d\varepsilon
_{0}d\varepsilon _{1}\text{,}
\end{equation*
implying by the first fundamental theorem and chain-rule tha
\begin{eqnarray}
&&\frac{\partial }{\partial a_{0}}q_{1}\left( \mathbf{a}\right) \notag \\
&=&-h_{0}^{\prime }\left( a_{0}\right) \int_{-\infty }^{\infty
}\int_{-\infty }^{\substack{ h_{1}\left( a_{1}\right) \\ -h_{2}\left(
a_{2}\right) +\varepsilon _{1}}}...\int_{-\infty }^{\substack{ h_{1}\left(
a_{1}\right) \\ -h_{J}\left( a_{J}\right) +\varepsilon _{1}}}g\left(
\begin{array}{c}
h_{1}\left( a_{1}\right) -h_{0}\left( a_{0}\right) +\varepsilon _{1}, \\
\varepsilon _{1},\varepsilon _{2},...\varepsilon _{J
\end{array
\right) d\varepsilon _{J}...d\varepsilon _{2}d\varepsilon _{1} \label{a} \\
&&\overset{(1)}{=}-h_{0}^{\prime }\left( a_{0}\right)
\begin{array}{c}
\int_{-\infty }^{\infty }\int_{-\infty }^{\substack{ s_{0}+h_{0}\left(
a_{0}\right) \\ -h_{2}\left( a_{2}\right) }}...\int_{-\infty
}^{s_{0}+h_{0}\left( a_{0}\right) -h_{J}\left( a_{J}\right) }g\left(
\begin{array}{c}
s_{0}, \\
s_{0}-h_{1}\left( a_{1}\right) +h_{0}\left( a_{0}\right) , \\
\varepsilon _{2},...\varepsilon _{J
\end{array
\right) \\
d\varepsilon _{J}...d\varepsilon _{2}ds_{0
\end{array}
\notag \\
&=&\frac{h_{0}^{\prime }\left( a_{0}\right) }{h_{1}^{\prime }\left(
a_{1}\right) }\frac{\partial }{\partial a_{1}}q_{0}\left( \mathbf{a}\right)
\text{, using (\ref{15}),} \notag
\end{eqnarray
where the second equality $\overset{(1)}{=}$ follows by substituting
s_{0}=h_{1}\left( a_{1}\right) -h_{0}\left( a_{0}\right) +\varepsilon _{1}$
in (\ref{a}).
The same argument can be repeated for any other pair of alternatives $l\neq
k $, to obtai
\begin{equation}
\frac{\frac{\partial }{\partial a_{k}}q_{l}\left( \mathbf{a}\right) }{\frac
\partial }{\partial a_{l}}q_{k}\left( \mathbf{a}\right) }=\frac
h_{k}^{\prime }\left( a_{k}\right) }{h_{l}^{\prime }\left( a_{l}\right)
\text{,} \label{b}
\end{equation
for all $\mathbf{a}$, and it is clear that the RHS\ of (\ref{b}) depends
only on $a_{k}$ and $a_{l}$, and thus satisfies condition (A) above. As an
aside, note that for the RHS of (\ref{b}) to be identically equal to 1 (the
Daly-Zachary condition), we must have that $h_{l}\left( a_{l}\right) =\beta
_{0}+\beta _{1}a_{l}$ for some $\beta _{0},\beta _{1}$. To see this, first
note that when evaluated at $a_{k}=a_{l}=c$, condition (\ref{b}) yields
\frac{h_{k}^{\prime }\left( c\right) }{h_{l}^{\prime }\left( c\right) }=1$
for all $c$, implying $h_{k}\left( c\right) =h_{l}\left( c\right) +k$ for
all $c$. Using this, we have tha
\begin{equation*}
1=\frac{h_{k}^{\prime }\left( a_{k}\right) }{h_{l}^{\prime }\left(
a_{l}\right) }=\frac{h_{l}^{\prime }\left( a_{k}\right) }{h_{l}^{\prime
}\left( a_{l}\right) }\Rightarrow h_{l}^{^{\prime \prime }}\left( a\right) =
\text{,}
\end{equation*
implying $h_{l}\left( a_{l}\right) =\beta _{0}+\beta _{1}a_{l}$, and thus
the choice-probabilities cannot display income-effects.\smallskip
\begin{remark}
Condition (\ref{b}) has no relation with the Independence of Irrelevant
Alternatives (IIA) property. Indeed, the model above will \textbf{not} have
the IIA property if the $\varepsilon _{j}$s are correlated across
alternatives (i.e. across $j$), but it will continue to satisfy (\ref{b}),
since uncorrelatedness of $\varepsilon $s was not used to derive (\ref{b
).\smallskip
\end{remark}
\textbf{Main Result}: We are now ready to state and prove our main result.
The result is that the counterpart of Slutsky symmetry stated above, plus
two shape-restrictions on $q_{j}\left( \mathbf{\cdot }\right) $'s are
jointly \textit{sufficient} for rationalizability, i.e., under those
restrictions on $q_{j}\left( \mathbf{\cdot }\right) $'s, we can find a set
of utility functions and a joint distribution of unobserved preference
heterogeneity, such that individual maximization of these utilities will
indeed produce the conditional choice-probabilities $\left\{ q_{j}\left(
\mathbf{\cdot }\right) \right\} $, $j=0,1,...,J$.\smallskip
To state and prove this result, we will use the following additional
notation: let $\mathbf{a}_{-j}$ denote the vector $\left(
a_{0},a_{1},...a_{j-1},a_{j+1},...a_{J}\right) $ and let $\lim_{\mathbf{a
_{-j}\downarrow -\infty }$ denote that each component of $\mathbf{a}_{-j}$
goes to $-\infty $.\smallskip
\begin{theorem}
Suppose that the following three conditions are satisfied by the
choice-probabilities $\left\{ q_{j}\left( \mathbf{a}\right) \right\} $:
(i) For each $j=0,1,...,J$, and each $\mathbf{a}$, $q_{j}\left( \mathbf{a
\right) $ is strictly increasing in $a_{j}$ and strictly decreasing in
a_{k} $ for $k\neq j$, continuously differentiable in each argument, and for
all $j $, $\lim_{\mathbf{a}_{-j}\downarrow -\infty }q_{j}\left( \mathbf{a
\right) =1 $ and $\lim_{a_{j}\downarrow -\infty }q_{j}\left( \mathbf{a
\right) =0=1-\lim_{a_{j}\uparrow \infty }q_{j}\left( \mathbf{a}\right) $;
(ii) There exists an alternative $m$ such that for any other alternatives
j\neq m$ and any $\mathbf{a}$ satisfying $\frac{\partial }{\partial a_{j}
q_{m}\left( \mathbf{a}\right) \neq 0$, the ratio $\frac{\partial }{\partial
a_{m}}q_{j}\left( \mathbf{a}\right) /\frac{\partial }{\partial a_{j}
q_{m}\left( \mathbf{a}\right) $ does not depend on $a_{k}$, for $k\notin
\left\{ m,j\right\} $, and has uniformly bounded derivatives with respect to
$a_{m}$ and $a_{j}$;
(iii) for each $r=0,1,...J$, the $J$th order cross partial derivatives
\frac{\partial ^{J}}{\partial a_{0}\partial a_{1}...\partial a_{r-1}\partial
a_{r+1}...\partial a_{J}}q_{r}\left( \mathbf{a}\right) $ exist, are
continuous, and satisfy $\left( -1\right) ^{J}\frac{\partial ^{J}}{\partial
a_{0}\partial a_{1}...\partial a_{r-1}\partial a_{r+1}...\partial a_{J}
q_{r}\left( \mathbf{a}\right) \geq 0$.
Then there exist random variables $\mathbf{V}=\left(
V_{0},V_{1},...,V_{m-1},V_{m+1},...,V_{J}\right) $ with support $\mathcal{V
\sqsubseteq $ $\mathcal{R}^{J}$, and functions $w_{j}\left( a,v_{j}\right)
\mathcal{R}\times \mathcal{V}_{j}\rightarrow \mathcal{R}$, such that
w_{j}\left( \cdot ,v_{j}\right) $ are strictly increasing and continuous,
w_{m}\left( a_{m},v_{m}\right) \equiv a_{m}$, and
\begin{equation*}
q_{j}\left( a_{0},a_{1},...,a_{J}\right) =\int_{\mathcal{V}}\cap _{k\neq
j}1\left\{ w_{j}\left( a,v_{j}\right) \geq w_{k}\left( a_{k},v_{k}\right)
\right\} f\left( \mathbf{v}\right) d\mathbf{v}
\end{equation*
here $f\left( \cdot \right) $ denotes the joint density function of $\mathbf
V}$ on $\mathcal{V}$. Thus the utility functions $\left\{ w_{j}\left(
a,v_{j}\right) \right\} $ and heterogeneity distribution $f\left( \cdot
\right) $ rationalize the choice probabilities $\left\{ q_{j}\left( \mathbf{
}\right) \right\} $.\smallskip
\end{theorem}
Condition (i) is intuitive, and corresponds to preferences being
non-satiated in the quantity of numeraire. Indeed, if choice probabilities
are generated by the structur
\begin{equation*}
q_{j}\left( \mathbf{a}\right) =\int_{\mathcal{V}}1\left\{ W_{j}\left(
a_{j},\eta \right) \geq \max_{r\in \left\{ 0,1,...J\right\} \backslash
\left\{ j\right\} }W_{r}\left( a_{r},\eta \right) \right\} f\left( \eta
\right) d\eta \text{,}
\end{equation*
where $W_{j}\left( ,\eta \right) $ are strictly increasing and continuous,
and their distributions sufficiently smooth, then condition (i) must hold.
Condition (iii) is related to the existence of a density function for
unobserved heterogeneity. For models with \textit{parametrically specified
heterogeneity distributions, condition (iii) was previously used to recover
underlying utility functions (c.f. McFadden, 1978, just above Eqn. 12). The
motivation for condition (ii) was discussed right before Theorem 1.\smallskip
\begin{proof}[Proof]
WLOG take $m=0$, and use condition (ii) of the theorem to defin
\begin{equation}
t_{j0}\left( a_{j},a_{0}\right) \equiv \frac{\partial }{\partial a_{0}
q_{j}\left( \mathbf{a}\right) /\frac{\partial }{\partial a_{j}}q_{0}\left(
\mathbf{a}\right) \geq 0\text{.} \label{s}
\end{equation
\qquad
Now, because $\sum_{j=0}^{J}q_{j}\left( \mathbf{a}\right) =1$,
differentiating both sides w.r.t. $a_{0}$ give
\begin{equation}
\frac{\partial }{\partial a_{0}}q_{0}\left( \mathbf{a}\right) +\sum_{j=1}^{J
\frac{\partial }{\partial a_{0}}q_{j}\left( \mathbf{a}\right) =0\text{.}
\label{1}
\end{equation
Substituting (\ref{s}) in (\ref{1}), we get the linear, homogeneous, partial
differential equation in $q_{0}\left( \cdot \right) $:
\begin{equation}
\frac{\partial }{\partial a_{0}}q_{0}\left( \mathbf{a}\right) +\sum_{j=1}^{J
\frac{\partial }{\partial a_{j}}q_{0}\left( \mathbf{a}\right) \times
t_{j0}\left( a_{j},a_{0}\right) =0\text{.} \label{t}
\end{equation}
This PDE can be solved via the method of characteristics (c.f. Courant,
1962, Chapter I.5 and II.2). The characteristic curve, i.e. the $J
-dimensional subspace on which $q_{0}\left( a\right) $ remains constant, can
be obtained as follows. Parametrize $a_{j}=a_{j}\left( r\right) $
j=0,1,...J $ and conside
\begin{equation*}
0=\frac{dq_{0}}{dr}=\frac{\partial q_{0}\left( \mathbf{a}\right) }{\partial
a_{0}}\frac{da_{0}\left( r\right) }{dr}+\sum_{j=1}^{J}\frac{\partial
q_{0}\left( \mathbf{a}\right) }{\partial a_{j}}\times \frac{da_{j}\left(
r\right) }{dr}\text{.}
\end{equation*
Comparing with (\ref{t}), we ge
\begin{equation*}
\frac{da_{0}\left( r\right) }{dr}=1,\text{ }\frac{da_{j}\left( r\right) }{dr
=t_{j0}\left( a_{j},a_{0}\right) \text{, }j=1,...J\text{,}
\end{equation*
implying the characteristic ordinary differential equations
\begin{equation}
\frac{da_{j}}{da_{0}}=t_{j0}\left( a_{j},a_{0}\right) \text{,} \label{v}
\end{equation
for $j=1,...,J$. Using Picard's theorem and the principle of solving linear
homogeneous PDEs (see Appendix), we obtain the general solutions of (\ref{v
) given by $\omega _{j}\left( a_{j},a_{0}\right) =cons$, where $\omega
_{j}\left( a_{j},a_{0}\right) $ is differentiable, strictly increasing in
a_{0}$ and strictly decreasing in $a_{j}$, and satisfie
\begin{equation}
\frac{\partial \omega _{j}\left( a_{j},a_{0}\right) }{\partial a_{0}}+\frac
\partial \omega _{j}\left( a_{j},a_{0}\right) }{\partial a_{j}}t_{j0}\left(
a_{j},a_{0}\right) =0\text{,} \label{4}
\end{equation
and also, using (\ref{s}
\begin{equation}
-\frac{\partial \omega _{j}\left( a_{j},a_{0}\right) }{\partial a_{0}}/\frac
\partial \omega _{j}\left( a_{j},a_{0}\right) }{\partial a_{j}}\equiv \frac
\partial }{\partial a_{0}}q_{j}\left( \mathbf{a}\right) /\frac{\partial }
\partial a_{j}}q_{0}\left( \mathbf{a}\right) \text{.} \label{2}
\end{equation
A general solution $q_{0}\left( \mathbf{a}\right) $ is therefore of the for
\begin{equation}
q_{0}\left( \mathbf{a}\right) =H_{0}\left( \omega _{1}\left(
a_{1},a_{0}\right) ,\omega _{2}\left( a_{2},a_{0}\right) ,...,\omega
_{J}\left( a_{J},a_{0}\right) \right) \text{,} \label{u}
\end{equation
where $H_{0}\left( \cdot \right) $ can be chosen to be strictly increasing
and $C^{1}$ in each argument, and with continuous $J$th order cross partial
derivatives. In particular, any $J$ dimensional continuously differentiable
C.D.F. $H_{0}\left( \cdot \right) $ would produce an admissible solution.
Since $q_{0}\left( \mathbf{a}\right) $ is observed, the exact functional
form of $H_{0}\left( \cdot \right) $ is pinned down by (\ref{u}), for any
set of solutions $\omega _{j}\left( \cdot ,\cdot \right) $ to the ODEs (\re
{v}). This corresponds to the so-called "initial condition" in the PDE
nomenclature. In particular, given any $a_{0}$, the value of $H_{0}\left(
x_{1},x_{2},...x_{J}\right) $ at any vector $\left(
x_{1},x_{2},...x_{J}\right) $ is given b
\begin{equation*}
H_{0}\left( x_{1},x_{2},...x_{J}\right) =q_{0}\left( a_{0},b_{1}\left(
x_{1},a_{0}\right) ,...b_{J}\left( x_{J},a_{0}\right) \right) \text{,}
\end{equation*
where $b_{j}\left( x_{j},a_{0}\right) $ is defined b
\begin{equation}
\omega _{j}\left( b_{j}\left( x_{j},a_{0}\right) ,a_{0}\right) =x_{j}
\label{e}
\end{equation
In this construction, the choice of $a_{0}$ is immaterial. That is, for two
choices $a_{0}\neq a_{0}^{\prime }$
\begin{eqnarray*}
&&q_{0}\left( a_{0},b_{1}\left( x_{1},a_{0}\right) ,...b_{J}\left(
x_{J},a_{0}\right) \right) \\
&=&H_{0}\left( \omega _{1}\left( b_{1}\left( x_{1},a_{0}\right)
,a_{0}\right) ,\omega _{2}\left( b_{2}\left( x_{2},a_{0}\right)
,a_{0}\right) ,...,\omega _{J}\left( b_{J}\left( x_{J},a_{0}\right)
,a_{0}\right) \right) \\
&=&H_{0}\left( x_{1},x_{2},...x_{J}\right) \\
&=&H_{0}\left( \omega _{1}\left( b_{1}\left( x_{1},a_{0}^{\prime }\right)
,a_{0}^{\prime }\right) ,\omega _{2}\left( b_{2}\left( x_{2},a_{0}^{\prime
}\right) ,a_{0}^{\prime }\right) ,...,\omega _{J}\left( b_{J}\left(
x_{J},a_{0}^{\prime }\right) ,a_{0}^{\prime }\right) \right) \\
&=&q_{0}\left( a_{0}^{\prime },b_{1}\left( x_{1},a_{0}^{\prime }\right)
,...b_{J}\left( x_{J},a_{0}^{\prime }\right) \right) \text{.}
\end{eqnarray*}
Having obtained the $\omega _{j}\left( \cdot ,\cdot \right) $'s from (\ref{v
) and (\ref{4}), for each $j=1,...J$, define the function $w_{j}\left(
a_{j},v\right) $ by inversion, i.e
\begin{equation}
w_{j}\left( a_{j},v\right) =\left\{ a_{0}:\omega _{j}\left(
a_{j},a_{0}\right) =v\right\} \text{.} \label{7}
\end{equation
Note that by construction, $w_{j}\left( a_{j},v\right) $ is strictly
increasing and continuous in $a_{j}$ for each $v$. The $w_{j}\left( \cdot
.\cdot \right) $'s will play the role of `utilities' in our proof of
integrability. Set $w_{0}\left( a_{0},v_{0}\right) \equiv a_{0}$.
We now show how to construct the distribution of heterogeneity. Let
\mathcal{\bar{V}}_{j}$ denote the co-domain of $\omega _{j}\left( \cdot
,\cdot \right) $, and le
\begin{equation*}
\mathcal{V}_{j}=\mathcal{\bar{V}}_{j}\cap \left\{ \omega _{j}\left(
a_{j},a_{0}\right) :\dprod_{j=1}^{J}\left\{ \frac{\partial }{\partial a_{0}
\omega _{j}\left( a_{j},a_{0}\right) \times \frac{\partial }{\partial a_{j}
\omega _{j}\left( a_{j},a_{0}\right) \right\} \neq 0\right\} \text{,}
\end{equation*
and let $\mathcal{V}\equiv \times _{j=1}^{J}\mathcal{V}_{j}$. Now, given any
vector $\mathbf{v}\equiv \left( v_{1},...,v_{J}\right) \in \mathcal{V}$,
define the cumulative distribution function at $\mathbf{v}$ a
\begin{equation*}
F\left( v_{1},...,v_{J}\right) =q_{0}\left( a_{0},a_{1},...,a_{J}\right)
\text{,}
\end{equation*
where the vector $\left( a_{0},a_{1},...,a_{J}\right) $ satisfies
v_{j}=\omega _{j}\left( a_{j},a_{0}\right) $, for each $j=1,...J$. It
follows from (\ref{u}) and (\ref{e}) that this function is well-defined. The
above CDF implies the density function $f:\mathcal{V}\rightarrow \mathcal{R
^{+}$
\begin{eqnarray}
&&f\left( v_{1},...,v_{J}\right) \notag \\
&=&\frac{\frac{\partial ^{J}}{\partial a_{1}...\partial a_{J}}q_{0}\left(
a_{0},a_{1},...,a_{J}\right) |_{v_{j}=\omega _{j}\left( a_{j},a_{0}\right)
\text{, }j=1,...J}}{\dprod_{j=1}^{J}\frac{\partial }{\partial a_{j}}\omega
_{j}\left( a_{j},a_{0}\right) |_{v_{j}=\omega _{j}\left( a_{j},a_{0}\right)
\text{, }j=1,...J}} \label{5} \\
&=&\frac{\frac{\partial ^{J-1}}{\partial a_{1}...\partial a_{k-1}\partial
a_{k+1}...\partial a_{J}}\frac{\partial }{\partial a_{k}}q_{0}\left(
a_{0},a_{1},...,a_{J}\right) |_{v_{j}=\omega _{j}\left( a_{j},a_{0}\right)
\text{, }j=1,...J}}{\dprod_{j=1}^{J}\frac{\partial }{\partial a_{j}}\omega
_{j}\left( a_{j},a_{0}\right) |_{v_{j}=\omega _{j}\left( a_{j},a_{0}\right)
\text{, }j=1,...J}}\text{, for any }k\in \left\{ 1,...,J\right\} \notag \\
&=&\frac{\frac{\partial ^{J-1}}{\partial a_{1}...\partial a_{k-1}\partial
a_{k+1}...\partial a_{J}}\left[ -\underset{\text{ does not depend on
a_{1}...a_{k-1},a_{k+1}...a_{J}\text{ }}{\underbrace{\frac{\frac{\partial }
\partial a_{k}}\omega _{k}\left( a_{k},a_{0}\right) }{\frac{\partial \omega
_{k}\left( a_{k},a_{0}\right) }{\partial a_{0}}}}}\times \frac{\partial }
\partial a_{0}}q_{k}\left( a_{0},a_{1},...,a_{J}\right) \right]
|_{v_{j}=\omega _{j}\left( a_{j},a_{0}\right) }}{\dprod_{j=1}^{J}\frac
\partial }{\partial a_{j}}\omega _{j}\left( a_{j},a_{0}\right)
|_{v_{j}=\omega _{j}\left( a_{j},a_{0}\right) \text{, }j=1,...J}}\text{,
from (\ref{2})} \notag \\
&=&-\frac{\frac{\partial ^{J-1}}{\partial a_{1}...\partial a_{k-1}\partial
a_{k+1}...\partial a_{J}}\frac{\partial }{\partial a_{0}}q_{k}\left(
a_{0},a_{1},...,a_{J}\right) |_{v_{j}=\omega _{j}\left( a_{j},a_{0}\right)
\text{, }j=1,...J}}{\frac{\frac{\partial }{\partial a_{0}}\omega _{k}\left(
a_{k},a_{0}\right) }{\frac{\partial }{\partial a_{k}}\omega _{k}\left(
a_{k},a_{0}\right) }\times \dprod_{j=1}^{J}\frac{\partial }{\partial a_{j}
\omega _{j}\left( a_{j},a_{0}\right) |_{v_{j}=\omega _{j}\left(
a_{j},a_{0}\right) \text{, }j=1,...J}} \notag \\
&=&-\frac{\frac{\partial ^{J-1}}{\partial a_{1}...\partial a_{k-1}\partial
a_{k+1}...\partial a_{J}}\frac{\partial }{\partial a_{0}}q_{k}\left(
a_{0},a_{1},...,a_{J}\right) |_{v_{j}=\omega _{j}\left( a_{j},a_{0}\right)
\text{, }j=1,...J}}{\frac{\partial }{\partial a_{0}}\omega _{k}\left(
a_{k},a_{0}\right) \times \dprod_{j=1,j\neq k}^{J}\frac{\partial }{\partial
a_{j}}\omega _{j}\left( a_{j},a_{0}\right) |_{v_{j}=\omega _{j}\left(
a_{j},a_{0}\right) \text{, }j=1,...J}}\text{.} \label{6}
\end{eqnarray
Since $\frac{\partial ^{J}}{\partial a_{0}\partial a_{1}...\partial
a_{k-1}\partial a_{k+1}\partial a_{J}}q_{k}\left(
a_{0},a_{1},...a_{J}\right) $ has sign $\left( -1\right) ^{J}$ and $\frac
\partial }{\partial a_{j}}\omega _{j}\left( a_{j},a_{0}\right) <0$, and
\frac{\partial }{\partial a_{0}}\omega _{j}\left( a_{j},a_{0}\right) >0$ on
\mathcal{V}$, each of the above expressions has numerator and denominator of
the same sign, and is thus non-negative. We verify below that this joint
density integrates to 1.
We now show that the above construction of $w_{j}\left( \cdot ,\cdot \right)
$ (c.f. (\ref{7})) and the joint density of heterogeneity (\ref{5}) and (\re
{6}) will indeed produce the original choice probabilities. To see this for
alternative 1, consider the integra
\begin{eqnarray*}
&&\int_{\mathcal{V}}1\left\{ w_{1}\left( a_{1},v_{1}\right) \geq \max_{k\in
\left\{ 0,2,...J\right\} }w_{k}\left( a_{k},v_{k}\right) \right\} f\left(
v_{1},v_{2},...,v_{1}\right) dv_{1}...dv_{J} \\
&=&\int_{\mathcal{V}}1\left[ v_{1}\geq \omega _{1}\left( a_{1},a_{0}\right)
,\cap _{k\in \left\{ 2,...J\right\} }1\left\{ v_{k}\leq \omega _{k}\left(
a_{k},w_{1}\left( a_{1},v_{1}\right) \right) \right\} \right] f\left(
v_{1},v_{2},...,v_{1}\right) dv_{1}...dv_{J}
\end{eqnarray*
Consider the substitution $\left( v_{1},v_{2},...v_{J}\right) \rightarrow
\left( x_{1},x_{2},...x_{J}\right) $ given by $v_{1}=\omega _{1}\left(
a_{1},x_{1}\right) $ (so that $x_{1}=w_{1}\left( a_{1},v_{1}\right) $), and
for $k=2,...,J$, $v_{k}=\omega _{k}\left( x_{k},x_{1}\right) $, which
transforms the above integral t
\begin{eqnarray}
&&\int_{a_{0}}^{\infty }\int_{a_{2}}^{\infty }...\int_{a_{J}}^{\infty }\left[
\begin{array}{c}
f\left( \omega _{1}\left( a_{1},x_{1}\right) ,\omega _{2}\left(
x_{2},x_{1}\right) ...,\omega _{J}\left( x_{J},x_{1}\right) \right) \\
\times \left\vert \frac{\partial \omega _{1}\left( a_{1},x_{1}\right) }
\partial x_{1}}\times \dprod_{k=2}^{J}\frac{\partial \omega _{j}\left(
x_{j},x_{1}\right) }{\partial x_{j}}\right\ver
\end{array
\right] dx_{J}...dx_{2}dx_{1} \notag \\
&=&\int_{a_{0}}^{\infty }\int_{a_{2}}^{\infty }...\int_{a_{J}}^{\infty }
\left[
\begin{array}{c}
f\left( \omega _{1}\left( a_{1},x_{1}\right) ,\omega _{2}\left(
x_{2},x_{1}\right) ...,\omega _{J}\left( x_{J},x_{1}\right) \right) \\
\times \left( -1\right) ^{J-1}\times \frac{\partial \omega _{1}\left(
a_{1},x_{1}\right) }{\partial x_{1}}\times \dprod_{k=2}^{J}\frac{\partial
\omega _{j}\left( x_{j},x_{1}\right) }{\partial x_{j}
\end{array
\right] dx_{J}...dx_{2}dx_{1} \notag \\
&=&\left( -1\right) ^{J-1}\times \int_{a_{0}}^{\infty }\int_{a_{2}}^{\infty
}...\int_{a_{J}}^{\infty }\left\{ -\frac{\partial ^{J}}{\partial
x_{1}\partial x_{2}...\partial x_{J}}q_{1}\left(
x_{1},a_{1},x_{2},...x_{J}\right) \right\} dx_{J}...dx_{2}dx_{1}\text{, by
\ref{6})} \notag \\
&=&\left( -1\right) ^{J}\times \int_{a_{0}}^{\infty }\int_{a_{2}}^{\infty
}...\int_{a_{J}}^{\infty }\left\{ \frac{\partial ^{J}}{\partial
x_{1}\partial x_{2}...\partial x_{J}}q_{1}\left(
x_{1},a_{1},x_{2},...x_{J}\right) \right\} dx_{J}...dx_{2}dx_{1} \notag \\
&=&\int_{\infty }^{a_{0}}\int_{\infty }^{a_{2}}...\int_{\infty
}^{a_{J}}\left\{ \frac{\partial ^{J}}{\partial x_{1}\partial
x_{2}...\partial x_{J}}q_{1}\left( x_{1},a_{1},x_{2},...x_{J}\right)
\right\} dx_{J}...dx_{2}dx_{1} \notag \\
&=&q_{1}\left( a_{0},a_{1},a_{2},...a_{J}\right) \text{.} \label{13}
\end{eqnarray}
Exactly analogous steps for $j=2,...J$, and using (\ref{6}), lead to the
conclusion that for all $j\geq 1$
\begin{eqnarray*}
&&\int 1\left\{ w_{j}\left( a_{j},v_{j}\right) \geq \max_{k\in \left\{
0,1,2,...J\right\} \backslash \left\{ j\right\} }w_{k}\left(
a_{k},v_{k}\right) \right\} f\left( v_{1},v_{2},...,v_{1}\right)
dv_{1}...dv_{J} \\
&=&q_{j}\left( a_{0},a_{1},a_{2},...a_{J}\right) \text{.}
\end{eqnarray*}
Also, note tha
\begin{eqnarray*}
&&\int 1\left\{ a_{0}\geq \max_{k\in \left\{ 1,2,...J\right\} }w_{k}\left(
a_{k},v_{k}\right) \right\} f\left( v_{1},v_{2},...,v_{J}\right)
dv_{1}...dv_{J} \\
&=&\int_{0}^{\omega _{1}\left( a_{1},a_{0}\right) }...\int_{0}^{\omega
_{J}\left( a_{J},a_{0}\right) }f\left( v_{1},v_{2},...,v_{J}\right)
dv_{J}...dv_{1} \\
\text{substitute }v_{j} &\rightarrow &x_{j}\text{ satisfying }v_{j}=\omega
_{j}\left( x_{j},a_{0}\right) \\
&=&\int_{a_{1}}^{\infty }\int_{a_{2}}^{\infty }...\int_{a_{J}}^{\infty
}f\left( \omega _{1}\left( x_{1},a_{0}\right) ,...,\omega _{J}\left(
x_{J},a_{0}\right) \right) \left\vert \frac{\partial \omega _{1}\left(
x_{1},a_{0}\right) }{\partial x_{1}}...\frac{\partial \omega _{J}\left(
x_{J},a_{0}\right) }{\partial x_{J}}\right\vert dx_{J}...dx_{1} \\
&=&\int_{a_{1}}^{\infty }\int_{a_{2}}^{\infty }...\int_{a_{J}}^{\infty
}\left( -1\right) ^{J}\times f\left( \omega _{1}\left( x_{1},a_{0}\right)
,...,\omega _{J}\left( x_{J},a_{0}\right) \right) \frac{\partial \omega
_{1}\left( x_{1},a_{0}\right) }{\partial x_{1}}...\frac{\partial \omega
_{J}\left( x_{J},a_{0}\right) }{\partial x_{J}}dx_{J}...dx_{1} \\
&=&\int_{\infty }^{a_{1}}...\int_{\infty }^{a_{J}}\frac{\partial ^{J}}
\partial \alpha _{1}...\partial \alpha _{J}}q_{0}\left( a_{0},\alpha
_{1},...\alpha _{J}\right) |_{\alpha _{1}=x_{1},...\alpha
_{J}=x_{J}}dx_{J}...dx_{1}\text{, by (\ref{5})} \\
&=&q_{0}\left( a_{0},a_{1},...a_{J}\right) \text{.}
\end{eqnarray*}
Finally, to show that the joint density (\ref{5}) integrates to 1, use
exactly the same substitution as the one leading to (\ref{13}), and observe
tha
\begin{eqnarray*}
&&\int f\left( v_{1},v_{2},...,v_{J}\right) dv_{1}...dv_{J} \\
&=&\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }...\int_{-\infty
}^{\infty }\left[
\begin{array}{c}
f\left( \omega _{1}\left( a_{1},x_{1}\right) ,\omega _{2}\left(
x_{2},x_{1}\right) ...,\omega _{J}\left( x_{J},x_{1}\right) \right) \\
\times \left\vert \frac{\partial \omega _{1}\left( a_{1},x_{1}\right) }
\partial x_{1}}\times \dprod_{k=2}^{J}\frac{\partial \omega _{j}\left(
x_{j},x_{1}\right) }{\partial x_{j}}\right\ver
\end{array
\right] dx_{2}...dx_{J}dx_{1} \\
&=&\left( -1\right) ^{J}\times \int_{-\infty }^{\infty }\int_{-\infty
}^{\infty }...\int_{-\infty }^{\infty }\left\{ \frac{\partial ^{J}}{\partial
x_{1}\partial x_{2}...\partial x_{J}}q_{1}\left(
x_{1},a_{1},x_{2},...x_{J}\right) \right\} dx_{2}...dx_{J}dx_{1} \\
&=&q_{1}\left( -\infty ,a_{1},-\infty ,...-\infty \right) \\
&=&1\text{,}
\end{eqnarray*
where $q_{1}\left( -\infty ,a_{1},-\infty ,...-\infty \right) $ denotes the
limit of the choice probability of alternative 1 when the $a$'s for all
other alternatives are tend to $-\infty $.
Thus we have shown that a population endowed with our constructed
w_{j}\left( \cdot ,v_{j}\right) $ as utilities, together with the joint
density of heterogeneity given by (\ref{5}) would indeed produce the choice
probabilities $\left\{ q_{j}\left( \cdot ,...\cdot \right) \right\} $ for
each $j=0,1,...J$.\smallskip
\end{proof}
\begin{remark}
The \textbf{utility}\textit{\ function} for each alternative $j$, viz.
w_{j}\left( a_{j},v_{j}\right) $, constructed in the proof of Theorem 1,
consists of a scalar heterogeneity $v_{j}$. However, the individual \textbf
demand}\textit{\ function} for alternative $j$ has $J$ separate sources of
heterogeneity, i.e
\begin{eqnarray*}
Q_{j}\left( \mathbf{a,v}\right) &=&1\left\{ w_{j}\left( a_{j},v_{j}\right)
\geq \max_{r\in \left\{ 0,1,...J\right\} \backslash \left\{ j\right\}
}w_{r}\left( a_{r},v_{r}\right) \right\} \\
&=&Q_{j}\left( a_{0},a_{1},...a_{J},\underset{J\text{ dimensional
heterogeneity}}{\underbrace{v_{1},v_{2},...,v_{J}}}\right)
\end{eqnarray*
Thus, we have rationalized a $\left( J+1\right) $ dimensional choice
probability function via a $J$-dimensional heterogeneity distribution.
\end{remark}
\section{Further Points}
\textbf{Identification}: Theorem 1 can also be used to identify utilities
and the heterogeneity distributions nonparametrically from
choice-probabilities observed in a dataset. Nonparametric identification of
multinomial choice models has been studied previously in the econometric
literature (c.f. Matzkin, 1993). Since our proof of rationalizability
presented in Theorem 1 is constructive, it provides an alternative and novel
way to obtain identification by solving PDEs.
Toward that end, suppose that the choice-probabilities are generated by
maximization of the utilities $u_{j}\equiv \left\{ h_{j}\left( a_{j}\right)
+\varepsilon _{j}\right\} $, $j=0,...,J$, where the $h_{j}\left( \cdot
\right) $ functions are strictly increasing and continuous, and hence
invertible. Observe that an observationally equivalent utility structure is
where utility for the $0$th alternative is $a_{0}$ and that for the $j$th
alternative is $h_{0}^{-1}\left( h_{j}\left( a_{j}\right) +\underset{v_{j}}
\underbrace{\varepsilon _{j}-\varepsilon _{0}}}\right) \equiv w_{j}\left(
a_{j},v_{j}\right) $, in that these utilities will produce exactly the same
choice probabilities as the $\left\{ u_{j}\right\} $s. We work under this
normalization from now on. We also note in passing that the $w_{j}\left(
a_{j},v_{j}\right) $ are not necessarily additive in the unobserved
heterogeneity $v_{j}$.
Let $\mathbf{a}$ and $q_{j}\left( \mathbf{a}\right) $ be as above. We can
use the proof of Theorem 1 to identify the $w_{j}\left( a_{j},v_{j}\right) $
functions and the joint distribution of $\left( v_{1},...,v_{J}\right) $
from the $\left\{ q_{j}\left( \mathbf{a}\right) \right\} $, as follows.
First, note that
\begin{equation*}
q_{0}\left( \mathbf{a}\right) =\Pr \left( \cap _{j\neq 0}\left\{
a_{0}>w_{j}\left( a_{j},v_{j}\right) \right\} \right) =\Pr \left[ \cap
_{j\neq 0}\left\{ v_{j}<\omega _{j}\left( a_{j},a_{0}\right) \right\} \right]
\text{,}
\end{equation*
so tha
\begin{equation}
\frac{\partial }{\partial a_{j}}q_{0}\left( \mathbf{a}\right) =\frac
\partial }{\partial a_{j}}\omega _{j}\left( a_{j},a_{0}\right) \times
F_{j}\left( \omega _{1}\left( a_{1},a_{0}\right) ,...,\omega _{J}\left(
a_{J},a_{0}\right) \right) \text{,} \label{14a}
\end{equation
where $F_{j}\left( \cdot \right) $ denotes the derivative of the joint
distribution function of $\mathbf{v}$ w.r.t. its $j$th element. On the other
hand
\begin{eqnarray*}
q_{j}\left( \mathbf{a}\right) &=&\Pr [w_{j}\left( a_{j},v_{j}\right)
>a_{0},w_{j}\left( a_{j},v_{j}\right) >w_{1}\left( a_{1},v_{1}\right)
,...w_{j}\left( a_{j},v_{j}\right) >w_{J}\left( a_{J},v_{J}\right) \\
&=&\Pr [v_{j}>\omega _{j}\left( a_{j},a_{0}\right) ,v_{1}<\omega _{1}\left(
a_{1},w_{j}\left( a_{j},v_{j}\right) \right) ,...v_{J}<\omega _{J}\left(
a_{J},w_{j}\left( a_{j},v_{j}\right) \right) \\
&=&\int_{\omega _{j}\left( a_{j},a_{0}\right) }^{\infty }\int_{-\infty
}^{\omega _{1}\left( a_{1},w_{j}\left( a_{j},v_{j}\right) \right)
}...\int_{-\infty }^{\omega _{J}\left( a_{J},w_{j}\left( a_{j},v_{j}\right)
\right) }f\left( v_{1},...,v_{J}\right) dv_{J}...dv_{1}dv_{j}\text{,}
\end{eqnarray*
and therefore, by chain-rule, the first fundamental theorem of calculus, and
using $w_{j}\left( a_{j},\omega _{j}\left( a_{j},a_{0}\right) \right) =a_{0}
, we have that
\begin{eqnarray}
\frac{\partial }{\partial a_{0}}q_{j}\left( \mathbf{a}\right) &=&-\frac
\partial }{\partial a_{0}}\omega _{j}\left( a_{j},a_{0}\right) \times
\int_{-\infty }^{\omega _{1}\left( a_{1},a_{0}\right) }...\int_{-\infty
}^{\omega _{J}\left( a_{J},a_{0}\right) }f\left( v_{1},...,v_{J}\right)
dv_{J}...dv_{1} \notag \\
&=&-\frac{\partial }{\partial a_{0}}\omega _{j}\left( a_{j},a_{0}\right)
\times F_{j}\left( \omega _{1}\left( a_{1},a_{0}\right) ,...,\omega
_{J}\left( a_{J},a_{0}\right) \right) \text{,} \label{14b}
\end{eqnarray
and thus from (\ref{14a}) and (\ref{14b}), we have tha
\begin{equation}
-\frac{\partial \omega _{j}\left( a_{j},a_{0}\right) }{\partial a_{0}}/\frac
\partial \omega _{j}\left( a_{j},a_{0}\right) }{\partial a_{j}}\equiv \frac
\partial }{\partial a_{0}}q_{j}\left( \mathbf{a}\right) /\frac{\partial }
\partial a_{j}}q_{0}\left( \mathbf{a}\right) \text{,} \label{11}
\end{equation
which is the same as (\ref{2}). The RHS of (\ref{11}) is observable from the
data, and under the hypothesis of the model, is solely a function of $a_{0}$
and $a_{j}$, which is a testable implication. If this implication is not
rejected, denote the RHS of (\ref{11}) as $t_{j}\left( a_{j},a_{0}\right) $
(this $t_{j}\left( \cdot ,\cdot \right) $\ can be estimated by, say a least
squares projection of $\frac{\partial }{\partial a_{0}}q_{j}\left( \mathbf{a
\right) /\frac{\partial }{\partial a_{j}}q_{0}\left( \mathbf{a}\right) $ on
a polynomial sieve in $a_{j},a_{0}$). Then solve the PD
\begin{equation*}
\frac{\partial \omega _{j}\left( a_{j},a_{0}\right) }{\partial a_{0}}+\frac
\partial \omega _{j}\left( a_{j},a_{0}\right) }{\partial a_{j}}t_{j0}\left(
a_{j},a_{0}\right) =0\text{,}
\end{equation*
for the $\omega _{j}\left( \cdot ,\cdot \right) $'s as outlined above in
\ref{v}), obtain the $w_{j}\left( a_{j},v_{j}\right) $ by inverting the
solution $\omega _{j}\left( a_{j},a_{0}\right) $'s w.r.t. $a_{0}$, and the
joint density of $\mathbf{v}$ using (\ref{5}).\medskip
\textbf{Incorporating Covariates}: In our discussion above, choice
probabilities $q_{j}\left( \cdot \right) $ defined in Section 2, correspond
to so-called \textquotedblleft structural\textquotedblright\ parameters in
Econometrics. Estimating these from a non-experimental dataset might be
non-trivial when observed budget sets (i.e. price and/or income) are
correlated with unobserved individual preferences across the cross-section
of consumers. A common empirical assumption is that budget sets and
preferences are independent, conditional on a set of observed covariates.
Hence it is useful to see how to adapt the above results to the presence of
covariates.\footnote
If this \textit{conditional} independence of preferences and budget sets is
also suspect, then one needs to employ a \textquotedblleft control
function\textquotedblright\ type strategy (c.f. Blundell and Powell, 2004)
to estimate the structural choice-probabilities. Indeed, our results above
explore the connection between random utility models and \textquotedblleft
structural\textquotedblright\ choice probabilities. So, given the extensive
econometric literature on estimating structural parameters under
endogeneity, we refrain from discussing the consistent estimation of
q_{j}\left( \mathbf{\cdot }\right) $ any further.}
Suppose in addition to price and income, we also observe a vector of
consumer characteristics, denoted by $s$, and a vector of characteristics
z_{j}$ for each alternative $j=1,...,J$. Assume that the
choice-probabilities are generated by maximization of the utilities
u_{0}\equiv \left\{ h_{0}\left( a_{0},s\right) +\varepsilon _{0}\right\} $,
and $u_{j}\equiv \left\{ h_{j}\left( a_{j},z_{j},s\right) +\varepsilon
_{j}\right\} $, $j=1,...,J$, where $h_{0}\left( a,s\right) $ and each
h_{j}\left( a,z,s\right) $ are strictly increasing and continuous in $a$,
and hence invertible. Then an observationally equivalent utility structure
is where utility for the $0$th alternative is $a_{0}$ and that for the $j$th
alternative is $h_{0}^{-1}\left( h_{j}\left( a_{j},z_{j},s\right) +\underset
v_{j}}{\underbrace{\varepsilon _{j}-\varepsilon _{0}}},s\right) \equiv
w_{j}\left( a_{j},z_{j},v_{j},s\right) $, which is in general not linear or
separable in $v_{j}$. Working off this normalization, and essentially
repeating the same steps as above holding $z_{j}$,$s$ fixed, lead to the
conclusion that for each $z_{j}$, $s$
\begin{equation}
-\frac{\partial \omega _{j}\left( a_{j},a_{0},z_{j},s\right) }{\partial a_{0
}/\frac{\partial \omega _{j}\left( a_{j},a_{0},z_{j},s\right) }{\partial
a_{j}}\equiv \frac{\partial }{\partial a_{0}}q_{j}\left( \mathbf{a,z,
s\right) /\frac{\partial }{\partial a_{j}}q_{0}\left( \mathbf{a,z,}s\right)
\text{.} \label{111}
\end{equation
The RHS of (\ref{111}) is observable from the data, and for each fixed
z_{j} $ and $s$, is solely a function of $a_{0}$, $a_{j}$, which is a
testable implication. If this implication is not rejected, denote the RHS of
(\ref{111}) as $t_{j}\left( a_{j},a_{0},z_{j},s\right) $, just as above.
Then for each each fixed $z_{j}$ and $s$, solve the PD
\begin{equation*}
\frac{\partial \omega _{j}\left( a_{j},a_{0},z_{j},s\right) }{\partial a_{0}
+\frac{\partial \omega _{j}\left( a_{j},a_{0},z_{j},s\right) }{\partial a_{j
}t_{j}\left( a_{j},a_{0},z_{j},s\right) =0\text{,}
\end{equation*
to obtain the $\omega _{j}\left( a_{j},a_{0},z_{j},s\right) $, invert w.r.t.
$a_{0}$ to obtain the utilities $w_{j}\left( a_{j},v_{j},z_{j},s\right) $
and the joint density of $\mathbf{v}$ using the analog of (\ref{5}), where
we utilize the inverse of $\omega _{j}\left( a_{j},a_{0},z_{j},s\right) $
w.r.t. $a_{j}$, analogous to (\ref{e}) above.
\begin{center}
{\Large References}
\end{center}
\begin{enumerate}
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theory of product differentiation. MIT press.
\item Armstrong, M. and Vickers, J., 2015. Which demand systems can be
generated by discrete choice?. Journal of Economic Theory, 158, pp.293-307.
\item Coddington EA. 1961. An introduction to ordinary differential
equations. Dover Publishing, New York.
\item Courant, Richard, 1962. Methods of Mathematical Physics, Vol. 2,
Interscience, New York.
\item Daly, A., and Zachary, S.,1978. Improved multiple choice models. In
Hensher,D., Dalvi,Q.(Eds.), Identifying and Measuring the Determinants of
Mode Choice, Teakfields, London.
\item Lewbel, A., 2001. Demand Systems with and without Errors. American
Economic Review, 91(3), pp.611-618.
\item Mas-Colell, A., Whinston, M.D. and Green, J.R., 1995. Microeconomic
theory. New York: Oxford university press.
\item Matzkin R., 1993. Nonparametric identification and estimation of
polychotomous choice models. Journal of Econometrics, 58(1-2):137-68.
\item McFadden, D., 1973. Conditional logit analysis of qualitative choice
behavior.
\item McFadden, D., 1978. Modeling the choice of residential location.
Transportation Research Record, (673).
\end{enumerate}
\section{Appendix}
Two basic ideas from the theory of partial and ordinary differential
equations are used to prove Theorem 1. We will use the notation $C^{1}$ to
indicate a function that is once continuously differentiable.\medskip
First consider the linear homogeneous PD
\begin{equation}
\frac{\partial \sigma \left( x,y,z\right) }{\partial x}+g_{2}\left(
x,y\right) \frac{\partial \sigma \left( x,y,z\right) }{\partial y
+g_{3}\left( x,z\right) \frac{\partial \sigma \left( x,y,z\right) }{\partial
z}=0\text{.} \label{9}
\end{equation
Suppose $g_{2}$ and $g_{3}$ are $C^{1}$ and do not vanish simultaneously.
Then a general solution to this equation is given b
\begin{equation*}
\sigma \left( x,y,z\right) =\phi \left( h_{2}\left( x,y\right) ,h_{3}\left(
x,z\right) \right) \text{,}
\end{equation*
where $\phi \left( \cdot \right) $ is \textit{any} arbitrary $C^{1}$
function, and $h_{2}\left( x,y\right) =c_{2}$ and $h_{3}\left( x,z\right)
=c_{3}$ are general solutions to the ordinary differential equation
\begin{equation}
\frac{dy}{dx}=g_{2}\left( x,y\right) \text{, }\frac{dz}{dx}=g_{3}\left(
x,z\right) \text{.} \label{8}
\end{equation
(See e.g. Courant, 1962, Chapter I.5, II.2). In particular, $\phi \left(
\cdot ,\cdot \right) $ can be chosen to be strictly increasing in both
arguments.
\bigskip
The ODE (\ref{8}) are referred to as the "characteristic equations" of the
linear PDE (\ref{9}), and existence of a solution to the PDE (\ref{9})
amounts to existence of a solution of the ODE (\ref{8}). The following lemma
restates a global version of the Picard-Lindel\"{o}f theorem that
establishes conditions for existence of a solution to a first-order ODE.
\medskip
\begin{lemma}[Picard-Lindel\"{o}f theorem]
Suppose that a function $g:\mathcal{R}\mathbb{\times }\mathcal{R\rightarrow
}$ is continuous, and on each strip $S_{a}=\left\{ \left( x,y\right)
:\left\vert x\right\vert \leq a,\text{ }\left\vert y\right\vert <\infty
\right\} $, $g\left( x,y\right) $ is Lipschitz in $y$. Then the ordinary
differential equation $n^{\prime }\left( x\right) =g\left( x,n\left(
x\right) \right) $, has a general solution $n\left( \cdot \right) :\mathcal{
}\rightarrow \mathcal{R}$ with $n\left( \cdot \right) $ being $C^{1}$. (See,
for instance, Coddington, 1961, Theorem 9 and corollary).
\end{lemma}
This result is proved by showing that under the assumptions of the lemma,
the map $n\left( \cdot \right) :\rightarrow \int_{x_{0}}^{x}g\left(
s,n\left( s\right) \right) ds$ for any arbitrary $x_{0}$ is a contraction,
thereby ensuring, via the Banach fixed point theorem, the existence of
n\left( \cdot \right) $ satisfyin
\begin{equation*}
n\left( x\right) =n\left( x_{0}\right) +\int_{x_{0}}^{x}g\left( s,n\left(
s\right) \right) ds\text{.}
\end{equation*}
\end{document}
|
{
"timestamp": "2019-03-01T02:15:50",
"yymm": "1902",
"arxiv_id": "1902.11017",
"language": "en",
"url": "https://arxiv.org/abs/1902.11017"
}
|
\section{Introduction}
Approximate computing is an emerging paradigm that allows to develop highly energy-efficient computing systems such as various hardware accelerators for image filtering, video processing and data mining. It capitalizes inherent error resilience of many applications to trade Quality of Result (QoR) with energy efficiency. At the circuit level, functional approximation is achieved by employing approximate implementations for carefully selected operations of the accelerator. Literature contains a good body of works dealing with automated design methods for approximate circuits, e.g., CGP~\cite{vasicek:TEC}, SALSA~\cite{Salsa}, SASIMI~\cite{sasimi}. Majority of these works focus on elementary approximate circuits such as approximate adders and multipliers because they are building blocks of many applications. Approximate implementations of arithmetic circuits can also be downloaded (at the level of synthesized netlist or C code) from open source libraries such as~\cite{mrazek:date16lib} or created using quality-configurable approximate structures (such as QuAd adders~\cite{hanif:quad}, GeAR adders~\cite{Shafique:gear2015}, structured multipliers~\cite{Rehman:2016} or Broken-array multipliers (BAM)~\cite{jiang,Mahdiani:2010}). All the approximate circuits available in these libraries are fully characterized in terms of electrical properties and various error metrics.
Because these libraries contain from tens to thousands of approximate implementations for each arithmetic operation, the user is provided with a broad set of implementation options to reach the best possible tradeoff between QoR and energy (or other hardware parameters) at the accelerator level. However, \textbf{it is intractable to find an optimal combination of approximate circuits} even for an accelerator consisting of a few operations. The problem addressed in this paper is to \textit{identify the most suitable replacement of arithmetic operations of target accelerator with approximate circuits available in the library}. As it is a multi-objective optimization problem, there is no single optimal solution, rather multiple ones typically exist. We are primarily interested in approximate circuits belonging to the \emph{Pareto frontier} that contains the so-called non-dominated solutions. Consider two objectives to be minimized, for example, the mean error and energy. Circuit C1 (Pareto) \emph{dominates} another circuit C2 if: 1) C1 is no worse than C2 in all objectives and 2) C1 is strictly better than C2 in at least one objective.
This problem resembles the binding step of the high-level synthesis (HLS) whose objective is to (i) map elementary operations of the algorithm to specific instances of components that are available in the component library, and (ii) optimize hardware parameters such as latency, area and power consumption. In the context of approximate circuits, the principal difference and \textbf{difficulty lies in the QoR evaluation} at the accelerator level. Except some very specific cases (e.g.~\cite{Mazahir:tc17mult,Mazahir:2017adders}), it is in general unknown how the errors propagate if two or more approximate circuits are connected in a more complex circuit. A common approach is to estimate the resulting error using either analytic or statistical techniques, but it usually is a very unreliable approach as seen in~\cite{Li:2015}. If the problem is simplified in such a way that the only approximation technique is truncation then an optimal number of bits to be approximated can be determined~\cite{Sengupta:dac17}.
\textbf{Proposed Methodology:} In this paper, our objective is to identify the most suitable replacement of arithmetic operations (of the original accelerator) with approximate circuits. It is assumed that approximate circuits available in a library are fully characterized (in terms of error and hardware parameters), but nothing is assumed about their internal structure (i.e., an arbitrary approximation technique can be used to build the elementary approximate circuit, not only truncation). As a huge number of candidate replacements exist, the key idea is to eliminate as many clearly sub-optimal solutions as possible without performing precise evaluation of QoR and time-consuming circuit synthesis at the accelerator level. In order to estimate QoR, we propose to build a computational model using the error metrics (which are pre-calculated for each approximate circuit in the library) and machine learning techniques. The error model is then used to estimate QoR of candidate designs during the design space exploration process. Similarly, another computational model is constructed and applied to estimate hardware parameters in the design space exploration process. A similar approach has already been applied for common circuits on FPGAs~\cite{dai:2018}. In the context of approximate computing, machine learning techniques were applied to estimate QoR in the design of approximate accelerators in which the approximations are based on using multiple voltage islands~\cite{zervakis:2018}. In our methodology, due to the enormous number of possible candidate solutions, the resulting Pareto frontier is identified using a hill climbing algorithm which works with estimated QoR and estimated hardware parameters.
\textbf{Novel Contributions:} In this paper, we propose a novel methodology for \textit{searching, selecting} and \textit{combining} the most suitable approximate circuits from a set of available libraries to generate an approximate accelerator for a given application. \textbf{To address the aforementioned scientific challenges, in this paper, we make the following key contributions.}
(i) A new QoR estimation technique is developed, which is based on computational models constructed using machine learning methods. This technique works with arbitrary approximate circuits, i.e., not only with those created by truncation or other well-understood methods.
(ii) A new heuristic Pareto frontier construction algorithm, based on proposed estimation techniques, is presented and evaluated.
(iii) The proposed methodology is evaluated using three case studies (Sobel edge detector, Gaussian filter with fixed coefficients and Generic Gaussian filter) in which approximate accelerators showing high-quality tradeoffs between QoR and hardware parameters are generated in a fully automated way using a library containing thousands of approximate circuits. The proposed method significantly reduces the number of design alternatives that have to be considered and evaluated.
\vspace{-0.6em}
\section{Proposed methodology}
\vspace{-0.3em}
\subsection{Overview}
The methodology requires the following input from the user: a hardware description of the chosen accelerator, corresponding software model and training (benchmark) data.
Hierarchical hardware as well as software models are expected in order to be able to replace relevant operations with their approximate versions, and to evaluate how this change affects the QoR. Approximate circuits are taken from a library, in which each of them is fully characterized and many approximate implementations exist for each operation.
Let the accelerator contain $n$ operations that can be implemented using some approximate circuits for the library. By \emph{configuration} we mean a particular assignment of approximate circuits from the library to $n$ operations of the accelerator. The goal of the methodology is to find a Pareto set of \textit{configurations} where the design objectives to be optimized are QoR (e.g., SSIM, PSNR etc.) and hardware cost (e.g., area, delay, power or energy).
\begin{figure}[tb]
\centering
\includegraphics[width=0.85\columnwidth]{arch2.pdf}\vspace{-1em}
\caption{Overview of the proposed \textit{autoAx} methodology.}\label{fig:arch}
\vspace{-7mm}
\end{figure}
The whole process consists of three steps as illustrated in Figure~\ref{fig:arch}.\\
\textit{Step 1:} The library of the approximate circuits is pre-processed in such a way that clearly irrelevant circuits are removed. The irrelevant circuits are identified on the basis of their quality (measured w.r.t. a particular application) and hardware cost.\\
\textit{Step 2:} Computational models enabling to estimate QoR and hardware cost are constructed by means of some machine learning algorithm.
A small (randomly selected) subset of possible configurations is used for learning of the computational models.\\
\textit{Step 3:} The Pareto frontier reflecting QoR and HW cost is constructed. To quickly remove as many low-quality solutions as possible, the construction algorithm employs the values estimated by the proposed models. The final Pareto front is then constructed using precisely computed QoR and hardware parameters by means of simulation and synthesis.
\subsection{Library pre-processing}
For each operation of the accelerator, a suitable subset of approximate circuits is separately identified in the library by means of benchmark data. For example, if $k$-th operation of the accelerator is 8-bit addition then the objective of this step is to identify approximate 8 bit adders that form the Pareto front with respect to a suitable error metric (score) and hardware cost.
We propose to base the selection on probability mass function (PMF) of the given operation which can be easily determined by simulation of the accelerator on benchmark data.
This process can be formalized as follows. Let $I$ denote a set of all possible combination of values from the benchmark data set that can occur on the input of $k$-th operation $M(x_1, x_2, \dots)$, $x \in I$, $k=1 \dots n$. Then, $D_k: I \rightarrow \mathbb{R}$ denoting the PMF of this operation is defined as
$D_k(i_1,i_2,\dots) = Pr(x_1 = i_1 \wedge x_2 = i_2 \wedge \dots)$.
This function is used to determine a score (weighted mean error distance) of an approximate circuit $\widetilde{M}$ implementing $k$-th operation as follows:
$
\mathrm{WMED_{k}(\widetilde{M})} = \sum_{\forall i \in I} D_k({i}) \cdot | M(i) - \widetilde{M}(i)|
$
For each operation of the accelerator, this score is then used together with hardware cost to identify only those approximate circuits (i.e., 8-bit adders in our example) that are lying on a Pareto frontier.
\subsection{Models construction}
Since the full synthesis and simulation are typically very time consuming processes, it is intractable to use them to perform the analysis of hardware cost and QoR for every possible configuration of the accelerator. To address this issue, we propose to construct two independent computational models, one for estimating QoR and a second for estimating hardware parameters. The estimation is based on the parameters of approximate circuits belonging to one selected configuration.
The models are constructed independently using a suitable supervised \textit{machine learning algorithm}. The learning process is based on providing example input--output pairs. In our case, each input--output pair corresponds with a particular configuration.
One input is represented by a vector, which contains a subset of hardware or quality parameters of each approximate circuit realizing one of operations as defined by the configuration. The output is a single value of QoR or hardware cost that is obtained by simulation and synthesis of the concrete accelerator with the given configuration. For learning, we have to generate a training set typically containing from hundreds or thousands of configurations.
The goal of this step is to obtain high-quality models. A set of configurations different from the training set is used to determine the quality of the model and avoid \textit{overfitting}\footnote{the estimated values correspond too closely or exactly to training output values, and the model may, therefore, therefore fail in fitting additional data}. Typically, the accuracy is optimized by the machine learning algorithms. However, as the models are used for determining a relation between two different configurations, it is not necessary to focus on the accuracy. We propose to consider \emph{fidelity} as the optimization criterion and maximize the fidelity of the model.
The fidelity tells us how often the estimated values are in the same relation ($<,=$ or $>$) as the real values for each pair of configurations.
If the fidelity of the constructed model is insufficient, we have to tune parameters of the chosen learning algorithm or select a different learning engine.
\subsection{Model-based design space exploration}
In this step, Pareto frontier containing those configurations that show the best tradeoffs between QoR and hardware cost is constructed. In order to avoid time-consuming simulation and synthesis, the construction is divided into two stages. In the first stage, the computational models that we have developed in the previous step are used to build a pseudo Pareto set of potentially good configurations.
In the second stage, based on the configurations forming the pseudo Pareto set, a set of approximate accelerators is determined, fully synthesized
and analyzed by means of a simulator and benchmark data. A real QoR and real hardware cost is assigned to each configuration. Finally, these real values are used to construct the final Pareto set.
\newlength{\textfloatsepsave} \setlength{\textfloatsepsave}{\textfloatsep}
\setlength{\textfloatsep}{5pt
\begin{algorithm}[t]
\caption{Pareto set construction}\label{hillclimb}\smaller%
\input{algorithmv2.tex}
\end{algorithm}
Although the first step reduced the number of possible configurations, the number of combinations may still be
enormous especially for complex problems consisting of tens of operations. Therefore, we proposed an iterative heuristic algorithm (Algorithm \ref{hillclimb}) to construct the pseudo Pareto set.
The algorithm is a variant of stochastic hill climbing which starts with a random configuration (denoted as $Parent$), selects a neighbor at random (denoted as $C$), and decides whether to move to that neighbor or to examine another. The neighbor configuration is derived from $Parent$ by modifying a randomly chosen item of the configuration (i.e., another circuit is picked from the library for a randomly chosen operation).
The quality and hardware cost parameters of $C$ ($e_{QoR}$ and $e_{HW}$) are estimated by means of appropriate estimation models. If the estimated values dominate those already present in Pareto set $P$, configuration $C$ is inserted to the set, the set is updated (operation \textsc{ParetoInsert}) and the candidate is used as the $Parent$ in the next iteration. In order to avoid getting stuck in a local optimum, restarts are used. If the $Parent$ remains unchanged for $k$ successive iterations, the $Parent$ is replaced by a randomly chosen configuration from $P$.
The quality of the resulting Pareto set depends on the fidelity of the estimation models and on the number of allowed iterations. The higher fidelity, the better results. The number of iterations depends on the chosen termination condition. It can be determined by the size of $P$, execution time, or the maximum allowed number of iterations.
\section{Experimental setup}
The proposed methodology is evaluated on three accelerators of different complexity that are typically used as benchmarks in the area of image processing. In particular, Sobel edge detector (Sobel ED), Gaussian filter with fixed coefficients (Fixed GF) and Generic Gaussian filter (Generic GF) working on 3x3 filter kernel were chosen. While the approximation of the first problem is solvable by an exhaustive enumeration of all possible configurations, the Generic GF consists of 17 operations and represents a non-trivial problem. The particular instances of the operations the chosen problems consist of are reported in Table~\ref{tab:circs}. In all cases, 8-bit gray-scale images are considered at the input. The problems were described in Verilog HDL which is used for synthesis (HW model) and in C++ (SW model) which is used for QoR analysis. The images consisting of $384\times256$ pixels from Berkeley Segmentation Dataset\footnote{\url{https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/}} are used as benchmark data.
To evaluate QoR, i.e., to determine the difference between the output of approximate and accurate implementations, we chose a commonly used measure known as the \textit{structural similarity} index (SSIM).
To determine the hardware cost, Synopsys Design Compiler targeting 45~nm ASIC technology was employed as a synthesis tool. The total \textit{area on the chip} was considered in this study as a cost parameter.
\setlength{\textfloatsep}{\textfloatsepsave}
\begin{table}[t]\vspace{-5pt}
\centering%
\caption{The number of operations in target accelerators}\vspace{-1em}\smaller%
{\setlength\tabcolsep{2pt}\begin{tabular}{p{2cm} | c c c | c c | c | c}\toprule
& \multicolumn{3}{c|}{\bf Adder} & \multicolumn{2}{c|}{\bf Subtractor} & \bf Multiplier & \bf Total \\
\bf Problem & 8-bit & 9-bit & 16-bit & 10-bit & 16-bit & 8-bit & \bf \#\\\midrule
Sobel ED & 2 & 2 & -- & 1 & -- & -- & 5 \\
Fixed GF & 4 & 2 & 4 & -- & 1 & -- & 11 \\
Generic GF & -- & -- & 8 & -- & -- & 9 & 17 \\\bottomrule
\end{tabular}}
\label{tab:circs}
\end{table}
The approximate circuits implementing each of six operations shown in Table~\ref{tab:circs} are
obtained from an extended version of EvoApprox~\cite{mrazek:date16lib} library. In addition to that, QuAd~\cite{hanif:quad} adders and BAM~\cite{Mahdiani:2010} multipliers are utilized.
The total number of various circuits that are available in our initial library is shown in Table~\ref{tab:libs}.
\begin{table}[htb]\vspace{-5pt}
\centering%
\caption{Approximate circuits included in the library}\vspace{-1em}\smaller%
\setlength\tabcolsep{2pt}\begin{tabular}{c | c c c | c c | c }\toprule
& \multicolumn{3}{c|}{\bf Adder} & \multicolumn{2}{c|}{\bf Subtractor} & \bf Multiplier \\
\bf instance & 8-bit & 9-bit & 16-bit & 10-bit & 16-bit & 8-bit \\\midrule
\bf \# implementations & 6979 & 332 & 884 & 365 & 460 & 29911\\\bottomrule
\end{tabular}
\vspace{-1em}
\label{tab:libs}
\end{table}
We implemented \textit{Sobel edge detector} for detecting vertical edges.
Its structure is shown in Figure \ref{fig:sobel}a. It consists of four adders, one subtractor and two shifts.
For QoR analysis, 24 images from the benchmark data set were employed.
\begin{figure}[th]
\centering
\includegraphics[width=0.9\columnwidth]{filters.pdf}\vspace{-1em}
\caption{Architecture of (a) Sobel edge detector; (b) fixed Gaussian filter}\vspace{-1.0em}
\label{fig:sobel}
\end{figure}
The filter kernel for the \textit{Fixed GF} was generated using the following parameters: $w=3,\sigma=2$.
Since the coefficients are constant, multiplierless constant multipliers (MCMs) can be employed. The architecture of this filter is shown in Figure~\ref{fig:sobel}b. The filter thus consists of adders, subtractors and shifts only. The optimum MCMs were obtained using \textit{SPIRAL tool}~\cite{spiral:2018}. For QoR analysis, 24 images from the benchmark data set were employed.
Contrasted to the fixed GF, the generic GF is, in fact, a common convolution filter with variable kernel coefficients. The hardware model consists of nine 8-bit multipliers whose results are summed. To evaluate QoR, we created a C++ model which considers $50$ different Gaussian kernels generated for $w=3$ and $\sigma$ ranging from $0.3$ to $0.8$.
Four images were selected from the bechmark dataset. In total, $200$ different simulations have been performed during QoR and the average SSIM was used as the quality indicator.
\newcommand{\hspace{3mm}}{\hspace{3mm}}
\section{Results}
The results are divided into two parts. Firstly, a detailed analysis of the results for the Sobel ED is provided to illustrate the principle of the proposed methodology. In the second part, only the final results are discussed due to the complexity of these problems and a limited space.
\subsection{Sobel edge detector}
\subsubsection{Library pre-processing}
To eliminate irrelevant circuits from the library, a score is calculated for each circuit in the library. Firstly, the target accelerator is profiled with a profiler which calculates the probability mass functions $D_k$ for all operations (Figure~\ref{fig:profile}). Note that $add_3$ (resp. $add_4$) has almost identical PMF with $add_1$ (resp. $add_2$). Figure~\ref{fig:profile} shows that operand values (neighbour pixels) are typically very close. In the plot dealing with $D_{add_2}$ one can see regular white stripes caused by shifting of the second operand.
\begin{figure}[ht]
\centering
\vspace{-1em}
\includegraphics[width=\columnwidth]{profile.pdf}\vspace{-1em}
\caption{Probability mass function of operations in the Sobel ED}
\label{fig:profile}
\vspace{-1em}
\end{figure}
Using the obtained probabilities, we calculated $WMED_k$ for all approximate circuits implementing $k$-th operation. Then we executed a component filtering process guided by $area$ and $WMED_k$ parameters of the isolated circuits and kept only Pareto optimal implementations.
At the end of this process, the number of circuits in reduced libraries is $|RL_{add_1}|=35, |RL_{add_2}|=32, |RL_{add_3}|=37, |RL_{add_4}|=33,$ and $|RL_{sub}|=36$.
\subsubsection{Model construction}
The next step in the methodology is to construct models estimating SSIM and hardware parameters using parameters of the circuits belonging to one selected configuration. We used $WMED$ of all employed circuits as the input vector for the QoR model. For the hardware model we used $power, area$ and $delay$ of all circuits as the input vector. Several learning engines were compared to identify the most suitable one for our methodology (1500 configurations for learning and 1500 configurations for testing were randomly generated using the reduced libraries).
The considered learning engines were the regression algorithms from \textit{scikit-learn} tool for Python. Additionally, we constructed na\"ive models for area ($M_a(C) = \sum_{\forall c \in C}{area(c)}$) and for SSIM ($M_{SSIM}(C) = -\sum_{\forall c \in C}{WMED_k(c)}$) to test if SSIM correlates with the cumulative arithmetic error and if the area correlates with the sum of areas of all employed circuits. These simple models were also considered in our comparisons.
\begin{table}[h]
\vspace{-0.5em}
\caption{The fidelity of models for Sobel edge detector constructed by different learning engines. }\label{tab:regression}\vspace{-1em}
\input{tab_regression.tex}
\vspace{-1em}
\end{table}
Table~\ref{tab:regression} shows the fidelities for all constructed models when evaluated on the training and testing data sets. The best result for the testing data sets are provided by random forest consisting of 100 different trees. The correlation between estimated and real area is shown in Figure~\ref{fig:regression}.
The na\"ive models exhibit unsatisfactory results especially for small resulting approximate accelerators. When we analyze some of these cases in detail we observe that the inaccuracy was typically caused by the last operation in the application (i.e., \textit{sub}). As this operation shows a big error, it is significantly simplified by the synthesis tool and as a consequence of that many other circuits are removed from the circuit because their outputs are no longer connected to any component. Hence the real area of these circuits was significantly smaller than the area calculated using the library. Due to this elimination, machine learning methods based on conditional structures (e.g., trees) exhibit better performance than methods primarily utilizing algebraic approaches (e.g., MLP NN).
\begin{figure}[b]
\centering\vspace{-1em}
\includegraphics[width=\columnwidth]{regression_hw.png}\vspace{-2em}
\caption{Correlation of estimated area and real area obtained by synthesis tool for the selected learning engines used in Sobel ED experiment.}
\label{fig:regression}\vspace{-1.5em}
\end{figure}
We tried to understand the impact of input parameters on the model quality. Including different error metrics such as the error variance did not improve the fidelity of QoR models. In contrast, omitting of power and delay in hardware modeling led to 2\% lower fidelities of these models in average.
\subsubsection{Model-based design space exploration}
In this part, the quality of proposed heuristic algorithm that we used for Pareto frontier construction is evaluated. Because of a low number of operations in Sobel ED, we are able to evaluate all possible configurations derivable from the reduced libraries $RL_k$ (i.e., $4.92 \cdot 10^7$ configurations in total). Note that the limit for stagnation detection was set to $50$ iterations in Alg. 1.
\begin{table}[bht]
\centering
\caption{Distances of the configurations identified by the proposed algorithm and random search from the optimal Pareto front. The lower value the better.}
\label{tab:distances} \vspace{-1em} \small
{\setlength\tabcolsep{2pt}\begin{tabular}{l c c |c c | c c }\toprule
\multirow{2}{*}{\bf Algorithm} & \multirow{2}{*}{\bf \#eval} & \multirow{2}{*}{\bf \#Pareto} & \multicolumn{2}{c|}{\bf To optimal} & \multicolumn{2}{c}{\bf From optimal} \\
& & & \it avg & \it max & \it avg & \it max \\\midrule
Optimal Pareto & $5\cdot10^7$ & 335 & --- & --- & --- & --- \\\midrule
& $10^3$ & 71 & 0.02538 & 0.07554 & 0.03318 & 0.08650\\
Proposed & $10^4$ & 177 & 0.00253 & 0.01328 & 0.00341 & 0.01690\\
& $10^5$ & \bf 324 & \bf 0.00001 & \bf 0.00095 & \bf 0.00009 & \bf 0.00657\\\midrule
& $10^3$ & 37 & 0.05276 & 0.10615 & 0.05616 & 0.11307\\
Random sampling & $10^4$ & 61 & 0.02631 & 0.08981 & 0.02875 & 0.07215\\
& $10^5$ & 82 & 0.01172 & 0.03770 & 0.01353 & 0.03820\\\bottomrule
\end{tabular}}
\end{table}
Pareto fronts created by means of the proposed algorithm were compared with Pareto fronts constructed using the random sampling (RS) algorithm and the optimal Pareto fronts.
The results are summarized in Table~\ref{tab:distances}. We can see that the proposed algorithm with $10^5$ evaluations allows us to get almost the same number of Pareto configurations as the optimal Pareto front contains. To show that obtained configurations $S$ are very close to the optimal configurations $P$, the distances of obtained configurations to the nearest optimal configuration $(\forall s \in S: \min_{\forall p \in P}|s - p|)$ and the distances from the optimal configuration to the nearest obtained configurations $(\forall p \in P: \min_{\forall s \in S}|s - p|)$ are analyzed. Both algorithms provided configurations that are typically close to the optimal one, but RS missed a lot of important configurations. Note that the distance is calculated from estimated QoR and HW parameters normalized to range \textit{<0,1>}.
\subsection{Gaussian filters}
The methodology was also applied to obtain approximate implementations of two versions of Gaussian image filter (fixed GF and generic GF). After profiling this accelerator and reducing the library of approximate circuits accordingly, random forest-based models of QoR and hardware parameters were created using 4000 training and 1000 testing randomly generated configurations. In the case of fixed GF, the fidelity of the area estimation model is 87\% for hardware parameters and 92\% for QoR. The fidelity of both models of generic GF is 89\%. If the synthesis and simulations run in parallel, the detailed analysis of one configuration takes $10$~s on average and the model-based estimation of one configuration takes $0.01$~s on average.
The Pareto construction algorithm evaluated $10^6$ candidate solutions. On average, $39$ iterations were undertaken to find a new candidate suitable for the Pareto front.
\begin{table}[tb!]
\centering
\caption{Size of the design space after performing particular steps of the proposed methodology}
\label{tab:space}\vspace{-1em}\small
{\setlength\tabcolsep{2pt}\begin{tabular}{l | c c c c}
\toprule \multirow{2}{*}{\bf Application} & \multicolumn{4}{c}{\bf \# configurations} \\
& \it all possible & \it lib. pre-processing & \it pseudo Pareto & \it final Pareto \\\midrule
Sobel ED & $1.96 \cdot 10^{15}$ & $4.92 \cdot 10^{7}$ & $335$ & $62$ \\
Fixed GF & $7.35 \cdot 10^{34}$ & $1.73 \cdot 10^{16}$ & $1166$ & $132$ \\
Generic GF & $7.15 \cdot 10^{63}$ & $3.75 \cdot 10^{23}$ & $946$ & $102$ \\\bottomrule
\end{tabular}\vspace{-1em}
}
\end{table}
Table~\ref{tab:space} shows the size of the design space after performing particular steps of the proposed methodology. For example, there are $7.15\cdot10^{63}$ configurations in the generic GF design space. The elimination of irrelevant circuits in the library reduced the number of configurations to $3.75\cdot 10^{23}$. The number of configurations is enormous because it would take $10^{17}$ years to analyze them. In contrast, the construction of 4000 random solutions for training of the models takes approximately 11 hours, $10^6$ iterations of the proposed Pareto construction algorithm employing the models takes $3$ hours and the remaining $1000$ configurations are analyzed in $3$ hours. Finally, approximately $100$ configurations that are Pareto optimal in terms of area, SSIM and energy are selected. In total, the proposed approach takes $17$ hours on a common desktop. Hypothetically, if we would use the analysis instead of the estimation model in the Pareto front construction, the analysis of $10^6$ configurations would take $115$ days.
\begin{figure}[b!]
\centering\vspace{-10pt}
\includegraphics[width=\columnwidth]{quality.png} \vspace{-2.5em}
\caption{Pareto fronts showing best tradeoffs between SSIM, area and energy obtained using three methods (orange -- the proposed method; blue -- RS; black -- uniform selection) for three approximate accelerators. }
\label{fig:quality}
\end{figure}
Figure~\ref{fig:quality} compares resulting Pareto fronts obtained using the proposed methodology (orange line), the RS-based Pareto front construction algorithm (blue line) and the uniform selection approach (black line). The uniform selection approach is a manual selection method which one would probably take if no automated design methodology is available. In this method, particular approximate circuits are deterministically selected to exhibit the same error WMED (relatively to the output range). Figure~\ref{fig:quality} shows that this method provides relevant results only for accelerators containing a few operations. The randomly generated configurations (blue points) were obtained from a 3 hour run of the random configuration generation-and-evaluation procedure. They are included to these plots in order to emphasize high quality solutions obtained by the proposed method.
\section{Conclusions}
We developed an automatic design space exploration and circuit approximation methodology which replaces operations in an original accelerator by their approximate versions taken from a library of approximate circuits. In order to accelerate the approximation process, QoR and hardware parameters are estimated using computational models created by means of machine learning methods. On three case studies we have shown that the proposed methodology provides approximate accelerators showing high-quality tradeoffs between QoR and hardware parameters. Our methodology paves a way towards a fully automated approximation of complex accelerators that are composed of approximate operations whose error models are in principle unknown.
\paragraph{Acknowledgments}
\footnotesize{
This work was supported by Czech Science Foundation project 19-10137S and by the Ministry of Education of Youth and Physical Training from the Operational Program Research, Development and Education project International Researcher Mobility of the Brno University
of Technology --- CZ.02.2.69/0.0/0.0/ 16\_027/0008371}
|
{
"timestamp": "2019-04-02T02:27:57",
"yymm": "1902",
"arxiv_id": "1902.10807",
"language": "en",
"url": "https://arxiv.org/abs/1902.10807"
}
|
\section{\label{sec:level1}First-level heading}
X-ray photon correlation spectroscopy (XPCS) is a powerful method for studying dynamics in disordered systems, giving access to length and time scales inaccessible by other techniques.
The typical time scales are much longer than those probed by inelastic X-rays scattering and the length scales much shorter than those investigated by visible-light techniques \cite{Grubel}.
XPCS requires the capability of third generation synchrotron radiation sources of producing coherent X-ray beams several orders of magnitude more intense than previously available \cite{Grubel}.
The pioneering papers by Sutton \textit{et al.} \cite{Sutton} and Brauer \textit{et al.} \cite{Brauer} laid the foundations of this research area.
XPCS has been used to study dynamics occurring on length scales $>$10 nm, for example, in small-angle scattering experiments on colloids and polymers, or near Bragg peaks of crystals \cite{Madsen}.
The capability to resolve single atomic motion in condensed matter with this technique has also been shown \cite{Stephenson, Leitner1}.
Much of the excitement about scattering with coherent X-rays, however, arises from the perspective to perform atomic resolution correlation experiments to study the complex dynamics of disordered
systems, whose archetypes are glasses. Their microscopic structure remains the object of active research.
Among the works that laid the groundwork in this field are those of Zachariasen \cite{Zach} and Warren \cite{Warren}.
The existence of short-range order in glasses was rather clear, while only the application of many different methods like neutron scattering, extended fine structure X-ray absorption and Raman spectroscopy \cite{Rao} has made it possible to provide hints into the medium-range order.
For what concerns the dynamics, the glassy state is described as arrested, with relaxation times too large to be observed on human time scales \cite{Ediger}. But what happens at the atomic scale?
The works by Ruta \textit{et al.} report the rather unexpected result that glasses display atomic rearrangements within few minutes in both metallic \cite{Ruta1} and silicate glasses \cite{Ruta2},
and this even in the deep glassy state. While the dynamics of metallic glasses is intrinsic \cite{Giordano}, recent investigations clarify that this is not the case for silicon and germanium dioxide glasses where the atomic motion in the glassy state is induced by the X-ray beam \cite{Ruta3}.
Here we utilize XPCS to shed light on the effect of hard X-rays on the dynamics at the atomic level in the network glass B$_{2}$O$_{3}$ across the glass transition.
We show that this beam-induced dynamics competes with the structural relaxation, is negligible in the undercooled liquid phase and dominant in the glass.
The artificial dynamics induced by the beam can be described as a sequence of structural rearrangements involving the collective motion of up to thousands of atoms.
XPCS measurements on the B$_2$O$_3$ glass-former were performed at beamline ID10 at the ESRF in Grenoble (F), see Supplemental Material \cite{SM} for more details on the setup and on the sample preparation. The measurements were conducted by varying the flux of the incident beam, $F$, on the sample by means of different attenuators.
Each attenuator, made out of Si, leads to a decrease of the beam flux by a factor $\sim1/e$.
In particular, the atomic dynamics of the B$_{2}$O$_{3}$ glass was measured for: i) no attenuator, corresponding to an incoming beam flux $F_{0}= 8.6\cdot10^{10}$ ph/s per 200 mA current in the storage ring;
ii) a single attenuator filter, corresponding to a flux $F_{1} =2.6\cdot10^{10}$ ph/s per 200 mA; iii) and a double attenuator filter corresponding to a flux $F_{2}=9.8\cdot10^{9}$ ph/s per 200 mA.
The intensity scattered by the B$_{2}$O$_{3}$ glass was collected for different temperatures in the 297--593 K range. At each temperature series up to 3,000 frames were taken with exposure times per frame,
$\Delta t_e$, in the range 2--7 s depending on the attenuator employed during the measurement. The recorded frames were subsequently analyzed by the multispeckle XPCS method \cite{Lumma, Chushkin} to obtain a set of temporal correlation functions. Fig. \ref{fig:func}(a) shows a series of normalized intensity autocorrelation functions, $g_{2}(Q,t)$,
measured in vitreous B$_{2}$O$_{3}$ by cooling the sample from the supercooled liquid phase to the glassy state ($T_{g}=526$ K) using the full beam flux $F_{0}$.
The dynamics becomes slower as the temperature is lowered down to 498 K, and shows very little temperature dependence at lower temperatures.
Fig. \ref{fig:func}(b) shows, moreover, that the atomic motion in the glassy state at $T = 413$ K strongly depends on the X-ray beam flux,
leading to an induced relaxation time that is shorter the higher is the incident beam flux, similar to what reported in Ref. \cite{Ruta3}. This effect is independent of the global dose released on the sample,
at least up to the maximum doses of $\simeq$ 2 GGy released on the same scattering volume during the measurements. This beam-induced effect is also not related to any visible structural damage: the scattered intensity, for instance, remains unaltered within about 2\% (see Supplemental Material \cite{SM}), and the beam-induced dynamics timescale reversibly changes with the incident flux, as also shown in Ref. \cite{Ruta3} for the case of SiO$_2$.
\begin{figure}
\includegraphics[width=0.5\textwidth]{Figfunzcorr.eps}
\caption{\label{fig:func} (a) Normalized intensity auto-correlation functions (symbols) measured at Q$_{max}=1.5$ $\text{\AA}^{-1}$ in B$_{2}$O$_{3}$ for different temperatures across $T_g = 526$ K together with the best fitting stretched exponential line shapes. These measurements have been carried out with full beam flux, $F_{0}$. (b) Normalized intensity auto-correlation functions (symbols) measured at $T = 413$ K and Q$_{max}=1.5$ $\text{\AA}^{-1}$ for different incoming beam fluxes, see legend, together with the best fitting stretched exponential line shapes.}
\end{figure}
The shape of the correlation functions can be quantified by fitting to the data the Kohlrausch-Williams-Watts (KWW) expression \cite{Williams, Goetze}
\begin{eqnarray}
g_{2}\left(Q,t\right)=1+\mathcal{C}\left(Q\right)\cdot \exp \left[ -2\left(t\!/\tau\right)^{\beta} \right],
\label{eq:one}
\end{eqnarray}
where $\mathcal{C}=\mathcal{B}\left(Q \right) f_{Q}^{2}$ is the product of the experimental contrast and the square of the non-ergodicity factor; $\beta$ is the shape parameter and $\tau$ is the characteristic decay time.
Only few curves show the full decay from $1+\mathcal{C}$ to 1: most of them show in fact only the tail of the curve with a decay time that is fast on the scale fixed by $\Delta t_e$.
The fitting analysis of the experimental curves using Eq. (\ref{eq:one}) has then been carried out using all free fitting parameters ($\mathcal{C}$, $\tau$, $\beta$) only for the curves with longer $\tau$.
For these curves the parameter $\mathcal{C}$ comes out to be only little scattered around a mean value $\mathcal{C}=\left(8.5\pm0.4\right) \times 10^{-3}$.
This value is lower than that observed in other glasses \cite{Ruta1,Ruta2}, because we had to use thicker samples in order to increase the scattered intensity (see Supplemental Material \cite{SM}).
Recalling that the non-ergodicity factor $f_{Q}$ in correspondence to the maximum of the structure factor is expected to display only a weak temperature dependence (e.g. see Ref. \cite{Ruta12}),
the fits to all experimental curves have been carried out using the previously mentioned fixed value for $\mathcal{C}$.
The temperature dependence of the decay time is reported in Fig. \ref{fig:model}. Macroscopic values (black squares) have been obtained from measurements carried out with dynamic light scattering \cite{Dallari}.
Different symbols for the XPCS data refer to different beam intensities, as reported in the legend. We highlight three main observations: i) Above $T_g$ the XPCS relaxation time, $\tau_{X}$, is very close to that measured in the visible range, and with very similar temperature dependence, confirming previous measurements on other systems~\cite{Ruta1,Ruta2,Evenson}.
ii) In the glassy state $\tau_{X}$ is almost temperature independent and remains in the 10--100 s range.
These findings are very similar to the behaviour recently observed in other network glasses \cite{Ruta2, Ruta3, Ross} at the atomic length--scale, contrary to the expectation of an almost arrested dynamics.
iii) While in the supercooled liquid region all of the XPCS data basically overlap, in the glassy state we observe different values of $\tau_{X}$ depending on the incident beam flux.
\begin{figure}
\includegraphics[width=0.5\textwidth]{Figtau.eps}
\caption{Decay time measured using XPCS in B$_{2}$O$_{3}$ at Q$_{max}=1.5$ $\text{\AA}^{-1}$. Different symbols for the XPCS data refer to different X-ray beam fluxes, as reported in the legend. The black pentagons are macroscopic data obtained using dynamic light scattering \cite{Dallari}. The solid lines are obtained using Eq. (\ref{eq:model}) and refer to the XPCS data measured with the lowest and highest beam flux. The dashed vertical line marks the position of $T_g$.}
\label{fig:model}
\end{figure}
However, there are no clear evidences of radiation--damage (meaning permanent damage): the beam flux simply fixes the time--scale of the dynamics~\cite{Ruta3}.
In order to discriminate the beam-induced dynamics from the equilibrium dynamics, we can use a simple model where the decorrelation time measured with XPCS is written as:
\begin{eqnarray}
\frac{1}{\tau_{X}}=\frac{1}{\tau}+\frac{1}{\tau_{ind}}
\label{eq:model}
\end{eqnarray}
where $\tau$ is the structural relaxation time of B$_2$O$_3$, and $\tau_{ind}$ is the beam--induced decorrelation time.
We can use for $\tau_{ind}$ a temperature-independent (but beam-flux dependent) value given by $\tau_{X}$ in the glass;
and for $\tau$ the values obtained by photon correlation in the visible range and extrapolated below $T_g$. Fig. \ref{fig:model}
shows that this simple model (solid lines) describes very well the measured $\tau_{X}$ data. The beam induced dynamics takes place
in parallel to the spontaneous sample dynamics: above the glass transition temperature the structural relaxation is the fastest process and
therefore dominates, while below $T_{g}$ it becomes completely irrelevant.
The shape parameter, $\beta$, extracted from the KWW fits, is shown in Fig. \ref{fig:par}.
\begin{figure}
\includegraphics[width=0.37\textwidth]{Figbeta.eps}
\caption{Temperature dependence of the shape parameter $\beta$ for B$_{2}$O$_{3}$ at Q$_{max}=1.5$ $\text{\AA}^{-1}$. The different symbols refer to different beam fluxes, see legend. The full line indicates the mean value obtained with visible light scattering in the range 500--550 K \cite{Dallari}.}
\label{fig:par}
\end{figure}
For the incoming beam flux $F_2$,
the obtained values for $\beta$ are basically temperature independent in the glass with a mean value of $\beta=0.97\pm0.04$.
At higher temperatures, they decrease to reach a value which is compatible with the average equilibrium value of $0.67\pm0.09$
obtained with visible light scattering \cite{Dallari} on a B$_2$O$_3$ sample prepared by exactly the same method as reported here. The reduction of $\beta$ in the XPCS data is therefore here another sign of the transition from a beam-induced dynamics in the glass to the equilibrium dynamics in the undercooled liquid.
The $\beta$ values at higher fluxes are affected by a considerable uncertainty because the decorrelation is faster and only a portion of the curve is measured.
Taking this into account, we conclude that the $\beta$ parameter does not show an appreciable dependence on the flux in the entire explored temperature range. It is however interesting to observe that, differently from the case of silica and germania \cite{Ruta3},
the shape parameter for the beam-induced decay corresponds to a simple exponential rather than a compressed ($\beta>1$) one.
In the glassy state the decay time obtained by XPCS is only little temperature dependent and clearly decreases on increasing the X-ray beam flux, as shown in Fig. \ref{fig:model}.
In particular, our data are compatible with the expression $\tau_{X} \propto \langle F \rangle ^{-1}$, see the points corresponding to 413 K in Fig. \ref{fig:flux}(a). Here $\langle F \rangle = F\cdot \Delta t_e\!/\Delta t_l$ is the
average X-ray flux arriving at the sample; and $\Delta t_l$ is the lagtime, i.e. the sum of the exposure time $\Delta t_e$ and the readout time $\Delta t_r= 1.4$ s.
Fig. \ref{fig:flux}(a) confirms the results already reported in Ref. \cite{Ruta3} for the case of the silica and germania glasses.
We can also rephrase the previous observation by stating that the relaxation time measured by XPCS is inversely proportional to the average number of photons absorbed by the B$_2$O$_3$ sample, \textit{i.e.} $\tau \propto \langle F \rangle_a^{-1}$, where $\langle F \rangle_a = \langle F \rangle \left[1-\exp\left(-\mu L\right)\right]$, $\mu$ is the attenuation coefficient for B$_{2}$O$_{3}$ at 8.1 keV and $L$ is the sample thickness. It is then easy to clarify the meaning of this relation.
In fact, from the definition of intensity autocorrelation function, we know that in a time $\tau_X$, $N_{tot}\!/e$ of B$_2$O$_3$ units move by a distance $1/Q$, where $N_{tot}$ is the number of units in the scattering volume.
\begin{figure}
\includegraphics[width=0.5\textwidth]{FigNu.eps}
\caption{(a) Decay time obtained from the XPCS measurements in B$_{2}$O$_{3}$ at Q$_{max}=1.5$ $\text{\AA}^{-1}$ as a function of the inverse of the average flux. Different symbols refer to different temperatures, see legend. The best fitting lines for $T=413$ K and $T =297$ K are also reported. (b) Temperature dependence of the number of B$_{2}$O$_{3}$ units that move following an X-ray absorption event, same symbols as in (a).}
\label{fig:flux}
\end{figure}
More precisely, in a time $\tau_X$, $f_Q N_{tot}\!/e $ of units move by a distance $1\!/Q$. However, $f_Q$ is very close to 1 when Q is close to the maximum of the structure factor (e.g. see Ref. \cite{Ruta12}), and therefore we can neglect its presence in what follows.
We also know that the number of photons absorbed in time $\tau_X$ is obviously $\langle F\rangle_a \cdot \tau_X$. Consequently, the number of units, $N_u$, that move after the absorption of one photon is the ratio of the number of units that move by a distance $1\!/Q$ in time $\tau_X$ and the number of photons absorbed in the same time:
\begin{eqnarray}
N_u=\frac{1}{e}\cdot\frac{N_{tot}}{\langle F\rangle_a \cdot \tau_X},
\label{eq:six}
\end{eqnarray}
where $N_u$ and $\tau_X$ can in principle be $Q$ dependent. The number $N_{tot}$ can be calculated using the sample mass density, $\rho$ = 1.83 g$\!/$cm$^{3}$, and the scattering volume defined by the beam spot size and the sample thickness.
The values for $N_u$ obtained in this way are reported in Fig. \ref{fig:flux}(b) as a function of temperature in the range where the observed dynamics is beam-induced,
i.e. for $T \leqslant 453$ K. It is interesting to remark that $N_u$ is large: $600 \pm 70$ B$_{2}$O$_{3}$ units, or $3000\pm200$ atoms. Eq. \ref{eq:six} is clearly a way to rationalize the flux dependence of the beam-induced decorrelation time measured in XPCS experiments: $N_u$ is the sample-dependent value that describes the proportionality of $\tau_X$ on the inverse average flux, and is the real outcome of XPCS measurements in beam induced conditions.
Note that the XPCS relaxation time depends on the scattering volume, being proportional to it. This simply reflects the fact that it takes longer to fluidize a larger amount of atoms.
It is interesting to explore the possibility that the $N_u$ units belong to the same volume $V_{c}$.
Assuming this volume being spherical, its radius $\xi$ will be related to $N_u$ by the relation $\xi = \sqrt[3]{3N_uv_{B_2O_3}\!/4\pi}$, where $v_{B_2O_3}$ corresponds to the volume of a B$_2$O$_3$ unit.
We obtain a value $\xi=2.3 \pm 0.1$ nm at $T=297$ K. It is suggestive to observe that this value is similar to those reported for the cooperativity length $\xi_{\alpha}$ at the
glass transition temperature ($\xi_{\alpha}$ = 2.0 nm \cite{Hong1} and 1.5 nm \cite{Hempel}).
We can hypothesise the following mechanism as responsible for the beam induced dynamics. The absorption of one photon generates a photoelectron which gives rise to a radiolysis-induced atomic displacement with a given probability~\cite{Griscom, Kinchin, Hobbs,Dapor}; however, since a glass is a metastable system characterized by internal stresses, this atomic displacement cannot be accommodated on its own and will be rather accompanied by the rearrangement of a larger region, corresponding to the cooperative volume.
A similar mechanism of stress release generated by random bond breaking has been recently exploited in numerical simulations of soft solids~\cite{DelGado} to probe their slow dynamics and was found responsible for the emergence of compressed correlation functions and of superdiffusivity.
It is also tempting to recognize, despite the large scattering of the present data, a temperature dependence for $N_u$, and thus for $\xi$, in Fig. 4b: $\xi$ is possibly decreasing on increasing $T$ as it is expected for the cooperative length measured in the liquid phase \cite{Adam, Donth}.
While this idea needs to be confirmed by experiments on more materials and validated in detail, it is clear that alternative schemes can also be imagined to explain the value that we obtain for $N_u$. Considering in fact that the primary electrons produced by photoelectric absorption have an energy of about 8 keV and assuming a few tens of eV as the average energy loss per inelastic collision of the primary electron \cite{Egerton}, we can estimate that up to a few hundreds inelastic collisions per absorption event have the potential to give rise to atomic displacement by radiolysis. While this number is an order of magnitude smaller than the number of atomic displacements corresponding to $N_u$, it is possible (though unlikely) that all of these inelastic collisions give rise to the \AA-long displacements detected here, and therefore that each radiolysis event leads to the displacement of about ten atoms. Also in this scenario then there is some cooperativity required for the atomic displacements due to radiolysis, though clearly on a different length scale as the one discussed above.
In summary, we have investigated in some detail the effect of a hard X-ray beam on a borate glass using XPCS. In the supercooled liquid we probe the spontaneous dynamics related
to the structural relaxation at the atomic length-scale; in the glassy state, instead, the X-ray beam gives rise to a beam-induced dynamics. The X-ray beam thus fluidizes the sample: it induces local changes
and the overall configuration is renewed after the decay time $\tau_{X}$. These results confirm and extend to a new class of oxide glasses those already reported in \cite{Ruta3}; differently from that case, however, the shape of the correlation functions remains basically exponential instead than compressed. Moreover, the beam induced and the structural relaxation characteristic times are here shown to compete with each other and the two processes take place in parallel, so that the shorter one dominates the observed dynamics. We also confirm the proportionality between the induced-dynamics characteristic time and the inverse of the
average flux of the X-ray beam impinging on the sample reported in \cite{Ruta3}. We show here that this proportionality can be interpreted in terms of a fixed amount of material that rearranges after
one photon absorption event. The obtained value for this amount of material turns out to be similar to that expected for dynamical heterogeneities, and actually rather close to the available estimates for B$_2$O$_3$ \cite{Hong1, Hempel}. This observation,
when confirmed for other glasses, would establish a useful connection between the X-ray beam-induced dynamics here observed and a property of large interest for glasses.
The XPCS data here reported have been collected during one experiment at the ESRF (proposal HC1735).
We thank C. Armellini for help during the preparation of the sample. We acknowledge the ESRF for provision of synchrotron radiation facilities, and thank Y. Chushkin and K. L'Hoste for assistance in using beamline ID10.
|
{
"timestamp": "2019-03-01T02:16:39",
"yymm": "1902",
"arxiv_id": "1902.11027",
"language": "en",
"url": "https://arxiv.org/abs/1902.11027"
}
|
\section{Introduction}
The separability problem in Quantum Information Theory asks for a deterministic criterion to distinguish the entangled states from the separable states \cite{Guhne}. This problem is known to be a hard problem even for bipartite mixed states \cites{gurvits2003, gurvits2004}.
The Schmidt number of a state ($SN(\gamma)$ - Definition \ref{def1}) is a measure of how entangled a state is \cites{Terhal, Sperling2011}. If its Schmidt number is 1 then the state is separable. If its Schmidt number is greater than 1 then the state is entangled. A method to compute the Schmidt Number is unknown.
Denote by $M_k$ the set of complex matrices of order $k$. The separability problem has been completely solved in $M_2\otimes M_2$. A state in $M_2\otimes M_2$ is separable if and only if it is positive under partial transposition or simply PPT (Definition \ref{def1}) \cites{peres,horodeckifamily}. Therefore, the Schmidt number of a PPT state in $M_2\otimes M_2$ is equal to 1. Recently, the Schmidt number of every PPT state of $M_3\otimes M_3$ has been proved to be less or equal to 2 \cites{Yang, Chen}.
The authors of \cite{Marcus} left an open problem to determine the best possible Schmidt number for PPT states.
They also presented a construction of PPT states in $M_k\otimes M_k$ whose Schmidt numbers are greater or equal to $\left\lceil\frac{k-1}{4}\right\rceil$.
This was the first explicit
example of a family of PPT states achieving a Schmidt number that scales linearly in the local dimension.
We investigate this matter. We present an explicit construction of PPT states in $M_k\otimes M_k$, whose Schmidt numbers are \textbf{equal} to $n$, for any given $1\leq n\leq \left\lceil\frac{k-1}{2}\right\rceil$. This is a new contribution to their open problem.
We manage to compute the Schmidt number of these PPT states using the following inequality \begin{equation}\label{equation1}
SN(\gamma)\geq \max\left\{\frac{SN(\gamma_S)}{2}, \frac{SN(\gamma_A)}{2}\right\},
\end{equation}
where $\gamma_S=(Id+F)\gamma (Id+F)$, $\gamma_A=(Id-F)\gamma(Id-F)$ and $F\in M_k\otimes M_k$ is the flip operator $($i.e., $F(a\otimes b)=b\otimes a,$ for every $a,b\in\mathbb{C}^k)$.
We believe this is one of the simplest constructions of an entangled PPT state made so far.
Another inequality that we present here extends a result that was previously known for separable states $($\cite[Theorem 1]{smolin}$)$ to every state of $ M_k\otimes M_m$. Denote by $\gamma_L$ and $\gamma_R$ the marginal states of a state $\gamma\in M_k\otimes M_m$ (Definition \ref{def1}).
We show that every state $\gamma$ of $M_k\otimes M_m$ satisfies\\
\begin{equation}\label{equation2}
\text{rank}(\gamma)SN(\gamma)\geq \max\{\text{rank}(\gamma_L), \text{rank}(\gamma_R)\}.
\end{equation}
\vspace{0.3 cm}
We can use this inequality to obtain a lower bound for the Schmidt number of low rank states.
Next, through a series of very technical results, the author of \cite{Cariello_LAA} obtained the following lower bounds for the $\text{rank}(\gamma_S)$ of any separable state $\gamma\in M_k\otimes M_k$ $$\text{rank}(\gamma_S)\geq \max\left\{\frac{r}{2}, \frac{2}{r} \text{rank}(\gamma_A)\right\},$$
where $r$ is the marginal rank of $\gamma+F\gamma F$.
These inequalities can be combined into one inequality: \\
\begin{center}
$\displaystyle\text{rank}(\gamma_S)\geq \frac{2}{r} \text{rank}(\gamma_A)\geq \frac{\text{rank}(\gamma_A)}{\text{rank}(\gamma_S)}.$
\end{center}
\vspace{0.3 cm}
Hence, $\text{rank}(\gamma_S)\geq \sqrt{\text{rank}(\gamma_A)}$ for every separable state $\gamma\in M_k\otimes M_k$.
Therefore, if $\text{rank}(\gamma_S)<\sqrt{\text{rank}(\gamma_A)}$ then $\gamma$ is entangled. \\
Next, we can combine equations \ref{equation1} and \ref{equation2} in order to obtain
$$\displaystyle SN(\gamma)\geq \frac{\text{rank}((\gamma_S)_L)}{2\ \text{rank}(\gamma_S)}\text{ and }\displaystyle SN(\gamma)\geq \frac{\text{rank}((\gamma_A)_L)}{2\ \text{rank}(\gamma_A)}.$$
We can easily create entangled states by satisfying $\displaystyle\frac{\text{rank}((\gamma_S)_L)}{\text{rank}(\gamma_S)}>2$ or $ \displaystyle \frac{\text{rank}((\gamma_A)_L)}{\text{rank}(\gamma_A)}>2$ and no correlation between $\gamma_S$ and $\gamma_A$ is required. \\
These two methods of creating entangled states are completely opposite. One depends on a correlation between $\gamma_S, \gamma_A$ and the other does not. They show how diverse is quantum entanglement.\\
This paper is organized as follows. \begin{itemize}
\item In Section II, we prove that $SN(\gamma)\geq \max\left\{\frac{SN(\gamma_S)}{2}, \frac{SN(\gamma_A)}{2}\right\}$ (Proposition \ref{proposition1}) and we construct a PPT state whose Schmidt number is equal to $n$, for any given $n\in\{1,\ldots,\left\lceil\frac{k-1}{2}\right\rceil\}$ (Proposition \ref{proposition2}).
\item In Section III, we prove our main inequality $\text{rank}(\gamma)SN(\gamma)\geq \max\{\text{rank}(\gamma_L), \text{rank}(\gamma_R)\}$ (Theorem \ref{theoremnew0}) and two corollaries $\displaystyle SN(\gamma)\geq \frac{\text{rank}((\gamma_S)_L)}{2\ \text{rank}(\gamma_S)}$ and $ \displaystyle SN(\gamma)\geq \frac{\text{rank}((\gamma_A)_L)}{2\ \text{rank}(\gamma_A)}$ (Corollaries \ref{corollary1} and \ref{corollary2}). \\
\end{itemize}
Notation: Given $x\in\mathbb{R}$, define $\lceil x\rceil=\min\{n\in\mathbb{Z}, n \geq x \}$. Identify $M_k\otimes M_m\simeq M_{km}$ and $\mathbb{C}^k\otimes \mathbb{C}^m\simeq \mathbb{C}^{km}$ via Kronecker product. Let us call a positive semidefinite Hermitian matrix of $M_{km}$ a (non-normalized bipartite) state of $M_k\otimes M_m$. Let $\Im(\delta)$ denote the image of $\delta\in M_k\otimes M_m$ within $\mathbb{C}^k\otimes \mathbb{C}^m$. Given $w\in \mathbb{C}^k\otimes \mathbb{C}^m$ denote by $SR(w)$ its Schmidt rank $($or tensor rank$)$. Let the trace of a matrix $A\in M_k$ be denoted by $tr(A)$. \\
\section{Preliminary Inequalities}
\vspace{0.3cm}
In this section we present two preliminary inequalities (Proposition \ref{proposition1}). They have independent interest as we can see in Proposition \ref{proposition2}. There we construct a family of PPT states in $M_k\otimes M_k$ whose members have Schmidt number equal to $n$, for any given $1\leq n\leq \left\lceil\frac{k-1}{2}\right\rceil$. \\
\begin{definition}\label{def1}
Given a state $\delta=\sum_{i=1}^nA_i\otimes B_i \in M_k\otimes M_m$, define
\begin{itemize}
\item the Schmidt number of $\delta$ as \\$\displaystyle SN(\delta)=\min\left\{\max_{ j}\left\{SR(w_j) \right\}, \ \delta=\sum_{j=1}^mw_j\overline{w_j}^t \right\}$
$($This minimum is taken over all decompositions of $\delta$ as $\sum_{j=1}^mw_j\overline{w_j}^t)$.
\item the partial transposition of $\delta$ as $\delta^{\Gamma}=\sum_{i=1}^nA_i\otimes B_i^t$ .
Moreover, let us say that $\delta$ is positive under partial transposition or simply a PPT state if and only if $\delta$ and $\delta^{\Gamma}$ are states.
\item the marginal states of $\delta$ as $\delta_L=\sum_{i=1}^nA_i tr(B_i)$ and $\delta_R=\sum_{i=1}^n B_i tr(A_i)$.
\\
\end{itemize}
\end{definition}
\begin{proposition}\label{proposition1} Every state $\gamma\in M_k\otimes M_k$ satisfies $\displaystyle SN(\gamma)\geq \max\left\{\frac{SN(\gamma_S)}{2}, \frac{SN(\gamma_A)}{2}\right\}$.
\end{proposition}
\begin{proof}
By definition \ref{def1}, there is a subset $\{w_1,\ldots,w_n\}\subset\mathbb{C}^k\otimes \mathbb{C}^k$ such that
$\gamma=\sum_{i=1}^nw_i\overline{w_i}^t$ and $SR(w_i)\leq SN(\gamma)$, for every $i$.\\
Therefore, $(Id\pm F)\gamma(I\pm F)=\sum_{i=1}^n v_i\overline{v_i}^t$, where $v_i=(Id\pm F)w_i$. Notice that, for every $i$, $$SR(v_i)=SR(w_i\pm Fw_i)\leq 2SR(w_i)\leq 2SN(\gamma).$$
Hence, $SN((Id\pm F)\gamma(I\pm F))\leq 2SN(\gamma)$.
\end{proof}
\vspace{0.3cm}
\begin{proposition}\label{proposition2} Let $v=\sum_{i=1}^na_i\otimes b_i$, where $\{a_1,\ldots,a_n,b_1,\ldots b_n\}$ is a linearly independent subset of $\mathbb{C}^k$. Define $$\gamma=Id+F+\epsilon(v\overline{v}^t)\in M_k\otimes M_k.$$
\begin{enumerate}
\item For every $\epsilon>0$, $SN(\gamma)=n$. Notice that $1\leq n\leq \left\lceil\frac{k-1}{2}\right\rceil$.
\item There is $\epsilon>0$ such that $\gamma$ is positive under partial transposition.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) Notice that $ \gamma_A=(Id-F)\gamma(Id-F)=\epsilon(a\overline{a}^t)$, where $a=\sum_{i=1}^na_i\otimes b_i-b_i\otimes a_i$.
Since
$\{a_1,\ldots,a_n,b_1,\ldots b_n\}$ is linearly independent then $SR(v)=n$ and $SR(a)=2n$. Hence, $$SN(\epsilon(v\overline{v}^t))=SR(v)=n\text{ and } SN(\gamma_A)=SR(a)=2n.$$
Thus, $\displaystyle SN(\gamma)\geq\frac{SN(\gamma_A)}{2}=n$, by Proposition \ref{proposition1}.\\
Next, the separability of $Id+F\in M_k\otimes M_k$ is a well known fact, therefore $SN(Id+F)=1$. \\
Finally, $SN(\gamma)\leq \max\{SN(Id+F),SN(\epsilon(v\overline{v}^t))\}= \max\{1,n\}=n$.
Therefore, $SN(\gamma)=n$.\\
(2) Notice that $(Id+F)^{\Gamma}=Id+uu^t$, where $u=\sum_{i=1}^ k e_i\otimes e_i$ and $\{e_1,\ldots, e_k\}$ is the canonical basis of $\mathbb{C}^k$. So $(Id+F)^{\Gamma}$ is positive definite and, for a small $\epsilon$, $(Id+F)^{\Gamma}+\epsilon(v\overline{v}^t)^{\Gamma}$ is positive definite too.
\end{proof}
\section{Main Inequality}
In this section, we present our main result (Theorem \ref{theoremnew0}) and two corollaries (Corollaries \ref{corollary1} and \ref{corollary2}).\\
\begin{theorem}\label{theoremnew0}If $\gamma\in M_k\otimes M_m$ is a state then $\text{rank}(\gamma)SN(\gamma)\geq \max\{\text{rank}(\gamma_L),\text{rank}(\gamma_R)\}$.
\end{theorem}
\begin{proof}
The proof is an induction on $\text{rank}(\gamma)$. The case $\text{rank}(\gamma)=0$ is trivial. If $\text{rank}(\gamma)=1$ then $SN(\gamma)=\text{rank}(\gamma_L)=\text{rank}(\gamma_R)$.\\
Let $\text{rank}(\gamma)>1$ and assume that this result is valid for states $\delta \in M_k\otimes M_m$ satisfying $\text{rank}(\delta)<\text{rank}(\gamma)$.\\
Since $\Im(\gamma)\subset\Im(\gamma_L\otimes\gamma_R)$ then $\gamma$ can be embedded in $M_{\text{rank}(\gamma_L)}\otimes M_{\text{rank}(\gamma_R)}$. The embedding does not change its rank or its Schmidt number. Thus, we can assume without loss of generality that $\text{rank}(\gamma_L)=k$ and $\text{rank}(\gamma_R)=m$.\\
Let $v\in\Im(\gamma)\setminus\{0\}$ be such that $SR(v)=SN(\gamma).$ \\
\begin{itemize}
\item If $k\geq m$ then choose $U\in M_k$ satisfying $\text{rank}(U)=k-SN(\gamma)$ and $(U\otimes Id)v=0$.\\ Define $\delta=(U\otimes Id)\gamma(U^*\otimes Id)$. Note that $\text{rank}(\delta)\leq \text{rank}(\gamma)-1$, since $\Im(\delta)\subset(U\otimes Id)(\Im(\gamma))$ and $(U\otimes Id)v=0$.\\
\item If $k<m$ then choose $U\in M_m$ satisfying $\text{rank}(U)=m-SN(\gamma)$ and $(Id\otimes U)v=0$.\\ Define $\delta=(Id\otimes U)\gamma(Id\otimes U^*)$. Note that $\text{rank}(\delta)\leq \text{rank}(\gamma)-1$, since $\Im(\delta)\subset(Id\otimes U)(\Im(\gamma))$ and $(Id\otimes U)v=0$.\\
\end{itemize}
In any case, by induction hypothesis, $ \text{rank}(\delta)SN(\delta)\geq\max\{\text{rank}(\delta_L), \text{rank}(\delta_R)\}$.\\
\begin{itemize}
\item If $k\geq m$ then $\delta_L=U\gamma_LU^*$. Since $\gamma_L$ is positive definite then $\text{rank}(\delta_L)=\text{rank}(U)=k-SN(\gamma).$\\
\item If $k<m$ then $\delta_R=U\gamma_RU^*$. Since $\gamma_R$ is positive definite then $\text{rank}(\delta_R)=\text{rank}(U)=m-SN(\gamma).$\\
\end{itemize}
Since $\text{rank}(\delta)\leq \text{rank}(\gamma)-1$ and $SN(\delta)\leq SN(\gamma)$ then\\
\begin{itemize}
\item$(\text{rank}(\gamma)-1)SN(\gamma)\geq k-SN(\gamma)$, if $k\geq m$. Therefore, $\text{rank}(\gamma)SN(\gamma)\geq k$. \\
\item$(\text{rank}(\gamma)-1)SN(\gamma)\geq m-SN(\gamma)$, if $k<m$. Therefore, $\text{rank}(\gamma)SN(\gamma)\geq m$. \\
\end{itemize}
The induction is complete.
\end{proof}
\vspace{0.3cm}
\begin{corollary}\label{corollary1} If $\gamma\in M_k\otimes M_k$ is a state then $\displaystyle SN(\gamma)\geq \frac{\text{rank}((\gamma_A)_L)}{2\ \text{rank}(\gamma_A)}$.
\end{corollary}
\begin{proof}
First, notice that $(\gamma_A)_L=(\gamma_A)_R$. Therefore, $\text{rank}((\gamma_A)_L)=\text{rank}((\gamma_A)_R)$.\\
Next, since $SN(\gamma_A)\leq 2SN(\gamma)$, by Proposition \ref{proposition1}, then $\displaystyle\text{rank}(\gamma_A)SN(\gamma)\geq\frac{1}{2}\text{rank}(\gamma_A)SN(\gamma_A)\geq \frac{\text{rank}((\gamma_A)_L)}{2}$, by Theorem \ref{theoremnew0}.
\end{proof}
\begin{corollary}\label{corollary2} If $\gamma\in M_k\otimes M_k$ is a state then $\displaystyle SN(\gamma)\geq \frac{\text{rank}((\gamma_S)_L)}{2\ \text{rank}(\gamma_S)}$.
\end{corollary}
\begin{proof}
First, notice that $(\gamma_S)_L=(\gamma_S)_R$. Therefore, $\text{rank}((\gamma_S)_L)=\text{rank}((\gamma_S)_R)$.\\
Since $SN(\gamma_S)\leq 2SN(\gamma)$, by Proposition \ref{proposition1}, then $\displaystyle \text{rank}(\gamma_S)SN(\gamma)\geq\frac{1}{2}\text{rank}(\gamma_S)SN(\gamma_S)\geq \frac{\text{rank}((\gamma_S)_L)}{2}$, by Theorem \ref{theoremnew0}.
\end{proof}
\vspace{0.3cm}
\section*{Summary and Conclusion}
We presented an inequality that relates the marginal ranks of any bipartite state of $M_k\otimes M_m$ to its rank and its Schmidt number . Using this inequality, we described a method of constructing entangled states which is not based on any correlation between $\text{rank}(\gamma_A)$ and $\text{rank}(\gamma_S)$.
This form of entanglement differs completely from the entanglement derived from the inequality $\text{rank}(\gamma_S)<\sqrt{\text{rank}(\gamma_A)}$.
We also constructed a family of PPT states whose members have Schmidt number equal to $n$, for any given $1\leq n\leq \left\lceil\frac{k-1}{2}\right\rceil$. This is a new contribution to the open problem of finding the best possible Schmidt number for PPT states.
\vspace{0.3cm}
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|
{
"timestamp": "2019-03-12T01:12:00",
"yymm": "1902",
"arxiv_id": "1902.11069",
"language": "en",
"url": "https://arxiv.org/abs/1902.11069"
}
|
\section{Introduction}
Any experiment witnessing or exploiting quantum coherent phenomena may be viewed as a test of whether quantum theory is complete at a fundamental level. While quantum mechanics is supported by all empirical observations up to date, all these observations are equally compatible with a number of alternative theories restoring macroscopic realism and resolving the measurement problem \cite{Leggett2002,Bassi2013}.
In recent years, various experiments demonstrated quantum superpositions or entanglement with mechanical objects of increasingly high masses and particle number, involving ever larger spatial delocalizations and coherence times. They include setups as diverse as
counter-propagating superconducting loop currents \cite{friedman2000,vanDerWal2000},
large path-separation atom interferometers \cite{peters1999,dimopoulos2007},
high-mass molecular near-field interferometers \cite{Gerlich2011,eibenberger2013},
trapped and freely-falling Bose-Einstein condensates \cite{Schmiedmayer2013,Kovachy2015},
de-localized states and Leggett-Garg tests in optical lattices \cite{Alberti2009,Robens2015},
entangled ion chains \cite{jurcevic2014,islam2015}, and nanomechanical oscillators \cite{Riedinger2018,ockeloen2018,Marinkovi2018}. While all these experiments establish variants of a Schr{\"o}dinger-cat-like state, an obvious question is the degree of macroscopicity (or `cattiness') reached.
There are many ways to assess the macroscopicity of a Schr{\"o}dinger cat realized in a quantum experiment \cite{frowis2018}. Most measures quantify the complexity of the quantum state based on information- or resource-theoretic concepts \cite{Korsbakken2007,Marquardt2008,Froewis2012,yadin2016general,Yadin2018}, or introduce suitable distance measures in Hilbert space \cite{Bjoerk2004,Lee2011}. While such abstract state vector ranking schemes may be used to compare experimental setups of similar kind, none can cover the entire variety of present-day superposition experiments \cite{friedman2000,vanDerWal2000,peters1999,dimopoulos2007,Gerlich2011,eibenberger2013,Schmiedmayer2013,Kovachy2015,Alberti2009,Robens2015,jurcevic2014,islam2015,Riedinger2018,ockeloen2018,Marinkovi2018}.
A viable alternative is to regard a Schr{\"o}dinger cat as more macroscopic than others if its demonstration is more at odds with the classical expectations shaped by our every-day experiences. In Ref.~\cite{Nimmrichter2013} this was cast into a macroscopicity measure by quantifying the extent to which a superposition experiment rules out a wide and natural class of objective modifications of quantum theory that predict classical behavior on the macroscale, so-called \emph{classicalizing modifications}.
Recent tests of nonlocality and macrorealism, demonstrating the violation of Bell and Leggett-Garg inequalities at unprecedented mass and time scales, call for a generalization of this measure for arbitrary quantum tests.
\begin{figure}
\centering
\includegraphics[width=0.39\textwidth]{fig1.pdf}
\caption{Scheme to compare the macroscopicity of two different quantum superposition tests: The experiments deliver raw data sets $D_1$ and $D_2$, which may be of arbitrary type and structure. They can be used to rule out modifications of standard quantum theory which classicalize the dynamics. Combining the data with the theoretical expectation yields a probability distribution for the classicalization timescale $\tau_e$, given the modification parameters $\sigma$ and the background information $I$. A quantum experiment is considered more macroscopic if the data rule out greater values of $\tau_e$, as inferred from the $5\,\%$ quantile $\tau_{\rm m}$.}
\label{fig:scheme}
\end{figure}
In this article, we present the most general framework for assigning the macroscopicity reached in quantum mechanical superposition experiments, based on non-informative Bayesian hypothesis testing, see Fig.~\ref{fig:scheme}. As the natural generalization of the measure presented in \cite{Nimmrichter2013}, it relies only on the empirical evidence (i.e. the raw measurement outcomes) delivered by a given superposition test. It thus accounts for the measurement imperfections independently of the chosen experimental figure of merit.
This measure of macroscopicity can be applied to assess any mechanical superposition experiment. It is unbiased by construction and it accounts naturally for experimental uncertainties and statistical fluctuations. These advantages come at the expense of a certain theoretical effort required for calculating the macroscopicity of a given experiment. Specifically, the time evolution of the quantum system must be calculated in presence of classicalizing modifications to obtain the probability distribution for all possible measurement outcomes. In the second part of this article we demonstrate how this task is accomplished for three superposition tests at the cutting-edge of quantum physics: double-well interference of number-squeezed Bose-Einstein condensates (BECs) \cite{Schmiedmayer2013}, Leggett-Garg inequality tests with atomic quantum random walks \cite{Robens2015}, and generation and witnessing of entanglement between two spatially separated nanomechanical oscillators \cite{Riedinger2018}.
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{fig2.pdf}
\caption{(a) Schematic illustration of the double-well BEC interference experiment \cite{Schmiedmayer2013}. The BEC is initially split into a superposition between slightly detuned left and right double-well states, then number squeezed, then let to freely evolve for a delay time, before a final $\pi/2$-pulse (recombiner) converts the phase difference between the states into an occupation difference. (b) Time-of-flight measurement data of the occupation imbalance versus delay time (from Ref.~\cite{Schmiedmayer2013}). (c) Posterior distribution of the classicalization timescale (red solid line) as obtained via Bayesian updating of the Jeffreys' prior (black dashed line) with the measurement data. The blue line is the intermediate distribution obtained by using only the blue data points up to one millisecond in (b). The shaded areas indicate the lowest five percent quantiles and all distributions are normalized to the same maximum value.}
\label{fig:2}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{fig3.pdf}
\caption{(a) Schematic illustration of the quantum random walk consisting of four consecutive steps. In each step the atom is coherently split into a left- and right-moving state, and the final populations are read-out after the fourth step. (b) Experimental data from Ref.\,\cite{Robens2015}. The blue solid line is the data from the total quantum random walk, while the red lines are conditioned on the first step being either left (dashed) or right (dotted). (c) Posterior distribution of the classicalization timescale (red solid line) as obtained via Bayesian updating of the Jeffreys' prior (black dashed line) with the measurement data. The blue line is the intermediate distribution obtained by using only the blue measurement runs in (b). The shaded areas indicate the lowest five percent quantiles and all distributions are normalized to the same maximum value.}
\label{fig:3}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{fig4.pdf}
\caption{(a) Stoke's scattering of a pump photon, prepared in a spatial superposition by the entrance beam splitter, generates entanglement between two nanomechanical oscillators, which is then read-out by anti-Stoke's scattering of a read photon. Entanglement is certified by a coincidence measurement of the scattered photons in the upper ($+$) or lower ($-$) detector behind the exit beam splitter. (b) Measurement data \cite{Riedinger2018} as a function of the tunable relative phase $\theta$ between the two interferometer arms ({\it phase sweep}). (c) Posterior distribution of the classicalization timescale (red solid line) as obtained via Bayesian updating of the Jeffreys' prior (black dashed line) with the measurement data. The blue line is the posterior obtained by taking only phase sweep data points into account, while the red line also accounts for measurements with variable time delay between pump and read [{\it time sweep}; not shown in (b)]. The shaded areas indicate the lowest five percent quantiles and all distributions are normalized to the same maximum value.}
\label{fig:4}
\end{figure*}
\section{Macroscopicity of three recent superposition tests} \label{sec:neue2}
Before presenting the formal framework of the proposed measure of macroscopicity, we illustrate its application to three recent superposition tests \cite{Schmiedmayer2013,Robens2015,Riedinger2018}. As a common theme, these experiments use derived quantities, such as visibilities, correlation functions, and entanglement witnesses, to certify the quantumness of their observations. One important advantage of the Bayesian approach advocated here is that it is independent of such data processing (and thus of secondary observables) and based exclusively on likelihoods associated with elementary measurement events. A theoretical derivation of the likelihoods required to assess the three mentioned experiments is presented in Secs. \ref{sec:3}--\ref{sec:5}.
The measure uses the experimental data $D$ to determine the posterior probability distribution $p(\tau_e|D,\sigma, I)$ of classicalization timescales $\tau_e$, given the modification parameters $\sigma$, and any background information $I$ required to model the experiment. To ensure that each experiment is rated without bias, the least informative prior is used for Bayesian updating to yield the final posterior distribution. Figures \ref{fig:2}--\ref{fig:4} show how disparate experimental measurement protocols and data sets \cite{Schmiedmayer2013,Robens2015,Riedinger2018} yield comparable posterior distributions, narrowly peaked around a definite modification timescale.
As an increasing number of data-points is included in the Bayesian updating procedure, the distributions shift to higher modification time scales, while their widths decrease. The lowest five percent quantile $\tau_{\rm m}(\sigma)$ of the posterior distribution determines the macroscopicity as
\begin{equation*}
\mu_{\rm m} = \max_\sigma \left [ \log_{10}\left(\frac{\tau_{\rm m}(\sigma)}{1\,{\rm s}}\right) \right ].
\end{equation*}
The value $\mu_{\rm m}$ thus quantifies the degree to which the quantum measurement data rules out a wide and natural class of classicalizing modifications of quantum theory.
The resulting macroscopicities of the experiments are: $\mu_{\rm m}=8.5$ for the BEC interferometer \cite{Schmiedmayer2013}, $\mu_{\rm m}=7.1$ for the atomic Leggett-Garg test \cite{Robens2015}, and $\mu_{\rm m}=7.8$ for the entangled nanobeams. That the BEC and the atomic random walk experiments exhibit comparable macroscopicities is due to the fact that they both witness single atom interference at a similar product of squared mass and coherence time. The macroscopicity associated with the entangled nanobeam experiment is roughly on the same order of magnitude on the logarithmic scale, despite the high mass and the large separation between the two beams and as well as coherence times of microseconds. This surprising result can be explained by the fact that the probed superposition state is delocalized merely by a few femtometers, and thus probes quantum theory only on sub-atomic scales.
Comparison of the three experiments also reveals that the convergence rate of the posterior distribution can vary strongly. In case of the Leggett-Garg test with an atomic quantum random walk \cite{Robens2015}, the data set consists of 627 walks which all end in one of five final lattice sites. Since the likelihood of two of those outcomes is independent of the modification they include no information for the hypothesis test, which slows the convergence of the Bayesian updating procedure. In contrast, the double-well BEC-interferometer \cite{Schmiedmayer2013} provides a distribution of measurement outcomes over a practically continuous range of values, so that each experimental run yields a high degree of information gain, implying that 1457 measured population imbalances lead to a relatively narrow posterior distribution. In the case of nanobeams only two of four possible coincidence outcomes have different likelihoods, and thus several thousand repetitions of the measurement protocol are required to make the posterior converge.
\section{Macroscopicity via hypothesis falsification} \label{sec:2}
\subsection{Empirical measure of macroscopicity}\label{sec:2A}
Classicalizing modifications of quantum theory propose an alternative (stochastic) evolution equation for the wavefunction. The observable consequences of these alternative theories are then encoded in the dynamics of the state operator $\rho_t$, which evolves according to a modified von Neumann equation
\begin{align} \label{eq:modvonneum}
\partial_t \rho_t=\mathcal{L}\rho_t+\frac{1}{\tau_e}\mathcal{M}_{\sigma}\rho_t.
\end{align}
Here ${\cal L}\rho_t$ denotes the time evolution according to standard quantum theory (including possible decoherence) and ${\cal M}_\sigma \rho_t / \tau_e$ describes the effect of the proposed modification, characterized by the time scale $\tau_e$ and the set of modification parameters $\sigma$.
Indeed, a wide class of modification theories are compatible with all observations up to date, and they restore realism on the macroscale. This class can be parametrized by imposing a few natural consistency requirements, such as Galilean and scale invariance and exchange symmetry \cite{Nimmrichter2013}. The parameters $\sigma=(\sigma_q,\sigma_s)$ then specify the length- and momentum-scale on which the modification acts by means of the distribution function $g_\sigma(q,s)$ with zero mean and widths $\sigma_q,\sigma_s$,
\begin{align}
\mathcal{M}_{\sigma}\rho_t=&\int d^3{\bf q}d^3{\bf s}\,g_\sigma(q,s)\left [
{\sf L}({\bf q},{\bf s})\rho_t{\sf L}^{\dagger}({\bf q},{\bf s}) \vphantom{\frac{1}{2}}\right. \nonumber \\
& \left. -\frac{1}{2}\left\{{\sf L}^{\dagger}({\bf q},{\bf s}){\sf L}({\bf q},{\bf s}),\rho_t
\right\}\right].
\label{eq:MIMpointparticle}
\end{align}
The Lindblad operators in second quantization,
\begin{align} \label{eq:lindblad}
{\sf L}({\bf q},{\bf s})=\sum_\alpha\frac{m_\alpha}{m_e}\int d^3{\bf p}e^{i{\bf p}\cdot m_e{\bf s}/m_\alpha\hbar}{\sf c}_\alpha^{\dagger}({\bf p}){\sf c}_\alpha({\bf p}-{\bf q})\,,
\end{align}
induce displacements in phase-space by means of the annihilation operator ${\sf c}_\alpha({\bf p})$ for momentum ${\bf p}$. They involve a sum over the different particle species $\alpha$ with mass $m_\alpha$, whose ratio over the electron mass $m_e$ effectively amplifies the strength of the modification for heavy particles, ensuring that macrorealism is restored \cite{Nimmrichter2013}. We take $g_\sigma$ to be Gaussian in the following. The modification \eqref{eq:MIMpointparticle} then reduces to the model of continuous spontaneous localization (CSL) \cite{Bassi2013} for fixed $\sigma_q$ and $\sigma_s=0$. As explained in Ref.~\cite{Nimmrichter2013}, the bounds $\hbar/\sigma_q\gtrsim 10\,$fm and $\sigma_s\lesssim 20\,$pm ensure that the modification does not drive the system into the regime of relativistic quantum mechanics. In what follows, we will define the empirical measure of macroscopicity as the extent to which a quantum experiment rules out such classicalizing modifications.
Since the modified evolution \eqref{eq:modvonneum} predicts deviations from standard quantum mechanics at some scale these modification theories are empirically falsifiable. Thus, any quantum experiment gathering measurement data $D$ can be considered as testing the hypothesis $H_{\tau_e^*}$:
\begin{quote}
\textit{Given a classicalizing modification \eqref{eq:MIMpointparticle} with parameters $\sigma$, the dynamics of the system state $\rho_t$ are determined by Eq.~\eqref{eq:modvonneum} with a modification time scale $ \tau_e \leq \tau_e^*$.}
\end{quote}
Note that greater values of $\tau_e$ imply weaker modifications.
The empirical data $D$ determine the Bayesian probability $P(H_{\tau_e^*} | D,\sigma, I)$ that $H_{\tau_e^*}$ is true, given the background information $I$. The latter includes all knowledge required for describing the experiment, such as the Hamiltonian, environmental decoherence processes, and the measurement protocol.
In order to compare $H_{\tau_e^*}$ with the complementary hypothesis $\overline{H}_{\tau_e^*}$, one defines the odds ratio \cite{von2014bayesian}
\begin{equation} \label{eq:oddsratio1}
o(\tau_e^* | D,\sigma, I) = \frac{P(H_{\tau_e^*} | D,\sigma, I)}{P(\overline{H}_{\tau_e^*} | D,\sigma, I)}.
\end{equation}
If the data implies that the odds ratio is less than a certain maximally acceptable value $o_{\rm m}$ we can favor $\overline{H}_{\tau_e^*}$ over $H_{\tau_e^*}$. Modifications of quantum theory with $\tau_e \leq \tau_e^*$ are then ruled out by the data at odds $o_{\rm m}$ .
In order to evaluate the odds ratio \eqref{eq:oddsratio1} we use Bayes' theorem and exploit that for the hypothesis test to be unbiased by earlier experiments, $H_{\tau_e^*}$ and $\overline{H}_{\tau_e^*}$ must be {\em a priori} equally probable. Further using that the hypothesis $H_{\tau_e^*}$ implies $\tau_e \leq \tau_e^*$ yields
\begin{align}
o(\tau_e^*|D,\sigma,I)=&\frac{\displaystyle \int_0^{\tau_e^*}d\tau_e\,P(D|\tau_e,\sigma,I)p(\tau_e|\sigma,I)}{\displaystyle \int_{\tau_e^*}^{\infty}d\tau_e\,P(D|\tau_e,\sigma,I)p(\tau_e|\sigma,I)},
\label{eq:oddsratio}
\end{align}
where $p(\tau_e|\sigma,I)$ is the prior distribution of $\tau_e$, whose choice will be discussed in Sec.~\ref{subsec:32}. The probabilities $P(D |\tau_e,\sigma,I)$ are independent of the hypothesis $H_{\tau_e^*}$; they can be calculated for any experiment by solving the modified evolution equation \eqref{eq:modvonneum} with classicalization time scale $\tau_e$ and parameters $\sigma$.
The data $D$ is usually gathered in $N$ consecutive independent runs, $D = \{D_1,D_2,\ldots,D_{N}\}$, where $D_k$ denotes the set of (possibly correlated) measurement outcomes of round $k$. The likelihood for the entire data set $D$ is then given by
\begin{align}
P(D|\tau_e,\sigma,I)=\prod_{k}P(D_k|\tau_e,\sigma,I).
\label{eq:likelihoodproduct}
\end{align}
Every additional experimental run thus refines the posterior probability density, according to Bayes' theorem
\begin{align} \label{eq:posterior}
p(\tau_e|D,\sigma,I) = \frac{P(D|\tau_e,\sigma,I)p(\tau_e|\sigma,I)}{P(D|\sigma,I)},
\end{align}
where the normalization constant $P(D|\sigma,I)$ plays no role for the odds ratio.
For sufficiently large data sets the posterior turns independent of the prior distribution $p(\tau_e|\sigma, I)$ under very general conditions \cite{schwartz1965bayes,ghosh2003springer}.
For what follows, we choose the threshold odds $o_{\rm m} = 1:19$, corresponding to the posterior probability
\begin{equation} \label{eq:taum}
P(\tau_e \leq \tau_{\rm m}|D,\sigma,I) \equiv \int_0^{\tau_{\rm m}} d \tau_e p(\tau_e \vert D,\sigma,I) = 5\,\%.
\end{equation}
This determines the greatest excluded modification time scale $\tau_{\rm m}$ (at odds $o_{\rm m}$) so that for all $\tau_e^* \leq \tau_{\rm m}$ the odds ratio \eqref{eq:oddsratio} is smaller than $o_{\rm m}$ for given modification parameters $\sigma$.
Given the greatest excluded modification time scale $\tau_{\rm m}(\sigma)$ as a function of the modification parameters $\sigma$, one defines the empirical measure of macroscopicity as
\begin{equation}
\mu_{\rm m} = \max_\sigma \left [ \log_{10}\left(\frac{\tau_{\rm m}(\sigma)}{1\,{\rm s}}\right) \right ],
\label{eq:Macroscopicity}
\end{equation}
where $\tau_{\rm m}(\sigma)$ [Eq.~\eqref{eq:taum}] is the extent to which the measurement data $D$ of a given quantum experiment rules out the class of modifications \eqref{eq:MIMpointparticle}. The value of $\mu_{\rm m}$ thus ranks superposition experiments against each other according to the degree to which they are at odds with our classical expectation.
We emphasize that this definition must only be used for experiments that undeniably show genuine quantum signatures. It cannot be used to \emph{certify} whether a given experiment observes a superposition state. This is due to the fact that the absence of modification-induced heating and momentum diffusion can be observed also in classical experiments. Even though quantum coherence plays no role in such setups, they can serve to exclude combinations of classicalization timescales and modification parameters \cite{Laloe2014,Nimmrichter2014,carlesso2016experimental,li2016discriminating,
goldwater2016testing, vinante2017, schrinski2017collapse,adler2018bulk,bahrami2018testing}.
Even in genuine quantum superposition experiments the observed absence of modification-induced heating may dominate the range of excluded modification parameters. In this case it is necessary to recombine the observables in such a way that they separate into a subset of random variables $d$ providing information about quantum coherence and a subset $d_{\rm heat}$ yielding only information about the energy gain. (For example, in the case of the double-well BEC interference experiment, where one measures the particle numbers in the two different wells, their difference shows interference based on quantum coherence, while their sum constraints particle loss due to heating.) For a fair assessment of the macroscopicity, the likelihood $P(d,D_{\rm heat}|\tau_e,\sigma, I)$ must be conditioned on the realized data $D_{\rm heat}$ restricting modification induced heating,
\begin{equation}\label{eq:10cl}
P(d|\tau_e,\sigma, I, D_{\rm heat}) = \frac{P(d,D_{\rm heat}|\tau_e,\sigma, I)}{ P(D_{\rm heat}|\tau_e,\sigma, I)}\,
\end{equation}
with $P(D_{\rm heat}|\tau_e,\sigma, I)=\sum_d P(d,D_{\rm heat}|\tau_e,\sigma, I)$. This way the witnessed lack of heating is effectively added to the background information $I$. (It also shows how to formally take into account the observation that the experiment could be executed at all, i.e.\ that the setup did not disintegrate due to modification-induced heating.) In Sect.~\ref{sec:3} we demonstrate how the conditioning on quantum observables works in practice by means of a nontrivial example.
\subsection{Jeffreys' prior} \label{subsec:32}
If the data set is not sufficiently large, the measure \eqref{eq:Macroscopicity} will in general depend on the prior distribution chosen to evaluate the odds ratio \eqref{eq:oddsratio}. It is therefore necessary to specify which prior distribution $p(\tau_e | \sigma,I)$ must be used to calculate the macroscopicity \eqref{eq:Macroscopicity}.
In order to ensure that the macroscopicity $\mu_{\rm m}$ does not have a bias towards a selected class of quantum superposition tests, the prior must be chosen in the most uninformative way, i.e.\ without including any {\it a priori} believes. For instance, this implies that it must not play a role whether we use the time scale $\tau_e$ or the rate $1/\tau_e$ to parametrize the class of modifications, which already excludes a uniform or piecewise-constant prior. Therefore, the natural choice is Jeffreys' prior \cite{jeffreys1998theory}. Given the likelihood $P(d|\tau_e, \sigma,I)$ associated with a random variable $d$, it is defined as the square root of the Fisher information,
\begin{align}
p(\tau_e|\sigma,I)=\sqrt{ \left \langle \left ( \frac{\partial}{\partial \tau_e}\log[P(d|\tau_e,\sigma,I)] \right )^2\right \rangle_d}\,.
\label{eq:JeffreysPrior}
\end{align}
The ensemble average $\langle \cdot \rangle_d$ is performed over the entire range of possible measurement outcomes $d$ with Probability $P(d |\tau_e,\sigma,I)$.
This prior coincides with the so-called reference prior, so that it maximizes the Kullback-Leibler-divergence between prior and posterior and thus the average information gain in the Bayesian updating process \eqref{eq:posterior} \cite{bernardo1979reference,ghosh2011objective}. In this sense, Jeffreys' prior can be considered as the least informative prior \cite{berger2009formal}. In addition, it is invariant under re-parametrizations of the model \cite{jeffreys1998theory}, implying that it is irrelevant whether we use the timescale $\tau_e$ or the rate $\lambda=1/\tau_e$ (as employed in the model of Continuous Spontaneous Localization \cite{Bassi2013}) or any other power of $\tau_e$ as the fundamental parameter of our model.
We demonstrate in App.\,\ref{app:beweis} that for all practical purposes Eq.~\eqref{eq:JeffreysPrior} yields a normalizable posterior distribution \eqref{eq:posterior} because the master equation \eqref{eq:modvonneum} and thus the likelihood $P(d|\tau_e,\sigma, I)$ are smooth functions of $\tau_e$.
If different measurement protocols are implemented, it is not obvious which likelihood should be used for calculating the prior since the Bayesian updating is independent of the order of measurement runs. A natural choice is to use the least favorable measurement protocol, i.e. the one yielding the lowest macroscopicity if no updating is performed. However, in experiments with sufficiently large data sets the posterior will in any case become independent of the prior distribution.
\subsection{General scheme for assigning macroscopicities}
The formal framework of how to assess the macroscopicity of arbitrary quantum mechanical superposition tests is now complete:
\begin{enumerate}
\item Determine the Hamiltonian, environmental decoherence channels, and quantum measurement protocol, and use these to calculate the likelihood $P(d|\tau_e, \sigma,I)$ in presence of the modification \eqref{eq:MIMpointparticle}. If appropriate use Eq.~\eqref{eq:10cl} to focus on data demonstrating quantum coherence.
\item Calculate Jeffreys' prior \eqref{eq:JeffreysPrior}. In case that more than one quantum measurement protocol was experimentally implemented, use the one yielding the smallest prior macroscopicity.
\item Determine the posterior distribution via Bayesian updating \eqref{eq:posterior} to extract $\tau_{\rm m}(\sigma)$ via \eqref{eq:taum}.
\item Find the maximum of the function $\tau_{\rm m}(\sigma)$, which determines the macroscopicity \eqref{eq:Macroscopicity}.
\end{enumerate}
This recipe prescribes how to calculate the macroscopicity based on the empirical evidence of a quantum experiment. It formalizes and generalizes the notion of macroscopicity introduced in Ref.~\cite{Nimmrichter2013}. The approximate expressions derived in Ref.~\cite{Nimmrichter2013} intrinsically assume that imperfections of a given experiment yield a definite value of $\tau_e<\infty$, corresponding to a delta-peaked posterior distribution. The Bayesian framework put forward here extends this to measurement schemes and data sets yielding a finite posterior distribution $p(\tau_e|D,\sigma,I)$. It is thus the natural extension for noisy data and arbitrary measurement strategies.
In practice, the most complicated part of the above scheme is calculating the likelihoods in step 1. This requires finding an appropriate and quantitative description of the quantum dynamics in presence of decoherence and the modification. Note that the macroscopicity is underestimated if relevant decoherence channels are neglected in the calculation of the likelihoods. The remainder of this article demonstrates how the likelihoods can be calculated for the three superposition tests discussed in Sec.~\ref{sec:neue2}.
\section{Ramsey interferometry with a number-squeezed BEC} \label{sec:3}
\subsection{Experimental Setting and Basics}
In the experiment reported in Ref.~\cite{Schmiedmayer2013} a ${}^{87}$Rb BEC is trapped in a double-well potential and made to interfere, see Fig.~\ref{fig:2}(a). The two involved modes $a,b$ form an effective two-level system described by the annihilation operators ${\sf c}_{a}$, ${\sf c}_{b}$. The state of the BEC can thus be represented by a collective pseudospin, defined by means of the (dimensionless) quasi angular momentum operators \cite{Arecchi1972}
\begin{align}
{\sf J}_x&=\frac{1}{2}\left({\sf c}^{\dagger}_a{\sf c}_b+{\sf c}^{\dagger}_b{\sf c}_a\right)\nonumber\\
{\sf J}_y&=\frac{1}{2i}\left({\sf c}^{\dagger}_a{\sf c}_b-{\sf c}^{\dagger}_b{\sf c}_a\right)\nonumber\\
{\sf J}_z&=\frac{1}{2}\left({\sf c}^{\dagger}_a{\sf c}_a - {\sf c}^{\dagger}_b{\sf c}_b\right).
\label{eq:AngularMomentumOperators}
\end{align}
They fulfill the angular momentum commutation relations $[{\sf J}_{\lambda},{\sf J}_{\mu}]=i\epsilon_{\lambda,\mu,\nu}{\sf J}_{\nu}$. The simultaneous eigenstates of ${\sf J}^2$ with eigenvalue $J(J+1)$ and ${\sf J}_z$ with eigenvalue $m$ is denoted by $\left|J,m\right\rangle$ (Dicke state), where $J=N/2$.
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{fig5.pdf}
\caption{(a)--(c) The dynamics of large collective spin states close the equator of a generalized Bloch sphere can be effectively described by first evolving the state in the local tangent plane and then wrapping it back around the sphere. (d)--(f) Exact simulations of the BEC number differences (red histograms) are in very good agreement with the analytical approximation \eqref{eq:PhaseFlipDistributionApprox} (black lines). The simulation was performed for $N=100$ particles and an initial variance of $\Delta{\sf J}^2_z=N/5$, reached by means of one-axis-squeezing \cite{Kitagawa1993}. The snapshots are taken at times (d) $t_0=0$, (e) $t_1=5.25\pi\hbar/\epsilon$, and (f) $t_2=400 \pi\hbar/\epsilon$ with $\Gamma_{\rm P}=0.002\epsilon/\hbar$ and $\zeta=0.002\epsilon/\hbar$. At time $t_2$ the distributions have practically converged towards the fully dephased steady state. (g) In absence of phase diffusion the distribution exhibits (partial) revivals, as illustrated in Panel (g) for time $t_2$. (A complete revival to the state shown in (a) would first be observed at $t=1000\pi\hbar/\epsilon$.) }
\label{fig:blochspheres}
\end{figure*}
The product of $N$ bosons being in a superposition state (coherent spin state; CSS) can be represented on a generalized Bloch sphere (see Fig.~\ref{fig:blochspheres}), whose polar angle $\theta$ indicates the relative population in $a$ and $b$, while the azimuth $\phi$ is the relative phase of the superposition state. Such a product state $\vert \theta, \phi \rangle$ can be expanded in terms of Dicke states as
\begin{align}
\left|\theta,\phi\right\rangle \equiv &\frac{1}{\sqrt{(2J)!}}\left(\cos\left(\frac{\theta}{2}\right){\sf c}^{\dagger}_a+e^{i\phi}\sin\left(\frac{\theta}{2}\right){\sf c}^{\dagger}_b\right)^{2J} \vert {\rm vac} \rangle \nonumber\\
=&\sum_{m=-J}^J{2J \choose J+m}\cos\left(\frac{\theta}{2}\right)^{J-m}\sin\left(\frac{\theta}{2}\right)^{J+m}\nonumber\\
&\times e^{-i(J+m)\phi}\left|J,m\right\rangle.
\end{align}
It has minimal and symmetric uncertainties, e.g.\ $\Delta {\sf J}_z^2=\Delta{\sf J}_y^2=|\left\langle{\sf J}_x\right\rangle/2|=J/2$ for $\theta = \pi/2$ and $\phi = 0$.
Applying a nonlinear squeezing operator turns the CSS into a squeezed spin state (SSS) \cite{Kitagawa1993,Ma2011}, which can be useful for metrology \cite{toth2012,toth2014quantum,hosten2016quantum} or robust against dephasing processes \cite{Javanainen1997PhaseDispersion,Schmiedmayer2013}. In addition, it has been demonstrated that the {\em depth of entanglement} increases with squeezing \cite{sorensen2001many,Sorensen2001,Toth2014}, as quantified by the squeezing parameter $\xi^2 = 2(\Delta {\sf J}_{\rm min})^2 / J$.
We note that according to the information-theoretic measure from Ref.~\cite{Froewis2012}
already the existence of such a state yields a large macroscopicity since squeezing increases the quantum Fisher information.
In terms of the depth of entanglement \cite{Sorensen2001,Toth2014} the non-classicality of SSS lies between a product state (CSS) and the maximally entangled NOON-state $\left|\psi\right\rangle\propto\left|N,0\right\rangle+\left|0,N\right\rangle$, a superposition of all particles being either in mode $a$ or mode $b$. Applying the modification on this NOON state yields a decoherence rate proportional to $N^2$, while that of a product state is proportional to $N$. It is thus easy to see that a NOON-state with stable phase could serve to exclude a large range of classicalization time scales \cite{bilardello2017collapse}, but they have not been generated experimentally thus far. In contrast, the modification-induced dynamics of SSS, which are frequently realized in experiments, is much more intricate, as discussed in the following.
The free time evolution ${\cal L}\rho = -i [{\sf H},\rho]/\hbar$ of the BEC is characterized by the energy difference $\epsilon$ between the two modes and by the interaction between the particles. Approximating the latter to leading order in ${\sf J}_z$, yields the Hamiltonian \cite{Javanainen1997PhaseDispersion}
\begin{equation}
{\sf H}=\epsilon {\sf J}_z +\hbar\zeta{\sf J}^2_z,
\label{eq:FreeDephasingHamiltonian}
\end{equation}
where $\zeta=d\tilde{\mu}/d (\hbar m)|_{m=0}$ is the change of chemical potential with the occupation difference $m$. Thus, the first term of the Hamiltonian describes rotations around the $z$-axis with angular frequency $\epsilon/\hbar$ on the Bloch sphere, while the second term leads to dispersion.
The experiment starts with the BEC in the state $|\theta=\pi/2,\phi=0\rangle$, which is then squeezed in $z$-direction and freely evolved for up to 20\,ms. Finally, a $\pi/2$-rotation around the $x$-axis converts the phase distribution into mode occupation differences, which are read-out by time-of-flight measurements, see Fig.\,\ref{fig:2}.
The likelihood required for the hypothesis test is the probability of observing a number difference of $m$ between the two modes,
\begin{align}
P(m | \tau_e,\sigma,I)= \sum_{J = 0}^{\infty} \left\langle J, m\right|e^{-i\pi{\sf J}_x/2}\rho_t e^{i\pi{\sf J}_x/2}\left| J, m\right\rangle\,,
\label{eq:PiHalfPulse}
\end{align}
where the sum over $J$ accounts for the possibility of modification-induced particle loss from the BEC during the experiment \cite{Laloe2014}. The modification parameters $\tau_e$ and $\sigma$ enter through the modified time evolution of the state $\rho_t$, which will be discussed next.
\subsection{Double-well potential: phase flips}
Expanding the momentum annihilation operators ${\sf c}({\bf p})$ in Eq.~\eqref{eq:lindblad} in the single-particle eigenmodes in presence of the potential, and neglecting particle loss for the moment (${\mathsf c}_a^\dagger{\mathsf c}_a+{\mathsf c}_b^\dagger{\mathsf c}_b=2J$), yields
\begin{align}
\mathcal{M}_\sigma\rho=&\frac{4 m_{\rm Rb}^2}{\tau_e m_e^2}\int d^3{\bf q}\,f_\sigma(q)\nonumber\\
&\times\left[
{\sf A}({\bf q})\rho{\sf A}^{\dagger}({\bf q})-\frac{1}{2}\{{\sf A}^{\dagger}({\bf q}){\sf A}({\bf q}),\rho\}
\right].
\end{align}
Here, we used that spatial displacements are negligible on the length scale of the experiment and thus $f_\sigma(q) = \int ds g_\sigma(q,s)$ depends only on $\sigma_q$. The Lindblad operators are given by
\begin{align}
{\sf A}({\bf q})=a_x({\bf q}){\sf J}_x+a_z({\bf q}){\sf J}_z\,,
\end{align}
with
\begin{align}
a_x({\bf q})=&\left\langle \psi_a\right|{\sf W}({\bf q})\left|\psi_b\right\rangle
\nonumber\\
a_z({\bf q})=&\left\langle \psi_a\right|{\sf W}({\bf q})\left|\psi_a\right\rangle\sin\left(\frac{\Delta_x q_x}{2\hbar}\right).
\label{eq:FlipCoefficients}
\end{align}
Here, $\left|\psi_a\right\rangle$ and $\left|\psi_b\right\rangle$ are the single-atom eigenstates of the two level system with real wavefunctions $\psi_b({\mathbf r})=\psi_a({\mathbf r}-\Delta_x {\mathbf e}_x)\in \mathbb{R}$ and ${\sf W}({\bf q})=\exp(i {\bf q}\cdot \boldsymbol{\mathsf{r}}/\hbar)$
is the momentum transfer operator.
The first part of the Lindblad operator describes rotations around the $x$-axis, or spin-flips, while the second one induces rotations around the $z$-axis, or phase-flips. Such flip operators are frequently used to describe disturbance channels in collective spin states \cite{wang2010sudden,Ma2011}. Since the spatial overlap between the two modes is negligible, $a_x({\bf q})\ll a_z({\bf q})$, the spin-flip contribution will be neglected in the following, implying that $\langle {\sf J}^2_z \rangle_t$ remains constant.
The expectation value of the perpendicular spin components decays as $\left\langle{\sf J}_y\right\rangle_t=e^{-\Gamma_{\rm P} t/2}\left\langle{\sf J}_y\right\rangle_{{\rm f}, t}$ with phase-flip rate (or dephasing rate)
\begin{align}
\quad\Gamma_{\rm P}=\frac{4 m_{\rm Rb}^2}{\tau_e m_e^2}\int d^3{\bf q}\,f_\sigma(q)|a_z({\bf q})|^2.
\label{eq:SqueezingTEFirstMoments}
\end{align}
Here $\left\langle{\sf J}_y\right\rangle_{{\rm f},t}$ denotes the free time evolution of the expectation value due to Eq.~\eqref{eq:FreeDephasingHamiltonian}; the same relation holds for $\langle {\sf J}_x \rangle_t$. Note that the phase-flip decay rate $\Gamma_{\rm P}$ is independent of the degree of squeezing.
The phase-flip operators induce diffusion in the azimuthal plane of the Bloch sphere. The second moment of ${\sf J}_y$ thus evolves as
\begin{align}
\left\langle{\sf J}^2_y\right\rangle_t=\frac{1}{2} \left\langle{\sf J}^2_x + {\sf J}^2_y\right\rangle_{{\rm f},t}
-
\frac{e^{-2\Gamma_{\rm P} t}}{2} \left\langle{\sf J}^2_x - {\sf J}^2_y\right\rangle_{{\rm f},t},
\label{eq:SqueezingTESecondMoments}
\end{align}
and similar for ${\sf J}^2_x$.
For sufficiently large $N$ the squeezing loss rate is again independent of the initial squeezing since $\left\langle{\sf J}^2_x\right\rangle_{{\rm f},t}\approx J^2$ (as long as oversqueezing is avoided).
Equations~\eqref{eq:SqueezingTEFirstMoments} and \eqref{eq:SqueezingTESecondMoments} show that squeezing has no direct implications for the sensitivity on modification-induced decoherence. In contrast to what might be expected intuitively, an increased {\em depth of entanglement} does therefore not improve substantially the macroscopicity of experiments that measure only the first two moments of the collective spin observables.
\subsection{Continuum approximation}
In order to calculate the likelihood \eqref{eq:PiHalfPulse}, we will utilize a continuum approximation on the tangent plane of the Bloch sphere, replacing the discrete probability $P(m|\tau_e,\sigma,I)$ by the continuous probability density $p(m|\tau_e,\sigma,I)$ for real $m$. For this sake, we use that the initial state is aligned with the $x$-axis, $\langle {\sf J}_x \rangle \approx J$, so that
\begin{equation}
[{\sf J}_y, {\sf J}_z] \approx i J\,,
\end{equation}
which is approximately constant (and not operator valued). Thus we locally replace the sphere by its flat tangent plane and may interpret ${\sf J}_y$ as a position and ${\sf J}_z$ as a momentum operator, see Fig.\,\ref{fig:blochspheres}. The Wigner function of the initial state is then approximated by a Gaussian distribution,
\begin{align} \label{eq:ini}
w_0(j_y,j_z) = &\frac{1}{\sqrt{4\pi^2\left\langle{\sf J}^2_y\right\rangle_0\left\langle{\sf J}^2_z\right\rangle_0}}\nonumber\\
&\times \exp\left[
-\frac{1}{2}\frac{j_y^2}{\left\langle{\sf J}^2_y\right\rangle_0}-\frac{1}{2} \frac{j_z^2}{\left\langle{\sf J}^2_z\right\rangle_0}
\right],
\end{align}
where $(j_y,j_z) \in \mathbb{R}^2$ are continuous variables in the flat tangent plane.
The time evolution of the initial state \eqref{eq:ini} contains the free rotation and dispersion described by Eq.~\eqref{eq:FreeDephasingHamiltonian}, as well as modification-induced dephasing. Representing the dynamics in quantum phase space, the quadratic term in the Hamiltonian \eqref{eq:FreeDephasingHamiltonian} induces shearing in $j_y$, while the linear term leads to a translation in $j_y$ with constant velocity. The phase flips induce diffusion in $j_y$, which increases the variance linearly with time. The corresponding time evolved state can thus be written as
\begin{align} \label{eq:wignert}
w_t(j_y,j_z) = &\frac{1}{\sqrt{4\pi^2\left(\left\langle{\sf J}^2_y\right\rangle_0+\Gamma_{\rm P} J^2 t\right)\left\langle{\sf J}^2_z\right\rangle_0}}\nonumber\\
&\times \exp\left[
-\frac{1}{2}\frac{(j_y-\epsilon t/\hbar - 2\zeta j_z t)^2}{\left\langle{\sf J}^2_y\right\rangle_0+\Gamma_{\rm P} J^2 t}-\frac{1}{2} \frac{j_z^2}{\left\langle{\sf J}^2_z\right\rangle_0}
\right],
\end{align}
implying that the marginal distribution of $j_z$ remains unaffected by the dynamics.
In order to calculate the likelihood $P_J(m|\tau_e,\sigma,I)=\langle J, m | e^{-i\pi{\sf J}_x/2}\rho_t e^{i\pi{\sf J}_x/2}\left| J, m\right\rangle$ at fixed $J$, we first perform the $\pi/2$-rotation around the $x$-axis, which exchanges $j_y$ and $j_z$ in Eq.~\eqref{eq:wignert}. The resulting distribution is then integrated over $j_y$, and $j_z$ is wrapped back onto the sphere by using $\sin(j_z) = m/J$ and the summation $\int dj_y\sum_k w_t(j_z+2\pi k,j_y)$. This way one obtains the continuous probability density approximating $P_J$,
\begin{widetext}
\begin{align}
p_J(m|\tau_e,\sigma,I)=&\frac{\Theta(J^2-m^2)}{2\pi \sqrt{J^2-m^2}}\left[
\vartheta_3\left(\frac{\arcsin(m/J)-\epsilon t/\hbar}{2},g(t)\right) + \vartheta_3\left(\frac{\pi-\arcsin(m/J)-\epsilon t/\hbar}{2},g(t)\right)
\right],
\label{eq:PhaseFlipDistributionApprox}
\end{align}
\end{widetext}
where
$\Theta(x)$ is the Heaviside function, $\vartheta_3$ denotes the Jacobi-theta functions of the third kind
\begin{align}
\vartheta_3(u,q)=\sum_{n=-\infty}^{\infty}q^{n^2}e^{2inu}\,,
\end{align}
and the dependence on the initial state is expressed by
\begin{align} \label{eq:gdouble}
g(t)=\exp\left[-\frac{\left\langle{\sf J}^2_y\right\rangle_0}{2J^2}-\frac{\Gamma_{\rm P}t}{2}-2\zeta^2t^2\left\langle{\sf J}^2_z\right\rangle_0\right]\,.
\end{align}
This analytic result captures the generic dephasing effect of random phase flips on a two-mode BEC. The comparison of Eq.~\eqref{eq:PhaseFlipDistributionApprox} with exact numerical calculations shows very good agreement, as demonstrated in Fig.\,\ref{fig:blochspheres}.
At this stage it might be tempting to use Eq.~\eqref{eq:PhaseFlipDistributionApprox} for Bayesian updating to calculate the macroscopicity. However, since the spatial distance between the two wells of the potential is not much greater than the extension of the modes, the resulting maximizing modification parameters $\sigma$ imply a moderate heating of the BEC. This must be taken into account for a consistent description. A brief discussion of the role of spin flips in single-well potentials will prepare this.
\subsection{Single-well potentials: spin flips}
The dynamics of a BEC in the two lowest eigenstates of a single-well potential, as studied in Ref.~\cite{Schmiedmayer2014}, is strongly affected by spin flips. This marked difference to the double well is due to the spatial overlap between the two modes, see Eq.~\eqref{eq:FlipCoefficients}. The resulting Lindblad operators do not commute with ${\sf J}_z $, but induce additional diffusion in $z$-direction. In combination with the Hamiltonian \eqref{eq:FreeDephasingHamiltonian} this leads to an enhanced dispersion.
If the free rotation frequency $\epsilon/\hbar$ exceeds the spin-flip diffusion rate
\begin{align}
\quad\Gamma_{\rm S}=\frac{4m_{\rm Rb}^2}{\tau_e m_e^2}\int d^3{\bf q}\,f_\sigma(q)|a_x({\bf q})|^2\,,
\end{align}
the average gain in the second moment of $\sf J_z$ can be easily calculated. For times much greater than the rotation period one obtains
\begin{align}
\left\langle{\sf J}^2_z\right\rangle_t\approx
\frac{\left\langle{\sf J}^2_z\right\rangle_0+J^2}{3}
+\frac{2\left\langle{\sf J}^2_z\right\rangle_0-J^2}{3}
e^{-3\Gamma_{\rm S} t/2}\,.
\label{eq:SpinFlipVarianceBroadening}
\end{align}
For single wells, spin flips will typically dominate the influence of the modification, and phase flips can safely be neglected.
Expanding Eq.~\eqref{eq:SpinFlipVarianceBroadening} for small $\Gamma_{\rm S} t$ and exploiting that $J^2\gg\left\langle {\sf J}^2_z\right\rangle$, yields in the continuum approximation (see App.~\ref{app:a})
\begin{align}
\Delta j_y^2(t)\approx\Delta j_y^2(0)+4\zeta^2 J^2 t^2\left[\left\langle{\sf J}^2_z\right\rangle_0+\frac{\Gamma_{\rm S}J^2 t}{6}\right].
\label{eq:AngleVariance}
\end{align}
Thus the random spin flips enhance dispersion so that the variance of $j_y$ increases with $t^3$. This results in the probability distribution \eqref{eq:PhaseFlipDistributionApprox} with
\begin{align}
g(t)=\exp\left[-\frac{\left\langle{\sf J}^2_y\right\rangle_0}{2J^2}-2\zeta^2t^2
\left(\left\langle{\sf J}^2_z\right\rangle_0
+\frac{\Gamma_{\rm S}J^2t}{6}
\right)\right].
\label{eq:SpinFlipGaussApprox}
\end{align}
In single-well BEC interferometers the modification thus strongly influences the final occupation difference, rendering them attractive for future superposition tests. As explained next, diffusion in the orthogonal $z$-direction is also caused by modification-induced particle loss. The above results can be directly transferred.
\subsection{Heating-induced particle loss}
In order to include modification-induced particle loss from the BEC, we assume that atoms leaving the two ground modes will never return. This assumption is well justified for a large modification parameter $\sigma_q$, where the particles have a negligible probability of being scattered back to the two lowest modes.
In this simplified scenario their populations decay exponentially,
\begin{align}
\left\langle{\sf c}^{\dagger}_a{\sf c}_a\right\rangle_t
=e^{-\Gamma_a t}
\left\langle{\sf c}^{\dagger}_a{\sf c}_a\right\rangle_0,\quad
\left\langle{\sf c}^{\dagger}_b{\sf c}_b\right\rangle_t
=e^{-\Gamma_b t}
\left\langle{\sf c}^{\dagger}_b{\sf c}_b\right\rangle_0,
\label{eq:GSpopulation}
\end{align}
with loss rates
\begin{align}
\Gamma_{a,b}=&\frac{m_{\rm Rb}^2}{\tau_e m_e^2}\int d^3{\bf q}\,f_\sigma(q)\left[1-\left|\left\langle \psi_{a,b}\right|{\sf W}({\bf q})\left|\psi_{a,b}\right\rangle\right|^2\right].
\end{align}
The radius of the generalized Bloch sphere thus decreases with time, and for $\Gamma_a \neq \Gamma_b$ the state is shifted towards one of the poles.
Also the coherences decay exponentially,
\begin{align}
\left\langle{\sf c}^{\dagger}_a{\sf c}_b\right\rangle_t
=e^{-\Gamma_{\rm C} t}
\left\langle{\sf c}^{\dagger}_a{\sf c}_b\right\rangle_0,\quad
\langle{\sf c}^{\dagger}_b{\sf c}_a\rangle_t
=e^{-\Gamma_{\rm C} t}
\langle{\sf c}^{\dagger}_b{\sf c}_a\rangle_0\,,
\label{eq:Coherences}
\end{align}
with
\begin{align}
\Gamma_{\rm C}=\frac{m_{\rm Rb}^2}{\tau_e m_e^2}\int d^3 {\bf q}\, f_\sigma(q)
\left[
1-\left\langle \psi_a\right|{\sf W}({\bf q})\left|\psi_a\right\rangle
\left\langle \psi_b\right|{\sf W}^{\dagger}({\bf q})\left|\psi_b\right\rangle
\right]\,.
\label{eq:CoherenceDecay}
\end{align}
In order to evaluate the effect of particle loss on the likelihood (\ref{eq:PhaseFlipDistributionApprox}) we use the result of Ref.~\cite{Ma2011} to determine how the variance of $\sf J_{\bf n}$, i.e.\ the angular momentum component in direction ${\bf n}$, changes due to particle loss. Using $J_0,J\gg 1$ one obtains
\begin{align} \label{eq:rescale}
\frac{\left\langle{\sf J}^2_{\bf n}\right\rangle_{J}}{J^2}\approx \frac{\left\langle{\sf J}^2_{\bf n}\right\rangle_{J_0}}{J_0^2}
+\frac{J_0-J}{2J_0J},
\end{align}
where $J$ ($J_0$) is the current (initial) collective spin after the loss of $2(J_0-J)$ particles, and angular brackets $\left\langle \dots\right\rangle_{J}$ denote expectation values after tracing out the lost particles. The second term shows that the rescaled second moment $\left\langle{\sf J}^2_{\bf n}\right\rangle_{J}/{J^2}$ increases due to the particle loss.
Combining Eq.~\eqref{eq:rescale} with Eq.~(\ref{eq:GSpopulation}), using that in the double-well $\Gamma_a=\Gamma_b \equiv \Gamma_{\rm L}$, expanding the result to linear order in $\Gamma_{\rm L}t$, and finally repeating the steps carried out in the previous section to account for simultaneous shearing and diffusion, yields the distribution (\ref{eq:PhaseFlipDistributionApprox}) with
\begin{align}
g(t)=&\exp\left[-\frac{\left\langle{\sf J}^2_y\right\rangle_0}{2J_0^2}
-\frac{\Gamma_{\rm P}t}{2}-\frac{\Gamma_{\rm L}t}{4J_0}
\right.\nonumber\\
&\left.-2\zeta^2t^2
\left(\left\langle{\sf J}^2_z\right\rangle_0
+\frac{\Gamma_{\rm L} J_0 t}{6}
\right)\right].
\label{eq:ContinuousDoubleWellGauss}
\end{align}
The enhancement of the dispersion looks similar to the single-well case \eqref{eq:SpinFlipGaussApprox}, but it is weaker by the (significant) factor $1/J_0$. Note that the dispersion rate $\zeta$ decreases with decreasing $J_0$, and the linear approximation of the chemical potential leading to the free Hamiltonian (\ref{eq:FreeDephasingHamiltonian}) will fail if too many particles are lost.
The distribution of the remaining particles turns out to be binomial \cite{schrinski2017sensing} given that $\Gamma_a=\Gamma_b\equiv \Gamma_{\rm L}(\tau_e,\sigma,I)$. The probability density for $m \in \mathbb{R}$, i.e. the continuous approximation of Eq.~\eqref{eq:PiHalfPulse}, therefore takes the final form
\begin{align}\label{eq:37}
p(m|\tau_e,\sigma,I) = & \sum_{J=0}^{J_0}{{J_0}\choose{J}}\left(1-e^{-\Gamma_{\rm L}t}\right)^{J}\left (e^{-\Gamma_{\rm L}t}\right)^{J_0-J} \nonumber \\
& \times p_J(m|\tau_e,\sigma,I),
\end{align}
where $p_J(m|\tau_e,\sigma,I)$ is given by Eqs.~\eqref{eq:PhaseFlipDistributionApprox} and \eqref{eq:ContinuousDoubleWellGauss} and $p_0(m|\tau_e,\sigma,I) = \delta(m)$. This equation can now be used for the Bayesian updating procedure \eqref{eq:likelihoodproduct} and for evaluating the macroscopicity \eqref{eq:Macroscopicity}.
\subsection{Experimental parameters}
The BEC reported in Ref.\ \cite{Schmiedmayer2013} consists of $N=2J_0\approx1200$ $^{87}$Rb atoms in a double-well configuration with a spatial separation of $\Delta_x\approx2\,\mu{\rm m}$ in $x$-direction and an initial number squeezing of $\Delta {\mathsf J}_z^2=0.41^2 J_0/2$. The trapping frequencies are $\omega_x/2\pi=1.44\,{\rm kHz}$, $\omega_y/2\pi=1.84\,{\rm kHz}$ and $\omega_z/2\pi=13.2\,{\rm Hz}$, so that the motion in $z$-direction is quasi-free. The two lowest energy levels of this potential have a gap of $\epsilon/\hbar=2.19\,{\rm kHz}$ and the first order corrections of the chemical potential are characterized by
$\zeta=4\,$Hz.
Approximating the ground states harmonically with the widths $\sigma_{x,y}=\sqrt{\hbar/2m_{\rm Rb}\omega_{x,y}}$ yields the phase-flip and loss rates
\begin{align}
\Gamma_{\rm P}= & \frac{m_{\rm Rb}^2}{\tau_e m_e^2} \frac{1-\exp[-\Delta_x^2\sigma_q^2/(4\sigma_q^2\sigma_x^2+2\hbar^2)]}{\sqrt{(1+2\sigma_q^2\sigma_x^2/\hbar^2)(1+2\sigma_q^2\sigma_y^2/\hbar^2)}} \\
\Gamma_{\rm L}= &\frac{m_{\rm Rb}^2}{\tau_e m_e^2}\left(1-\frac{1}{\sqrt{(1+2\sigma_q^2\sigma_x^2/\hbar^2)(1+2\sigma_q^2\sigma_y^2/\hbar^2)}}\right).
\label{eq:Gl}
\end{align}
For the experimental parameters given above, the particle loss rate $\Gamma_{\rm L}$ cannot be neglected compared to the phase-flip rate $\Gamma_{\rm P}$ in the entire parameter regime of $\sigma$. This is due to the fact that the widths of the ground state modes $\sigma_{x,y}$ are comparable to the spatial separation of the wells $\Delta_x$. Consequently, it cannot be excluded that the observed lack of particle loss due to modification-induced heating may significantly affect the hypothesis test, even though confirming the conservation of particle number does not verify quantum coherence.
As a remedy, we condition the likelihood \eqref{eq:37} on the observed particle number, as explained at the end of Sect.~\ref{sec:2A}. This makes the overall atom number part of the experimental background information, and we can separately assess the modification-induced loss of interference visibility \emph{given that} a certain particle number was detected. The conditioned likelihood \eqref{eq:10cl} is obtained by dividing the likelihood \eqref{eq:37} by the probability
\begin{align}
P(D_{\rm heat}|\tau_e,\sigma, I) = & \sum_{J=\lfloor 0.9J_0\rfloor}^{J_0}{{J_0}\choose{J}}\left(e^{\Gamma_{\rm L}t}-1\right)^{J}e^{-J_0\Gamma_{\rm L}t}\,
\end{align}
that not more than 10\% of the particles are lost,
$D_{\rm heat}:=\{J\geq 0.9J_0\}$.
This threshold value is taken as a conservative estimate given that the number of the trapped particles fluctuates by at most 10\% between the individual experimental runs.
All information is now available to perform the Bayesian hypothesis test, as described in Sect.~\ref{sec:2} using the 1438 data points presented in Fig.~\ref{fig:2}(b). Numerical maximization of $\tau_{\rm m}(\sigma)$ yields a macroscopicity value of $\mu_{\rm m}=8.5$.
The maximum of $\tau_{\rm m}(\sigma)$ is attained for the modification parameter $\sigma_q\simeq\hbar/0.77\,{\rm mm}$. As one would expect, this
roughly corresponds to the parameter value where the phase-flip rate is maximized (at $\Gamma_{\rm P}=1.7/\tau_e$), implying that dephasing is most pronounced. The corresponding particle loss rate is an order of magnitude lower ($\Gamma_{\rm L}=0.11/\tau_e$).
The macroscopicity attained in the double-well BEC interferometer is comparable to the value expected for an atom interferometer operating single Rubidium atoms on the same timescale. For instance, using the estimate in \cite{Nimmrichter2013} with an interference visibility $f=0.2$ after $t=20\,$ms, one would also obtain $\mu=8.5$. This close match might be expected for an unsqueezed BEC, where all atoms are uncorrelated. That the number squeezed BEC discussed here does not reach an appreciably higher macroscopicity, despite its large depth of entanglement, can be attributed to the fact that single-particle observables are measured. They are not sensitive to many-particle correlations that are potentially destroyed by the classicalizing modification.
In contrast, if the modification had induced spin flips, as in a single-well interferometer scenario \cite{Schmiedmayer2014}, the resulting destruction of number-squeezing could be observed
due to the interplay between the modification effect and the intrinsic dispersion caused by atom-atom interactions,
see Eq.~\eqref{eq:SpinFlipGaussApprox}.
\section{Leggett-Garg test with an atomic quantum random walk} \label{sec:4}
\subsection{Setup}
Reference~\cite{Robens2015} describes a test of the Leggett-Garg inequality with single atoms performing a quantum random walk in an optical lattice formed by two circularly polarized laser beams. The form of the lattice potential depends on the hyperfine state of the atoms, so that by preparing single $^{133}$Cs atoms in a superposition of two hyperfine states and displacing the two lattices in opposite directions, one can prepare the atom in a superposition of left- and right-directed movements. We denote the displacement length of a single step by $d$, and the associated time required to displace the lattices by $T_{\rm d}$.
The quantum random walk (Fig.\,\ref{fig:3}) is performed by first applying a $\pi/2$-pulse over the duration $T_{\rm r}$, which prepares the atom in a superposition of the hyperfine states and then transforming this into a spatial superposition by displacing the lattices for the duration $T_{\rm d}$. This scheme is iterated four times and finally a position measurement of the atom is performed, collapsing its position into a definite lattice site. Since no $\pi/2$-pulse is applied after the fourth step, atoms which do not end up in the same hyperfine state are excluded by the measurement protocol. This means that all paths which contribute to the interference must recombine after the third step.
Representing the two-level internal degree of freedom by a spinor, the action of a single step in the quantum random walk is given by the unitary operator
\begin{align} \label{eq:trafo}
{\sf S}=\frac{1}{\sqrt{2}}
\begin{pmatrix}
{\sf U}_d & -{\sf U}_d \\
{\sf U}_{d}^\dagger & {\sf U}_{d}^\dagger
\end{pmatrix},
\end{align}
with the translation operator ${\sf U}_d=\exp (-i{\sf p}d/\hbar )$. A straight-forward calculation shows that in addition to the classical random-walk trajectories, involving no coherences, there are only two classes of trajectories contributing to the interference pattern, see Fig. \ref{fig:QRWSkizze}: (i) the atomic wavefunction is split and recombines immediately in the following step; (ii) the atomic wavefunction is split in the first step, then both parts are displaced either to the left or the right in the second step, and they recombine in the third step. To model the experimental outcome, one has to determine the likelihood
\begin{equation} \label{eq:qrwlike}
P(\ell | \tau_e,\sigma,I) = {\rm tr}_{\rm spin} \left(\langle \ell | \rho | \ell \rangle\right ),
\end{equation}
where $\ell\in \{-2,-1,0,1,2\}$ labels the lattice sites that can be reached in four steps and $\rho$ is the final state evolved under influence of the modification \eqref{eq:MIMpointparticle} with parameters $\tau_e$ and $\sigma$.
\subsection{Impact of the modification}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{fig6.pdf}
\caption{(a) Examples of the two classes of coherently split trajectories contributing to the quantum random walk: (i) the atomic wavefunction splits in the first or second step and recombines afterwards; (ii) the atomic wavefunction splits in the first step, then both parts are move one step in parallel, and recombine in the third step. (b) Quantum-to-classical transition of the quantum random walk with decreasing classicalization timescale $\tau_e$. The diagrams depict the final-site probabilities \eqref{eq:QRW4} for modification parameters $\hbar/\sigma_q=d/10$ and $\tau_e m_e^2/m_{\rm Cs}^2=1\,\mu{\rm s},50\,\mu{\rm s},100\,\mu{\rm s},10\,{\rm ms}$ from left to right.}
\label{fig:QRWSkizze}
\end{figure}
Since the separation between neighboring lattice sides is $d = 433$\,nm, spatial displacements can be neglected in the modification \eqref{eq:MIMpointparticle}, i.e.\ we can set $\sigma_s =0$. The influence of the modification on a superposition of momentum states can be calculated by drawing on the results in Ref.~\cite{schrinski2017sensing}, where the momentum superposition of a non-interacting BEC in the limit of a high number of atoms was approximated by a macroscopic wave function (obeying the single particle Schr\"odinger equation). One can directly carry over these results to the present case of a single Cesium atom. As a result, the likelihood \eqref{eq:qrwlike} can be calculated with the help of the dimensionless coherence reduction factor
\begin{align} \label{eq:redfac}
R(t) = & \exp\left[
-\frac{2 T_{\rm d}m_{\rm Cs}^2}{\tau_e m_e^2} \left (1 - \frac{\sqrt{\pi} \hbar}{\sqrt{2} d \sigma_q}\erf \left (\frac{d \sigma_q}{\sqrt{2} \hbar } \right ) \right ) \right ] \nonumber \\
& \times \exp\left[
-\frac{t m_{\rm Cs}^2}{\tau_e m_e^2} \left (1
- \exp \left ( -\frac{d^2 \sigma_q^2}{2 \hbar^2} \right ) \right ) \right],
\end{align}
where $t$ is the time over which the superposition state is maintained at a constant distance of $d$. Thus, in the case of the path (i) $t = T_{\rm r}$, and in case of path (ii) $t = T_{\rm d} + 2 T_{\rm r}$.
Initializing the random walk in the upper hyperfine state,
one can identify all contributing trajectories by applying Eq.~\eqref{eq:trafo} four times. After weighting these with the appropriate reduction factors \eqref{eq:redfac}, the trace \eqref{eq:qrwlike} finally yields the probability distribution\footnote{Starting with the lower hyperfine state one obtains the mirrored version of the distribution \eqref{eq:QRW4}.}
\begin{subequations}
\label{eq:QRW4}
\begin{align}
P(-2|\tau_e,\sigma,I)=&\frac{1}{16}, \\
P(-1|\tau_e,\sigma,I)= & \frac{1}{4} + \frac{1}{4}R(T_{\rm r})+\frac{1}{8}R(T_{\rm d} + 2T_{\rm r})\\
P(0|\tau_e,\sigma,I)=&\frac{3}{8}- \frac{1}{4}R(T_{\rm r}),\\
P(1|\tau_e,\sigma,I)=& \frac{1}{4}- \frac{1}{8}R(T_{\rm d} + 2T_{\rm r}),\\
P(2|\tau_e,\sigma,I)= & \frac{1}{16}.
\end{align}
\end{subequations}
These results reflect what is to be expected from a classicalizing modification applied to the quantum random walk: The classical random walk probabilities are retrieved in the limit $\tau_e \to 0$, where $R(t) = 0$, while the opposite limit $\tau_e \to \infty$, i.e.\ $R(t) = 1$, yields the
ideal quantum random walk probabilities. The gradual transition between classical and quantum behavior is depicted in Fig.~\ref{fig:QRWSkizze}.
In the Leggett-Garg test of Ref.~\cite{Robens2015} additional measurement results were postselected conditioned on whether the walker moves in the first step to the left or to the right. In this case the random walk effectively starts one step later, and thus only trajectories of type (i) contribute to the interference. The resulting probabilities can be determined as above,
\begin{subequations}
\label{eq:QRW3}
\begin{eqnarray}
P_{\rm L}(-2|\tau_e,\sigma,I)& = & P_{\rm R}(2|\tau_e,\sigma,I)=\frac{1}{8}, \\
P_{\rm L}(-1|\tau_e,\sigma,I)& = & P_{\rm R}(1|\tau_e,\sigma,I)= \frac{3}{8}+ \frac{1}{4} R(T_{\rm r}), \\
P_{\rm L}(0|\tau_e,\sigma,I)&= & P_{\rm R}(0|\tau_e,\sigma,I)= \frac{3}{8} - \frac{1}{4} R(T_{\rm r}), \\
P_{\rm L}(1|\tau_e,\sigma,I)&= &P_{\rm R}(-1|\tau_e,\sigma,I)= \frac{1}{8},\\
P_{\rm L}(2|\tau_e,\sigma,I) &= &P_{\rm R}(-2|\tau_e,\sigma,I)=0.
\end{eqnarray}
\end{subequations}
The subscripts L or R denote that the first step was performed to the left or right.
For completeness, we note that the Leggett-Garg inequality studied in \cite{Robens2015} reads as
\begin{align}
\sum_{\ell=-2}^2 {\rm sgn}(\ell)\left (P(\ell) - \frac{1}{2} \left [ P_{\rm L}(\ell)+P_{\rm R}(\ell) \right ] \right )
\leq 0,
\end{align}
where we dropped the parameters $\tau_e,\sigma,I$ for brevity. This Leggett-Garg inequality can be rewritten in terms of the modification parameters through the reduction factor \eqref{eq:redfac} by inserting Eqs. (\ref{eq:QRW4}) and (\ref{eq:QRW3}),
\begin{align}
R(T_{\rm r}) + R(T_{\rm d} + 2 T_{\rm r}) \le 0.
\label{eq:LGTInequalityMIM}
\end{align}
This inequality is always violated unless $\tau_e $ vanishes, but the left-hand side approaches zero exponentially with decreasing $\tau_e$. Note that our assessment of macroscopicity is not based on such a derived quantity, but on the raw data of detection clicks.
\subsection{Experimental parameters}
In the experiment the displacement and resting time are $T_{\rm d}=21\,\mu{\rm s}$ and $T_{\rm r}=5\,\mu{\rm s}$ and the distance between each lattice site is $d=433\,{\rm nm}$. Maximizing the effect of the modification we note that the reduction factor \eqref{eq:redfac} decreases with increasing $\sigma_q$ and that the five percent quantile $\tau_{\rm m}(\sigma)$ saturates for $\hbar/\sigma_q \ll d$. To assess the macroscopicity, we take the value $\hbar/\sigma_q \approx d/10$, where $\tau_{\rm m}(\sigma)$ already takes the saturated value, yielding $\mu_{\rm m}=7.1$.
Finally, since we neglected possible effects of modification-induced heating so far, we have to verify that this is justified here, i.e. at the stated value of $\sigma_q$ and for the relevant range of classicalization time scales $\tau_{e }$. This can be done conservatively by calculating the heating rate with the 5\% quantile of the Jeffreys' prior ($\tau_{e} = 16.75\,\mu{\rm s}$). It serves as an upper bound (see Fig.~\ref{fig:3}) due to Bayesian updating. The resulting temperature increase of $\Delta T\approx 5.6\,\mu{\rm K}$ over the duration of the whole experiment is moderate, amounting to 1/16 of the potential depth. It thus renders particle loss negligible, so that no explicit conditioning on a likelihood which accounts for heating is required to arrive at \eqref{eq:QRW4} and \eqref{eq:QRW3}.
In summary, the macroscopicity of the atomic Leggett-Garg test is dominated by the timescale on which the experiment was performed, i.e.\ the ramp- and waiting-time between random walk steps. Since only neighboring trajectories contribute to interference, the relevant length scale of the superposition state is given by the lattice spacing $d$ rather than by the spatial extension of the final state. This could be enhanced by implementing a $\pi/2$-pulse after the fourth step, or by performing more steps, so that also trajectories separated by more distant sites contribute to the interference pattern.
\section{Mechanical entanglement of photonic crystals} \label{sec:5}
\subsection{Measurement protocol}
The observation of entanglement between two nanomechanical oscillators reported in Ref.~\cite{Riedinger2018} is based on a coincidence measurement of Stokes- and anti-Stokes photons created in photonic crystal nanobeams placed in the two arms of a Mach-Zehnder interferometer, see Fig.~\ref{fig:4}. In the first step (pump), a photon is sent through the entrance beam splitter, excites a single phonon in one of the two nanobeams, thereby creating entanglement in their mechanical excitation. The Stokes-scattered photon is detected behind the exit beam splitter. In the second step (read), a further photon enters the interferometer through the entrance beam splitter, leading to stimulated emission in the photonic crystal. The resulting anti-Stokes scattered photon, which serves to read out the entanglement, is also detected behind the exit beam splitter.
We denote the measurement outcomes of the Stokes and the anti-Stokes photon detectors by $\pm_{1,2}$, where $+$ ($-$) refers to the upper (lower) detector behind the exit beam splitter and the index refers to the pump and read photon, respectively. The likelihood for a certain coincidence measurement is
\begin{equation} \label{eq:likenano}
P(\pm_1,\pm_2|\tau_e,\sigma,I) = {\rm tr} \left (| \pm_1, \pm_2 \rangle\langle \pm_1, \pm_2| \rho_{\rm fin} \right )
\end{equation}
where $\rho_{\rm fin}$ is the total final state of both oscillators and both photons. The modification parameters $\tau_e$ and $\sigma$ only enter through their influence on the dynamics of the nanomechanical oscillators.
In each nanobeam a single mechanical mode contributes to the measurement signal of the experiment. Even though the pump photon can excite this mode only once, we will in the following allow for arbitrary phonon occupations $|k, \ell \rangle$ of the two oscillators to account for modification-induced heating.
Given that the two relevant oscillator modes are initially in the ground state, the total wave function of the system after the pump photon traversed the exit beam splitter reads
\begin{align}
\left|\psi\right\rangle_{t=0} =&
\frac{1}{2}\left[
\left|+\right\rangle_1\left(\left|1,0\right\rangle+e^{i\phi}\left|0,1\right\rangle\right)\right.\nonumber\\
&\left.+\left|-\right\rangle_1\left(\left|1,0\right\rangle-e^{i\phi}\left|0,1\right\rangle\right)
\right] | {\rm vac} \rangle_2,
\label{eq:MHOInitialStateSP}
\end{align}
where $\phi$ is the initial relative phase. The state \eqref{eq:MHOInitialStateSP} now evolves freely according to the modified master equation \eqref{eq:modvonneum} into the mixed state $\rho_t$ until the read photon passes the interferometer.
The measurement with the read photon can be described through application of the read operator ${\sf R}$, as $\rho_{\rm fin}={\sf R} \rho_t {\sf R}^\dagger/\mathcal{N}$. Here, the factor $\mathcal{N}={\rm tr}({\sf R}^\dagger{\sf R} \rho_t )$ accounts for the conditioning on coincident detections of Stokes and anti-Stokes photons. The read operator ${\sf R}$ first annihilates a phonon in one of the two oscillators and simultaneously creates a read photon in the corresponding interferometer arm,
with the relative phase $\theta$ between the two arms determined by the experimental setup. In a second step, the thus created photon traverses again the beam splitter, yielding in total
\begin{align} \label{eq:read}
{\sf R} |\pm\rangle_1|k, \ell \rangle |{\rm vac}\rangle_2 = & \frac{|\pm\rangle_1}{\sqrt{2k+2\ell}} \left [\sqrt{k} |k-1,\ell \rangle \left ( |+\rangle_2 + |-\rangle_2 \right ) \right. \nonumber \\
& \left. + e^{i \theta} \sqrt{\ell} |k,\ell-1\rangle \left ( |+\rangle_2 - |-\rangle_2 \right ) \right ]
\end{align}
for $(k,\ell) \neq (0,0)$. By in addition setting ${\sf R}|\pm\rangle_1 |0, 0 \rangle |{\rm vac}\rangle_2=0 $ we account for the fact that the phonon ground state (which may be populated by modification-induced transitions) cannot lead to a coincidence detection involving an anti-Stokes photon.
The probability \eqref{eq:likenano} can be written as due to a generalized measurement, $P(\pm_1,\pm_2|\tau_e,\sigma,I) = {\rm tr} ({\sf F}_{\pm_2} \rho^{(\pm_1)}_t )/\mathcal{N}$. Here, the oscillator state
\begin{align}\label{eq:rhopm}
\rho^{(\pm_1)}_t=\langle \pm|_1\langle{\rm vac}|_2\rho_t|{\rm vac}\rangle_2| \pm\rangle_1
\end{align}
is conditioned on the detection of the Stokes photon, and ${\sf F}_{\pm_2} = {\rm tr}_1 (\langle {\rm vac} |_2 {\sf R}^\dagger | \pm \rangle_2 \langle \pm |_2 {\sf R} | {\rm vac} \rangle_2)$ describes the measurement of the anti-Stokes photon,
\begin{align}
{\sf F}_{\pm_2} = & \frac{1}{2}\left(\sum_{k=1,\ell=0}^{\infty}\frac{k}{k+\ell}\left|k,\ell\right\rangle\left\langle k,\ell\right|+\sum_{k=0,\ell=1}^{\infty}\frac{\ell}{k+\ell}\left|k,\ell\right\rangle\left\langle k,\ell\right|\right.\nonumber \\
&\left.+\sum_{k=1,\ell=0}^{\infty}e^{i\theta}\frac{\sqrt{k(\ell+1)}}{k+\ell}\left|k,\ell\right\rangle\left\langle k-1,\ell+1\right|\right.\nonumber\\
&\left.+\sum_{k=0,\ell=1}^{\infty}e^{-i\theta}\frac{\sqrt{(k+1)\ell}}{k+\ell}\left|k,\ell\right\rangle\left\langle k+1,\ell-1\right|\right) \,.
\label{eq:MHORiedingerProjectorPositive}
\end{align}
To prepare the calculation of the likelihoods, we now determine the influence of the modification on the initial oscillator state \eqref{eq:rhopm}.
\subsection{Impact of the modification}
To handle the elastic deformation of a single nanomechanical beam, we
first note that all atoms in the solid can be treated as distinguishable. One can therefore use the Lindblad operators \eqref{eq:lindblad} in first quantization,
\begin{align}\label{eq:lindblad2}
{\sf L}({\bf q},{\bf s})
=&\sum_n\frac{m_n}{m_e}\exp\left[-i\frac{\textbf{\textsf{r}}_n\cdot{\bf q}-\textbf{\textsf{p}}_n\cdot{\bf s}}{\hbar}\right]\,.
\end{align}
To express this in terms of the mode variables, we expand the position operator $\textbf{\textsf{r}}_n$ of each individual atom around its equilibrium position ${\bf r}_n^{(0)}$,
\begin{align}
\textbf{\textsf{r}}_n={\bf r}_n^{(0)}+{\bf w}({\bf r}_n^{(0)}){\sf Q}\,,
\end{align}
in terms of the classical mode function \cite{madelung2012introduction,fetter2003theoretical} of the relevant displacement mode ${\bf w}({\bf r})$ and its operator-valued amplitude ${\sf Q}$. The latter can also be written using the mode creation and annihilation operators ${\sf a}^\dagger$ and ${\sf a}$,
\begin{align} \label{eq:displfield}
{\sf Q}
=\sqrt{\frac{\hbar}{2\varrho V_{\rm m}\omega}}\left({\sf a}+{\sf a}^{\dagger}\right),
\end{align}
where $\varrho$ is the mass density of the material, $\omega$ the mechanical frequency, and $V_{\rm m}$ the mode volume, see App.\ \ref{app:b}.
Accordingly, the momentum operator in \eqref{eq:lindblad2} takes the form
\begin{align} \label{eq:momfield}
\textbf{\textsf{p}}_n =\frac{m_n}{\varrho V_{\rm m}}{\bf w}({\bf r}^{(0)}_n){\sf P}= i \sqrt{\frac{\hbar \omega_k m_n^2}{2\varrho V_{\rm m}}}{\bf w}({\bf r}^{(0)}_n)\left({\sf a}^{\dagger}-{\sf a}\right).
\end{align}
This equation implies that the modification-induced spatial displacement $\mathbf{s}$ in \eqref{eq:lindblad2} scales with the mass of the atom divided by the effective mass of the mechanical mode, which is on the order of the nanobeam mass. The spatial displacement is therefore negligible for all scenarios that lead to observable decoherence,
allowing us to approximate the Lindblad operators as
\begin{align}
&{\sf L}({\bf q})
\simeq\sum_n\frac{m_n}{m_e}\exp\left[-\frac{i}{\hbar}\left({\bf r}_n^{(0)}+\sum_k{\bf w}_k({\bf r}^{(0)}_n){\sf Q}_k\right)\cdot{\bf q}\right]
\nonumber\\
&= \frac{1}{m_e}\int d^3{\bf r}\,\varrho({\bf r}) \exp\left[-\frac{i}{\hbar}\left({\bf r}+\sum_k{\bf w}_k({\bf r}){\sf Q}_k\right)\cdot{\bf q}\right],
\label{eq:HarmOscFullLindbladOp}
\end{align}
where $k$ is a mode index, and $\varrho({\bf r})=\sum_n m_n\delta({\bf r}-{\bf r}^{(0)}_n)$ denotes the mass density of the oscillator. The latter can be replaced by a continuous, homogeneous mass density provided the characteristic length scale $\hbar/\sigma_q$ is much greater than the lattice spacing of the crystal structure.
The Lindblad operators \eqref{eq:HarmOscFullLindbladOp} may be expanded to first order in the relevant mode amplitude ${\sf Q}$ as long as $\sigma_q \ll \sqrt{2 \varrho V_{\rm m} \omega\hbar}$. This decouples the different modes and we have
\begin{align}
{\sf L}({\bf q})=-\frac{i}{\hbar}\left[\widetilde{{\bf w}}_{\varrho}({\bf q})\cdot{\bf q}\right]{\sf Q},
\label{eq:HarmOscDiffLO}
\end{align}
where we introduced
\begin{align}
\widetilde{{\bf w}}_{\varrho}({\bf q})&=
\frac{1}{m_e}\int d^3{\bf r}\,\varrho({\bf r}){\bf w}({\bf r})e^{-i{\bf r}\cdot{\bf q}/\hbar}.
\end{align}
The total master equation including the free harmonic Hamiltonian and the Lindblad operators \eqref{eq:HarmOscDiffLO} of both oscillators can be solved analytically with the help of the characteristic function
\begin{align}
\chi_t({\bf Q},{\bf P})=\int d^2{\bf Q}'\,e^{i{\bf P}\cdot{\bf Q}'/\hbar}\left\langle {\bf Q}'+\frac{{\bf Q}}{2}\right|\rho_t\left|{\bf Q}'-\frac{{\bf Q}}{2}\right\rangle,
\end{align}
where ${\bf Q} = (Q_1,Q_2)$ and ${\bf P} = (P_1,P_2)$ are the joint position and momentum coordinates of both oscillators. The evolution equation for the characteristic function reads
\begin{align}
\partial_t\chi_t({\bf Q},{\bf P})=&
\left(- \frac{1}{\varrho V_{\rm m}}{\bf P}\cdot \nabla_{\bf Q}+\varrho V_{\rm m}{\bf Q}\cdot{\rm \Omega}^2\nabla_{\bf P}-\frac{U(\sigma) {\bf Q}^2}{\tau_e}\right)\nonumber\\
&\times\chi_t({\bf Q},{\bf P}),
\label{eq:HarmOscDiffCharTE}
\end{align}
where ${\rm \Omega}$ is the diagonal matrix containing the two slightly detuned frequencies of both oscillators and
\begin{align}
U(\sigma)=&\frac{1}{2\hbar^2}\int d^3 {\bf q}\, f_\sigma(q) \left|\widetilde{{\bf w}}_{\varrho}({\bf q})\cdot {\bf q}\right|^2 .
\label{eq:HarmOscDiffFT}
\end{align}
Here we exploited that the separation of the two oscillators is much greater than $\hbar/\sigma_q$.
The time evolved characteristic function is given by
\begin{align} \label{eq:soldifflim}
\chi_t({\bf Q},{\bf P})=&
\exp\left[
-\frac{U(\sigma)}{\tau_e}\int_0^t dt' {\bf Q}^2_{t'} \right]\chi_0({\bf Q}_t,{\bf P}_t),
\end{align}
with
\begin{align}
{\bf Q}_t=&\cos({\rm \Omega t}){\bf Q}+\frac{1}{\varrho V_{\rm m}}{\rm \Omega}^{-1}\sin({\rm \Omega t}){\bf P}\nonumber\\
{\bf P}_t=&\cos({\rm \Omega t}){\bf P}-\varrho V_{\rm m}{\rm \Omega}\sin({\rm \Omega t}){\bf Q}.
\label{eq:MarmoOscFreeQP}
\end{align}
Calculating the initial characteristic function of the state \eqref{eq:rhopm} and evaluating \eqref{eq:HarmOscDiffFT} for a given mode function ${\bf w}({\bf r})$ allows one to determine analytically the likelihoods \eqref{eq:likenano}.
\subsection{Particle loss}
For increasing $\sigma_q$ the energy gain induced by momentum translations due to the Lindblad operators \eqref{eq:lindblad2} can exceed the binding energy of the silicon atoms in the crystal. Thus, the modification may induce particle loss already deep in the diffusive regime. The solution \eqref{eq:soldifflim} of the mode dynamics cannot capture this because the mode expansion assumes the atoms to reside in infinitely extended harmonic potentials. Due to the finiteness of the real binding potential there is a critical momentum transfer $q_c$ beyond which the sole effect of the modification is a reduction of the atom number in the crystal.
To account for this particle loss, we split Eq.~\eqref{eq:MIMpointparticle} into a part $\mathcal{M}_\sigma^{<}$ with momentum transfers $|{\bf q}|<q_c$ that will most likely leave the atoms in the crystal, and into the part $\mathcal{M}_\sigma^{>}$ with $|{\bf q}|>q_c$ removing them into the vacuum,
\begin{align}\label{eq:LindbladSum}
\mathcal{M}_\sigma\rho_t&=\int\displaylimits_{q<q_C} d^3 {\bf q}\,f_\sigma(q)\left[{\sf L}({\mathbf q})\rho {\sf L}^\dagger({\mathbf q})-\frac{1}{2}\left\{{\sf L}^\dagger({\mathbf q}){\sf L}({\mathbf q}),\rho\right\}\right]\nonumber\\
&+\int\displaylimits_{q>q_C} d^3 {\bf q}\,f_\sigma(q)\left[{\sf L}({\mathbf q})\rho {\sf L}^\dagger({\mathbf q})-\frac{1}{2}\left\{{\sf L}^\dagger({\mathbf q}){\sf L}({\mathbf q}),\rho\right\}\right].
\end{align}
A Dyson expansion shows that the final state can be written as a sum
\begin{align}
\rho_t=\exp\left[\frac{t}{i\hbar}\mathcal{H}+\frac{t}{\tau_e}\mathcal{M}_\sigma^{<}\right]\rho_0+\widetilde{\rho}
\end{align}
where only the first term is consistent with the coincidence measurement \eqref{eq:likenano}.
Its reduced trace can be absorbed in the normalization $\mathcal{N}$ reflecting the conditioning on the coincidence measurements.
The time evolution under the modification $\mathcal{M}_\sigma^{<}/{\tau_e}$ can now be treated as in the previous section, yielding Eq.~\eqref{eq:soldifflim} with
$U(\sigma)$ replaced by
\begin{align}
U_{<}(\sigma)=&\frac{1}{2\hbar^2}\int_{q<q_c} d^3 {\bf q}\, f_\sigma(q) \left|\widetilde{{\bf w}}_{\varrho}({\bf q})\cdot {\bf q}\right|^2 .
\end{align}
\subsection{Experimentally achieved macroscopicity}
The two oscillators in Ref.~\cite{Riedinger2018} are characterized by the effective mass $\varrho V_{\rm m} \approx 9\times10^{-17}$\,kg \cite{PrivateCommRiedinger} and the mechanical frequency $\omega \approx 2\pi\times 5\,{\rm GHz}$. The exact displacement field depends on the precise geometry of the photonic crystal, and is only numerically accessible. Since the details of the mode function are expected to be of minor relevance we approximate the shape of the oscillator by an elastic silicon cuboid containing only those atoms of the nanobeam that contribute to the elastic deformation. The dimension of this cuboid is set by the effective mass and frequency of the oscillator, yielding for its ground mode $L_x\times L_y\times L_z \approx 0.31\,\mu{\rm m}\times0.31\,\mu{\rm m}\times0.84\,\mu{\rm m}$, using the speed of sound $v = 8433$\,m/s and density $\varrho = 2300$\,kg/m$^3$ of silicon.
The resulting displacement field of the simplest longitudinal mode has the form
\begin{align}\label{eq:SineMode}
{\bf w}({\bf r})={\bf e}_z \sin\left(\frac{\pi z}{L_z}\right),
\end{align}
for $-L_z/2 \leq z \leq L_z/2$. This can now be used to calculate the Lindblad operators \eqref{eq:HarmOscFullLindbladOp}.
The likelihood \eqref{eq:likenano} can be calculated with the characteristic function \eqref{eq:soldifflim} of the state \eqref{eq:rhopm} as a phase space integral
\begin{align}
P(\pm_1,\pm_2|\tau_e,\sigma,I)=\int d^2{\bf Q}d^2{\bf P}
\chi^{\pm_1}_t({\bf Q},{\bf P})\eta^{\pm_2}({\bf Q},{\bf P}),
\label{eq:MHOProbabilities}
\end{align}
where $\eta^{\pm_2}({\bf Q},{\bf P})$ is the characteristic function of the operator \eqref{eq:MHORiedingerProjectorPositive}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{fig7.pdf}
\caption{ The maximally excluded time parameter $\tau_{\rm m}$ as defined by the five percent quantile obtained via Bayesian updating with Eq.~\eqref{eq:Ppm222}. The local maximum to the right is assumed for values of $\hbar/\sigma_q$ roughly equal to the spatial extension of the crystal mode $L_{x,z}$. The global maximum is achieved at $\hbar/\sigma_q\simeq\sqrt{\hbar^2/2m_{\rm Si}E_{\rm b}}$ where the momentum transfers become sufficiently strong to remove particles from the crystal. The fading of the graph indicates where the analytical descriptions derived in App.~\ref{app:b} fail: First, when $\hbar/\sigma_q$ is on the order of several \AA ngstr\"om so that the mass density can no longer be approximated as continuous, and second, when $\hbar/\sigma_q$ is on the order of femtometers where the diffusive regime ceases to be valid.}
\label{fig:BeyondContinuum}
\end{figure}
This expression can now be simplified by noting that the oscillator frequency is large on the timescale of the experiment, $\omega t \gg 1$, so that the time-averaged phase space coordinates \eqref{eq:MarmoOscFreeQP} can be used in the exponent of \eqref{eq:soldifflim},
\begin{align}
\chi_t({\bf Q},{\bf P}) \approx &
\exp\left[
-\frac{U_<(\sigma) t}{2 \tau_e} \left ( {\bf Q}^2 + \varrho^2 V_{\rm m}^2 (\Omega^{-1} {\bf P})^2 \right )\right]
\nonumber\\
&\times\chi_0({\bf Q}_t,{\bf P}_t)\,.
\end{align}
Moreover, the modification cannot create coherences between the oscillator states. In Eq.~\eqref{eq:MHORiedingerProjectorPositive} one can therefore keep only the diagonal terms and the initial coherences between ground state and first excited states,
\begin{align}
{\sf F}_{\pm_2} = \frac{1}{2} \left (\mathbb{1} - \left|0,0\right\rangle\left\langle 0,0\right|+e^{i\theta} \left|1,0\right\rangle\left\langle 0,1\right| + e^{-i\theta}\left|0,1\right\rangle\left\langle 1,0 \right|\right) .
\label{eq:MHORiedingerProjectorPositive2}
\end{align}
The corresponding characteristic function is given in App.~\ref{app:b}, together with the characteristic function of the state \eqref{eq:rhopm}.
The integral Eq.~\eqref{eq:MHOProbabilities} yields the likelihood in its final form,
\begin{align}\label{eq:Ppm222}
&P(\pm_1,\pm_2|\tau_e,\sigma,I)=\nonumber\\
&\frac{1}{4\mathcal{N}}
+\frac{(\pm_1)(\pm_2) 4\cos (\theta- \Delta \Omega t )-2\xi t/\tau_{ e}-\xi^2 t^2/\tau^2_{ e}}{\mathcal{N}(2+\xi t/\tau_{ e})^4},
\end{align}
where $\Delta \Omega = 2\pi \times 45$\,MHz is the frequency mismatch between the oscillators, and we defined the dimensionless parameter $\xi = 2 U_<(\sigma) \hbar/\varrho V_{\rm m} \omega$ characterizing the sensitivity of the relevant nanobeam mode to the modification parameter $\sigma_q$. The geometric factor $U$, as defined in Eq.~\eqref{eq:HarmOscDiffFT}, is evaluated in App.~\ref{app:b}.
The phase- or time-sweep measurement protocols performed in \cite{Riedinger2018} are described by varying $\theta$ and $t$, respectively. The (unreported) initial phase is deduced to be $ \phi\approx1.8\, {\rm rad}-\Delta\Omega\times123\,{\rm ns}$ by optimization. In order to obtain the achieved macroscopicity, we perform Bayesian updating to determine the posterior \eqref{eq:posterior} and maximize over $\sigma_q$. The resulting $\tau_{\rm m}$ is plotted in Fig.~\ref{fig:BeyondContinuum} for $q_c = \sqrt{2 m_{\rm Si} E_b}$ with $E_{\rm b}=4.6\,$eV \cite{Farid1991}. It exhibits a global maximum of $\tau_{\rm m}=6.6\times 10^{7}\,$s at $\hbar/\sigma_q\simeq\sqrt{\hbar^2/2m_{\rm Si}E_{\rm b}}$, yielding a macroscopicity value of $\mu_{\rm m}=7.8$.
Given the relatively high mass of the nanomechanical oscillators and the fairly long coherence time achieved, one might expect the entangled nanobeams to be characterized by a higher degree of macroscopicity. That this is not the case can be traced back to the fact that the superposition state is delocalized only on the scale of femtometers. For such small spatial delocalizations, the sole influence of the modification is to add momentum diffusion to the nanobeam dynamics, leading to weakest possible form of spatial decoherence.
\section{ Conclusion}
The empirical measure discussed in this article serves to quantify the macroscopicity reached in quantum mechanical superposition experiments by the degree to which they rule out classicalizing modifications of quantum theory. We showed how the framework of Bayesian hypothesis testing allows one to assess diverse experiments based on their raw data, thus accounting appropriately for all measurement uncertainties. The fact that measurement errors are fundamentally unavoidable, ensures that the macroscopicity $\mu_{\rm m}$ will always converge to a finite value, even if quantum mechanics holds on all scales. For sufficiently large data sets, when statistical errors tend to be negligible, the here presented measure will approach the one given in Ref. \cite{Nimmrichter2013} for interferometric superposition tests. Equation \eqref{eq:Macroscopicity} is thus the natural generalization of the latter.
A great benefit of the formalism is that it allows one to straightforwardly combine independent parts of an experiment, e.g. quantum random walks of different lengths (Sec.\,\ref{sec:4}) or different measurement protocols for entangled nanobeams (Sec.\,\ref{sec:5}). Moreover, the Bayesian updating process naturally allows for correlated observables to be taken into account, as for instance the total atom number and the population imbalance in BEC interferometers (see Sec.\,\ref{sec:3}). Finally, the use of
Jeffreys' prior ensures that the macroscopicity measure is solely determined by the experimental data at hand, irrespective of prior believes. In particular, using this least informative prior prevents the macroscopicity measure to favor any one type of quantum test against others. We showed that Jeffreys' prior exists for all physically relevant situations, where the likelihood is a smooth function of the modification parameters.
These advantages come at the cost that the required likelihoods are in general considerably more difficult to determine than e.g.\ specific coherences of the statistical operator. It requires one to capture appropriately how the relevant quantum degrees of freedom are affected by the master equation \eqref{eq:modvonneum} describing the impact of the modification on the many-particle system state.
We explained in Secs.~\ref{sec:3}-\ref{sec:5} how this works in practice for three rather different quantum superposition tests.
We reemphasize that a naive application of the macroscopicity measure may yield a finite value even for experiments demonstrating no quantum superposition, because already the absence of observed heating can constrain the classicalization parameters. To be on the safe side, one must identify those observations that yield information only about modification-induced heating and use this data to condition the likelihoods as described at the end of Sect.~\ref{sec:2A}. In most quantum tests this is not necessary because the conditioning is already implemented in the measurement protocol.
The measure of macroscopicity put forward in this article
can be used for any superposition test, provided a mechanical degree of freedom is involved, be it the electronic excitation of an atom or the motion of a kilogram-scale mirror. As such it does not apply to quantum tests involving only spins or photons. It seems natural to generalize the macroscopicity measure to pure photon experiments by drawing on a minimal class of classicalizing modifications of QED, but it is still an open problem how to get hold of the latter.
Beyond the assessment of macroscopicity,
the Bayesian hypothesis testing presented in Sec.\,\ref{sec:2}, can also be used for a proper
statistical description of tests of specific modification models, e.g.\ the various extensions of the Continuous Spontaneous Localization model \cite{Bassi2013}, but also of environmental decoherence mechanisms.
Finally, it goes without saying that the macroscopicity $\mu_{\rm m}$ attributed to a given superposition test serves to highlight a single aspect of the experiment, albeit an important one. It must not be taken as a proxy for the overall significance of an experimental finding.
\acknowledgements
We thank Andrea Alberti, Tarik Berrada, and Ralf Riedinger for helpful comments on their experiments, and the authors of Ref.~\cite{Riedinger2018} for providing us with the unpublished raw data reported in Fig.~\ref{fig:4}. BS thanks Gilles Kratzer for helpful discussions on the topic of Bayesian statistics. This work was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- 298796255.
|
{
"timestamp": "2019-03-01T02:20:16",
"yymm": "1902",
"arxiv_id": "1902.11092",
"language": "en",
"url": "https://arxiv.org/abs/1902.11092"
}
|
\section{Introduction}
We propose and demonstrate a semi-supervised learning algorithm to support clinical decisions in congestive heart failure (CHF) by quantifying pulmonary edema. Limited ground truth labels are one of the most significant challenges in medical image analysis and many other machine learning applications in healthcare. It is of great practical interest to develop machine learning algorithms that take advantage of the entire data set to improve the performance of strictly supervised classification or regression methods. In this work, we develop a Bayesian model that learns probabilistic feature representations from the entire image set with limited labels for predicting edema severity.
Chest x-ray images are commonly used in CHF patients to assess pulmonary edema, which is one of the most direct symptoms of CHF~\cite{mahdyoon1989radiographic}. CHF causes pulmonary venous pressure to elevate, which in turn causes the fluid to leak from the blood vessels in the lungs into the lung tissue. The excess fluid in the lungs is called pulmonary edema. Heart failure patients have extremely heterogenous responses to treatment~\cite{francis2014heterogeneity}. The assessment of pulmonary edema severity will enable clinicians to make better treatment plans based on prior patient responses and will facilitate clinical research studies that require quantitative phenotyping of the patient status~\cite{chakko1991clinical}. While we focus on CHF, the quantification of pulmonary edema is also useful elsewhere in medicine. Quantifying pulmonary edema in a chest x-ray image could be used as a surrogate for patient intravascular volume status, which would rapidly advance research in sepsis and other disease processes where volume status is critical.
Quantifying pulmonary edema is more challenging than detection of pathologies in chest x-ray images~\cite{wang2017chestx, rajpurkar2017chexnet} because grading of pulmonary edema severity relies on much more subtle image findings (features). Accurate grading of the pulmonary edema severity is challenging for medical experts as well~\cite{hammon2014improving}. Our work is grounded in a large-scale clinical dataset that includes approximately 330,000 frontal view chest x-ray images and associated radiology reports, which serve as the source of the severity labels. Of these, about 30,000 images are of CHF patients. Labels extracted from radiology reports via keyword matching are available for about 6,000 images. Thus our image set includes a large number of images, but only a small fraction of images is annotated with edema severity labels.
We use variational auto-encoder (VAE) to capture the image distribution from both unlabeled and labeled images to improve the accuracy of edema severity grading. Auto-encoder neural networks have shown promise for representational modeling~\cite{kingma2013auto}. Earlier work attempted to learn a separate VAE for each label category from unlabeled and labeled data~\cite{kingma2014semi}. We argue and demonstrate in our experiments that this structure does not fit our application well, because pulmonary edema severity score is based on subtle image features and should be represented as a continuous quantity. Instead, we learn one VAE from the entire image set. By training the VAE jointly with a regressor, we ensure it captures compact feature representations for scoring pulmonary edema severity. Similar setups have also been employed in computer vision~\cite{kamnitsas2018semi}. The experimental results show that our method outperforms the multi-VAE approach~\cite{kingma2014semi}, the entropy minimization based self-learning approach~\cite{grandvalet2005semi}, and strictly supervised learning. To the best of our knowledge, this paper demonstrates the first attempt to employ machine learning algorithms to automatically and quantitatively assess the severity of pulmonary edema from chest x-ray images.
\section{Methods}
\label{sec:methods}
\begin{figure*}[t]
\centerline{
\includegraphics[width=0.8\textwidth]{model_illustration.png}
}
\centerline{
\hskip0.4in
(a)
\hskip1.65in
(b)
\hskip 2.0in
}
\caption{Graphical model and inference algorithm. (a): Probabilistic graphical model, where $x$ represents chest x-ray image, $z$ represents latent feature representation, and $y$ represents pulmonary edema severity. (b): Our computational model. We use neural networks for implementing the encoder, decoder, and regressor. The dashed line (decoder) is used in training only. The network architecture is provided in the supplementary material.}
\label{fig:model_illustration}
\end{figure*}
Let $x\in\mathbb{R}^{n\times{n}}$ be a 2D x-ray image and $y\in{\{0,1,2,3\}}$ be the corresponding edema severity label. Our dataset includes a set of $N$ images $\mathbf{x}=\{x_{i}\}_{i=1}^{N}$ with the first $N_{\text{L}}$ images annotated with severity labels $\mathbf{y}=\{y_{i}\}_{i=1}^{N_\text{L}}$. Here, we derive a learning algorithm that constructs a compact probabilistic feature representation $z$ that is learned from all images and used to predict pulmonary edema severity. Fig.~\ref{fig:model_illustration} illustrates the Bayesian model and the inference algorithm.
\paragraph{\textbf{Learning.}}
The learning algorithm maximizes the log probability of the data with respect to parameters~$\theta$:
\begin{align}
\log p(\mathbf{x}, \mathbf{y}; \theta)=\sum_{i=1}^{N_\text{L}} \log p(\text{x}_{i}, \text{y}_{i}; \theta) + \sum_{i=N_\text{L}+1}^{N} \log p(\text{x}_{i}; \theta).
\label{eq:joint_factorization}
\end{align}
We model $z$ as a continuous latent variable with a prior distribution~$p(z)$, which generates images and predicts pulmonary edema severity. Unlike~\cite{kingma2014semi} that constructs a separate encoder $q(z|x,y)$ for each value of discrete label $y$, we use a single encoder $q(z|x)$ to capture image structure relevant to labels. Distribution $q(z|x)$ serves as a variational approximation for $p(z|x, y)$ for the lower bound:
\begin{align}
\mathcal{L}_{1}({\theta}; \text{x}_{i}, \text{y}_{i}) = & \log p(\text{x}_{i}, \text{y}_{i}; \theta) - D_{KL}(q(\text{z}_{i}|\text{x}_{i}; \theta)||p(\text{z}_{i}|\text{x}_{i}, \text{y}_{i})), \nonumber \\
= & \E_{q(\text{z}_{i}|\text{x}_{i}; \theta)}\big[\log p(\text{x}_{i}, \text{y}_{i}; \theta) + \log p(\text{z}_{i}|\text{x}_{i}, \text{y}_{i}) - \log q(\text{z}_{i}|\text{x}_{i}; \theta) \big] \nonumber \\
= & \E_{q(\text{z}_{i}|\text{x}_{i}; \theta)}\big[\log p(\text{x}_{i}, \text{y}_{i}| \text{z}_{i}; \theta) + \log p(\text{z}_{i}) - \log q(\text{z}_{i}|\text{x}_{i}; \theta) \big]\nonumber \\
= & \E_{q(\text{z}_{i}|\text{x}_{i}; \theta)}\big[\log p(\text{x}_{i}, \text{y}_{i}| \text{z}_{i}; \theta)\big] - D_{KL}(q(\text{z}_{i}|\text{x}_{i}; \theta)||p(\text{z}_{i})). \nonumber
\end{align}
We assume that $x$, $z$, and $y$ form a Markov chain, i.e., $y \mathrel{\perp\mspace{-10mu}\perp} x \mid z$, and therefore
\begin{align}
\mathcal{L}_{1}({\theta}; \text{x}_{i}, \text{y}_{i})
= & \E_{q(\text{z}_{i}|\text{x}_{i}; \theta_{\text{E}})}\big[\log p(\text{x}_{i}|\text{z}_{i}; \theta_{\text{D}})\big] + \E_{q(\text{z}_{i}|\text{x}_{i}; \theta_{\text{E}})}\big[\log p(\text{y}_{i}|\text{z}_{i}; \theta_{\text{R}}) \big]\nonumber \\
& -D_{KL}(q(\text{z}_{i}|\text{x}_{i}; \theta_{\text{E}})||p(\text{z}_{i})),
\label{eq:bound_labelled}
\end{align}
where $\theta_{\text{E}}$ are the parameters of the encoder, $\theta_{\text{D}}$ are the parameters of the decoder, and $\theta_{\text{R}}$ are the parameters of the regressor. Similarly, we have a variational lower bound for~$\log p(\text{x}_{i}; \theta)$:
\begin{align}
\mathcal{L}_{2}({\theta}; \text{x}_{i})
= \E_{q(\text{z}_{i}|\text{x}_{i}; \theta_{\text{E}})}\big[\log p(\text{x}_{i}|\text{z}_{i}; \theta_{\text{D}})\big]-D_{KL}(q(\text{z}_{i}|\text{x}_{i}; \theta_{\text{E}})||p(\text{z}_{i})).
\label{eq:bound_unlabelled}
\end{align}
By substituting Eq.~(\ref{eq:bound_labelled}) and Eq.~(\ref{eq:bound_unlabelled}) into Eq.~(\ref{eq:joint_factorization}), we obtain a lower bound for the log probability of the data and aim to minimize the negative lower bound:
\begin{align}
\mathcal{J}({\theta}; \mathbf{x}, \mathbf{y})
= & - \sum_{i=1}^{N_\text{L}} \mathcal{L}_{1}({\theta}; \text{x}_{i}, \text{y}_{i}) - \sum_{i=N_\text{L}+1}^{N} \mathcal{L}_{2}({\theta}; \text{x}_{i}) \nonumber \\
= & \sum_{i=1}^{N}{D_{KL}(q(\text{z}_{i}|\text{x}_{i}; \theta_{\text{E}})||p(\text{z}_{i}))}
-\sum_{i=1}^{N_\text{L}}{ \E_{q(\text{z}_{i}|\text{x}_{i}; \theta_{\text{E}})}\big[\log p(\text{y}_{i}|\text{z}_{i}; \theta_{\text{R}})\big]} \nonumber \\
& -\sum_{i=1}^{N}{ \E_{q(\text{z}_{i}|\text{x}_{i}; \theta_{\text{E}})}\big[\log p(\text{x}_{i}|\text{z}_{i}; \theta_{\text{D}})\big]}.
\label{eq:objective_function}
\end{align}
\paragraph{\textbf{Latent Variable Prior~$p(z)$.}}
We let the latent variable prior~$p(z)$ be a multivariate normal distribution, which serves to regularize the latent representation of images.
\paragraph{\textbf{Latent Representation~$q(z|x)$.}}
We apply the reparameterization trick used in~\cite{kingma2013auto}. Conditioned on image~$\text{x}_i$, the latent representation becomes a multivariate Gaussian variable, $\text{z}_{i}|\text{x}_{i}\sim \mathcal{N}(\text{z}_{i}; \mu_{i},\,\Lambda_{i})$, where~$\mu_{i}$ is a $D$-dimensional vector~$[\mu_{ik}]_{k=1}^{D}$ and~$\Lambda_{i}$ is a diagonal covariance matrix represented by its diagonal elements as~$[\lambda_{ik}^2]_{k=1}^{D}$. Thus, the first term in Eq.~(\ref{eq:objective_function}) becomes:
\begin{align}
\mathcal{J}_{KL}(\theta_{\text{E}}; \text{x}_{i}) = -\frac{1}{2} \sum_{k=1}^{D} \left(\log \lambda_{ik}^2 - \mu_{ik}^2 - \lambda_{ik}^2 \right) + \text{const.}
\label{eq:ae_kl}
\end{align}
We implement the encoder as a neural network~$f_\text{E}(x; \theta_{\text{E}})$ that estimates the mean and the variance of~$z|x$. Samples of~$z$ can be readily generated from this estimated Gaussian distribution. We use one sample per image for training the model.
\paragraph{\textbf{Ordinal Regression~$p(y|z)$.}}
In radiology reports, pulmonary edema severity is categorized into four groups: no/mild/ moderate/severe. Our goal is to assess the severity of pulmonary edema as a continuous quantity. We employ ordinal representation to capture the ordering of the categorical labels. We use a 3-bit representation $\text{y}_{i}=[\text{y}_{ij}]_{j=1}^{3}$ for the four severity levels. The three bits represent the probability of any edema, of moderate or severe edema, and of severe edema respectively (i.e., ``no" is~$[0,0,0]$, ``mild" is~$[1,0,0]$, ``moderate" is~$[1,1,0]$, and ``severe" is~$[1,1,1]$). This encoding yields probabilistic output, i.e., both the estimate of the edema severity and also uncertainty in the estimate. The three bits are assumed to be conditionally independent given the image:
\begin{equation}
p(\text{y}_{i}|\text{z}_{i}; \theta_{\text{R}}) = \prod_{j=1}^{3} f_\text{R}^{j}(\text{z}_{i}; \theta_{\text{R}})^{\text{y}_{ij}}\left(1-f_\text{R}^{j}(\text{z}_{i}; \theta_{\text{R}})\right)^{1-{\text{y}_{ij}}}, \nonumber \\
\end{equation}
where $\text{y}_{ij}$ is a binary label and $f_{\text{R}}^{j}(\text{z}_{i}; \theta_{\text{R}})$ is interpreted as the conditional probability $p(\text{y}_{ij}=1|\text{z}_{i})$. $f_\text{R}(\cdot)$ is implemented as a neural network. The second term in Eq.~(\ref{eq:objective_function}) becomes the cross entropy:
\begin{align}
\mathcal{J_\text{R}}(\theta_{\text{E}}, \theta_{\text{R}}; \text{y}_{i}, \text{z}_{i})
= & - \sum_{j=1}^{3} {\text{y}_{ij}}\log f_\text{R}^{j}(\text{z}_{i}; \theta_{\text{R}})
- \sum_{j=1}^{3} (1-\text{y}_{ij})\log \left(1-f_\text{R}^{j}(\text{z}_{i}; \theta_{\text{R}})\right).
\label{eq:clf_distribution}
\end{align}
\paragraph{\textbf{Decoding~$p(x|z)$.}}
We assume that image pixels are conditionally independent (Gaussian) given the latent representation. Thus, the third term in Eq.~(\ref{eq:objective_function}) becomes:
\begin{align}
\mathcal{J_\text{D}}(\theta_{\text{E}}, \theta_{\text{D}}; \text{x}_{i}, \text{z}_{i})
& = - \log \mathcal{N}(\text{x}_{i}; f_{\text{D}}(\text{z}_{i}; \theta_{\text{D}}), \Sigma_{i}) \nonumber \\
& = \dfrac{1}{2}(\text{x}_{i}-f_{\text{D}}(\text{z}_{i}; \theta_{\text{D}}))^{T}\Sigma_{i}^{-1}(\text{x}_{i}-f_{\text{D}}(\text{z}_{i}; \theta_{\text{D}}))+\text{const.},
\label{eq:ae_distribution}
\end{align}
where $f_\text{D}(\cdot)$ is a neural network decoder that generates an image implied by the latent representation~$z$, and $\Sigma_{i}$ is a diagonal covariance matrix.
\paragraph{\textbf{Loss Function.}}
Combining Eq.~(\ref{eq:ae_kl}), Eq.~(\ref{eq:clf_distribution}) and Eq.~(\ref{eq:ae_distribution}), we obtain the loss function for training our model:
\begin{align}
\mathcal{J} (\theta_{\text{E}}, \theta_{\text{R}}, \theta_{\text{D}}; \mathbf{x}, \mathbf{y})
= & \sum_{i=1}^{N} \mathcal{J}_{KL}(\theta_{\text{E}}; \text{x}_{i}) + \sum_{i=1}^{N_\text{L}} \mathcal{J_\text{R}}(\theta_{\text{E}}, \theta_{\text{R}}; \text{y}_{i}, \text{z}_{i}) + \sum_{i=1}^{N} \mathcal{J_\text{D}}(\theta_{\text{E}}, \theta_{\text{D}}; \text{x}_{i}, \text{z}_{i}) \nonumber \\
= & -\frac{1}{2} \sum_{i=1}^{N} \sum_{k=1}^{D} \left(\log \lambda_{ik}^2 - \mu_{ik}^2 - \lambda_{ik}^2 \right) \nonumber \\
& - \sum_{i=1}^{N_\text{L}} \left( \sum_{j=1}^{3} {\text{y}_{ij}}\log f_\text{R}^{j}(\text{z}_{i}; \theta_{\text{R}}) + \sum_{j=1}^{3} (1-\text{y}_{ij})\log \left(1-f_\text{R}^{j}(\text{z}_{i}; \theta_{\text{R}})\right) \right) \nonumber \\
& + \dfrac{1}{2} \sum_{i=1}^{N} \left(\text{x}_{i}-f_{\text{D}}(\text{z}_{i}; \theta_{\text{D}}))^{T}\Sigma_{i}^{-1}(\text{x}_{i}-f_{\text{D}}(\text{z}_{i}; \theta_{\text{D}})\right).
\label{eq:loss_function}
\end{align}
We employ the stochastic gradient-based optimization procedure Adam~\cite{kingma2014adam} to minimize the loss function. Our training procedure is outlined in the supplementary materiel. The pulmonary edema severity category extracted from radiology reports is a discrete approximation of the actual continuous severity level. To capture this, we compute the expected severity:
\begin{align}
\hat{y}
=0 \times (1-\hat{y}_1) + 1\times (\hat{y}_1-\hat{y}_2)+2\times (\hat{y}_2-\hat{y}_3) +3 \times \hat{y}_3 =\hat{y}_1+\hat{y}_2+\hat{y}_3. \nonumber
\end{align}
\section{Implementation Details}
The size of the chest x-ray images in our dataset varies and is around 3000$\times$3000 pixels. We randomly rotate and translate the images (differently at each epoch) on the fly during training and crop them to 2048$\times$2048 pixels as part of data augmentation. We maintain the original image resolution to preserve subtle differences between different levels of pulmonary edema severity.
The encoder is implemented as a series of residual blocks~\cite{he2016deep}. The decoder is implemented as a series of transposed convolutional layers, to build an output image of the same size as the input image (2048$\times$2048). The regressor is implemented as a series of residual blocks with an averaging pooling layer followed by two fully connected layers. The regressor output $\hat{y}$ has 3 channels. The latent representation~$z$ has a size of 128$\times$128. During training, one sample is drawn from~$z$ per image. The KL-loss (Eq.~(\ref{eq:ae_kl})) and the image reconstruction error (Eq.~(\ref{eq:ae_distribution})) in the loss function are divided by the latent feature size and the image size respectively. The variances in Eq.~(\ref{eq:ae_distribution}) are set to 10, which gives a weight of 0.1 to the image reconstruction error. The learning rate for the Adam optimizer training is 0.001 and the minibatch size is 4. The model is trained on a training dataset and evaluated on a separate validation dataset every few epochs during training. The model checkpoint with the lowest error on the validation dataset is used for testing. The neural network architecture is provided in the supplementary material.
\section{Experiments}
\paragraph{\textbf{Data.}}
Approximately 330,000 frontal view x-ray images and their associated radiology reports were collected as part of routine clinical care in the emergency department of Beth Israel Deaconess Medical Center and subsequent in-hospital stay. A subset of the image set has been released~\cite{johnson2019mimic}.
\begin{figure*}[t]
\centerline{
\hfill
\includegraphics[width=1.1 in]{0_xray.png}
\hskip0.02in
\includegraphics[width=1.1 in]{1_xray.png}
\hskip0.02in
\includegraphics[width=1.1 in]{2_xray.png}
\hskip0.02in
\includegraphics[width=1.1 in]{3_xray.png}
\hfill
}
\vskip0.03in
\centerline{
\hfill
\begin{minipage}[t]{1.1in}
\centering
No edema
\end{minipage}
\hskip0.02in
\begin{minipage}[t]{1.1in}
\centering
Mild edema
\end{minipage}
\hskip0.02in
\begin{minipage}[t]{1.1in}
\centering
Moderate edema
\end{minipage}
\hskip0.02in
\begin{minipage}[t]{1.1in}
\centering
Severe edema
\end{minipage}
\hfill
}
\caption{Representative chest x-ray images with varying severity of pulmonary edema.}
\vskip-0.1in
\label{fig:example_images}
\end{figure*}
In this work, we extracted the pulmonary edema severity labels from the reports by searching for keywords that are highly correlated with a specific stage of pulmonary edema. Due to the high variability of wording in radiology reports, the same keywords can mean different clinical findings in varying disease context. For example, perihilar infiltrate means moderate pulmonary edema for a heart failure patient, but means pneumonia in a patient with a fever. To extract meaningful labels from the reports using the keywords, we limited our label extraction to a CHF cohort. This cohort selection yielded close to 30,000 images, of which 5,771 images could be labeled via our keyword matching. Representative images of each severity level are shown in Fig.~\ref{fig:example_images}. The data details are summarized in the supplementary material.
\paragraph{\textbf{Evaluation.}}
We randomly split the images into training (4,537 labeled images, 334,664 unlabeled images), validation (628 labeled images), and test (606 labeled images) image sets. There is no patient overlap between the sets. Unlabeled images of the patients in the validation and test sets are excluded from training. The labeled data split is 80$\%$/10$\%$/10$\%$ into training/validation/test respectively.
We evaluated four methods: (i)~\textit{Supervised}: purely supervised training that uses labeled images only; (ii)~\textit{EM}: supervised training with labeled images that imputes labels for unlabeled images and minimizes the entropy of predictions~\cite{grandvalet2005semi}; (iii)~\textit{DGM}: Semi-supervised training with deep generative models (multiple VAEs) as described in~\cite{kingma2014semi}; (iv)~\textit{VAE}\_\textit{R}: Our method that learns probabilistic feature representations from the entire image set with limited labels. For the baseline supervised learning method, we investigated different neural network architectures previously demonstrated for chest x-ray images~\cite{wang2017chestx, rajpurkar2017chexnet} and did not find the network architecture changes the supervised learning results significantly.
We evaluate the methods on the test image set using the root mean squared (RMS) error and Pearson correlation coefficient (CC).
\begin{figure*}[t]
\centerline{
\includegraphics[width=1.3in]{table.png}
\hskip0.5in
\includegraphics[width=2.1in]{prediction_bar.png}
}
\vskip-0.1in
\caption{Summary of the results. Left table: Root mean squared errors and Pearson correlation coefficients for each method. Right plot: Predicted edema severity scores on each label category.}
\vskip-0.1in
\label{fig:results}
\end{figure*}
\paragraph{\textbf{Results.}}
Fig.~\ref{fig:results} summarizes the prediction performance of the four methods. The method that jointly learns probabilistic feature representations outperforms the other three models.
\section{Conclusions}
In this paper, we demonstrated a regression model augmented with a VAE trained on a large image dataset with a limited number of labeled images. Our results suggest that it is difficult for a generative model to learn distinct data clusters for the labels that rely on subtle image features. In contrast, learning compact feature representations jointly from images and limited labels can help inform prediction by capturing structure shared by the image distribution and the conditional distribution of labels given images.
We demonstrated the first attempt to employ machine learning algorithms to automatically and quantitatively assess the severity of pulmonary edema from chest x-ray images. Our results suggest that granular information about a patient's status captured in medical images can be extracted by machine learning algorithms, which promises to enable clinicians to deliver better care by quantitatively summarizing an individual patient's medical history, for example response to different treatments. This work also promises to enable clinical research studies that require quantitative summarization of patient status.
\bibliographystyle{splncs04}
|
{
"timestamp": "2019-04-11T02:05:40",
"yymm": "1902",
"arxiv_id": "1902.10785",
"language": "en",
"url": "https://arxiv.org/abs/1902.10785"
}
|
\section{Introduction}\label{sec:intro}
A $k$-photon state of a light field means that the light field contains {\it exactly} $k$ photons. Due to their highly quantum nature, photon states hold promising applications in quantum communication, quantum computing, quantum metrology and quantum simulations. Recently, there has been rapidly growing interest in the generation and manipulation (e.g., pulse shaping) of various photon states. A new and important problem in the field of quantum control engineering is: How to analyze and synthesize quantum systems driven by photon states to achieve pre-specified control performance? In this survey we study single-photon states from a control-theoretic perspective. For single photon generation and detection, please refer to the physics literature \cite{LHA+01,YKS+02, MBB+04, BC09, SFY10, BRV12, PHC+14,OOM+16,GKM+17} and references therein.
The rest of this article is organized as follows. Open quantum systems are briefly introduced in Section \ref{sec:system}. single-photon states are presented in Section \ref{sec:photon_state}. The response of a quantum linear system to a single-photon input is discussed in Section \ref{sec:linear_resp}. In Section \ref{sec:shaping} it is shown how to use a linear coherent feedback network to shape the temporal pulse of a single photon. A single-photon filter is presented in Section \ref{sec:filtering}. Several possible future research problems are given in Section \ref{sec:Con}.
\emph{Notation.} $\ket{0}$ denotes the vacuum field state. Given a column vector of operators or complex numbers $X=[x_1,\cdots,x_n]^\top$, the adjoint operator or complex conjugate of $X$ is denoted by $X^\#=[x_1^\ast,\cdots,x_n^\ast]^\top$. Let $X^\dagger=(X^\#)^\top$ and $\breve{X} = [X^\top \ X^\dag]^\top$. The commutator between operators $A$ and $B$ is defined to be $[A,B]=AB-BA$. Given operators $L,H,X,\rho$, two superoperators are
\begin{eqnarray*}
&\mathrm{Lindbladian}:&\mathcal{L}_GX \triangleq -i[X,H]+\mathcal{D}_LX,\\
&\mathrm{Liouvillian}:&\mathcal{L}_G^\star\rho\triangleq -i[H,\rho]+\mathcal{D}_L^\star\rho,
\end{eqnarray*}
where $\mathcal{D}_LX=L^\dagger XL-\frac{1}{2}(L^\dagger LX+XL^\dagger L)$, and $\mathcal{D}_L^\star\rho=L\rho L^\dagger-\frac{1}{2}(L^\dagger L\rho+\rho L^\dagger L)$. Finally, $\delta_{jk}$ is the Kronecker delta function and $\delta(t-r)$ is the Dirac delta function.
\section{Open quantum systems}\label{sec:system}
In this section, we briefly introduce open quantum systems. Interested readers may refer to the well-known books \cite{GZ00,BP02,WM09} for mored detailed discussions.
\begin{figure}[tbph]
\centering
\includegraphics[width=0.4\textwidth]{Sigma.eps}
\caption{A quantum system $G$ driven by $m$ input fields.}
\label{fig:sys}
\end{figure}
The open quantum system, as shown in Fig.~\ref{fig:sys}, can be described in the so-called $(S,L,H)$ formalism \cite{GJ09,TNP+11,ZJ12,CKS17,ZLW+17}.
In this formalism, $S,L,H$ are all operators on the Hilbert space for the system $G$. Specifically, $S$ is a scattering operator that satisfies $S^\dagger S=SS^\dagger=I$ (identity operator), $L$ describes how the system $G$ is coupled to its surrounding environment, and the self-adjoint operator $H$ denotes the inherent system Hamiltonian. The quantum system $G$ is driven by $m$ input fields. Denote the annihilation operator of the $j$-th boson input field by $b_j(t)$ and the creation operator, the adjoint operator of $b_j(t)$, by $b_j^\ast(t)$, $j=1,\ldots,m$. These input fields satisfy the following singular commutation relations:
\begin{equation} \label{eq:SCR}
\left[b_j(t),b_k^\ast(r)\right]=\delta_{jk}\delta(t-r), \ \ \ j,k=1, \ldots, m.
\end{equation}
Moreover, as $b_j(t)$ annihilates photons, and $\ket{0}$ is the vacuum state (no photon at all) of the field, we have $b_j(t)\ket{0}=0$. Denote $b(t) = [ b_1(t) \ \ \cdots \ \ b_m(t)]^\top$.
The integrated input annihilation, creation, and gauge processes are given by
\begin{eqnarray}\label{eq:gauge}
B(t)=\int_{-\infty}^tb(s)ds, ~ B^\#(t)=\int_{-\infty}^tb^\#(s)ds, ~ \Lambda(t)=\int_{-\infty}^tb^\#(s)b^\top(s)ds
\end{eqnarray}
respectively. Due to (\ref{eq:SCR}), these quantum stochastic processes satisfy
\begin{eqnarray*}
&&dB_j(t)dB_k^\ast(t)=\delta_{jk}dt, \ \ dB_j(t)d\Lambda_{kl}(t)=\delta_{jk}dB_l(t),
\nonumber
\\
&&d\Lambda_{jk}(t)dB_l^\ast(t)=\delta_{kl}dB_j^\ast(t), \ \ d\Lambda_{jk}(t)d\Lambda_{lm}(t)=\delta_{kl}d\Lambda_{jm}(t).
\label{table}
\end{eqnarray*}
According to quantum mechanics, the system in Fig. \ref{fig:sys} evolves in a unitary manner. Specifically, there is a unitary operator $U(t)$ on the tensor product $\mathrm{System}\otimes\mathrm{Field}$ Hilbert space that governs the dynamical evolution of this quantum system. It turns out that the unitary operator $U(t)$ is the solution to the quantum stochastic differential equation (QSDE)
\begin{equation}\label{QSDE}
dU(t)=\Bigg\{-\left(iH+\frac{1}{2}L^\dagger L\right)dt+LdB^\dagger(t)-L^\dagger SdB(t)+{\rm Tr}[S-I]d\Lambda(t)\Bigg\}U(t), \ \ t> t_0
\end{equation}
with the initial condition $U(t_0)=I$ (identity operator). In particular, if $L=0$ and $S=I$, then (\ref{QSDE}) reduces to
\begin{equation*}\label{QSDE:Schrodinger}
i\dot{U}=HU,
\end{equation*}
which is the well-known Schr\"{o}dinger equation for an isolated quantum system with Hamiltonian $H$. ($\hbar$ is set to 1 in this paper.)
Using the unitary operator $U(t)$ in (\ref{QSDE}), the dynamical evolution of system operators and the environment can be obtained in the Heisenberg picture. Indeed, the time evolution of the system operator $X$, denoted by
\begin{equation*}\label{eq:X}
j_t(X)\equiv X(t)=U^\dagger(t)(X\otimes I_{\mathrm{field}})U(t),
\end{equation*}
follows the QSDE
\begin{eqnarray*}
dj_t(X)=j_t(\mathcal{L}_GX)dt+dB^\dagger(t)j_t(S^\dagger[X,L])+j_t([L^\dagger,X]S)dB(t)+{\rm Tr}[j_t(S^\dagger XS-X)d\Lambda(t)].
\end{eqnarray*}
On the other hand, the dynamical evolution of the output field is given by
\begin{eqnarray*}
dB_{\mathrm{out}}(t)&=&L(t)dt+S(t)dB(t),
\label{B_out}
\\
d\Lambda_{\mathrm{out}}(t)&=&L^\#(t)L^\top(t)dt+S^\#(t)dB^\#(t)L^\top(t)
\nonumber
\\
&&+L^\#(t)dB^\top(t)S^\top(t)+S^\#(t)d\Lambda(t)S^\top(t),
\label{Lambda_out}
\end{eqnarray*}
where
\begin{eqnarray*}
B_{\mathrm{out}}(t)&=&U^\dagger(t)(I_{\mathrm{system}}\otimes B(t))U(t),
\\
\Lambda_{\mathrm{out}}(t)&=&U^\dagger(t)(I_{\mathrm{system}}\otimes \Lambda(t))U(t)
\end{eqnarray*}
are the integrated output annihilation operator and gauge process, respectively.
\begin{example}[Optical cavity]\label{ex:cavity}
Let $G$ be an optical cavity. Here we consider the simplest case: the cavity has a single internal mode (a quantum harmonic oscillator represented by its annihilation operator $a$) which interacts with an external light field represented by its annihilation operator $b$. Because $a$ is an internal mode, it and its adjoint operator $a^\ast$ satisfy the canonical commutation relation $[a,a^\ast]$=1, in contrast to the singular commutation relation (\ref{eq:SCR}) for free propagating fields. Let $\omega_c$ be the detuned frequency between the resonant frequency of the internal mode $a$ and the central frequency of the external light field $b$. Let $\kappa $ be the half linewidth of the cavity. In the $(S,L,H)$ formalism, we have
$S=1$, $L=\sqrt{\kappa}a$, and $H= \omega_c a^\ast a$. Then, the dynamics of this system can be described by
\begin{eqnarray*}
d a(t) &=& -(i\omega_c+\frac{\kappa}{2}) a(t)dt - \sqrt{\kappa} dB(t) ,
\\
dB_{\rm out} (t) &=& \sqrt{\kappa} a(t)dt + dB(t).
\end{eqnarray*}
\end{example}
\begin{example}[Two-level system]\label{ex:atom}
A two-level system resides in a chiral nanophotonic waveguide may be parametrized by $
S=1$, $L=\sqrt{\kappa}\sigma_-$, and $H=\frac{\omega_a}{2}\sigma_z $.
The two-level system has two energy states: the ground state $\ket{g}$ and excited state $\ket{e}$. Then $\sigma_- = \ket{g}\bra{e}$ and $\sigma_z = \ket{e}\bra{e}-\ket{g}\bra{g}$. The scalar $\omega_a$ is the detuning frequency between the transition frequency of the two-level system (between $\ket{g}$ and $\ket{e}$) and the the central frequency of the external light field, and $\kappa$ is the decay rate of the two-level system. The dynamics of the system is described by
\begin{eqnarray*}
d\sigma_-(t)&=&-(i\omega_a+\frac{\kappa}{2})\sigma_-(t)dt+\sqrt{\kappa}\sigma_z(t)dB(t) ,
\label{stwo1}
\\
dB_{\rm out}(t)&=&\sqrt{\kappa}\sigma_-(t)dt+dB(t).
\label{stwo2}
\end{eqnarray*}
\end{example}
\begin{remark}
In this section, the dynamics of a quantum system is given directly in terms of the system operators $S,L,H$. This is unlike the traditional way where the starting point is a total Hamiltonian for the joint system plus field system. It fact, the $(S,L,H)$ formalism originates from and is a simplified version of the traditional approach. A demonstrating example can be found in \cite[Example 1]{SZX16}.
\end{remark}
\section{Single-photon states}\label{sec:photon_state}
In this section, we introduce single-photon states of light lieds. A continuous-mode single-photon state $\ket{1_\xi}$ of a light field can be defined to be
\begin{equation*}
\ket{1_\xi} \equiv {\bf B}^\ast(\xi) \triangleq\int_{-\infty}^\infty b^\ast (t)\xi(t)dt\ket{0},
\end{equation*}
where $\xi(t)$ is an $L^2$ integrable function and satisfies $\int_{-\infty}^\infty |\xi(t)|^2 dt =1$. Under the state $\ket{1_\xi} $, the field operator $b(t)$, which is a quantum stochastic process, has zero mean, and the covariance function
\begin{eqnarray*}\label{eq:R-photon}
R(t,r)\triangleq
\langle 1_\xi | \breve b(t) \breve b^\dag(r)| 1_\xi\rangle
=\delta(t-r) \left[\begin{array}{ll}
1 & 0 \\
0 & 0
\end{array}
\right] +
\left[ \begin{array}{ll}
\xi(r)^\ast \xi(t) & 0
\\
0 & \xi(r)\xi(t)^\ast
\end{array} \right].
\end{eqnarray*}
By (\ref{eq:gauge}), the gauge process is $\Lambda(t) = \int_{-\infty}^t n(r) dr$, where $n(t) \triangleq b^\ast(t) b(t)$ is the number operator for the field. In the case of the single-photon state $\ket{1_\xi}$, the intensity is the mean $\bar n(t) = \langle 1_\xi \vert n(t) \vert 1_\xi \rangle = \vert \xi(t) \vert^2$, which is an important physical quantity that determines the probability of photodetection per unit time. Clearly, $\int_{-\infty}^\infty \bar n(t)dt=1$, i.e., there is one photon in the field.
Next, we look at three commonly used single-photon states. Firstly, when $\xi(t)$ is an exponentially decaying pulse shape
\begin{equation}\label{31}
\xi(t)=\left\{\begin{array}{ll}
\sqrt{\beta}e^{-\frac{\beta}{2} t}, & t\geq0, \\
0, & t<0,
\end{array}\right.
\end{equation}
the state $\ket{1_\xi}$ can describe a single photon emitted from an optical cavity with damping rate $\beta$ or a two-level atom with atomic decay rate $\beta$ \cite{WM08,RL00}.
Secondly, if $\xi(t)$ is a rising exponential pulse shape
\begin{equation}\label{50}
\xi(t)=\left\{\begin{array}{ll}
-\sqrt{\beta}e^{\frac{\beta}{2} t}, & t\leq0, \\
0, & t>0,
\end{array}\right.
\end{equation}
then the single-photon state $\ket{1_\xi}$ is able to fully excite a two-level system if $\beta = \kappa$, where $\kappa$ is the decay rate as introduced in Example \ref{ex:atom}, see, e.g., \cite{SAL09, WMS+11, YJ14, PZJ16}. The single photon with pulse shape (\ref{31}) or (\ref{50}) has Lorentzian lineshape function with FWHM $\beta$ \cite{BR04, RL00}, which in the {\it frequency domain} is
\[
|\xi[i\omega]|^2 = \frac{1}{2\pi}\frac{\beta}{\omega^2 + \left(\frac{\beta}{2}\right)^2}.
\]
Finally, the Gaussian pulse shape of a single-photon state $\ket{1_\xi}$ can be given by
\begin{equation} \label{51}
\xi(t)=\left(\frac{\Omega^2}{2\pi}\right)^{\frac{1}{4}}\exp\left(-\frac{\Omega^2}{4}(t-\tau)^2\right),
\end{equation}
where $\tau$ is the photon peak arrival time. Applying Fourier transform to $\xi(t)$ in (\ref{51}) we get
\[
|\xi[i\omega]|^2 =\frac{ 1}{\sqrt{2\pi}\; (\Omega/2) }\exp \left(-\frac{\omega ^2}{2(\Omega/2) ^2}\right).
\]
Hence, $\Omega$ is the frequency bandwidth of the single-photon wavepacket. In contrast to the full excitation of a two-level atom by a single photon of rising exponential pulse shape, the maximal excitation probability of a two-level atom by a single photon of Gaussian pulse shape is around 0.8 which is achieved at $\Omega = 1.46\kappa$, see, e.g., \cite{SAL09, WMS+11, YJ14, PZJ16}.
We end this section with a final remark.
\begin{remark}
It should be noted that a continuous-mode single-photon state $\ket{1_\xi}$ discussed above is different from a continuous-mode single-photon {\it coherent} state $\ket{\alpha_\xi}$ which can be defined as
\begin{equation}
\ket{\alpha_\xi} \triangleq \exp(\alpha {\bf B}^\ast(\xi) -\alpha^\ast {\bf B}(\xi) ),
\end{equation}
where $\alpha = e^{i\theta}\in \mathbb{C}$. For $\ket{\alpha_\xi}$, although the mean photon number is $\langle \alpha_\xi | {\bf B}^\ast(\xi){\bf B}(\xi) | \rangle = |\alpha|^2=1$, the mean amplitude is $\langle \alpha_\xi | {\bf B}(\xi)| \alpha_\xi\rangle = \alpha$. In contrast, the mean amplitude of the single-photon state $\ket{1_\xi}$ is $\langle 1_\xi | {\bf B}(\xi)| 1_\xi\rangle =0$. More discussions can be found in \cite{DZA16}.
\end{remark}
\section{Linear response to single-photon states} \label{sec:linear_resp}
Let the system $G$ in Fig.~\ref{fig:sys} be linear and driven by $m$ photons, one in each input field. In this section, we present the state of the output fields.
Given two constant matrices $U$, $V\in \mathbb{C}^{r\times k}$, a doubled-up matrix $\Delta\left(U,V\right) $ is defined as
\begin{equation}
\Delta\left(U,V\right)\triangleq\left[
\begin{array}{ll}
U & V \\
V^{\#} & U^{\#}%
\end{array}
\right] .
\end{equation}
In the linear case, the system $G$ can be used to model a collection of $n$ quantum harmonic oscillators that are driven by $m$ input fields. Denote $a(t) = [a_1(t) \ \ \cdots \ \ a_n(t)]^\top$, where $a_j(t)$ is the annihilation operator for the $j$th harmonic oscillator, $j=1,\ldots, n$. In the $(S,L,H)$ formalism, the inherent
system Hamiltonian is given by
$H=(1/2)a^{\dag }\Omega a$, where
$a
=
[a^{\top } \; (a^{\#})^{\top}]^{\top }$, and
$\Omega =\Delta (\Omega _{-},\Omega _{+})\in \mathbb{C}^{2n\times 2n}$
is a Hermitian matrix with $\Omega _{-},\Omega _{+}\in
\mathbb{C}^{n\times n}$. The coupling between the system and the fields is
described by the operator
$L=[C_{-} \ C_{+}]a$,
with $C_{-},C_{+}\in \mathbb{C}^{m\times n}$. For simplicity, we assume the scattering operator $S$ is an $m\times m$ identity matrix. The dynamics of the open quantum linear system in Fig.~\ref{fig:sys} is
described by the following QSDEs (\cite[Eq. (26)]{GJN10a}, \cite[Eqs. (14)-(15)]{ZJ13}, \cite[Eqs. (5)-(6)]{ZGPG18})
\begin{equation}\label{eq:sys_a}
\begin{split}
d\breve{a}(t)
=&\; Aa(t)dt+Bd\breve{B}(t),
\\
d\breve{B}_{\mathrm{out}}(t)
=&\; Ca(t)dt+d\breve{B}(t),
\ \ t\geq t_0,
\end{split}
\end{equation}
where the constant system matrices are parametrized by the physical parameters $\Omega_{-},\Omega_{+},C_{-},C_{+}$ and satisfy
\begin{subequations}
\begin{eqnarray}
A+A^{\flat }+BB^{\flat } &=& 0,
\label{eq:PR_a}
\\
B &=&-C^{\flat }.
\label{eq:PR_b}
\end{eqnarray}
\end{subequations}
Eq. (\ref{eq:PR_a}) is equivalent to
\begin{equation}
[\breve{a}(t), \breve{a}^\dag(t)] \equiv [\breve{a}(t_0), \breve{a}^\dag(t_0)] = J_n, \ \ \forall t\geq t_0,
\end{equation}
where $J_n = [I_n \ 0_n; 0_n \ I_n]$.
That is, the system variables preserve commutation relations. On the other hand,
Eq. (\ref{eq:PR_b}) is equivalent to
\begin{equation}
[\breve{a}(t), \breve{b}_{\rm out}^\dag (r)]=0, \ \ t\geq r\geq t_0.
\end{equation}
That is, the system variables and the output satisfy the non-demolition condition. In the quantum control literature, equations (\ref{eq:PR_a})-(\ref{eq:PR_b}) are called physical realization conditions. Roughly speaking, if these conditions are met, the mathematical model \eqref{eq:sys_a} could in principle be physically realized (\cite{JNP08}, \cite{NJD09}).
As in classical linear systems theory, the \emph{impulse response function} for the system $G$ is defined as
\begin{equation}\label{eq:gg}
g_{G}(t)\triangleq\left\{
\begin{array}{ll}
\delta(t)I_m-Ce^{At}C^{\flat}, & t\geq 0, \\
0, & t<0.
\end{array}
\right.
\end{equation}
It is easy to show that $g_{G}(t)$ defined in (\ref{eq:gg}) is in the form of
\begin{equation*}
g_{G}(t)=\Delta\left( g_{G^{-}}(t),g_{G^{+}}(t)\right), \label{eq:impulse}
\end{equation*}
where
\begin{eqnarray*}
g_{G^{-}}(t)&\triangleq&\left\{
\begin{array}{ll}
\delta(t)S_{-}-[%
\begin{array}{ll}
C_{-} & C_{+}%
\end{array}
]e^{At}\left[
\begin{array}{c}
C_{-}^{\dag} \\
-C_{+}^{\dag}%
\end{array}
\right], & t\geq 0 \\
0, & t<0 %
\end{array}
\right., \nonumber \\
\ g_{G^{+}}(t)&\triangleq&\left\{
\begin{array}{ll}
-[%
\begin{array}{ll}
C_{-} & C_{+}%
\end{array}
]e^{At}\left[
\begin{array}{c}
-C_{+}^{T} \\
C_{-}^{T}%
\end{array}
\right] , & t\geq 0 \\
0 , & t<0 %
\end{array}
\right. . \label{eq:io}
\end{eqnarray*}
Given a function $f(t)$ in the time domain, its two-sided Laplace transform \cite[Chapter 10]{WRL61} is defined as
\begin{equation}
F[s] = \mathscr{L}_b \{f(t)\}(s) \triangleq \int_{-\infty}^\infty e^{-st} f(t) dt.
\end{equation}
Applying the two-sided Laplace transform to the impulse response function (\ref{eq:gg}) yields the transfer function
\begin{equation*}\label{eq:G_omega}
\Xi_G[s] = \Delta( \Xi_{G^-}[s], \Xi_{G^+}[s] ),
\end{equation*}
where $\Xi_{G^-}[s]= \mathscr{L}_b \{g_{G^-}(t)\}(s)$ and $\Xi_{G^+}[s]= \mathscr{L}_b \{g_{G^+}(t)\}(s)$.
If $C^+=0$ and $\Omega^+=0$, the resulting quantum linear system is said to be {\it passive}. In this case
$\Xi_{G^+}[s] \equiv 0 $. For example, for the optical cavity discussed in Example \ref{ex:cavity}, it is easy to show that
\begin{equation*}
\Xi_{G^-}[s] = \frac{s+i\omega_c-\frac{\kappa}{2}}{s+i\omega_c+\frac{\kappa}{2}}, \ \ \ \Xi_{G^+}[s]\equiv 0.
\end{equation*}
Let the linear system $G$ be initialized in the state $\ket{\eta}$ and the input field be initialized in the vacuum state $\ket{0}$. Then the initial joint system-field state is $\rho_{0g}\triangleq \ket{\eta}\bra{\eta}
\otimes \ket{0}\bra{0}$ in the form of a density matrix. Denote
\begin{equation*}\label{eq:rho_inf_g}
\rho_{\infty g} = \lim_{t\rightarrow\infty,t_{0}\rightarrow-\infty}U\left(
t,t_{0}\right) \rho_{0g}U\left(t,t_{0}\right) ^{\ast}.
\end{equation*}
Here, $t_0\to -\infty$ indicates that the interaction starts in the remote past and $t\to\infty$ means that we are interested in the dynamics in the far future. In other words, we look at the steady-state dynamics.
Define
\begin{equation}\label{eq:rho_field}
\rho_{\rm field,g}\triangleq \langle \eta |\rho_{\infty g}| \eta \rangle.
\end{equation}
In other words, the system is traced off and we focus on the steady-state state of the output field.
\begin{theorem} \cite[Proposition 2]{ZJ13} \label{prop:out-state-photon}
Assume there is one input field which is in the single photon state $ \vert 1_\xi \rangle$. Then the steady-state output field state for the linear quantum system in Fig.~\ref{fig:sys} is
\begin{equation*}
\rho_{\rm out} = ( {\bf B}^\ast( \xi^-_{\rm out} ) - {\bf B}(\xi^+_{\rm out}) ) \rho_{\rm field,g} ( {\bf B}^\ast( \xi^-_{\rm out} ) - {\bf B}(\xi^+_{\rm out}) )^\ast ,
\label{eq:out-state-photon}
\end{equation*}
where
\begin{equation*}
\Delta ( \xi^-_{\rm out}[s] , \xi^+_{\rm out}[s] ) = \Xi_{G}[s] \Delta ( \xi[s] , 0 ) ,
\end{equation*}
and $\rho_{\rm field, g}$, defined in Eq. (\ref{eq:rho_field}), is the density operator for the output field with zero mean and covariance function
\begin{equation*} \label{eq:R_out}
R_{\rm out}[i\omega] = \Xi_G[ i\omega] R_{\rm in}[i\omega] \Xi_G[i\omega]^\dag
\end{equation*}
with
\begin{equation*} \label{R_in_vacuum}
R_{\rm in}[i\omega] = \left[
\begin{array}{ll}
1 & 0 \\
0 & 0
\end{array}
\right].
\end{equation*}
In particular, if the linear system $G$ is passive and initialized in the vacuum state, then $\xi^+_{\rm out}[s] \equiv 0$ and $
R_{\rm out} [i\omega] \equiv R_{\rm in}[i\omega]$.
In other words, the steady-state output is a single-photon state $\ket{1_{\xi^-_{\rm out}}}$.
\end{theorem}
The multichannel version of Theorem \ref{prop:out-state-photon} is given in \cite[Theorem 5]{ZJ13}.
\begin{example}\label{ex:xi_out}
Let the optical cavity introduced in Example \ref{ex:cavity} be initialized in the vacuum state. Then, by Theorem \ref{prop:out-state-photon}, the steady-state output field state is also a single-photon state $\ket{1_{\xi^-_{\rm out}}}$ with the pulse shape
\begin{equation*}\label{48}
\xi^-_{\rm out}[i\omega]=\frac{i(\omega+\omega_c)-\frac{\kappa}{2}}{i(\omega+\omega_c)+\frac{\kappa}{2}}\xi[i\omega].
\end{equation*}
\end{example}
\begin{remark}
It has been shown in \cite{PZJ16} that the output field of a two-level atom driven by a single-photon field $\ket{1_\xi}$ is also a single-photon state $\ket{1_{\xi^-_{\rm out}}}$. Thus, although the dynamics of a two-level atom is bilinear, see Example \ref{ex:atom}, in the single-photon input case it can be fully characterized by a linear systems theory.
\end{remark}
If the linear system $G$ is not passive, or is not initialized in the vacuum state, the steady-state output field state $\rho_{\rm out}$ in general is not a single-photon state. This new type of states has been named ``photon-Gaussian'' states in \cite{ZJ13}. Moreover, it has been proved in \cite{ZJ13} that the class of ``photon-Gaussian'' states is invariant under the steady-state action of a linear quantum system.
In what follows we present this result.
Let the $k$th input channel be in a single photon state $\vert 1_{\nu_k} \rangle$, $k=1,\ldots, m$. Thus, the state of the $m$-channel input is given by the tensor product
\begin{equation*}\label{eq:multichannel}
\vert \Psi_{\nu} \rangle = \vert 1_{\nu_1} \rangle \otimes \cdots \otimes \vert 1_{\nu_m} \rangle .
\end{equation*}
\begin{definition}\cite[Definition 1]{ZJ13} \label{def:F}
A state $\rho_{\xi, R}$ is said to be a \emph{photon-Gaussian} state if it belongs to the set
\begin{eqnarray*}
\mathcal{F} &\triangleq& \left\{\rho_{\xi, R} = \prod\limits_{k=1}^{m}\sum_{j=1}^{m}\left(B_j^\ast (\xi_{jk}^-) -B_j(\xi_{jk}^+) \right)\rho_R\left(\prod\limits_{k=1}^{m}\sum_{j=1}^{m}\left(B_j^\ast (\xi_{jk}^-) -B_j(\xi_{jk}^+) \right)\right)^\ast \right.
\label{class_F}
\\
& & \ \ \ \ \ \ \ \ \ \ \ \left. : \mathrm{function~} \xi=\Delta(\xi^-, \xi^+) \mathrm{~and~ density~matrix~} \rho_R \mathrm{~satisfy ~} \mathrm{Tr}[\rho_{\xi, R}] = 1 \right\}.
\nonumber
\end{eqnarray*}
\end{definition}
\begin{theorem}\cite[Theorem 5]{ZJ13} \label{thm:main}
Let $\rho_{\xi_{\rm in}, R_{\rm in}} \in \mathcal{F}$ be a photon-Gaussian input state. Then the linear quantum system $G$ produces in steady state a photon-Gaussian output state
$\rho_{\xi_{\rm out}, R_{\rm out}} \in \mathcal{F}$, where
\begin{eqnarray*}
\xi_{\rm out}[s] &=& \Xi_{G}[s] \xi_{\rm in}[s] , \label{eq:xi_out} \\
R_{\rm out}[i\omega] &=& \Xi_G[ i\omega] R_{\rm in}[i\omega] \Xi_G[i\omega]^\dag . \label{eq:R_out_gnr}
\end{eqnarray*}
\end{theorem}
Response of quantum linear systems to multi-photon states has been studied in \cite{Z14, Z17}.
\section{Single-photon pulse shaping via coherent feedback}\label{sec:shaping}
In this section, we show how a coherent feedback network can be constructed to manipulate the temporal pulse shape of a single-photon state.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{fig_34b.png}
\caption{Linear quantum feedback network composed of an optical cavity and a beamsplitter.}
\label{fig_34}
\end{figure}
If an optical cavity, as given in Example \ref{ex:cavity}, is driven by a single-photon state $\ket{1_\xi}$, by Example \ref{ex:xi_out} the output pulse shape in the frequency domain is
\begin{equation}\label{48_feb10}
\eta_1[i\omega]= \frac{i(\omega+\omega_c)-\frac{\kappa}{2}}{i(\omega+\omega_c)+\frac{\kappa}{2}}\xi[i\omega].
\end{equation}
Now we put the cavity into a feedback network closed by a beamsplitter, as shown in Fig.~\ref{fig_34}. Let the beamsplitter be
\begin{equation}
S_{\rm BS}=\left[
\begin{array}{cc}
\sqrt{\gamma} & e^{-i\phi}\sqrt{1-\gamma} \\
-e^{i\phi}\sqrt{1-\gamma} & \sqrt{\gamma} \\
\end{array}
\right], \ 0\leq\gamma\leq1.
\end{equation}
The input-output relation is
\[\left[
\begin{array}{c}
b_3\\
b_1
\end{array}
\right] =S_{\rm BS} \left[
\begin{array}{c}
b_0\\
b_2
\end{array}
\right].
\]
The whole system from input $b_0$ to output $b_3$ in Fig.~\ref{fig_34} is still a quantum linear system that is driven by the single-photon state $|1_{\xi}\rangle$ for the input $b_0$. By the development in Section \ref{sec:linear_resp}, we can get the pulse shape for the output field $b_3$, which is
\begin{equation}\label{52}
\eta_3[i\omega]=\frac{-\displaystyle\frac{1-\sqrt{\gamma}}{1+\sqrt{\gamma}}(\omega+\omega_c)i+\frac{\kappa}{2}}
{\displaystyle\frac{1-\sqrt{\gamma}}{1+\sqrt{\gamma}}(\omega+\omega_c)i+\frac{\kappa}{2}}\xi[i\omega].
\end{equation}
Fix $\beta = 2$ for the rising exponential single-photon state, and $\omega_c = 0$ and $\kappa =2$ for the optical cavity.
When $\xi(t)$ is of an exponentially decaying pulse shape (\ref{31}), the {\it temporal} pulse shapes $\xi(t)$, $\eta_1(t)$ and $\eta_3(t)$ are plotted in Fig.~\ref{fig_43}.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fig_43.png}
\caption{$|\xi(t)|^2$ denotes the detection probability of input photon, $|\eta_1(t)|^2$ denotes the detection probability of output photon in the case of the optical cavity alone, $|\eta_3(t)|^2$ are the detection probabilities of output photon in the coherent feedback network (Fig.~\ref{fig_34}) with different beamsplitter parameters $\gamma$.}
\label{fig_43}
\end{figure}
Fix $\tau = 0$ and $\Omega=2.92$ for a Gaussian single-photon state, and $\omega_c = 0$ and $\kappa =1$ for the optical cavity.
When $\xi(t)$ is of a Gaussian pulse shape (\ref{51}), the {\it temporal} pulse shapes $\xi(t)$, $\eta_1(t)$ and $\eta_3(t)$ are plotted in Fig.~\ref{fig_gaussian}.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{gaussian_pulse.png}
\caption{$|\xi(t)|^2$ denotes the detection probability of input photon, $|\eta_1(t)|^2$ denotes the detection probability of output photon in the case of the optical cavity alone, $|\eta_3(t)|^2$ are the detection probabilities of output photon in the coherent feedback network (Fig.~\ref{fig_34}) with different beamsplitter parameters $\gamma$.}
\label{fig_gaussian}
\end{figure}
\section{Single-photon filtering}\label{sec:filtering}
As discussed in Section \ref{sec:photon_state}, a single-photon light field has statistical properties. Hence, it makes sense to study the filtering problem of a quantum system driven by a single-photon field. Single-photon filters were first derived in \cite{GJN+12, GJN+12b}, and their multi-photon version was developed in \cite{SZX16}. In this section, we focus on the single-photon case. The basic setup is given in Fig.~\ref{fig_3}.
The output field of an open quantum system can be continuously measured, and based on the measurement data a quantum filter can be built to estimate some quantity of the system. For example, we desire to know which state a two-level atom is in, the ground state $\ket{g}$ or the excited state $\ket{e}$. Unfortunately, it is not realistic to measure the state of the atom directly. Instead, a light field may be impinged on the atom and from the scattered light we estimate the state of the atom. Homodyne detection and photon-counting measurements are the two most commonly used measurement methods in quantum optical experiments. In this survey, we focus on Homodyne detection. In Fig.~\ref{fig_3}, $G$ is a quantum system which is driven a single photon of pulse shape $\xi$.
After interaction, the output field, represented by its integrated annihilation operator $B_{\rm out}$ and creation operator $B_{\rm out}^\ast$, is also in a single-photon state with pules shape $\eta$. Due to measurement imperfection (measurement inefficiency), the output field $\ket{1_\eta}$ may be contaminated \cite{SKH13,RR15}. This is usually mathematically modeled by mixing $\ket{1_\eta}$ with an additional quantum vacuum through a beam splitter, as shown in Fig.~\ref{fig_3}. The beam splitter in Fig.~\ref{fig_3} is of a general form
\begin{equation} \label{S_b}
S_{\rm BS}=\left[
\begin{array}{ll}
s_{11} & s_{12} \\
s_{21} & s_{22} \\
\end{array}
\right]
\end{equation}
where $s_{ij}\in \mathbb{C}$. As a result, there are two final output fields, which are
\[\left[
\begin{array}{c}
B_{1,\mathrm{out}}\\
B_{2,\mathrm{out}}
\end{array}
\right] =S_{\rm BS} \left[
\begin{array}{c}
B_{\rm out}\\
B_v
\end{array}
\right],
\]
where $B_v$ is the integrated annihilation operator for the additional quantum noise channel. The quadratures of the outputs are continuously measured by homodyne detectors, which are given by
\begin{equation}\label{filter:output}
Y_{1}(t)=B_{1,\mathrm{out}}(t)+B_{1,\mathrm{out}}^\ast(t),~~Y_{2}(t)=-i(B_{\rm 2, out}(t)-B_{\rm 2, out}^\ast(t)).
\end{equation}
In other words, the amplitude quadrature of the first output field is measured, while for the second output field the phase quadrature is monitored. $Y_{i}(t)$ ($i=1,2$) enjoy the self-non-demolition property
\begin{equation*}
[Y_i(t),Y_j(r)]=0,~~t_0\leq r\leq t, ~~ i,j=1,2,
\end{equation*}
and the non-demolition property
\begin{equation*}
[X(t),Y_i(r)]=0,~~t_0\leq r\leq t, ~~ i=1,2,
\end{equation*}
where $t_0$ is the time when the system and field start interaction. The quantum conditional expectation is defined as
\begin{equation*}
\hat{X}(t)\equiv\pi_t(X)\triangleq\mathbb{E}[j_t(X)|\mathcal{Y}_t],
\end{equation*}
where $\mathbb{E}$ denotes the expectation, and the commutative von Neumann algebra $\mathcal{Y}_t$ is generated by the past measurement observations $\{Y_1(s), Y_2(s): t_0\leq s\leq t\}$. The {\it conditioned} system density operator $\rho(t)$ can be obtained by means of $\pi_t(X)=\mathrm{Tr}\left[\rho(t)^\dagger X\right]$. It turns out that $\rho(t)$ is a solution to a system of stochastic differential equations, which is called the {\it quantum filter} in the quantum control community or {\it quantum trajectories} in the quantum optics community. quantum filtering theory was initialized by Belavkin in the early 1980s \cite{B80}. More developments can be found in \cite{belavkin1989nondemolition,PK98,vHSM05,bouten2007introduction,barchielli2009quantum,WM09,GJN+12b,RR15,SZX16,CKS17} and references therein.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{fig_3.eps}
\caption{Single-photon filtering.}
\label{fig_3}
\end{figure}
In the extreme case that $S_{\rm BS}$ is a 2--by-2 identity matrix,
then the single-photon state $\ket{1_\eta}$ and the vacuum noise are not mixed and $\ket{1_\eta}$ is directly measured by ``Measurement 1''. This is the case that the output of the two-level system $G$ is perfectly measured. In this scenario, a quantum filter constructed based on ``Measurement 1'' is sufficient for the estimation of conditioned system dynamics, as constructed in \cite{GJN+12, GJN+12b}. However, for a general beam splitter of the form (\ref{S_b}), the output of the two-level system $G$ is contaminated by vacuum noise, using two measurements may improve estimation efficiency, as investigated in \cite{DZA18}.
The single-photon filter for the set-up in Fig.~\ref{fig_3} is given by the following result.
\begin{theorem}\cite[Corollary 6.1]{DZA18}\label{corollary1}
Let the quantum system $G=(S,L,H)$ in Fig.~\ref{fig_3} be initialized in the state $\ket{\eta}$ and driven by a single-photon input field $\ket{1_\xi}$. Assume the output fields are under two homodyne detection measurements (\ref{filter:output}). Then the quantum filter in the Schr\"{o}dinger picture is given by
\begin{eqnarray}\label{qpsch}
\begin{aligned}
d\rho^{11}(t)=&\left\{\mathcal{L}_G^\star\rho^{11}(t)+[S\rho^{01}(t),L^\dag]\xi(t)+[L,\rho^{10}(t)S^\dag]\xi^\ast(t)
+(S\rho^{00}(t)S^\dag-\rho^{00}(t))|\xi(t)|^2\right\}dt\\
&+\left[s_{11}^\ast\rho^{11}(t)L^\dag+s_{11}L\rho^{11}(t)+s_{11}^\ast\rho^{10}(t)S^\dag\xi^\ast(t)+
s_{11}S\rho^{01}(t)\xi(t)-\rho^{11}(t)k_1(t)\right]dW_1(t)\\
&+\left[is_{21}^\ast\rho^{11}(t)L^\dag-is_{21}L\rho^{11}(t)+is_{21}^\ast\rho^{10}(t)S^\dag\xi^\ast(t)-
is_{21}S\rho^{01}(t)\xi(t)-\rho^{11}(t)k_2(t)\right]dW_2(t),\\
d\rho^{10}(t)=&\left\{\mathcal{L}_G^\star\rho^{10}(t)+[S\rho^{00}(t),L^\dag]\xi(t)\right\}dt\\
&+\left[s_{11}^\ast\rho^{10}(t)L^\dag+s_{11}L\rho^{10}(t)+
s_{11}S\rho^{00}(t)\xi(t)-\rho^{10}(t)k_1(t)\right]dW_1(t)\\
&+\left[is_{21}^\ast\rho^{10}(t)L^\dag-is_{21}L\rho^{10}(t)-
is_{21}S\rho^{00}(t)\xi(t)-\rho^{10}(t)k_2(t)\right]dW_2(t),\\
d\rho^{00}(t)=&\mathcal{L}_G^\star\rho^{00}(t)dt+\left[s_{11}^\ast\rho^{00}(t)L^\dag+s_{11}L\rho^{00}(t)-\rho^{00}(t)k_1(t)\right]dW_1(t)\\
&+\left[is_{21}^\ast\rho^{00}(t)L^\dag-is_{21}L\rho^{00}(t)-\rho^{00}(t)k_2(t)\right]dW_2(t),\\
\rho^{01}(t)=&(\rho^{10}(t))^\dagger,
\end{aligned}
\end{eqnarray}
where $dW_j(t) = dY_j(t)-k_j(t)dt$ with
\begin{align*}
k_1(t)=&s_{11}\mathrm{Tr}[L\rho^{11}(t)]+s_{11}^\ast\mathrm{Tr}[L^\dag\rho^{11}(t)]
+s_{11}\mathrm{Tr}[S\rho^{01}(t)]\xi(t)+s_{11}^\ast\mathrm{Tr}[S^\dag\rho^{10}(t)]\xi^\ast(t),
\\
k_2(t)=&-is_{21}\mathrm{Tr}[L\rho^{11}(t)]+is_{21}^\ast\mathrm{Tr}[L^\dag\rho^{11}(t)]
-is_{21}\mathrm{Tr}[S\rho^{01}(t)]\xi(t)+is_{21}^\ast\mathrm{Tr}[S^\dag\rho^{10}(t)]\xi^\ast(t).
\end{align*}
The initial conditions are $\rho^{11}(t_0)=\rho^{00}(t_0)=|\eta\rangle\langle\eta|$, $\rho^{10}(t_0)=\rho^{01}(t_0)=0$.
\end{theorem}
\begin{remark}
If the beam splitter $S_{\rm BS}$ is a 2--by-2 identity matrix, the single-photon filter (\ref{qpsch}) in Theorem \ref{corollary1} reduces to
\begin{eqnarray}\label{qpsch_b}
\begin{aligned}
d\rho^{11}(t)
=&
\left\{\mathcal{L}_G^\star\rho^{11}(t)+[S\rho^{01}(t),L^\dag]\xi(t)+[L,\rho^{10}(t)S^\dag]\xi^\ast(t)+(S\rho^{00}(t)S^\dag-\rho^{00}(t))|\xi(t)|^2\right\}dt
\\
&+\left[\rho^{11}(t)L^\dag+L\rho^{11}(t)+\rho^{10}(t)S^\dag\xi^\ast(t)+
S\rho^{01}(t)\xi(t)-\rho^{11}(t)k_1(t)\right]dW_1(t)
\\
d\rho^{10}(t)
=&
\left\{\mathcal{L}_G^\star\rho^{10}(t)+[S\rho^{00}(t),L^\dag]\xi(t)\right\}dt\\
&+\left[\rho^{10}(t)L^\dag+L\rho^{10}(t)+S\rho^{00}(t)\xi(t)-\rho^{10}(t)k_1(t)\right]dW_1(t),
\\
d\rho^{00}(t)
=&
\mathcal{L}_G^\star\rho^{00}(t)dt+\left[\rho^{00}(t)L^\dag+L\rho^{00}(t)-\rho^{00}(t)k_1(t)\right]dW_1(t),
\\
\rho^{01}(t)=&(\rho^{10}(t))^\dagger,
\end{aligned}
\end{eqnarray}
where $dW_1(t)$, $k_1(t)$ and the initial conditions are the same as those in Theorem \ref{corollary1}.
The filter (\ref{qpsch_b}) is the quantum single-photon filter, first proposed in \cite{GJN+12}.
\end{remark}
Quantum filters describe the joint system-field dynamics conditioned on measurement outputs, while master equations describe the system dynamics itself. In this sense, master equations can be regarded as unconditional system dynamics, see e.g., \cite{barchielli2009quantum,WM09,GJN+12}. In this paper, the master equations we used are Lindblad master equations (also called ensemble average dynamics), which can be directly obtained by tracing out the field from the filtering equations in any case above. Indeed, setting $S=I$ and tracing out the noise terms in the quantum filter \eqref{qpsch} or (\ref{qpsch_b}), we end up with the single-photon master equation for the quantum system $G$ in the Schr\"{o}dinger picture
\begin{eqnarray*}\label{mesch}\begin{aligned}
\dot{\varrho}^{11}(t)=&\mathcal{L}^\star_G\varrho^{11}(t)+[\varrho^{01}(t),L^\dag]\xi(t)+[L,\varrho^{10}(t)]\xi^\ast(t),\\
\dot{\varrho}^{10}(t)=&\mathcal{L}^\star_G\varrho^{10}(t)+[\varrho^{00}(t),L^\dag]\xi(t),\\
\dot{\varrho}^{00}(t)=&\mathcal{L}^\star_G\varrho^{00}(t),\\
\varrho^{01}(t)=&(\varrho^{10}(t))^\dagger
\end{aligned}
\end{eqnarray*}
with initial conditions $\varrho^{11}(t_0)=\varrho^{00}(t_0)=|\eta\rangle\langle\eta|$, $\varrho^{10}(t_0)=\varrho^{01}(t_0)=0$.
The interaction between a two-level atom and a single photon of Gaussian pulse shape has been studied intensively in the literature. In the single-photon case, when the photon has a Gaussian pulse shape (\ref{51}) with $\Omega=1.46\kappa$, it is shown that the maximal excitation probability is around $0.8$, see, e.g., \cite{SAL09}, \cite{RSF10}, \cite[Fig. 1]{WMS+11}, \cite[Fig. 8]{GJN+12}, and \cite[Fig. 2]{BCB+12}. Recently, the analytical expression of the pulse shape of the output single photon has been derived in \cite{PZJ16}, which is exactly $\eta_1$ is (\ref{48_feb10}). Assume the pulse shape $\xi(t)$ of the input photon is of the form (\ref{51}) with photon peak arrival time $\tau=3$ and frequency bandwidth $\Omega=1.46\kappa$. Denote the pulse shape of the output photon by $\eta(t)$. Then it can be easily verified that $\int_{-\infty}^{{4}}\left(|\xi(\tau)|^2-|\eta(\tau)|^2\right)d\tau=0.8$. Interestingly, the excitation probability achieves its maximum 0.8 at time $t=4$ (the upper limit of the above integral). Hence, the filtering result is consistent with the result of input-output response.
\section{Concluding remarks}\label{sec:Con}
In this section, we discuss several possible future research directions.
In Section \ref{sec:shaping}, we have discussed the problem of single-photon pulse shaping by using a very simple example, see Fig.~\ref{fig_34}, where the system $G$ is an optical cavity. Clearly, a passive quantum linear controller $K$ can be added into the network, see Fig.~\ref{fig_34c}. If both the system $G$ and the controller $K$ are linear time invariant, that is, all their parameters $\Omega_-$ and $C_-$ as discussed in Section \ref{sec:linear_resp} are constant matrices, then the overall system from $b_0$ to $b_3$ is still a passive quantum linear system with constant system matrices. In this case, the output channel $b_3$ contains a single photon whose temporal pulse shape can be derived by Theorem \ref{thm:main}. Clearly, the output pulse shape is a function of the physical parameters of the passive quantum controller $K$. Thus adding a passive quantum linear controller may increase flexibility of temporal pulse shaping of the input single photon. There may be one of the future research directions.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{fig_34c.png}
\caption{Linear quantum feedback network}
\label{fig_34c}
\end{figure}
In \cite{milburn08}, Milburn investigated the response of an optical cavity to a continuous-mode single photon, where frequency modulation applied to the cavity is used to engineer the temporal output pulse shape. As frequency modulation involves a time-varying function, the transfer function approach in \cite{ZJ13} is not directly applicable. However, it appears that the general procedure outlined in the proof of Proposition 2 and Theorem 5 \cite{ZJ13} can be generalized to the time-varying, yet still linear, case. For example, if the quantum linear passive controller $K$ in Fig.~\ref{fig_34c} is allowed to be time-varying, can the output single-photon pulse shape can be engineered satisfactorily? There may be another future research direction.
In this survey, the single photons are characterized in terms of their temporal pulse shapes, which are $L^2$ integral functions. In this sense, these photon states are continuous-variable (CV) states. In additional to CV photonic states, there are photonic discrete-variable (DV) states, for example, number states and polarization states. In \cite{ZJ13,Z14}, the linear systems theory summarized in Section \ref{sec:linear_resp} was applied to derive the output of a quantum linear system driven by a coherent state and a single-photon state (or a multi-photon state). If the pulse information is ignored and only the number of photons is counted, the results reduces to the main equation, Equation (1) in \cite{BDS+12}; see Example 3 in \cite{Z14} for detail. It can be easily shown that the general framework still works if the coherent state is replaced by a squeezed-vacuum state. Nonetheless, it is not clear to what extent that the mathematical methods proposed for photonic CV states also work for DV photonic states.
In measurement-based feedback control of quantum systems, {\it real-time} implementation of a quantum filter is essential. In the case of a two-level system driven by a vacuum field, a system of 3 stochastic differential equations (SDEs) is sufficient for filtering. However, for a two-level system driven by a single photon, the single-photon filter (\ref{mesch}) consists of a system of 9 SDEs, and a two-photon filter consists of 30 SDEs, \cite{SZX16}. Thus, numerically efficient and reliable implementation of single- or multi-photon filters is an important problem in the measurement-based feedback control of quantum systems driven by photons. There is presently very few pieces of work in the direction, \cite{RR15,GZP19}.
{\bf Acknowledgement.}
I wish to thank financial support from the Hong Kong Research Grant Council under grants 15206915 and 15208418.
|
{
"timestamp": "2019-03-01T02:12:48",
"yymm": "1902",
"arxiv_id": "1902.10961",
"language": "en",
"url": "https://arxiv.org/abs/1902.10961"
}
|
\section{Introduction}
Multiplication is one of the most fundamental computational problems and the simple ``long multiplication'' $O(n^2)$-time algorithm for multiplying two $n$-digit numbers is taught to elementary school pupils around the world. Despite its centrality, the true complexity of multiplication remains elusive. In 1960, Kolmogorov conjectured that the thousands of years old $O(n^2)$-time algorithm is optimal and he arranged a seminar at Moscow State University with the goal of proving this conjecture. However only a week into the seminar, the student Karatsuba came up with an $O(n^{\lg_2 3}) \approx O(n^{1.585})$ time algorithm~\cite{Karatsuba:1962:MoMDNbAC}. The algorithm was presented at the next seminar meeting and the seminar was terminated. This sparked a sequence of improved algorithm such as the Toom-Cook algorithm~\cite{Toom:1963:TCoaSoFERtMoI,Cook:1966:OtmCToF} and the Sch\"{o}nhage-Strassen algorithm~\cite{Schonhage:1971:SMgZ}. The Sch\"{o}nhage-Strassen algorithm, as well as the current fastest algorithm by F\"{u}rer~\cite{Furer:2009:FIM}, are both based on the Fast Fourier Transform (FFT). F\"{u}rer's algorithm can be shown to run in time $O(n \lg n \cdot 4^{\lg^* n})$ when multiplying two $n$-bit numbers~\cite{Harvey:2018:FIMuSLV}. It can even be implemented as a constant degree Boolean circuit of the same size. Here $\lg^* n$ is the very slowly growing iterated logarithm.
But what is the true complexity of multiplying two $n$-bit numbers? Can it be done via e.g. a Boolean circuit of size $O(n)$ like addition? Or is multiplication strictly harder? Our main contribution is to show a connection between multiplication and a central conjecture by Li and Li~\cite{lili} in the area of \emph{network coding}. Our results show that if the conjecture by Li and Li~\cite{lili} is true, then any constant degree Boolean circuit for computing the product of two $n$-bit numbers must have size $\Omega(n \lg n)$. This establishes a conditional lower bound for multiplication that comes within a $4^{\lg^*n}$ factor of F\"{u}rer's upper bound and implies that multiplication is strictly harder than addition.
Before diving into the details of our results, we first give a brief introduction to network coding.
\paragraph{Network Coding.}
Network coding studies communication problems in graphs. Given a graph $G$ with capacity constraints on the edges and $k$ data streams, each with a designated source-sink pair of nodes $(s_i,t_i)$ in $G$, what is the maximum rate at which data can be transmitted concurrently between the source-sink pairs? One solution is to just forward the data, which reduces the problem to a {\em multicommodity flow} problem. The central question in network coding is whether one can achieve a higher rate by using coding/bit tricks. This question is known to have a positive answer in directed graphs, where the rate increase may be as high as a factor $\Omega(|G|)$ (by sending XOR's of carefully chosen input bits), see e.g.~\cite{Adler:soda}. However the question remains wide open for undirected graphs where there are no known examples for which network coding can do better than the multicommodity flow rate. A central conjecture in network coding, due to Li an Li~\cite{lili}, says that coding yields no advantage in undirected graphs.
\begin{conjecture}[Undirected $k$-pairs Conjecture~\cite{lili}]
\label{con:undirected}
The coding rate is equal to the Multicommodity-Flow rate in undirected graphs.
\end{conjecture}
Despite the centrality of this conjecture, it has heretofore resisted all attempts at either proving or refuting it.
Conjecture~\ref{con:undirected} has been used twice before for proving lower bounds for computational problems. Adler \etal~\cite{Adler:soda} were the first to initiate this line of study. They presented conditional lower bounds for computing the transpose of a matrix via an \emph{oblivious algorithm}. Here oblivious means that the memory access pattern is fixed and independent of the input. Since a circuit is oblivious, they also obtain circuit lower bounds for matrix transpose. Very recently Farhadi \etal~\cite{FHLS18} showed how to remove the \emph{obliviousness} assumption for external memory problems. Their main result was a tight lower bound for external memory integer sorting, conditioned on Conjecture~\ref{con:undirected} being true.
\subsection{Our Results}
Our main result is an exciting new connection between network coding and the complexity of multiplication. Formally, we prove the following theorem:
\begin{theorem} \label{th:multiplicationLB}
Assuming Conjecture~\ref{con:undirected}, every boolean circuit with arbitrary gates and bounded in and out degrees that computes the product of two numbers given as two $n$-bit strings has size $\Omega(n \lg n)$.
\end{theorem}
In fact, we prove our $\Omega(n \lg n)$ lower bound for an even simpler problem than multiplication, namely the \emph{shift problem}: In the shift problem, we are given an $n$-bit string $x$ and an index $j \in [n]$. The goal is to construct a circuit that outputs the $2n$-bit string $y$ whose $i$th bit equals the $(i-j+1)$th bit of $x$ for every $j \le i \le j + n - 1$. Here we think of the index $j$ as being given in binary using $\lceil \lg n \rceil$ bits. We prove the following result:
\begin{theorem} \label{th:shiftLB}
Assuming Conjecture~\ref{con:undirected}, every boolean circuit with arbitrary gates and bounded in and out degrees that computes the shift problem has size $\Omega(n \lg n)$.
\end{theorem}
Theorem~\ref{th:multiplicationLB} follows as a corollary of Theorem~\ref{th:shiftLB} by observing that shifting $x$ by $j$ positions is equivalent to multiplication by $2^j$. Moreover, it is not hard to see that there is a linear sized circuit that has $\lceil \lg n \rceil$ input gates and $n$ output gates, where on an index $j \in [n]$, it outputs the number $2^j$ in binary (i.e. a single $1$-bit at position $j$).
We find it quite fascinating that even a simple instruction such as shifting requires circuits of size $\Omega(n \lg n)$, at least if we believe Conjecture~\ref{con:undirected}.
\paragraph{Valiant's Depth Reduction and Circuit Complexity Lower Bounds.}
In addition to our main lower bound results for multiplication, we also demonstrate that the network coding conjecture sheds new light on some fundamental conjectures by Valiant.
In a 1977 survey
Valiant~\cite{ValiantGraph} outlined potentially plausible attacks on the problem of
proving a lower bound for the size of any circuit that can
compute a permutation or even shifts of a given input.
The goal was to prove that achieving both $O(n)$ size and $O(\lg n)$ depth
for such circuits is impossible.
While most of his attacks were rebuffed due to existence of
complex and highly connected graphs that only had $O(n)$ edges (superconcentrators),
Valiant outlined one last potential approach that could still be fruitful.
His main brilliant idea was to start with a circuit of some depth and by applying graph theoretical approaches
reducing the depth of the circuit while eliminating only a small number of edges.
The hope was that information theoretical approaches could finish the job once the depth of the circuit was very low and once the (graph theoretical) complexity of the circuit was peeled away.
More formally, Valiant showed that for every circuit $C$ with $n$ input and output gates, of size $O(n)$, depth $O(\lg n)$ and fan-in $2$, and for every $\varepsilon > 0$, the function computed by $C$ can be computed by a boolean circuit with arbitrary gates $C'$ of depth $3$ with $n$ input and output gates and $\varepsilon n$ extra nodes. Moreover, the number of input gates directly connected to an output gate is bounded. That is, if we denote the set of input and output gates by $X$ and $Y$ respectively, then for every $y \in Y$, there are at most $O(n^{\varepsilon})$ wires connecting $y$ and $X$.
In turn, this reduction shows that it is enough to prove lower bounds on such depth $3$ circuits.
Almost 20 years later and based on these ideas, Valiant~\cite{ValiantWhy} put forward several conjectures that if resolved could open the way
for proving circuit complexity lower bounds. Loosely speaking, Valiant conjectured that if $\varepsilon \le 1/2$ then such depth $3$ circuits cannot compute cyclic-shift permutation. Before discussing Valiant's conjectures more formally, we first state our second main result, which essentially shows that Conjecture~\ref{con:undirected} implies one of Valiant's conjectures, albeit with a smaller (but still constant) bound on $\varepsilon$.
\begin{theorem} \label{th:depth3MultiplicationLB}
Let $C$ be a depth $3$ circuit that computes multiplication such that the following holds.
\begin{enumerate}
\item The number of gates in the second layer of $C$ is at most $\varepsilon n$ for $\varepsilon \le 1/300$; and
\item for every output gate $y$ of $C$, the number of input gates directly connected to $y$ is at most $c$.
\end{enumerate}
Then assuming Conjecture~\ref{con:undirected}, $c = \Omega\left(\frac{\lg n}{\lg \lg n}\right)$.
\end{theorem}
As with Theorem~\ref{th:multiplicationLB}, we prove Theorem~\ref{th:depth3MultiplicationLB} on an even restricted set of circuits, namely circuits that compute the shift function. We now turn to give a formal description of Valiant's Conjectures, and demonstrate how Theorem~\ref{th:depth3MultiplicationLB} brings us closer to settling them.
\paragraph{Valiant's Conjectures.}
Let $\Gamma$ be a bipartite graph on two independent sets $X$ and $Y$ such that
$X = \left\{x_1, \dots, x_n\right\}$ denotes a set of inputs and
$Y = \left\{y_1, \ldots, y_n\right\}$ denotes a set of outputs.
Furthermore assume, let $f_1, \ldots f_{\varepsilon n}$ be $\varepsilon n$ extra nodes and connect them by edges to all the nodes in $\Gamma$.
Denoting the resulting graph by $G$ consider all possible boolean circuits with arbitrary gates whose underlying topology is $G$.
We say such a circuit computes a permutation $\pi\from Y \to X$ if for every assignment $x_1,\ldots,x_n \in \{0,1\}^n$ to the input gates, after the evaluation of the circuit $y_j$ is assigned $\pi(y_j)$ for every $j \in [n]$.
Valiant conjectured that this should be impossible if $\varepsilon$ is too small or if $\Gamma$ has too few edges.
In particular, he proposed the following.
\begin{conjecture}\label{con:v1}
If $\Gamma$ has maximum degree at most 3 and if $\varepsilon \le 1/2$, then there exists a permutation $\pi$ such that no circuit that has $G$ as its underlying topology can compute the permutation $\pi$. Moreover, there exists such $\pi$ that is a cyclic shift.
\end{conjecture}
Theorem~\ref{th:depth3MultiplicationLB} shows that conditioned on Conjecture~\ref{con:undirected}, if $\varepsilon \le 1/300$ then Valiant's first conjecture holds. We note that our proof for Theorem~\ref{th:depth3MultiplicationLB} continues to hold even if the gates' boolean functions are fixed after the shift offset is given. That is, if only the topology is fixed in advance. This coincides exactly with the formulation of Valiant's conjecture.
Valiant further conjectured the following.
\begin{conjecture}\label{con:v2}
If $\Gamma$ has at most $n^{2-\delta}$ edges for some constant $\delta>0$, and if $\varepsilon \le 1/2$,
then there exists a permutation $\pi$ such that no circuit that has $G$ as its underlying topology can compute the permutation $\pi$. Moreover, there exists such $\pi$ that is a cyclic shift.
\end{conjecture}
\subsection{Related Work}
\paragraph{Lower Bounds for Multiplication.}
There are a number of previous lower bounds for multiplication in various restricted models of computation.
Clifford and Jalsenius~\cite{CJ11} considered a streaming variant of multiplication, where one number is fixed and the other is revealed one digit at a time. They require that a digit of the output is reported before the next digit of the input is revealed. In this streaming setting, they prove an $\Omega((\delta /w)n \lg n)$ lower bound, where $\delta$ is the number of bits in a digit and $w$ is the word size. For $\delta=1$ and $w=O(1)$, this is $\Omega(n \lg n)$. Ponzio~\cite{Ponz98} considered multiplication via read-once branching programs, i.e. programs that have bounded working memory and may only read each input bit exactly once. He proved that any read-once branching program for computing the middle bit of the product of two $n$-bit numbers, must use $\Omega(\sqrt{n})$ bits of working memory. Finally, we also mention the work of Morgenstern~\cite{Morgenstern:1973:NoaLBotLCotFFT} who proved lower bounds for computing the related FFT. Morgenstern proved an $\Omega(n \lg n)$ lower bound for computing the \emph{unnormalied} FFT via an arithmetic circuit when all constants used in the circuit are bounded. Unfortunately this doesn't say anything about the complexity of multiplying two $n$-bit numbers.
\paragraph{Valiant's Conjectures.}
Despite their importance, Valiant's conjectures are still mostly open. One interesting development by Riis~\cite{Riis2007}, shows that Conjecture~\ref{con:v2} as stated is incorrect. Riis proved that all cyclic shifts are realizable for $\varepsilon = \tfrac{1}{2} - \tfrac{1}{2n^{1-\delta}}$ where $n^{1 + \delta}$ is the total number of edges of $\Gamma$. Riis further conjectured that replacing the bound on $\varepsilon$ by a slightly stricter bound should result in a correct conjecture. Specifically, Riis suggest bounding $\varepsilon = \Theta\left(\frac{1}{\lg \lg n}\right)$.
\section{Preliminaries}
\label{sec:prelim}
We now give a formal definition of Boolean circuits with arbitrary gates, followed by definitions of the $k$-pairs communication problem, the multicommodity flow problem. In the two latter problems we reuse some of the definitions used by Farhadi \etal~\cite{FHLS18}, which have been simplified a bit compared to the more general definition by Adler \etal~\cite{Adler:soda}. In particular, we have forced communication networks to be directed acyclic graphs. This is sufficient to prove our lower bounds and simplifies the definitions considerably.
\paragraph{Boolean Circuits with Arbitrary Gates.}
A {\em Boolean Circuit with Arbitrary Gates} with $n$ source or input nodes and $m$ target or output nodes is a directed acyclic graph $C$ with $n$ nodes of in-degree $0$, which are called {\em input gates}, and are labeled with input variables $X=\{x_i\}_{i \in [n]}$ and $m$ nodes out-degree $0$, which are called {\em output gates} and are labeled with output variables $Y=\{y_i\}_{i \in [m]}$. All other nodes are simply called {\em gates}. For every gate $u$ of in-degree $k \ge 1$, $u$ is labeled with an arbitrary function $f_u:\{0,1\}^k \to \{0,1\}$. The circuit is also equipped with a topological ordering $v_1,\ldots,v_t$ of $C$, in which $v_i=x_i$ for $i \in [n]$ and $v_{t-i+1}=y_{m-i+1}$ for all $i \in [m]$. The {\em depth} of a circuit $C$ is the length of the longest path in $C$.
An {\em evaluation} of a circuit on an $n$ bit input $x=(x_1,\ldots,x_n) \in \{0,1\}^n$ is conducted as follows. For every $i \in [n]$, assign $x_j$ to $v_j$. For every $j \ge n+1$, assign to $v_j$ the value $f_{v_j}(u_1,\ldots,u_k)$, where $u_1,\ldots,u_k$ are the nodes of $C$ with edges going into $v_j$ in the order induced by the topological ordering. The {\em output} of $C$ on an $n$ bit input $x=(x_1,\ldots,x_n)$, denoted $C(x_1,\ldots,x_n)$ is the value assigned to $(y_1,\ldots,y_m)$ in the evaluation. We say a circuit computes a function $f : \{0,1\}^n \to \{0,1\}^m$ if for every $x=(x_1,\ldots,x_n) \in \{0,1\}^n$, $f(x_1,\ldots,x_n)=C(x_1,\ldots,x_n)$.
For every $j \in [t]$ and $b \in \{0,1\}$, we {\em hardwire} $b$ for $v_j$ in $C$ by removing $v_j$ and all adjacent edges from $C$, and replacing $v_j$ for $b$ in the evaluation of $f_{v_i}$ for every $i > j$ such that $v_jv_i$ is an edge in $C$.
\paragraph{$k$-Pairs Communication Problem.}
The input to the $k$-pairs communication problem is a directed acyclic graph $G=(V,E)$ where each edge $e \in E$ has a capacity $c(e) \in \R^+$. There are $k$ sources $s_1,\dots,s_k \in V$ and $k$ sinks $t_1,\dots,t_k \in V$.
Each source $s_i$ receives a message $A_i$ from a predefined set of messages $A(i)$. It will be convenient to think of this message as arriving on an in-edge. Hence we add an extra node $S_i$ for each source, which has a single out-edge to $s_i$. The edge has infinite capacity.
A network coding solution specifies for each edge $e \in E$ an alphabet $\Gamma(e)$ representing the set of possible messages that can be sent along the edge. For a node $v \in V$, define $\In(u)$ as the set of in-edges at $u$. A network coding solution also specifies, for each edge $e=(u,v) \in E$, a function $f_e : \prod_{e' \in \In(u)} \Gamma(e') \to \Gamma(e)$ which determines the message to be sent along the edge $e$ as a function of all incoming messages at node $u$. Finally, a network coding solution specifies for each sink $t_i$ a decoding function $\sigma_i : \prod_{e \in \In(t_i)} \Gamma(e) \to M(i)$. The network coding solution is correct if, for all inputs $A_1,\dots,A_k \in \prod_i A(i)$, it holds that $\sigma_i$ applied to the incoming messages at $t_i$ equals $A_i$, i.e. each source must receive the intended message.
In an execution of a network coding solution, each of the extra nodes $S_i$ starts by transmitting the message $A_i$ to $s_i$ along the edge $(S_i,s_i)$. Then, whenever a node $u$ has received a message $a_e$ along all incoming edges $e=(v,u)$, it evaluates $f_{e'}(\prod_{e \in \In(u)} a_e)$ on all out-edges and forwards the message along the edge $e'$.
We define the \emph{rate} of a network coding solution as follows: Let each source receive a uniform random and independently chosen message $A_i$ from $A(i)$. For each edge $e$, let $A_e$ denote the random variable giving the message sent on the edge $e$ when executing the network coding solution with the given inputs. The network coding solution achieves rate $r$ if:
\begin{itemize}
\item $H(A_i) \geq r$ for all $i$.
\item For each edge $e \in E$, we have $H(A_e) \leq c(e)$.
\end{itemize}
Here $H(\cdot)$ denotes binary Shannon entropy. The intuition is that the rate is $r$, if the solution can handle sending a message of entropy $r$ bits between every source-sink pair.
\paragraph{Multicommodity Flow.}
A multicommodity flow problem in an undirected graph $G=(V,E)$ is specified by a set of $k$ source-sink pairs $(s_i,t_i)$ of nodes in $G$. We say that $s_i$ is the source of commodity $i$ and $t_i$ is the sink of commodity $i$. Each edge $e \in E$ has an associated capacity $c(e) \in \R^+$.
\
A (fractional) solution to the multicommodity flow problem specifies for each pair of nodes $(u,v)$ and commodity $i$, a flow $f^i(u,v) \in [0,1]$. Intuitively $f^i(u,v)$ specifies how much of commodity $i$ that is to be sent from $u$ to $v$. The flow satisfies \emph{flow conservation}, meaning that:
\begin{itemize}
\item For all nodes $u$ that is not a source or sink, we have $\sum_{w \in V} f^i(u,w) - \sum_{w \in V} f^i(w,u) = 0$.
\item For all sources $s_i$, we have $\sum_{w \in V} f^i(s_i,w) - \sum_{w \in V}f^i(w,s_i) = 1$.
\item For all sinks we have $\sum_{w \in V} f^i(w,t_i) - \sum_{w \in V} f^i(t_i,w) = 1$.
\end{itemize}
The flow also satisfies that for any pair of nodes $(u,v)$ and commodity $i$, there is only flow in one direction, i.e. either $f^i(u,v)=0$ or $f^i(v,u)=0$. Furthermore, if $(u,v)$ is not an edge in $E$, then $f^i(u,v) = f^i(v,u)=0$. A solution to the multicommodity flow problem achieves a rate of $r$ if:
\begin{itemize}
\item For all edges $e=(u,v) \in E$, we have $r \cdot \sum_i (f^i(u,v) + f^i(v,u)) \leq c(e)$.
\end{itemize}
Intuitively, the rate is $r$ if we can handle a demand of $r$ for every commodity.
\paragraph{The Undirected $k$-Pairs Conjecture.} Conjecture~\ref{con:undirected} implies the following for our setting: Given an input to the $k$-pairs communication problem, specified by a directed acyclic graph $G$ with edge capacities and a set of $k$ source-sink pairs, let $r$ be the best achievable network coding rate for $G$. Similarly, let $G'$ denote the undirected graph resulting from making each directed edge in $G$ undirected (and keeping the capacities and source-sink pairs). Let $r'$ be the best achievable flow rate in $G'$. Conjecture~\ref{con:undirected} implies that $r \leq r'$.
Having defined coding rate and flow rate formally, we also mention that a result of Braverman \etal~\cite{braverman2016network} implies that if there exists a graph $G$ where the network coding rate $r$, and the flow rate $r'$ in the corresponding undirected graph $G'$, satisfies $r \geq (1+\eps)r'$ for a constant $\eps>0$, then there exists an infinite family of graphs $\{G^*\}$ for which the corresponding gap is at least $(\lg |G^*|)^c$ for a constant $c>0$. So far, all evidence suggest that no such gap exists, as formalized in Conjecture~\ref{con:undirected}.
\section{Key Tools and Techniques}
The main idea in the heart of both proofs is the simple fact that in a graph with $t$ vertices and maximum degree at most $c$, most node pairs lie far away from one another. Specifically, for every node $u$ in $G$, at least $t - \sqrt{t}$ nodes have distance $\ge \tfrac{1}{2}\log_c t$ from $u$. While this key observation is almost enough to prove Theorem~\ref{th:shiftLB}, the proof of Theorem~\ref{th:depth3MultiplicationLB} requires a much more subtle approach, as there is no bound on the maximum degree in the circuits in question. The only bound we have is on the number of wires going directly between from input gates into output gates. Specifically, every two nodes in the underlying undirected graph are at distance $\le 3$ (see figure~\ref{fig:circuit}).
In order to overcome this obstacle, we present a construction of a communication network based on the circuit $C$ that essentially eliminates the middle layer in the depth-$3$ circuit $C$, thus leaving a bipartite graph with bounded maximum degree.
To this end, we observe that since the size of the middle layer is bounded by $\varepsilon n$, then there exists a large set ${\cal F}$ of inputs in $\{0,1\}^n$ such that on all inputs from ${\cal F}$, the gates $f_1,\ldots,f_{\varepsilon n}$ attain the same values. By hardwiring these values to the circuit, we can evaluate the circuit for all inputs in ${\cal F}$ on a depth-$2$ circuit $\Gamma$ obtained from $C$ by removing $f_1,\ldots, f_{\varepsilon n}$. We next turn to construct the communication network.
Employing ideas recently presented by Farhadi \etal~\cite{FHLS18}, we "wrap" the depth-$2$ circuit by adding source and target nodes. In order to cope with inputs that do not belong to ${\cal F}$, we add a designated {\em supervisor} node $u$ (see figure~\ref{fig:network}). Loosely speaking, the source nodes transmit their input to $u$, and $u$ sends back the information needed to "edit" the input string $x$ and construct an input string $x' \in {\cal F}$, which is then transferred to the circuit $\Gamma$ as blackbox.
\paragraph{The Correction Game.} In order to bound the edge capacities of the network $G$ in a way that the supervisor node can transmit enough information to achieve a high communication rate, but then again not allow to much flow to go through the supervisor when considering $G$ as a multicommodity flow instance, Farhadi \etal~\cite{FHLS18} defined a game between a set of $m$ players and a supervisor, where given a fixed set ${\cal F} \subseteq \{0,1\}^n$ and a random string $\beta \in \{0,1\}^n$ given as a concatenation of $m$ strings $\beta_1,\ldots,\beta_m$ of length $n/m$ each, the goal is to "correct" $x$ and produce a string $\chi \in \{0,1\}^n$ such that $\beta \oplus \chi \in {\cal F}$. The caveat is that the only communication allowed is between the players and the supervisor. That is, no communication, and thus no cooperation, is allowed between the $m$ players. Formally, the game is defined as follows.
\begin{definition}
Let ${\cal F} \subseteq \{0,1\}^n$. The {\em ${\cal F}$-correction game} with $m+1$ players is defined as follows. The game is played by $m$ ordinary players $p_1,\ldots, p_m$ and one designated {\em supervisor} player $u$. The supervisor $u$ receives $m$ strings $\beta_1,\ldots,\beta_m \in \{0,1\}^{n/m}$ chosen independently at random. For every $\ell \in [m]$, $u$ then sends $p_\ell$ a message $R_\ell$. Given $R_\ell$, the player $p_\ell$ produces a string $\chi_\ell \in \{0,1\}^{n/m}$ such that $(\beta_1 \oplus \chi_1)\circ(\beta_2 \oplus \chi_2)\circ(\beta_m \oplus \chi_m) \in {\cal F}$.
\end{definition}
Farhadi \etal additionally present a protocol for the ${\cal F}$-correction game in which the supervisor player sends prefix-free messages to the $m$ players, and moreover, they give a bound on the amount of communication needed as a function of the number of players and the size of ${\cal F}$.
\begin{lemma} [\cite{FHLS18}]\label{l:protocolBound}
If $|{\cal F}| \ge 2^{(1-\varepsilon)n}$, then there exists a protocol for the ${\cal F}$-correction game with $m+1$ players such that the messages $\{R_\ell\}_{\ell \in [m]}$ are prefix-free and
$$\sum_{\ell \in [m]}{\mathbb{E}[|R_\ell|]} \le 3m + 2m\lg\left(\sqrt{\frac{\varepsilon}{2}}\cdot \frac{n}{m} + 1\right) + \sqrt{\frac{\varepsilon}{8}}\cdot n \lg\frac{2}{\varepsilon} \;,$$
\end{lemma}
\section{A Lower Bound for Boolean Circuits Computing Multiplication}
In this section we show that conditioned on Conjecture~\ref{con:undirected}, every bounded degree circuit computing multiplication must have size at least $\Omega(n \lg n)$, thus proving Theorems~\ref{th:multiplicationLB} and~\ref{th:shiftLB}.
In fact, we will prove something slightly stronger. Define the shift function $s : \{0,1\}^n\times[n] \to \{0,1\}^{2n}$ as follows. For every $x=(x_1,\ldots,x_n) \in \{0,1\}^n$ and $\ell \in [n]$, $s(x,\ell) = (y_1,\ldots,y_{2n})$ where $y_j = x_{j-\ell+1}$ if $\ell \le j \le \ell+n-1$ and $y_j=0$ otherwise. We will show that every circuit with bounded in and out degrees that computes the shift function on $n$-bit numbers has size $\Omega(n \lg n)$. Clearly, a circuit that can compute the product of two $n$-bit numbers can also compute the shift function. Let $c$ denote the maximum in and out degree in $C$, and let $j \in [n]$. Then in the undirected graph induced by $C$, there are at most $\sqrt{n}$ nodes whose distance from $x_j$ is at most $\frac{1}{2}\log_{2c}n$. Therefore among $y_j,\ldots,y_{j+n-1}$, at least $n - \sqrt{n} - 1 \ge n - 2\sqrt{n}$ are at distance at least $\frac{1}{2}\log_{2c}n$. In other words, $\Pr_{\ell \in [n]}[d_{\hat{C}}(x_j, y_{j+\ell-1}) \ge \frac{1}{2}\log_{2c}n] \ge 1-\frac{2}{\sqrt{n}}$, where $\hat{C}$ denotes the undirected graph induced by $C$ (by removing edge directions). Therefore there exists a shift $\ell_0 \in [n]$ such that $|\{j \in [n] : d_{\bar{C}}(x_j, y_{j+\ell_0-1}) \ge \frac{1}{2}\log_{2c}n\}| \ge n-2\sqrt{n} \ge n/2$.
Fixing $\ell_0$, let consider the following communication problem. For each $j \in [n]$, $s_j=x_j \in_R \{0,1\}$ and $t_j = y_{j+\ell_0-1}$. The circuit $C$ equipped with $1$-uniform edge capacities is a network coding solution to this problem with rate $r \ge 1$. By the undirected $n$-pairs conjecture, there is a multicommodity flow in $\hat{C}$ that transfers one unit of flow from each source to its corresponding sink. For every $j$, let $f^j : E \to [0,1]$ be the flow associated with commodity $j$. Then
$$|E| = \sum_{e \in E}{c_e} \ge \sum_{e \in E}{\sum_{j \in [n]}{f^j(e)}} \ge \Omega(n \log_c n) \;.$$
\section{A Lower Bound for Depth \texorpdfstring{$3$}{3} Boolean Circuits Computing Multiplication}
\begin{figure}[t]
\centering
\includegraphics[scale=0.7]{circuit}
\caption{The depth $3$ circuit $C$.}
\label{fig:circuit}
\hrule
\end{figure}
Let $C$ be a depth $3$ circuit that computes multiplication such that the number of gates in the second layer of $C$ is at most $\varepsilon n$ for some small $\varepsilon \in (0,1)$ and for every $u \in Y$, $deg_{\bar{C}[X \cup Y]}(u) \le c$, where once again $\bar{C}$ denotes the undirected graph induced by $C$, and $\bar{C}[X \cup Y]$ is the subgraph of $\bar{C}$ induced by $X \cup Y$. By slightly increasing $c$ and $\varepsilon$ (by a small constant factor) and without loss of generality, we can assume that this applies for all $u \in X$ as well.
Denote the input and output gates of $C$ by $X = \{x_1,\ldots,x_n,\hat{x}_1,\ldots,\hat{x}_n\}$ and $Y = \{y_1,\ldots,y_{2n}\}$ respectively, and denote the set of the middle-layer gates by $F = \{f_1,\ldots,f_{\varepsilon n}\}$ (see Figure~\ref{fig:circuit}).
\begin{figure}[t]
\centering
\includegraphics[scale=0.7]{network}
\caption{Given the $2$-layer circuit $\Gamma$ spanned by $x_1,\ldots,x_n,y_1,\ldots,y_n$, we construct the communication network graph $G$.}
\label{fig:network}
\hrule
\end{figure}
As before, we focus on computing the shift function, thus limiting the input to $(\hat{x}_1,\ldots,\hat{x}_n)$ to have exactly one $1$-entry.
We next partition $(x_1,\ldots,x_n)$ into consecutive blocks of size $k=20$ bits each. For every $\ell \in [n/k]$ let $B_\ell = \{k(\ell-1)+1,\ldots,k\ell\}$ be the set of indices belonging to the $\ell$th block.
\begin{definition}
For every $\alpha \in [n]$ and $\ell \in [n/k]$, we say $B_\ell$ is {\em far from all targets} (with respect to $\alpha$) if for all sources in the block are at distance at least $\frac{1}{2}\log_{2c}n$ from all respective destinations in $\bar{C}[X \cup Y]$. That is for every $u,v \in B_\ell$, $d_{\bar{C}[X \cup Y]}(x_u,y_{v+\alpha-1}) \ge \frac{1}{2}\log_{2c}n$.
\end{definition}
Let $\alpha \in_R [n]$. By the constraint on the degrees, for every $j \in [n]$, there are at most $\sqrt{n}$ nodes whose distance from $x_j$ is at most $\frac{1}{2}\log_{2c}n$ in $\bar{C}[X \cup Y]$. Therefore for every $\ell \in [n/k]$,
$$\Pr_{\alpha \in_R[n]}\left[B_\ell \; \text{is far from all targets}\right] \ge 1 - \frac{k^2}{\sqrt{n}} \;.$$
By averaging we get that for large enough $n$ there is some $\alpha_0 \in [n]$ such that there are at least $\frac{n}{k} - k\sqrt{n} \ge \frac{9n}{10k}$ blocks which are far from all targets. Without loss of generality, we may assume for ease of notation that $\alpha_0=1$.
By hardwiring $1$ for $\alpha_0$ into the circuit $C$, the circuit now simply transfers $(x_1,\ldots,x_n)$ to $(y_1,\ldots,y_n)$.
\paragraph{Reduction to Network Coding.} Let $x = (x_1,\ldots,x_n)$ and $i \in [\varepsilon n]$. By slightly abusing notation, we denote the value of the gate $f_i$ when evaluating the circuit by $f_i(x_1,\ldots,x_n)$.
By averaging, there exist a string $(\hat{f}_1, \ldots, \hat{f}_{\varepsilon n})$ and a set ${\cal F} \subseteq \{0,1\}^n$ such that $|{\cal F}| \ge 2^{(1-\varepsilon) n}$ and such that for every $x = (x_1,\ldots,x_n) \in {\cal F}$ and $i \in [\varepsilon n]$, $f_i(x_1,\ldots,x_n) = \hat{f}_i$.
By hardwiring $(\hat{f}_1, \ldots, \hat{f}_{\varepsilon n})$ for $(f_1,\ldots,f_n)$ into the circuit $C$, we get a new circuit denoted $\Gamma$ that contains only the input and output gates of $C$, and transfers $(x_1,\ldots,x_n)$ to $(y_1,\ldots,y_n)$ for every $(x_1,\ldots,x_n) \in {\cal F}$. Moreover, the set of edges between $X$ and $Y$ in $\Gamma$ is equal to the set of edges between $X$ and $Y$ in $C$.
Next, we construct a communication network $G$ by adding some nodes and edges to $\Gamma$, as demonstrated also in Figure~\ref{fig:network}. We add a new set of nodes $\{s_j,a_j,t_j\}_{j=1}^{n/k} \cup \{u\}$. For every $\ell \in [n/k]$, add edges $s_{\ell}a_{\ell}$ and $s_{\ell}u$ of capacity $k$ and edges $ua_{\ell}$ and $ut_{\ell}$ of capacity $c_{\ell} = \mathbb{E}[|R_\ell|]$, where $R_\ell$ is the message sent to player $p_\ell$ by the supervisor player in the ${\cal F}$-correction game protocol for $n/k+1$ players guaranteed in Lemma~\ref{l:protocolBound}. In addition, for every $\ell \in [n/k]$ and every $j \in B_{\ell}$ add edges $a_{\ell}x_j$ and $y_jt_{\ell}$ of capacity $1$. All edges of $\Gamma$ are assigned capacity of $1$.
\paragraph{Transmitting Data.} In what follows, we will lower bound the communication rate of the newly constructed network $G$.
\begin{lemma}\label{l:comRate}
There exists a network coding solution on $G$ that achieves rate $k$.
\end{lemma}
To this end, let $A_1,\ldots,A_{n/k} \in \{0,1\}^k$ be independent uniform random variables. We next give a protocol by which the sources $s_1,\ldots,s_{n/k}$ transmit $A_1,\ldots,A_{n/k}$ to the targets $t_1,\ldots,t_{n/k}$. The protocol employs as a an intermediate step the correction game protocol guaranteed by Lemma~\ref{l:protocolBound}.
\begin{enumerate}
\item For every $\ell \in [n/k]$, $s_\ell$ sends $A_\ell$ to $a_\ell$ over the edge $s_\ell a_\ell$ and to $u$ over the edge $s_\ell u$.
\item Employing the ${\cal F}$-correction game protocol with $n/k+1$ players, for every $\ell \in [n/k]$, $u$ sends a message $R_\ell$ to $a_\ell$ over the edge $ua_\ell$ and to $t_\ell$ over the edge $ut_\ell$. Following the correction game protocol, for every $\ell$, given $R_\ell$, $a_\ell$ and $t_\ell$ produce a string $\chi_\ell$ satisfying that $(A_1\oplus \chi_1)\circ\ldots\circ(A_{n/k}\oplus \chi_{n/k}) \in {\cal F}$.
\item For every $\ell \in [n/k]$ and every $i \in [k]$, $a_\ell$ transmits the $i$th bit of $A_\ell \oplus \chi_\ell$ to the $i$th gate in the $\ell$th block, namely $x_{(\ell-1)k+i}$. Note that $(x_1,\ldots,x_n) = (A_1\oplus \chi_1)\circ\ldots\circ(A_{n/k}\oplus \chi_{n/k}) \in {\cal F}$.
\item Next, the communication network employs the circuit $\Gamma$ and transmits $(x_1,\ldots,x_n)$ to $(y_1,\ldots,y_n)$. For every $\ell \in [n/k]$ and every $i \in B_\ell$, $y_i$ transmits $x_i$ to $t_\ell$.
\item Finally, for every $\ell \in [n/k]$, $t_\ell$ now holds both $A_\ell \oplus \chi_\ell$ and $\chi_\ell$. Therefore $t_\ell$ can recover $A_\ell$.
\end{enumerate}
By invoking the protocol described above, every one of the $n/k$ sources sends $k$ bits to the corresponding target. For every edge $e \in G$, let $A_e$ denote the random variable giving the message sent on the edge $e$ when executing the protocol.
\begin{claim}
For every $e \in G$, $H(A_e) \le c_e$.
\end{claim}
\begin{proof}
First note that for every $\ell \in [n/k]$, every edge $e$ leaving $s_\ell$ has capacity $k$ and transmits $A_\ell$. Therefore $H(A_\ell) = k \le c_e$.
Every edge $e$ that is not leaving any source nor $u$ has capacity $1$ and transmits exactly one bit (not necessarily uniformly random) of information. Therefore $c_e = 1 \ge H(A_e)$.
Finally, let $e$ be an edge leaving $u$. Then there exists some $\ell \in [n/k]$ such that $e=ua_\ell$ or $e=ut_\ell$. In both cases the message transmitted on $e$ is $R_\ell$ and the capacity $c_e$ of $e$ satisfies $c_e = c_\ell = \mathbb{E}[|R_\ell|] \ge H(R_\ell)$, where the last inequality follows from Shannon's Source Coding theorem, as all messages are prefix-free.
\end{proof}
We can therefore conclude that the network $G$ achieves rate $\ge k$, and the proof of Lemma~\ref{l:comRate} is complete.
\paragraph{Deriving the Lower Bound.} By Conjecture~\ref{con:undirected}, the underlying undirected graph $\bar{G}$ achieves a multicommodity-flow rate $\ge k$. Therefore there exists a multicommodity flow $\{f^{\ell}\}_{\ell \in [n/k]} \subseteq [0,1]^{E(\bar{G})}$ that achieves rate $k$. We first observe that at most a constant fraction of the flow can go through the supervisor node $u$. To see this, we note that as $|{\cal F}| \ge 2^{(1-\varepsilon)n}$, then by Lemma~\ref{l:protocolBound} the expected total information sent by the supervisor in the ${\cal F}$-correction game with $n/k$ players is at most
\begin{equation}
\frac{3n}{k} + \frac{2n}{k}\lg\left(k\sqrt{\frac{\varepsilon}{2}} + 1\right) + \sqrt{\frac{\varepsilon}{8}} \cdot n \lg\frac{2}{\varepsilon} \le \frac{5n}{k}
\label{eq:protocol}
\end{equation}
Therefore by the definition of the capacities $\{c_\ell\}_{\ell \in [n/k]}$ we get that for small enough (constant) $\varepsilon$,
\begin{equation}
\sum_{\ell \in [n/k]}{c_{ua_\ell}} = \sum_{\ell \in [n/k]}{c_{ut_\ell}} = \sum_{\ell \in [n/k]}{c_{\ell}} \le \frac{5n}{k}
\label{eq:capacities}
\end{equation}
Since $\{f^{\ell}\}_{\ell \in [n/k]}$ achieves rate $k$ we conclude that
\begin{equation*}
\begin{split}
k \cdot\sum_{v \in V(\bar{G}) : uv \in E(\bar{G})}{ \sum_{\ell \in [n/k]}{(f^{\ell}(u,v) + f^{\ell}(v,u))}} &\le \sum_{v \in V(\bar{G}) : uv \in E(\bar{G})}{c_e}\\
&= \sum_{\ell \in [n/k]}{c_{us_\ell}} + \sum_{\ell \in [n/k]}{(c_{ua_\ell}+c_{ut_\ell})}\le n + \frac{10n}{k}\;,
\end{split}
\label{eq:uCapacity}
\end{equation*}
and therefore
\begin{equation}
\sum_{v \in V(\bar{G}) : uv \in E(\bar{G})}{ \sum_{\ell \in [n/k]}{(f^{\ell}(u,v) + f^{\ell}(v,u))}} \le \frac{n}{k} + \frac{10n}{k^2} \le 1.5\frac{n}{k} \;.
\label{eq:uTotal}
\end{equation}
By the flow-conservation constraint, we know that
therefore the total amount of flow that can go through $u$ is $\le 0.75 \frac{n}{k}$. By averaging, at least a $1/6$ fraction of the sources send at least $1/10$ units of flow through $\bar{G} - u$. By the choice of $\alpha_0$, in $\bar{G}-u$, at least a $1/15$ of the sources are at least $\frac{1}{2}\log_{2c}(n)$ away from their targets. Without loss of generality, assume these are the first $\tfrac{n}{15k}$ sources. We conclude that
\begin{equation}
\begin{split}
cn \ge |E[X \cup Y]| &= \sum_{e \in E[X \cup Y]}{c_e} \ge k \cdot \sum_{e = vw \in E[X \cup Y]}{\sum_{\ell \in [n/k]}{f^{\ell}(v,w)+f^{\ell}(w,v)}} \\
&\ge k \cdot \sum_{\ell \in [n/15k]}{\sum_{e = vw \in E[X \cup Y]}{f^{\ell}(v,w)+f^{\ell}(w,v)}} \ge \frac{n}{30}\log_{2c}(n) \;,
\end{split}
\end{equation}
and therefore $c \ge \Omega\left(\frac{\lg n}{\lg \lg n}\right)$, and the proof of Theorem~\ref{th:depth3MultiplicationLB} is now complete.
\subsection{Remarks and Extensions}
For sake of fluency, some minor remarks and extensions were intentionally left out of the text, and will be discussed now.
\paragraph{Circuits with Bounded Average Degree.} Our results still hold if we relax the second requirement of Theorem~\ref{th:depth3MultiplicationLB} and require instead that the number of edges in $\bar{C}[X \cup Y]$ is at most $cn$. That is, the average degree in $\bar{C}[X \cup Y]$ is at most $c$. To see this, note that under this assumption, there are at most $0.001n$ gates in $X \cup Y$ whose degree in $\bar{C}[X \cup Y]$ is larger than $1000c$. For each such gate $v$, add a new node $f$ in the middle layer, and connect $v$ and all the neighbours of $v$ in $\bar{C}[X \cup Y]$ to $f$. Then delete all the edges adjacent to $v$ in $\bar{C}[X \cup Y]$. The number of nodes added to the middle layer is at most $0.001n$, and the degree of all nodes in $\bar{C}[X \cup Y]$ is now bounded by $1000c$. The rest of our proof continues as before.
\paragraph{Shifts vs. Cyclic Shifts.} In order to prove lower bounds for circuits computing multiplication, our results are stated in terms of shifts (which are a special case of products, as mentioned). This is in contrast to Valiant's conjectures, which are stated in terms of cyclic shifts. However, we draw the readers attention to the fact that our proofs work for cyclic shifts as well. The exact same arguments apply, and the proofs remain unchanged.
\newcommand{\etalchar}[1]{$^{#1}$}
|
{
"timestamp": "2019-03-01T02:11:24",
"yymm": "1902",
"arxiv_id": "1902.10935",
"language": "en",
"url": "https://arxiv.org/abs/1902.10935"
}
|
\part{Supplementary Material}
\section{Analytic Results for the Two-Pulse}
Here we provide explicit formulas for the vector potential of the recombined two-pulse $a_\perp(\varphi)$, its envelope $a(\varphi)$, and local frequency $\omega_L(\varphi)$. First, by calculating the inverse Fourier transformation of the modulated spectrum,
Eq.~(5) of the main text,
we immediately find the complex scalar amplitude
\begin{align}
a_\perp & = \frac{a_0}{2 \sqrt[4]{1+B_2^2}}
e^{-i \omega_0 \varphi }
\left(
e^{- \frac{c_-}{2} (1+iB_2) + i B_0 + \frac{i}{2} \arctan B_2 }
+
e^{- \frac{c_+}{2} (1-iB_2) - iB_0 - \frac{i}{2} \arctan B_2}
\right)
\end{align}
with
\begin{align}
c_\pm &= \frac{(B_1 \pm \varphi \Delta \omega_L)^2}{1+B_2^2} \,,
\end{align}
and $\varphi = t-z $.
The real vector potential of a circularly polarized laser pulse is related to $a_\perp$ by
$\vec a_\perp = \Re [ a_\perp \vec \epsilon ]$ with $\vec \epsilon = \vec e_x + i \vec e_y$.
The squared envelope of the laser pulse is given by
\begin{align} \label{eq:a2}
a^2(\varphi) &= \vec a_\perp^2 = | a_\perp |^2
= \frac{a_0^2}{2\sqrt{1+B_2^2}} e^{-\xi}
\left[ \cosh 2d_0\varphi
+ \cos \zeta
\right]
\end{align}
where we introduced the following abbreviations
\begin{align}
d_0 & = \frac{B_1 \Delta \omega}{1+B_2^2}\,, \qquad
\xi = \frac{B_1^2 + \varphi^2 \Delta\omega^2_L}{1+B_2^2} \,, \qquad
\zeta = 2 B_0 - B_2 \xi + \arctan B_2 \,.
\end{align}
An important quantity is the infinite integral over the squared vector potential envelope,
\begin{align} \label{total_energy_eq}
\intop_{-\infty}^\infty \! \mathrm{d} \varphi \: a^2(\varphi) =
\frac{a_0^2\sqrt{\pi}}{2}\frac{1}{\Delta\omega_L}
\left[
1
+
\frac{e^{-\frac{B_1^2}{1+B_2^2}}}{(1+B_2^2)^\frac{1}{4}}
\,
\cos \left(
{2B_0 - \frac{B_1^2B_2}{1+B_2^2} + \frac{1}{2} \arctan B_2 } \right)
\right] \,.
\end{align}
When the cosine term is maximised, i.e.~its argument a multiple of $2\pi$, then
the interference between the two sub-pulses is mostly constructive.
Because $a_\perp = a \, e^{- i \Phi(\varphi)}$ we can write
$\Phi = i \log \frac{a_\perp}{a}$, and
\begin{align} \label{eq:omega-analytic}
\omega_L(\varphi) &= \frac{\mathrm{d} \Phi }{\mathrm{d}\varphi}
= - \frac{{\rm Im}\,[ a_\perp'a_\perp^* ] }{a^2}
= \omega_{0} - d_0 B_2 + d_0 \: \frac{\varphi \frac{\Delta \omega B_2}{B_1} \sinh 2d_0 \varphi - \sin \zeta}{\cosh 2d_0\varphi + \cos \zeta} \,.
\end{align}
\section{Defining the Matched Gaussian Pulse}
We match the a Gaussian with constant frequency $\omega_0$ to the two-pulse, having the same effective peak amplitude and total energy in the pulse,
\begin{align}
a_\mathrm{matched}(\varphi) = a_\mathrm{eff} \, e^{-i\omega_0\varphi} \, e^{-\varphi^2/2\Delta \varphi_\mathrm{eff}^2}\,.
\end{align}
First, the amplitude is matched by evaluating Eq.~\eqref{eq:a2} at $\varphi=0$ with the approximation $\zeta \to 0$,
yielding
\begin{align}
a_\mathrm{eff}^2 &= a_0^2 \: \frac{e^{- \frac{B_1^2}{1+B_2^2}}}{\sqrt{1+B^2_2}} \,.
\end{align}
Second, the pulse duration is matched by the requirement that both the chirped two-pulse and the matched Gaussian
have the same energy,
\begin{align}
w = \int \mathrm{d} \varphi \left| \frac{ \mathrm{d} a_\perp }{\mathrm{d} \varphi } \right|^2 \,,
\end{align}
i.e.
\begin{align}
\Delta\varphi_\mathrm{eff} = \frac{w}{\sqrt{\pi} \omega_0^2 a_\mathrm{eff}^2} \,.
\end{align}
\section{Derivation of the Formula for the on-axis Spectrum}
Under the assumption that we can use classical electrodynamics to calculate the radiation spectrum,
the energy and angular differential photon distribution is given by \cite{book:Jackson}
\begin{align} \label{N1}
\frac{\mathrm{d} N}{\mathrm{d} \omega' \mathrm{d} \Omega} = \frac{\omega'}{4\pi^2} | \vec n' \times (\vec n' \times \vec j)|^2 \,,
\end{align}
with the Fourier transformed electron current
\begin{align}
\vec j (\omega',\vec n') = - e \int \! \mathrm{d} s \, \vec u(s) \, e^{i \omega' [ t(s) - \vec n' \cdot \vec x(s)]} \,,
\end{align}
and $\vec n'$ the direction under which the radiation is observed.
Here, $s$ denotes the electron's proper time, parametrizing the electron orbits $t(s),\vec x(s)$ and four-velocity components $\gamma(s) = \mathrm{d} t/\mathrm{d} s = \sqrt{1+\vec u^2}$ and $\vec u = \mathrm{d} \vec x/\mathrm{d} s$, which is a solution of the Lorentz force equation
\begin{align}
\frac{\mathrm{d} \vec u}{\mathrm{d} s} = \frac{e}{m} ( \gamma \vec E + \vec u\times \vec B ) \,.
\end{align}
For on-axis radiation, $\vec n' = - \vec e_z$, the double vector product can be simplified to
\begin{align}
| \vec n' \times (\vec n' \times \vec j)|^2 = | \vec j_\perp |^2 \,,
\end{align}
and with $\vec u_\perp = -\vec a_\perp$ and changing integration variables from proper time to laser phase
$\varphi = t-z$ via $ \mathrm{d} \varphi / \mathrm{d} s \approx 2\gamma $ (for initial value of $\gamma\gg1$),
\eqref{N1} turns into
\begin{align}
\left.\frac{\mathrm{d} N}{\mathrm{d} \omega' \mathrm{d} \Omega}\right|_\mathrm{on-axis}
= \frac{e^2 \omega'}{4\pi^2 (2 \gamma)^2} \: \left| \int \! \mathrm{d} \varphi \, \vec a_\perp \: e^{i\omega' [ t(\varphi) + z(\varphi) ] } \right|^2 \,.
\end{align}
By noting that $t(\varphi) + z(\varphi) = (2\gamma)^{-1} \int \! \mathrm{d} \varphi [ \gamma(\varphi) + u_z(\varphi) ] $,
with $\gamma(\varphi) + u_z(\varphi) = ( 1 + \vec a_\perp^2)/(2\gamma)$, the definition of the normalized frequency of the emitted photon, $y = \omega' / (4\gamma^2 \omega_{0})$
we eventually arrive at Eq.~(6) of the main text.
In Fig.~\ref{fig:onaxis} we show the on-axis spectra for optimized laser pulses of various intensities and bandwidth $\Delta\omega_L/\omega_0=0.1$.
\begin{figure}
\begin{center}
\includegraphics[width=0.6\columnwidth]{figureS1.pdf}
\end{center}
\caption{On-axis photon spectra from optimized laser pulses with the chirping parameters $\vec B$ calculated using the model Eqns.~(8) with $\chi=1$.}
\label{fig:onaxis}
\end{figure}
\end{widetext}
\end{document}
|
{
"timestamp": "2019-03-01T02:03:09",
"yymm": "1902",
"arxiv_id": "1902.10777",
"language": "en",
"url": "https://arxiv.org/abs/1902.10777"
}
|
\section{ Introduction and main results}\label{sect1}
\setcounter{equation}{0}
The convergence of the sums of independent random variables are well-studied. For example, it is well-known that, if $\{X_n;n\ge 1\}$ is a sequence of independent random variables on a probability space $(\Omega, \mathcal{F}, \pr)$, then that the infinite series $\sum_{n=1}^{\infty}X_n$ is convergent almost surely, that it is convergent in probability and that it is convergent in distribution are equivalent. In this paper, we consider this elementary equivalence under the sub-linear expectations.
The general framework of the sub-linear expectation is introduced by Peng \cite{PengG-Expectation06,peng2008a,peng2009survey} in a general function space by relaxing the linear property of the classical linear expectation
to the sub-additivity
and positive homogeneity (cf. Definition~\ref{def1.1} below).
The sub-linear expectation does not depend on the probability measure, provides
a very flexible framework to model distribution uncertainty problems and produces many interesting properties different from those of the linear expectations. Under Peng's framework, many limit theorems have been being gradually established recently, including the central limit theorem and weak law of large numbers (cf. Peng \cite{peng2008a,peng2010}), the small derivation and Chung's law of the iterated logarithm (cf. Zhang \cite{Zhang Donsker}), the strong law of large numbers (cf. Chen \cite{chen2016strong}, Chen et al \cite{Chen Z 2013}, Hu \cite{Hu C 2018}, Zhang \cite{Zhang Rosenthal}, Zhang and Lin \cite{ZhangLin}), and the law of the iterated logarithm (cf. Chen \cite{Chen2014LIL}, Zhang \cite{Zhang Exponential}).
For the convergence of the infinite series $\sum_{n=1}^{\infty}X_n$, Xu and Zhang \cite{XuZhang2018} gave sufficient conditions of the almost sure convergence for independent random variables under the sub-linear expectation via a three-series theorem, recently. In this paper, we will consider the necessity of these conditions and the equivalence of the almost sure convergence, the convergence in capacity and the convergence in distribution. In the classical probability space, the Levy maximal inequalities are basic to the study of the almost sure behavior of sums of independent random variables and a key to show that the convergence in probability of $\sum_{n=1}^{\infty} X_n$ implies its almost sure convergence. We will establish Levy type inequalities under the sub-linear expectation. For showing that the convergence in distribution of $\sum_{n=1}^{\infty} X_n$ implies its convergence in probability, the characteristic function is a basic tool. But, under the sub-linear expectation, there is no such tools. We will find a new way to show a similar implication under the sub-linear expectations basing on a Komlogorov type maximal inequality.
As for the central limit theorem, it is well-known that the finite variances and mean zeros are sufficient and necessary for $\frac{\sum_{k=1}^n X_k}{\sqrt{n}}$ to converge in distribution to a normal random variable if $\{X_n;n\ge 1\}$ is a sequence of independent and identically distributed random variables on a classical probability space $(\Omega, \mathcal{F}, \pr)$. Under the sub-linear expectation, Peng \cite{peng2008a, peng2010} proved the cental limit theorem under the finite $(2+\alpha)$-th moment. By applying a moment inequality and the truncation method, Zhang \cite{Zhang Exponential} and Lin and Zhang \cite{LinZhang2017} showed that the moment condition can be weakened to the finite second moment. A nature question is whether the finite second moment is necessary. In this paper, by applying the maximal inequalities, we will obtain the sufficient and necessary conditions for the central limit theorem.
In the remainder of the section, we state some natation. In the next section, we will establish the maximal inequalities for random variables under the sub-linear expectation. The results on the convergence of the infinite series of random variables will given in Section \ref{Sect Convergece}. The sufficient and necessary conditions for the central limit theorem are given in Section \ref{Sect CLT}.
We use the framework and notations of Peng \cite{peng2008a}. Let $(\Omega,\mathcal F)$
be a given measurable space and let $\mathscr{H}$ be a linear space of real functions
defined on $(\Omega,\mathcal F)$ such that if $X_1,\ldots, X_n \in \mathscr{H}$ then $\varphi(X_1,\ldots,X_n)\in \mathscr{H}$ for each
$\varphi\in C_{l,Lip}(\mathbb R_n)$, where $C_{l,Lip}(\mathbb R_n)$ denotes the linear space of local Lipschitz
functions $\varphi$ satisfying
\begin{eqnarray*} & |\varphi(\bm x) - \varphi(\bm y)| \le C(1 + |\bm x|^m + |\bm y|^m)|\bm x- \bm y|, \;\; \forall \bm x, \bm y \in \mathbb R_n,&\\
& \text {for some } C > 0, m \in \mathbb N \text{ depending on } \varphi. &
\end{eqnarray*}
$\mathscr{H}$ is considered as a space of ``random variables''. In this case we denote $X\in \mathscr{H}$. In the paper, we also denote $C_{b,Lip}(\mathbb R_n)$ the space of bounded Lipschitz
functions, $C_b(\mathbb R_n)$ the space of bounded continuous functions, and $C_b^{1}(\mathbb R_n)$ the space of bounded continuous functions with bounded continuous derivations on $\mathbb R_n$.
\begin{definition}\label{def1.1} A sub-linear expectation $\Sbep$ on $\mathscr{H}$ is a function $\Sbep: \mathscr{H}\to \overline{\mathbb R}$ satisfying the following properties: for all $X, Y \in \mathscr H$, we have
\begin{description}
\item[\rm (a)] Monotonicity: If $X \ge Y$ then $\Sbep [X]\ge \Sbep [Y]$;
\item[\rm (b)] Constant preserving: $\Sbep [c] = c$;
\item[\rm (c)] Sub-additivity: $\Sbep[X+Y]\le \Sbep [X] +\Sbep [Y ]$ whenever $\Sbep [X] +\Sbep [Y ]$ is not of the form $+\infty-\infty$ or $-\infty+\infty$;
\item[\rm (d)] Positive homogeneity: $\Sbep [\lambda X] = \lambda \Sbep [X]$, $\lambda\ge 0$.
\end{description}
Here $\overline{\mathbb R}=[-\infty, \infty]$. The triple $(\Omega, \mathscr{H}, \Sbep)$ is called a sub-linear expectation space. Give a sub-linear expectation $\Sbep $, let us denote the conjugate expectation $\cSbep$of $\Sbep$ by
$$ \cSbep[X]:=-\Sbep[-X], \;\; \forall X\in \mathscr{H}. $$
\end{definition}
From the definition, it is easily shown that $\cSbep[X]\le \Sbep[X]$, $\Sbep[X+c]= \Sbep[X]+c$ and $\Sbep[X-Y]\ge \Sbep[X]-\Sbep[Y]$ for all
$X, Y\in \mathscr{H}$ with $\Sbep[Y]$ being finite. Further, if $\Sbep[|X|]$ is finite, then $\cSbep[X]$ and $\Sbep[X]$ are both finite.
\begin{definition}\label{def1.2}
\begin{description}
\item[ \rm (i)] ({\em Identical distribution}) Let $\bm X_1$ and $\bm X_2$ be two $n$-dimensional random vectors defined
respectively in sub-linear expectation spaces $(\Omega_1, \mathscr{H}_1, \Sbep_1)$
and $(\Omega_2, \mathscr{H}_2, \Sbep_2)$. They are called identically distributed, denoted by $\bm X_1\overset{d}= \bm X_2$ if
$$ \Sbep_1[\varphi(\bm X_1)]=\Sbep_2[\varphi(\bm X_2)], \;\; \forall \varphi\in C_{b,Lip}(\mathbb R_n), $$
where $C_{b,Lip}(\mathbb R_n)$ is the space of bounded Lipschitz functions.
\item[\rm (ii)] ({\em Independence}) In a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$, a random vector $\bm Y =
(Y_1, \ldots, Y_n)$, $Y_i \in \mathscr{H}$ is said to be independent to another random vector $\bm X =
(X_1, \ldots, X_m)$ , $X_i \in \mathscr{H}$ under $\Sbep$ if
$$ \Sbep [\varphi(\bm X, \bm Y )] = \Sbep \big[\Sbep[\varphi(\bm x, \bm Y )]\big|_{\bm x=\bm X}\big], \;\; \forall \varphi\in C_{b,Lip}(\mathbb R_m \times \mathbb R_n). $$
Random variables $\{X_n; n\ge 1\}$
are said to be independent, if $X_{i+1}$ is independent to $(X_1,\ldots, X_i)$ for each $i\ge 1$.
\end{description}
\end{definition}
In Peng \cite{peng2008a,peng2010,peng2010b}, the space of the test function $\varphi$ is $C_{l,Lip}(\mathbb R_n)$.
Here, the test function $\varphi$ in the definition is limit in the space of bounded Lipschitz functions. When the considered random variables have finite moments of each order, i.e., $\Sbep[|X|^p]<\infty$ for each $p>0$, then the space of test functions $ C_{b,Lip}(\mathbb R_n)$ can be equivalently extended to $C_{l,Lip}(\mathbb R_n)$.
A function $V:\mathcal{F}\to [0,1]$ is called a capacity if $V(\emptyset)=0$, $V(\Omega)=1$ and $V(A\cup B)\le V(A)+V(B)$ for all $A, B\in \mathcal{F}$. Let $(\Omega, \mathscr{H}, \Sbep)$ be a sub-linear space. We denote a pair $(\Capc,\cCapc)$ of capacities by
$$ \Capc(A):=\inf\{\Sbep[\xi]: I_A\le \xi, \xi\in\mathscr{H}\}, \;\; \cCapc(A):= 1-\Capc(A^c),\;\; \forall A\in \mathcal F, $$
where $A^c$ is the complement set of $A$.
Then
$$
\Sbep[f]\le \Capc(A)\le \Sbep[g], \;\;\cSbep[f]\le \cCapc(A) \le \cSbep[g],\;\;
\text{ if } f\le I_A\le g, f,g \in \mathscr{H}.
$$
It is obvious that $\Capc$ is sub-additive, i.e., $\Capc(A\bigcup B)\le \Capc(A)+\Capc(B)$. But $\cCapc$ and $\cSbep$ are not. However, we have
$$
\cCapc(A\bigcup B)\le \cCapc(A)+\Capc(B) \;\;\text{ and }\;\; \cSbep[X+Y]\le \cSbep[X]+\Sbep[Y]
$$
due to the fact that $\Capc(A^c\bigcap B^c)=\Capc(A^c\backslash B)\ge \Capc(A^c)-\Capc(B)$ and $\Sbep[-X-Y]\ge \Sbep[-X]-\Sbep[Y]$.
Further,
if $X$ is not in $\mathscr{H}$, we define $\Sbep[X]$ by
$$ \Sbep[X]=\inf\{\Sbep[Y]: X\le Y, \; Y\in \mathscr{H}\}. $$
Then $\Capc(A)=\Sbep[I_A]$.
\begin{definition}\label{def1.3}
(I) A function $V: \mathcal{F}\to [0, 1]$ is called to be countably sub-additive if
\[
V\Big(\bigcup_{n=1}^{\infty}A_n\Big)\leq \sum_{n=1}^{\infty}V(A_n),\ \ \forall A_n\in \mathcal{F}.
\]
(II) A function $V: \mathcal{F}\to [0, 1]$ is called to be continuous if it satisfies:\\
(i) Continuity from below: $V(A_n)\uparrow V(A)$ if $A_n\uparrow A$, where $A_n, A\in \mathcal{F}$.\\
(ii) Continuity from above: $V(A_n)\downarrow V(A)$ if $A_n \downarrow A$, where $A_n, A\in \mathcal{F}$.
\end{definition}
It is easily seen that a continuous capacity is countably sub-additive.
\section{Maximal inequalities}\label{Sect Inequality}
\setcounter{equation}{0}
In this section, we establish several inequalities on the maximal sums. The first one is the Levy maximal inequality.
\begin{lemma}\label{LevyIneq} Let $X_1,\cdots, X_n$ be independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$, $S_k=\sum_{i=1}^k X_i$, and $0<\alpha<1$ be a real number. If there exist real constants $\beta_{n, k}$ such that
$$ \Capc\left(S_k-S_n\ge \beta_{n,k}+\epsilon\right)\le \alpha, \text{ for all } \epsilon>0 \text{ and } k=1,\cdots ,n, $$
then
\begin{equation}\label{eqLIQ1} (1-\alpha) \mathbb{V}\left(\max_{k\le n}(S_k -\beta_{n,k})> x+\epsilon\right)\le \mathbb{V}\left(S_n>x\right), \text{ for all }x>0, \epsilon>0.
\end{equation}
If there exist real constants $\beta_{n, k}$ such that
$$ \Capc\left(|S_k-S_n|\ge \beta_{n,k}+\epsilon\right)\le \alpha, \text{ for all } \epsilon>0 \text{ and } k=1,\cdots ,n, $$
then
\begin{equation}\label{eqLIQ2} (1-\alpha) \mathbb{V}\left(\max_{k\le n}(|S_k| -\beta_{n,k})> x+\epsilon\right)\le \mathbb{V}\left(|S_n|>x\right), \text{ for all }x>0, \epsilon>0.
\end{equation}
\end{lemma}
{\bf Proof.} We only give the proof of (\ref{eqLIQ1}) since the proof of (\ref{eqLIQ2}) is similar.
Let $g_{\epsilon}(x)$ be a function with
\begin{equation}\label{eqproofLIQ.1} g_{\epsilon} \in C_b^1(\mathbb R) \; \text{ and } I_{\{x\ge \epsilon \}}\leq g_{\epsilon}(x)\leq I_{\{x\ge \epsilon/2\}} \text{ for all } x,
\end{equation}
where $0<\epsilon<1/2$, $ C_b^1(\mathbb R)$ is the space of bounded continuous function having bounded continuous derivations. Denote $Z_k=g_{\epsilon}\left(S_k-\beta_{n,k}-x\right)$, $Z_0=0$ and $\eta_k=\prod_{i=1}^k(1-Z_i)$. Then $S_n-S_m$ is independent to $(Z_1,\ldots,Z_m)$, and
\begin{align*}
&(1-\alpha) I\{\max_{k\le n}(S_k -\beta_{n,k})> x+\epsilon\}\\
= & (1-\alpha)\left[1-\prod_{k=1}^n I\{ S_k-\beta_{n,k}- x\le \epsilon\}\right] \\
\le & (1-\alpha)\left[1-\eta_n\right]=(1-\alpha)\left[\sum_{m=1}^n \eta_{m-1} Z_m\right]\\
=& \sum_{m=1}^n \eta_{m-1} Z_m I\{S_m-S_n<\beta_{n,m}+\epsilon/2\}\\
& +\sum_{m=1}^n \eta_{m-1}Z_m\left[1-\alpha-I\{S_m-S_n<\beta_{n,m}+\epsilon/2\}\right]\\
\le & \sum_{m=1}^n \eta_{m-1}Z_m I\{S_n>x\}
+\sum_{m=1}^n \eta_{m-1}Z_m\left[-\alpha+I\{S_m-S_n\ge \beta_{n,m}+\epsilon/2\}\right]\\
=& I\{ S_n >x\}
+\sum_{m=1}^n \eta_{m-1}Z_m\left[-\alpha+g_{\epsilon/2}\left( S_m-S_n-\beta_{n,m} \right) \right],
\end{align*}
where the second inequality above is due to the fact that on the event $\{Z_m\ne 0\}$ and $\{S_m-S_n< \beta_{n,m}+\epsilon/2\}$ we have $S_n\ge S_m-(S_m-S_n)>x$. Note
$$ \Sbep\left[g_{\epsilon/2}\left(S_m-S_n-\beta_{n,m}\right) \right]
\le \mathbb{V}\left(S_m-S_n\ge \beta_{n,m}+\epsilon/4\right)\le \alpha. $$
By the independence,
\begin{align*}
& \Sbep\left[\eta_{m-1}Z_m\left[-\alpha+g_{\epsilon/2}\left(S_m-S_n-\beta_{n,m}\right) \right]\right]\\
=&\Sbep\left[\eta_{m-1}Z_m\left\{-\alpha+\Sbep\left[g_{\epsilon/2}\left(S_m-S_n-\beta_{n,m}\right) \right]\right\}\right]\le 0.
\end{align*}
By the sub-additivity of $\Sbep$, it follows that
\begin{align*}
& (1-\alpha) \mathbb{V}\left(\max_{k\le n}(S_k -\beta_{n,k})> x+\epsilon\right)\\
\le & \mathbb{V}\left(S_n>x\right)
+\sum_{m=1}^n \Sbep\left[ \eta_{m-1}Z_m\left[-\alpha+g_{\epsilon/2}\left( S_m-S_n-\beta_{n,m} \right) \right]\right]
\\
\le & \mathbb{V}\left(S_n>x\right).
\end{align*}
The proof is completed. $\Box$
The second lemma is on the Kolmogorov type inequality.
\begin{lemma}\label{KolIneq} Let $X_1,\cdots, X_n$ be independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$. Let $S_k=\sum_{i=1}^k X_i$.
\begin{description}
\item[\rm (i) ] Suppose $|X_k|\le c$, $k=1,\cdots, n$. Then
\begin{equation}\label{eqKIQ1} \mathbb{V}\left(\max_{k\le n}|S_k|> x \right)\ge 1 -\frac{(x+c)^2+2x\sum_{k=1}^n \big\{\big(\Sbep[X_k]\big)^++\big(\Sbep[-X_k]\big)^+\big\}}{\sum_{k=1}^n \Sbep[X_k^2]},
\end{equation}
for all $x>0$.
\item[\rm (ii)] Suppose $ X_k \le c$, $\Sbep[X_k]\ge 0$, $k=1,\cdots, n$. Then
\begin{equation}\label{eqKIQ2.1} \mathbb{V}\left(\max_{k\le n} S_k>x \right)\ge 1 -\frac{ x+c }{\sum_{k=1}^n \Sbep[X_k]}\; \text{ for all }x>0.
\end{equation}
\end{description}
\end{lemma}
{\bf Proof.} (i) Let $g_{\epsilon}$ be defined as in (\ref{eqproofLIQ.1}). Denote $Z_k=g_{\epsilon}\left(|S_k|-x\right)$, $Z_0=0$, $\eta_k=\prod_{i=1}^k(1-Z_i)$.
Then $I\{|S_k|\ge x+\epsilon\}\le Z_k\le I\{|S_k|> x\}$. Also, $|S_{k-1}|< x+\epsilon$ and $|S_k|< |S_{k-1}|+|X_k|\le x+\epsilon+c$ on the event $\{\eta_{k-1}\ne 0\}$. So
\begin{align*}
S_{k-1}^2 \eta_{k-1} + 2S_{k-1}X_k \eta_{k-1} +X_k^2 \eta_{k-1}
=& S_k^2 \eta_k +S_k^2 \eta_{k-1} Z_k \\
\le &S_k^2 \eta_k +(x+\epsilon+c)^2\left[\eta_{k-1} -\eta_k \right].
\end{align*}
Taking the summation over $k$ yields
\begin{align*}
&\left(\sum_{k=1}^n \Sbep[X_k^2]\right)\eta_n +\sum_{k=1}^n\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1}
\le \sum_{k=1}^nX_k^2 \eta_{k-1} \\
\le & S_n^2 \eta_n +(x+\epsilon+c)^2\left[1-\eta_n \right]-2\sum_{k=1}^n S_{k-1}X_k \eta_{k-1} \\
\le &(x+\epsilon)^2 \eta_n +(x+\epsilon+c)^2\left[1-\eta_n \right]-2\sum_{k=1}^n S_{k-1}X_k \eta_{k-1} \\
\le & (x+\epsilon+c)^2 -2\sum_{k=1}^n S_{k-1}X_k \eta_{k-1} .
\end{align*}
Write $B_n^2=\sum_{k=1}^n \Sbep[X_k^2]$. It follows that
\begin{align*} & 1-\frac{ (x+\epsilon+c)^2 }{B_n^2}+\frac{ \sum_{k=1}^n\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1} }{B_n^2}\\
\le & 1-\eta_n +\frac{2}{B_n^2}\sum_{k=1}^n \left[X_k S_{k-1}^-\eta_{k-1} -X_kS_{k-1}^+\eta_{k-1} \right].
\end{align*}
Note $$\Sbep[X_k S_{k-1}^-\eta_{k-1} ]=\Sbep[X_k]\Sbep[ S_{k-1}^-\eta_{k-1} ]\le (x+\epsilon) \big(\Sbep[X_k]\big)^+,$$
$$\Sbep[-X_k S_{k-1}^+\eta_{k-1} ]=\Sbep[-X_k]\Sbep[ S_{k-1}^+\eta_{k-1} ]\le (x+\epsilon) \big(\Sbep[-X_k]\big)^+$$
and
\begin{align}\label{eqpppoofKIQ1}
& \Sbep\left[\sum_{k=1}^n\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1} \right] \nonumber\\
= & \Sbep\left[\Sbep\left[\sum_{k=1}^n\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1} \Big|X_1,\cdots,X_{n-1} \right]\right] \nonumber\\
= & \Sbep\left[\sum_{k=1}^{n-1}\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1} +\eta_{n-1}\Sbep[X_n^2-\Sbep[X_n^2] ] \right]\nonumber\\
=& \Sbep\left[\sum_{k=1}^{n-1}\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1} \right]=\cdots =0.
\end{align}
It follows that
\begin{align*}
& 1-\frac{ (x+\epsilon+c)^2 }{B_n^2}-\frac{ 2(x+\epsilon)\sum_{k=1}^n \big\{\big(\Sbep[X_k]\big)^++\big(\Sbep[-X_k]\big)^+\big\} }{B_n^2} \\
& \;\; \le \Sbep\left[ 1-\eta_n\right]\le \Capc\left(\max_{k\le n} |S_k|> x\right).
\end{align*}
By letting $\epsilon\to 0$, we obtain (\ref{eqKIQ1}). The proof of (i) is completed.
(ii) Redefine $Z_k$ and $\eta_k$ by $Z_k=g_{\epsilon}\left(S_k-x\right)$, $Z_0=0$, $\eta_k=\prod_{i=1}^k(1-Z_i)$.
Then $I\{S_k\ge x+\epsilon\}\le Z_k\le I\{S_k> x\}$. Also, $ S_{k-1}< x+\epsilon$ and $ S_k = S_{k-1} + X_k < x+\epsilon+c$ on the event $\{\eta_{k-1}\ne 0\}$. So
$$
S_{k-1} \eta_{k-1} + X_k \eta_{k-1}
= S_k \eta_k +S_k \eta_{k-1} Z_k
\le S_k \eta_k +(x+\epsilon+c) \eta_{k-1} Z_k.
$$
Taking the summation over $k$ yields
\begin{align*}
&\left(\sum_{k=1}^n \Sbep[X_k ]\right)\eta_n +\sum_{k=1}^n\left(X_k -\Sbep[X_k ]\right) \eta_{k-1} \\
\le & \sum_{k=1}^nX_k \eta_{k-1}
\le S_n \eta_n +(x+\epsilon+c) \left[1-\eta_n \right] \\
\le &(x+\epsilon) \eta_n +(x+\epsilon+c) \left[1-\eta_n \right]
\le (x+\epsilon+c) .
\end{align*}
Write $e_n =\sum_{k=1}^n \Sbep[X_k]$. It follows that
\begin{align*} 1-\frac{ (x+\epsilon+c) }{e_n}+\frac{ \sum_{k=1}^n\left(X_k-\Sbep[X_k]\right) \eta_{k-1} }{e_n}
\le 1-\eta_n.
\end{align*}
Note
\begin{align*}
\Sbep\left[\sum_{k=1}^n\left(X_k -\Sbep[X_k ]\right) \eta_{k-1} \right]
= \Sbep\left[\sum_{k=1}^{n-1}\left(X_k -\Sbep[X_k ]\right) \eta_{k-1} \right]=\cdots =0,
\end{align*}
similar to (\ref{eqpppoofKIQ1}). It follows that
$$ 1-\frac{ x+\epsilon+c }{e_n } \le \Sbep\left[ 1-\eta_n \right]\le \Capc\left(\max_{k\le n} S_k > x\right). $$
By letting $\epsilon\to 0$, we obtain (\ref{eqKIQ1}). The proof is completed. $\Box$
The following lemma on the bounds of the capacities via moments will be used in the paper.
\begin{lemma}[ \cite{Zhang Exponential}]\label{moment_v}
Let $X_1, X_2, \ldots, X_n$ be independent
random variables in $(\Omega, \mathscr{H}, \Capc)$. If $\Sbep[X_{k}] \leq 0$, $k=1,\ldots, n$, then there exists a constant $C>0$ such that
\begin{equation*}
\Capc(S_n\geq x)\leq C\frac{\sum_{k=1}^{n}\Sbep[X_k^2]}{x^2} \;\text{ for all } \; \forall x>0.
\end{equation*}
\end{lemma}
\section{The convergence of infinite series}\label{Sect Convergece}
\setcounter{equation}{0}
Our results on the convergence of the series $\sum_{n=1}^{\infty}$ are stated as three theorems. The first one gives the equivalency between the almost sure convergence and the convergence in capacity.
\begin{theorem}\label{th1} Let $\{X_n;n\geq1\}$ be a sequence of independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$, $S_n=\sum_{k=1}^n X_k$, and $S$ be a random variable in the measurable space $(\Omega, \mathcal{F})$.
\begin{description}
\item[(i)] If $\Capc$ is countably sub-additive, and
\begin{equation}\label{eqth1.1} \Capc\left(|S_n-S|\ge \epsilon\right)\to 0 \text{ as } n\to \infty \text{ for all } \epsilon>0,
\end{equation}
then
\begin{equation}\label{eqth1.2} \Capc\left(\left\{\omega: \lim_{n\to \infty} S_n(\omega)\ne S(\omega)\right\}\right)=0.
\end{equation}
When (\ref{eqth1.2}) holds, we call that $\sum_{n=1}^{\infty}X_n$ is almost surely convergent in capacity, and when (\ref{eqth1.1}) holds, we call that $\sum_{n=1}^{\infty}X_n$ is convergent in capacity.
\item[(ii)] If $\Capc$ is continuous, then (\ref{eqth1.2}) implies (\ref{eqth1.1}).
\end{description}
\end{theorem}
The second theorem gives the equivalency between the convergence in capacity and the convergence in distribution.
\begin{theorem}\label{th2} Let $\{X_n;n\geq1\}$ be a sequence of independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$, $S_n=\sum_{k=1}^n X_k$.
\begin{description}
\item[(i)] If there is a random variable $S$ in the measurable space $(\Omega, \mathcal{F})$ such that
\begin{equation}\label{eqth2.1} \Capc\left(|S_n-S|\ge \epsilon\right)\to 0 \text{ as } n\to \infty \text{ for all } \epsilon>0,
\end{equation}
and $S$ is tight under $\Sbep$, i.e., $\Sbep\left[I_{\{|S|\le x\}^c}\right]=\Capc(|S|> x)\to 0$ as $x\to \infty$,
then
\begin{equation}\label{eqth2.2} \Sbep\left[\phi(S_n)\right]\to \Sbep\left[\phi(S)\right],\;\; \phi\in C_b(\mathbb{R}),
\end{equation}
where $C_b(\mathbb R)$ is the space of bounded continuous functions on $\mathbb R$. When (\ref{eqth2.2}) holds, we call that $\sum_{n=1}^{\infty}X_n$ is convergent in distribution.
\item[(ii)] Suppose that there is a sub-linear space $(\widetilde{\Omega}, \widetilde{\mathscr{H}}, \widetilde{\mathbb E})$ and a random variable $\widetilde{S}$ on it such that $\widetilde{S}$ is tight under $\widetilde{\mathbb E}$, i.e., $\widetilde{\Capc}(|\widetilde{S}|> x)\to 0$ as $x\to \infty$, and
\begin{equation}\label{eqth2.3} \Sbep\left[\phi(S_n)\right]\to \widetilde{\mathbb E} \left[\phi(\widetilde{S})\right],\;\; \phi\in C_{b}(\mathbb{R}),
\end{equation}
then $S_n$ is a Cauchy sequence in capacity $\Capc$, namely
\begin{equation}\label{eqth2.4} \Capc\left(|S_n-S_m|\ge \epsilon \right)\to 0 \text{ as } n,m\to \infty \text{ for all } \epsilon>0.
\end{equation}
Furthermore, if $\Capc$ is countably sub-additive, then on the measurable space $(\Omega, \mathcal F)$ there is a random variable $S$ which is tight under $\Sbep$, such that (\ref{eqth1.1}) and (\ref{eqth1.2}) hold.
\end{description}
\end{theorem}
Recently, Xu and Zhang \cite{XuZhang2018} gave sufficient conditions for $\sum_{n=1}^{\infty} X_n$ to be convergent almost surely in capacity via three series theorem. The third theorem of us gives the sufficient and necessary conditions for $S_n$ to be a Cauchy sequence in capacity.
For any random variable $X$ and constant $c$, we denote $X^c=(-c)\vee(X\wedge c)$.
\begin{theorem}\label{th4} Let $\{X_n;n\geq1\}$ be a sequence of independent random variables in $(\Omega, \mathscr{H}, \Sbep)$, $S_n=\sum_{k=1}^n X_k$. Then $S_n$ will be a Cauchy sequence in capacity $\Capc$ if the following three conditions hold for some $c>0$.
\begin{description}
\item[\rm (S1) ] $\sum\limits_{n=1}^{\infty}\Capc(|X_n|>c)<\infty$,
\item[\rm (S2) ] $\sum\limits_{n=1}^{\infty}\Sbep[X_n^c]$ and $ \sum\limits_{n=1}^{\infty}\Sbep[-X_n^c]$ are both convergent,
\item[\rm (S3) ] $\sum\limits_{n=1}^{\infty}\Sbep\left[ \big(X_n^c-\Sbep[X_n^c]\big)^2\right] <\infty$ or/and $\sum\limits_{n=1}^{\infty}\Sbep\left[ \big(X_n^c+\Sbep[-X_n^c]\big)^2\right] <\infty$.
\end{description}
Conversely, if $S_n$ is a Cauchy sequence in capacity $\Capc$, then (S1),(S2) and (S3) will hold for all $c>0$.
\end{theorem}
From Theorem \ref{th4}, we have the following three series theorem on the sufficient and necessary conditions for the almost sure convergence of $\sum_{n=1}^{\infty} X_n$.
\begin{corollary}\label{three series} Let $\{X_n;n\geq1\}$ be a sequence of independent random variables in $(\Omega, \mathscr{H}, \Sbep)$. Suppose that $\Capc$ is countably sub-additive. Then $\sum_{n=1}^{\infty}X_n$ will converge almost surely in capacity if the three conditions (S1),(S2) and (S3) in Theorem \ref{th4} hold for some $c>0$.
Conversely, if $\Capc$ is continuous and $\sum_{n=1}^{\infty}X_n$ is convergent almost surely in capacity, then (i),(ii) and (iii) will hold for all $c>0$.
\end{corollary}
The sufficiency of (S1), (S2) and (S3) is proved by Xu and Zhang \cite{XuZhang2018}, and also follows from Theorem \ref{th4} and the second part of conclusion of Theorem \ref{th2} (ii). The necessity follows from Theorem \ref{th4} and Theorem \ref{th1} (ii).
\bigskip
The prove Theorems \ref{th1} and \ref{th2}. We need some more lemmas.
The first lemma is a version of Theorem 9 of Peng \cite{peng2010b}.
\begin{lemma}\label{lem1} Let $\{\bm Y_n; n\ge 1\}$ be a sequence of $d$-dimensional random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$. Suppose that $\bm Y_n$ is asymptotically tight, i.e.,
$$ \limsup_{n\to\infty}\Sbep\left[I_{\{\bm Y_n\|\le x\}^c}\right]=\limsup_{n\to\infty} \Capc\left(\|\bm Y_n\|> x\right)\to 0 \; \text{ as } x\to \infty. $$
Then for any subsequence $\{\bm Y_{n_k}\}$ of $\{\bm Y_n\}$, there exist further a subsequence $\{\bm Y_{n_{k^{\prime}}}\}$ of $\{\bm Y_{n_k}\}$ and a sub-linear expectation space $(\overline{\Omega}, \overline{\mathscr{H}}, \overline{\mathbb E})$ with a $d$-dimensional random variable $\bm Y$ on it such that
$$ \Sbep\left[\phi\left(\bm Y_{n_{k^{\prime}}}\right)\right]\to \overline{\mathbb E}\left[\phi(\bm Y)\right] \text{ for any } \phi\in C_b(\mathbb R^d) $$
and $\bm Y$ is tight under $\overline{\mathbb E}$.
\end{lemma}
{\bf Proof.} Let
$$ \mathbb E\left[\phi\right]=\limsup_{n\to \infty}\Sbep\left[\phi(\bm Y_n)\right], \;\; \phi\in C_b(\mathbb R^d). $$
Then $\mathbb E$ is a sub-linear expectation on the function space $C_b(\mathbb R^d)$ and is tight in sense that for any $\epsilon>0$, there is a compact set $K=\{\bm x:\|\bm x\|\le M\}$ for which
$\mathbb E\left[I_{K^c}\right]<\epsilon$. With the same argument as in the proof of Theorem 9 of Peng \cite{peng2010b}, there is a countable subset $\{\varphi_j\}$ of $C_b(\mathbb R^d)$ such that for each $\phi\in C_b(\mathbb R^d)$ and any $\epsilon>0$ one can find a $\varphi_j$ satisfying
\begin{equation}\label{eqprooflem1.1}\mathbb E\left[|\phi-\varphi_j|\right]<\epsilon. \end{equation}
On the other hand, for each $\varphi_j$, the sequence $\Sbep\left[\varphi_j(\bm Y_n)\right]$ is bounded and so there is a Cauchy subsequence. Note that the set $\{\varphi_j\}$ is countable. By the diagonal choice method, one can find a sequence $\{n_k\}\subset \{n\}$ such that $\Sbep\left[\varphi_j(\bm Y_{n_k})\right]$ is a Cauchy sequence for each $\varphi_j$. Now, we show that $\Sbep\left[\phi(\bm Y_{n_k})\right]$ is a Cauchy sequence for any $\phi\in C_b(\mathbb R^d)$. For any $\epsilon>0$, choose a $\varphi_j$ such that (\ref{eqprooflem1.1}) holds. Then
\begin{align*}
& \left|\Sbep\left[\phi(\bm Y_{n_k})\right]-\Sbep\left[\phi(\bm Y_{n_l})\right]\right| \\
\le & \left|\Sbep\left[\varphi_j(\bm Y_{n_k})\right]-\Sbep\left[\varphi_j(\bm Y_{n_l})\right]\right| \\
&+ \Sbep\left[\left|\phi(\bm Y_{n_k}) - \varphi_j(\bm Y_{n_k})\right|\right] + \Sbep\left[\left|\phi(\bm Y_{n_l}) -\varphi_j(\bm Y_{n_l})\right|\right].
\end{align*}
Taking the limits yields
$$
\limsup_{k,l\to\infty} \left|\Sbep\left[\phi(\bm Y_{n_k})\right]-\Sbep\left[\phi(\bm Y_{n_l})\right]\right|
\le 0+ 2\mathbb E\left[|\phi-\varphi_j|\right]<2\epsilon.
$$
Hence $\Sbep\left[\phi(\bm Y_{n_k})\right]$ is a Cauchy sequence for any $\phi\in C_b(\mathbb R^d)$, and then
\begin{equation}\label{eqprooflem1.2} \lim_{k\to \infty} \Sbep\left[\phi(\bm Y_{n_k})\right] \; \text{ exists and is finite for any } \phi\in C_b(\mathbb R^d).
\end{equation}
Now, let $\overline{\Omega}=\mathbb R^d$, $\overline{\mathscr{H}}=C_{l,lip}(\mathbb R^d)$. Define
$$ \overline{\mathbb E}\left[\varphi\right]=\limsup_{k\to \infty} \Sbep\left[\varphi(\bm Y_{n_k})\right], \;\; \varphi\in C_{l,lip}(\mathbb R^d). $$
Then $(\overline{\Omega}, \overline{\mathscr{H}},\overline{\mathbb E})$ is a sub-linear expectation space. Define the random variable $\bm Y$ by $\bm Y(\bm x)=\bm x$, $\bm x\in \overline{\Omega}$.
From (\ref{eqprooflem1.2}) it follows that
$$ \lim_{k\to \infty} \Sbep\left[\phi(\bm Y_{n_k})\right]=\overline{\mathbb E}\left[\phi(\bm Y)\right] \text{ for any } \phi\in C_b(\mathbb R^d).
$$
The proof is completed. $\Box$
\begin{lemma}\label{lem2} Let $X$ and $Y$ be random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$. Suppose that $Y$ and $X$ are independent ($Y$ is independent to $X$, or $X$ is independent to $Y$), and $X$ is tight, i.e. $\Capc(|X|\ge x)\to 0$ as $x> \infty$. If $X+Y\overset{d}=X$, then $\Capc(|Y|\ge \epsilon)=0$ for all $\epsilon>0$.
\end{lemma}
{\bf Proof.} Without loss of generality, we assume that $Y$ is independent to $X$. We can find a sub-linear expectation space $(\Omega^{\prime}, \mathscr{H}^{\prime}, \Sbep^{\prime})$ on which there are independent random variables $X_1, Y_1,Y_2, \cdots, Y_n,\cdots$ such that $X_1\overset{d}=X$, $Y_i\overset{d}=Y$, $i=1,2,\cdots,$. Without loss of generality, assume $(\Omega^{\prime}, \mathscr{H}^{\prime}, \Sbep^{\prime})=(\Omega, \mathscr{H}, \Sbep)$. Let $S_k=\sum_{j=1}^k Y_k$. Then $X_1+S_k\overset{d}=X$. So,
\begin{align}\label{eqprooflem2.1} \max_{k\le n} \Capc(|S_k|>x_0)\le & \max_{k\le n} \Capc(|X_1+S_k|>x_0/2)+ \Capc(|X_1|>x_0/2) \nonumber
\\
\le & \Sbep\left[g_{1/2}\left(\frac{|X_1+S_k|}{x_0}\right)\right]+ \Sbep\left[g_{1/2}\left(\frac{|X_1|}{x_0}\right)\right]\nonumber\\
=& 2 \Sbep\left[g_{1/2}\left(\frac{|X|}{x_0}\right)\right]\le 2\Capc(|X|\ge x_0/4)<1/4
\end{align}
for $x_0$ large enough, where $g_{\epsilon}$ is defined as in (\ref{eqproofLIQ.1}).
By Lemma \ref{LevyIneq},
\begin{equation}\label{eqprooflem2.2} \Capc(\max_{k\le n}|S_k|>2x_0+\epsilon)\le \frac{4}{3} \max_n \Capc(|S_n|>x_0) \le \frac{4}{3} \cdot 2 \Capc(|X|\ge x_0/4)<\frac{1}{3} .
\end{equation}
It follows that for any $\epsilon>0$,
$$ \Capc(\max_{k\le n}|Y_k|>4x_0+2\epsilon) <\frac{1}{3}. $$
Let $Z_k=g_{\epsilon} (|Y_k|-4x_0-2\epsilon)$, where $g_{\epsilon}$ is defined as in (\ref{eqproofLIQ.1}). Denote $q=\Capc(|Y_1|>4x_0+3\epsilon)$. Then $Z_1,Z_2\cdots, Z_n$ are independent and identically distributed with $\{|Y_k|>4x_0+3\epsilon\}\le Z_k\le \{|Y_k|>4x_0+2\epsilon\}$ and $\Sbep[Z_1]\ge \Capc(|Y_1|>4x_0+3\epsilon)=q$. Then by Lemma \ref{KolIneq} (ii),
\begin{equation} \label{eqprooflem2.3} \frac{1}{3}> \Capc\left(\sum_{k=1}^n Z_k\ge 1\right)
\ge 1-\frac{1+1}{\sum_{k=1}^n \Sbep[Z_k]}\ge 1-\frac{2}{nq}.
\end{equation}
The above inequality holds for all $n$, which is impossible unless $q=0$. So we conclude that
$$\Capc(|Y_1|>4x_0+\epsilon) = 0 \;\; \text{ for any } \epsilon>0. $$
Now, let $\widetilde{Y}_k=(-5x_0)\vee Y_k \wedge (5x_0)$, $\widetilde{S}_k=\sum_{i=1}^k \widetilde{Y}_i$. Then $\widetilde{Y}_1, \cdots, \widetilde{Y}_n$ are independent and identically distributed bounded random variables, $\Capc(\widetilde{Y}_k\ne Y_k)=0$ and $\Capc(\widetilde{S}_k\ne S_k)=0 $. If $\Sbep[\widetilde{Y}_1]>0$, then by Lemma \ref{KolIneq} (ii) again,
$$
\Capc(\max_{k\le n}S_k\ge 3x_0)=\Capc(\max_{k\le n}\widetilde{S}_k\ge 3x_0)\ge 1 -\frac{3x_0+5x_0}{n \Sbep[\widetilde{Y}_1]}, $$
which contradicts to (\ref{eqprooflem2.2}) when $n> 12 x_0/\Sbep[\widetilde{Y}_1]$. Hence, $\Sbep[\widetilde{Y}_1]\le 0$. Similarly, $\Sbep[-\widetilde{Y}_1]\le 0$. We conclude that $\Sbep[\widetilde{Y}_1]=\Sbep[-\widetilde{Y}_1]= 0$. Now, if $ \Sbep[\widetilde{Y}_1^2]\ne 0$, then by Lemma \ref{KolIneq} (i) we have
$$
\Capc(\max_{k\le n}|S_k|\ge 3x_0)\ge 1 -\frac{(3x_0+5x_0)^2}{n \Sbep[\widetilde{Y}_1^2]}, $$
which contradicts to (\ref{eqprooflem2.2}) when $n> 96x_0^2/\Sbep[\widetilde{Y}_1^2]$. We conclude that $\Sbep[\widetilde{Y}_1^2]=0$.
Finally, for any $\epsilon>0$ ($\epsilon<5x_0$),
$$ \Capc\left(|Y|\ge \epsilon\right)\le \frac{\Sbep[Y^2\wedge(5x_0)^2]}{\epsilon^2} = \frac{\Sbep[\widetilde{Y}_1^2]}{\epsilon^2}=0. $$
The proof is completed. $\Box$
\bigskip
{\bf Proof of Theorem \ref{th1}.} (i) Let $\epsilon_k=1/2^k$, $\delta_k=1/4^k$. By (\ref{eqth1.1}), there exits a sequence $n_1<n_2<\cdots<n_k\to \infty$, such that
\begin{equation}\label{eqproofth1.1} \max_{n\ge n_k} \Capc\left(|S_n-S|\ge \epsilon_k\right)<\delta_k.
\end{equation}
By the countably sub-additivity of $\Capc$, we have
\begin{align*}
&\Capc\left(\limsup_{k\to \infty}|S_{n_k}-S|>0\right)
\le
\Capc\left(\bigcap_{m=1}^{\infty} \bigcup_{k=m}^{\infty} \{|S_{n_k}-S|\ge \epsilon_k\}\right)\\
\le & \sum_{k=m}^{\infty}\Capc\left( |S_{n_k}-S|\ge \epsilon_k \right)\le \sum_{k=m}^{\infty}\delta_k \to 0 \text{ as } m\to \infty.
\end{align*}
By (\ref{eqproofth1.1}), $\max_{n\ge n_k} \Capc\left(|S_n-S_{n_{k+1}}|\ge 2\epsilon_k\right)<2\delta_k<1/2$. Apply the Levy inequality (\ref{eqLIQ2}) yields
\begin{equation}\label{eqproofth1.2} \Capc\left(\max_{n_k\le n\le n_{k+1}}|S_n-S_{n_k}|> 5\epsilon_k\right)\le 2 \Capc\left(|S_{n_{k+1}}-S_{n_k}|> 2\epsilon_k\right)<4 \delta_k.
\end{equation}
By the countably sub-additivity of $\Capc$ again,
\begin{align*}
&\Capc\left(\limsup_{k\to \infty}\max_{n_k\le n\le n_{k+1}}|S_n-S_{n_k}|>0\right)\\
\le &
\Capc\left(\bigcap_{m=1}^{\infty} \bigcup_{k=m}^{\infty} \{\max_{n_k\le n\le n_{k+1}}|S_n-S_{n_k}|\ge 5\epsilon_k\}\right)\\
\le & \sum_{k=m}^{\infty}\Capc\left( \max_{n_k\le n\le n_{k+1}}|S_n-S_{n_k}|\ge 5\epsilon_k \right)\le 4\sum_{k=m}^{\infty}\delta_k \to 0 \text{ as } m\to \infty.
\end{align*}
It follows that
\begin{align*}
& \Capc\left(\limsup_{n\to \infty}|S_n-S|>0\right)\\
\le & \Capc\left(\limsup_{k\to \infty}|S_{n_k}-S|>0\right)+\Capc\left(\limsup_{k\to \infty}\max_{n_k\le n\le n_{k+1}}|S_n-S_{n_k}|>0\right)=0.
\end{align*}
(\ref{eqth1.2}) follows.
(ii) From (\ref{eqth1.2}) and the continuity of $\Capc$, it follows that for any $\epsilon>0$,
\begin{align*}
0\ge \Capc\left(\bigcap_{n=1}^{\infty}\bigcup_{m=n}^{\infty}\{|S_m-S|\ge \epsilon\}\right)= & \lim_{n\to \infty}\Capc\left(\bigcup_{m=n}^{\infty}\{|S_m-S|\ge \epsilon\}\right) \\
\ge & \limsup_{n\to \infty}\Capc\left( |S_n-S|\ge \epsilon \right).
\end{align*}
(\ref{eqth1.1}) follows. The proof is completed. $\Box$
{\bf Proof of Theorem \ref{th2}.} (i) We first show that (\ref{eqth2.2}) holds for any bounded uniformly continuous function $\phi$. For any $\epsilon>0$, there is a $\delta>0$ such that $|\phi(x)-\phi(y)|<\epsilon$ when $|x-y|<\delta$. It follows that
$$ \left|\Sbep\left[\phi(S_n)\right]-\Sbep\left[\phi(S)\right]\right|\le \epsilon+ 2\sup_x|\phi(x)|\Capc\left(|S_n-S|\ge \delta\right). $$
By letting $n\to \infty$ and the arbitrariness of $\epsilon>0$, we obtain (\ref{eqth2.2}). Now, suppose that $\phi$ is a bounded continuous function. Then for any $N>1$, $\phi((-N)\vee x\wedge N)$ is a bounded uniformly continuous function. Hence
$$ \lim_{n\to \infty}\Sbep\left[\phi((-N)\vee S_n\wedge N)\right]=\Sbep\left[\phi((-N)\vee S\wedge N)\right]. $$
On the other hand,
$$\left|\Sbep\left[\phi((-N)\vee S\wedge N)\right]-\Sbep\left[\phi( S )\right]\right|\le 2\sup_x\big|\phi(x)\big|\Capc\left(|S|>N\right)\to 0 \text{ as } N\to \infty, $$
and
\begin{align*}
&\limsup_{n\to \infty} \left|\Sbep\left[\phi((-N)\vee S_n\wedge N)\right]-\Sbep\left[\phi( S_n )\right]\right|\\
\le & 2\sup_x|\phi(x)|\limsup_{n\to \infty} \Capc\left(|S_n|\ge N\right)\le 2\sup_x\big|\phi(x)\big|\limsup_{n\to \infty} \Sbep\left[g_1\left(\frac{|S_n|}{N}\right)\right]\\
=& 2\sup_x|\phi(x)| \Sbep\left[g_1\left(\frac{|S|}{N}\right)\right]\le 2\sup_x\big|\phi(x)\big|\limsup_{n\to \infty} \Capc\left(|S|\ge N/2\right)\to 0 \text{ as } N\to \infty,
\end{align*}
where $g_{\epsilon}$ is defined as in (\ref{eqproofLIQ.1}). Hence, (\ref{eqth2.2}) holds for a bounded continuous function $\phi$.
(ii) Note
$$ \Capc\left(|S_n-S_m|\ge 2x\right)\le \Capc\left(|S_n|\ge x\right)+\Capc\left(|S_m|\ge x\right). $$
It follows that
\begin{align*}
&\limsup_{m\ge n\to \infty}\Capc\left(|S_n-S_m|\ge 2x\right)\le 2\limsup_{n\to \infty} \Capc\left(|S_n|\ge x\right) \\
\le & 2\limsup_{n\to \infty} \Sbep\left[g_1\left(\frac{|S_n|}{x}\right)\right]=\widetilde{\mathbb E} \left[g_1\left(\frac{|\widetilde{S}|}{x}\right)\right]
\le 2\Capc\left(|\widetilde{S}|\ge x/2\right) \to 0 \text{ as } x\to \infty.
\end{align*}
Write $\bm Y_{n,m}=(S_n, S_m-S_n)$, then the sequence $\{\bm Y_{n,m}; m\ge n\}$ is asymptotically tight, i.e.,
$$\limsup_{m\ge n\to \infty} \Capc\left(\|\bm Y_{n,m}\|\ge x\right)\to 0\; \text{ as } x \to \infty. $$
By Lemma \ref{lem1}, for any subsequence $(n_k,m_k)$ of $(n,m)$, there is further a subsequence $(n_{k^{\prime}},m_{k^{\prime}})$ of $(n_k,m_k)$ and a sub-linear expectation space $(\overline{\Omega}, \overline{\mathscr{H}}, \overline{\mathbb E})$ with a random vector $\bm Y=(Y_1,Y_2)$ such that
\begin{equation}\label{eqproofth2.3} \Sbep\left[\phi\left(\bm Y_{n_{k^{\prime}},m_{k^{\prime}}}\right)\right]\to \overline{\mathbb E}\left[\phi(\bm Y)\right],\;\;\phi\in C_b(\mathbb R^2).
\end{equation}
Note that $S_{m_{k^{\prime}}}- S_{n_{k^{\prime}}}$ is independent to $S_{n_{k^{\prime}}}$. By Lemma 4.4 of Zhang \cite{Zhang Donsker}, $Y_2$ is independent to $Y_1$ under $ \overline{\mathbb E}$.
Let $\phi\in C_{b,Lip}(\mathbb R)$. By (\ref{eqproofth2.3}),
\begin{equation}\label{eqproofth2.4}\Sbep\left[\phi\left(S_{m_{k^{\prime}}}\right)\right]\to \overline{\mathbb E}\left[\phi(Y_1+Y_2)\right], \;\; \Sbep\left[\phi\left(S_{n_{k^{\prime}}}\right)\right]\to \overline{\mathbb E}\left[\phi(Y_1)\right]
\end{equation}
and
\begin{equation}\label{eqproofth2.5} \Sbep\left[\phi\left(S_{m_{k^{\prime}}}-S_{n_{k^{\prime}}}\right)\right]\to \overline{\mathbb E}\left[\phi(Y_2)\right].
\end{equation}
On the other hand, by (\ref{eqth2.3}),
\begin{equation}\label{eqproofth2.6} \Sbep\left[\phi\left(S_{m_{k^{\prime}}}\right)\right] \to \widetilde{\mathbb E} \left[\phi(\widetilde{S})\right]
\; \text{ and } \Sbep\left[\phi\left(S_{n_{k^{\prime}}}\right)\right] \to \widetilde{\mathbb E} \left[\phi(\widetilde{S})\right].
\end{equation}
Combing (\ref{eqproofth2.4}) and (\ref{eqproofth2.6}) yields
$$ \overline{\mathbb E}\left[\phi(Y_1+Y_2)\right] =\overline{\mathbb E}\left[\phi(Y_1)\right]=\widetilde{\mathbb E} \left[\phi(\widetilde{S})\right],\;\;\phi\in C_{b,Lip}(\mathbb R). $$
Hence, by Lemma \ref{lem2}, we obtain $\overline{\mathbb V}(|Y_2|\ge \epsilon)=0$ for all $\epsilon>0$. By choosing $\phi\in C_{b,Lip}(\mathbb R)$ such that $I_{|x|\ge \epsilon}\le \phi(x)\le I_{|x|\ge \epsilon/2}$ in
(\ref{eqproofth2.5}), we have
$$
\limsup_{k^{\prime}\to \infty} \Capc\left(\left|S_{m_{k^{\prime}}}-S_{n_{k^{\prime}}}\right|\ge \epsilon\right)\le \overline{\mathbb V}(|Y_2|\ge \epsilon/2)=0.
$$
So, we conclude that for any subsequence $(n_k,m_k)$ of $(n,m)$, there is a further a subsequence $(n_{k^{\prime}},m_{k^{\prime}})$ of $(n_k,m_k)$ such that
$$ \Capc\left(\left|S_{m_{k^{\prime}}}-S_{n_{k^{\prime}}}\right|\ge \epsilon\right)\to 0 \text{ for all }\epsilon>0. $$
Hence (\ref{eqth2.4}) is proved.
Next, suppose that $\Capc$ is countably sub-additive. Let $\epsilon_k=1/2^k$, $\delta_k=1/3^k$. By (\ref{eqth2.4}), there is a sequence $n_1<n_2<\cdots<n_k<\cdots$ such that
$$ \Capc\left(|S_{n_{k+1}}-S_{n_k}|\ge \epsilon_k\right)\le \delta_k. $$
Let $A=\{\omega: \sum_{k=1}^{\infty}|S_{n_{k+1}}-S_{n_k}|<\infty\}$. Then
\begin{align*} \Capc\left(A^c\right)\le & \Capc\left(\sum_{k=K}^{\infty}|S_{n_{k+1}}-S_{n_k}|\ge \sum_{k=K}^{\infty}\epsilon_k\right)\\
\le & \sum_{k=K}^{\infty} \Capc\left(|S_{n_{k+1}}-S_{n_k}|\ge \epsilon_k\right)\le \sum_{k=K}^{\infty}\delta_k \to 0 \text{ as } K\to \infty.
\end{align*}
Define $S=\lim_{k\to \infty} S_{n_k}$ on $A$, and $S=0$ on $A^c$. Then
\begin{align*} \Capc\left(|S-S_{n_k}|\ge 1/2^{k-1}\right)\le &\Capc(A^c)+ \Capc\left(A, \sum_{i=k}^{\infty}|S_{n_{i+1}}-S_{n_i}|\ge \sum_{i=k}^{\infty}\epsilon_i\right)\\
\le & \sum_{i=k}^{\infty} \Capc\left(|S_{n_{i+1}}-S_{n_i}|\ge \epsilon_i\right)\le \sum_{i=k}^{\infty}\delta_i \to 0 \text{ as } k\to \infty.
\end{align*}
On the other hand, by (\ref{eqth2.4}),
$$\Capc\left(|S_n-S_{n_k}|\ge \epsilon\right)\to 0 \text{ as } n, n_k\to \infty. $$
Hence
$$\Capc\left(|S_n-S|\ge \epsilon\right)\le \Capc\left(|S_n-S_{n_k}|\ge \epsilon/2\right)+\Capc\left(|S-S_{n_k}|\ge \epsilon/2\right)\to 0. $$
(\ref{eqth1.1}) is proved. Further,
\begin{align*}
\Capc\left(|S|\ge 2M\right)\le & \limsup_n \Capc\left(|S_n|\ge M\right)+\limsup_n \Capc\left(|S_n-S|\ge M\right)\\
\le & \widetilde{\mathbb V}\left(|\widetilde{S}|\ge M/2\right)\to 0
\text{ as } M\to \infty.
\end{align*}
So, $S$ is tight.
Finally, (\ref{eqth1.2}) follows from Theorem \ref{th1}. $\Box$
\bigskip
For showing Theorem \ref{th4}, we need a more lemma.
\begin{lemma}\label{lem3} Let $\{X_n; n\ge 1\}$ be a sequence of independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$ with $ |X_k| \le c$, $\Sbep[X_k]\ge 0$ and $\Sbep[-X_k]\ge 0$, $k=1,2,\cdots$. Let $S_k=\sum_{i=1}^k X_i$. Suppose
\begin{equation}\label{eqlem3.1} \lim_{x\to \infty}\lim_{n\to \infty} \mathbb{V}\left(\max_{k\le n} |S_k|>x \right)<1.
\end{equation}
Then $\sum_{n=1}^{\infty}\Sbep[X_n]$, $\sum_{n=1}^{\infty}\Sbep[-X_n]$ and $\sum_{n=1}^{\infty}\Sbep[X_n^2]$ are convergent.
\end{lemma}
{\bf Proof.} By (\ref{eqlem3.1}), there exist $0<\beta<1$, $x_0>0$ and $n_0$, such that
$$\mathbb{V}\left(\max_{k\le n} |S_k|>x \right) <\beta, \;\;\text{for all } x\ge x_0, \; n\ge n_0. $$
By (\ref{eqKIQ2.1}),
$$ \sum_{k=1}^n \Sbep[X_k]\le \frac{x+c}{1-\beta},\;\; \text{for all } x\ge x_0, \; n\ge n_0. $$
So $\sum_{k=1}^{\infty} \Sbep[X_k]$ is convergent. Similarly, $\sum_{k=1}^{\infty} \Sbep[-X_k]$ is convergent.
Now, by (\ref{eqKIQ1}),
\begin{align*} \sum_{k=1}^n \Sbep[X_k^2] \le & \frac{(x+c)^2+2x\sum_{k=1}^n \big\{\big(\Sbep[X_k]\big)^++\big(\Sbep[-X_k]\big)^+\big\}}{1-\beta} \\
\le & \frac{(x+c)^2+2x\sum_{k=1}^{\infty} \big\{ \Sbep[X_k] + \Sbep[-X_k] \big\}}{1-\beta},\;\; \text{for all } x\ge x_0, \; n\ge n_0.
\end{align*}
So $\sum_{n=1}^{\infty} \Sbep[X_n^2]$ is convergent. The proof is completed.
$\Box$
\bigskip
{\bf Proof of Theorem \ref{th4}.} (i) By Lemma \ref{moment_v} and the condition (S3),
\begin{align*}
& \Capc\left(S_n-S_m-\sum_{k=m+1}^n \Sbep[X_k]\ge \epsilon\right)\\
\le & C\frac{\sum_{k=m+1}^n\Sbep\left[(X_k-\Sbep[X_k])^2\right]}{\epsilon^2}\to 0 \text{ as } n\ge m\to \infty.
\end{align*}
The convergence of $\sum_{n=1}^{\infty} \Sbep[X_n] $ implies $\sum_{k=m+1}^n \Sbep[X_k]\to 0$. It follows that
$$ \lim_{n\ge m\to \infty}\Capc\left(S_n-S_m \ge \epsilon\right)=0\; \text{ for all }\epsilon>0. $$
On the other hand, note $\Sbep[X_k]+\Sbep[-X_k]\ge 0$. The condition (S2) implies $\sum_{n=1}^{\infty}\left(\Sbep[X_k]+\Sbep[-X_k]\right)<\infty$, and then $\sum_{n=1}^{\infty}\left(\Sbep[X_k]+\Sbep[-X_k]\right)^2<\infty$. Hence, by the condition (S3) and the fact that $\Sbep\left[(-X_k-\Sbep[-X_k])^2\right]\le \Sbep\left[(X_k-\Sbep[X_k])^2\right]+(\Sbep[X_k]+\Sbep[-X_k])^2$,
$$\sum_{n=1}^{\infty}\Sbep\left[(-X_n-\Sbep[-X_n])^2\right]<\infty. $$
By considering $-X_n$ instead of $X_n$, we have
$$ \lim_{n\ge m\to \infty}\Capc\left(-S_n+S_m \ge \epsilon\right)=0\; \text{ for all }\epsilon>0. $$
It follows that (\ref{eqth2.4}) holds, i.e., $S_n$ is a Cauchy sequence in capacity $\Capc$.
(ii) Suppose that $S_n$ is a Cauchy sequence in capacity $\Capc$. Similar to (\ref{eqproofth1.2}), by applying the Levy inequality (\ref{eqLIQ2}) we have
\begin{equation}\label{eqproofth4.1} \lim_{n\ge m\to \infty} \Capc\left(\max_{m\le k\le n} |S_k-S_m|> \epsilon \right)=0 \text{ for all } \epsilon>0.
\end{equation}
Then
\begin{equation}\label{eqproofth4.2}\lim_{n\ge m\to \infty} \Capc\left(\max_{m\le k\le n} |X_k|\ge c \right)=0 \text{ for all } c>0.
\end{equation}
Write $v_k=\Capc\left( |X_k|\ge 2c \right)$. Similar to (\ref{eqprooflem2.3}), we have for $m_0$ large enough and all $n\ge m\ge m_0$,
\begin{align*}
\frac{1}{3}>\Capc\left(\max_{m\le k\le n} |X_k|\ge c \right)\ge 1-\frac{2}{ \sum_{k=m+1}^n v_k}.
\end{align*}
It follows that $\sum_{k=1}^{\infty} v_k<\infty$. The condition (S1) is satisfied for all $c>0$.
Next, we consider (S3). Write $X_n^c=(-c)\vee X_n\wedge c$ and $S_n^c=\sum_{k=1}^n X_k^c$. Note on the event $\{\max_{m\le k\le n} |X_k|< c\}$, $X_k^c=X_k$, $k=m+1,\cdots, n$. By (\ref{eqproofth4.1}) and (\ref{eqproofth4.2}),
\begin{equation}\label{eqproofth4.3} \lim_{n\ge m\to \infty} \Capc\left(\max_{m\le k\le n} |S_k^c-S_m^c|> \epsilon \right)=0 \text{ for all } \epsilon>0.
\end{equation}
Let $Y_1, Y_1^{\prime}, Y_2, Y_2^{\prime},\cdots, Y_n, Y_n^{\prime}, \cdots $ be independent random variables under the sub-linear expectation $\Sbep$ with $Y_k\overset{d}=Y_k^{\prime}\overset{d}= X_k^c$, $k=1,2,\cdots$. Then
$$\{Y_{m+1},\cdots, Y_n \} \overset{d}=\{Y_{m+1}^{\prime},\cdots, Y_n^{\prime}\} \overset{d}=\{X_{m+1}^c,\cdots, X_n^c\}. $$
Let $T_k=\sum_{i=1}^k Y_i$ and $T_k^{\prime}=\sum_{i=1}^k Y_i^{\prime}$. By (\ref{eqproofth4.3}),
\begin{align}\label{eqproofth4.4} & \lim_{n\ge m\to \infty} \Capc\left(\max_{m\le k\le n} |T_k-T_m|> \epsilon \right)\nonumber \\
= &\lim_{n\ge m\to \infty} \Capc\left(\max_{m\le k\le n} |T_k^{\prime}-T_m^{\prime}|> \epsilon \right)=0 \text{ for all } \epsilon>0.
\end{align}
Write $\widetilde{Y}_n=Y_n-Y_n^{\prime}$ and $\widetilde{T}_n=\sum_{k=1}^n \widetilde{Y}_k$. Then $\{\widetilde{Y}_n;n\ge 1\}$ is a sequence of independent random variables with $\Capc(|\widetilde{Y}_n|> 3c)=0$. Without loss of generality, we can assume $|\widetilde{Y}_n|\le 3c$ for otherwise we can replace $\widetilde{Y}_n$ by $(-3c)\vee \widetilde{Y}_n \wedge(3c)$. By (\ref{eqproofth4.4}),
$$ \lim_{n\ge m\to \infty} \mathbb{V}\left(\max_{m\le k\le n} |\widetilde{T}_k-\widetilde{T}_m|>2\epsilon \right)=0 \text{ for all } \epsilon>0.$$
Note $\Sbep[-\widetilde{Y}_k]=\Sbep[\widetilde{Y}_k]=(\Sbep[X_k^c]+\Sbep[-X_k^c])/2\ge 0$. By Lemma \ref{lem3},
$$ \sum_{n=1}^{\infty} \left(\Sbep[X_n^c]+\Sbep[-X_n^c]\right)\;\text{ and } \sum_{n=1}^{\infty} \Sbep[\widetilde{Y}_n^2] \text{ are convergent}. $$
Note
\begin{align*}
\Sbep\left[\widetilde{Y}_n^2|Y_n\right]\ge & \big(Y_n-\Sbep[Y_n]\big)^2+\Sbep\left[\big(Y_n^{\prime}-\Sbep[Y_n^{\prime}]\big)^2\right]\\
&+2\big(Y_n-\Sbep[Y_n]\big)^-\cSbep\left[Y_n^{\prime}-\Sbep[Y_n^{\prime}] \right].
\end{align*}
So
\begin{align*}
\Sbep\left[\widetilde{Y}_n^2\right]\ge & 2\Sbep\big[ \big(X_n^c-\Sbep[X_n^c]\big)^2\big]-2 \{\Sbep[X_n^c]+\Sbep[-X_n^c]\} \Sbep\left[\big(X_n^c-\Sbep[X_n^c]\big)^-\right]
\\
\ge & 2\Sbep\big[ \big(X_n^c-\Sbep[X_n^c]\big)^2\big]-2c \{\Sbep[X_n^c]+\Sbep[-X_n^c]\}.
\end{align*}
It follows that
\begin{equation} \label{eqproofth4.5}
\sum_{n=1}^{\infty}\Sbep\big[ \big(X_n^c-\Sbep[X_n^c]\big)^2\big]<\infty. \end{equation}
Since $\Sbep\big[ \big(-X_n^c-\Sbep[-X_n^c]\big)^2\big]\le \Sbep\big[ \big(X_n^c-\Sbep[X_n^c]\big)^2\big]+ \big(\Sbep[X_n^c+\Sbep[-X_n^c]\big)^2$, we also have
$$ \sum_{n=1}^{\infty}\Sbep\big[ \big(-X_n-\Sbep[-X_n]\big)^2\big]<\infty. $$
The condition (S3) is proved.
Finally, we consider (S2). For any $\epsilon>0$, when $m,n$ are large enough, $\sum_{k=m+1}^n \big(\Sbep[X_n^c]+\Sbep[-X_n^c]\big)<\epsilon$. By (\ref{eqproofth4.5}) and Lemma \ref{moment_v},
\begin{align*}
&\Capc\left(S_n^c-S_m^c-\sum_{k=m+1}^n \frac{\Sbep[X_k^c]-\Sbep[-X_k^c]}{2}\ge \epsilon\right)\\
= & \Capc\left(S_n^c-S_m^c-\sum_{k=m+1}^n \Sbep[X_k^c] \ge \epsilon-\sum_{k=m+1}^n \frac{\Sbep[-X_k^c]+\Sbep[X_k^c]}{2}\right)\\
\le & C \frac{\sum_{k=m+1}^n \Sbep\big[ \big(X_k^c-\Sbep[X_k^c]\big)^2\big]}{(\epsilon/2)^2}\to 0 \text{ as } n\ge m\to \infty.
\end{align*}
Similarly, by considering $-X_k^c$ instead of $X_k^c$ we have
\begin{align*}
\Capc\left(-S_n^c+S_m^c-\sum_{k=m+1}^n \frac{\Sbep[-X_k^c]-\Sbep[X_k^c]}{2}\ge \epsilon\right) \to 0 \text{ as } n\ge m\to \infty.
\end{align*}
It follows that, for any $\epsilon>0$,
$$
\Capc\left(\left|S_n^c-S_m^c-\sum_{k=m+1}^n \frac{\Sbep[X_k^c]-\Sbep[-X_k^c]}{2}\right|\ge \epsilon\right) \to 0 \text{ as } n\ge m\to \infty,
$$
which, together with (\ref{eqproofth4.3}), implies
$$ \sum_{k=m+1}^n \frac{\Sbep[X_k^c]-\Sbep[-X_k^c]}{2} \to 0 \text{ as } n\ge m\to \infty. $$
Hence, $\sum_{n=1}^{\infty} \big(\Sbep[X_k^c]-\Sbep[-X_k^c]\big)$ is convergent. Note that $\sum_{n=1}^{\infty} \big(\Sbep[X_k^c]+\Sbep[-X_k^c]\big)$ is convergent. We conclude that both
$\sum_{n=1}^{\infty} \Sbep[X_k^c]$ and $\sum_{n=1}^{\infty} \Sbep[-X_k^c]$ are convergent. The proof of (ii) is completed. $\Box$.
\section{Central limit theorem}\label{Sect CLT}
\setcounter{equation}{0}
In this section, we consider the sufficient and necessary conditions for the central limit theorem. We first recall the definition of G-normal random variables which is introduced by Peng \cite{peng2008a, peng2010}.
\begin{definition}\label{def4.1} ({\em G-normal random variable})
For $0\le \underline{\sigma}^2\le \overline{\sigma}^2<\infty$, a random variable $\xi$ in a sub-linear expectation space $(\widetilde{\Omega}, \widetilde{\mathscr H}, \widetilde{\mathbb E})$ is called a normal $N\big(0, [\underline{\sigma}^2, \overline{\sigma}^2]\big)$ distributed random variable (written as $\xi \sim N\big(0, [\underline{\sigma}^2, \overline{\sigma}^2]\big)$ under $\widetilde{\mathbb E}$), if for any $\varphi\in C_{l,Lip}(\mathbb R)$, the function $u(x,t)=\widetilde{\mathbb E}\left[\varphi\left(x+\sqrt{t} \xi\right)\right]$ ($x\in \mathbb R, t\ge 0$) is the unique viscosity solution of the following heat equation:
\begin{equation}\label{eqheatequation}\partial_t u -G\left( \partial_{xx}^2 u\right) =0, \;\; u(0,x)=\varphi(x),
\end{equation}
where $G(\alpha)=\frac{1}{2}(\overline{\sigma}^2 \alpha^+ - \underline{\sigma}^2 \alpha^-)$.
\end{definition}
That $\xi$ is a normal distributed random variable is equivalent to that, if $\xi^{\prime}$ is an independent copy of $\xi$ (i.e., $\xi^{\prime}$ is independent to $\xi$ and $\xi\overset{d}=\xi^{\prime})$, then
\begin{equation}\label{eqnormal} \widetilde{\mathbb E}\left[\varphi(\alpha \xi+\beta \xi^{\prime})\right]
=\widetilde{\mathbb E}\left[\varphi\big(\sqrt{\alpha^2+\beta^2}\xi\big)\right], \;\;
\forall \varphi\in C_{l,Lip}(\mathbb R) \text{ and } \forall \alpha,\beta\ge 0,
\end{equation}
(cf. Definition II.1.4 and Example II.1.13 of Peng \cite{peng2010}). We also write $\eta\overset{d}= N\big(0, [\underline{\sigma}^2, \overline{\sigma}^2]\big)$ if $\eta\overset{d}=\xi$ (as defined in Definition \ref{def1.2} (i)) and $\xi \sim N\big(0, [\underline{\sigma}^2, \overline{\sigma}^2]\big)$ (as defined in Definition \ref{def4.1}). By definition, $\eta\overset{d}=\xi$ if and only if for any $\varphi\in C_{b,Lip}(\mathbb R)$, the function $u(x,t)=\Sbep\left[\varphi\left(x+\sqrt{t} \eta\right)\right]$ ($x\in \mathbb R, t\ge 0$) is the unique viscosity solution of the equation (\ref{eqheatequation}). In the sequel, without loss of generality, we assume that the sub-linear expectation spaces $(\widetilde{\Omega}, \widetilde{\mathscr H}, \widetilde{\mathbb E})$ and $(\Omega, \mathscr{H},\Sbep)$ are the same.
Let $\{X_n; n\ge 1\}$ be a sequence of independent and identically distributed random variables in a sub-linear expectation space $(\Omega, \mathscr{H},\Sbep)$, $S_n=\sum_{k=1}^nX_k$. Peng \cite{peng2008a, peng2010} proved that, if $\Sbep[X_1]=\Sbep[-X_1]=0$ and $\Sbep[|X_1|^{2+\alpha}]<\infty$ for some $\alpha>0$, then
\begin{equation}\label{cltpeng}
\lim_{n\to \infty} \Sbep\left[\varphi\left(\frac{S_n}{\sqrt{n}}\right)\right]=\Sbep\left[\varphi(\xi )\right], \forall \varphi\in C_b(\mathbb R),
\end{equation}
where $\xi\sim N\left(0,[\underline{\sigma}^2,\overline{\sigma}^2]\right)$, $\overline{\sigma}^2=\Sbep[X_1^2]$ and $\underline{\sigma}^2=\cSbep[X_1^2]$. Zhang \cite{Zhang Exponential} showed that
$\Sbep[|X_1|^{2+\alpha}]<\infty$ can be weakened to $\Sbep[(X_1^2-c)^+]\to 0$ as $c\to\infty$ by applying the moment inequalities of sums of independent random variables and the truncation method. A nature question is whether $\Sbep[X_1^2]<\infty$ and $ \Sbep[X_1]=\Sbep[-X_1]=0$ are sufficient and necessary for (\ref{cltpeng}). The following theorem is our main result.
\begin{theorem}\label{thclt} Let $\{X_n; n\ge 1\}$ be a sequence of independent and identically distributed random variables in a sub-linear expectation space $(\Omega, \mathscr{H},\Sbep)$, $S_n=\sum_{k=1}^nX_k$. Suppose that
\begin{description}
\item[\rm (i) ] $\lim_{c\to\infty} \Sbep[X_1^2\wedge c]$ is finite;
\item[\rm (ii)] $x^2\Capc\left(|X_1|\ge x\right)\to 0$ as $x\to \infty$;
\item[\rm (iii)] $\lim_{c\to \infty}\Sbep\left[(-c)\vee X_1\wedge c)\right]=\lim_{c\to \infty}\Sbep\left[(-c)\vee (- X_1)\wedge c)\right]=0$.
\end{description}
Write $\overline{\sigma}^2=\lim_{c\to\infty} \Sbep[X_1^2\wedge c]$ and $\underline{\sigma}^2=\lim_{c\to\infty} \cSbep[X_1^2\wedge c]$. Then for any $\varphi\in C_b(\mathbb R)$,
\begin{equation}\label{clt1}
\lim_{n\to \infty} \Sbep\left[\varphi\left(\frac{S_n}{\sqrt{n}}\right)\right]=\Sbep\left[\varphi(\xi )\right],
\end{equation}
where $\xi\sim N\left(0,[\underline{\sigma}^2,\overline{\sigma}^2]\right)$.
Conversely, if (\ref{clt1}) holds for any $\varphi\in C_b^1(\mathbb R)$ and a random variable $\xi$ with $x^2\Capc\left(|\xi|\ge x\right)\to 0$ as $x\to \infty$, then
(i),(ii) and (iii) hold and $\xi\overset{d}= N\left(0,[\underline{\sigma}^2,\overline{\sigma}^2]\right)$.
\end{theorem}
Before prove the theorem, we give some remarks on the conditions. Note that $\Sbep[X_1^2\wedge c]$ and $\cSbep[X_1^2\wedge c]$ are non-decreasing in $c$. So, $\overline{\sigma}^2$ and $\underline{\sigma}^2$ are well-defined and nonnegative, and are finite if the condition (i) is satisfied. It is easily seen that, for $c_1>c_2>0$,
\begin{equation}\label{eqproofclt6}\left|\Sbep[X_1^{c_1}]-\Sbep[X_1^{c_2}]\right|\le \Sbep[(|X_1|\wedge c_1-c_2)^+]\le \frac{\overline{\sigma}^2}{c_2}.
\end{equation}
So, the condition (i) implies that $ \lim_{c\to \infty}\Sbep[X_1^{c}]$ and $ \lim_{c\to \infty}\Sbep[-X_1^{c}]$ exist and are finite.
If $\Sbep$ is a continuous sub-linear expectation, i.e., $\Sbep[X_n]\nearrow \Sbep[X]$ whenever $0\le X_n\nearrow X$, and $\Sbep[X_n]\searrow 0$ whenever $X_n\searrow 0$, $\Sbep[X_n]<\infty$,
then (i) is equivalent to $\Sbep[X_1^2]<\infty$, (iii) is equivalent to $\Sbep[X_1]=\Sbep[-X_1]=0$, and (ii) is automatically implied by $\Sbep[X_1^2]<\infty$. In general, the condition $\Sbep[X_1^2]<\infty$ and (i) with (ii) do not imply each other. However, it is easily verified that, if $\Sbep[(X_1^2-c)^+]\to 0$ as $c\to \infty$, then (i) and (ii) are satisfied and (iii) is equivalent to $\Sbep[X_1]=\Sbep[-X_1]=0$.
\bigskip
To prove Theorem \ref{thclt}, we need a more lemma.
\begin{lemma}\label{lem4.1} Let $X_{n1},\cdots X_{nn}$ be independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H},\Sbep)$ with
$$ \frac{1}{\sqrt{n}}\sum_{k=1}^n \left\{\left|\Sbep[X_{nk}]\right|+\left|\Sbep[-X_{nk}]\right|\right\}\to 0, $$
$$ \frac{1}{n}\sum_{k=1}^n \left\{\big|\Sbep[X_{nk}^2]-\overline{\sigma}^2\big|+\big|\cSbep[X_{nk}^2]-\underline{\sigma}^2\big|\right\}\to 0 $$
and
$$ \frac{1}{n^{3/2}}\sum_{k=1}^n \Sbep[|X_{nk}|^3]\to 0. $$
Then
$$\lim_{n\to \infty} \Sbep\left[\varphi\left(\frac{\sum_{k=1}^n X_{nk}}{\sqrt{n}}\right)\right]=\Sbep\left[\varphi(\xi )\right], \forall \varphi\in C_b(\mathbb R), $$
where $\xi\sim N\left(0,[\underline{\sigma}^2,\overline{\sigma}^2]\right)$.
\end{lemma}
This lemma can be proved by refining the arguments of Li and Shi \cite{LiShi10} and can also follow from the Lindeberg central limit theorem \cite{Zhang Lindeberg}. We omit the proof here.
\bigskip
{\bf Proof of Theorem \ref{thclt}. } We first prove the sufficient part, i.e., (i),(ii) and (iii) $\implies$ (\ref{clt1}). Let $X_{nk}= (-\sqrt{n})\vee X_k \wedge \sqrt{n}$. Then
for any $\epsilon>0$,
$$\frac{1}{n^{3/2}} \sum_{k=1}^n \Sbep[|X_{nk}|^3]=\frac{1}{n^{1/2}} \Sbep[|X_{n1}|^3]\le \epsilon \overline{\sigma}^2+n\Capc\left(|X_1|\ge \epsilon\sqrt{n}\right)\to 0
$$
as $n\to \infty$ and then $\epsilon\to 0$,
by the condition (ii). Also,
\begin{align*}
& \frac{1}{n}\sum_{k=1}^n \left\{\big|\Sbep[X_{nk}^2]-\overline{\sigma}^2\big|+\big|\cSbep[X_{nk}^2]-\underline{\sigma}^2\big|\right\}\\
=& \big|\Sbep\left[X_1^2\wedge n\right]
-\overline{\sigma}^2\big|+\big|\cSbep\left[X_1^2\wedge n\right]
-\underline{\sigma}^2\big|\to 0,
\end{align*}
by (i). Note by (ii) and (i),
\begin{align*}
&\frac{1}{\sqrt{n}} \sum_{k=1}^n \left|\Sbep[X_{nk}]\right|=\sqrt{n} \left|\Sbep[X_{n1}]\right|\\
= & \sqrt{n} \lim_{c\to \infty} \left|\Sbep[X_{n1}]-\Sbep\left[(-c\sqrt{n})\vee X_1 \wedge (c\sqrt{n})\right]\right]\\
\le & \sqrt{n}\lim_{c\to \infty} \Sbep\left[ \left(|X_1| \wedge (c\sqrt{n})-x\sqrt{n}\right)^+\right]+ \sqrt{n}\Sbep\left[ \left(|X_1| \wedge (x\sqrt{n})- \sqrt{n}\right)^+\right]\\
\le & \frac{\overline{\sigma}^2}{x}+ x n\Capc\left(|X_1|\ge \sqrt{n}\right)\to 0 \; \text{ as } n\to \infty \text{ and then } x\to \infty,
\end{align*}
and similarly,
$$ \frac{1}{\sqrt{n}}\sum_{k=1}^n \left|\Sbep[-X_{nk}]\right|\to 0. $$
The conditions in Lemma \ref{lem4.1} are satisfied. We obtain
$$\lim_{n\to \infty} \Sbep\left[\varphi\left(\frac{\sum_{k=1}^n X_{nk}}{\sqrt{n}}\right)\right]=\Sbep\left[\varphi(\xi )\right]. $$
It is obvious that
$$\Sbep\left[\left|\varphi\left(\frac{\sum_{k=1}^n X_{nk}}{\sqrt{n}}\right)-\varphi\left(\frac{S_n}{\sqrt{n}}\right)\right|\right]\le \sup_x|\varphi(x)|n\Capc\left(|X_1|\ge \sqrt{n}\right) \to 0. $$
(\ref{clt1}) is proved.
Now, we consider the necessary part. Letting $\varphi=g_{\epsilon}\big(|x|-t\big)$ yields
$$ \limsup_{n\to \infty}\Capc\left(\frac{|S_n|}{\sqrt{n}}\ge t+\epsilon\right)\le \Capc\left(|\xi|\ge t \right) \text{ for all } t>0, \epsilon>0. $$
So
$$ \limsup_{n\ge m\to \infty}\max_{m\le k,l\le n} \Capc\left(\frac{|S_k-S_l|}{\sqrt{n}}\ge 2t+\epsilon\right)\le 2\Capc\left(|\xi|\ge t \right) \text{ for all } t>0, \epsilon>0. $$
Choose $t_0$ such that $\Capc\left(|\xi |\ge t_0\right)<1/(32)$. Applying the Levy maximal inequality (\ref{eqLIQ2}) yields
\begin{equation}\label{eqproofclt1} \limsup_{n\ge m\to \infty} \Capc\left(\frac{\max_{m\le k\le n}|S_k-S_m|}{\sqrt{n}}\ge 4t\right)< \frac{64}{31}\Capc\left(|\xi|\ge t \right) \text{ for all } t>t_0. \end{equation}
Hence
\begin{equation}\label{eqproofclt2} \limsup_{n\ge m\to \infty} \Capc\left(\frac{\max_{m\le k\le n}|X_k|}{\sqrt{n}}\ge 8t\right)< \frac{64}{31}\Capc\left(|\xi|\ge t \right) \text{ for all } t>t_0.
\end{equation}
Let $t_1>t_0$ and $m_0$ such that
\begin{equation}\label{eqproofclt3} \Capc\left(\frac{\max_{m\le k\le n}|S_k-S_m|}{\sqrt{n}}> 4t_1\right)< \frac{2}{31} \text{ for all } m\ge m_0
\end{equation}
and
\begin{equation}\label{eqproofclt4} \Capc\left(\frac{\max_{m\le k\le n}|X_k|}{\sqrt{n}}> 8t_1\right)< \frac{4}{31} \text{ for all } m\ge m_0.
\end{equation}
Write $Y_{nk}=(-8t_1)\vee\left(\frac{X_k}{\sqrt{n}}\right)\wedge(8t_1)$. Then by (\ref{eqproofclt3}) and (\ref{eqproofclt4}),
\begin{equation}\label{eqproofclt5} \Capc\left( \max_{m\le k\le n}\big|\sum_{j=m+1}^k Y_{nj}\big| > 4t_1\right)< \frac{2}{31} + \frac{4}{31}<\frac{1}{5} \text{ for all } m\ge m_0
\end{equation}
If $\Sbep[Y_{n1}]>0$, then by Lemma \ref{KolIneq} (ii),
$$\frac{1}{5}>1-\frac{4t_1+8t_1}{(n-m) \Sbep[Y_{n1}]}. $$
Hence $(n-m) \big(\Sbep[Y_{n1}])^+\le 15t_1$. Similarly, $(n-m) \big(\Sbep[-Y_{n1}])^+\le 15t_1$. Hence, by Lemma \ref{KolIneq} (i), it follows that
$$ \frac{1}{5}> 1-\frac{ (4t_1+8t_1)^2+8t_1 \big\{(n-m) \big(\Sbep[Y_{n1}])^++(n-m) \big(\Sbep[-Y_{n1}])^+\big\}}{ (n-m) \Sbep[Y_{n1}^2]}. $$
We conclude that $(n-m)\Sbep[Y_{n1}^2]\le \frac{5}{4}(12^2+240) t_1^2$. Choose $m=n/2$ and let $n\to \infty$. We have
$$ \lim_{c\to \infty}\Sbep[X_1^2\wedge c]=\lim_{n\to \infty} n \Sbep[Y_{n1}^2]\le \frac{5}{2}(12^2+240) t_1^2. $$
(i) is proved. Note that (i) implies that $ \lim_{c\to \infty}\Sbep[X_1^{c}]$ exists and is finite. Then
$$ \lim_{c\to \infty}\Sbep[X_1^{c}]=\limsup_{n\to \infty}\sqrt{n}\Sbep[Y_{n1}] \le \limsup_{n\to \infty}\frac{30t_1}{\sqrt{n}}=0. $$
Similarly, $ \lim_{c\to \infty}\Sbep[-X_1^{c}]$ exists, is finite and not positive. Note $\Sbep[-X_1^{c}]+\Sbep[X_1^{c}]\ge 0$.
Hence (iii) follows.
Finally, we show (ii). For any given $0<\epsilon<1/2$, by the condition $x^2\Capc(|\xi|\ge x)\to 0$, one can choose $t_1>t_0$ such that
$\frac{64}{31}\Capc(|\xi|\ge t_1)\le \frac{\epsilon}{9^3t_1^2}<1/2$. Then by (\ref{eqproofclt2}), there is $m_0$ such that
$$ \Capc\left(\frac{\max_{m\le k\le n}|X_k|}{\sqrt{n}}\ge 8t_1\right)< \frac{\epsilon}{9^3t_1^2}, \; n\ge m\ge m_0. $$
Choose $Z_k=g_{\epsilon}\big(\frac{|X_k|}{8t_1\sqrt{n}}-1\big)$ such that $I\{|X_k|\ge 9t_1\sqrt{n}\}\le Z_k\le I\{|X_k|\ge 8t_1\sqrt{n}\}$. Let $q_n=\Capc\left(|X_1|\ge 9t_1\sqrt{n}\right)$. Then
\begin{align*}
\Capc\left(\frac{\max_{m\le k\le n}|X_k|}{\sqrt{n}}\ge 8t_1\right)\ge & \Sbep\left[1-\prod_{k=m+1}^n(1-Z_k)\right]\\
=& 1-\prod_{k=m+1}^n(1-\Sbep[Z_k])\ge 1-e^{-(n-m)q_n}.
\end{align*}
It follows that
$$ n\Capc\left(|X_1|\ge 9t_1\sqrt{n}\right)\le 2(n-m)q_n< 2\times 2\times \frac{\epsilon}{9^3t_1^2}\text{ for } m=[n/2]\ge m_0. $$
Hence
$$ (9t_1\sqrt{n})^2 \Capc\left(|X_1|\ge 9t_1\sqrt{n}\right)< \frac{4 \epsilon}{9}, \;\; n\ge 2m_0. $$
When $x\ge 9t_1 \sqrt{2m_0}$, there is $n$ such that $9t_1\sqrt{n}\le x\le 9t_1\sqrt{n+1}$. Then
$$ x^2\Capc\left(|X_1|\ge x\right)\le (9t_1\sqrt{n+1})^2 \Capc\left(|X_1|\ge 9t_1\sqrt{n}\right)\le \frac{8 \epsilon}{9}. $$
It follows that $\limsup_{x\to \infty}x^2\Capc\left(|X_1|\ge x\right)<\epsilon$. (ii) is proved. The proof is now completed. $\Box$
\begin{remark} From the proof, we can find that
$$ \lim_{x\to \infty}\limsup_{n\to \infty}\Capc\left(\frac{|S_n|}{\sqrt{n}}\ge x\right)=0 $$
implies (i) and (ii). One may conjecture that,
\begin{description}
\item[\rm C1] if (\ref{clt1}) holds for any $\varphi\in C_b^1(\mathbb R)$ and a tight random variable $\xi$ (i.e., $\Capc\left(|\xi|\ge x\right)\to 0$ as $x\to \infty$), then (i), (ii) and (iii) holds and $\xi\overset{d}= N\left(0,[\underline{\sigma}^2,\overline{\sigma}^2]\right)$.
\end{description}
An equivalent conjecture is that,
\begin{description}
\item[\rm C2] if $\xi$ and $\xi^{\prime}$ are independent and identically distributed tight random variables, and
\begin{equation}\label{eqnormal2} \Sbep\left[\varphi(\alpha \xi+\beta \xi^{\prime})\right]
=\Sbep\left[\varphi\big(\sqrt{\alpha^2+\beta^2}\xi\big)\right], \;\;
\forall \varphi\in C_b(\mathbb R) \text{ and } \forall \alpha,\beta\ge 0,
\end{equation}
then $\xi\overset{d}= N\big(0,[\underline{\sigma}^2,\overline{\sigma}^2]\big)$, where $\overline{\sigma}^2=\lim_{c\to \infty}\Sbep[\xi^2\wedge c]$ and $\underline{\sigma}^2=\lim_{c\to \infty}\cSbep[\xi^2\wedge c]$.
\end{description}
It should be noted that the conditions (\ref{eqnormal}) and (\ref{eqnormal2}) are different. The condition (\ref{eqnormal}) implies that $\xi$ have finite moments of each order, but non information about the moments of $\xi$ is hidden in (\ref{eqnormal2}). As Theorem \ref{thclt}, the conjecture C2 is true when $x^2\Capc\left(|\xi|\ge x\right)\to 0$ as $x\to \infty$. In fact, let $X_1,X_2, \cdots, $ be independent random variables with $X_k\overset{d}=\xi$. Then by (\ref{eqnormal2}), $\frac{S_n}{\sqrt{n}}\overset{d}=\xi$. By the necessary part of Theorem \ref{thclt}, the conditions (i), (ii) and (iii) are satisfied. Then by the sufficient part of the theorem, $\xi\overset{d}=N\big(0,[\underline{\sigma}^2,\overline{\sigma}^2]\big)$. We don't known whether
conjectures C1 and C2 are true without assuming any moment conditions. It is very possible that they are not true in general. But finding a counterexample is not an easy task.
\end{remark}
\bigskip
|
{
"timestamp": "2019-03-01T02:08:39",
"yymm": "1902",
"arxiv_id": "1902.10872",
"language": "en",
"url": "https://arxiv.org/abs/1902.10872"
}
|
\section{Introduction}
The ability estimate position is key to most, if not all, robotics application involving mobility. In this paper we focus on the problem of visual odometry (VO), i.e. relative motion estimation based on visual information. This is the corner stone in vision-based SLAM systems, which will be the setting we will demonstrate our work in. As in our previous work, \cite{GCN}, we estimate the motion using only an RGB-D sensor, and our target platform is a drone operating in an indoor environment. The RGB-D sensor makes scale directly observable without the need for visual-inertial integration or the computational cost of inferring depth using a neural network as in \cite{Left-right,Single-stereo,SFMLearner}. This increases robustness, which is a key property when used on a drone and in particular for indoor environments where the margin for error is small and the there is typically less obvious textures than in outdoor environments. Our method is designed to be applicable to any system simply by adding an RGB-D sensor, without the need for complicated calibration and synchronisation routines with additional sensors, such as cameras or IMUs. Fusion can instead take place at a lower rate and with less need for precise timing, which, for example, makes integration with the flight control system of a drone simpler.
\begin{figure}
\centering
\resizebox{\hsize}{!}{
\includegraphics[width=0.5\textwidth]{uav.pdf}
}
\caption{The top figure shows our drone preforming position hold using GCN-SLAM. The figures below show the intermediate output for comparison of binary features, ORB and GCNv2, in ORB-SLAM2 and GCN-SLAM respectively. GCNv2 (left) tends to predict more repeatable and evenly distributed features compared with ORB (right.)}
\label{fig:digest}
\end{figure}
Like many other areas of research, there is a trend in SLAM to investigate deep learning-based methods. In \cite{SuperPoint} a keypoint detector and descriptor called SuperPoint is presented. Experimental results show that this CNN-based method has more distinctive descriptors than classical descriptors such as SIFT, and a detector on par with them. However, evaluation on homography estimation in \cite{SuperPoint} shows that it is only working on par with other keypoint extractors, classical or learning-based. In our previous work~\cite{GCN}, presented before SuperPoint, we introduced the Geometric Correspondence Network, GCN, specifically tailored for producing keypoints for camera motion estimation, achieving better accuracy than classical methods. However, due to the computation and structure limitation of GCN, it is difficult to achieve real-time performance in a fully-operational system, e.g. on board a drone. Both keypoint extraction and matching are computationally too expensive. Furthermore, due to the multi-frame matching setup, integrating GCN into existing SLAM systems becomes non-trivial.
In this paper we introduce GCNv2, based on the conclusions from~\cite{GCN}, to improve computational efficiency while still maintaining the high precision of GCN. Our contributions are:
\begin{itemize}
\item GCNv2 maintains the accuracy of GCN, providing significant improvements in motion estimation in comparison to related deep learning-based feature extraction methods as well as classical methods.
\item The inference of GCNv2 can run on embedded low-power hardware, such as a Jetson TX2, compared to GCN which requires a desktop GPU for real-time inference.
\item We design GCNv2 to have the same descriptor format as the ORB feature so that it can be drop-in substituted as the keypoint extractor in SLAM systems like ORB-SLAM2~\cite{ORBSLAM2-TRO17} or SVO2~\cite{SVO-TRO17}.
\item We demonstrate the effectiveness and robustness of our work by using GCN-SLAM\footnote{Built on ORB-SLAM2 with ORB substituted for GCNv2} on a real drone for control, and show that it handles situations where ORB-SLAM2 fails.
\end{itemize}
\section{Related Work}
\label{sec:related}
In this section we cover related work in two areas. First VO and SLAM methods are covered and then we focus specifically on work on deep learning-based methods for image correspondence.
\subsection{VO and SLAM}
In direct methods for VO and SLAM, motion is estimated by aligning frames based directly on the pixel intensities, with~\cite{Comport07} being an early example. DVO (Direct Visual Odometry), presented in~\cite{DVO-Kerl13}, adds a pose graph to reduce the error. DSO~\cite{DSO} is a direct and sparse method that adds joint optimisation of all model parameters. An alternative to the frame-to-frame matching is to match each new frame to a volumetric representation as in KinectFusion~\cite{KinectFusion}, Kintinous~\cite{Kintinous-IJRR14} and ElasticFusion~\cite{ElasticFusion}.
In indirect methods, the first step in a typical pipeline is to extract keypoints, which are then matched to previous frames to estimate the motion. The matching is based on the keypoint descriptors and geometric constraints. The state-of-the-art in this category is still defined by ORB-SLAM2~\cite{ORBSLAM-TRO15, ORBSLAM2-TRO17}. The ORB descriptor is a binary vector allowing high-performance matching.
Somewhere between direct and indirect methods we find the semi-direct approaches. SVO2~\cite{SVO-TRO17} is a sparse method in this category, and can run at hundreds of Hertz. There are also semi-dense methods, in which category LSD-SLAM~\cite{LSD-SLAM} was one of the first. RGBDTAM~\cite{RGBDTAM} combines both semi-dense photometric and dense geometric errors for pose estimation.
There are a number of recent deep learning-based mapping systems like \cite{CNN-SLAM, DVSO}. The focus in these methods is deep learning-based single view depth estimation to reduce the scale drift inherent in monocular systems. CNN-SLAM~\cite{CNN-SLAM} feeds the depth into LSD-SLAM. In DVSO~\cite{DeepVO}, depth is predicted in a similar way to \cite{Left-right}, using a virtual stereo view. CodeSLAM \cite{code-slam} learns an optimizable representation from conditioned auto-encoding for 3D reconstruction. In S2D~\cite{S2D}, we build on DSO~\cite{DSO} and exploit both depth and normals predicted by a jointly optimised CNN. Some work on end-to-end training for motion estimation also exist. Image reconstruction loss is used for unsupervised learning in~\cite{SFMLearner, Undeepvo}. However, geometry-based optimization methods still outperform end-to-end systems as shown in~\cite{DeepVO}.
\subsection{Deep Correspondence Matching}
There is a an abundance of recent works that deploy variants of metric learning for training deep features for finding image correspondences~\cite{feat_ref1, feat_ref2, feat_ref3, feat_ref4, feat_ref5, feat_ref6, LIFT, UCN, SuperPoint}.
Works in~\cite{cd_ref1, cd_ref2} focus on improving learning-based detection with better invariances.
Aimed at a different aspect, \cite{cd_ref3, cd_ref4, cd_ref5} use synthetic samples generated in a self-supervised manner to improve general feature matching.
Among the aforementioned methods, LIFT~\cite{LIFT} in particular uses a patch-based method to perform both keypoint detection and descriptor extraction. SuperPoint~\cite{SuperPoint} predicts the keypoints and descriptors using a single network together with the self-supervised strategy in~\cite{cd_ref5}. Notably, \cite{SuperPoint} shows that \cite{SuperPoint, LIFT, UCN} work on par with classical methods like SIFT for motion estimation.
In GCN~\cite{GCN}, we show that by learning keypoints and descriptors specifically targeting motion estimation, performance is improved -- contrary to what is reported for other more general deep learning-based keypoint extractor systems~\cite{SuperPoint,UCN}. In this paper we introduce an extension to GCN, GCNv2. We demonstrate the applicability of these keypoints for SLAM and build on ORB-SLAM2 as it offers a comprehensive multi-threaded state-of-the-art indirect SLAM system with support for monocular as well as RGB-D cameras. ORB-SLAM2 complements the tracking front-end with a back-end that does both pose graph optimisation using g2o~\cite{g2o} and loop closure detection using a binary bag of words representation~\cite{Galves-BoW-TRO12}. To simplify this integration, we design the GCNv2 descriptor to have the same format as that of ORB.
\section{Geometric Correspondence Network}
In this section, we present the design of GCNv2, aimed at making it suitable for real-time SLAM applications running on embedded hardware rather than a powerful desktop computer. We first introduce the modifications to the network structure. Then, we present the training scheme for the binarized feature descriptor and keypoint detector.
\subsection{Network Structure}
The original GCN structure, proposed in~\cite{GCN}, consists of two major parts: an FCN~\cite{FCN} structure with a ResNet-50 backbone and a bidirectional convolutional network. The FCN is adopted for dense feature extraction and the bidirectional network is used for temporal keypoint prediction. Although impressive tracking performance has been achieved using GCN compared with existing methods, GCN has practical limitations when it comes to the use in a real-time SLAM system with limited hardware.
The network architecture requires relatively powerful computational hardware which renders it unable to run in real-time on board e.g. the Jetson TX2 used on our drone.
Furthermore, the bidirectional structure requires matching between two or more frames at the same time. This significantly increases computational complexity for a window-based SLAM method, since keyframes then are updated dynamically based on the current camera position.
Inspired by SuperPoint~\cite{SuperPoint}, which uses a simple structure to perform detection using a single frame,
we deploy a network with even fewer parameters and working on a lower scale than SuperPoint. Intuitively, the network performs an individual prediction for each grid cell of size $16\times16$ pixels in the original image.
In GCNv2 all the pooling layers are replaced by convolutions with kernel size $4\times4$, stride ${2}$ and padding $1$. As in SuperPoint, the network takes $320\times240$ images as input. This is also the image size we used later for SLAM. Further details on the GCNv2 network specifics can be found in our publicly available source code\footnote{\url{https://github.com/jiexiong2016/GCNv2_SLAM}}.
GCN-SLAM with GCNv2 runs at 20 Hz on Jetson TX2 and runs at around 80 Hz on a laptop with Intel i7-7700HQ and mobile version NVIDIA 1070. To achieve even higher frame rates we introduce a smaller version of GCNv2, called GCNv2-tiny, where we reduce the number of feature maps by half from the second layer and onward. GCN-SLAM with GCNv2-tiny runs at 40 Hz on a TX2 and is therefore well-suited for deployment on a drone.
\subsection{Feature Extractor}
\textbf{Binarized Descriptor}
We trained the features of GCNv2 to be binary for accelerating the matching procedure and to match those of ORB. To binarize the features, we add a binary activation layer on top the final output. It is essentially a hard sign function and is therefore not differentiable. The challenge is how to back-propagate the loss properly through this layer of the network. We used the method proposed in~\cite{Binarization}. The binary activation layer can be written as follows:
\begin{equation} \label{eq:bin_activation}
\begin{split}
\text{Forward: }\,\bm{b}(\bm{x}) &= \operatorname{sign}(\bm{f}(\bm{x})) =
\left\{
\begin{array}{c l}
+1 & \bm{f}(\bm{x}) \ge 0\\
-1 & \text{otherwise}
\end{array}\right. \\
\text{Backward: }\,\frac{\partial \bm{b}}{\partial \bm{f}} &= \frac{\partial L_{metric}}{\partial \bm{b}} \bm{1}_{|\bm{f}| \le 1}
\end{split}
\end{equation}
where $\bm{x} = (u, v)$ is the 2D coordinates in the image and $\bm{f}(\cdot)$ is the feature vector at a given location. $L_{metric}$ is the loss for metric learning. $\bm{1}$ is the indicator function.
We found that it is more efficient to train the network with the above method rather than forcing the network to directly predict a binary output by minimizing quantification loss as in \cite{Lin-DeepBit-CVPR16}. One possible reason is that forcing the value to be clustered around $
\{+1,-1\}$ conflicts with the metric learning that follows, which uses distance as margin making the training unstable.
The number of feature maps $\bm{f}$ is set to $256$ to make the descriptor have the same bit width as ORB features so that the descriptor can be directly incorporated into existing ORB-based visual tracking systems.
\textbf{Nested Metric Learning} The pixel-wise metric learning is used for training the descriptor in a nearest-neighbour manner. The triplet loss for binarized features is as follows:
\begin{equation} \label{eq:triplet}
\begin{split}
L_{metric} &= \sum_{i} \operatorname{max} (0, d_{\bm{x}_i, \bm{x}_{i,+}} - d_{\bm{x}_i, \bm{x}_{i,-}} + m) \\
d_{\bm{x}_1, \bm{x}_2} &= (\operatorname{sign}(\bm{f}_1 ( {\bm{x}_1} ) ) - \operatorname{sign}(\bm{f}_2 ( {\bm{x}_2} ) ) )^2
\end{split}
\end{equation}
where $m$ is the distance margin for the truncation. $d$ is equivalent to the squared Hamming distance for the 32-byte descriptor. We use the squared distance since we found it leads to faster and better convergence for training. $(\bm{x}_i, \bm{x}_{i,+})$ is a matching pair obtained using the ground truth camera poses from the training data as follows:
\begin{equation} \label{eq:cor_gt}
\bm{x}_{i,+} = \pi^{-1}(\mathbf{R}_{gt} \cdot \pi(\bm{x}_i, d_i) + \bm{t}_{gt})
\end{equation}
where $\mathbf{R}\in \mathbb{R}^{3 \times 3}$ is the rotation matrix and $\bm{t} \in \mathbb{R}^3$ is the translation vector. $(\bm{x}_i, \bm{x}_{i,-})$ is a non-matching pair retrieved by negative sample mining. The mining procedure is described in \cref{alg:ENSM}. The exhaustive search will further penalize the already matched features with the relaxed criteria described in~\cite{GCN}. The relaxed criteria is used to increase the tolerance to potentially noisy data.
\addtolength{\topmargin}{2mm}
\begin{algorithm}[!t]
\caption{Exhaustive Negative Sample Mining}
\label{alg:ENSM}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{Feature in 1st frame: $ \bm{f}_1 ( \bm{x}_{i} )$, \\
Features in 2nd frame: $ \{\bm{f}_2 ( \bm{x}_{j} ) \, | \, j\in[0,N] \} $, \\
Correspondence Ground Truth: $\bm{x}_{i, +}$, \\
Relaxed Criteria: $\bm{c} = (u_c, v_c)$}
\Output{Negative sample $\bm{f}_2 ( \bm{x}_{i,-} ) $ \textbf{or} None}
\textbf{k-NN Matching}: Find top $k$ matched $\bm{x}_j$ using Hamming distance\\
\For{$n=1;n<=k;n\text{\scshape++}$}{
Query the $n$th nearest distance $\bm{x}_{j}^{n}$ \\
\If{$\operatorname{abs}( \bm{x}_{j}^{n}- \bm{x}_{i}) > c $}{
\Return $\bm{f}_2 ( \bm{x}_{j}^{n} )$
}
}
\Return None
\end{algorithm}
\subsection{Distributed Keypoint Detector}
The training loss for keypoint detection is the same as in the original GCN. It is computed using two consecutive frames as follows:
\begin{equation} \label{eq:NLL}
\begin{gathered}
\resizebox{0.55\hsize}{!}{$L_{mask} = L_{ce}(\bm{o}_1, \bm{x}) + L_{ce}(\bm{o}_2, \bm{x}_{+})$} \\
\resizebox{1.0\hsize}{!}{
$L_{ce}(\bm{o}, \bm{x}) = - \sum_{i} ( \alpha_1 c_{\bm{x}_i} \mathrm{log}(\bm{o}({\bm{x}_i})) +
\alpha_2 (1-c_{\bm{x}_i}) \mathrm{log}(1-\bm{o}({\bm{x}_i})) $}
\end{gathered}
\end{equation}
A notable difference from SuperPoint is that GCNv2 specifically targets motion estimation. SuperPoint tries to accomplish SIFT-like corner detection and its performance gain can chiefly be attributed to its superior descriptor. However, a CNN is capable of generating more representative features with a larger receptive field than classical methods. We therefore generate the ground truth by detecting Shi-Tomasi corners in a $16x16$ grid and warp them to the next frame using \cref{eq:cor_gt}. This leads to better distribution of keypoints and the objective function directly reflects the ability to track the keypoints based on texture.
\subsection{Training Details}
The triplet loss in \cref{eq:triplet} and cross entropy in \cref{eq:NLL} are weighted by ${100, 1}$ during training to provide a coarse normalization of the two terms. The learning rate for the adaptive gradient descent method, ADAM~\cite{ADAM}, used for training is started from $10^{-4}$ and halved every 40 epoch for a total of 100 training epochs. The weights of GCNv2 are randomly initialized.
We mapped the squared Hamming distance to the $L_2$ unit sphere to perform the fast nearest neighbour matching as in \cite{SuperPoint}. The margin for the triplet loss is set to $1$.
The weights $[\alpha_1, \alpha_2]$ of the weighted cross entropy is set to $[0.1, 1.0]$.
\section{GCN-SLAM}
\begin{figure}
\centering
\includegraphics{orbvsgcnpipe.pdf}
}
\caption{Illustration of the original ORB-SLAM2 keypoint extraction process on the left side, and our method on the right. The keypoint extraction in GCN-SLAM is comparatively simple, in large part because it relies on 2D convolutions and matrix multiplication which is off-loaded to the GPU.}
\label{fig:piplines}
\end{figure}
One of the most important design choices for a keypoint-based SLAM system is the choice of keypoint extractor. The keypoints are often re-used at multiple stages in such systems. ORB features \cite{ORB}, the namesake of ORB-SLAM2, are a particularly well-suited candidate as ORBs are invariant to rotation and scale, cheap to compute compared to other keypoint detectors with equivalent properties such SIFT or SURF, and have a binary feature vector to cater for fast matching.
As previously shown~\cite{GCN}, GCN used in a naive motion estimation pipeline performs better than or on par with ORB-SLAM2~\cite{ORBSLAM2-TRO17}. Notably, this is without higher-order SLAM functionality such as pose graph optimization, global bundle adjustment, or loop detection. Incorporating GCN into a system with such functionality would therefore be likely to yield better results. However, as mentioned, GCN is prohibitively expensive for real-time use on embedded hardware which we target in this work. In what follows we show how we modify ORB-SLAM2 to incorporate GCNv2, in a system we call GCN-SLAM.
ORB-SLAM2's motion estimation is based on frame-to-frame keypoint tracking and feature-based bundle adjustment. We will briefly describe the detection and description of these features. ORB-SLAM2 employs a \textit{scale pyramid} where the input image is iteratively scaled down to enable multi-scale feature detection by running single-scale algorithms on the multiple rescaled images. For each scale level, the FAST corner detector is applied in a $30\times{}30$ grid. If no detections are found in a cell, FAST is run again with a decreased threshold. After all detections have been gathered from all cells at a given level in the scale pyramid, a space partitioning algorithm is used to cull the keypoints first by their image coordinates, then by detection score. Finally, once typically 1000 keypoints have been selected in total, the viewing angle of each keypoint is computed, then each pyramid scale level is filtered with Gaussian blur, and the 256-bit ORB descriptor for each keypoint at each level is computed from the blurred image.
Our method computes both keypoint locations and descriptors simultaneously in a single forward-pass of the network, and as stated before, its end result is designed to be a drop-in replacement for the ORB feature extractor outlined above. In GCNv2, we input a single grey image frame to the network which outputs two matrices: a $1\times{}320\times{}240$ keypoint mask, and a $256\times{}320\times{}240$ feature descriptor matrix. The keypoint mask is thresholded to obtain a set of keypoint locations, their confidences, and their corresponding 256-bit feature descriptors. As in \cite{SuperPoint}, we apply non-maximum suppression with a grid size of $8\times{}8$. As it is not possible to know the orientation of the detected features, we set the angle to zero. The two keypoint methods are illustrated in \cref{fig:piplines}.
Once keypoints and their respective descriptors are found, ORB-SLAM2 relies primarily on two methods for frame-to-frame tracking: first, by assuming constant velocity and projecting the previous frame's keypoints into the current frame, and if that fails, by matching the keypoints of the current frame to the last-created keyframe using bag-of-words similarity. We have disabled the former so as to use only use the latter keypoint-based reference frame tracking. We have also replaced the matching algorithm with a standard nearest-neighbor search in our experiments. These modifications are made to examine the performance of our keypoint extraction method, rather than that of ORB-SLAM2's other tracking heuristics.
Finally, we have left ORB-SLAM2's loop closure and pose graph optimization intact, apart from having regenerated the bag-of-words vocabulary to suit GCNv2 feature descriptors by computing them on the training dataset presented in \cref{sec:trainingdata}.
\section{Experimental Results}
\addtolength{\topmargin}{2mm}
\begin{table*}[!ht]
\caption{ATE USING FRAME TO FRAME TRACKING}\label{tab:ate_open}
\centering
\begin{tabular}{ l | c | c | c | c || c | c | c | c}
\hline
Dataset (200 Frames) & GCN & ORB & SIFT & SURF & SuperPoint & GCNv2 & GCNv2-tiny & GCNv2-large\\
\hline \hline
fr1\textunderscore floor & \textbf{0.015m} & 0.080m & 0.073m & 0.074m & - & - & -& -\\ \hline
fr1\textunderscore desk & \textbf{0.037m} & 0.151m & 0.144m & 0.148m & 0.166m & 0.049m & 0.084m & 0.038m \\ \hline
fr1\textunderscore 360 & \textbf{0.059m} & 0.278m & 0.305m & 0.279m & - & - & - & 0.097m \\ \hline
fr3\textunderscore long\textunderscore office& 0.061m & 0.090m & 0.076m & 0.070m & 0.105m & \textbf{0.046m} & 0.085m & 0.067m\\ \hline
fr3\textunderscore large\textunderscore cabinet & 0.073m & 0.097m & 0.091m & 0.143m & 0.195m & 0.064m & 0.067m & \textbf{0.056m} \\ \hline
fr3\textunderscore nst & 0.020m & 0.061m & 0.036m & 0.030m & 0.055m & \textbf{0.018m} & 0.024m & 0.021m\\ \hline
fr3\textunderscore nnf & \textbf{0.221m} & - & - & - & - & - & - & -\\ \hline
\end{tabular}
\end{table*}
In this section, we present experimental results to justify our conclusions regarding the performance of our keypoint extraction method, and its embodiment in the GCN-SLAM system. We first introduce our training dataset, then four datasets on which we examined our method's performance and compare to some related methods, and finally we outline the quantitative and qualitative conclusions of these results.
Note that our aim in this section is not to show that GCN-SLAM is better that ORB-SLAM2 but to show that GCNv2 is: i) better suited for accurate motion estimation, ii) computationally efficient, and iii) providing robustness for a SLAM system.
In the results below, evaluations on datasets were performed on a laptop with an Intel i7-7700HQ and a mobile version of NVIDIA 1070. For real-world experiments we used an NVIDIA Jetson TX2 embedded computer for processing and an Intel RealSense D435 RGB-D camera sensor on our drone (see \cref{fig:digest}.)
\subsection{Training Data}
\label{sec:trainingdata}
The original GCN was trained using the TUM dataset from sensor fr2. It provides accurate pose through a motion capturing system. In GCNv2, we trained the network using a subset of the SUN-3D~\cite{SUN-3D} dataset we created in our recent work~\cite{S2D}. SUN-3D contains millions of real-world recorded RGB-D images in various typical indoor environments. A total $44,624$ frames were extracted by roughly one frame per second from the videos. It is very large and can potentially produce a more generalized network. However, the ground truth poses provided by SUN-3D are estimated by visual tracking with loop closure and so are relatively accurate globally, but have misalignments at frame level. To account for this local error, we extract SIFT features and use the provided poses as initial guesses for bundle adjustment to update the relative pose of each frame pair. In this sense, the training of GCNv2 is using self-annotated data with only vision information.
\subsection{Quantitative Results}
For comparison with the original GCN, we select the same sequences of the TUM datasets as in~\cite{GCN} and evaluating tracking performance with an open and a closed loop system. We use the Absolute trajectory error~(ATE) as the metric.
Since we trained GCNv2 on a different dataset than the original GCN~\cite{GCN}, we also show results using the original recurrent structure for comparison. We have therefore also created GCNv2-large, with ResNet-18 as the backbone and deconvolutional up-sampling for the feature maps. The bidirectional feature detector is moved to the lowest scale as the other two versions of GCNv2.
\begin{table}[ht]
\centering
\caption{ATE USING CLOSED LOOP SYSTEM}\label{tab:ate_closed}
\resizebox{\hsize}{!}{\begin{tabular}{ l | c | c | c | c || c }
\hline
\multirow{2}{*}{Dataset} & \multirow{2}{*}{GCN} & ORB & Elastic & RGBD & GCN \\
& & SLAM2 & Fusion & TAM & SLAM\\
\hline \hline
fr1\textunderscore floor & 0.038m & 0.036m & - & - & \textbf{0.021m} \\ \hline
fr1\textunderscore desk & 0.029m & \textbf{0.016m} & 0.020m & 0.027m & 0.031m \\ \hline
fr1\textunderscore 360 & \textbf{0.069m} & 0.213m & 0.108m & 0.101m & 0.155m \\ \hline
fr3\textunderscore long\textunderscore office & 0.040m & \textbf{0.010m} & 0.017m & 0.027m & 0.021m \\ \hline
fr3\textunderscore large\textunderscore cabinet & 0.097m & - & 0.099m & \textbf{0.070m} & \textbf{0.070m} \\ \hline
fr3\textunderscore nst & 0.020m & 0.019m & 0.016m & \textbf{0.010m} & 0.014m\\ \hline
fr3\textunderscore nnf & \textbf{0.064m} & - & - & - & 0.086m \\ \hline
\end{tabular}}
\end{table}
Frame-to-frame tracking results are shown in \cref{tab:ate_open}. The first columns, before SuperPoint, are from~\cite{GCN} where 640x480 images were used and GCN was trained on TUM fr2 data only. All versions of GCNv2 use the same image resolution as SuperPoint, i.e. 320x240. The results are consistent with the results reported in \cite{SuperPoint}, SuperPoint performs on par with classical method like SIFT. GCNv2 has a performance close to GCN, and like GCN, significantly better than both SuperPoint and classical keypoints. GCNv2 performance is on par with GCN, and even slightly better in two cases -- likely due to using a much larger dataset for training. The exceptions are fr1\textunderscore floor and fr1\textunderscore 360. These sequences require fine details, and the detection and descriptor extraction in GCNv2 is performed with a lower scale feature map for efficiency, though GCNv2-large handles one of these sequences. The smaller version of GCNv2, GCNv2-tiny, is only slightly less accurate than GCNv2.
In \cref{tab:ate_closed}, we compare the closed loop performance of GCN-SLAM with our previous work, as well as ORB-SLAM2, Elastic Fusion, and RGBDTAM. GCN-SLAM successfully tracks the position in all sequences with an error similar to that of GCN, whereas ORB-SLAM2 fails on two sequences. GCNv2 has significantly reduced drift error compared to ORB-SLAM2 in the fast rotations of fr1\textunderscore360. It is also noteworthy that for this particular sequence, the original GCN does significantly better than both ORB-SLAM2 and GCN-SLAM. ORB-SLAM2 is tracking well in all other sequences, and the errors of both GCN-SLAM and ORB-SLAM2 are small.
\subsection{Qualitative Results}
\begin{figure*}[hp]
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{cor_traj}
\caption{\textit{Corridor}: indoor, handheld.}
\label{fig:trajs_cor}
\end{subfigure
\hfill
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{park_traj}
\caption{\textit{Parking lot}: outdoor, handheld.}
\label{fig:trajs_park}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{atrium_traj.pdf}
\caption{\textit{Alcove}: indoor, flying with primitive optical flow sensor.}
\label{fig:trajs_atrium}
\end{subfigure
\hfill
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{kitchen_traj.pdf}
\caption{\textit{Kitchen}: indoor, flying with GCN-SLAM for positioning.}
\label{fig:trajs_kitchen}
\end{subfigure
\caption{Estimated trajectory on each of the four datasets using the GCN-SLAM pipeline and illustrating the difference between GCNv2 and ORB features. Note that these trajectories do not have any ground truth but we mark track lost with a cross and we see that this happened in all four cases when using ORB features.}
\label{fig:trajs}
\end{figure*}
\begin{figure*}[hp]
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{cor}
\end{subfigure
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{park}
\end{subfigure}
\caption{Keypoint extractor performance in terms of tracking shown for the \textit{Corridor} (left) and \textit{Kitchen} datasets (right.) Lines indicate the total number of keypoints detected per frame. The filled-in area under the lines shows the fraction of keypoints that were successfully used for local map tracking, i.e. contribute to tracking and is plotted against the right-hand axis.}
\label{fig:inliers}
\end{figure*}
To further verify the robustness of using GCNv2 in a real-world SLAM system, we show results on datasets collected in our environment under different conditions: a) going up a corridor, turning 180 degrees and walking back with a handheld camera, b) walking in a circle on an outdoor parking lot with a handheld sensor in daylight, c) flying in an alcove with windows and turning 180 degrees, and d) flying in a kitchen and turning 360 degrees while using GCN-SLAM for positioning.
\begin{figure*}[ht]
\centering
\begin{subfigure}[c]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{room_crop.png}
\end{subfigure
\centering
\begin{subfigure}[c]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{rpl_crop.png}
\end{subfigure}
\begin{subfigure}[c]{0.49\textwidth}
\centering
\caption{\textit{Lab room}: handheld.}
\end{subfigure
\centering
\begin{subfigure}[c]{0.49\textwidth}
\centering
\caption{\textit{Lab corridor}: handheld.}
\end{subfigure}
\caption{Mesh reconstruction results using GCN-SLAM as input to TSDF volume integration from Open3D. Loop closure detection in GCN-SLAM was disabled to demonstrate the accuracy in the tracking alone.}
\label{fig:mesh}
\end{figure*}
Since they are without ground truth, the results are only qualitative.
These datasets were chosen to show that our method handles difficult scenarios, is robust, and can be used for positioning of a real drone. \Cref{fig:trajs} shows the estimated trajectory of GCN-SLAM using ORB versus GCNv2 as keypoints. Note that both features are evaluated in exactly the same tracking pipeline for fair comparison, i.e. GCNv2 or ORB features is the \textit{only} differences. Refer to the source-code for further details.
using GCN-SLAM as a basis for drone control improves performance as can be seen by comparing \cref{fig:trajs_atrium,fig:trajs_kitchen}. In \cref{fig:trajs_atrium}, the position is estimated using only an optical flow sensor whereas \cref{fig:trajs_kitchen} uses GCN-SLAM as a source of position, and it is clear that the drone is able to hold its position better as there is significantly less jitter in this trajectory. In all four datasets, tracking is maintained with GCNv2, but lost with ORB. We used a remote control to send setpoints to the flight control unit on the drone for control, using the built-in position holding mode.
In \cref{fig:inliers} we compare the performance of our keypoint extractor to the original ORB keypoint extractor. We plot the number of inliers during tracking of the local map for our adapted SLAM system, first with ORB keypoints, and then with GCNv2 keypoints. As the figure illustrates, while there are many more ORB features, our method has a higher percentage of inliers. In addition, as shown in \cref{fig:digest}, GCNv2 results in better distributed features compared with ORB.
\Cref{fig:mesh} shows the mesh reconstruction using GCN-SLAM output poses from two additional sequences. The left sequence was from an office and the right was acquired walking between two floors using stairs. TSDF volume integration from Open3D\footnote{\url{http://www.open3d.org}} was used to create the mesh. To show the accuracy of our method, the loop closure detection of GCN-SLAM is disabled.
\section{Conclusions}
In our previous work~\cite{GCN}, we showed that our method, GCN, achieves better performance in visual tracking compared with existing deep learning and classical methods. However, it cannot be directly deployed into a real-time SLAM system in an efficient way due to its deep recurrent structure. In this paper, we addressed these issues by proposing a smaller, more efficient version of GCN, called GCNv2, that is readily adaptable to existing SLAM systems. We showed that GCNv2 can be effectively used in a modern feature-based SLAM system to achieve state-of-the-art tracking performance. The robustness and performance of the method was verified by incorporating GCNv2 into GCN-SLAM and using it on-board for positioning on our drone.
\textbf{Limitations}
GCNv2 is trained mainly for projective geometry and not generic feature matching.
As always with learning-based methods generalization is an important factor. GCNv2 works relatively well for outdoor scenes, as demonstrated in the experiments (Cf. \cref{fig:trajs_park}). However, since no outdoor data was used for training, further improvements can likely be made. Our target here is an indoor setting and we did not investigate this further.
\textbf{Future work}
In this paper, we mainly improved the efficiency of GCN and achieve stable tracking perform on our platform, a drone with NVIDIA Jetson TX2. However, since the original recurrent structure is removed, there is trade-off between the accuracy and running speed. In the future, we would like to further investigate to use the self-supervised learning to improve our system.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2019-03-26T01:13:51",
"yymm": "1902",
"arxiv_id": "1902.11046",
"language": "en",
"url": "https://arxiv.org/abs/1902.11046"
}
|
\section{Introduction}
Facial gestures are an effective medium of non-verbal communication, and communication becomes more appealing through 3D animated characters. This has led to extensive research \cite{DDEregression,exprgen,hsieh2015unconstrained} in developing techniques to retarget human facial motion to 3D animated characters. The standard approach is to model human face by a 3D morphable model (3DMM)\cite{3DMM} and learn the weights of a linear combination of blendshapes that fits to the input face image. The learned ``expression" weights and ``head pose" angles are then directly mapped to semantically equivalent blendshapes of the target 3D character rig to drive the desired facial animation. Previous methods, such as \cite{DDEregression}, formulate 3DMM fitting as an optimization problem of regressing the 3DMM parameters from the input image. However, these methods require significant pre-processing or post-processing operations to get the final output.
Using deep convolution neural networks, recent works have shown remarkable accuracy in regressing 3DMM parameters from a 2D image. However, while 3DMM fitting with deep learning is frequently used in related domains like 2D face alignment\cite{3ddfa,adrian3Dfacealignment}, 3D face reconstruction\cite{CNN3DMMsynth,prn,dff,mobileface} etc., it hasn't been proven yet as an effective approach for facial motion retargeting. This is because 1) face alignment methods focus more on accurate facial landmark localization while face reconstruction methods focus more on accurate 3D shape and texture reconstruction to capture the fine geometric details. In contrast, facial retargeting to an arbitrary 3D character only requires accurate transfer of facial expression and head pose. However, due to the ambiguous nature of this ill-posed problem of extracting 3D face information from 2D image, both facial expression and head pose learned by those methods are generally \textit{sub-optimal} as they are not well decoupled from other information like identity. 2) Unlike alignment and reconstruction, retargeting often requires real-time tracking and transfer of the facial motion. However, existing methods for alignment and reconstruction are highly memory intensive and often involve complex rendering of the 3DMM as intermediate steps, thereby making these methods difficult to deploy on light-weight hardware like mobile phones.
It is important to note that all previous deep learning based 3DMM fitting methods work on a single face image assuming face is already detected and cropped. To support multiple faces in a single image, a straightforward approach is to run a face detector on the image first to detect the all face regions and then perform the retargeting operations on each face individually. Such an approach, however, requires additional execution time for face detection and the computational complexity increases linearly with the number of faces in the input image. Additionally, tracking multiple faces with this approach becomes difficult when people move in and out from the frame or occlude each other. In the literature of joint face detection and alignment, existing methods \cite{JointDA_cascade, JointDA_cascade_mtl, Jointalignment} either use a random forest to predict the face bounding boxes and landmarks or adopt an iterative two-step approach to generate region proposals and predict the landmark locations in the proposed regions. However, these methods are primarily optimized for regressing accurate landmark locations rather than 3DMM parameters.
To this end, we divide our work into two parts. In the first part, we propose a multitask learning network to directly regress the 3DMM parameters from a well-cropped 2D image with a single face; we call this as Single Face Network (SFN). Our 3DMM parameters are grouped into: a) identity parameters that contain the face shape information, b) expression parameters that captures the facial expression, c) pose parameters that include the 3D rotation and 3D translation of the head and d) scale parameters that links the 3D face with the 2D image. We have observed that pose and scale parameters require global information while identity and expression parameters require different level of information, so we propose to emphasize on high level image features for pose and scale and the multi-scale features for identity and expression. Our network architecture is designed such that different layers embed image features at different resolutions, and these multi-scale features help in disentangling the parameter groups from each other. In the second part, we propose a single end-to-end trainable network to jointly detect the face bounding boxes and regress the 3DMM parameters for multiple faces in a single image. Inspired by YOLO\cite{yolo} and its variants\cite{yolov2,yolov3}, we design our Multiple Face Network (MFN) architecture that takes a 2D image as input and predicts the centroid position and dimensions of the bounding box as well as the 3DMM parameters for each face in the image. Unfortunately, existing publicly available multi-face image datasets provide ground truth for face bounding boxes only and not 3DMM parameters. Hence, we leverage our SFN to generate the weakly labelled ``ground truth" for 3DMM parameters for each face to train our MFN. Experimental results show that our MFN not only performs well for multi-face retargeting but also improves the accuracy of face detection. Our main contributions can be summarized as follows:
\begin{enumerate}
\itemsep0em
\item We design a multitask learning network, specifically tailored for facial motion retargeting by casting the scale prior into a novel network topology to disentangle the representation learning. Such network has been proven to be crucial for both single face and multiple face 3DMM parameters estimation.
\item We present a novel top-down approach using an end-to-end trainable network to jointly learn the face bounding box locations and the 3DMM parameters from an image having multiple faces with different poses and expressions.
\item Our system is easy to deploy into practical applications without requiring separate face detection for pose and expression retargeting. Our joint network can be run in real-time on mobile devices without engineering level optimization, e.g. only 39ms on Google Pixel 2.
\end{enumerate}
\vspace{-10pt}
\section{Related Work}
\subsection{2D Face Alignment and 3D Face Reconstruction}
Early methods like \cite{alignmentbytrees} used a cascade of decision trees or other regressors to directly regress the facial landmark locations from a face image. Recently, the approach of regressing 3DMM parameters using CNNs and fitting 3DMM to the 2D image has become popular. While Jourabloo et al. \cite{facealign3D2} use a cascade of CNNs to alternately regress the shape (identity and expression) and pose parameters, Zhu et al. \cite{3ddfa,pncc_new} perform multiple iterations of a single CNN to regress the shape and pose parameters together. These methods use large networks and require 3DMM in the network during testing, thereby requiring large memory and execution time. Regressing 3DMM parameters using CNNs is also popular in face reconstruction \cite{CNN3DMM,CNN3DMMsynth,cnnreconst,Tewari}. Richardson et al. \cite{Richardson} uses a coarse-to-fine approach to capture fine details in addition to face geometry. However, reconstruction methods also regress texture and focus more on capturing fine geometric details. For joint face alignment and reconstruction, \cite{prnet} regresses a position regression map from the image and \cite{nonlinear3dmm} regresses the parameters of a nonlinear 3DMM using an unsupervised encoder-decoder network. For joint face detection and alignment, recent methods either use a mixture of trees \cite{afwdatapaper} or a cascade of CNNs \cite{JointDA_cascade, JointDA_cascade_mtl}. In \cite{Jointalignment}, separate networks are trained to perform different tasks like proposing regions, classifying and regressing the bounding boxes from the regions, predicting the landmark locations in those regions etc. In \cite{hyperface}, region proposals are first generated with selective search algorithm and bounding box and landmark locations are regressed for each proposal using a multitask learning network. In contrast, we use a single end-to-end network to do join face detection and 3DMM fitting for face retargeting purposes.
\subsection{Performance-Based Animation}
Traditional performance capture systems (using either depth cameras or 3D scanners for direct mesh registration with depth data) \cite{AAM&depth,onlinemodeling,avataranimation_blendshapes} require complex hardware setup that is not readily available. Among the methods which use 2D images as input, the blendshape interpolation technique \cite{DDEregression,avataranimation_landmarks} is most popular. However, these methods require dense correspondence of facial points \cite{mocap} or user-specific adaptations \cite{realtime:animation:onthefly,Cao3d} to estimate the blendshape weights. Recent CNN based approaches either require depth input \cite{realtime:cnn:animation,SelfsupervisedCF} or regress character-specific parameters with several constraints \cite{exprgen}. Commercial software products like Faceshift \cite{faceshift}, Faceware \cite{faceware} etc. perform realtime retargeting but with poor expression accuracy \cite{exprgen}.
\subsection{Object Detection and Keypoint Localization}
In the literature of multiple object detection and classification, Fast RCNN \cite{rcnn} and YOLO \cite{yolo} are the two most popular methods with state-of-the-art performance. While \cite{rcnn} uses a region proposal network to get candidate regions before classification, \cite{yolo} performs joint object location regression and classification. Keypoint localization for multiple objects is popularly used for human pose estimation \cite{multiposenet,multipose} or object pose estimation \cite{object6d}. In case of faces, landmark localization for multiple faces can be done in two approaches: \textit{top-down approach} where landmark locations are detected after detecting face regions and \textit{bottom-up approach} where the facial landmarks are initially predicted individually and then grouped together into face regions. In our method, we adopt the top-down approach.
\section{Methodology}
\subsection{3D Morphable Model}
The 3D mesh of a human face can be represented by a multilinear 3D Morphable Model (3DMM) as
\begin{equation}
\mathcal{M} = \mathcal{V} \times \text{b}_{\text{id}} \times \text{b}_{\text{exp}}
\end{equation}
where $\mathcal{V}$ is the mean neutral face, $\text{b}_{\text{id}}$ are the identity bases and $\text{b}_{\text{exp}}$ are the expression bases. We use the face tensor provided by FacewareHouse \cite{facewarehouse} as 3DMM, where $\mathcal{V} \in \mathbb{R}^{11510 \times 3}$ denotes $11,510$ 3D co-ordinates of the mesh vertices, $\text{b}_{\text{id}}$ denotes 50 shape bases obtained by taking PCA over 150 identities and $\text{b}_{\text{exp}}$ denotes 47 bases corresponding to 47 blendshapes (1 neutral and 46 micro-expressions). To reduce the computational complexity, we manually mark 68 vertices in $\mathcal{V}$ as the facial landmark points based on \cite{afwdatapaper} and create a reduced face tensor $\hat{\mathcal{M}} \in \mathbb{R}^{204 \times 50 \times 47}$ for use in our networks. Given a set of identity parameters $w_\text{id} \in \mathbb{R}^{50 \times 1}$, expression parameters $w_\text{exp} \in \mathbb{R}^{47 \times 1}$, 3D rotation matrix $\mathbf{R} \in \mathbb{R}^{3 \times 3}$, 3D translation parameters $\mathbf{t} \in \mathbb{R}^{3 \times 1}$ and a scale parameter (focal length) $f$, we use weak perspective projection to get the 2D landmarks $\mathbf{P_{lm}} \in \mathbb{R}^{68\times2}$ as:
\begin{equation}
\mathbf{P_{lm}} = \begin{bmatrix}
f & 0 & 0 \\
0 & f & 0
\end{bmatrix} [ \mathbf{R} * (\hat{\mathcal{M}}*w_\text{id}*w_\text{exp}) + \mathbf{t}]
\label{lm_eq}
\end{equation}
where $w_\text{exp}[1] = 1 - \sum_{i=2}^{47} w_\text{exp}[i]$ and $w_\text{exp}[i] \in [0,1], i = 2, \ldots, 47$. We use a unit quaternion $\mathbf{q} \in \mathbb{R}^{4 \times 1}$ \cite{pncc_new} to represent 3D rotation and convert it into rotation matrix for use in equation \ref{lm_eq}. Please note that, for retargeting purposes, we omit the learning of texture and lighting in the 3DMM.
\subsection{Multi-scale Representation Disentangling}
A straightforward way of holistically regressing all the 3DMM parameters together through a fully connected layer on top of one shared representation will not be optimal particularly for our problem where each group of parameters has strong semantic meanings. Intuitively speaking, head pose learning does not require detailed local face representations since it is fundamentally independent of skin texture and subtle facial expressions, which has also been observed in recent work on pose estimation \cite{tcdcn}. However, for identity learning, a combination of both local and global representations would be necessary to differentiate among different persons. For example, some persons have relatively small eyes but fat cheek while others have big eyes and thin cheek, so both the local features around the eyes and the overall face silhouette would be important to approximate the identity shape. Similarly, expression learning possibly requires even fine-grained granularity of different scales of representations. Single eye wink, mouth grin and big laugh clearly require three different levels of representations to differentiate them from other expressions.
Another observation is, given the 2D landmarks of an image, there exist multiple combinations of 3DMM parameters that can minimize the 2D landmark loss. This ambiguity would cause additional challenges to the learning to favor the semantically more meaningful combinations. For examples, as shown in Fig. \ref{fig:decouple}, we can still minimize the 2D landmark loss by rotating the head and using different identity coefficients to accommodate the jaw left even without a strong jaw left expression coefficient. Motivated by both the multi-scale prior and the ambiguity nature of this problem, we designed a novel network structure that is specifically tailored for facial retargeting applications as illustrated in Fig. \ref{fig:teaser}, where pose is only learned through the final global features while expression learning depends on the concatenation of multi-layer representations.
\vspace{-10pt}
\paragraph{Disentangled Regularization} In addition to the above network design, we add a few regularization during the training to further enforce the disentangled representation learning. For example, for each face image, we can augment it by random translation/rotation perturbation to ask their resulting output to have the same identity and expression coefficients. Using image warping technique, we can re-edit the face image to slightly change the facial expression without hurting the pose and identity. Fig. \ref{fig:synthesis} shows a few such synthesized examples where their identity parameters need to be the same.
\begin{figure}[t]
\centering
\includegraphics[width=0.32\linewidth]{d1.jpg}
\includegraphics[width=0.32\linewidth]{d2.jpg}
\includegraphics[width=0.32\linewidth]{d3.jpg}
\caption{\textbf{left}: landmark projection from both meshes are exactly the same, \textbf{mid}: mesh with maximum jaw left, \textbf{right}: mesh without jaw left, but larger roll angle}
\label{fig:decouple}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.19\linewidth]{s0.jpg}
\includegraphics[width=0.19\linewidth]{s1.jpg}
\includegraphics[width=0.19\linewidth]{s2.jpg}
\includegraphics[width=0.19\linewidth]{s4.jpg}
\includegraphics[width=0.19\linewidth]{s6.jpg}
\caption{Synthesis image for regularization}
\label{fig:synthesis}
\end{figure}
\subsection{Single Face Retargeting Network}\label{sec:sfn}
When the face bounding box is given, we can train a single face retargeting network to output 3DMM parameters for each cropped face image using the above proposed network structure. Fortunately, many public datasets \cite{300w, facewarehouse, LFWTech, 3ddfa} already provide bounding boxes along with 68 2D facial landmark points. To encourage disentangling, we fit 3DMM parameters for each cropped single face image using the optimization method of \cite{adrian3Dfacealignment} and treat them as ground truth for our network in addition to the landmarks. Although individual optimization may result in over-fitting and noisy ground truth, our network can intrinsically focus more on the global common patterns from the training data. To achieve this, we initially train with a large weight on the L1 loss with respect to the ground truth ($g$), and then gradually decay this weight to trust more on the 2D landmarks loss, as shown in the following loss function:
\begin{multline}
\tau * \Bigg\{\frac{1}{50}\sum_{i = 1}^{50}|w_{\text{id}_i} - {w}^g_{\text{id}_i}|
+\frac{1}{46}\sum_{i = 1}^{46}|w_{\text{exp}_i} - {w}^g_{\text{exp}_i}| \\
+\frac{1}{4}\sum_{i = 1}^{4}|\mathbf{R}_i - \mathbf{R}^g_i|\Bigg\}
+ \sqrt{\frac{1}{68}\sum_{i = 1}^{68}(\mathbf{P}_{\text{lm}_i} - \mathbf{P}^g_{\text{lm}_i})^2}
\label{eq:sfn_loss}
\end{multline}
where $\tau$ denotes decay parameter with respect to epoch. We choose $\tau$ = 10/epoch across all experiments. Note that, although we drop the 3D translation and scale ground truth loss to allow 2D translation and scaling augmentation, the translation and scale parameters can still be learned by the 2D landmark loss.
\subsection{Joint Face Detection and Retargeting}
Our goal is to save computation cost by performing both face detection and 3DMM parameter estimation simultaneously instead of sequentially running a separate face detector and then single face retargeting network on each face separately. The network could potentially also benefit from the cross domain knowledge, especially for detection task, where introducing 3DMM gives the prior on how the face should look like in 3D space which complements the 2D features in separate face detection framework.
Inspired by YOLO \cite{yolo}, our joint network is designed to predict 3DMM parameters for each anchor point in additional to bounding box displacement and objectness. We divide the input image into $9 \times 9$ grid and predict a vector of length $4 + 1 + (50 + 46 + 4 + 3 + 1) = 109$ for a bounding box in each grid cell. Here 4 denotes 2D co-ordinates of the centroid, width and height of the face bounding box, 1 denotes the confidence score for the presence of a face in that cell and the rest denote the 3DMM parameters for the face in the cell. We also adopt the method of starting with 5 anchor boxes as bounding box priors. Our final loss function is the summation of equation \ref{eq:sfn_loss} across all grids and anchors, as shown in the following:
\begin{multline}
\tau * \Bigg\{\frac{1}{50}\sum_{j = 1}^{9^2} \sum_{k = 1}^{5}\sum_{i = 1}^{50}\mathbbm{1}_{ijk}|w_{\text{id}_{ijk}} - {w}^g_{\text{id}_{ijk}}| \\
+\frac{1}{46}\sum_{j = 1}^{9^2} \sum_{k = 1}^{5}\sum_{i = 1}^{46}\mathbbm{1}_{ijk}|w_{\text{exp}_{ijk}} - {w}^g_{\text{exp}_{ijk}}| \\
+\frac{1}{4}\sum_{j = 1}^{9^2} \sum_{k = 1}^{5}\sum_{i = 1}^{4}\mathbbm{1}_{ijk}|\mathbf{R}_{ijk} - \mathbf{R}^g_{ijk}|\Bigg\} \\
+ \sqrt{\frac{1}{68}\sum_{j = 1}^{9^2} \sum_{k = 1}^{5}\sum_{i = 1}^{68}\mathbbm{1}_{ijk}(\mathbf{P}_{\text{lm}_{ijk}} - \mathbf{P}^g_{\text{lm}_{ijk}})^2}\
\end{multline}
\label{eq:yolo_loss}
where $\mathbbm{1}_{ijk}$ denotes whether a $k$th bounding box predictor in cell $j$ contains a face. Since there are no publicly available multi-face datasets that provide both bounding box location and 3DMM parameters for each face, for proof-of-concept, we obtain the 3DMM ground truth by running our single face retargeting network on each face separately. The $x,y$ co-ordinates of the centroid and the width and height of a bounding box are calculated in the same manner as in \cite{yolo} and we use the same loss functions for these values.
\section{Experimental Setup}
\subsection{Datasets}
For single face retargeting, we combine multiple datasets to have a good training set for accurate prediction of each group of 3DMM parameters. 300W-LP contains many large poses and Facewarehouse is a rich dataset for expressions. The ground truth 68 2D landmarks provided by these datasets are used to obtain 3DMM ground truth by \cite{adrian3Dfacealignment}. LFW and AFLW2000-3D are used as test sets for static images and 300VW is used as test set for tracking on videos. For multiple face retargeting, AFW has ground truth bounding boxes, pose angles and 6 landmarks and is used as a test set for static images, while FDDB and WIDER only provide bounding box ground truth and are therefore used for training (WIDER test set is kept separate for testing). Music videos dataset is used to test our MFN performance on videos. We remove all images with more than 20 faces and also remove faces whose bounding box dimensions are \textless2\% of the image dimensions from both the training and test sets. This mainly includes faces in the background crowd with size less than 5$\times$5 pixels. The reason is that determining the facial expressions for such small faces is ambiguous even for human eyes and hence retargeting is not meaningful.
More dataset details are summarized in Table \ref{tab:dataset}. We use an 80-20 split of the training set for training and validation. To measure the performance of expression accuracy, we manually collect an expression test set by selecting those extreme expression images (Fig. \ref{fig:result4}). The number of images in each of the expression categories are: eye close: 185, eye wide: 70, brow raise: 124, brow anger: 100, mouth open: 81, jaw left/right: 136, lip roll: 64, smile: 105, kiss: 143, total: 1008 images.
\begin{table}[t]
\begin{tabular}{|l|l|c|c|}
\hline
\multicolumn{2}{|c|}{\textbf{Dataset}} & \multicolumn{1}{c|}{\textbf{\#images}} & \multicolumn{1}{c|}{\textbf{\#faces}} \\ \hline
\multirow{5}{*}{SFN} & 300W-LP \cite{300w,3ddfa} & 61225 & 61225 \\ \cline{2-4}
& FacewareHouse \cite{facewarehouse} & 5000 & 500 \\ \cline{2-4}
& LFW \cite{LFWTech} & 12639 & 12639 \\ \cline{2-4}
& AFLW2000-3D \cite{3ddfa} & 2000 & 2000 \\ \cline{2-4}
& 300VW \cite{300VW} & 114 (videos) & 218K\\
\hline
\multirow{4}{*}{MFN} & FDDB \cite{fddb} & 2845 & 5171 \\ \cline{2-4}
& WIDER \cite{wider} & 11905 & 56525 \\ \cline{2-4}
& AFW \cite{afwdatapaper} & 205 & 1000 \\ \cline{2-4}
& Music videos \cite{musicvideo} & 8 (videos) & - \\
\hline
\end{tabular}
\caption {Number of images or videos and faces for each dataset used in training and testing of our networks.}
\label{tab:dataset}
\end{table}
\subsection{Evaluation Metrics}
We use 4 metrics: 1) average precision (AP) with different intersection-over-union thresholds as defined in \cite{yolov3} to evaluate our MFN performance for face detection, 2) normalized mean error (NME) defined as the Euclidean distance between the predicted and ground truth 2D landmarks averaged over 68 landmarks and normalized by the bounding box dimensions, 3) area under the curve (AUC) of the Cumulative Error Distribution curve for landmark error normalized by the diagonal distance of ground truth bounding box \cite{Jointalignment}, and 4) expression metric defined as the mean absolute distance between the predicted expression parameters with respect to the ground truth, which is 1 in our case following the practice of \cite{facewarehouse}.
\subsection{Implementation Details}
\subsubsection{Training Configuration}
Our networks are implemented in Keras \cite{chollet2015keras} with Tensorflow backend and trained using Adam optimizer with batch size 32. The initial learning rate ($10^{-3}$ for SFN and $10^{-4}$ for MFN) is decreased by 10 times (up to $10^{-6}$) when the validation loss does not change over 5 consecutive epochs. Training takes about a day on a Nvidia GTX 1080 for each network. For data augmentation, we use random scaling in the range [0.8,1.2], random translation of 0-10\%, color jitter and in-plane rotation. These augmentation techniques improve the performance of SFN and also help in generating more accurate ground truth for individual faces for MFN.
\subsubsection{Single Face Retargeting Architecture}
Our network takes 128x128 resized image as input. In the first layer, we use a $7 \times 7$ convolution layer with 64 filters and stride 2 followed by a $2\times2$ maxpooling layer to capture the fine details in the image. The following layers are made up of Fire Modules(FM) of SqueezeNet \cite{SqueezeNet} (with 16 and 64 filters in squeeze and expand layers respectively) and squeeze-and-excite modules(SE) of \cite{SEnet} in order to compress the model size and reduce the model execution time without compromising the accuracy. At the end of network, we use a global average pooling layer followed by fully connected (FC) layers to generate the parameters. The penultimate FC layers each has 64 units with ReLU activation and sigmoid activation is used at the end of the last expression branch's FC layer to restrict the values between 0 and 1. To realize the multiscale prior and the disentangled learning, we concatenate the features at different scales and form separate branches for each group of parameters. The extra branches are built with the same blocks as the main branch, but we reduce the channel size by half to restrict the extra computation cost.
\subsubsection{Joint Detection and Retargeting Architecture}
Our joint detection and retargeting network architecture is similar to Tiny DarkNet \cite{yolo} with the final layer changed to predict a tensor of size $9\times 9 \times 5 \times 109$. However, since we only have one object class (face) in our problem, we reduce the number of filters in each layer to a quarter of their original values. For multi-scale version, we change the input image size to $288 \times 288$ and extend the multi-scale backbone for single face retargeting by changing the output of each branch to accommodate grid output (Fig. \ref{fig:teaser}). The pose branch outputs change from 4 ($R$) + 3 ($T$) + 1 ($f$) = 8 to $9\times 9 \times 5 \times 8$. The expression branch outputs change from 46 to $9\times 9 \times 5 \times 46$, and identity branch outputs change from 50 to $9\times 9 \times 5 \times 50$. One extra branch is also added to output objectness and bounding box location ($9\times 9 \times 5 \times (4+1)$). In total, multi-scale version outputs the same dimension as single-scale, but the output channels are split with respect to each type of branch.
\begin{figure}[t]
\centering
\begin{subfigure}[t]{1.0\linewidth}
\centering
\begin{subfigure}[t]{0.19\textwidth}
\centering
\includegraphics[width = 0.95\textwidth, height = 0.065\textheight]{images/layer1.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.19\textwidth}
\centering
\includegraphics[width = 0.95\textwidth, height = 0.065\textheight]{images/ss1.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.19\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.065\textheight]{images/exp1.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.19\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.065\textheight]{images/id1.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.19\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.065\textheight]{images/pose1.jpg}
\end{subfigure}
\end{subfigure}
\begin{subfigure}[t]{1.0\linewidth}
\centering
\begin{subfigure}[t]{0.19\textwidth}
\centering
\includegraphics[width = 0.95\textwidth, height = 0.065\textheight]{images/layer2.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.19\textwidth}
\centering
\includegraphics[width = 0.95\textwidth, height = 0.065\textheight]{images/ss2.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.19\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.065\textheight]{images/exp2.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.19\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.065\textheight]{images/id2.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.19\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.065\textheight]{images/pose2.jpg}
\end{subfigure}
\end{subfigure}
\caption{Visualization of learned features. From left to right in each row: input image, features for single scale SFN, features for expression branch of multi-scale SFN, features for identity branch of multi-scale SFN, features for pose branch of multi-scale SFN.}
\label{fig:layeroutputs}
\end{figure}
\section{Results}
\subsection{Importance of Multi-Scale Representation}
Our multi-scale network design reduces the load on the network to learn complex features by allowing the network to concentrate on different image features to learn different parameters unlike the single scale design. In Fig. \ref{fig:layeroutputs}, we see that single scale network learns generic facial features that combines the representations for identity, expression and pose. On the other hand, multi-scale network learns different levels of representation (pixel-level detailed features for expression, region level features for identity and global aggregate features for pose). We have randomly chosen only 25 filter outputs at level 3 of our SFN for clearer visualization. Table \ref{table:full} shows that our multi-scale design not only reduces NME for single face images using SFN but also improves the performance of MFN in terms of both NME (by generating a better weakly supervised ground truth) and AP for detection. Clearly, different feature representations are crucial to accurately learn different groups of parameters. By reducing the network load, this design also allows model compression so that multi-scale networks can be of comparable size with respect to single scale networks while having better accuracy. Fig. \ref{fig:howfarcomparison} shows that the multi-scale design predicts more accurate expression parameters (first row has correct landmarks for closed eyes) and identity parameters (second row has correct landmarks that fit the face shape) while being robust to large poses (second row), illumination (first row) and occlusion (third row).
\begin{table}[t]
\small
\begin{tabular}{|l|c|c|c|c|}
\hline
\multicolumn{1}{|c|}{\multirow{3}{*}{Model}} & \multicolumn{4}{c|}{Evaluation} \\
\cline{2-5} \multicolumn{1}{|c|}{} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}NME\\ (\%)\end{tabular}} & \multicolumn{3}{c|}{Multi Face} \\
\cline{3-5} \multicolumn{1}{|c|}{} & & \multicolumn{1}{l|}{AP} & \multicolumn{1}{l|}{AP50} & \multicolumn{1}{l|}{AP75} \\ \hline
(1) MFN (detection only) & - & 92.1 & 99.2 & 94.3\\ \hline
(2) Single scale SFN & 2.16 & - & - & -\\ \hline
(3) Multi-scale SFN & 1.91 & - & - & - \\ \hline
(4) SS-MFN + GT from (2) & 2.89 & 97.5 & 99.8 & 98.1 \\ \hline
(5) SS-MFN + GT from (3) & 2.65 & 98.2 & 100 & 98.9 \\ \hline
(6) MS-MFN + GT from (3) & 2.23 & 98.8 & 100 & 99.3\\ \hline
\end{tabular}
\caption{Quantitative evaluation of our SFN and MFN. SS-MFN and MS-MFN denote single scale and multi-scale MFN respectively. NME values are calculated for LFW (single faces) and AP values are calculated for AFW.}
\label{table:full}
\end{table}
\begin{figure}[t]
\centering
\begin{subfigure}[t]{1.0\linewidth}
\centering
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res3.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res31.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage1res3.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage2res3.jpg}
\end{subfigure}
\end{subfigure}
\begin{subfigure}[t]{1.0\linewidth}
\centering
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res5.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res51.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage1res5.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage2res5.jpg}
\end{subfigure}
\end{subfigure}
\begin{subfigure}[t]{1.0\linewidth}
\centering
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res6.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res61.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage2res6.jpg}e
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage2res6.jpg}
\end{subfigure}
\end{subfigure}
\caption{2D Face Alignment results for AFLW2000-3D. First column: original image with ground truth landmarks; Second column: results using \cite{adrian3Dfacealignment}; Third column: our single scale SFN; Last column: our multi-scale SFN.}
\label{fig:howfarcomparison}
\end{figure}
\subsection{Comparison with 2D Alignment Methods}
Even though we aim to predict the 3DMM parameters for retargeting applications, our model can naturally serve the purpose for 2D face alignment (via 3D). Therefore, we can evaluate our model from the performance of 2D alignment perspective. Table \ref{table:aflw_nme} compares the performance of our single scale and multi-scale SFN with state-of-the-art 2D face alignment methods (compared under the same settings). As can be seen, our model achieves much smaller errors compared to most of the methods that are dedicated for precise landmark localization. While PRN \cite{prn} has lower NME, its network size is 80 times bigger than ours and takes 9.8ms on GPU compared to $<$1ms required by our network. In addition to evaluations on static images, we also measure the face tracking performance in a video using our SFN. We set the bounding box of the current frame using the boundaries of the 2D landmarks detected in the previous frame and perform retargeting on a frame-by-frame basis. Table \ref{table:300vw_auc} compares the AUC values on 300VW dataset for three scenarios categorized by the dataset (compared under the same settings). Our method performs significantly better than other methods (about 9\% improvement over the second best method for Scenario 3) with negligible failure rate because extensive data augmentation helps our tracking algorithm to quickly recover from failures.
\begin{table}[t]
\small
\centering
\begin{tabular}{|c||c|c|c|c|}
\hline
Method & [0\degree,30\degree] & [30\degree,60\degree] & [60\degree,90\degree] & Mean\\
\hline
SDM \cite{sdm} & 3.67 & 4.94 & 9.67 & 6.12 \\
\hline
3DDFA \cite{3ddfa} & 3.78 & 4.54 & 7.93 & 5.42\\
\hline
3DDFA2 \cite{3ddfa} & 3.43 & 4.24 & 7.17 & 4.94\\
\hline
Yu et al. \cite{Yu2017LearningDF} & 3.62 & 6.06 & 9.56 & 6.41\\
\hline
3DSTN \cite{FasterTR} & 3.15 & 4.33 & 5.98 & 4.49\\
\hline
DFF \cite{dff} & 3.20 & 4.68 & 6.28 & 4.72 \\
\hline
PRN \cite{prn} & 2.75 & 3.51 & 4.61 & 3.62\\
\hline
SS-SFN (ours) & \textbf{3.09} & \textbf{4.27} & \textbf{5.59} & \textbf{4.31}\\
\hline
MS-SFN (ours) & \textbf{2.91} & \textbf{3.83} & \textbf{4.94} & \textbf{3.89}\\
\hline
\end{tabular}
\caption{Comparison of NME(\%) for 68 landmarks for AFLW2000-3D (divided into 3 groups based on yaw angles). 3DDFA2 refers to 3DDFA+SDM \cite{3ddfa}.}
\label{table:aflw_nme}
\end{table}
\begin{table}[t]
\small
\centering
\begin{tabular}{|c||c|c|c|}
\hline
Method & Scenario 1 & Scenario 2 & Scenario 3\\
\hline
Yang et al. \cite{yangetal} & 0.791 & 0.788 & 0.710\\
\hline
Xiao et al. \cite{xiaoetal} & 0.760 & 0.782 & 0.695 \\
\hline
CFSS \cite{cfss} & 0.784 & 0.783 & 0.713 \\
\hline
MTCNN \cite{MTCNN} & 0.748 & 0.760 & 0.726\\
\hline
MHM \cite{Jointalignment} & 0.847 & 0.838 & 0.769\\
\hline
MS-SFN (ours) & \textbf{0.901} & \textbf{0.884} & \textbf{0.842}\\
\hline
\end{tabular}
\caption{Landmark localization performance of our method on videos in comparison to state-of-the-art face tracking methods. The values are reported in terms of Area under the Curve (AUC) for Cumulative Error Distribution of the 2D landmark error for 300VW test set.}
\label{table:300vw_auc}
\end{table}
\begin{figure*}[h!]
\centering
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/1130084326.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/24795717.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/5082623456.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/4_Dancing_Dancing_4_357.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/35_Basketball_basketballgame_ball_35_993.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/39_Ice_Skating_Ice_Skating_39_696.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/1130084326_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/24795717_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/5082623456_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/4_Dancing_Dancing_4_357_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/35_Basketball_basketballgame_ball_35_993_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/39_Ice_Skating_Ice_Skating_39_696_3d.jpg}
\end{subfigure}
\caption{Testing results of our joint detection and retargeting model.
Column \textbf{1-3}: Sampled from AFW; Column \textbf{4-6}: Sampled from WIDER. We show both the predicted bounding boxes and the 3D meshes constructed from 3DMM parameters. }
\label{fig:result1}
\end{figure*}
\begin{table*}[t]
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{1}{|c|}{Model} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Eye\\ Close\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Eye\\ Wide\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Brow\\ Raise\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Brow\\ Anger\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Mouth\\ Open\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Jaw\\ L/R\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Lip\\ Roll\end{tabular}} & Smile & Kiss & \multicolumn{1}{c|}{Avg} \\ \hline
(1) Single scale SFN & 0.082 & 0.265 & 0.36 & 0.451 & 0.373 & 0.331 & 0.359 & 0.223 & 0.299 & 0.305 \\ \hline
(2) Multi-scale SFN & 0.016 & 0.257 & 0.216 & 0.381 & 0.334 & 0.131 & 0.204 & 0.245 & 0.277 & 0.229 \\ \hline
(3) MS-MFN + GT from (2) & 0.117 & 0.407 & 0.284 & 0.405 & 0.284 & 0.173 & 0.325 & 0.248 & 0.349 & 0.288 \\ \hline
\end{tabular}
\caption{Quantitative evaluation of expression accuracy (measured by the expression metric) on our expression test set. Lower error means the model performs better for extreme expressions when required.}
\label{table:exp}
\end{table*}
\begin{figure*}[h!]
\centering
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/2.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/3.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/5.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/7.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/21.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/25.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/27.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/28.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/12.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/13.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/15.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/17.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/31.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/35.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/37.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/38.jpg}
\end{subfigure}
\caption{Results by applying MS-SFN on our expression test set.}
\label{fig:result4}
\end{figure*}
\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\linewidth]{images/retargetingresults.jpg}
\caption{Retargeting from face(s) to 3D character(s).}
\label{fig:retargeting}
\end{figure*}
\subsection{Importance of Joint Training}
Joint regression of both face bounding box locations and 3DMM parameters forces the network to learn exclusive facial features that characterize face shape, expression and pose in addition to differentiating face regions from the background. This helps in more precise face detection in-the-wild by leveraging both 2D information from bounding boxes and 3D information from 3DMM parameters. Table \ref{table:full} shows that average precision (AP) is improved by a large margin with joint training compared to when the same network is trained to only regress bounding box locations. The retargeting accuracy for MFN is also comparable to that of SFN and the slight decrease in NME is because of training MFN on multi-face images and testing on single face images. Nevertheless, we observe improved performance in terms of both NME and AP by using better ground truth generated by multi-scale model. Our detection accuracy (mAP: 98.8\%) outperforms Hyperface \cite{hyperface} (mAP: 97.9\%) and Faceness-Net \cite{facenessnet} (mAP: 97.2\%) on the entire AFW dataset when compared under the same settings. Results of our MFN on multi-face images are illustrated in Fig. \ref{fig:result1}.
\subsection{Evaluation of Expressions}
Our expression evaluation results in Table \ref{table:exp} emphasizes the improvement of multi-scale design on SFN. MS-MFN performs much better than SS-SFN for all expressions except the eye expressions. This is because eye patches are usually small compared to the entire image for MFN whereas they are zoomed in on cropped images for SFN. Attention network for emphasizing small eyes region could be a future work for our MFN. However, MS-MFN shows less accuracy compared to MS-SFN because it is being tested on single face images while being trained on multi-face images. For the multi-person test set images, we found similar visual results by applying MS-MFN on the whole image and by applying MS-SFN on each face individually cropped from the image. This is expected because MFN is trained with ground truth from SFN. The performance of MS-SFN on our expression test set is shown in Fig. \ref{fig:result4}. We also conducted live performance capture experiments to evaluate the efficiency our system in retargeting facial motion from one or more faces to one or more 3D characters. Fig. \ref{fig:retargeting} shows some screenshots recorded during the experiments.
\subsection{Computational Complexity}
Excluding the IO time, SFN can run at 15ms/frame on Google Pixel 2 (assuming single face and excluding face detector runtime). Face detection with our compressed detector model is 34ms, so separate face detection and retargeting requires 49ms for 1 face, 109ms for 5 faces and 184ms for 10 faces. On the other hand, our proposed MFN performs joint face detection and retargeting at 39ms on any number of faces. The model sizes for compressed face detector is 11.5MB and SFN is 2MB, so the combination is 13.5MB, while our MFN is only 13MB. Hence our joint network reduces both memory requirement and execution time.
\section{Conclusion}
We propose a lightweight multitask learning network for joint face detection and facial motion retargeting on mobile devices in real time. The lack of 3DMM training data for multiple faces is tackled by generating weakly supervised ground truth from a network trained on images with single faces. We carefully design the network structure and regularization to enforce disentangled representation learning inspired by key observations. Extensive results have demonstrated the effectiveness of our design.
\textbf{Acknowledgements} We thank the anonymous reviewers for their constructive feedback, Muscle Wu, Xiang Yan, Zeyu Chen and Pai Zhang for helping, and Linda Shapiro, Alex Colburn and Barbara Mones for valuable discussions.
\iffalse
{\small
\bibliographystyle{ieee}
\section{Introduction}
Facial gestures are an effective medium of non-verbal communication, and communication becomes more appealing through 3D animated characters. This has led to extensive research \cite{DDEregression,exprgen,hsieh2015unconstrained} in developing techniques to retarget human facial motion to 3D animated characters. The standard approach is to model human face by a 3D morphable model (3DMM)\cite{3DMM} and learn the weights of a linear combination of blendshapes that fits to the input face image. The learned ``expression" weights and ``head pose" angles are then directly mapped to semantically equivalent blendshapes of the target 3D character rig to drive the desired facial animation. Previous methods, such as \cite{DDEregression}, formulate 3DMM fitting as an optimization problem of regressing the 3DMM parameters from the input image. However, these methods require significant pre-processing or post-processing operations to get the final output.
Using deep convolution neural networks, recent works have shown remarkable accuracy in regressing 3DMM parameters from a 2D image. However, while 3DMM fitting with deep learning is frequently used in related domains like 2D face alignment\cite{3ddfa,adrian3Dfacealignment}, 3D face reconstruction\cite{CNN3DMMsynth,prn,dff,mobileface} etc., it hasn't been proven yet as an effective approach for facial motion retargeting. This is because 1) face alignment methods focus more on accurate facial landmark localization while face reconstruction methods focus more on accurate 3D shape and texture reconstruction to capture the fine geometric details. In contrast, facial retargeting to an arbitrary 3D character only requires accurate transfer of facial expression and head pose. However, due to the ambiguous nature of this ill-posed problem of extracting 3D face information from 2D image, both facial expression and head pose learned by those methods are generally \textit{sub-optimal} as they are not well decoupled from other information like identity. 2) Unlike alignment and reconstruction, retargeting often requires real-time tracking and transfer of the facial motion. However, existing methods for alignment and reconstruction are highly memory intensive and often involve complex rendering of the 3DMM as intermediate steps, thereby making these methods difficult to deploy on light-weight hardware like mobile phones.
It is important to note that all previous deep learning based 3DMM fitting methods work on a single face image assuming face is already detected and cropped. To support multiple faces in a single image, a straightforward approach is to run a face detector on the image first to detect the all face regions and then perform the retargeting operations on each face individually. Such an approach, however, requires additional execution time for face detection and the computational complexity increases linearly with the number of faces in the input image. Additionally, tracking multiple faces with this approach becomes difficult when people move in and out from the frame or occlude each other. In the literature of joint face detection and alignment, existing methods \cite{JointDA_cascade, JointDA_cascade_mtl, Jointalignment} either use a random forest to predict the face bounding boxes and landmarks or adopt an iterative two-step approach to generate region proposals and predict the landmark locations in the proposed regions. However, these methods are primarily optimized for regressing accurate landmark locations rather than 3DMM parameters.
To this end, we divide our work into two parts. In the first part, we propose a multitask learning network to directly regress the 3DMM parameters from a well-cropped 2D image with a single face; we call this as Single Face Network (SFN). Our 3DMM parameters are grouped into: a) identity parameters that contain the face shape information, b) expression parameters that captures the facial expression, c) pose parameters that include the 3D rotation and 3D translation of the head and d) scale parameters that links the 3D face with the 2D image. We have observed that pose and scale parameters require global information while identity and expression parameters require different level of information, so we propose to emphasize on high level image features for pose and scale and the multi-scale features for identity and expression. Our network architecture is designed such that different layers embed image features at different resolutions, and these multi-scale features help in disentangling the parameter groups from each other. In the second part, we propose a single end-to-end trainable network to jointly detect the face bounding boxes and regress the 3DMM parameters for multiple faces in a single image. Inspired by YOLO\cite{yolo} and its variants\cite{yolov2,yolov3}, we design our Multiple Face Network (MFN) architecture that takes a 2D image as input and predicts the centroid position and dimensions of the bounding box as well as the 3DMM parameters for each face in the image. Unfortunately, existing publicly available multi-face image datasets provide ground truth for face bounding boxes only and not 3DMM parameters. Hence, we leverage our SFN to generate the weakly labelled ``ground truth" for 3DMM parameters for each face to train our MFN. Experimental results show that our MFN not only performs well for multi-face retargeting but also improves the accuracy of face detection. Our main contributions can be summarized as follows:
\begin{enumerate}
\itemsep0em
\item We design a multitask learning network, specifically tailored for facial motion retargeting by casting the scale prior into a novel network topology to disentangle the representation learning. Such network has been proven to be crucial for both single face and multiple face 3DMM parameters estimation.
\item We present a novel top-down approach using an end-to-end trainable network to jointly learn the face bounding box locations and the 3DMM parameters from an image having multiple faces with different poses and expressions.
\item Our system is easy to deploy into practical applications without requiring separate face detection for pose and expression retargeting. Our joint network can be run in real-time on mobile devices without engineering level optimization, e.g. only 39ms on Google Pixel 2.
\iffalse
\item We design a multitask learning framework that combines image features at different scales to disentangle and accurately predict different groups of 3DMM parameters.
\item We present a novel top-down approach to jointly learn the face bounding boxes and the 3DMM parameters from an image with multiple faces with different poses and expressions.
\item We present a weakly supervised method to generate ground truth 3DMM parameters for multiple faces to train our multi-face network.
\item We evaluate our networks on both image and video datasets and show that our method enables face tracking in videos with large inter-frame motion without the need for any additional face detection step.
\fi
\end{enumerate}
\section{Related Work}
\subsection{2D Face Alignment and 3D Face Reconstruction}
Early methods like \cite{alignmentbytrees} used a cascade of decision trees or other regressors to directly regress the facial landmark locations from a face image. Recently, the approach of regressing 3DMM parameters using CNNs and fitting 3DMM to the 2D image has become popular. While Jourabloo et al. \cite{facealign3D2} use a cascade of CNNs to alternately regress the shape (identity and expression) and pose parameters, Zhu et al. \cite{3ddfa,pncc_new} perform multiple iterations of a single CNN to regress the shape and pose parameters together. These methods use large networks and require 3DMM in the network during testing, thereby requiring large memory and execution time. Regressing 3DMM parameters using CNNs is also popular in face reconstruction \cite{CNN3DMM,CNN3DMMsynth,cnnreconst,Tewari}. Richardson et al. \cite{Richardson} uses a coarse-to-fine approach to capture fine details in addition to face geometry. However, reconstruction methods also regress texture and focus more on capturing fine geometric details. For joint face alignment and reconstruction, \cite{prnet} regresses a position regression map from the image and \cite{nonlinear3dmm} regresses the parameters of a nonlinear 3DMM using an unsupervised encoder-decoder network. For joint face detection and alignment, recent methods either use a mixture of trees \cite{afwdatapaper} or a cascade of CNNs \cite{JointDA_cascade, JointDA_cascade_mtl}. In \cite{Jointalignment}, separate networks are trained to perform different tasks like proposing regions, classifying and regressing the bounding boxes from the regions, predicting the landmark locations in those regions etc. In \cite{hyperface}, region proposals are first generated with selective search algorithm and bounding box and landmark locations are regressed for each proposal using a multitask learning network. In contrast, we use a single end-to-end network to do join face detection and 3DMM fitting for face retargeting purposes.
\subsection{Performance-Based Animation}
Traditional performance capture systems (using either depth cameras or 3D scanners for direct mesh registration with depth data) \cite{AAM&depth,onlinemodeling,avataranimation_blendshapes} require complex hardware setup that is not readily available. Among the methods which use 2D images as input, the blendshape interpolation technique \cite{DDEregression,avataranimation_landmarks} is most popular. However, these methods require dense correspondence of facial points \cite{mocap} or user-specific adaptations \cite{realtime:animation:onthefly,Cao3d} to estimate the blendshape weights. Recent CNN based approaches either require depth input \cite{realtime:cnn:animation,SelfsupervisedCF} or regress character-specific parameters with several constraints \cite{exprgen}. Commercial software products like Faceshift \cite{faceshift}, Faceware \cite{faceware} etc. perform realtime retargeting but with poor expression accuracy \cite{exprgen}.
\subsection{Object Detection and Keypoint Localization}
In the literature of multiple object detection and classification, Fast RCNN \cite{rcnn} and YOLO \cite{yolo} are the two most popular methods with state-of-the-art performance. While \cite{rcnn} uses a region proposal network to get candidate regions before classification, \cite{yolo} performs joint object location regression and classification. Keypoint localization for multiple objects is popularly used for human pose estimation \cite{multiposenet,multipose} or object pose estimation \cite{object6d}. In case of faces, landmark localization for multiple faces can be done in two approaches: \textit{top-down approach} where landmark locations are detected after detecting face regions and \textit{bottom-up approach} where the facial landmarks are initially predicted individually and then grouped together into face regions. In our method, we adopt the top-down approach.
\section{Methodology}
\subsection{3D Morphable Model}
The 3D mesh of a human face can be represented by a multilinear 3D Morphable Model (3DMM) as
\begin{equation}
\mathcal{M} = \mathcal{V} \times \text{b}_{\text{id}} \times \text{b}_{\text{exp}}
\end{equation}
where $\mathcal{V}$ is the mean neutral face, $\text{b}_{\text{id}}$ are the identity bases and $\text{b}_{\text{exp}}$ are the expression bases. We use the face tensor provided by FacewareHouse \cite{facewarehouse} as 3DMM, where $\mathcal{V} \in \mathbb{R}^{11510 \times 3}$ denotes $11,510$ 3D co-ordinates of the mesh vertices, $\text{b}_{\text{id}}$ denotes 50 shape bases obtained by taking PCA over 150 identities and $\text{b}_{\text{exp}}$ denotes 47 bases corresponding to 47 blendshapes (1 neutral and 46 micro-expressions). To reduce the computational complexity, we manually mark 68 vertices in $\mathcal{V}$ as the facial landmark points based on \cite{afwdatapaper} and create a reduced face tensor $\hat{\mathcal{M}} \in \mathbb{R}^{204 \times 50 \times 47}$ for use in our networks. Given a set of identity parameters $w_\text{id} \in \mathbb{R}^{50 \times 1}$, expression parameters $w_\text{exp} \in \mathbb{R}^{47 \times 1}$, 3D rotation matrix $\mathbf{R} \in \mathbb{R}^{3 \times 3}$, 3D translation parameters $\mathbf{t} \in \mathbb{R}^{3 \times 1}$ and a scale parameter (focal length) $f$, we use weak perspective projection to get the 2D landmarks $\mathbf{P_{lm}} \in \mathbb{R}^{68\times2}$ as:
\begin{equation}
\mathbf{P_{lm}} = \begin{bmatrix}
f & 0 & 0 \\
0 & f & 0
\end{bmatrix} [ \mathbf{R} * (\hat{\mathcal{M}}*w_\text{id}*w_\text{exp}) + \mathbf{t}]
\label{lm_eq}
\end{equation}
where $w_\text{exp}[1] = 1 - \sum_{i=2}^{47} w_\text{exp}[i]$ and $w_\text{exp}[i] \in [0,1], i = 2, \ldots, 47$. We use a unit quaternion $\mathbf{q} \in \mathbb{R}^{4 \times 1}$ \cite{pncc_new} to represent 3D rotation and convert it into rotation matrix for use in equation \ref{lm_eq}. Please note that, for retargeting purposes, we omit the learning of texture and lighting in the 3DMM.
\subsection{Multi-scale Representation Disentangling}
A straightforward way of holistically regressing all the 3DMM parameters together through a fully connected layer on top of one shared representation will not be optimal particularly for our problem where each group of parameters has strong semantic meanings. Intuitively speaking, head pose learning does not require detailed local face representations since it is fundamentally independent of skin texture and subtle facial expressions, which has also been observed in recent work on pose estimation \cite{tcdcn}. However, for identity learning, a combination of both local and global representations would be necessary to differentiate among different persons. For example, some persons have relatively small eyes but fat cheek while others have big eyes and thin cheek, so both the local features around the eyes and the overall face silhouette would be important to approximate the identity shape. Similarly, expression learning possibly requires even fine-grained granularity of different scales of representations. Single eye wink, mouth grin and big laugh clearly require three different levels of representations to differentiate them from other expressions.
Another observation is, given the 2D landmarks of an image, there exist multiple combinations of 3DMM parameters that can minimize the 2D landmark loss. This ambiguity would cause additional challenges to the learning to favor the semantically more meaningful combinations. For examples, as shown in Fig. \ref{fig:decouple}, we can still minimize the 2D landmark loss by rotating the head and using different identity coefficients to accommodate the jaw left even without a strong jaw left expression coefficient. Motivated by both the multi-scale prior and the ambiguity nature of this problem, we designed a novel network structure that is specifically tailored for facial retargeting applications as illustrated in Fig. \ref{fig:teaser}, where pose is only learned through the final global features while expression learning depends on the concatenation of multi-layer representations.
\vspace{-10pt}
\paragraph{Disentangled Regularization} In addition to the above network design, we add a few regularization during the training to further enforce the disentangled representation learning. For example, for each face image, we can augment it by random translation/rotation perturbation to ask their resulting output to have the same identity and expression coefficients. Using image warping technique, we can re-edit the face image to slightly change the facial expression without hurting the pose and identity. Fig. \ref{fig:synthesis} shows a few such synthesized examples where their identity parameters need to be the same.
\begin{figure}[t]
\centering
\includegraphics[width=0.32\linewidth]{d1.jpg}
\includegraphics[width=0.32\linewidth]{d2.jpg}
\includegraphics[width=0.32\linewidth]{d3.jpg}
\caption{\textbf{left}: landmark projection from both meshes are exactly the same, \textbf{mid}: mesh with maximum jaw left, \textbf{right}: mesh without jaw left, but larger roll angle}
\label{fig:decouple}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.19\linewidth]{s0.jpg}
\includegraphics[width=0.19\linewidth]{s1.jpg}
\includegraphics[width=0.19\linewidth]{s2.jpg}
\includegraphics[width=0.19\linewidth]{s4.jpg}
\includegraphics[width=0.19\linewidth]{s6.jpg}
\caption{Synthesis image for regularization}
\label{fig:synthesis}
\end{figure}
\section{Methodology}
\subsection{Single Face Retargeting Network}\label{sec:sfn}
When the face bounding box is given, we can train a single face retargeting network to output 3DMM parameters for each cropped face image using the above proposed network structure. Fortunately, many public datasets \cite{300w, facewarehouse, LFWTech, 3ddfa} already provide bounding boxes along with 68 2D facial landmark points. To encourage disentangling, we fit 3DMM parameters for each cropped single face image using the optimization method of \cite{adrian3Dfacealignment} and treat them as ground truth for our network in addition to the landmarks. Although individual optimization may result in over-fitting and noisy ground truth, our network can intrinsically focus more on the global common patterns from the training data. To achieve this, we initially train with a large weight on the L1 loss with respect to the ground truth ($g$), and then gradually decay this weight to trust more on the 2D landmarks loss, as shown in the following loss function:
\begin{multline}
\tau * \Bigg\{\frac{1}{50}\sum_{i = 1}^{50}|w_{\text{id}_i} - {w}^g_{\text{id}_i}|
+\frac{1}{46}\sum_{i = 1}^{46}|w_{\text{exp}_i} - {w}^g_{\text{exp}_i}| \\
+\frac{1}{4}\sum_{i = 1}^{4}|\mathbf{R}_i - \mathbf{R}^g_i|\Bigg\}
+ \sqrt{\frac{1}{68}\sum_{i = 1}^{68}(\mathbf{P}_{\text{lm}_i} - \mathbf{P}^g_{\text{lm}_i})^2}
\label{eq:sfn_loss}
\end{multline}
where $\tau$ denotes decay parameter with respect to epoch. We choose $\tau$ = 10/epoch across all experiments. Note that, although we drop the 3D translation and scale ground truth loss to allow 2D translation and scaling augmentation, the translation and scale parameters can still be learned by the 2D landmark loss.
\iffalse
The network has two stages. The first stage takes a 2D human face image of size $128 \times 128$ pixels as input and predicts the identity, expression, pose and scale parameters of the 3DMM as output. The optional second stage is a light-weight extension that takes the eye and mouth regions of the image as input and refines the expression parameters predicted by the first stage.
\baoyuan{I think we should move this single-scale network to the implementation details, only discuss the multi-scale network}
\fi
\iffalse
\begin{figure}[]
\centering
\includegraphics[width=1.0\linewidth]{images/single_scale_model.jpg}
\caption{Single scale face retargeting network. The final feature map is connected to fully connected layer to respect the dimension of each 3DMM parameters.}
\label{fig:sfn}
\end{figure}
\fi
\subsection{Joint Face Detection and Retargeting}
Our goal is to save computation cost by performing both face detection and 3DMM parameter estimation simultaneously instead of sequentially running a separate face detector and then single face retargeting network on each face separately. The network could potentially also benefit from the cross domain knowledge, especially for detection task, where introducing 3DMM gives the prior on how the face should look like in 3D space which complements the 2D features in separate face detection framework.
Inspired by YOLO \cite{yolo}, our joint network is designed to predict 3DMM parameters for each anchor point in additional to bounding box displacement and objectness. We divide the input image into $9 \times 9$ grid and predict a vector of length $4 + 1 + (50 + 46 + 4 + 3 + 1) = 109$ for a bounding box in each grid cell. Here 4 denotes 2D co-ordinates of the centroid, width and height of the face bounding box, 1 denotes the confidence score for the presence of a face in that cell and the rest denote the 3DMM parameters for the face in the cell. We also adopt the method of starting with 5 anchor boxes as bounding box priors. Our final loss function is the summation of equation \ref{eq:sfn_loss} across all grids and anchors, as shown in the following:
\iffalse
\begin{equation*}
b_{\text{id}} = t_{v_{1-50}}; \; b_{\text{exp}} = \sigma (t_{v_{51-97}})
\end{equation*}
\begin{equation*}
b_{\textbf{R}} = t_{v_{98-101}}; \; b_{\textbf{t}} = t_{v_{102-104}}; \; b_{\text{f}} = \sigma (t_{v_{105}})
\end{equation*}
\begin{equation*}
b_{\text{lm}_x} = b_x + b_w * b_{\Hat{\text{lm}}_x}; \; b_{\text{lm}_y} = b_y + b_h * b_{\Hat{\text{lm}}_y}
\end{equation*}
where $\sigma$ denotes sigmoid function and $b_{\Hat{\text{lm}}}$ are the initial landmarks obtained using equation \ref{lm_eq}. The loss function for the 3DMM parameters is:
\fi
\begin{multline}
\tau * \Bigg\{\frac{1}{50}\sum_{j = 1}^{9^2} \sum_{k = 1}^{5}\sum_{i = 1}^{50}\mathbbm{1}_{ijk}|w_{\text{id}_{ijk}} - {w}^g_{\text{id}_{ijk}}| \\
+\frac{1}{46}\sum_{j = 1}^{9^2} \sum_{k = 1}^{5}\sum_{i = 1}^{46}\mathbbm{1}_{ijk}|w_{\text{exp}_{ijk}} - {w}^g_{\text{exp}_{ijk}}| \\
+\frac{1}{4}\sum_{j = 1}^{9^2} \sum_{k = 1}^{5}\sum_{i = 1}^{4}\mathbbm{1}_{ijk}|\mathbf{R}_{ijk} - \mathbf{R}^g_{ijk}|\Bigg\} \\
+ \sqrt{\frac{1}{68}\sum_{j = 1}^{9^2} \sum_{k = 1}^{5}\sum_{i = 1}^{68}\mathbbm{1}_{ijk}(\mathbf{P}_{\text{lm}_{ijk}} - \mathbf{P}^g_{\text{lm}_{ijk}})^2}\
\end{multline}
\label{eq:yolo_loss}
where $\mathbbm{1}_{ijk}$ denotes whether a $k$th bounding box predictor in cell $j$ contains a face. Since there are no publicly available multi-face datasets that provide both bounding box location and 3DMM parameters for each face, for proof-of-concept, we obtain the 3DMM ground truth by running our single face retargeting network on each face separately. The $x,y$ co-ordinates of the centroid and the width and height of a bounding box are calculated in the same manner as in \cite{yolo} and we use the same loss functions for these values.
\section{Experimental Setup}
\subsection{Datasets}
\iffalse
The authors of \cite{DDEregression} provide the corresponding 68 landmarks for both the datasets.
Front camera images captured by mobile phones have lower resolution and image quality and are strikingly different from rear camera images because of perspective distortion due to proximity of the face to the camera. As a result, networks trained on in-the-wild face images do not perform well with the front camera. To overcome this issue, we captured the facial performance of several users using front camera and annotated each frame of the captured videos with 68 landmarks using the method proposed in \cite{adrian3Dfacealignment}.\fi
For single face retargeting, we combine multiple datasets to have a good training set for accurate prediction of each group of 3DMM parameters. 300W-LP contains many large poses and Facewarehouse is a rich dataset for expressions. The ground truth 68 2D landmarks provided by these datasets are used to obtain 3DMM ground truth by \cite{adrian3Dfacealignment}. LFW and AFLW2000-3D are used as test sets for static images and 300VW is used as test set for tracking on videos. For multiple face retargeting, AFW has ground truth bounding boxes, pose angles and 6 landmarks and is used as a test set for static images, while FDDB and WIDER only provide bounding box ground truth and are therefore used for training (WIDER test set is kept separate for testing). Music videos dataset is used to test our MFN performance on videos. We remove all images with more than 20 faces and also remove faces whose bounding box dimensions are \textless2\% of the image dimensions from both the training and test sets. This mainly includes faces in the background crowd with size less than 5$\times$5 pixels. The reason is that determining the facial expressions for such small faces is ambiguous even for human eyes and hence retargeting is not meaningful.
More dataset details are summarized in Table \ref{tab:dataset}. We use an 80-20 split of the training set for training and validation. To measure the performance of expression accuracy, we manually collect an expression test set by selecting those extreme expression images (Fig. \ref{fig:result4}). The number of images in each of the expression categories are: eye close: 185, eye wide: 70, brow raise: 124, brow anger: 100, mouth open: 81, jaw left/right: 136, lip roll: 64, smile: 105, kiss: 143, total: 1008 images.
\begin{table}[t]
\begin{tabular}{|l|l|c|c|}
\hline
\multicolumn{2}{|c|}{\textbf{Dataset}} & \multicolumn{1}{c|}{\textbf{\#images}} & \multicolumn{1}{c|}{\textbf{\#faces}} \\ \hline
\multirow{5}{*}{SFN} & 300W-LP \cite{300w,3ddfa} & 61225 & 61225 \\ \cline{2-4}
& FacewareHouse \cite{facewarehouse} & 5000 & 500 \\ \cline{2-4}
& LFW \cite{LFWTech} & 12639 & 12639 \\ \cline{2-4}
& AFLW2000-3D \cite{3ddfa} & 2000 & 2000 \\ \cline{2-4}
& 300VW \cite{300VW} & 114 (videos) & 218K\\
\hline
\multirow{4}{*}{MFN} & FDDB \cite{fddb} & 2845 & 5171 \\ \cline{2-4}
& WIDER \cite{wider} & 11905 & 56525 \\ \cline{2-4}
& AFW \cite{afwdatapaper} & 205 & 1000 \\ \cline{2-4}
& Music videos \cite{musicvideo} & 8 (videos) & - \\
\hline
\end{tabular}
\caption {Number of images or videos and faces for each dataset used in training and testing of our networks.}
\label{tab:dataset}
\end{table}
\iffalse
\subsection{Face Tracking for SFN}
For a video, we initially feed the entire image into the network and keep shrinking the bounding box over time until it fits the face. Then we set the bounding box of the current frame using the boundaries of the 2D landmarks of the previous frame and perform retargeting on a frame-by-frame basis. Hence, unlike other methods, our method does not require a separate face detection module at any instant. Finally, we smooth the predicted parameters using a Kalman filter.
\fi
\subsection{Evaluation Metrics}
We use 4 metrics: 1) average precision (AP) with different intersection-over-union thresholds as defined in \cite{yolov3} to evaluate our MFN performance for face detection, 2) normalized mean error (NME) defined as the Euclidean distance between the predicted and ground truth 2D landmarks averaged over 68 landmarks and normalized by the bounding box dimensions, 3) area under the curve (AUC) of the Cumulative Error Distribution curve for landmark error normalized by the diagonal distance of ground truth bounding box \cite{Jointalignment}, and 4) expression metric defined as the mean absolute distance between the predicted expression parameters with respect to the ground truth, which is 1 in our case following the practice of \cite{facewarehouse}.
\iffalse
\begin{table}[h]
\begin{tabular}{l|l|l|l|l|l|}
\cline{2-6}
& \multicolumn{1}{c|}{\textbf{\begin{tabular}[c]{@{}c@{}}Eye\\ Close\end{tabular}}} & \multicolumn{1}{c|}{\textbf{\begin{tabular}[c]{@{}c@{}}Eye\\ Wide\end{tabular}}} & \multicolumn{1}{c|}{\textbf{\begin{tabular}[c]{@{}c@{}}Brow\\ Raise\end{tabular}}} & \multicolumn{1}{c|}{\textbf{\begin{tabular}[c]{@{}c@{}}Brow\\ Anger\end{tabular}}} & \multicolumn{1}{c|}{\textbf{\begin{tabular}[c]{@{}c@{}}Mouth\\ Wide\end{tabular}}} \\ \hline
\multicolumn{1}{|l|}{\textbf{\#images}} & 100 & 100 & 100 & 100 & 100 \\ \hline
\end{tabular}
\end{table}
\begin{table}[h]
\begin{tabular}{l|l|l|l|l|l|}
\cline{2-6}
& \multicolumn{1}{c|}{\textbf{Jaw}} & \multicolumn{1}{c|}{\textbf{LipRoll}} & \multicolumn{1}{c|}{\textbf{Smile}} & \multicolumn{1}{c|}{\textbf{Kiss}} & \multicolumn{1}{c|}{\textbf{Total}} \\ \hline
\multicolumn{1}{|l|}{\textbf{\#images}} & 100 & 100 & 100 & 100 & 100 \\ \hline
\end{tabular}
\end{table}
\fi
\subsection{Implementation Details}
\subsubsection{Training Configuration}
Our networks are implemented in Keras \cite{chollet2015keras} with Tensorflow backend and trained using Adam optimizer with batch size 32. The initial learning rate ($10^{-3}$ for SFN and $10^{-4}$ for MFN) is decreased by 10 times (up to $10^{-6}$) when the validation loss does not change over 5 consecutive epochs. Training takes about a day on a Nvidia GTX 1080 for each network. For data augmentation, we use random scaling in the range [0.8,1.2], random translation of 0-10\%, color jitter and in-plane rotation. These augmentation techniques improve the performance of SFN and also help in generating more accurate ground truth for individual faces for MFN.
\subsubsection{Single Face Retargeting Architecture}
Our network takes 128x128 resized image as input. In the first layer, we use a $7 \times 7$ convolution layer with 64 filters and stride 2 followed by a $2\times2$ maxpooling layer to capture the fine details in the image. The following layers are made up of Fire Modules(FM) of SqueezeNet \cite{SqueezeNet} (with 16 and 64 filters in squeeze and expand layers respectively) and squeeze-and-excite modules(SE) of \cite{SEnet} in order to compress the model size and reduce the model execution time without compromising the accuracy. At the end of network, we use a global average pooling layer followed by fully connected (FC) layers to generate the parameters. The penultimate FC layers each has 64 units with ReLU activation and sigmoid activation is used at the end of the last expression branch's FC layer to restrict the values between 0 and 1. To realize the multiscale prior and the disentangled learning, we concatenate the features at different scales and form separate branches for each group of parameters. The extra branches are built with the same blocks as the main branch, but we reduce the channel size by half to restrict the extra computation cost.
\subsubsection{Joint Detection and Retargeting Architecture}
Our joint detection and retargeting network architecture is similar to Tiny DarkNet \cite{yolo} with the final layer changed to predict a tensor of size $9\times 9 \times 5 \times 109$. However, since we only have one object class (face) in our problem, we reduce the number of filters in each layer to a quarter of their original values. For multi-scale version, we change the input image size to $288 \times 288$ and extend the multi-scale backbone for single face retargeting by changing the output of each branch to accommodate grid output (Fig. \ref{fig:teaser}). The pose branch outputs change from 4 ($R$) + 3 ($T$) + 1 ($f$) = 8 to $9\times 9 \times 5 \times 8$. The expression branch outputs change from 46 to $9\times 9 \times 5 \times 46$, and identity branch outputs change from 50 to $9\times 9 \times 5 \times 50$. One extra branch is also added to output objectness and bounding box location ($9\times 9 \times 5 \times (4+1)$). In total, multi-scale version outputs the same dimension as single-scale, but the output channels are split with respect to each type of branch.
\section{Results}
\subsection{Importance of Multi-Scale Representation}
Our multi-scale network design reduces the load on the network to learn complex features by allowing the network to concentrate on different image features to learn different parameters unlike the single scale design. In Fig. \ref{fig:layeroutputs}, we see that single scale network learns generic facial features that combines the representations for identity, expression and pose. On the other hand, multi-scale network learns different levels of representation (pixel-level detailed features for expression, region level features for identity and global aggregate features for pose). We have randomly chosen only 25 filter outputs at level 3 of our SFN for clearer visualization. Table \ref{table:full} shows that our multi-scale design not only reduces NME for single face images using SFN but also improves the performance of MFN in terms of both NME (by generating a better weakly supervised ground truth) and AP for detection. Clearly, different feature representations are crucial to accurately learn different groups of parameters. By reducing the network load, this design also allows model compression so that multi-scale networks can be of comparable size with respect to single scale networks while having better accuracy. Fig. \ref{fig:howfarcomparison} shows that the multi-scale design predicts more accurate expression parameters (first row has correct landmarks for closed eyes) and identity parameters (second row has correct landmarks that fit the face shape) while being robust to large poses (second row), illumination (first row) and occlusion (third row).
\begin{table}[t]
\small
\begin{tabular}{|l|c|c|c|c|}
\hline
\multicolumn{1}{|c|}{\multirow{3}{*}{Model}} & \multicolumn{4}{c|}{Evaluation} \\
\cline{2-5} \multicolumn{1}{|c|}{} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}NME\\ (\%)\end{tabular}} & \multicolumn{3}{c|}{Multi Face} \\
\cline{3-5} \multicolumn{1}{|c|}{} & & \multicolumn{1}{l|}{AP} & \multicolumn{1}{l|}{AP50} & \multicolumn{1}{l|}{AP75} \\ \hline
(1) MFN (detection only) & - & 92.1 & 99.2 & 94.3\\ \hline
(2) Single scale SFN & 2.16 & - & - & -\\ \hline
(3) Multi-scale SFN & 1.91 & - & - & - \\ \hline
(4) SS-MFN + GT from (2) & 2.89 & 97.5 & 99.8 & 98.1 \\ \hline
(5) SS-MFN + GT from (3) & 2.65 & 98.2 & 100 & 98.9 \\ \hline
(6) MS-MFN + GT from (3) & 2.23 & 98.8 & 100 & 99.3\\ \hline
\end{tabular}
\caption{Quantitative evaluation of our SFN and MFN. SS-MFN and MS-MFN denote single scale and multi-scale MFN respectively. NME values are calculated for LFW (single faces) and AP values are calculated for AFW.}
\label{table:full}
\end{table}
\begin{figure}[t]
\centering
\begin{subfigure}[t]{1.0\linewidth}
\centering
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res3.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res31.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage1res3.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage2res3.jpg}
\end{subfigure}
\end{subfigure}
\begin{subfigure}[t]{1.0\linewidth}
\centering
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res5.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res51.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage1res5.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage2res5.jpg}
\end{subfigure}
\end{subfigure}
\begin{subfigure}[t]{1.0\linewidth}
\centering
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res6.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/res61.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage2res6.jpg}e
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.08\textheight]{images/stage2res6.jpg}
\end{subfigure}
\end{subfigure}
\caption{2D Face Alignment results for AFLW2000-3D. First column: original image with ground truth landmarks; Second column: results using \cite{adrian3Dfacealignment}; Third column: our single scale SFN; Last column: our multi-scale SFN.}
\label{fig:howfarcomparison}
\end{figure}
\subsection{Comparison with 2D Alignment Methods}
Even though we aim to predict the 3DMM parameters for retargeting applications, our model can naturally serve the purpose for 2D face alignment (via 3D). Therefore, we can evaluate our model from the performance of 2D alignment perspective. Table \ref{table:aflw_nme} compares the performance of our single scale and multi-scale SFN with state-of-the-art 2D face alignment methods (compared under the same settings). As can be seen, our model achieves much smaller errors compared to most of the methods that are dedicated for precise landmark localization. While PRN \cite{prn} has lower NME, its network size is 80 times bigger than ours and takes 9.8ms on GPU compared to $<$1ms required by our network. In addition to evaluations on static images, we also measure the face tracking performance in a video using our SFN. We set the bounding box of the current frame using the boundaries of the 2D landmarks detected in the previous frame and perform retargeting on a frame-by-frame basis. Table \ref{table:300vw_auc} compares the AUC values on 300VW dataset for three scenarios categorized by the dataset (compared under the same settings). Our method performs significantly better than other methods (about 9\% improvement over the second best method for Scenario 3) with negligible failure rate because extensive data augmentation helps our tracking algorithm to quickly recover from failures.
\begin{table}[t]
\small
\centering
\begin{tabular}{|c||c|c|c|c|}
\hline
Method & [0\degree,30\degree] & [30\degree,60\degree] & [60\degree,90\degree] & Mean\\
\hline
SDM \cite{sdm} & 3.67 & 4.94 & 9.67 & 6.12 \\
\hline
3DDFA \cite{3ddfa} & 3.78 & 4.54 & 7.93 & 5.42\\
\hline
3DDFA2 \cite{3ddfa} & 3.43 & 4.24 & 7.17 & 4.94\\
\hline
Yu et al. \cite{Yu2017LearningDF} & 3.62 & 6.06 & 9.56 & 6.41\\
\hline
3DSTN \cite{FasterTR} & 3.15 & 4.33 & 5.98 & 4.49\\
\hline
DFF \cite{dff} & 3.20 & 4.68 & 6.28 & 4.72 \\
\hline
PRN \cite{prn} & 2.75 & 3.51 & 4.61 & 3.62\\
\hline
SS-SFN (ours) & \textbf{3.09} & \textbf{4.27} & \textbf{5.59} & \textbf{4.31}\\
\hline
MS-SFN (ours) & \textbf{2.91} & \textbf{3.83} & \textbf{4.94} & \textbf{3.89}\\
\hline
\end{tabular}
\caption{Comparison of NME(\%) for 68 landmarks for AFLW2000-3D (divided into 3 groups based on yaw angles). 3DDFA2 refers to 3DDFA+SDM \cite{3ddfa}.}
\label{table:aflw_nme}
\end{table}
\begin{table}[t]
\small
\centering
\begin{tabular}{|c||c|c|c|}
\hline
Method & Scenario 1 & Scenario 2 & Scenario 3\\
\hline
Yang et al. \cite{yangetal} & 0.791 & 0.788 & 0.710\\
\hline
Xiao et al. \cite{xiaoetal} & 0.760 & 0.782 & 0.695 \\
\hline
CFSS \cite{cfss} & 0.784 & 0.783 & 0.713 \\
\hline
MTCNN \cite{MTCNN} & 0.748 & 0.760 & 0.726\\
\hline
MHM \cite{Jointalignment} & 0.847 & 0.838 & 0.769\\
\hline
MS-SFN (ours) & \textbf{0.901} & \textbf{0.884} & \textbf{0.842}\\
\hline
\end{tabular}
\caption{Landmark localization performance of our method on videos in comparison to state-of-the-art face tracking methods. The values are reported in terms of Area under the Curve (AUC) for Cumulative Error Distribution of the 2D landmark error for 300VW test set.}
\label{table:300vw_auc}
\end{table}
\begin{figure*}[h!]
\centering
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/1130084326.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/24795717.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/5082623456.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/4_Dancing_Dancing_4_357.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/35_Basketball_basketballgame_ball_35_993.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/39_Ice_Skating_Ice_Skating_39_696.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/1130084326_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/24795717_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/5082623456_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/4_Dancing_Dancing_4_357_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/35_Basketball_basketballgame_ball_35_993_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/39_Ice_Skating_Ice_Skating_39_696_3d.jpg}
\end{subfigure}
\caption{Testing results of our joint detection and retargeting model.
Column \textbf{1-3}: Sampled from AFW; Column \textbf{4-6}: Sampled from WIDER. We show both the predicted bounding boxes and the 3D meshes constructed from 3DMM parameters. }
\label{fig:result1}
\end{figure*}
\begin{table*}[t]
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{1}{|c|}{Model} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Eye\\ Close\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Eye\\ Wide\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Brow\\ Raise\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Brow\\ Anger\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Mouth\\ Open\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Jaw\\ L/R\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Lip\\ Roll\end{tabular}} & Smile & Kiss & \multicolumn{1}{c|}{Avg} \\ \hline
(1) Single scale SFN & 0.082 & 0.265 & 0.36 & 0.451 & 0.373 & 0.331 & 0.359 & 0.223 & 0.299 & 0.305 \\ \hline
(2) Multi-scale SFN & 0.016 & 0.257 & 0.216 & 0.381 & 0.334 & 0.131 & 0.204 & 0.245 & 0.277 & 0.229 \\ \hline
(3) MS-MFN + GT from (2) & 0.117 & 0.407 & 0.284 & 0.405 & 0.284 & 0.173 & 0.325 & 0.248 & 0.349 & 0.288 \\ \hline
\end{tabular}
\caption{Quantitative evaluation of expression accuracy (measured by the expression metric) on our expression test set. Lower error means the model performs better for extreme expressions when required.}
\label{table:exp}
\end{table*}
\begin{figure*}[h!]
\centering
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/2.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/3.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/5.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/7.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/21.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/25.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/27.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/28.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/12.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/13.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/15.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/17.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/31.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/35.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/37.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/38.jpg}
\end{subfigure}
\caption{Results by applying MS-SFN on our expression test set.}
\label{fig:result4}
\end{figure*}
\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\linewidth]{images/retargetingresults.jpg}
\caption{Retargeting from face(s) to 3D character(s).}
\label{fig:retargeting}
\end{figure*}
\subsection{Importance of Joint Training}
Joint regression of both face bounding box locations and 3DMM parameters forces the network to learn exclusive facial features that characterize face shape, expression and pose in addition to differentiating face regions from the background. This helps in more precise face detection in-the-wild by leveraging both 2D information from bounding boxes and 3D information from 3DMM parameters. Table \ref{table:full} shows that average precision (AP) is improved by a large margin with joint training compared to when the same network is trained to only regress bounding box locations. The retargeting accuracy for MFN is also comparable to that of SFN and the slight decrease in NME is because of training MFN on multi-face images and testing on single face images. Nevertheless, we observe improved performance in terms of both NME and AP by using better ground truth generated by multi-scale model. Our detection accuracy (mAP: 98.8\%) outperforms Hyperface \cite{hyperface} (mAP: 97.9\%) and Faceness-Net \cite{facenessnet} (mAP: 97.2\%) on the entire AFW dataset when compared under the same settings. Results of our MFN on multi-face images are illustrated in Fig. \ref{fig:result1}.
\subsection{Evaluation of Expressions}
\iffalse
Our expression evaluation result in Table \ref{table:exp} emphasizes the improvement of multi-scale design on SFN. Since many applications can be created from expression (such as face rig avatar, emotion detection, etc.), it is important to maximize the expression coefficient when it should do. If the model itself can not predict large expression, the common approach is to apply post-processing by multiplying with heuristic number. This post-processing, however, can cause the retargeting result to shake. Most of the expressions are significantly better, however, MFN suffers from the small eye size (in comparison with the entire image). SFN, on the other hand, works on a cropped image, so eyes are zoomed in much further. Attention network for emphasizing small eyes region could be a future work for our MFN.
\fi
Our expression evaluation results in Table \ref{table:exp} emphasizes the improvement of multi-scale design on SFN. MS-MFN performs much better than SS-SFN for all expressions except the eye expressions. This is because eye patches are usually small compared to the entire image for MFN whereas they are zoomed in on cropped images for SFN. Attention network for emphasizing small eyes region could be a future work for our MFN. However, MS-MFN shows less accuracy compared to MS-SFN because it is being tested on single face images while being trained on multi-face images. For the multi-person test set images, we found similar visual results by applying MS-MFN on the whole image and by applying MS-SFN on each face individually cropped from the image. This is expected because MFN is trained with ground truth from SFN. The performance of MS-SFN on our expression test set is shown in Fig. \ref{fig:result4}. We also conducted live performance capture experiments to evaluate the efficiency our system in retargeting facial motion from one or more faces to one or more 3D characters. Fig. \ref{fig:retargeting} shows some screenshots recorded during the experiments.
\subsection{Computational Complexity}
Excluding the IO time, SFN can run at 15ms/frame on Google Pixel 2 (assuming single face and excluding face detector runtime). Face detection with our compressed detector model is 34ms, so separate face detection and retargeting requires 49ms for 1 face, 109ms for 5 faces and 184ms for 10 faces. On the other hand, our proposed MFN performs joint face detection and retargeting at 39ms on any number of faces. The model sizes for compressed face detector is 11.5MB and SFN is 2MB, so the combination is 13.5MB, while our MFN is only 13MB. Hence our joint network reduces both memory requirement and execution time.
\iffalse
\begin{table}[h]
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Method} & \multicolumn{2}{c|}{\textbf{NME (\%)}} & \multicolumn{2}{c|}{\textbf{MAE}} \\ \hline
{} & S1 & S2 & S1 & S2 \\ \hline
Single-scale & 2.21 & - & 0.218 & - \\ \hline
Multi-scale & 2.17 & - & 0.216 & - \\ \hline
Proposed S1 & 1.91 & - & 0.215 & - \\ \hline
S1 + S2 (sep) & 1.91 & 1.62 & 0.215 & 0.209 \\ \hline
S1 + S2 (e2e) & 1.47 & 1.82 & 0.212 & 0.222 \\ \hline
S1 + S2 (sep) + finetune & 1.53 & \textbf{1.25 } & 0.207 & \textbf{0.203} \\ \hline
\end{tabular}
\caption {Normalized Mean Error of 68 landmarks and Mean Absolute Error of 46 expression parameters for LFW dataset (12639 images). S1 and S2 refers to Stage 1 and Stage 2 respectively.}
\end{table}
\begin{table}[h]
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Backbone} & \textbf{Size (MB)} & \textbf{GFLOP} & \textbf{Error} \\ \hline
State-of-the-art 1 & & & \\ \hline
State-of-the-art 2 & & & \\ \hline
State-of-the-art 3 & & & \\ \hline
Ours & & & \\ \hline
\end{tabular}
\caption {Generalization to 29 Expression Basis Dataset}
\end{table}
\fi
\iffalse
\begin{figure}[t]
\centering
\begin{subfigure}[t]{1.0\linewidth}
\centering
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{img1.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{img2.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{img3.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{img4.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{img5.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{img6.jpg}
\end{subfigure}
\caption{Input human images}
\end{subfigure}
\begin{subfigure}[t]{1.0\linewidth}
\centering
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{mesh1.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{mesh2.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{mesh3.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{mesh4.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{mesh5.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\centering
\includegraphics[width = 1.0\textwidth, height = 0.055\textheight]{mesh6.jpg}
\end{subfigure}
\caption{Output 3D meshes}
\end{subfigure}
\caption{
Facial retargeting results. This example shows (a) six different human images with different expressions and different head poses (b) retargeted 3D face meshes.}
\label{mh2c}
\end{figure}
\fi
\iffalse
\subsubsection{Importance of multi-scale features}
Recent works on pose estimation have shown satisfactory performance using just the eye, nose and mouth corner landmarks, which indicates that pose estimation requires global features. Identity or shape of the face depends on the face contour landmarks along with the corners, hence they require region-based features. However, estimation of expression parameters requires a dense arrangement of landmarks and hence requires local features from the input image. This motivated us to combine features at different scales for different parameters. The low level layers of a CNN learn the local features, whereas the high level layers capture the global information.
\subsubsection{Importance of decoupling}
When we used only 2D landmark loss for optimization, we found that the combination of identity, expression and pose parameters learned by the model to fit the landmarks is different from the ground truth, although the output landmarks looked correct. This proves that due to sparsity of the 2D landmarks and the weak perspective projection approximation, the learned parameters may not be ideally distributed unless constrained individually.
\subsubsection{Importance of aligned refinement}
\begin{itemize}
\itemsep0em
\item In-plane rotation of the images before the refinement stage reduces the ambiguity in the data and helps the small network to quickly learn the details about the expression without worrying about the head pose.
\item Training the RefineNet with separate branches for the eye region and the mouth region helps in disentangling the relationship between the eye and mouth blendshapes apart from making it easier for the small network to concentrate on a smaller region.
\item Estimating only the update value of the expression parameters instead of predicting them from scratch helps in faster convergence.
\end{itemize}
We tried finetuning the expression branch in the CoarseNet by fixing all other branches after convergence, but we did not observe noticeable improvement.
\subsection{Comparison with state-of-the-art methods}
\subsubsection{Importance of having our new dataset(for expression)}
1.state of the art technique + old training set + old testing set\\
2. state of the art technique + old training set + our testing set\\
3. state of the art technique + our training set + our testing set\\
4. our technique + old training set + old testing set\\
5. our technique + our training set + our testing set\\
4VS1\& 2 - our technique is about the same for alignment purposes
5VS3 - our technique is designed to tailor the expression transfer
no worries about the complexity of model size, can we come up a compression technique
\subsection{train for tracking}
\subsubsection{Importance of our new DNN design(end2end comparison with state of the art)}
\subsubsection{2D Face Alignment Methods}
3DDFA and SDM\\
Evaluation metric: Normalized Mean Error, normalized by the face bounding box size.
Data-sets: AFLW and AFLW-20003D,
What to compare? : 68 and 21 facial landmark errors
\subsubsection{3D Face Reconstruction Methods}
Richardson (supervised) and Tewari (unsupervised).\\
Evaluation metric: Normalized Average Depth Error (NADE), normalized by the maximum depth of the neutral 3D mesh (generally a heatmap diagram) and Mean Absolute Error (MAE) of all the 3D vertices.
\subsection{Performance Analysis}
Our model is super light-weight, only occupying 1.8 MB of memory. The processing speed of our model is about 25 ms per image on a standard Android mobile device.
\fi
\section{Conclusion}
We propose a lightweight multitask learning network for joint face detection and facial motion retargeting on mobile devices in real time. The lack of 3DMM training data for multiple faces is tackled by generating weakly supervised ground truth from a network trained on images with single faces. We carefully design the network structure and regularization to enforce disentangled representation learning inspired by key observations. Extensive results have demonstrated the effectiveness of our design.
\textbf{Acknowledgements} We thank the anonymous reviewers for their constructive feedback, Muscle Wu, Xiang Yan, Zeyu Chen and Pai Zhang for helping, and Linda Shapiro, Alex Colburn and Barbara Mones for valuable discussions.
\section{Appendix}
Our goal is to perform live facial motion retargeting from multiple faces in a frame to 3D characters on mobile devices. We propose a lightweight multi-scale network architecture to disentangle the 3DMM parameters so that only the expression and pose parameters can be seamlessly transferred to any 3D character. Furthermore, we avoid the performance overhead of running a separate face detector (as in \cite{DDEregression}) by integrating the face detection task with the parameter estimation task.
\subsection{Network Topology}
The architecture of our single scale single face retargeting network is shown in Fig. \ref{fig:ss-sfn}. The details of each block are given in our paper. As can be seen, the resolution (scale) of the image feature maps is reduced by 2 after every block, resulting in a $8 \times 8$ feature map at the end before global average pooling. The network then needs to learn pose, expression and identity parameters from the same $8\times8$ feature map, hence the term single scale.
In retargeting applications, it is necessary to disentangle expression and rotation parameters from the rest (identity, translation and scale) and accurately predict each group of parameters. Besides, we have argued how low-level features are important for expression and high-level features are important for head pose. Keeping these two requirements in mind, we designed our multi-scale single face retargeting network to learn different groups of parameters from separate branches that represent image features at different scales.
\subsection{Multi-face Retargeting Network Outputs}
As mentioned in our paper, the multi-face retargeting network divides the input image into $9\times9$ grid and predicts 5 bounding boxes for each grid cell. Each bounding box $b$ has the following co-ordinates: $t_x$, $t_y$, $t_w$, $t_h$, $t_o$ and $t_{v_{1-104}}$. The final outputs ($b_x, b_y$ - $x,y$ co-ordinates of the box centroid, $b_w, b_h$ - width and height of the box, $b_o$ - objectness score, $b_{\text{id}}, b_{\text{exp}}, b_{\textbf{R}}, b_{\textbf{t}}, b_{\text{f}}$ - 3DMM parameters and $b_{\text{lm}}$ - corresponding 2D landmarks) are then given by:
\begin{equation*}
b_x = \sigma (t_x) + c_x;\; b_y = \sigma (t_y) + c_y
\end{equation*}
\begin{equation*}
b_w = p_w * e^{t_w}; \; b_h = p_h * e^{t_h}
\end{equation*}
\begin{equation*}
b_o = Pr(\text{face}) * IOU(b,\text{face}) = \sigma (t_o)
\end{equation*}
\begin{equation*}
b_{\text{id}} = t_{v_{1-50}}; \; b_{\text{exp}} = \sigma (t_{v_{51-97}})
\end{equation*}
\begin{equation*}
b_{\textbf{R}} = t_{v_{98-101}}; \; b_{\textbf{t}} = t_{v_{102-104}}; \; b_{\text{f}} = \sigma (t_{v_{105}})
\end{equation*}
\begin{equation*}
b_{\text{lm}_x} = b_x + b_w * b_{\Hat{\text{lm}}_x}; \; b_{\text{lm}_y} = b_y + b_h * b_{\Hat{\text{lm}}_y}
\end{equation*}
where $\sigma$ denotes sigmoid function, $(c_x,c_y)$ is the offset of the grid cell containing $b$ from the top left corner of the image, $(p_w,p_h)$ are the dimensions of the bounding box prior (anchor box), $b_{\Hat{\text{lm}}}$ are the initial landmarks obtained using and IOU denotes intersection over union. As evident from the equation, the landmark loss puts additional constraints on the bounding box locations and dimensions. This contributes to our observation that joint face detection and regression of 3DMM parameters improves the accuracy of face detection compared to simple face detection.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{images/single_scale_model.jpg}
\caption{Our single scale single face retargeting network}
\label{fig:ss-sfn}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{images/facedetection.jpg}
\caption{Network output for an image with multiple small faces from the AFW dataset.}
\label{fig:facedetection}
\end{figure}
\begin{figure*}[h!]
\centering
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/(1).jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/(6).jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/(8).jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/(26).jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new1.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new3.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new5.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new7.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/(11).jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/(16).jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/(18).jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/(36).jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new2.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new4.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new6.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new8.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new9.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new11.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new13.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new15.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new17.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new19.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new23.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new21.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new10.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new12.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new14.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new16.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new18.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new20.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new24.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.118\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.09\textheight]{images/showcase/new22.jpg}
\end{subfigure}
\caption{More results from our own expression test set using our single face retargeting network.}
\label{fig:result_sfn}
\end{figure*}
\begin{figure*}[h!]
\centering
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/45092961.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/2353849.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/3944399031.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/2580479214.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/2233672250.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/156474078.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/45092961_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/2353849_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/3944399031_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/2580479214_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/2233672250_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/156474078_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/5144909700.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/5301057994.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/435926861.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/79378097.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/719902933.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/4239974048.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/5144909700_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/5301057994_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/435926861_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/79378097_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/719902933_3d.jpg}
\end{subfigure}
\begin{subfigure}[t]{0.16\linewidth}
\includegraphics[width = 1.0\textwidth, height = 0.1\textheight]{images/showcase/4239974048_3d.jpg}
\end{subfigure}
\caption{More results from AFW dataset \cite{afwdatapaper} using our joint detection and retargeting model.}
\label{fig:result_mfn}
\end{figure*}
\subsection{Accuracy of Face Detection}
Our network can detect multiple small faces of reasonable size even though it is not trained on images more than 20 faces. Figure \ref{fig:facedetection} shows our network outputs for an image with more than 20 faces in the AFW dataset. The blue rectangles denote the predicted bounding boxes and the red points denote the 2D landmarks (for better viewing) projected from the predicted 3DMM parameters.
\subsection{Video Results}
The performance of our networks on videos is shown in a video\footnote{\url{https://homes.cs.washington.edu/~bindita/cvpr2019.mp4}}. In the first half, we show the results of retargeting from a single face video to a generic 3D human face model using our single face retargeting network. The face bounding box for the current frame is obtained from the boundaries of the 2D landmarks predicted in the previous frame. In the second half, we show the results of retargeting with videos having multiple faces (Music video dataset \cite{musicvideo}) using our multi-face retargeting network (only frames with multiple faces are shown).
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\bibliographystyle{acmsiggraph}
\nocite{*}
|
{
"timestamp": "2019-03-01T02:01:30",
"yymm": "1902",
"arxiv_id": "1902.10744",
"language": "en",
"url": "https://arxiv.org/abs/1902.10744"
}
|
\section{Introduction}
\IEEEPARstart{T}{he} da Vinci Research Kit (dVRK) is an open-source surgical robotic system whose mechanical components are obtained from the first generation of the da Vinci Surgical Robot\textregistered \cite{kazanzides2014open}. It has made the research on surgical robotics more accessible. To date, researchers from over 30 institutes around the world are using the physical dVRK \cite{dVRK_wiki}, and some other researchers are using dVRK simulations \cite{kazanzides2014open}.
Simulations and model-based control require the dynamic model of the dVRK. \citeauthor{fontanelli2017modelling} \cite{fontanelli2017modelling} has obtained the dynamic model of the dVRK using dynamic model identification techniques. However, the dynamic parameters obtained in \cite{fontanelli2017modelling} are base parameters (also called lumped parameters) \cite{gautier1991numerical}, a minimum set of dynamic parameters that can sufficiently describe the dynamic model of a robot. Although base parameters are adequate to represent the dynamics of a robot in dynamic equations, standard parameters are required for the efficient recursive Newton-Euler-based dynamic algorithms. Towards this end, several dynamics libraries utilize standard parameters, such as
Rigid Body Dynamics Library (RBDL) \cite{Felis2016} and Kinematics and Dynamics Library (KDL) \cite{kdl-url}.
The dynamic parameters vary between different robots of the same make and model due to manufacturing and assembly variances. Furthermore, the assembly components of the robots are subject to deformation and wear \& tear along their life cycle which can potentially alter the dynamic model. As such, the dynamic model identification is required before implementation of any robust model-based control algorithm. This requirement drives the need for a robust open-source dynamic model identification package.
There are existing software packages for the dynamic model identification of generic open-chain manipulators such as SymPybotics \cite{https://doi.org/10.5281/zenodo.11365} and FloBaRoID \cite{bethge2017flobaroid}. However, these packages lack the capability of modeling closed-loop kinematic chains, springs, counterweights, and tendon couplings, which are inherent to dVRK's mechanical design. To address this, we utilize and extend the convex optimization-based method proposed by \citeauthor{sousa2014physical} \cite{sousa2014physical} to obtain the physically-feasible standard dynamic parameters of the dVRK arms.
\begin{figure}
\centering
\includegraphics[width=0.32\textwidth]{procedure.pdf}
\caption{\label{fig:identification_workflow} Workflow of dynamic model identification}
\end{figure}
This paper is structured into seven sections as the workflow of dynamic model identification in Fig. \ref{fig:identification_workflow}. Sections \ref{sec_kinematics} and \ref{sec_dynamics} explain the mathematical formulation of the kinematic and dynamic modeling of the Master Tool Manipulator (MTM) and Patient Side Manipulator (PSM). Section \ref{sec_trajectory} describes the trajectory optimization method to improve parameter identification quality. Section \ref{sec_identification} presents the identification method to obtain the standard dynamic parameters with physical feasibility considered.The experimental results which validate the proposed approaches are presented in section \ref{sec_identification}. Finally, the concluding arguments are entailed in section \ref{sec_conclusion}.
\section{Kinematic Modeling of the dVRK}\label{sec_kinematics}
\begin{table*}[!t]
\caption{Modeling Description of the MTM. Links 1 to 7 Correspond to the Links Described in Fig. \ref{fig:mtm_frame_def}. $M_4$ Corresponds to the Friction and Inertia Modeling of Motor 4. The Dimensions are Shown in Fig. \ref{fig:mtm_frame_def}}
\label{table:mtm_geometry}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|}
\hline
link & joint type & prev & succ & $a_{i-1}$ & $\alpha_{i-1}$ & $d_i$ & $\theta_i$ & link inertia & motor inertia & friction & spring\\
\hline
\hline
1 & R & 0 & $2, 3'$ & 0 & 0 & $-l_{base2pitch}$ & $q_1$ & \cmark & \xmark & \cmark & \xmark\\
\hline
2 & R & 1 & 3 & 0 & $-\pi/2$ & 0 & $q_2 + \pi/2$ & \cmark & \xmark & \cmark & \xmark\\
\hline
3 & R & 2 & 4 & $l_{arm}$ & 0 & 0 & $q_3 + \pi/2$ & \cmark & \xmark & \cmark & \xmark\\
\hline
$3'$ & R & 1 & $3''$ & 0 & $-\pi/2$ & 0 & $q_2 + q_3 + \pi$ & \cmark & \xmark & \cmark & \xmark\\
\hline
$3''$ & R & $3'$ & $3'''$ & $l_{back2front}$ & 0 & 0 & $-q_3 - \pi/2$ & \cmark & \xmark & \cmark & \xmark\\
\hline
4 & R & 3 & 5 & $l_{forearm}$ & $-\pi/2$ & $h$ & $q_4$ & \cmark & \xmark & \cmark & \xmark\\
\hline
5 & R & 4 & 6 & 0 & $\pi/2$ & 0 & $q_5$ & \cmark & \xmark & \cmark & \cmark\\
\hline
6 & R & 5 & 7 & 0 & $-\pi/2$ & 0 & $q_6 + \pi/2$ & \cmark & \xmark & \cmark & \xmark\\
\hline
7 & R & 6 & - & 0 & $-\pi/2$ & 0 & $q_7 + \pi$ & \cmark & \xmark & \cmark & \xmark\\
\hline
$M_4$ & R & - & - & 0 & 0 & 0 & $q^d_4$ & \xmark & \cmark & \cmark & \xmark\\
\hline
\end{tabular}
\end{center}
\end{table*}
The dVRK-ROS package \cite{kazanzides2014open} employs the (Denavit-Hartenberg) DH convention based on kinematic frames located on the joint axes, whereas almost all the joints are actuated off axes using a combination of cams, links, or cables and pulleys. Consequently, we need to define additional coordinate axes to relate the joint motions - as defined in the dVRK-ROS package - with the motor torques.
To build the relationship between the robot joint motion in the dVRK-ROS package \cite{kazanzides2014open} and the torque of each motor, several types of joint coordinates are defined. $\boldsymbol{q}^d$ is the joint coordinate used in the dVRK-ROS package. $\boldsymbol{q} = \begin{bmatrix}
(\boldsymbol{q}^b)^{\top} & \boldsymbol{(q}^a)^{\top}
\end{bmatrix}^{\top}$ is the joint coordinate used in the kinematic modeling in this work, where $\boldsymbol{q}^b$ is the basis joint coordinate which can adequately represent the kinematics of the robot, and $\boldsymbol{q}^a$ is the additional joint coordinate which represents the joint coordinate of the passive joints in the parallel mechanism and can be represented by the linear combination of $\boldsymbol{q}^b$. Since both the MTM and PSM have seven actuated degrees of freedom (DOF), the basis joint coordinate can be represented by $\boldsymbol{q}^b =
\begin{bmatrix}
q_1 & q_2 & \hdots & q_7
\end{bmatrix}^\top$. $\boldsymbol{q}^{m}$ is the equivalent motor coordinate which is considered to happen at joints, with the reduction ratio caused by gearboxes and tendons included for most motors unless explicitly specified. Finally,
$\boldsymbol{q}^{c} = \begin{bmatrix}
\boldsymbol{q}^\top & (\boldsymbol{q}^m)^\top
\end{bmatrix}^\top$ defines the complete joint coordinate.
The relation between these joint coordinates is illustrated for both the MTM and PSM in this section.
\subsection{Kinematic Modeling of the MTM}
The left and right MTMs are identical to each other, except the last four joints being mirrored to each other. Consequently, the two MTMs can be modeled in a similar fashion.
The frame definition is shown in Fig. \ref{fig:mtm_frame_def}, and the kinematic parameters of the MTM are described in Table \ref{table:mtm_geometry}. The kinematics of the MTM can be described as
\begin{itemize}
\item Joint 1 rotates around the Z-axis of the base frame, $z_0$.
\item Joints 2, 3, $3'$, $3''$, and $3'''$ construct a parallelogram, which is actuated by joints 2 and $3'$.
\item Joints 4, 5, 6, and 7 form a 4-axis non-locking gimbal.
\end{itemize}
The kinematics of the MTM is fully described by the basis joint coordinates $\boldsymbol{q}^b$, which are equal to the dVRK joint coordinate $\boldsymbol{q}^d$, $ \boldsymbol{q}^b = \boldsymbol{q}^d$.
The additional joints $\boldsymbol{q}^a$ can be described as the linear combination of $\boldsymbol{q}^b$ by
\begin{equation}\label{eq:mtm_coupling_model2closeloop}
\boldsymbol{q}^a =
\begin{bmatrix}
q_{3'} &
q_{3''} &
q_{3'''}
\end{bmatrix}^\top =\begin{bmatrix}
q_2 + q_3 &
-q_3 &
q_3
\end{bmatrix}^\top
\end{equation}
\begin{figure}[!t]
\centering
\includegraphics[width=0.29\textwidth]{mtm_crop.pdf}
\caption{\label{fig:mtm_frame_def}MTM frame definition using modified DH convention. The dimensions (in mm) are referred from the user guide of the dVRK and measured manually if not available}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.26\textwidth]{coupled_tendon_croped.pdf}
\caption{\label{fig:mtm_tendon}Modeling of the tendon coupling of the MTM}
\end{figure}
Joints 1, 5, 6, and 7 are independently driven, and thus the motion of these joints is equivalent to their corresponding driven motors, $\boldsymbol{q}^d_{1, 5-7}$ = $\boldsymbol{q}^m_{1, 5-7}$. The motion of $q^d_4$ depends on both $q^m_4$ and $q^d_3$ and can be described by
\begin{equation}
q^d_4 = q^m_4 - {r_3}/{r_4}\cdot q^d_3
\end{equation}
where $r_3$ and $r_4$ are the radii of the pulleys shown in Fig. \ref{fig:mtm_tendon}.
Based on the user guide of the dVRK, $r_3 \approx 14.01$ mm and $r_4 \approx 20.92$ mm, and thus $r_3/r_4 \approx 0.6697$.
\begin{table*}[!ht]
\caption{Modeling Description of the PSM.
Links 1 to 7 Correspond to the Links Described in Fig. \ref{fig:psm_frame_def}. $M_6$ and $M_7$ Correspond to the Friction and Inertia Modeling of Motor 6 and Motor 7. $F_{67}$ Corresponds to the Friction Modeling between Link 6 and Link 7. The Dimensions are Shown in Fig. \ref{fig:psm_size_def}.}
\label{table:psm_geometry}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|}
\hline
link & \thead{joint \\ type}& prev & succ & $a_{i-1}$ & $\alpha_{i-1}$ & $d_i$ & $\theta_i$ & \thead{link \\ inertia} & \thead{motor \\ inertia} & friction & spring\\
\hline
\hline
1 & R & 0 & 2 & 0 & $\pi/2$ & 0 & $q_1+\pi/2$& \cmark & \xmark & \cmark & \xmark\\
\hline
2 & R & 1 & $2'$, $3'$ & 0 & $-\pi/2$ & 0 & $q_2 - \pi/2$& \cmark & \xmark & \cmark & \xmark\\
\hline
$2'$ & - & 2 & $2''$, $2'''$ & $l_{2L3}$ & 0 & 0 & $\pi/2$& \xmark & \xmark & \xmark & \xmark\\
\hline
$2''$ & R & $2'$ & $2''''$, $2'''''$, 3 & $l_{2H1}$ & 0 & 0 & $ -q_2 + \pi/2$ & \cmark & \xmark & \xmark & \xmark\\
\hline
$2'''$ & R & $2'$ & - & $l_{2H1}+l_{2H2}$ & 0 & 0 & $ -q_2 + \pi/2$& \cmark & \xmark & \xmark & \xmark\\
\hline
$2''''$ & R & $2''$ & 3 & $l_{2L2}$ & 0 & 0 & $q_2$& \cmark & \xmark & \xmark & \xmark\\
\hline
$2'''''$ & R & $2''$ & - & $l_{2L1}$ & 0 & 0 & $ q_2 + \pi$& \cmark & \xmark & \xmark & \xmark\\
\hline
3 & P & $2''''$ & 4 & $l_3$ & $-\pi/2$ & $q_3 - l_{RCC} + l_{2H1}$ & 0 & \cmark & \xmark & \cmark & \xmark\\
\hline
$3'$ & P & 2 & - & $l_{2L3}$ & $-\pi/2$ & $q_3 $ & 0 & \cmark & \xmark & \xmark & \xmark\\
\hline
4 & R & 3 & 5 & 0 & 0 & $l_{tool}$ & $q_4$& \xmark & \cmark & \cmark & \cmark\\
\hline
5 & R & 4 & 6, 7 & 0 & $\pi/2$ & 0 & $q_5 +\pi/2$ & \xmark & \cmark & \cmark & \xmark\\
\hline
6 & R & 5 & - & $l_{pitch2yaw}$ & $-\pi/2$ & 0 & $q_6+\pi/2$& \xmark & \xmark & \cmark & \xmark\\
\hline
7 & R & 5 & - & $l_{pitch2yaw}$ & $-\pi/2$ & 0 & $q_7+\pi/2$& \xmark & \xmark & \cmark & \xmark\\
\hline
$M_6$ & R & - & - & 0 & 0 & 0 & $q^m_6$ & \xmark & \cmark & \cmark & \xmark\\
\hline
$M_7$ & R & - & - & 0 & 0 & 0 & $q^m_7$ & \xmark & \cmark & \cmark & \xmark\\
\hline
$F_{67}$ & R & - & - & 0 & 0 & 0 & $q_6 - q_7$ & \xmark & \xmark & \cmark & \xmark\\
\hline
\end{tabular}
\end{center}
\end{table*}
The coupling between the dVRK-ROS joint coordinates $\boldsymbol{q}^d_{2-4}$ and motor joint coordinates $\boldsymbol{q}^m_{2-4}$ due to the parallelogram and tendon is resolved by the coupling matrix $\boldsymbol{A}^d_m$ as
\begin{equation}
\boldsymbol{q}^d_{2-4} = \boldsymbol{A}^d_m \boldsymbol{q}^m_{2-4}
\end{equation}
where $\boldsymbol{A}^m_d = \small \begin{bmatrix}
1 & 0 & 0\\ -1 & 1 & 0\\ 0.6697 & -0.6697 & 1 \end{bmatrix} \normalsize$, based on the user guide of the dVRK.
\subsection{Kinematic Modeling of the PSM}
The frame definition and kinematic dimensions of the PSM are shown in Fig. \ref{fig:psm_geo}, and the corresponding parameters are shown in Table \ref{table:psm_geometry}. The kinematics of the PSM can be concluded as
\begin{itemize}
\item The first two revolute joints form a remote-center-of-motion (RCM) point which remains fixed in Cartesian space. This RCM is achieved via a double four-bar linkage with six links actuated by a single motor.
\item The third joint is prismatic and provides insertion of the instrument through the RCM. The first three joints allow the 3-DOF Cartesian space motion.
\item Revolute joints 4 and 5 construct the roll and pitch motion of the wrist to reorient the end-effector.
\item The last two joints construct the yaw motion of the end-effector, as well as the opening and closing of the gripper.
\end{itemize}
\begin{figure}[!t]
\centering
\subfloat[\label{fig:psm_frame_def}Frame definition of the PSM using modified DH convention]{\includegraphics[width=0.98\columnwidth]{psm_frame_def.pdf}}\\
\subfloat[\label{fig:psm_size_def}Dimension definition of the PSM. The dimensions (in mm) are from the user guide of the dVRK and measured manually if not available]{\includegraphics[width=0.98\columnwidth]{psm_size_def.pdf}}
\caption{\label{fig:psm_geo}
PSM frame and dimension definition}
\end{figure}
\begin{figure}[!htbp]
\centering
\subfloat[\label{fig:psm_gripper_motion_modeling}Modeling of the motion of the PSM gripper]{\includegraphics[width=0.37\columnwidth]{psm_coupling.pdf}}
\qquad
\subfloat[\label{fig:psm_gripper_friction_modeling}Modeling of the frictions of the PSM gripper]{\includegraphics[width=0.35\columnwidth]{psm_friction.pdf}}
\caption{\label{fig:psm_gripper}Modeling of the gripper of the PSM}
\end{figure}
We model the first five joints of the PSM identical to the dVRK-ROS package, that is $\boldsymbol{q}_{1-5} = \boldsymbol{q}^d_{1-5}$. The dVRK-ROS package models the last two joints as $q^d_6$, the angle from the insertion direction to the bisector of the two jaw tips, and $q^d_7$, the angle between the two jaw tips. However, the gripper jaws are designed and actuated as two seperate links which we consider in our model. As shown in Fig. \ref{fig:psm_gripper_motion_modeling}, the relation between the dVRK-ROS joint coordinates $\boldsymbol{q}^d_{6-7}$, and the joint coordinates $\boldsymbol{q}_{6-7}$ in our model is described by
\begin{equation}\label{eq:psm_coupling_model2dvrk}
\boldsymbol{q}^d_{6-7} =
\begin{bmatrix}
q^d_{6} &
q^d_{7}
\end{bmatrix}^\top =\begin{bmatrix}
0.5q_{6} + 0.5q_{7} &
-q_{6}+q_{7}
\end{bmatrix}^\top
\end{equation}
Since the first four joints are independently driven, the equivalent motor motion is considered to occur at joints and thus is the same as joint motion, $\boldsymbol{q}^d_{1-4} = \boldsymbol{q}^m_{1-4}$. Based on the user guide of the dVRK, the coupling of the wrist joint actuation can be resolved by the coupling matrix $\boldsymbol{A}^d_m$ mapping the motor joint coordinates $\boldsymbol{q}^m_{5-7}$ to the dVRK-ROS joint coordinates $\boldsymbol{q}^d_{5-7}$ by
\begin{equation}\label{eq:psm_coupling}
\boldsymbol{q}^d_{5-7} = \boldsymbol{A}^d_m \boldsymbol{q}^m_{5-7}
\end{equation}
where $\boldsymbol{A}^d_m = \small \begin{bmatrix}
1.0186 & 0 & 0\\
-0.8306 & 0.6089 & 0.6089\\
0 & -1.2177 & 1.2177
\end{bmatrix} \normalsize$.
\section{Dynamic Modeling of the dVRK}\label{sec_dynamics}
In this section, the dynamic parameters are described first. The dynamic equation is then formulated based on Euler-Lagrange equation. Finally, the dynamic modeling of the MTM and PSM is introduced based on the formulation.
\subsection{Dynamic Parameters}
The formulation of the dynamic parameters is modified from \cite{sousa2014physical} and tailored towards the mechanical design of the dVRK arms.
Each link $k$ is characterized by the mass $m_k$, the center of mass (COM) relative to the link frame $k$, $
\boldsymbol{r}_k$,
and the inertia tensor about the COM, $\boldsymbol{I}_k$.
To express the equations of motion as a linear form of dynamic parameters, we use the so-called barycentric parameters \cite{maes1989linearity}, in which the mass $m_k$ of link $k$ is first used, followed by the first moment of inertia $\boldsymbol{l}_k$.
\begin{equation} \label{eq:mr2l}
\boldsymbol{l}_k =m_k \boldsymbol{r}_k
\end{equation}
Finally, the inertia tensor $\boldsymbol{L}_k$ about frame $k$ is used \cite{khalil2004modeling}. $\boldsymbol{L}_k$ is calculated via the parallel axis theorem
\begin{equation}\label{eq:Iml2L}
\boldsymbol{L}_k = \boldsymbol{I}_k + m_k \boldsymbol{S}(\frac{\boldsymbol{l}_k}{m_k})^\top\boldsymbol{S}(\frac{\boldsymbol{l}_k}{m_k}) =
\small
\begin{bmatrix}
L_{kxx} & L_{kxy} & L_{kxz}\\
L_{kxy} & L_{kyy} & L_{kyz}\\
L_{kxz} & L_{kyz} & L_{kzz}
\end{bmatrix}
\normalsize
\end{equation}
where $\boldsymbol{S}(\cdot)$ is the skew-symmetric operator.
The aforementioned inertial parameters of link $k$ are grouped into a vector $\boldsymbol{\delta}_{Lk} \in \mathbb{R}^{10}$ as
\begin{equation}
\label{eq:inertia}
\boldsymbol{\delta}_{Lk} = \small [\begin{matrix} L_{kxx} & L_{kxy} & L_{kxz} & L_{kyy} & L_{kyz} & L_{kzz} &
\boldsymbol{l}_{k}^{\top} & m_k \end{matrix}]^\top \normalsize
\end{equation}
Besides the inertial parameters of link $k$, the corresponding joint friction coefficients, motor inertia $I_{mk}$, and spring stiffness $K_{sk}$ are grouped as additional parameters
\begin{equation}
\boldsymbol{\delta}_{Ak} = \begin{bmatrix}
F_{vk} & F_{ck} & F_{ok} & I_{mk} & K_{sk}
\end{bmatrix}^\top
\end{equation}
where $F_{vk}$ and $F_{ck}$ are the viscous and Coulomb friction constants, and $F_{ok}$ is the Coulomb friction offset of joint $k$.
Eventually, all the parameters of $n$ joints are grouped together as the dynamic parameters $\boldsymbol{\delta}$ of the robot.
\begin{equation}
\label{eq:parameters}
\boldsymbol{\delta} =
\begin{bmatrix}
\boldsymbol{\delta}_{L1}^\top & \boldsymbol{\delta}_{A1}^\top & ... & \boldsymbol{\delta}_{Ln}^\top & \boldsymbol{\delta}_{An}^\top
\end{bmatrix}^{\top}
\end{equation}
\subsection{Dynamic Model Formulation}
The Euler-Lagrange equation for closed-chain robots \cite{nakamura1989dynamics} is used to model the dynamics of the dVRK. The Lagrangian is calculated by the difference of the kinetic energy $K$ and potential energy $P$ of the robot, $L = K - P$. Motor inertias and springs are not included in $L$ and are modeled separately.
The relation from motor motion $\boldsymbol{q}^m$ to the torque of each motor $i$ caused by link inertia is then computed as
\begin{equation}
\tau^m_{LIi} = \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial \dot{q}^m_{i}} - \frac{\partial L}{\partial q^m_{i}}
\end{equation}
The friction torques of all the joints $\boldsymbol{q}^{c}$ are considered as
\begin{equation} \label{eq:friction}
\boldsymbol{\tau}^{c}_f(\dot{\boldsymbol{q}}^{c}) = \boldsymbol{F}_v \dot{\boldsymbol{q}}^{c} + \boldsymbol{F}_c \boldsymbol{\mathrm{sgn}}(\dot{\boldsymbol{q}}^{c}) + \boldsymbol{F}_o
\end{equation}
where $\boldsymbol{F}_v$ and $\boldsymbol{F}_c$ are diagonal matrices encapsulating the viscous and Coulomb friction constants, and $\boldsymbol{F}_o$ is the vector of the Coulomb friction offset constants corresponding to the joint coordinate $\boldsymbol{q}^c$.
The torques caused by motor inertia are defined as
\begin{equation}
\boldsymbol{\tau}^m_{MI}(\ddot{\boldsymbol{q}}^m) = \boldsymbol{I}_m \ddot{\boldsymbol{q}}^m
\end{equation}
For spring $k$, we only model the stiffness constant $K_{sk}$ as its parameter, which results into the spring torques
\begin{equation}
\boldsymbol{\tau}^c_s(\boldsymbol{q}^{c}) = \boldsymbol{K}_s {\boldsymbol{\Delta l}_s}
\end{equation}
where $\boldsymbol{K}_s$ is a diagonal matrix of the stiffness constants of the springs, and $\boldsymbol{\Delta l}_s$ is their corresponding equivalent prolongation vector.
The joint torques caused by springs and frictions can be projected onto the motor joints, using the Jacobian matrix of their corresponding joint coordinate with respect to the motor joint angle $\boldsymbol{q}^m$ \cite{nakamura1989dynamics}. Thus, the motor torques $\boldsymbol{\tau}^m$ with link inertia, springs, frictions, motor inertia, and motion couplings considered are given by
\begin{equation} \label{eq:close_chain_torque}
\boldsymbol{\tau}^m = \boldsymbol{\tau}^m_{LI} + \boldsymbol{\tau}^m_{MI}(\ddot{\boldsymbol{q}}^m) + \frac{\partial{\boldsymbol{q}^{c}}}{\partial\boldsymbol{q}_m}(\boldsymbol{\tau}^c_s(\boldsymbol{q}^{c}) + \boldsymbol{\tau}^{c}_f(\dot{\boldsymbol{q}}^{c}))
\end{equation}
To identify $\boldsymbol{\delta}$, \eqref{eq:close_chain_torque} is rewritten into \eqref{eq:close_chain_torqe_linear} by the linear parameterization.
\begin{equation} \label{eq:close_chain_torqe_linear}
\boldsymbol{\tau}^m = \boldsymbol{H}(\boldsymbol{q}^m, \dot{\boldsymbol{q}}^m, \ddot{\boldsymbol{q}}^m) \boldsymbol{\delta}
\end{equation}
QR decomposition with pivoting \cite{gautier1991numerical} is used to calculate the base parameters.
With this method, we get a permutation matrix $\boldsymbol{P}_\mathrm{b} \in \boldsymbol{\mathbb{R}}^{n\times b}$, where $n$ is the number of standard dynamic parameters and $b$ is the number of base parameters.
The base parameters $\boldsymbol{\delta}_{\mathrm{b}}$ and the corresponding regressor $\boldsymbol{H}_\mathrm{b}$ can be calculated by
\begin{equation} \label{eq:base_parameters}
\boldsymbol{\delta}_{\mathrm{b}} = \boldsymbol{P}_\mathrm{b}^{\top}\boldsymbol{\delta},\,\,\,\, \boldsymbol{H}_{\mathrm{b}} = \boldsymbol{H}\boldsymbol{P}_\mathrm{b}
\end{equation}
\subsection{Dynamic Modeling of the Master Tool Manipulator (MTM)}
The dynamic modeling description for each link of the MTM is shown in Table \ref{table:mtm_geometry}. All the nine links are modeled with link inertia. The frictions of all the modeling joints $\boldsymbol{q}$ are considered, except joint $3'''$ since joint $3'''$ and joint $3''$ share the same joint coordinate. For an independently driven joint, the friction from the joint and its driven motor is coupled together and thus impossible to distinguish from each other. All the motors except the $\nth{4}$ one have their corresponding independently driven joints which have already been modeled with link inertia and joint friction. Therefore, only motor 4 is modeled with motor inertia and motor friction.
The electrical cable along joint 4 (Fig. \ref{fig:mtm_cable}) affects its joint torque significantly. The joint torque data of joint 4, $\tau_{4}^+$ and $\tau^-_4$, is collected, with joint 4 rotating at $\pm 0.4$ rad/s and other joints being stationary, as shown in Fig. \ref{fig:mtm_cable_modeling}. We exclude the joint acceleration to record data at constant joint velocities, which explicitly removes any torque due to inertia. Moreover, due to the friction model in \eqref{eq:friction}, the frictions with the joint velocity at $\pm 0.4$ rad/s should be opposite to each other if the Coulomb friction offset is not considered. Thus finally, if we computed the mean of $\tau_{4}^+$ and $\tau^-_4$, the viscous friction and Coulomb friction terms will be canceled, and the joint friction offset and torque applied to the joint from the cable physically acting on it $\tau^m_{c4}(q_4)$ will be left.
To get $\tau^m_{c4}(q_4)$, we first fit the joint torque data at $\pm 0.4$ rad/s using $\nth{7}$ order polynomial functions of $q_4$, respectively, as shown in Fig. \ref{fig:mtm_cable_modeling}. Next, the mean of the obtained coefficients of the two polynomials $\boldsymbol{p}_4^+$ and $\boldsymbol{p}_4^-$ is calculated as the coefficients of the polynomial that represents $\tau^m_{c4}(q_4)$.
\begin{figure}[!t]
\centering
\subfloat[\label{fig:mtm_cable}Cable on joint 4 of the MTM]{\includegraphics[width=0.35\columnwidth]{mtm_cable4.pdf}}
\qquad
\subfloat[\label{fig:mtm_cable_modeling}Test and modeling of the joint torque from the Cable on joint 4]{\includegraphics[width=0.48\columnwidth]{mtm_cable_torque.pdf}}
\caption{Test and modeling of the joint torque from the cable on joint 4 of the MTM}
\end{figure}
\begin{figure}[!t]
\centering
\subfloat[\label{fig:mtm_right_spring_picture}Spring on joint 5]{\includegraphics[width=0.36\columnwidth]{mtm_spring.pdf}}
\qquad
\subfloat[\label{fig:mtm_right_spring_modeling}Modeling of the spring]{\includegraphics[width=0.38\columnwidth]{mtm_spring_modeling.pdf}}
\caption{\label{fig:mtm_right_spring}Spring on joint 5 of the MTM and its modeling}
\end{figure}
In addition, on joint 5 of the MTM, there is a spring to balance the gravitational force (Fig. \ref{fig:mtm_right_spring_picture}).
Due to the model of the spring shown in Fig. \ref{fig:mtm_right_spring_modeling}, the joint torque from the spring is given by
\begin{equation}
\tau_{s5} = f_{s} \cdot d_{s} = K_{s5}(l_s - l_r) \cdot d_{s} = K_{s5}\Delta l_{s5}
\end{equation}
where $l_s$ is the length between the two axes connecting the spring, which can be calculated using the law of sines as
\begin{equation}
l_s = \sqrt[]{h_s^2 + r_s^2 - 2 h_s r_s \cos(\pi + q_o - q_5)}
\end{equation}
and $l_r \approx 61.3$ mm by measurement is the value of $l_s$ when the spring is relaxed.
Based on basic trigonometry, the moment arm $d_{s}$ can be calculated by
\begin{equation}
d_{s} = h_s r_s \sin(\pi + q_o - q_5) / l_s
\end{equation}
where $h_s$, $r_s$ and $q_o$ are constants shown in Fig. \ref{fig:mtm_right_spring_modeling}.
Thus, $\Delta l_{s5} = (l_s - l_r)d_{s}$.
\subsection{Dynamic Modeling of the PSM}
The dynamic modeling description of the PSM is shown in Table \ref{table:psm_geometry}. Inertia is considered for all the links contributing to the Cartesian motion, including the counterweight, link $3'$. The motor inertia of these joints is ignored since it is not significant compared to their link inertia. The inertia of the wrist and gripper links is minimal, and thus infeasible to identify. Therefore, we only model the inertia of motors for the wrist and gripper, corresponding to the motion of $\boldsymbol{q}^m_{4-7}$.
Since joints $2$, $2''$, $2'''$, $2''''$, and $2'''''$ are all driven by a single motor, their friction can be represented by the friction of one joint for simplicity. Thus, among these joints, only joint 2 is modeled with friction. Similarly, only joint 3 is modeled with friction out of joints 3 and $3'$. Because of the contact between links 5 and 6, and between links 5 and 7 as shown in Fig. \ref{fig:psm_gripper_friction_modeling}, the frictions on joints 6 and 7 are modeled, corresponding to the motion of $q_6$ and $q_7$. Moreover, the friction between link 6 and link 7 due to the contact between the two jaw tips is considered, corresponding to the motion of $q_7-q_6$. Additionally, the frictions on the motor sides of the last four joints are also modeled, corresponding to the motor motion of $\boldsymbol{q}^m_{4-7}$.
The torsional spring on joint 4 which rotates the joint back to its home position is modeled as
\begin{equation}
\tau_{s4} = K_{s4}(-q_4) = K_{s4}\Delta l_{s4}
\end{equation}
\section{Excitation Trajectory Optimization}\label{sec_trajectory} \label{sec:excitation_trajectory}
A periodic excitation trajectory based on Fourier series \cite{swevers1997optimal} is used to generate data for dynamic model identification. This trajectory minimizes the condition number of the regression matrix $\boldsymbol{W}_b$ for the base parameters $\boldsymbol{\delta}_\mathrm{b}$ that decide the dynamic behavior of a robot.
\begin{equation}
\label{eq:reg_base}
\boldsymbol{W}_b =
\small
\begin{bmatrix}
\boldsymbol{H}_b(\boldsymbol{q}^m_1, \dot{\boldsymbol{q}}^m_1, \ddot{\boldsymbol{q}}^m_1)\\
\boldsymbol{H}_b(\boldsymbol{q}^m_2, \dot{\boldsymbol{q}}^m_2, \ddot{\boldsymbol{q}}^m_2)\\
\vdots\\
\boldsymbol{H}_b(\boldsymbol{q}^m_{S}, \dot{\boldsymbol{q}}^m_{S}, \ddot{\boldsymbol{q}}^m_S)\\
\end{bmatrix}
\normalsize
\end{equation}
where $\boldsymbol{q}^m_{i}$ is the motor joint coordinate at $i^{\mathrm{th}}$ sampling points and $S$ is the sampling point number.
The joint position $q_{ak}$ of joint $k$ can be calculated by
\begin{equation}
q^m_{ak}(t) = q^m_{ok} + \sum_{l=1}^{n_\mathrm{H}} \frac{a_{lk}}{\omega_f l} \sin(\omega_f l t) - \frac{b_{lk}}{\omega_f l}\cos(\omega_f l t)
\end{equation}
where $\omega_f = 2\pi f_f$ is the angular component of the fundamental frequency $f_f$, $n_\mathrm{H}$ is the harmonic number of Fourier series, $a_{lk}$ and $b_lk$ are the amplitudes of the $l^{\mathrm{th}}$-order sine and cosine functions of joint $k$, $q^m_{ok}$ is the position offset of motor joint $k$, and $t$ is the time.
The motor joint velocity $\dot{q}^m_k(t)$ and acceleration $\ddot{q}^m_k(t)$ can be calculated easily by the differentiation of $q^m_k(t)$. And the trajectory must satisfy the following constraints:
\begin{itemize}
\item The joint position $\boldsymbol{q}$ is between the lower bound $\boldsymbol{q}_l$ and the upper bound $\boldsymbol{q}_u$, $\boldsymbol{q}_l \le \boldsymbol{q} \le \boldsymbol{q}_u$.
\item The robot is confined in its work space. The Cartesian position $\boldsymbol{p}_k$ of frame $k$ is within its lower bound $\boldsymbol{p}_l$ and upper bound $\boldsymbol{p}_u$, $\boldsymbol{p}_l \le \boldsymbol{p} \le \boldsymbol{p}_u$.
\end{itemize}
Finally, pyOpt \cite{perez2012pyopt} is used to solve this constrained nonlinear optimization problem.
\section{Parameter Identification}\label{sec_identification}
To identify the dynamic parameters, we use the excitation trajectory described in Section \ref{sec:excitation_trajectory} to move the robot. Data is collected at each sampling time to obtain the regression matrix $\boldsymbol{W}$ and the dependent variable vector $\boldsymbol{\omega}$:
\begin{equation}
\label{eq:reg}
\boldsymbol{W} =
\small \begin{bmatrix}
\boldsymbol{H}(\boldsymbol{q}^m_1, \dot{\boldsymbol{q}}^m_1, \ddot{\boldsymbol{q}}^m_1)\\
\boldsymbol{H}(\boldsymbol{q}^m_2, \dot{\boldsymbol{q}}^m_2, \ddot{\boldsymbol{q}}^m_2)\\
\vdots\\
\boldsymbol{H}(\boldsymbol{q}^m_\mathrm{S}, \dot{\boldsymbol{q}}^m_S, \ddot{\boldsymbol{q}}^m_S)\\
\end{bmatrix} \normalsize,\:
\boldsymbol{\omega} =
\small \begin{bmatrix}
\boldsymbol{\tau}^m_{1} \\
\boldsymbol{\tau}^m_{2} \\
\vdots \\
\boldsymbol{\tau}^m_{S}\\
\end{bmatrix}
\normalsize
\end{equation}
where $S$ is the sampling point number, and $\boldsymbol{q}^m_i$ and $\boldsymbol{\tau}^m_{i}$ are the motor joint position and torque at the $i$th sampling point.
The identification problem can then be formulated into an optimization problem which minimizes the squared residual error $||\boldsymbol{\epsilon}||^2$ with respect to the decision vector $\boldsymbol{\delta}$.
\begin{equation} \label{eq:residual_error}
||\boldsymbol{\epsilon}||^2 = ||\boldsymbol{W} \boldsymbol{\delta} - \boldsymbol{\omega}||^2.
\end{equation}
To get more realistic dynamic parameters and avoid overfitting to the identification data, we utilized the physical feasibility constraints for the dynamic parameters:
\begin{itemize}
\item The mass for each link $k$ is positive, $m_k > 0$.
\item The inertia matrix of each link $k$ is positive definite, $\boldsymbol{I}_k \succ \boldsymbol{0}$ \cite{yoshida2000verification}, and its eigenvalues $Y_x$, $Y_y$ and $Y_z$ should follow the socalled triangle inequality conditions \cite{traversaro2016identification}, $Y_x + Y_y > Y_z$, $Y_y + Y_z > Y_x$, and $Y_z + Y_x > Y_y$.
\item The COM of link $k$, $\boldsymbol{r}_k$, is inside its convex hull, $m_k\boldsymbol{r}_{lk} - \boldsymbol{l}_k \le 0$ and $m_k\boldsymbol{r}_{uk} + \boldsymbol{l}_k \le 0$, where $\boldsymbol{r}_{lk}$ and $\boldsymbol{r}_{uk}$ are the lower and upper bound of $\boldsymbol{r}_{k}$, respectively \cite{sousa2014physical}.
\item The viscous and Coulomb friction coefficients for each joint $i$ are positive, $F_{vi} > 0$ and $F_{ci} > 0$.
\item The inertia of motor $k$ is positive, $I_{mk} > 0$.
\item The stiffness of spring $j$ is positive, $K_j > 0$.
\end{itemize}
The first two constraints regarding the inertia properties of each link $k$ can be derived into an equivalent \cite{wensing2018linear} as
\begin{equation}
\boldsymbol{\bar{D}}_k(\boldsymbol{\delta}_{Lk}) = \small \begin{bmatrix}
\frac{1}{2}\mathrm{tr}(\boldsymbol{L}_{k})\cdot \boldsymbol{1} - \boldsymbol{L}_{k} & \boldsymbol{l}_k\\
\boldsymbol{l}_k^{\top} & m_k
\end{bmatrix} \normalsize \succ \boldsymbol{0}
\end{equation}
We can also add the lower and upper bounds to $m_k$, $F_{vi}$, $F_{ci}$ and $K_j$ when we have more knowledge about them.
Finally, we use the CVXPY package \cite{cvxpy} with the SCS solver \cite{scs} to solve this convex optimization problem. With the identified barycentric parameters, the standard inertia parameters are computed by solving \eqref{eq:mr2l} and \eqref{eq:Iml2L}.
\begin{table}[!t]
\caption{Joint Constraints of the MTM. The Units are $\degree$ for $q$ and rad/s for $\dot{q}$}
\label{table:joint_const_mtm}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
& $q_1$ & $q_2$ & $q_3$ & $q_{3'}$ & $q_4$ & $q_5$ & $q_6$ & $q_7$ \\
\hline
\hline
$q_\mathrm{min}$ & -57 & -10 & -30 & -9 & -40 & -87 & -40 & -460 \\
\hline
$q_\mathrm{max}$ & 29 & 60 & 30 & 39 & 195 & 180 & 38 & 450 \\
\hline
$\dot{q}_\mathrm{min}$ & -2.8 & -3.1 & -3.1 &-6.2 & -6.2 & -3.1 & -3.1 & -12.6 \\
\hline
$\dot{q}_\mathrm{max}$ & 2.8 & 3.1 & 3.1 & 6.2 & 6.2 & 3.1 & 3.1 & 12.6 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[!t]
\caption{Joint Constraints of the PSM. The Units are $\degree$ or m for $q$ and rad/s or m/s for $\dot{q}$}
\label{table:joint_const_psm}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
& $q_1$ & $q_2$ & $q_3$ & $q_4$ & $q_6$ & $q_7$ & $q^d_5$ & $q^d_7$ \\
\hline
\hline
$q_\mathrm{min}$ & -85 & -43 & 0.07 & -86 & -86 & -86 & -80 & 8 \\
\hline
$q_\mathrm{max}$ & 85 & 46 & 0.235 & 86 & 86 & 86 & 80 & 172 \\
\hline
$\dot{q}_\mathrm{min}$ & -1.7 & -1.7 & -0.35 & -2 & -2 & -2 & -2 & - 3 \\
\hline
$\dot{q}_\mathrm{max}$ & 1.7 & 1.7 & 0.35 & 2 & 2 & 2 & 2 & 3 \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Experimental results}\label{sec_result}
This section presents the experimental results of the dynamic model identification conducted on the dVRK arms.
\subsection{Excitation Trajectory Generation and Robot Excitation}
Two independent excitation trajectories are generated for each of the MTM and PSM. One is for identification, and the other is for test. The harmonic number $n_H$ is set to 6. The fundamental frequency $f_f$ of the MTM and PSM are 0.1 Hz and 0.18 Hz, respectively. The joint position and velocity are constrained in the optimization, as in Table \ref{table:joint_const_mtm} and \ref{table:joint_const_psm}.
Since links $2''$ and $2'''$ of the PSM have similar motion and are very close to each other, it is hard to get a trajectory with a low condition number of the regression matrix $\boldsymbol{W}_b$ when both links $2''$ and $2'''$ are considered. Links 2 and $2'''''$ have the similar problem. Therefore, the trajectory optimization of the PSM is based on the model without links $2'''$ and $2'''''$. The obtained optimal excitation trajectories for identification of the MTM and PSM are shown in Fig. \ref{fig:identification_traj}, with the condition number of 211 and 362, respectively. The trajectories for test can be found in the open-source package and not shown here.
\begin{figure}[!ht]
\centering
\subfloat[\label{fig:mtm_identification_traj}Identification trajectory of the MTM]{\includegraphics[width=0.995\columnwidth]{mtm_train_traj_pos_change_font.pdf}}\\
\subfloat[\label{fig:psm_identification_traj}Identification trajectory of the PSM]{\includegraphics[width=0.995\columnwidth]{psm_train_traj_pos_change_font.pdf}}
\caption{\label{fig:identification_traj}Identification trajectories of the MTM and PSM}
\end{figure}
When the robot moves along these trajectories, the trajectory amplitudes $\boldsymbol{a}$ and $\boldsymbol{b}$ are increased gradually from zero to their nominal values in the first five seconds, which ensures the continuity of velocity and acceleration. The joint position, velocity, and torque are then collected at 200 Hz with the robots running in position control mode.
The joint position and velocity are collected directly, and the joint acceleration is obtained by the second-order numerical differentiation of the velocity. A $\nth{6}$ order low-pass Butterworth filter is used to filter all the data with the cutoff frequencies of 1.8 Hz for the MTM and 5.4 Hz for the PSM. To achieve zero phase delay, we apply this filter in both the forward and backward directions.
\subsection{Identification}
To get uniformly precise identification results for all joints, the residual error $\boldsymbol{\epsilon}_i$ of each motor joint $i$ in \eqref{eq:residual_error} is weighted by $w_i = 1/(\max\{\boldsymbol{\tau}^m_i\} - \min\{\boldsymbol{\tau}^m_{i}\})$.
The identified dynamic parameters $\hat{\boldsymbol{\delta}}$ from identification trajectories are used to predict the motor joint torque on test trajectories, $\hat{\boldsymbol{\omega}} = \boldsymbol{W}\hat{\boldsymbol{\delta}}$. The relative root mean squared error is used as the relative prediction error to assess the identification quality,
$\boldsymbol{\epsilon} = ||\boldsymbol{\omega} - \hat{\boldsymbol{\omega}}||_2/{||\boldsymbol{\omega}||_2}$.
\begin{table}[!htb]
\caption{Relative Prediction Error on Test Trajectories}
\label{table:relative_predict_error}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
& $\tau^m_1$ & $\tau^m_2$ & $\tau^m_3$ & $\tau^m_4$ & $\tau^m_5$ & $\tau^m_6$ & $\tau^m_7$\\
\hline
\hline
MTM (\%) & 7.3 & 15.1 & 16.2 & 22.3 & 27.0 & 23.3 & 34.0 \\
\hline
PSM (\%) & 9.1 & 17.9 & 18.9 & 13.4 & 23.9 & 21.2 & 26.5\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[!ht]
\hfil
\hfil
\includegraphics[width=0.46\textwidth]{mtm_cvx_test_change_font.pdf}
\caption{\label{fig:mtm_cvx_test} Measured and predicted torque on the test trajectory for the MTM }
\end{figure}
\begin{figure}[!ht]
\hfil
\hfil
\includegraphics[width=0.46\textwidth]{psm_cvx_test_change_font.pdf}
\caption{\label{fig:psm_cvx_test} Measured and predicted torque on the test trajectory for the PSM}
\end{figure}
Fig. \ref{fig:mtm_cvx_test} and \ref{fig:psm_cvx_test} show the comparison of the measured motor torque and predicted motor torque on the test trajectories using the identified dynamic parameters for the MTM and PSM, respectively. The relative prediction error of each motor joint is shown in Table \ref{table:relative_predict_error}.
The maximum relative prediction error of the MTM occurs on motor joint 7, which is $34.0\%$. The relative prediction error of the first three motor joints is less than $16.3\%$, which correspond to the Cartesian motion and most of the link inertia of the MTM. The large backlash caused by the gearboxes and the small link inertia of the last four joints make it hard to identify their dynamic parameters accurately. Hence, the relative prediction error of the last four motor joints is relatively higher. The overall identification performance of the MTM is better than \cite{fontanelli2017modelling}, in which the largest relative prediction error is $43.5\%$ for all the joints and $39.1\%$ for the first three joints.
The maximum relative prediction error of the PSM occurs on motor joint 7, which is $26.5\%$. The relative prediction error of the first three motor joints is less than $18.9\%$, which correspond to the Cartesian motion and most of the link inertia of the arm. The relative prediction error of the last four motor joints is relatively larger since they are only modeled with motor inertia and frictions and the magnitudes of the joint torques are very small. The overall identification performance of the PSM is also better than \cite{fontanelli2017modelling}, in which the largest relative prediction error is $45.3\%$ for all the joints and $31.6\%$ for the first three joints.
\section{Conclusion}\label{sec_conclusion}
In this work, an open-source software package for the dynamic model identification of the full standard dynamic parameters of the dVRK is presented (\url{https://github.com/WPI-AIM/dvrk_dynamics_identification}). Link inertia, joint friction, springs, tendon couplings, cable force, and closed-chains are incorporated in the modeling. Fourier series-based trajectories are used to excite the dynamics of the dVRK, with the condition number of the regression matrix minimized. A convex optimization-based method is used to get the full standard dynamic parameters subject to physical feasibility constraints, which are more suitable for the fast recursive Newton-Euler computation. The identification performance is improved significantly compared to previous work \cite{fontanelli2017modelling}. The package is written in Python under Jupyter Notebooks with free dependent software modules, which makes it easy to read and replicate. Although this software package is initially developed for the dVRK, it is easy to use it for the dynamic model identification of other robots.
Future work will be devoted to further reducing the error by precisely modeling the cables and gears, and applying the identified dynamic model in simulations and model-based control.
\section*{Acknowledgment}
This work has been supported by the National Science Initiative NSF grant: \textbf{IIS-1637759}.
\bibliographystyle{IEEEtranN}
|
{
"timestamp": "2019-03-12T01:22:49",
"yymm": "1902",
"arxiv_id": "1902.10875",
"language": "en",
"url": "https://arxiv.org/abs/1902.10875"
}
|
\section{Introduction}
The strong electron correlation effect is known to bear exotic quantum states of matter~\cite{Mott1990, Imada1998}.
Representative examples are the high-temperature superconductivity in cuprates and iron-based materials~\cite{Keimer2015,Hosono2015}, and the multiferroics with simultaneous presence of more than two ferroic order parameters~\cite{Schmid2008}.
Although both of the high-temperature superconductivity and the multiferroics share common scientific backgrounds~\cite{Liang2013,Kruger2009,Scagnoli696}, the relationship between these two states is rarely elucidated from both theoretical and experimental viewpoints.
Within this context, BaFe$_2$Se$_3$ is a nice platform, since it is theoretically predicted to be in a multiferroic state at the ambient pressure, and it is experimentally shown to become superconducting under pressure~\cite{Luo2013, Dong2014}.
The detailed study on BaFe$_2$Se$_3$ is considered to be a good starting point for exploring a fertile research field of the superconducting multiferroics~\cite{Kanasugi2018}.
The crystal structure of BaFe$_2$Se$_3$ is composed of edge-shared FeSe$_4$ tetrahedra forming a quasi-one-dimensional ladder structure of Fe atoms (Fig. 1(a)).
The structure can be regarded as the case one-third of the Fe atom stripes are removed from the two-dimensional square lattice of Fe atoms, reminding us of a close relationship with iron-based superconductors~\cite{Caron2011, Lei2011, Krzton-Maziopa2012,Lv2013}.
Superconductivity has been observed quite recently under high pressures in BaFe$_2$Se$_3$~\cite{Ying2017, Zhang2018}.
The material is frequently argued in comparison with a related material BaFe$_2$S$_3$, which also becomes superconducting under pressures~\cite{Takahashi2015, Yamauchi2015, Arita2015, Suzuki2015, Patel2016}; however these two compounds have a striking difference in the crystal structure.
Whereas BaFe$_2$S$_3$ has the $Cmcm$ space group in the whole temperature range measured (Fig. 1 (b)), BaFe$_2$Se$_3$ exhibits a structural phase transition at 660 K (= $T_{s1}$) from the high-temperature $Cmcm$ space group to low-temperature $Pnma$ space group~\cite{Svitlyk2013}, which induces small distortions both in intra and inter ladder structures (Fig. 1 (c)).
Below $T_{s1}$, Fe-Fe bonds along the leg direction become staggered in the anti-phase manner between the two adjacent legs, and the ladder planes are slightly rotated from the principal axes of the orthorhombic structure.
At around 400 K (= $T_{s2}$), the second structural transition occurs, which is evidenced by a shift of ($h00$) reflection in the X-ray diffraction profile and a sharp peak in the differential scanning calorimetry signal~\cite{Svitlyk2013}.
However, the nature of this structural transition is uncovered, which is the central question of this study.
With further cooling BaFe$_2$Se$_3$, an antiferromagnetic order develops below the N\'{e}el temperature ($T_{\rm N}$) of 220$\sim$255 K.
The magnetic structure was determined to be a block-type one with the magnetic easy axis along the layer direction, which is distinct from the stripe-type magnetic ordering with the magnetic easy axis of the rung direction in BaFe$_2$S$_3$~\cite{Luo2013,Lovesey2015}.
Block-type magnetically ordered phase is theoretically predicted to have large ferroelectric polarizations reflecting the broken inversion symmetry~\cite{Dong2014}.
The related important point is that the block-type magnetic structure within $Pnma$ involves basis functions over separate irreducible representations (irreps) to form the corepresentation~\cite{Nambu2012}, which puzzles us in the light of the second-order fashion of the magnetic transition.
Therefore, the actual crystal structure is expected to have lower symmetry than $Pnma$ even in the paramagnetic phase ($T_{\rm N}<T<T_{s2}$).
In this study, we performed the optical second harmonic generation (SHG) measurements for BaFe$_2$Se$_3$.
Together with re-analysis on powder neutron diffraction profiles, we show that the phase transition at 400 K is the structural transition from unpolar $Pnma$ to polar $Pmn2_1$ space group.
The transition is triggered by block-type lattice distortions through the spin-lattice coupling, and the resultant $Pmn2_1$ structure can compatibly accommodate the block-type magnetic structure.
Indeed linear spin-wave calculations with assuming this space group well account for observed magnetic excitations~\cite{Mourigal2015}.
Relationship between our experimental findings and electronic nematic transitions widely observed in the iron-based superconductors will be discussed.
High-quality single crystals of BaFe$_2$Se$_3$ were grown by the melt-growth method.
Stoichiometric amounts of elemental Ba shots, Fe powders, and Se powders in a carbon crucible were sealed into an evacuated quartz ampoule.
The ampoule was slowly heated up to 1373 K, kept for 48 hrs, and slowly cooled to room temperature.
The powder X-ray diffraction using the Cu-$K_{\alpha}$ radiation indicates no trace of impurity phases.
The electrical resistivity was measured by the standard four-probe method.
The magnetic susceptibility was collected by using a superconducting quantum interference device magnetometer with applying magnetic field along the $a$ (leg direction) and $b$ (layer direction) axes below room temperature, and only along the $a$ axis above room temperature.
For the optical measurements, the oriented crystals were polished by a sand-paper and Al$_2$O$_3$ fine powders.
The SHG experiments were performed in the experimental setup shown in Fig. 3(a).
Incident pulsed light with a wavelength of 800 nm (1.55 eV) and the pulse duration of 130 fs at a repetition rate of 1 kHz is generated by using a Ti:sapphire regenerative amplifier system, and is irradiated on the sample.
The laser power is $\sim$1 mW and the spot size is typically $\sim$200 $\mu$m.
The reflected light with twice energy is detected by a photomultiplier tube.
Optical measurements at low temperatures were performed by using He and N$_2$ flow cryostats.
\begin{figure}[h]
\centering
\includegraphics[width=7cm]{Figure1.pdf}
\caption{(a) Crystal structure of BaFe$_2$Se$_3$ in the $Pmn2_1$ phase. The principle axes ($a$, $b$, and $c$) are defined under the $Pmn2_1$ notation. Red arrows indicate the electric polarization induced below $T_{s2}$. (b-d) Schematic drawings of the local ladder structure with the space group of (b) $Cmcm$, (c) $Pnma$, and (d) $Pmn2_11^{\prime}$. Preserved symmetry operations are also shown. The block-type magnetic structure is also shown in (d) with white circles (up spins) and crosses (down spins).}
\label{Fig1}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=7cm]{Figure2.pdf}
\caption{Temperature dependence of physical properties of BaFe$_2$Se$_3$. (a) Electrical resistivity ($\rho$) with applying the current parallel to the leg direction (left), and the temperature derivative of log$\rho$ (right). (b) Magnetic susceptibility $(\chi)$ under the external magnetic field $(H)$ of 1 T applied along the layer (open) and leg (filled) directions. (c) Second harmonic generation (SHG) intensity in the experimental setup shown in Figs. 3(a) and 3(b). Below the structural transition temperature of $T_{s2} \sim $ 400 K, the SHG intensity due to the breaking of the spatial inversion symmetry develops.}
\label{Fig2}
\end{figure}
Figure 2 summarizes fundamental physical properties of BaFe$_2$Se$_3$.
The electrical resistivity ($\rho$) shown in Fig. 2(a) exhibits the Arrhenius-type temperature dependence with small excitation energy of 0.2 eV, which is consistent with the energy gap observed in the optical conductivity spectra shown in Supplemental Materials~\cite{supple}.
The $\rho$ curve shows an anomaly at the second structural transition temperature of $T_{s2}$, as can be clearly seen from the dip-like feature in the temperature derivative of log$\rho$, indicating that the insulating behavior is enhanced in the low-temperature phase.
Temperature dependence of the magnetic susceptibility ($\chi$) taken under the magnetic field of 1 T is shown in Fig. 2(b).
Besides the reported anomaly corresponding to the antiferromagnetic transition at $T_{\rm N}$ = 220 K, we observe a clear anomaly at $T_{s2}$, suggesting a large change in electronic states is likely across $T_{s2}$.
To obtain further insights into the structural transition at $T_{s2}$, we performed optical SHG experiments.
The SHG is the second-order nonlinear optical phenomena, in which photons with twice the energy of initial photons are generated from samples.
The SHG intensity ($I$) is represented by $I(2\omega) \propto |\mu_0\frac{\partial ^2\vec{P}}{\partial t^2} + \mu_0 (\nabla \times \frac{\partial \vec{M}}{\partial t})|^2 $, which consists of an electric dipole contribution $P_i(2\omega) = \varepsilon_0\chi^{{\rm polar}}_{ijk} E_j(\omega)E_k(\omega)$ and a magnetic dipole contribution $M_i(2\omega) = \varepsilon_0c\chi^{{\rm axial}}_{ijk} E_j(\omega)E_k(\omega)$, where $P$, $M$, and $E$ stand for the electric polarization, magnetic dipole moment and electric field, respectively~\cite{Fiebig05}.
The tensors $\chi^{\rm polar}_{ijk}$ and $\chi^{\rm axial}_{ijk}$ are the third-rank SHG tensors having polar and axial characteristics, respectively.
The SHG tensors are governed by the point group of the system, and the particularly important is that polar tensors can be finite only in noncentrosymmetric systems.
For example, in the centrosymmetric $mmm$ symmetry (the point group of $Cmcm$ and $Pnma$ space groups), all the components in $\chi^{{\rm polar}}$ should be zero, whereas $\chi^{\rm axial}_{abc}$, $\chi^{\rm axial}_{bca}$, and $\chi^{\rm axial}_{cab}$ components of $\chi^{\rm axial}$ can be finite.
We first performed experiments in a configuration shown in Figs. 3(a) and 3(b): the incident laser is irradiated on the (010) surface and the reflected SHG light normal to the surface is detected; the polarization of the initial light ($\phi_{\omega}$) and that of the reflected SHG light ($\phi_{2\omega}$) were analyzed by linear polarizers.
The advantage of this setup is that magnetic-dipole contributions ($\chi^{\rm axial}_{abc}$, $\chi^{\rm axial}_{bca}$, and $\chi^{\rm axial}_{cab}$) expected for the $mmm$ point group are undetectable due to the symmetrical reason.
Nevertheless, as can be seen from Fig. 2(c), when $\phi_{\omega}$ = 90$^{\circ}$ and $\phi_{2\omega}$ = 0$^{\circ}$, we observe that the SHG intensity begins to develop below $T_{s2}$, strongly indicating the inversion symmetry being broken in the low-temperature phase.
Indeed, the detailed analysis presented in the following revealed that the observed signal reflects the $\chi^{\rm polar}_{caa}$ component in the $mm2$ point group.
\begin{figure*}[t!]
\centering
\includegraphics[width=12cm]{Figure3.pdf}
\caption{Polarization dependence of the SHG intensity for BaFe$_2$Se$_3$. (a) Experimental geometry of SHG measurements. The incident laser beam with the linear polarization is irradiated normal to the (010) or (011) surface, and the reflected SHG is detected by a photomultiplier tube after analyzing the polarization.
(b, g) Relationship between crystal axes and the polarization angle of the incident light $\phi_{\omega}$ and that of SHG $\phi_{2\omega}$. Polarization dependence of (c, e) simulated and (d, f) observed SHG signals at 200 and 450 K for the (010) plane. Polarization dependence of (h, j) simulated and (i, k) observed SHG signals at 200 and 450 K for the (011) plane. In the simulations, the $mmm$ and $mm2$ point groups are respectively assumed for 450 and 200 K data. In the right panel of (f) and (k), $\phi_{2\omega}$ dependences of the SHG intensity collected at $\phi_{\omega}$ = 90$^{\circ}$ are shown.}
\label{Fig3}
\end{figure*}
Here, we discuss the point group below $T_{s2}$ by analyzing the polarization dependence of the SHG signal.
Figures 3(d) and 3(f) show the SHG signal in the $\phi_{\omega}$ - $\phi_{2\omega}$ plane, which are collected in the experimental configuration of Figs. 3(a) and 3(b).
At 450 K, as shown in Fig. 3(d), there is negligibly small SHG signal observed in accordance with the centrosymmetric $Pnma$ symmetry (Fig. 3(c)).
At 200 K, on the other hand, one can see clear polarization dependences: the strong signal is discernible at ($\phi_{\omega}$, $\phi_{2\omega}$) = (0$^{\circ}$, 0$^{\circ}$) and (90$^{\circ}$, 0$^{\circ}$) (Fig. 3(f)).
We here postulate that the point group of the low-temperature phase is $mm2$, which is the subgroup of $mmm$.
The symmetry operation lost in lowering the point group is the inversion operation, so that $mm2$ is a noncentrosymmetric point group.
Then, in $mm2$, $\chi^{\rm polar}_{caa}$, $\chi^{\rm polar}_{ccc}$, and $\chi^{\rm polar}_{aac}$ components can be finite, whereas axial tensors have the same component as in the case of $mmm$ ($\chi^{\rm axial}_{abc}$, $\chi^{\rm axial}_{bca}$, and $\chi^{\rm axial}_{cab}$ can be excited in the present configuration).
Taking care that axial tensors cannot be detected in the current experimental setup, one can simulate the SHG pattern by changing the sign and magnitude of the polar tensor components.
The simulated pattern with $\chi^{\rm polar}_{caa}$ : $\chi^{\rm polar}_{ccc}$ : $\chi^{\rm polar}_{aac}$ = −2.0 : 1.0 : 0 well reproduces the experimentally observed polarization dependence (Fig. 3(e)).
We next performed similar experiments for the (011) surface: the polarization of the initial light ($\phi_{\omega}$) and that of the reflected SHG light ($\phi_{2\omega}$) are defined as shown in Fig. 3(g).
In contrast to the results for the (010) surface, the SHG signals exhibit characteristic polarization dependences even at 450 K (Fig. 3(i)), which is above the second structural transition temperature.
This SHG pattern is well reproduced by putting the axial tensor components in the $mmm$ point group to be $\chi_{bca}^{\rm axial}$ + $\chi_{cab}^{\rm axial}$: $\chi_{abc}^{\rm axial}$ = $-$1.0 : 1.0 (Fig. 3(h)).
We stress here that the axial tensor contributions can be allowed even in centrosymmetric crystals.
When the temperature is decreased down to 200 K, the strong signal appears at ($\phi_{\omega}$, $\phi_{2\omega}$) = (90$^{\circ}$, 0$^{\circ}$) in addition to signals observed at 450 K (Fig. 3(k)).
The observed SHG pattern can be well reproduced by simulations under the assumption of the point group of $mm2$ with the rung direction as the two-fold rotation axis (Fig. 3 (j)); here, we set $\chi^{\rm axial}_{abc}$ : $\chi^{\rm axial}_{bca}$ +$\chi^{\rm axial}_{cab}$ : $\chi^{\rm polar}_{caa}$ : $\chi^{\rm polar}_{ccc}$ : $\chi^{\rm polar}_{cbb}$ + $\chi^{\rm polar}_{bcb}$ = 1 : $-$1 : $-$0.78 : 0.39 : $-$0.67.
We thus conclude that the phase below $T_{s2}$ has the point group of $mm2$ with the polar axis as the rung direction.
Herein, let us identify the crystal structure below $T_{s2}$.
To do this, we first list up possible space groups.
Since any discontinuity or thermal hysteresis is not observed in physical quantities across the transition, the phase transition at $T_{s2}$ is considered to be of the second-order; therefore, we consider the maximal subgroup of $Pnma$.
We then list up three candidate space groups, $Pmc2_1$, $Pmn2_1$, and $Pna2_1$; these respectively lose one of three orthogonal mirror/glide operations associated with the layer, rung, and leg directions.
Among them, $Pmn2_1$ is the most plausible one, because the SHG results indicate that the polar axis is along the rung direction.
We then re-analyze the powder neutron diffraction profiles collected at $T$ = 300 K temperature, yielding better convergence in the $Pmn2_1$ model than in others.
The obtained structural parameters are shown in Table I in the Supplemental Materials~\cite{supple}.
Importantly, the $Pmn2_1$ model is compatible with the block-type magnetic structure below $T_{\rm N}$.
Actually, the powder neutron diffraction profiles below $T_{\rm N}$ can be well fitted by the $Pmn2_1$ structure model and the block-type magnetic structure model with the magnetic wave vector of (1/2, 1/2, 1/2).
Here, the magnetic structure is represented by a single irrep $\Gamma_1$ (Table II of Supplemental Materials~\cite{supple}) in consistent with the second-order nature of the magnetic transition.
The resultant magnetic space group is $Pmn2_11^{\prime}$, so that the time-reversal symmetry is preserved by combining the translation operation even below $T_{\rm N}$ (Fig.1 (d)).
Therefore, the polar-$c$ and axial-$c$ tensors do not become finite upon the magnetic order, justifying the analysis of the SHG results.
The observed polar state in BaFe$_2$Se$_3$ is almost in accordance with the theoretical proposals by Dong, $et$ $al$~\cite{Dong2014}.
According to the theory, the block-type magnetic ordering is stabilized by an electronic origin in a realistic band structures~\cite{Luo2013}.
As a result, there appears uniform displacements of Se atoms perpendicular to the local ladder plane, bearing the macroscopic polarization along the rung direction, which is in consistent with our experimental results.
The microscopic mechanism of this magneto-elastic coupling is argued to be the so-called exchange striction.
The mechanism is known to affect in the up-up-down-down-type antiferromagnetic phase of a conventional multiferroic material $o$-YMnO$_3$ ~\cite{Sergienko2006, Picozzi2007}.
The large lattice distortions induced at around $T_{\rm N}$ shown in ref.~\cite{Nambu2012} strongly support this scenario.
However, there is an apparent incompatibility between the theory and our experimental results.
Whereas the theory predicts emergence of polarization below $T_{\rm N}$, our results indicate that polar lattice distortions emerge below $T_{s2}$, which is much higher than $T_{\rm N}$.
This kind of successive structural and magnetic phase transitions, which are closely related with each other from the symmetry point of view, are reminiscent of the iron-based superconductors including Co-doped BaFe$_2$As$_2$~\cite{Chu2009, Fernandes2014}.
Among them, the high temperature structural transition has been considered to be originating from an orbital order or electronic nematic order, since the rotational symmetry is broken in the electronic system~\cite{Chu2012, Kasahara2012, Fernandes2014, Sun2016}.
It is possible that a similar mechanism is involved in BaFe$_2$Se$_3$, in which prominent quantum fluctuations inherent in the reduced spatial dimensionality results in a large separation of the two transition temperatures.
We also note that the isostructural compound BaFe$_2$S$_3$ exhibits a broad anomaly in the electrical resistivity at around 200 K, which is higher than the antiferromagnetic transition temperature of 120 K.
This could be also due to the possible orbital order~\cite{Yamauchi2015}.
In the iron-based superconductors, not only antiferromagnetic fluctuations but also structural (orbital) fluctuations are expected to have key roles for the formation of superconducting state.
Such possibility shall be also pursued for the superconducting states under pressure in BaFe$_2$S$_3$ and BaFe$_2$Se$_3$, which will open a research field of the superconducting multiferroics.
In conclusion, we observed the optical second harmonic generation signals below the structural transition temperature of 400 K in quasi-one-dimensional ladder material BaFe$_2$Se$_3$.
Combined with neutron diffraction results, we uncovered the detailed crystal structure in the low temperature phase, which has the polar $Pmn2_1$ space group.
In the viewpoint of the symmetry, BaFe$_2$Se$_3$ is therefore multiferroics with both polarity and antiferromagnetic order.
The structural phase transition is driven by the block-type lattice distortions through the magneto striction mechanism.
Considerable change in the electrical resistivity and magnetic susceptibility across the polar-nonpolar phase transition suggests that electronic states are largely modified by this structural instability.
\begin{acknowledgments}
We are grateful to T. Yamauchi, D. Okuyama, F. Du, T. J. Sato, M. Avdeev, and T. Hawai for experimental help and fruitful discussion.
The present work is financially supported by JSPS KAKENHI Nos. 16K17732,
16H01062, 18H04302,
17H05474,
16H04019,
18H01159,
16H04007, 17H05473, 17H06137,
and 17H04844, Research Foundation for Opto-Science and Technology,
Murata Science Foundation,
and Mitsubishi Foundation.
This work was partially performed using facilities of the Institute for Solid State Physics, the University of Tokyo.
\end{acknowledgments}
\section{Optical reflectivity spectra}
\begin{figure}[h]
\centering
\includegraphics[width=10cm]{sFig1.pdf}
\caption{Optical reflectivity spectra and optical conductivity ($\sigma$) spectra for BaFe$_2$Se$_3$. (a-b) Polarizaiton dependence of the optical reflectivity and optical conductivity spectra at 300 K. The red and green triangles indicate the wavelength of the incident and emitted lights in second harmonic generation measurements. (c) Schematic band diagrams of BaFe$_2$Se$_3$. LHB and UHB stand for the lower Hubbard and upper Hubbard band, respectively.
(d-e) Temperature dependence of the optical reflectivity and optical conductivity spectra when the light polarization ($E$) is along the leg direction. }
\label{sF1}
\end{figure}
We performed optical reflectivity measurements for the (010) and (001) surfaces of BaFe$_2$Se$_3$, which are polished with Al$_{2}$O$_{3}$ powders.
A Fourier-transform infrared spectrometer and a grating spectrometer were used for measurements in the energy range of 0.1-0.9 and 0.7-4.3 eV, respectively.
The incident light was linearly polarized along three principal axes of the orthorombic crystal structure.
Optical conductivity spectra were obtained from the reflectivity spectra by the Kramers-Kronig transformation.
The optical reflectivity and conductivity spectra taken at 300 K (Figs. S1(a) and S1(b)) show a large polarization dependence.
The optical conductivity spectra ($\sigma$) have large spectral weights, when the light polarization $E$ is along the leg direction, indicating the quasi-one-dimensional nature of charge dynamics reflecting the ladder structure.
The key feature is the absence of Drude components in any direction.
This is because the system becomes a Mott insulator due to the prominence of the electron correlation effect in the low-dimensional system.
The optical gap is estimated to be 0.25 eV from the spectra with $E$ along the leg direction, which is consistent with the charge gap deduced from the electrical resistivity measurements.
One can clearly see two optical modes at around 0.6 and 1.7 eV, which are respectively marked as A and B in the figure.
The A mode is considered to be a Mott excitation of Fe-3$d$ orbitals hybridized with Se-4$p$ orbitals.
On the other hand, the B mode is considered to be a charge transfer excitation, which is the transition from the deeper Se-4$p$ orbitals to Fe-3$d$ orbitals.
The schematic picture of transitions is shown in Fig. S1(c).
These optical modes are analogous with that of iron-deficient two-dimensional iron chalcogenide superconductor K$_{2}$Fe$_{4}$Se$_{5}$, which also has block-type magnetism~\cite{Ye_2011, Charnukha2014}.
The SHG experiments described in the main text were performed with the incident energy of 1.55 eV and the emitted energy of 3.10 eV, which are indicated by arrows in Fig. S1. The relevant electronic states in these energy scale are Fe-3$d$ and Se-4$p$ orbitals.
Temperature dependences of reflectivity and optical conductivity spectra with $E$ parallel to the leg direction are shown in Figs. S1(d) and S1(e).
The spectral weights at the low-energy side become smaller with decreasing temperature in accordance with the insulating behavior of the electrical resistivity.
\section{Simulations of second harmonic generation signals}
The polarization dependences of the optical second harmonic generation (SHG) are simulated.
SHG is the second-order nonlinear optical phenomena, in which the incident light with the frequency $\omega$ are converted into the frequency-doubled (2$\omega$) light.
The SHG intensity $I$(2$\omega$) is written by the source term $S$(2$\omega$) as
\begin{eqnarray}
I(2\omega) &\propto& |S(2\omega)|^2 \\
S(2\omega) &=& \mu_0\frac{\partial ^2\vec{P}}{\partial t^2} + \mu_0 (\nabla \times \frac{\partial \vec{M}}{\partial t}) \\
P_i(2\omega) &=& \varepsilon_0\chi^{{\rm polar}}_{ijk} E_j(\omega)E_k(\omega) \\
M_i(2\omega) &=& \varepsilon_0c\chi^{{\rm axial}}_{ijk} E_j(\omega)E_k(\omega)
\end{eqnarray}
The SHG tensors $\chi^{\rm polar}_{ijk}$ and $\chi^{\rm axial}_{ijk}$ depends on the symmetry of the material~\cite{Fiebig05}.
Since the time reversal symmetry is preserved even in the antiferromagnetic state in BaFe$_2$Se$_3$, the SHG tensors should be $i$-tensors.
In $mmm$, which is the point group of the high temperature phase of BaFe$_2$Se$_3$, the axial-$i$ tensor can be finite though polar-$i$ tensor should be zero.
\begin{equation}
\chi^{\rm axial} = \left[
\begin{array}{rrrrrr}
0 & 0 & 0 & \chi^{\rm axial}_{abc} & 0 & 0\\
0 & 0 & 0 & 0 & \chi^{\rm axial}_{bca} & 0\\
0 & 0 & 0 & 0 & 0 & \chi^{\rm axial}_{cab}
\end{array}
\right]
.
\end{equation}
\begin{equation}
\chi^{\rm polar} = \left[
\begin{array}{rrrrrr}
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0
\end{array}
\right]
.
\end{equation}
In $mm2$, which is the most plausible point group of the low temperature phase of BaFe$_2$Se$_3$, several components of the axial-$i$ tensors and polar-$i$ tensors can be finite as follow~\cite{R1963}.
\begin{equation}
\chi^{{\rm axial}} = \left[
\begin{array}{rrrrrr}
0 & 0 & 0 & \chi^{\rm axial}_{abc} & 0 & 0\\
0 & 0 & 0 & 0 & \chi^{\rm axial}_{bca} & 0\\
0 & 0 & 0 & 0 & 0 & \chi^{\rm axial}_{cab}
\end{array}
\right]
.
\end{equation}
\begin{equation}
\chi^{{\rm polar}} = \left[
\begin{array}{rrrrrr}
0 & 0 & 0 & 0 & \chi^{\rm polar}_{aac} & 0\\
0 & 0 & 0 & \chi^{\rm polar}_{bbc} & 0 & 0\\
\chi^{\rm polar}_{caa} & \chi^{\rm polar}_{cbb} & \chi^{\rm polar}_{ccc} & 0 & 0 & 0
\end{array}
\right]
.
\label{E_3}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=10cm]{sFig2.pdf}
\caption{Simulated results of the polarization dependences of the optical second harmonic generation for the (010) and (011) surfaces shown in Fig. 3 in the main text. }
\label{sF2}
\end{figure}
Figure S\ref{sF2} shows the calculated results of polarization dependence of the SHG signals with the (010) and (011) configurations shown in Fig. 3 in the main text.
The marks in the table ($\circ$, $\bullet$, $\times$) indicate the corresponding SHG patterns shown in the lower panels.
If several components can be finite, they are expected to interference with each other.
\clearpage
\section{Powder neutron diffraction profiles}
As noted in the main text, the transition across $T_{s2}$ is in the second-order fashion, the space group is therefore assumed to be among the maximal subgroup of $Pnma$.
Three subgroups, $Pmc2_1$, $Pmn2_1$, and $Pna2_1$ out of all subgroups are possible, because they only possess point groups consistent with results inferred from the SHG experiments.
We then tested and evaluated the space group for the data taken at 300 K through the Rietveld refinement on neutron powder diffraction data.
Neutron powder diffraction data were collected on the high-resolution ECHIDNA diffractometer at the Australian Nuclear Science and Technology Organisation (ANSTO) with $\lambda=2.4395$ {\AA}~\cite{nambu2012}.
The fits yield $R_{wp}=4.40$ for $Pmc2_1$, 4.38 for $Pmn2_1$, and 4.41 for $Pna2_1$.
The difference in the outcomes finds very tiny; however, all the evaluation factors such as $R_p$ and the reduced $\chi^2$ are in the same fashion.
Therefore we concluded the crystal structure below $T_{s2}$ should have the space group $Pmn2_1$ (Table I).
\begin{table}[h!]
\caption{Atomic positions within $Pmn2_1$ of BaFe$_2$Se$_3$ at $T=300$ K determined by Rietveld analysis ($\chi^2=2.45$). Lattice constants are $a=5.43748(4)$ \AA, $b=11.93178(12)$ \AA, and $c=9.16473(9)$ \AA. Isotropic Debye-Waller factor ($B_{\rm iso}$) is employed.}
\label{atomic}
\begin{ruledtabular}
\begin{tabular}{lccccc}
Atom & Site & $x$ & $y$ & $z$ & $B_{\rm iso}$ ({\AA}$^2$) \\
\hline
Ba1 & $2a$ & 0 & 0.4397(16) & 0.7246(21) & 1.169(150) \\
Ba2 & $2a$ & 0 & 0.9337(15) & 0.2582(21) & 1.169(150) \\
Fe1 & $4b$ & 0.2393(10) & 0.7473(7) & 0.8817(30) & 1.048(38) \\
Fe2 & $4b$ & 0.2405(12) & 0.7585(6) & 0.5855(30) & 1.048(38) \\
Se1 & $2a$ & 0 & 0.6034(11) & 0.9950(27) & 1.200(41) \\
Se2 & $2a$ & 0 & 0.1159(10) & 0.9557(25) & 1.200(41) \\
Se3 & $2a$ & 0 & 0.8780(15) & 0.7425(31) & 1.200(41) \\
Se4 & $2a$ & 0 & 0.3779(15) & 0.2272(28) & 1.200(41) \\
Se5 & $2a$ & 0 & 0.6566(12) & 0.4101(26) & 1.200(41) \\
Se6 & $2a$ & 0 & 0.1445(12) & 0.5402(25) & 1.200(41) \\
\end{tabular}
\end{ruledtabular}
\end{table}
Additional magnetic reflections appear in the data below $T_{\rm N}$, and all magnetic-peak positions can be indexed by the magnetic wave vector, $\vec{q}_{\rm m}=(1/2,1/2,1/2)$.
We employed group theoretical analysis to identify the magnetic structure that is allowed by symmetry.
Basis vectors (BVs) of the irreducible representations (irreps) for the wave vector were obtained using the SARA$h$ code~\cite{Wills2000}.
There is only one irrep, and it consists of 12 BVs giving either parallel or antiparallel relationship between uniaxial magnetic moments along one crystallographic axis for two sites out of all the four iron positions per one Wyckoff site (Table II).
To describe magnetic structure, two BVs are required to join to let all the four iron atoms have magnetic moments.
In the case of moments along one crystallographic axis, there are two choices for two combinations, namely four patterns.
Two Wyckoff sites for irons reflect 16 patterns in total.
We assumed that the moment is parallel or antiparallel to the one axis and has the same coefficient of four BVs, and sorted out all 48 patterns to determine the magnetic structure by comparing the $R$-factor.
The best fit with $R_{\rm mag}=5.47$ is the combination of BVs, $\psi_5$ and $\psi_{11}$ for both Fe1 and Fe2, and the second comes $\psi_2$ and $\psi_8$ for $R_{\rm mag}=5.50$.
Both describes magnetic moments perpendicular to the ladder plane.
Note that the cases for moments along the ladder and rung directions poorly describe the data with $R_{\rm mag}>20$.
The optimized refinement of the magnetic structure with $\Phi_{5}$ and $\Phi_{11}$ reproduces the previous result~\cite{nambu2012}, where magnetic moments (2.67(2) $\mu_{\rm B}$/Fe at 5 K) are arranged to form a Fe$_4$ ferromagnetic unit, and it stacks antiferromagnetically along the ladder direction.
The obtained magnetic structure has magnetic point group $mm21^{\prime}$, and belongs to magnetic space group $Pmn2_1 1^{\prime}$ (Fig. S3).
\begin{table}[h!]
\caption{Basis vectors (BVs) of irreducible representations (irreps) for the space group $Pmn2_1$ with the magnetic wave vector $\vec{q}_{\rm m}=(1/2,1/2,1/2)$. Superscripts show the moment direction. Columns for positions represent \#1: $(x,y,z)$, \#2: $(-x+1/2,-y,z+1/2)$, \#3: $(-x,y,z)$, and \#4: $(x+1/2,-y,z+1/2)$ for both Fe1 and Fe2 sites.}
\label{irrep}
\begin{ruledtabular}
\begin{tabular}{cccccccccc}
irrep & BV & Fe1 (\#1) & Fe1 (\#2) & Fe1 (\#3) & Fe1 (\#4) & Fe2 (\#1) & Fe2 (\#2) & Fe2 (\#3) & Fe2 (\#4) \\
\hline
\multirow{12}{*}{$\Gamma_1$} & $\psi_1$ & $1^a$ & 0 & $-1^a$ & 0 & $1^a$ & 0 & $-1^a$ & 0 \\
& $\psi_2$ & $1^b$ & 0 & $1^b$ & 0 & $1^b$ & 0 & $1^b$ & 0 \\
& $\psi_3$ & $1^c$ & 0 & $1^c$ & 0 & $1^c$ & 0 & $1^c$ & 0 \\
& $\psi_4$ & 0 & $-1^a$ & 0 & $1^a$ & 0 & $1^a$ & 0 & $-1^a$ \\
& $\psi_5$ & 0 & $-1^b$ & 0 & $-1^b$ & 0 & $1^b$ & 0 & $1^b$ \\
& $\psi_6$ & 0 & $1^c$ & 0 & $1^c$ & 0 & $-1^c$ & 0 & $-1^c$ \\
& $\psi_7$ & 0 & $1^a$ & 0 & $1^a$ & 0 & $-1^a$ & 0 & $-1^a$ \\
& $\psi_8$ & 0 & $1^b$ & 0 & $-1^b$ & 0 & $-1^b$ & 0 & $1^b$ \\
& $\psi_9$ & 0 & $-1^c$ & 0 & $1^c$ & 0 & $1^c$ & 0 & $-1^c$ \\
& $\psi_{10}$ & $1^a$ & 0 & $1^a$ & 0 & $1^a$ & 0 & $1^a$ & 0 \\
& $\psi_{11}$ & $1^b$ & 0 & $-1^b$ & 0 & $1^b$ & 0 & $-1^b$ & 0 \\
& $\psi_{12}$ & $1^c$ & 0 & $-1^c$ & 0 & $1^c$ & 0 & $-1^c$ & 0 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}[h!]
\centering
\includegraphics[width=10cm]{sFig3.pdf}
\caption{High-resolution neutron powder diffraction pattern of BaFe$_2$Se$_3$ taken at 5 K obtained on ECHIDNA with the Rietveld refinement (solid lines). The calculated positions of nuclear and magnetic reflections are indicated (green ticks). The bottom line gives the difference between observed and calculated intensities.}
\label{sF3}
\end{figure}
\clearpage
|
{
"timestamp": "2019-03-01T02:09:36",
"yymm": "1902",
"arxiv_id": "1902.10900",
"language": "en",
"url": "https://arxiv.org/abs/1902.10900"
}
|
\section{Introduction}
We consider \emph{online convex optimization} (OCO) of a sequence of convex
functions $f_1,\ldots,f_T$ over a given bounded convex domain, which
become available one by one over the course of $T$ rounds
\citep{ShalevShwartz2011,HazanOCOBook2016}. Typically $f_t(\w) =
\textsc{loss}(\w,\x_t,y_t)$ represents the \emph{loss}
of predicting with parameters $\w$ on the $t$-th data point $(\x_t,y_t)$ in a machine
learning task. At the start of each round $t$, a learner has to predict
the best parameters $\w_t$ for the function $f_t$ before finding out
what $f_t$ is, and the goal is to minimize the \emph{regret}, which is
the difference in the sum of function values between the learner's
predictions $\w_1,\ldots,\w_T$ and the best fixed oracle parameters $\u$
that could have been chosen if all the functions had been given in
advance. A special case of OCO is prediction with expert advice
\citep{cesa06}, where the functions $f_t(\w) = \w^\top
\vloss_t$ are convex combinations of the losses $\vloss_t =
(\loss_{t,1},\ldots,\loss_{t,K})^\top$ of $K$ expert predictors and the
domain is the probability simplex.
Central results in these settings show that it is possible to control the
regret with almost no prior knowledge at all about the functions. For
instance, knowing only an upper bound $G$ on the $\ell_2$-norms of the
gradients $\grad_t = \nabla f_t(\w_t)$, the online gradient descent
(OGD) algorithm guarantees $O(G \sqrt{T})$ regret by tuning its learning
rate hyperparameter $\eta_t$ proportional to $1/(G\sqrt{t})$
\citep{Zinkevich2003}, and in the case of prediction with expert advice
the Hedge algorithm achieves regret $O(L\sqrt{T\ln K})$ knowing only an
upper bound $L$ on the range $\max_k \ell_{t,k} - \min_k \ell_{t,k}$ of
the expert losses \citep{FreundSchapire1997}. Here $G$ is the
$\ell_2$-Lipschitz constant of the learning task\footnote{We slightly
abuse terminology here, because the standard definition of a Lipschitz
constant requires an upper bound on the gradient norms for any
parameters $\w$, not just for $\w = \w_t$, and may therefore be
larger.}, and $L/2$ is the $\ell_1$-Lipschitz constant over the
probability simplex.
The above guarantees are tight if we make no further assumptions about
the functions $f_t$ \citep{HazanOCOBook2016,CesaBianchiEtAl1997}, but
they can be significantly improved if the functions have additional
special structure that makes the learning task easier. The literature on
online learning explores multiple orthogonal dimensions in which tasks
may be significantly easier in practice (see `related work' below). Here
we focus on the following regret guarantees that are known to exploit
multiple types of easiness at the same time:
\begin{align}
\text{OCO:}& &O\left(\sqrt{V_T^\u d \log T}\right)
\text{ for all $\u$,}
\quad \text{with $V_T^\u = \sum_{t=1}^T ((\w_t - \u)^\top
\grad_t)^2$,}\label{eqn:ourmetagradbound}\\
\text{Experts:}& &O\left(\sqrt{\E_{\rho(k)}[V_T^k]
\KL(\rho\|\pi)}\right)
\text{ for all $\rho$,}
\quad \text{with $V_T^k = \sum_{t=1}^T ((\w_t - \e_k)^\top
\vloss_t)^2$,}\label{eqn:oursquintbound}
\end{align}
where $d$ is the number of parameters and $\KL(\rho\|\pi) = \sum_{k=1}^K
\rho(k) \ln \rho(k)/\pi(k)$ is the Kullback-Leibler divergence of a
data-dependent distribution $\rho$ over experts from a fixed prior
distribution~$\pi$.
The OCO guarantee is achieved by the MetaGrad algorithm
\citep{Erven2016}, and implies regret that grows at most logarithmic in
$T$ both in case the losses are curved (exp-concave, strongly convex)
and in the stochastic case whenever the losses are independent, identically
distributed samples with variance controlled by the Bernstein condition
\citep{Erven2016,koolen2016}. The guarantee for the expert case is
achieved by the Squint algorithm \citep{koolen2015,squintPAC}. It also
exploits special structure along two dimensions simultaneously, because
the $V_T^k$ term is much smaller than $L^2 T$ in many cases
\citep{GaillardStoltzVanErven2014,koolen2016} and the so-called
\emph{quantile bound} $\KL(\rho\|\pi)$ is much smaller than the worst
case $\ln K$ when multiple experts make good predictions
\citep{ChaudhuriFreundHsu2009,ChernovVovk2010}. Squint and MetaGrad are
both based on the same technique of tracking the empirical performance
of \emph{multiple learning rates} in parallel over a quadratic
approximation of the original loss. A computational difference though is that Squint
is able to do this by a continuous integral that can be evaluated in
closed form, whereas MetaGrad uses a discrete grid of learning rates.
Unfortunately, to achieve \eqref{eqn:ourmetagradbound} and
\eqref{eqn:oursquintbound}, both MetaGrad and Squint need knowledge of
the Lipschitz constant ($G$ or $L$, respectively). Overestimating $G$ or
$L$ by a factor of $c > 1$ has the effect of reducing the effective
amount of available data by the same factor $c$, but underestimating the
Lipschitz constant is even worse because it can make the methods fail
completely. In fact, the ability to adapt to $G$ has been credited
\citep{WardWuBottou2018} as one of the main reasons for the practical
success of the AdaGrad algorithm
\citep{DuchiHazanSinger2011,McMahanStreeter2010}. Thus getting the
Lipschitz constant right makes the difference between having practical
algorithms and having promising theoretical results.
For OCO, an important first step towards combining Lipschitz adaptivity
to $G$ with regret bounds of the form \eqref{eqn:ourmetagradbound} was
taken by \citet{cutkosky2017}, who aimed for
\eqref{eqn:ourmetagradbound} but had to settle for a weaker result with
$G \sum_{t=1}^T \|\grad_t\|_2 \|\w_t - \u\|_2^2$ instead of $V_T^\u$.
Although not sufficient to adapt to the Bernstein condition, they do
provide a series of stochastic examples where their bound already leads
to fast $O(\ln^4 T)$ rates. For the expert setting,
\citet{Wintenberger2017} has made significant progress towards a version
of \eqref{eqn:oursquintbound} without the quantile bound improvement,
but he is left with having to specify an initial guess $L_\text{guess}$
for $L$ that enters as $O(\ln \ln (L/L_\text{guess}))$ in his bound, which may yet be arbitrarily large when the initial guess is on the
wrong scale.
\paragraph{Main Contributions}
Our main contributions are that we complete the process began by
\citet{cutkosky2017} and \citet{Wintenberger2017} by showing that it is
indeed possible to achieve \eqref{eqn:ourmetagradbound} and
\eqref{eqn:oursquintbound} without prior knowledge of $G$ or $L$. In
fact, for the expert setting we are able to adapt to the tighter
quantity $B \geq \max_k |(\w_t - \e_k)^\top \vloss_t|$. We achieve these
results by dynamically updating the set of active learning rates in
MetaGrad and Squint depending on the observed Lipschitz constants. In
both cases we encounter a similar tuning issue as
\citet{Wintenberger2017}, but we avoid the need to specify any initial
guess using a new restarting scheme, which restarts the algorithm when
the observed Lipschitz constant increases too much. In addition to these
main results, we remove the need to specify the number of rounds $T$ in
advance for MetaGrad by adding learning rates as $T$ gets larger, and we
improve the computational efficiency of how it handles constraints on
the domain of prediction: by a minor extension of the black-box reduction for projections of \citet{cutkosky2018}, we incur only the
computational cost of projecting on the domain of interest in
\emph{Euclidean} distance. This should be contrasted with the usual
projections in time-varying Mahalanobis distance for second-order methods like
MetaGrad.
\paragraph{Related Work}
If adapting to the Lipschitz constant were our only goal, a well-known
way to achieve it for OCO would be to change the learning rate in OGD to
$\eta_t \propto 1/\sqrt{\sum_{s\leq t} \|\grad_s\|_2^2}$, which leads to
$O(\sqrt{\sum_{t\leq T} \|\grad_t\|_2^2}) = O(G \sqrt{T})$ regret. This
is the approach taken by AdaGrad (for each dimension separately)
\citep{DuchiHazanSinger2011,McMahanStreeter2010}. In prediction with
expert advice, Lipschitz adaptive methods are sometimes called
\text{scale-free} and have previously been obtained by
\citet{cbms07,rooij14} with generalizations to OCO by
\citet{OrabonaPal2015}. In addition, the first two of these works obtain
a data-dependent variance term that is different from $V_T^k$ in \eqref{eqn:oursquintbound}, but no
quantile bounds are known for the former. Results for the latter have previously been obtained
by \citet{GaillardStoltzVanErven2014,Wintenberger2014Arxiv} without quantile bounds,
and with a slightly weaker notion of variance by \citet{AdaNormalHedge}.
Quantile bounds without variance adaptivity were introduced by
\citet{ChaudhuriFreundHsu2009,ChernovVovk2010}. These may be interpreted
as measures of the complexity of the comparator $\rho$. The
corresponding notion in OCO is to adapt to the norm of $\u$, which has
been achieved in various different ways, see for instance
\citep{McMahanAbernethy2013,cutkosky2018}. For curved functions,
existing results achieve fast rates assuming that the degree of
curvature is known \citep{HazanAgarwalKale2007}, measured online
\citep{BartlettHazanRakhlin2007,Do2009} or entirely unknown
\citep{Erven2016,cutkosky2018}. Fast rates are also possible for
slowly-varying linear functions and, more generally, optimistically
predictable gradient sequences
\citep{hazan2010extracting,GradualVariationInCosts2012,RakhlinSridharan2013}.
We view our results as a step towards developing algorithms that
automatically adapt to multiple relevant measures of difficulty at the
same time. It is not a given that such combinations are always possible.
For example, \citet{CutkoskyBoahen2017Impossible} show that Lipschitz
adaptivity and adapting to the comparator complexity in OCO, although
both achievable independently, cannot both be realized at the same time
(at least not without further assumptions). A general framework to study
which notions of task difficulty do combine into achievable bounds is
provided by \citet{FosterRakhlinSridharan2015}.
\citet{FosterRakhlinSridharan2017} characterize the achievability of
general data-dependent regret bounds for domains that are balls in
general Banach spaces.
\paragraph{Outline}
We add Lipschitz adaptivity to Squint for the expert setting in
Section~\ref{Squint2}. Then, in Section~\ref{MetaC}, we do the same
for MetaGrad in the OCO setting. The developments are analogous at a
high level but differ in the details for computational reasons. We
highlight the differences along the way. Section~\ref{MetaC} further
describes how to avoid specifying $T$ in advance for MetaGrad. Then, in
Section~\ref{four}, we add efficient projections for MetaGrad, and
finally Section~\ref{sec:conclusion} concludes with a discussion of
directions for future work.
\iffalse
section{Introduction}
Any source on prediction with expert advice will start with the celebrated minimax regret bound of the following form
\[
R_T^k
~\le~
\sqrt{\frac{T}{2} \ln K}
\qquad
\text{for each expert $k \in \{1,\ldots,K\}$}
,
\]
and follow up with the remark that its multiplicative constant is optimal in the limit of large $T,K$ \citep[Theorem 3.7]{cesa06}. Despite the mathematical strength and elegance, matching minimax algorithms are found to underwhelm in practice, whereas simple heuristics shine. This observation spurred multiple lines of research into adaptive algorithms with individual-sequence regret bounds with refined dependencies on the data and the comparator that hold under possibly relaxed assumptions.
Properties of bounds that have been identified as desirable are
\begin{itemize}
\item \textbf{quantile} bounds improve when multiple experts are good, a necessity when dealing with continuous expert spaces. These bounds also typically allow guarantees that are non-uniform across experts (adapting e.g.\ to available prior knowledge)
\item
\textbf{first-order} bounds improve when the loss $L^*$ of the comparator is small.
\item
\textbf{second-order} bounds improve when some measure of variance is small.
The literature distinguishes two flavours of variance, namely the variance of the loss \citep{hazan10} and the variance of the excess loss \citep{Gaillard2014}. The latter is particularly interesting because it can be shown that algorithms with guarantees for squared excess losses have constant regret in many statistical cases \citep{koolen2016}.
\item
\textbf{scale-free} bounds are for settings where no a-priori range of the losses can be assumed. Scale-free algorithms are unaffected by scaling the losses, while scale-free bounds scale along with any scaling of the losses imposed. \citet{cbms07} call algorithms/bounds that are also invariant under translations of the losses \textbf{fundamental}.
\item
\textbf{timeless} algorithms/bounds, as advocated by \citet{rooij14}, are unaffected when rounds are inserted with all-identical losses. Timelessness was proposed as a sanity-check to ``protest'' crudely measuring the complexity of the problem by its number of rounds. An algorithm can always be made timeless by simply ignoring any all-identical-loss round, but this is clunky and discontinuous, calling for prediction rules that are naturally smoothly timeless.
\end{itemize}
Taking stock (see Table~\ref{tab:stock}), we see that no algorithm currently has all desirable features. The closest candidates are \textsc{AdaHedge} by \citet{rooij14}, which is fundamental second-order timeless but not quantile, and the later
\textsc{Squint} by \citet{koolen2015} which is second-order quantile timeless but not scale-free.\footnote{%
Note that \textsc{AdaHedge} is timeless, and so are its refined bounds. Similarly, \textsc{Squint} is timeless (regardless of the prior on the learning rate $\eta$), and so is its CV bound but not its improper bound (which features a $\ln \ln T$).
}
Moreover, second-order bounds are by nature fundamental and timeless. This strongly suggests that it is possible to obtain everything. However, this seems to require a new idea in terms of algorithm design.
That is where we come in.
\begin{table}[h]
\centering
\begin{tabular}{llll}
& \textsc{AdaHedge} & \textsc{Squint} & \textsc{Squint2} \\
& \citet{rooij14} & \citet{koolen2015} & this paper \\
\midrule
Quantile & \XSolidBrush & \Checkmark & \Checkmark\\
Second-order & \Checkmark & \Checkmark & \Checkmark\\
Scale-free & \Checkmark & \XSolidBrush & \Checkmark\\
Translation-invariant & \Checkmark & \Checkmark & \Checkmark\\
Timeless & \Checkmark & \Checkmark & \Checkmark
\end{tabular}
\caption{State of algorithms on the above two dimensions.}
\label{tab:stock}
\end{table}
\todo[inline]{If one does not aim for second order (like e.g.\ coin betting) then adaptivity is perhaps more easy, e.g.\ doubling trick on $B_T$?}
\todo[inline]{Explain why a simple doubling trick on $B_T$ does not work for second order stuff.}
\todo[inline]{Also relate to the latest BOA algorithm of \cite{Wintenberger2017}.
Answer: BOA has second-order guarantees with non-quantile (but with non-uniform prior) with range adaptivity at the cost of a $\ln \ln \frac{B_T}{B_1}$ for some estimated $B_1$. Wintenberger either is not aware or is carefully hiding the fact that picking $B_1$ ``negligible'' explodes his bound.
}
\begin{table}[h]
\centering
\begin{tabular}{p{6.5cm}p{2cm}p{2cm}l}
& \textsc{MetaGrad} & \textsc{FreeRex-}\textsc{Momentum} & \textsc{MetaGrad+C}{} \\
& \citet{Erven2016} & \citet{cutkosky2017} & this paper \\
\midrule
No prior knowledge of the gradient range & \XSolidBrush & \Checkmark &\Checkmark \\
No prior knowledge of the horizon &\XSolidBrush & \Checkmark & \Checkmark \\
Log. regret under exp-concavity & \Checkmark & \XSolidBrush & \Checkmark \\
Log. regret under the Bernstein condition & \Checkmark & \XSolidBrush\footnote{\citet{cutkosky2017} show that their algorithm still achieves a regret of order $O(\ln^4 T)$ under a condition they name $\alpha$-acute convexity. The link of the latter to the more common Bernstein condition is unclear.} & \Checkmark \\
Worst-case time complexity per round in the constrained OCO setting & $O(C_{\mathcal{U}} \ln T$) & $O(C_{\mathcal{U}})$\footnote{While the original algorithm is designed for unbounded OCO, their algorithm can still be used for bounded $\mathcal{U}$ via a simple reduction proposed by \citep{cutkosky2018}} & $O(C_{\mathcal{U}} + d^2 \ln T)$
\end{tabular}
\caption{State of algorithms on the above two dimensions. $C_{\mathcal{U}}$ denotes the worst-case cost of performing a projection into the set $\mathcal{U}\subset \mathbb{R}^d$.
\todo[inline]{On the features (rows) in this table, it seems FreeRex is not better than gradient-norm adaptive GD. Add features or omit FreeRex. We want to see the Parato frontier of algorithms.}
}
\label{tab:metagrad}
\end{table}
\subsection{Related Work}
\subsubsection{Adaptive Expert Algorithms}
\begin{itemize}
\item First-order adaptivity: Auer et al: $L^*$
\item Second-order: Adahedge/CBMS, modified prod (Gaillard,Stoltz,Van
Erven)
\item Quantile bounds: NormalHedge, Chernov-Vovk, (precursor:
Poland\&Hutter for non-uniform prior)
\item Second-order+quantile: AdaNormalHedge, Squint
\item Optimistic (predictible/slowly varying/...)
\end{itemize}
\subsubsection{Adaptive OCO Algorithms}
\begin{itemize}
\item Lipschitz-adaptivity: GD with $\eta_t =
\frac{D}{\sqrt{\epsilon + \sum_{s=1}^t \|\bm{g}_s\|^2}}$ [Don't know
reference for this; \citet{McMahan2017} discusses it for $d=1$], diagonal version of
AdaGrad \citep{DuchiHazanSinger2011,McMahanStreeter2010} uses this per
dimension. \citet{McMahan2017} reviews data-dependent regularizers and
has a good discussion of AdaGrad (explicitly interpreted as per-coordinate
learning rates); he claims that you need an
initial guess $\epsilon = G$ for the FTRL-version of AdaGrad, but you
can get away with $\epsilon = 0$ for the Mirror-Descent version of
AdaGrad; otherwise not much novelty in \citep{McMahan2017}, so don't
cite too much.
\citet{WardWuBottou2018} (who call this AdaGrad even though it is not)
prove Lipschitz-adaptivity for this method in non-convex stochastic
optimization with a bound $c_G^2$ on the \emph{expected} gradient norm
squared. They actually get an extra term $\log \frac{c_G^2}{B_0}$ in
their convergence rate.
\citet{cutkosky2018} cite \citep{SrebroEtAl2010} for the claim that a
regret bound in terms of $\|\bm{g}_t\|^2$ implies fast rates for
smooth losses. This is not precise: \citet{SrebroEtAl2010} do not
directly say anything about regret in this case, but their Lemma~3.1
says that for smooth, \emph{non-negative} loss the dual norm of the
gradients can be bounded in terms of the square root of the function
value. This might be small if the algorithm is converging to the
minimum function value and this minimum value is 0, but showing that takes some work that is not
done in \citep{SrebroEtAl2010}.
\item Adapt to $G$ and $D$ simultaneously: not possible
\citep{CutkoskyBoahen2017Impossible} (actually, Manfred and Wojtek
have a paper submitted to ICML where they show you can get around this
impossibility result if you know $\bm{x}_t$ before making a prediction)
\citet{OrabonaPal2015} do have an ``adaptive'' bound $O(D^2\sqrt{\sum_{t=1}^T
\|\bm{g}_t\|^2})$, which they achieve by generalizing AdaHedge to
FTRL, but it has suboptimal dependence on $D$ (no square root) because
they just omit it when tuning the learning rate. They also try to hide
this suboptimality in their discussion, which is not good. They do
give a table with overview of `scale-free' algorithms: notably,
AdaGrad is only scale-free in its MD version and not in its FTRL
version.
\item \cite{cutkosky2017} adapt to $G$ and a class of stochastic settings. They say they
would like to combine adaptivity to $G$ with the MetaGrad regret
bound, but instead they settle for $R_T = O\left(\sqrt{G_T \sum_{t=1}^T
\|\bm{g}_t\| \|\bm{w}_t - \bm{w}^*\|^2}\right)$, which implies
``logarithmic regret'' $O(\log^4(T))$ for ``$\alpha$-acutely-convex''
functions. $\alpha$-acutely convex is a stochastic condition, which
implies the $(B,\beta)$-Bernstein condition with $B=G_T/\alpha$ and
$\beta=1$, so even the constants are what we would expect
(\citet{cutkosky2017}
have a factor $G_T$ in their regret bound, which we incur via the
Bernstein constant).
\item Adapt to curvature: \citet{BartlettHazanRakhlin2007} adapt to
strong convexity (need to observe strong convexity per round);
\citet{cutkosky2018} adapt to strong convexity (in Banach spaces)
without needing to observe strong convexity per round, but they lose
logarithmic factors; MetaGrad adapts to exp-concavity and strong
convexity without needing to observe the strong convexity per round,
but loses a factor $d$ for strong convex losses [this should be
fixable, but we have not done it]
\item Other well-known second-order methods in the mistake-bound
model: AROW and the second-order perceptron \todo{TIM: I still need to
look at these, but they are not going to be very relevant.}
\item
(Offset-) Rademacher complexity and its empirical versions.
\end{itemize}
Notes: if you need a reference showing that $O(DG \sqrt{T})$ is the
optimal rate in some sense, then \citet{cutkosky2017} refers to Jacob
Abernethy, Peter L Bartlett, Alexander Rakhlin, and Ambuj Tewari, COLT
2008. I could check this out.
\fi
\section{Problem Setting and Notation}
In OCO, a learner repeatedly chooses actions $\w_t$ from a closed convex
set $\mathcal{U} \subseteq \mathbb{R}^d$ during rounds $t=1,\ldots,T$, and
suffers losses~$f_t(\w_t)$, where $f_t: \mathcal{U} \to \mathbb{R}$ is a convex
function. The learner's goal is to achieve small \emph{regret}
$R_T^\u = \sum_{t=1}^T f_t(\w_t) - \sum_{t=1}^T f_t(\u)$
with respect to any comparator action $\u \in \mathcal{U}$, which measures
the difference between the cumulative loss of the learner and the
cumulative loss they could have achieved by playing the oracle
action~$\u$ from the start. A special case of OCO is prediction with
expert advice, where $f_t(\w) = \w^\top \vloss_t$ for $\vloss_t \in
\mathbb{R}^K$ and the domain $\mathcal{U}$ is the probability simplex
$\simplex_K = \{(w_1,\ldots,w_K) : w_i \geq 0, \sum_i w_i = 1\}$.
In this context we will further write $\p$ instead of $\w$ for the
parameters to emphasize that they represent a probability distribution.
We further define $[K] = \{1,\ldots,K\}$.
\section{Adaptive Second-order Quantile Method for Experts}
\label{Squint2}
In this section, we present an extension of the \textsc{Squint}
algorithm that adapts automatically to the loss range in the setting of
prediction with expert advice.
Throughout this section, we denote $r^k_t \coloneqq \inner{\what{\bm{p}}_t- \bm{e}_k}{\bm{\ell}_t}$ and $v^k_t \coloneqq (r^k_t)^2$, where $\bh{p}_t \in \triangle_K$ is the weight vector played by the algorithm at round $t$ and $\bm{\ell}_t$ is the observed loss vector. The cumulative regret with respect to expert $k$ is given by $\tmp{R}^k_t\coloneqq \sum_{s=1}^t r^k_s$. We use $\tmp{V}^k_t \coloneqq \sum_{s=1}^t v^k_s$ to denote the cumulative squared excess loss (which can be regarded as a measure of variance) of expert $k$ at round $t$. In the next subsection, we review the \textsc{Squint} algorithm.
\subsection{The \textsc{Squint} Algorithm}
\label{AdaptiveSquint}
We first describe the original
\textsc{Squint} algorithm, as introduced by \cite{koolen2015}. Let $\pi$
and $\gamma$ be prior distributions with supports on $[K]$ and $\left]0,
\frac{1}{2}\right]$, respectively. Then
\textsc{Squint} outputs predictions
\begin{gather}
\label{Squintforcaster}
\p_{t+1} \propto \underset{\pi(k)\gamma(\eta)}{\mathbb{E}}\left[ \eta e^{- \sum_{s=1}^t f_s(k,\eta)} \bm{e}_k \right],
\shortintertext{where $f_t(k,\eta)$ are quadratic \emph{surrogate losses} defined by}
\label{surrogatesquint0}
f_t(k,\eta) \coloneqq - \eta \inner{\bh{p}_t-\bm{e}_k}{\bm{\ell}_t} + \eta^2 \inner{\bh{p}_t-\bm{e}_k}{\bm{\ell}_t}^2.
\end{gather}
\cite{koolen2015} propose to use the \emph{improper} prior $\gamma(\eta)
= \frac{1}{\eta}$ which does not integrate to a finite value over its
domain, but because of the weighting by $\eta$ in
\eqref{Squintforcaster} the predictions $\p_{t+1}$ are still
well-defined. The benefit of the improper prior is that it allows
calculating $\p_{t+1}$ in closed form \citep{koolen2015}.
For any distribution $\rho \in \simplex_K$, \textsc{Squint} achieves the following bound:
\begin{align}
\tmp{R}^{\rho}_T = O\left(\sqrt{\tmp{V}^{\rho}_T\left( \KL(\rho || \pi ) +
\ln \ln T\right)}\right), \label{Squintbound}
\end{align}
where $R_T^{\rho} = \mathbb{E}_{\rho(k)}\left[R_T^{k} \right]$ and
$V_T^{\rho} = \mathbb{E}_{\rho(k)}\left[V_T^{k} \right]$. This version
of Squint assumes the loss range $\max_k \ell_{t,k} - \min_k \ell_{t,k}$
is at most $1$, and can fail otherwise. In the next subsection, we
present an extension of \textsc{Squint} which does not need to know the
Lipschitz constant.
\subsection{Lipschitz Adaptive Squint}
\let\scale\bar
We first design a version of \textsc{Squint}, called \textsc{Squint+C}{}, that still requires an
initial estimate $B > 0$ of the Lipschitz constant.
The next section will be devoted to setting this parameter online. For now, we consider it fixed. In addition to this, the algorithm takes a prior distribution $\pi \in \triangle_K$. In a sequence of rounds $t = 1, 2, \ldots$ the algorithm predicts with $\hat\p_t \in \triangle_K$, and receives a loss vector $\vloss_t^k \in \mathbb R^K$. We denote the \emph{instantaneous regret of expert $k$ in round $t$} by $r_t^k \df \tuple{\hat \p_t - \e_k, \vloss_t}$. We denote the observed Lipschitz constant in round $t$ at point $\hat \p_t$ by
$
b_t
\df
\max_k \abs{r_t^k}$, and we denote its running maximum by $B_t \df B \lub \max_{s \le t} b_s$, and we use the convention that $B_0=B$. In addition, we will also require a clipped version of the loss vector $\scale{\vloss_t} = \frac{B_{t-1}}{B_t} \vloss_t$, and we denote by $\scale{r}_t^k = \tuple{\hat \p_t - \e_k, \scale \vloss_t}$ the rescaled instantaneous regret. We will use that $\abs{\scale r_t^k} \le B_{t-1}$. It suffices to control the regret for the clipped loss, because the cumulative difference is a negligible lower-order constant\footnote{We learned this technique from Ashok Cutkosky}:
\begin{equation}\label{eq:ashok}
R_T^k
-
\scale R_T^k
~\df~
\sum_{t=1}^T \del*{r_t^k - \scale r_t^k}
~=~
\sum_{t=1}^T \del*{B_t - B_{t-1}} \frac{r_t^k}{B_t}
~\le~
B_T - B_0
.
\end{equation}
This means we can focus on regret for $\scale \vloss_t$, for which the range bound $\abs{\scale r_t^k} \le B_{t-1}$ is available \emph{ahead} of each round $t$. To motivate \textsc{Squint+C}{}, we define the potential function after $T$ rounds by
\begin{equation}\label{eq:sq.pot}
\Phi_T
\df
\sum_k \pi_k \int_0^\frac{1}{2 B_{T-1}} \frac{e^{\eta \scale{R}_T^k - \eta^2 \scale{V}_T^k} -1}{\eta} \dif \eta
\quad
\text{where}
\quad
\scale R_T^k \df \sum_{t=1}^T \scale r_t^k
~~
\text{and}
~~
\scale V_T^k \df \sum_{t=1}^T (\scale r_t^k)^2
.
\end{equation}
We also define $\Phi_0 = 0$ (due to the integrand being zero), even though it involves the meaningless $B_{-1}$ in the upper limit. The algorithm is now derived from the desire to keep this potential under control. As we will see in the analysis, this requirement uniquely forces the choice of weights
\begin{equation}\label{eq:sq.weights}
\hat p_{T+1}^k
~\propto~
\pi_k \int_0^\frac{1}{2 B_T} e^{\eta \scale{R}_T^k - \eta^2 \scale{V}_T^k} \dif \eta
.
\end{equation}
Like the original \textsc{Squint}, we see that the weights $\hat \p_{t+1}$ can be evaluated in closed form using Gaussian CDFs. The regret analysis consists of two parts. First, we show that the algorithm keeps the potential small.
\begin{lemma}\label{lem:pot.is.small}
Given parameter $B\geq0$, \textsc{Squint+C}{} ensures $\Phi_T \le \ln \frac{B_{T-1}}{B}$.
\end{lemma}
The next step of the argument is to show that small potential is useful. The argument here follows \cite{koolen2015}, specifically the version by \cite{squintPAC}. We have
\begin{lemma}\label{lem:small.is.good}
Definition \eqref{eq:sq.pot} implies that for any comparator distribution $\rho \in \triangle_K$ the regret is at most
\begin{gather}
\scale R_T^\rho
~\le~
\sqrt{2 \scale V_T^\rho} \del*{
1+
\sqrt{2 C_T^{\rho}}
}
+
5 B_{T-1}
\del*{C_T^{\rho}+ \ln 2},
\quad \text{where,} \\
C^{\rho}_T ~\df~
\KL \delcc*{\rho}{\pi}
+ \ln \del*{
\Phi_T
+ \frac{1}{2}
+ \ln \left(2+ \sum_{t=1}^{T-1} \frac{b_t}{B_t} \right)
}
\end{gather}
\end{lemma}
Keeping only the dominant terms, this reads
\[
\scale R_T^\rho
~=~
O\del*{
\sqrt{\scale V_T^\rho \del*{\KL \delcc*{\rho}{\pi} +
\ln \Phi_T + \ln \ln T}}
}
.
\]
The significance of \eqref{eq:ashok}, Lemmas~\ref{lem:pot.is.small} and ~\ref{lem:small.is.good} is that we obtain a bound of the form
\[
R_T^\rho
~=~
O \del*{
\sqrt{V_T^\rho \del*{\KL \delcc*{\rho}{\pi} +
\ln \ln \frac{TB_{T-1}}{B}}}
+
5 B_T \del*{\KL \delcc*{\rho}{\pi} +
\ln \ln \frac{TB_{T-1}}{B} }
}
.
\]
However, there does not seem to be any safe a-priori way to tune $B=B_0$. If we set it too small, the factor $\ln \ln \frac{B_{T-1}}{B}$ explodes. If we set it too large, the lower-order contribution $B_{T-1} \ge B$ blows up. It does not appear possible to bypass this tuning dilemma within the current construction. Fortunately, we are able to resolve it using restarts. Algorithm~\ref{bb1alg}, which applies to both \textsc{Squint+C}{} and \textsc{MetaGrad+C}{} (presented in the next section), monitors a condition of the sequences $(b_t)$ and $(B_t)$ to trigger restarts.
\begin{algorithm}[tbp]
\caption{Restarts to make {\textsc{Squint+C}} or {\textsc{MetaGrad+C}} scale-free}
\label{bb1alg}
\begin{algorithmic}[1]
\REQUIRE {\textsc{Alg}} is either {\textsc{Squint+C}} or {\textsc{MetaGrad+C}}, taking as
input parameter an initial scale $B$
\STATE Play $\w_1$ until the first time $t=\tau_1$ that $b_t
\neq 0$.
\STATE \label{line:runmetagrad} Run {\textsc{Alg}}
with input $B = B_{\tau_1}$ until the first time $t=\tau_2$ that
$\displaystyle \frac{B_t}{B_{\tau_1}} >
\sum_{s=1}^t \frac{b_s}{B_s}$.\\
\STATE Set $\tau_1 = \tau_2$ and goto line \ref{line:runmetagrad}.
\end{algorithmic}
\end{algorithm}
\begin{theorem}
\label{blackboxreduction0}
Let \textsc{Squint+L}{} be the result of applying
Algorithm~\ref{bb1alg} to \textsc{Squint+C}{} (as \textsc{Alg}). \textsc{Squint+L}{} guarantees, for any comparator $\rho\in \triangle_K$,
\begin{align}
R_T^\rho
~\le~
2\sqrt{ V_T^\rho} \del*{
1+
\sqrt{2 \Gamma_T^{\rho}}
}
+
10 B_{T}
\del*{
\Gamma_T^{\rho}
+ \ln 2} + 4 B_T,
\end{align}
where $ \Gamma_T^{\rho} ~\df~\KL \delcc*{\rho}{\pi}+ \ln \del*{\ln \left(\sum_{t=1}^{T-1} \frac{b_{t}}{B_t}\right) + \frac{1}{2}+ \ln \left(2+\sum_{t=1}^{T-1} \frac{b_{t}}{B_t}\right)}$.
\end{theorem}
Theorem \ref{blackboxreduction0} shows that the bound on the regret of \textsc{Squint+L}{} has a term of order $O(\ln \ln \sum_{t=1}^{T-1} \frac{b_{t}}{B_t})=O(\ln \ln T)$, which does not depend on the initial guess $B_0$ anymore.
\section{Adaptive Method for Online Convex Optimization}
\label{MetaC}
We consider the Online Convex Optimization (OCO) setting where at each round $t$, the learner predicts by playing $\what{\bm{u}}_t$ in a closed convex set $\mathcal{U} \subset \mathbb{R}^d$, then the environment announces a convex function $\ell_t : \mathcal{U}\rightarrow [0,+\infty[$ and the learner suffers loss $\ell_t(\what{\bm{u}}_t)$. The goal of the learner is to minimize the regret with respect to the single best action $\bm{u}\in \mathcal{U}$ in hindsight (after $T$ rounds); that is, minimizing $\tmp{R}^{\bm{u}}_T \coloneqq \sum_{t=1}^T \ell_t(\what{\bm{u}}_t) - \sum_{t=1}^T \ell_t(\bm{u})$ for the worst case $\bm{u}\in \mathcal{U}$. Since the losses are convex, it suffices to bound the sum of linearized losses $\tilde{R}^{\bm{u}}_T \df \sum_{t=1}^T \inner{\what{\bm{u}}_t - \bm{u}}{\bm{g}_t}$, where $\bm{g}_t \coloneqq \nabla \ell_t(\what{\bm{u}}_t)$. We will assume that the set $\mathcal{U}$ is bounded and let $D\in ]0,+\infty[$ be its diameter
\begin{align} \label{rad}D \coloneqq \sup_{\bm{u}, \bm{v}\in \mathcal{U}} \norm{\bm{u} - \bm{v}}_2.\end{align}
Our main contribution in this section is to devise a simple modification of \textsc{MetaGrad} --- \textsc{MetaGrad+C}{} --- which, without prior knowledge of the maximum value of the gradient range $G \coloneqq \max_{t \le T} \norm{\nabla \ell_t(\what{\bm u}_t)}$, guarantees the following regret bound
\begin{align}
\forall \bm{u}\in \mathcal{U}, \quad R_T^\u \leq \tilde{R}^{\bm{u}}_T
~=~
O \del*{\sqrt{\tmp{V}^{\bm{u}}_T d\ln T } + B_T d \ln T}, \label{mresult}
\end{align}
where $\tmp{V}^{\bm{u}}_T \coloneqq \sum_{t=1}^T \inner{\bh{u}_t - \bm{u}}{\bm{g}_t}^2$. Consequently, this algorithm inherits the fast convergence results of standard \textsc{MetaGrad} \citep{Erven2016}. In particular, it was shown that due to the form of the bound in \eqref{mresult}, \textsc{MetaGrad} achieves a logarithmic regret when the sequence of losses are exp-concave \citep{Erven2016}. Furthermore, when the sequence of gradient functions $(\nabla \ell_t)$ are i.i.d distributed with common distribution $\mathbb{P}$ and satisfy the ($B, \beta$)-Bernstein condition for $B > 0$ and $\beta \in [0, 1]$ with
respect to the risk minimizer $\bm{u}^* =\argmin_{\bm{u}\in \mathcal{U}}\mathbb{E}_{f \sim \mathbb{P}}[f(\bm{u})]$, then \textsc{MetaGrad} (and thus \textsc{MetaGrad+C}{}) achieves the expected regret
\begin{align*}
\mathbb{E}\left[\tmp{R}^{\bm{u}^*}_T\right]
~=~
O \del*{ (d \ln T)^{\frac{1}{2-\beta}} T^{\frac{1-\beta}{2-\beta}} + d \ln T}.
\end{align*}
See \citep{koolen2016} for more detail.
\subsection{The \textsc{MetaGrad} Algorithm}
The \textsc{MetaGrad} algorithm runs several sub-algorithms at each round; namely, a set of slave algorithms, which learn the best action in $\mathcal{U}$ given a learning rate $\eta$ in some predefined grid $\mathcal{G}$, and the master algorithm, which learns the best learning rate. The goal of \textsc{MetaGrad} is to maximize the sum of payoff functions $\sum_{t=1}^T f_t(\bm{u},\eta)$ over all $\eta \in \mathcal{G}$ and $\bm{u}\in \mathcal{U}$ simultaneously, where
\begin{align}
\label{surrmeta}
f_t(\bm{u},\eta) \coloneqq - \eta \inner{\bh{u}_t - \bm{u}}{\bm{g}_t} + \eta^2 \inner{\bh{u}_t - \bm{u}}{\bm{g}_t}^2,\quad t\in [T],
\end{align}
and $\bh{u}_t$ is the master prediction at round $t\geq 1$. Each slave algorithm takes as input a learning rate from a finite, exponentially-spaced grid $\mathcal{G}$ (with $\ceil{\log_2 \sqrt{T}}$ points) within the interval $\left[\frac{1}{5DG\sqrt{T}}, \frac{1}{5DG}\right]$, where $G$ is an upper bound on the norms of the gradients. In this case, the bound $G$ must be known in advance. In what follows, we let $\mathbf{M}_t \coloneqq \sum_{s=1}^t \bm{g}_s\bm{g}_s^{\T}$, for $ t \geq 0$.
\paragraph{Slave predictions.}
Every slave $\eta \in \mathcal G$ starts with $\bh{u}_1^\eta = \bm{0}$. At the end of round $t \ge 1$, it receives the master prediction $\bh{u}_t$ and updates the prediction in two steps
\begin{gather}
{\bm{u}}^{\eta}_{{t+1}} \coloneqq \bh{u}^{\eta}_t - \eta \mathbf{\Sigma}^{\eta}_{{t+1}} \bm{g}_t \left(1 + 2 \eta \left( \bh{u}^{\eta}_t -\bh{u}_t \right)^{\T}\bm{g}_t\right) , \text{ where }\ \mathbf{\Sigma}^{\eta}_{{t+1}} \coloneqq \left( \tfrac{\mathbf{I}}{D^2} +2\eta^2\mathbf{{M}}_t \right)^{-1},
\label{quadprog0}\\
\label{gaussian}
\text{ and } \ \ \bh{u}^{\eta}_{{t+1}} = \argmin_{\bm{u}\in \mathcal{U}} \left(\bm{u}_{{t+1}}^{\eta}- \bm{u} \right)^{\T}\left( \mathbf{\Sigma}^{\eta}_{{t+1}}\right)^{-1} \left(\bm{u}_{{t+1}}^{\eta}- \bm{u} \right) ,
\end{gather}
\paragraph{Master predictions}
After receiving the slaves predictions, $\left(\bh{u}^{\eta}_t\right)_{\eta \in \mathcal{G}}$, the master algorithm aggregates them and outputs $\bh{u}_t\in \mathcal{U}$ according to:
\begin{align}
\bh{u}_{t} \coloneqq \frac{\sum_{\eta\in \mathcal{G}} \eta w^{\eta}_t \bh{u}^{\eta}_{t} }{\sum_{\eta \in \mathcal{G}}\eta w^{\eta}_t };\quad w^{\eta}_t \coloneqq e^{- \sum_{s=1}^{t-1} f_s(\bh{u}^{\eta}_s,\eta)}, \label{masterpred}
\end{align}
As mentioned earlier, the \textsc{MetaGrad} algorithm requires the knowledge of the maximum value of the gradient range $G$ and the horizon $T$ in advance. These are needed to define the grid of the slave algorithms. In the analysis of \textsc{MetaGrad}, it is crucial for the $\eta$'s to be in the right interval in order to invoke a Gaussian exp-concavity result for the surrogate losses in \eqref{surrmeta} (see e.g.\ \citep[Lemma 10]{Erven2016}). In the next subsection, we explore a natural extension of \textsc{MetaGrad} which does not require the knowledge of the gradient range or the horizon $T$.
\subsection{An Extension of \textsc{MetaGrad} for Unknown Gradient Range and Horizon}
We present a natural extension of \textsc{MetaGrad}, called \textsc{MetaGrad+C}{}, which does not assume any knowledge on the gradient range or the horizon. Contrary to the original \textsc{MetaGrad} which requires knowledge of the horizon $T$ to define the grid for the slaves, \textsc{MetaGrad+C}{} circumvents this by defining an infinite grid $\mathcal{G}$, in which, at any given round $t\geq1$, only a finite number of slaves (up to $\log_2 t$ many) output a prediction (see Remark \ref{numslaves}). Each slave $\eta$ in this grid receives a prior weight $\pi(\eta) \in[0,1]$, where $\sum_{\eta\in \mathcal{G}} \pi(\eta) =1$. The expressions of $\mathcal{G}$ and $\pi$ are given by
\begin{align}
\mathcal{G} \coloneqq \left\{ \eta_i \coloneqq \tfrac{2^{-i}}{5 B}: i \in \mathbb{N} \right\} \label{Ggrid} ;\quad
\pi(\eta_i) \coloneqq \tfrac{1}{(i+1)(i+2)}, \ i\in \mathbb{N}.
\end{align}
where $B>0$ is the input to \textsc{MetaGrad+C}.
\subsubsection{Algorithm Description}
\paragraph{Preliminaries.} As in the previous subsection, we let $\bh{u}_t$ and $\bh{u}^{\eta}_{t}$ be the predictions of the master and slave $\eta$, respectively, at round $t\geq1$ (we give their explicit expressions further below). Let $(b_t)$ and $(B_t)$ be the sequences in $\mathbb{R}_{\geq0}$ defined by
\begin{align}
b_t \coloneqq D \norm{\bm{g}_t}_2; \quad \quad \quad B_t \coloneqq B \vee \max_{s\in [t]} b_s, \quad t\in[T], \label{littleb}
\end{align}
where $B$ is the input of \textsc{MetaGrad+C}, and we use the convention that $B_0 = B$.
Using the sequence $(B_t)$, we define the clipped gradients $\bar{\bm{g}}_t \coloneqq \frac{B_{t-1}}{B_t} \bm{g}_t$, and $\forall \bm{u}\in\mathcal{U},\forall t\geq 1, \forall \eta >0$, we let
\begin{align}
\bar{r}^{\bm{u}}_t \coloneqq \inner{\bh{u}_t-\bm{u}}{\bar{\bm{g}}_t},\quad \quad \bar{f}_t(\bm{u},\eta)\coloneqq - \eta \bar{r}^{\bm{u}}_t + \left(\eta \bar{r}^{\bm{u}}_t\right)^2,\quad \quad \bar{\mathbf{M}}_t\coloneqq \sum_{s=1}^t \bar{\bm{g}}_s \bar{\bm{g}}_s^{\T}. \label{clippedstuff}
\end{align}
For each slave $\eta\in \mathcal{G}$, we define the time $s_\eta$ to be
\begin{align}
\label{threshold}
s_\eta
~\df~
\min\setc*{t \ge 0}{
\eta \geq \frac{1}{D \sum_{s=1}^t \norm{\bar{\bm{g}}_s}_2 + B_t}
},
\end{align}
and define the set $\mathcal{A}_t$ of ``active'' slaves by
\begin{align}
\mathcal{A}_t \coloneqq \{ \eta \in \mathcal{G}_t : s_\eta < t \}, \quad \text{where} \quad \mathcal{G}_t \coloneqq \mathcal{G} \cap \left[0, \tfrac{1}{5B_{t-1}}\right] , \quad t\geq 1.
\end{align}
\paragraph{Slaves' predictions.} A slave $\eta \in \mathcal{A}_t$ issues its first prediction $\bh{u}_t^\eta = \bm{0}$ in round $t=s_\eta+1$. From then on, it receives the master prediction $\bh{u}_t$ as input and updates in two steps as
\begin{gather}
\bm{u}^{\eta}_{{t+1}} \coloneqq \bh{u}^{\eta}_t - \eta \mathbf{\Sigma}^{\eta}_{{t+1}} \bar{\bm{g}}_t \left(1 + 2 \eta \left( \bh{u}^{\eta}_t -\bh{u}_t \right)^{\T}\bar{\bm{g}}_t\right),
\text{ where }\
\mathbf{\Sigma}^{\eta}_{{t+1}} \coloneqq \left( \tfrac{\mathbf{I}}{D^2} +2\eta^2\left(\bar{\mathbf{M}}_t -\bar{\mathbf{M}}_{s_{\eta}}\right) \right)^{-1},
\nonumber
\\
\text{ and }\ \ \bh{u}^{\eta}_{{t+1}} = \argmin_{\bm{u}\in \mathcal{U}} \left(\bm{u}_{{t+1}}^{\eta}- \bm{u} \right)^{\T}\left( \mathbf{\Sigma}^{\eta}_{{t+1}}\right)^{-1} \left(\bm{u}_{{t+1}}^{\eta}- \bm{u} \right).
\label{quadprog}
\end{gather}
Slaves that are outside the set $\mathcal{A}_t$ at round $t$ are irrelevant to the algorithm\footnote{The predictions of the slaves outside $\mathcal{A}_t$ do not appear anywhere in the description or analysis of the algorithm. Alternatively, we may think of each slave $\eta$ as operating with $\eta_t=0$ in the first $s_\eta$ rounds and with $\eta_t=\eta$ afterwards. The presence of the factor $\eta$ in \eqref{masterpred} renders the master oblivious to inactive slaves.
}. Note that restricting the slaves to the set $\mathcal{G}_t$ is similar to clipping the upper integral range in the \textsc{Squint+C}{} case.
\paragraph{Master predictions.} At each round $t\geq1$, the master algorithm receives the slaves predictions $(\bh{u}_t^{\eta})_{t\in \mathcal{A}_{t}}$ and outputs the $\widehat{\bm{u}}_t$:
\begin{align}
\label{newmaster}
\bh{u}_t = \frac{\sum_{\eta \in \mathcal{A}_{t}}\eta w^{\eta}_t \bh{u}_t^{\eta}}{\sum_{\eta \in \mathcal{A}_{t}} \eta w^{\eta}_t }; \quad w^{\eta}_t \coloneqq \pi(\eta) e^{- \sum_{s=s_{\eta}+1}^{t-1} \bar{f}_s(\bh{u}^{\eta}_s,\eta)}, \quad t\geq 1.
\end{align}
\begin{remark}[Number of active slaves] \label{numslaves} At any round $t\geq 1$, the number of active slaves is at most $\floor{\log_2 t}$. In fact, if $\eta \in \mathcal{A}_t$, then by definition $\eta \geq 1/(D\sum_{s=1}^{s_{\eta}}\norm{\bm{g}_s}_2 + B_{s_{\eta}}) \geq 1/(t B_{t-1})$ (since $s_{\eta}\leq t-1$), and thus $\mathcal{A}_t \subset [1/(tB_{t-1}), 1/(5B_{t-1})]$.
Since $\mathcal{A}_t$ is an exponentially-spaced grid with base $2$, there are at most $\floor{\log_2 t}$ slaves in $\mathcal{A}_t$.
\end{remark}
\subsubsection{Analysis}
To analyse the performance of \textsc{MetaGrad+C}, we consider the potential function \begin{align}
\label{masterpot}
\Phi_t \coloneqq \pi(\mathcal{G}_t\setminus \mathcal{A}_t) + \sum_{\eta\in \mathcal{A}_t} \pi(\eta) e^{-\sum_{s=s_{\eta}+1}^t \bar{f}_s(\bh{u}^{\eta}_s,\eta)}, \quad t\geq 0.\end{align}
For $\bm{u}\in \mathcal{U}$, we define the pseudo-regret $\tilde{R}^{\bm{u}}_T \coloneqq \sum^T_{t=1} \inner{\bh{u}_t - \bm{u}}{{\bm{g}}_t}$ and its clipped version $\cliplinregret^{\bm{u}}_T \coloneqq \sum^T_{t=1} \inner{\bh{u}_t - \bm{u}}{\bar{\bm{g}}_t}$. The following analogue to \eqref{eq:ashok} relates these two regrets.
\begin{lemma}
\label{relatingtheregret}
Let $(b_t)$ and $(B_t)$ be as in \eqref{littleb}, respectively, then for all $\bm{u}\in \mathcal{U}$,
\begin{align}
\label{clippedrel}
\tilde{R}^{\bm{u}}_{T} \leq \cliplinregret^{\bm{u}}_{T} +B_T.
\end{align}
\end{lemma}
Similarly to the \textsc{Squint} case, one can use the prod-bound to control the growth of this potential function as shown in the proof of the following lemma (see Appendix \ref{MetaGrad2proofs}):
\begin{lemma}
\label{lemmameta}
\textsc{MetaGrad+C}{} guarantees that $\Phi_T \leq \dots \leq \Phi_0 = 1$, for all $T \in \mathbb{N}$.
\end{lemma}
We now give a bound on the clipped regret $\cliplinregret^{\u}_T$ in terms of the clipped variance $\clipvar^{\u}_T \coloneqq \sum_{t=1}^T (\bar{r}^{\u}_t)^2$:
\begin{theorem}
\label{naivemeta}
Given input $B>0$, the clipped pseudo-regret for {\textsc{MetaGrad+C}} is bounded by
\begin{equation}
\cliplinregret_T^\u\leq
3\sqrt{\clipvar_T^\u C_T} + 15 B_T C_T
\quad \text{for any $\u \in \mathcal{U}$,} \label{naivebound}
\end{equation}
where $C_T \coloneqq d\ln\left(1 + \frac{2 \sum_{t=1}^{T-1}
b_t^2 + 2 B^2_{T-1}}{25 d B^2_{T-1}}\right) + 2 \ln \left( \log^+_2
\frac{\sqrt{\sum_{t=1}^Tb^2_t }}{B} +3 \right) + 2$ and $\log_2^+ = 0 \vee \log_2 $.
\end{theorem}
\begin{remark}
\label{truebound}
We can relate the clipped pseudo-regret to the ordinary regret
via $R_T^\u \leq \linregret_T^\u \leq \cliplinregret_T^\u + B_T$ (see
\eqref{clippedrel}) and on the
right-hand side we can also use that $\clipvar_T^\u \leq V_T^\u$.
\end{remark}
An important thing to note from the result of Theorem \ref{naivemeta} is that the ratio $\sqrt{\sum_{t=1}^Tb^2_t}/B$, could in principle be arbitrarily large if the input $B$ is too small compared to the actual regret range. To resolve this issue, one can use the same restart trick as in the \textsc{Squint} case:
\begin{theorem}
\label{blackboxreduction1}
Let {\textsc{MetaGrad+L}} be the result of applying Algorithm~\ref{bb1alg} to
{\textsc{MetaGrad+C}}. Then the regret for {\textsc{MetaGrad+L}} is bounded by
\begin{align}
\linregret_T^\u
\leq 3\sqrt{V_T^\u \Gamma_T} + 15 B_T \Gamma_T + 4 B_T
\quad \text{for all $\u \in \mathcal{U}$,}\label{bbbound}
\end{align}
where $\Gamma_T \coloneqq 2 d\ln\left(\frac{27}{25} + \frac{2}{25d} \sum_{t=1}^{T} \frac{b_t^2}{B^2_{t}}\right) + 4 \ln \left( \log^+_2
\sqrt{\sum_{t=1}^T (\sum_{s=1}^t \frac{b_s}{B_s})^2} +3 \right)+ 4 = O(d
\ln T)$.
\end{theorem}
In Theorem \ref{blackboxreduction1}, we have replaced the ratio
$\sqrt{\sum_{t=1}^Tb^2_t} /B$ appearing in the (clipped)
pseudo-regret bound of \textsc{MetaGrad+C}{} by the term $\sigma_T
\coloneqq \sqrt{\sum_{t=1}^T (\sum_{s=1}^t \frac{b_s}{B_s})^2}$ which is
always smaller than $T^{\frac{3}{2}}$, but this is acceptable since $\sigma_T$
appears inside a $\ln \ln$. From the bound of Theorem \ref{blackboxreduction1} on can easily recover an ordinary regret bound, i.e. a bound on $R^{\bm{u}}_t, \bm{u}\in \mathcal{U}$ (see Remark \ref{truebound}).\\
\section{Efficient Implementation Through a Reduction to the Sphere}
\label{four}
Using \textsc{MetaGrad+C}{} (or \textsc{MetaGrad}), the computation of
each vector $\bh{u}^{\eta}_t$ requires a (Mahalanobis) projection step onto an
arbitrary convex set $\mathcal{U}$. Numerically, this typically
requires $O(d^p)$ floating point operations (flops), for some $p \in
\mathbb{N}$ which depends on the topology of the set $\mathcal{U}$.
Since $p$ can be large in many applications,
evaluating $\bh{u}^{\eta}_{t}$ at each grid point $\eta$ can become computationally prohibitive, especially when the number of grid points grows with $T$ --- in the case or \textsc{MetaGrad+C}{} there can be at most $\floor{\log_2 T}$ slaves at round $T\geq1$ (see Remark \ref{numslaves}).
\subsection{An efficient implementation of \textsc{MetaGrad+C}{} on the ball}
\label{ballefficient}
In this subsection, we assume that $\mathcal{U}$ is the ball of diameter $D$, i.e.\ $\mathcal{U}=\mathcal{B}_{D} \coloneqq \left\{\bm{u} \in \mathbb{R}^d \colon \norm{\bm{u}}_2 \leq D/2 \right\}$. In order to compute the slave prediction $\bh{u}^{\eta}_{t+1}$, for $t\geq 1$ and $\eta \in \mathcal{A}_t$, the following quadratic program needs to be solved:
\begin{align}
\bh{u}^{\eta}_{t+1} = \argmin_{\bm{u}\in \mathcal{U}} \left(\bm{u}_{t+1}^{\eta}- \bm{u} \right)^{\T}\left( \mathbf{\Sigma}^{\eta}_{t+1}\right)^{-1} \left(\bm{u}_{t+1}^{\eta}- \bm{u} \right), \label{quadprog2}
\end{align}
where $\bm{u}^{\eta}_{t+1}$ (the unprojected prediction) and $\mathbf{\Sigma}^{\eta}_{t+1}$ (the co-variance matrix) are defined in \eqref{quadprog}.
Since $\mathcal{U}$ is a ball, \eqref{quadprog} can be solved efficiently using the result of following lemma:
\begin{lemma}
\label{redquad}
Let $t\geq 1$, $\eta \in \mathcal{A}_t$, and $\bm{v}^{\eta}_{t+1}\coloneqq \left(\tfrac{\mathbf{I}}{D^2} +2\eta^2\left(\bar{\mathbf{M}}_t -\bar{\mathbf{M}}_{s_{\eta}}\right)\right) \bm{u}^{\eta}_{t+1}$. Let $\mathbf{Q}_t$ be an orthogonal matrix which diagonalizes $\bar{\mathbf{M}}_{t}$, and $\mathbf{\Lambda}_t \coloneqq \left[\lambda^i_t\right]_{i=1}^t$ the diagonal matrix which satisfies $\mathbf{Q}_t \bar{\mathbf{M}}_t \mathbf{Q}^{\T}_t = \mathbf{\Lambda}_t$. The solution of \eqref{quadprog2} is given by $\bh{u}^{\eta}_{t+1}=\bm{u}^{\eta}_{t+1}$, if $\bm{u}^{\eta}_{t+1} \in \mathcal{U}$; and otherwise, $\bh{u}^{\eta}_{t+1} = \mathbf{Q}_t^{\T} (x_t^{\eta}\mathbf{I} +2 \eta^2 (\mathbf{\Lambda}_t- \mathbf{\Lambda}_{s_{\eta}} ))^{-1} \mathbf{Q}_t \bm{v}^{\eta}_{t+1}$, where $x_{t}^{\eta}$ is the unique solution of
\begin{align}
\rho_t^{\eta}(x) \coloneqq \sum_{i=1}^{d} \frac{\inner{\bm{e}_i}{\mathbf{Q}_t \bm{v}^{\eta}_{t+1}}^2}{(x+2\eta^2 (\lambda^i_t -\lambda^i_{s_{\eta}}))^2} =\frac{D^2}{4}, \label{proxyfun}
\end{align}
\end{lemma}
The proof of the lemma is in Appendix \ref{fourproof}. Note that since the matrix $\bar{\mathbf{M}}_t$ is symmetric for all $t\geq 1$, the existence of the matrices $\mathbf{Q}_t$ and $\mathbf{\Lambda}_t$ in Lemma \ref{redquad} is always guaranteed. Since $\rho_t^{\eta}$ in \eqref{proxyfun} is strictly convex and decreasing, one can use the Newton method to efficiently solve $\rho_t^{\eta}(x)=D^2/4$. Thus, since the computation of $\mathbf{Q}_t \bm{v}^{\eta}_{t+1}$ only involves matrix-vector products, Lemma \ref{redquad} gives an efficient way of solving \eqref{quadprog2}.
\begin{algorithm}[tbp]
\begin{algorithmic}
\REQUIRE: A bounded convex set $\mathcal{U}\in \mathbb{R}^d$ with diameter $D$, and a fast \textsc{MetaGrad+C}{} implementation on the ball $\mathcal{B}_{D}$, \textsc{MetaGrad+C}{}, taking input $B$.
\FOR{$t=1$ \KwTo $T$}
\STATE Get $\bh{u}_t$ from \textsc{MetaGrad+C}{}
\STATE Play $\bh{w}_t = \Pi_{\mathcal{U}}(\bh{u}_t)$, receive $\mathring{\bm{g}}_t = \nabla \ell_t(\bh{w}_t)$
\STATE Set $\bm{g}_t \in \tfrac{1}{2} \left( \mathring{\bm{g}}_t +\norm{\mathring{\bm{g}}_t} \partial \op{d}_{\mathcal{U}}(\bh{u}_t) \right)$
\STATE Send $\bm{g}_t$ to \textsc{MetaGrad+C}{}
\ENDFOR
\caption{Fast implementation of \textsc{MetaGrad+C}{} on any bounded convex set $\mathcal{U}$ via reduction to the ball.}
\label{OCOGeneral}
\end{algorithmic}
\end{algorithm}
\paragraph{Implementation on the ball.}
At round $t\geq 1$, the implementation of \textsc{MetaGrad+C}{} on the ball $\mathcal{B}_{D}$ keeps in memory the orthogonal matrix $\mathbf{Q}_{t-1}$ which diagonalizes $\bar{\mathbf{M}}_{t-1}$. In this case, since $\bar{\mathbf{M}}_t = \bar{\mathbf{M}}_{t-1}+ \bar{\bm{g}}_t \bar{\bm{g}}_t^{\T}$ it is possible to compute the new matrices $\mathbf{Q}_t$ and $\mathbf{\Lambda}_t$ in $O(d^2)$ flops \citep{stor2015}. Note that this operation only needs to be performed once --- the diagonalization does not depend on $\eta$. Therefore, computing $\mathbf{Q}_t \bm{v}^{\eta}_{t+1}$ (and thus $\bh{u}^{\eta}_{t+1}$) can be performed in only $O(d^2)$ flops. Thus, aside from the matrix-vector products, the time complexity involved in computing $\bh{u}_{t+1}^{\eta}$ for a given $\eta\in \mathcal{A}_t$ is of the same order as that involved in solving $\rho_t^{\eta}(x)=D^2/4$.
\subsection{A Reduction to the ball} In this subsection, we make use of a recent technique by \cite{cutkosky2018} that reduces constrained optimization problems to unconstrained ones, to reduce any OCO problem on an arbitrary bounded convex set $\mathcal{U}\subset\mathbb{R}^d$ to an OCO problem on a ball, where we can apply \textsc{MetaGrad+C}{} efficiently. Let $D$ be the diameter of $\mathcal{U}\in \mathbb{R}^d$ as in \eqref{rad}, so that the ball $\mathcal{B}_D$ of radius $D/2$ encloses $\mathcal{U}$. For $\bm{u}\in \mathcal{U}$, we denote $\op{d}_{\mathcal{U}}(\bm{u}) = \min_{\bm{w} \in \mathcal U} \norm{\bm{u}-\bm{w}}_2$ the \emph{distance function} from the set $\mathcal{U}$, and we define $\Pi_{\mathcal{U}}(\u)\coloneqq \{\w\in \mathcal{U}: \norm{\bm{w}-\u}_2 = \op{d}_{\mathcal{U}}(\u) \}$.
Algorithm \ref{OCOGeneral} reduces the OCO problem on the set $\mathcal{U}$ to one on the ball $\mathcal{B}_{D}$, where \textsc{MetaGrad+C}{} is used as a black-box to solve it efficiently. As a result, Algorithm \ref{OCOGeneral} (including its \textsc{MetaGrad+C}{} subroutine) only performs a single Euclidean projection (as opposed to the projection in Mahalanobis distance as in \eqref{quadprog2}) onto the set $\mathcal{U}$, which is applied to the output of \textsc{MetaGrad+C}{} --- the \textsc{MetaGrad+C}{} subroutine only performs projections onto the ball $\mathcal{B}_D$, which can be done efficiently as described in the previous subsection.
Let $\mathring{\tmp{R}}^{\bm{u}}_T \coloneqq \sum_{t=1}^T \inner{\bh{w}_t - \bm{u}}{\mathring{\grad}_t}$ and $\mathring{\tmp{V}}^{\bm{u}}_T \coloneqq \sum_{t=1}^T \inner{\bh{w}_t - \bm{u}}{\mathring{\grad}_t}^2$ be the pseudo-regret and variance of Algorithm \ref{OCOGeneral}. The following theorem, whose proof is in Appendix \ref{fourproof}, shows how the regret guarantee of \textsc{MetaGrad+C}{} readily transfers to Algorithm \ref{OCOGeneral}:
\begin{theorem}
\label{reductionbound}
Algorithm \ref{OCOGeneral}, which uses \textsc{MetaGrad+C}{} as a black-box, guarantees:
\begin{align}
\sum_{t=1}^T \left(\ell_t(\bh{w}_t) - \ell_t(\bm{u})\right) \leq \mathring{\tmp{R}}^{\bm{u}}_T \leq 3 \sqrt{\mathring{\tmp{V}}^{\bm{u}}_T {\Gamma}_T} + 24 B_T {\Gamma}_T +B_T, \ \text{ for }\u\in \mathcal{U}, \label{alg1bound}
\end{align}
where $ {\Gamma}_T \coloneqq d\ln\left(\frac{27}{25} + \frac{2 \sum_{t=1}^{T-1}
{b}_t^2}{25 d{B}^2_{T-1}}\right) + 2 \ln \left( \log^+_2
\frac{\sqrt{\sum_{t=1}^T {b}_t^2}}{B} +3 \right) + 2= O(d \ln T)$, and
\begin{align}
{b}_t \coloneqq D \norm{{\bm{g}}_t}_2; \quad \quad \quad {B}_t \coloneqq B \vee \max_{s\in [t]} {b}_s, \quad t\in[T]. \label{littlebbre}
\end{align}
\end{theorem}
Note that Algorithm \ref{OCOGeneral}, guarantees the same type of regret as \textsc{MetaGrad+C}, and thus can also adapt to exp-concavity of the losses $(\ell_t)$ and the Bernstein condition.
\section{Conclusion}
\label{sec:conclusion}
We present algorithms that adapt to the Lipschitz constant of the loss
for OCO and experts. Stepping back, we see that an interesting
combination of problem complexity dimensions can be adapted to, with
hardly
any overhead in either regret or computation. The main question for future work is to obtain a better understanding of the landscape of interactions between measures of problem complexity and their algorithmic reflection.
One surprising conclusion from our work, which provides a curious
contrast with incompatibility of Lipschitz adaptivity with comparator
complexity adaptivity in general OCO
\citep{CutkoskyBoahen2017Impossible}, is
the following observation. Our results for the expert setting, which we
phrased for a finite set of $K$ experts, in fact generalise unmodified
to priors with infinite support. Considering a countable set of experts,
we find a scenario where the comparator complexity $\KL(\rho\|\pi)$ is
unbounded, yet our Squint strategy adapts to the Lipschitz constant of the loss without inflating the regret compared to an a-priori known complexity by more than a constant.
A final very interesting question is when it is possible to exploit
scenarios with large ranges that occur only very infrequently. A example
of this is found in statistical learning with heavy-tailed loss
distributions. Martingale methods for such scenarios that are related to
our potential functions suggest that it may be necessary to replace the
``surrogate'' negative quadratic term $f_t(\u,\eta)$ that our algorithms include in the exponent by another function appropriate for the specific distribution \cite[Table~3]{linecrossing}. It is not currently clear what individual sequence analogues can be obtained.
\DeclareRobustCommand{\VAN}[3]{#3}
|
{
"timestamp": "2019-03-01T02:04:13",
"yymm": "1902",
"arxiv_id": "1902.10797",
"language": "en",
"url": "https://arxiv.org/abs/1902.10797"
}
|
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